COLD MOLECULES Theory, Experiment, Applications
© 2009 by Taylor and Francis Group, LLC
COLD MOLECULES Theory, Experiment, Applications Edited by
Roman V. Krems William C. Stwalley Bretislav Friedrich
CRC Press Taylor & Francis Croup Boca Raton London New York
CRC Press is an imprint of the Taylor & Francis Group, an informs business
© 2009 by Taylor and Francis Group, LLC
CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4200-5903-8 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Cold molecules: theory, experiment, applications / [edited by] Roman Krems, Bretislav Friedrich, and William C. Stwalley. p. cm. Includes bibliographical references and index. ISBN 978-1-4200-5903-8 (hard back: alk. paper) 1. Collisions (Nuclear physics) 2. Low temperatures. 3. Quantum solids. 4. Quantum liquids. 5. Cold gases. 6. Molecular dynamics. I. Krems, Roman. II. Friedrich, Bretislav. III. Stwalley, William C, 1942- IV. Title. QC794.6.C6C635 2009 530.4-dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
© 2009 by Taylor and Francis Group, LLC
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Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi A Guided Tour of the Monograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix Part I
Cold Collisions
Chapter 1
Theory of Cold Atomic and Molecular Collisions . . . . . . . . . . . . . . . . . 3 Jeremy M. Hutson
Chapter 2
Electric Dipoles at Ultralow Temperatures . . . . . . . . . . . . . . . . . . . . . . 39 John L. Bohn
Chapter 3
Inelastic Collisions and Chemical Reactions of Molecules at Ultracold Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Goulven Quéméner, Naduvalath Balakrishnan, and Alexander Dalgarno
Chapter 4
Effects of External Electromagnetic Fields on Collisions of Molecules at Low Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Timur V. Tscherbul and Roman V. Krems
Part II
Photoassociation
Chapter 5
Ultracold Molecule Formation by Photoassociation . . . . . . . . . . . . . 169 William C. Stwalley, Phillip L. Gould, and Edward E. Eyler
Chapter 6
Molecular States Near a Collision Threshold . . . . . . . . . . . . . . . . . . . 221 Paul S. Julienne
Chapter 7
Prospects for Control of Ultracold Molecule Formation via Photoassociation with Chirped Laser Pulses . . . . . . . . . . . . . . . . . . . 245 Eliane Luc-Koenig and Françoise Masnou-Seeuws
Chapter 8
Adiabatic Raman Photoassociation with Shaped Laser Pulses . . . . 291 Evgeny A. Shapiro and Moshe Shapiro v
© 2009 by Taylor and Francis Group, LLC
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Part III
Few- and Many-Body Physics
Chapter 9
Ultracold Feshbach Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Francesca Ferlaino, Steven Knoop, and Rudolf Grimm
Chapter 10
Molecular Regimes in Ultracold Fermi Gases . . . . . . . . . . . . . . . . . 355 Dmitry S. Petrov, Christophe Salomon, and Georgy V. Shlyapnikov
Chapter 11
Theory of Ultracold Feshbach Molecules . . . . . . . . . . . . . . . . . . . . . 399 Thomas M. Hanna, Hugo Martay, and Thorsten Köhler
Chapter 12
Condensed Matter Physics with Cold Polar Molecules . . . . . . . . . 421 Guido Pupillo, Andrea Micheli, Hans-Peter Büchler, and Peter Zoller
Part IV
Cooling and Trapping
Chapter 13
Cooling, Trap Loading, and Beam Production Using a Cryogenic Helium Buffer Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Wesley C. Campbell and John M. Doyle
Chapter 14
Slowing, Trapping, and Storing of Polar Molecules by Means of Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 Sebastiaan Y.T. van de Meerakker, Hendrick L. Bethlem, and Gerard Meijer
Part V
Tests of Fundamental Laws
Chapter 15
Preparation and Manipulation of Molecules for Fundamental Physics Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 Michael R. Tarbutt, Jony J. Hudson, Ben E. Sauer, and Edward A. Hinds
Chapter 16
Variation of Fundamental Constants as Revealed by Molecules: Astrophysical Observations and Laboratory Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 Victor V. Flambaum and Mikhail G. Kozlov
Part VI
Quantum Computing
Chapter 17
Quantum Information Processing with Ultracold Polar Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 Susanne F. Yelin, Dave DeMille, and Robin Côté
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Part VII Chapter 18
Cold Molecular Ions Sympathetically Cooled Molecular Ions: From Principles to First Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 Bernhard Roth and Stephan Schiller
© 2009 by Taylor and Francis Group, LLC
Foreword This volume offers an exhilarating journey to a compelling frontier of molecular physics. It is a special pleasure to applaud the authors and editors. Intrepid explorers of the frontier, they are also adept guides for both recruits and curious visitors. The result is a lucid and lively evangelical survey of seminal results, experimental and theoretical methods, prospects and challenges. My benediction will simply touch on some kinships, past or future, with more civilized fields. Any physical chemist of my vintage is delighted to see molecules now ardently embraced by physicists. The oft-quoted dictum: “A diatomic molecule has one atom too many” has been disavowed. The new converts to molecular physics are emulating illustrious ancestors from early decades of the 20th century, who found that molecules presented challenging questions for the then fledgling quantum theory. Today, much of the “hot” appeal of molecules in the “cold” (<1 K) and “ultracold” (<10−3 K) regimes does not stem from attaining low temperatures as such. Rather, a chief aim is to gain access to dramatic quantum phenomena that emerge when de Broglie wavelengths become comparable to or longer than the size or the separation of molecules. This key issue of wavelength versus interaction range was first exemplified over 80 years ago in the celebrated prediction of Bose–Einstein condensation (BEC). However, it soon after also became prominent in nuclear physics. For collisions of protons or neutrons at kinetic energy of ∼1 MeV (equivalent to ∼1010 K), the de Broglie wavelength (∼ 30 × 10−13 cm) is considerably longer than the range of nuclear forces. Consequently, s-wave scattering, resonances, and tunneling became major features in probing nuclear physics. That may soon become so for molecular physics too, although it has proved much more difficult to reach the corresponding long wavelength realm. On entering that realm, molecular physicists will enjoy some great advantages over their nuclear predecessors. Both intramolecular and intermolecular interactions can be strongly influenced by applying external electric or magnetic fields. Also, for molecules, coherent excitation processes induced by laser light provide another particularly powerful tool. Moreover, at least for prototype systems, molecular theory and electronic structure calculations often can offer guidance in design and interpretation of experiments. The “hot” appeal of the cold and ultracold molecular regimes has another fundamental aspect. Attaining these regimes requires liberation from thermodynamics. BEC was long presumed to be attainable only for liquid helium. It was expected that, before the temperature and density required to form a degenerate quantum gas could be reached, thermodynamics would impose the mundane process of ordinary condensation. Now BEC has been obtained at or below 10−6 K for many atomic and a few molecular vapors. That required finding pathways along which the kinetics becomes much too slow to reach equilibrium states. This is the same principle governing much of conventional chemical synthesis. Hordes of organic molecules, many ix © 2009 by Taylor and Francis Group, LLC
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crucial for biology, are thermodynamically unstable but are made by exploiting kinetically dominated pathways. This also must be how sizable organic molecules appear in the interstellar medium, despite the low cosmic abundance of carbon. Viewed in this context, cold molecule research may embolden even pragmatic chemists to seek more extreme means to evade thermodynamics. Many among the wide range of techniques and concepts found in this monograph have roots entwined with the pioneering molecular beam work conducted by Otto Stern. As Einstein’s first postdoctoral fellow, Stern acquired a taste for Gedanken experiments, later fulfilled in his laboratory. In his day, “beams” probably would have seemed a misleadingly robust term. Rather, he spoke of “molecular rays.” That now seems prescient in an era striving for long de Broglie wavelengths. In the preface for a 1931 book by Fraser titled Molecular Rays, Stern emphasized the characteristics of the method as “Its directness and (in principle at least) its primitiveness.” For the present volume, equally apt is his praise of “… that beauty and peculiar charm which so firmly captivates physicists working in this field.” Dudley Herschbach Cambridge, Massachusetts
© 2009 by Taylor and Francis Group, LLC
Acknowledgments The editors wish to thank John Doyle, Gerard Meijer, and Françoise Masnou-Seeuws for their help with developing this monograph, and the Institute for Theoretical Atomic, Molecular and Optical Physics at the Harvard-Smithsonian Center for Astrophysics for partial support. The European Union Research Training Network on Cold Molecules, coordinated by Françoise Masnou-Seeuws and Olivier Dulieu, catalyzed much of the research reported in this book.
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A Guided Tour of the Monograph COLD MOLECULES ARE HOT This monograph was prompted by the need for a comprehensive and comprehensible account of the research that has been amassed on translationally (and otherwise) cold molecules. Having started, in the mid-1990s, as a challenge to a handful of enthusiasts who sought to widen the range of cold-matter research to include gaseous molecules, the venture picked up speed quickly and underwent a strongly nonlinear growth. Today, nearly forty research groups worldwide are engaged in cold molecule research, and produce more than 100 papers annually. The leading protagonists of the field, in the midst of many mutual encounters and interactions, are now meeting on the pages of this monograph to jointly survey the state of the art of their field. It was the task of the coeditors to help to present their material in a consistent and concise manner. The hope of all of us involved in bringing this monograph to life has been that it will serve as a text- or handbook to students entering the field and weathered researchers alike, and will help to foster their identity within academic institutions and the physics and chemistry communities at large. Cold molecule research owes much of its original inspiration to the field of cold atoms. However, it pursued its own aims. Molecules are not only more complicated than atoms, but in some respects also more interesting: apart from possessing vibrational and rotational degrees of freedom, molecules may carry dipole (and higher) electric and magnetic moments. These moments lend properties to molecules on which new operational principles can be based (e.g., quantum computing) or that may allow researchers to investigate novel physics (e.g., quantum degenerate dipolar systems, whose scope is currently limited to the quantum gas of chromium atoms). This introduction is a preview of what the chapters in this monograph have to offer. The methods for producing translationally cold (<1 K) molecules can be generally divided into two categories: direct and indirect ones. The indirect methods rely on linking together ultracold (<1 mK) atoms through magnetically tunable Feshbach resonances (magnetoassociation) or by photoassociation. The molecules that ensue are translationally as ultracold as the atoms they were formed from. The direct methods slow or select preexisting molecules. This is effected either by decelerating a supersonic molecular beam by time-dependent electric, magnetic, or radiative fields or by selecting out a slow fraction of a molecular ensemble. Preexisting molecules can also be sympathetically cooled in a buffer gas of precooled atoms. Once slowed/cooled down to the cold regime, the preexisting molecules can be subject to further sympathetic cooling via a thermal contact with an ultracold gas of atoms or, if the molecules were confined in a trap, to evaporative cooling. xiii © 2009 by Taylor and Francis Group, LLC
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We note that, of the above, a phase-space density increase is concomitant only with the processes of sympathetic and evaporative cooling. In all other cases, phase-space density remains at most constant. A drop in phase-space density takes the gaseous sample further away from quantum degeneracy.
COLLISIONS OF COLD AND ULTRACOLD MOLECULES When hot molecules are placed in a bath of cold atoms, their translational energy is quenched by elastic collisions with the buffer-gas atoms. The evaporative cooling is likewise based on elastic collisions leading to energy transport and reequilibration of the translational temperature of a molecular ensemble as the trap depth is gradually reduced. If the molecules are initially prepared in metastable excited states, they may undergo inelastic and reactive as well as elastic collisions. Inelastic and reactive collisions release energy and accelerate the colliding particles. This results in trap loss. Because the density of molecules is usually the largest near the center of a trap, where the temperature is the coldest, collisional trap loss removes some of the coldest atoms and molecules and leads to heating. Ideally, a cooling experiment should begin with an ensemble of molecules prepared in the absolute ground state to avoid inelastic collisions and their deleterious effects. However, this is most often not feasible. Electrostatic and magnetic traps widely used to isolate molecular gases from the thermal environment confine molecules in excited Zeeman and Stark states. Molecules generated by linking ultracold atoms are normally produced in highly excited vibrational states. In order to establish the possibility of cooling specific molecules to ultracold temperatures, it is therefore critically important to understand the efficiency of elastic scattering and inelastic and reactive collisions of these molecules at cold and ultracold temperatures. Theoretical analysis of molecular collisions at low temperatures has played an important role in the development of the research field of cold molecules. In particular, quantum-mechanical calculations carried out in the early days of cold molecule research demonstrated that rovibrationally inelastic collisions and chemical reactions of molecules at ultracold temperatures may be very fast. Theoretical work showed that collision-induced Stark relaxation of molecules confined in an electrostatic trap is prohibitively fast but molecules in magnetic traps may be more stable against collisional trap loss. Quantum-mechanical calculations had elucidated the mechanisms of inelastic collisions of molecules in electrostatic and magnetic traps and demonstrated that collisional properties of cold and ultracold molecules can be effectively tuned by external fields. This work identified a range of molecules that may be amenable to collisional cooling in magnetic traps and stimulated the development of new traps that could confine molecules in the absolute ground state. These and other results of the theoretical work on molecular collisions at cold and ultracold temperatures are described in Chapters 1 to 4 of this monograph. To set the stage for further dispositions, Jeremy Hutson describes in Chapter 1 (entitled “Theory of Cold Atomic and Molecular Collisions”) the basic theory of atomic and molecular scattering at low temperatures. This chapter introduces the concepts of the cross-section, the rate coefficient, and the scattering length, and describes the main ingredients of quantum collision theory. It is shown that the © 2009 by Taylor and Francis Group, LLC
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scattering wavefunctions and the bound-state wavefunctions can be obtained from the calculations based on the same principles and numerical methods. A particular emphasis is placed on the extension of the collision theory to molecular scattering in the presence of external fields and the discussion of scattering resonances that can be tuned by external fields. These scattering resonances can be used to vary the interaction properties of ultracold molecules, which may be necessary for a number of applications of ultracold molecules discussed in further chapters. A major thrust of recent experimental work has been to create ultracold polar molecules, as ultracold dipolar gases offer exciting opportunities for new fundamental discoveries. More than ten research groups demonstrated the possibility of creating ultracold polar molecules by photoassociation of ultracold alkali-metal atoms. Chapter 2 (“Electric Dipoles at Ultralow Temperatures”), written by John Bohn, presents a comprehensive and rigorous discussion of classical and quantummechanical dipoles. With meticulous attention to detail, this chapter describes the effects of molecular structure such as the presence of the rotational angular momentum, the electron spin, or Λ-doubling on interactions between molecular dipoles. The discussion is conveniently formulated in terms of spherical tensors, which makes the mathematical expressions for the matrix elements of the dipole–dipole interaction operator elegant and transparent. Chapter 3 (“Inelastic Collisions and Chemical Reactions of Molecules at Ultracold Temperatures”), written by Goulven Quéméner, Naduvalath Balakrishnan, and Alexander Dalgarno, is a comprehensive account of recent theoretical work on rovibrationally inelastic and reactive scattering of molecules at cold and ultracold temperatures. The chapter includes a compilation of zero-temperature rate constants for inelastic and reactive collisions of various molecules. The authors describe both neutral and ionic systems and collision processes involving atom–molecule and molecule–molecule scattering. This chapter contains 25 diagrams illustrating various physical phenomena such as near-resonant energy transfer in cold collisions, the effect of near-threshold virtual states on chemical reactivity of ultracold molecules, threshold collision laws, Feshbach and shape resonances, and the effects of rotational degrees of freedom on vibrational relaxation of cold and ultracold molecules. The authors analyze tunneling-dominated chemical reactions and insertion chemical reactions that occur without an activation barrier and discuss the challenges for the theory of ultracold chemical dynamics. Controlling chemical reactions with electromagnetic fields has been a long-sought goal of researchers. External field control of chemical reactions will not only allow chemists to selectively produce desired species, but will also reveal the mechanisms of chemical reactions, yield information on interactions determining chemical reactions, and elucidate the role of nonadiabatic and relativistic effects in chemical dynamics. External control of bimolecular reactions is, however, complicated by thermal motion of molecules, which randomizes molecular encounters and diminishes the effects of external fields on molecular collisions. The effects of the thermal motion can be reduced by cooling molecular gases to low temperatures. Electromagnetic fields may influence molecular collisions significantly only when the translational energy of the molecules is smaller than the perturbations due to interactions with the external fields. Static magnetic and electric fields (up to 5 T and 200 kV/cm, respectively), © 2009 by Taylor and Francis Group, LLC
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as well as off-resonant laser fields (1012 W/cm2 ) readily available in the laboratory, shift molecular energy levels by up to a few Kelvin. As a result, external field control of gas-phase molecular dynamics may be most easily achieved at cold and ultracold temperatures less than 1 K. In Chapter 4, entitled “Effects of External Electromagnetic Fields on Collisions of Molecules at Low Temperatures,” Timur Tscherbul and Roman Krems discuss the effects of external fields on the dynamics of molecular collisions at cold and ultracold temperatures and outline the prospects for new discoveries in the research of molecule–field interactions at low temperatures. The first part of Chapter 4 focuses on collision-induced Zeeman and Stark relaxation relevant for molecular cooling experiments, the discussion of tunable shape and Feshbach resonances, and the scattering of molecular dipoles in an electric field. The results presented exemplify what is described in general terms in Chapters 1 and 2. The second part of this chapter describes new mechanisms for external field control of atom–molecule and intermolecular interactions by combined electric and magnetic fields. Here, the authors discuss the effects of electric fields on collisions of magnetically aligned molecules, collisions in nonparallel electric and magnetic fields, and differential scattering of cold molecules in electric fields. In the third part of the chapter, the authors discuss inelastic and chemically reactive collisions of ultracold molecules confined by laser fields to move in quasi-two-dimensional geometry. The experiments with such systems may probe the effects of external space symmetry on binary interactions of molecules and the role of long-range intermolecular interactions in determining chemical reactivity of ultracold molecules. Chapter 4 concludes with a discussion of prospects for cold controlled chemistry.
PHOTOASSOCIATION OF ULTRACOLD ATOMS The formation of ultracold molecules starting with readily produced ultracold atoms is an attractive and relatively simple method for obtaining such molecules. There are two widely employed methods for such formation: photoassociation (discussed in this section) and magnetoassociation by tuning a magnetic field to convert a Feshbach resonance between two colliding atoms into a very weakly bound molecular level (discussed in the next section). Chapter 5 by William Stwalley, Phillip Gould, and Edward Eyler, entitled “Ultracold Molecule Formation by Photoassociation,” provides an overview of experimental results and their theoretical interpretation. After a brief introduction to the various approaches to formation of ultracold molecules, the process of photoassociation itself is introduced as an important component of atomic line broadening, especially on the wings of an atomic line. However, at room temperature and above, a broadened atomic line has relatively little information content because of the wide range of collision energies and angular momenta of collisions contributing to the lineshape. This is in stark contrast to photoassociation of ultracold atoms, typically at 100 μK, where only a few angular momenta of collision lead to penetration to short distances where photoassociation at significant detunings from atomic resonance can occur. This ultracold photoassociation leads to sharp line-like spectra, where the observed lines correspond to transitions from a colliding pair of ground-state atoms © 2009 by Taylor and Francis Group, LLC
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to excited molecular rovibronic levels with specific quantum numbers for each line; that is, very nearly a bound–bound molecular electronic spectrum. Photoassociation of like atoms to form homonuclear molecules, emphasizing the widely studied alkali metals, is treated first. The simple one-color experiments are described in detail, including the variety of techniques used for detection: trap loss (decrease in atomic fluoresence), direct detection of excited molecule ionization, detection of fragments by resonance-enhanced multiphoton ionization, and detection of ground or metastable molecules (formed by decay of the upper photoassociation level) by resonance-enhanced multiphoton ionization. Two-color experiments are discussed, both the ladder scheme for reaching highly excited states and the lambda (Raman) scheme for reaching bound molecular levels correlating with two ground-state atoms. The latter scheme is, of course, relevant for forming molecules in the ground or lowest metastable electronic state (the X and the a states for the alkali-metal dimers). Chapters 7 and 8 of this monograph use ultrafast lasers for this lambda scheme, as opposed to the cw lasers used in the experiments described in Chapter 5. Photoassociation of unlike atoms to form heteronuclear molecules (with dipole moments) is treated next, with emphasis on the extensively studied case of KRb. The differences between like and unlike atom photoassociation is then discussed in some detail. Next, the promise of atom–molecule and molecule–molecule photoassociation (not yet experimentally realized) is briefly mentioned. The rapidly expanding fields of photoassociation in a quantum degenerate gas, in an electomagnetic field, and in an optical lattice are briefly discussed as a partial introduction to later chapters. The characterization of the X and a state levels of homonuclear and heteronuclear alkali dimer molecules formed by decay of the upper levels formed by photoassociation is discussed next. Resonance-enhanced multiphoton ionization is shown to be a powerful technique for establishing the population of vibrational levels formed in the X and a states near the dissociation limit. A new ion depletion technique for observing rotational and hyperfine structure as well for these levels near dissociation is also discussed. To reach lower levels, especially in the ground X state, it is useful to select specific photoassociation approaches such as double minimum excited-state potentials, resonant coupling of two (or more) excited state potentials, and stimulated Raman transfer from levels near dissociation to low levels (e.g., the collisionally stable v = 0, J = 0 level). Finally connections to ultracold molecular ions, to tests of fundamental laws, to quantum computing with ultracold polar molecules, to ultracold collisions and chemistry, and to more speculative areas are mentioned. Chapter 6 by Paul Julienne, entitled “Molecular States Near a Collision Threshold,” provides a thorough introduction to the region near dissociation dominated by long-range forces. It provides, on the one hand, a connection to the collision dynamics discussions of Chapters 1–4 and, on the other hand, to the photoassociation of Chapters 5–8 and the magnetoassociation (Feshbach molecules) of Chapters 9–11. Of particular interest is the elegant long-range scaling of properties for both collisions at ultracold energies and of bound vibrational levels just below dissociation (longrange molecules) for the case of a single internuclear potential energy curve. The case © 2009 by Taylor and Francis Group, LLC
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of multiple interacting potentials is then introduced and the resulting modifications of the formalism discussed. Chapter 7 by Eliane Luc-Koenig and Francoise Masnou-Seeuws, entitled “Prospects for Control of Ultracold Molecule Formation via Photoassociation with Chirped Laser Pulses,” provides an intriguing set of case studies on the theme “where ultrafast meets ultracold.” A series of calculations are described for two-color lambda ultrafast photoassociation of the cesium dimer into the a state via the long-range well of a double minimum state. With an emphasis on developing some physical intuition concerning ultrafast photoassociation, the authors develop insightful concepts such as resonance window, photoassociation window, dynamical hole, momentum kick, compression effect, and integrated mass current to understand as well as present their interesting results. Chapter 8 by Evgeny Shapiro and Moshe Shapiro, entitled “Adiabatic Raman Photoassociation with Shaped Laser Pulses,” provides an interesting complementary approach to the ultrafast/ultracold two-color photoassociation scheme. They study the Raman adiabatic passage of an incoming wavepacket of two colliding atoms to form a translationally ultracold diatomic molecule. This wavepacket approach provides an interesting alternative to the theory described in the previous chapter. In addition, the authors of Chapter 8 pursue both analytical theory and numerical simulations for KRb and Rb2 , which shows how a photoassociation measurement can be used to probe the wavefunction of the colliding atoms.
FEW- AND MANY-BODY PHYSICS WITH COLD MOLECULES Recent theoretical work has shown that ultracold molecules trapped in optical lattices can be used to simulate condensed-matter systems, engineer novel phases with topological order, and explore many-body interactions. Polar molecules in external field traps may form chains, which can be used to study rheological phenomena with nonclassical behavior. The creation of a Bose–Einstein condensate of molecules may enable the study of Bose-enhanced chemistry and the effects of symmetry breaking on chemical interactions at ultracold temperatures. The realization of these proposals as well as the creation of dense ensembles of ultracold molecules depends critically on the possibility of controlling the scattering length of molecules by external electromagnetic fields. The prediction and observation of magnetic Feshbach resonances in atomic collisions opened the door to many groundbreaking experiments with ultracold gases such as the realization of the BEC–BCS crossover, the observation of quantum phase transitions, and the creation of ultracold molecules using time-varying magnetic fields. Extension of this work to molecular collisions may similarly lead to the development of new research fields such as cold controlled chemistry, quantum coherent control, and quantum condensed-matter physics with molecular condensates. Chapter 9, “Ultracold Feshbach Molecules,” written by Francesca Ferlaino, Steven Knoop, and Rudolf Grimm, introduces the concept of magnetically tunable Feshbach resonances and describes the experimental techniques for the creation of ultracold atomic gases, making ultracold molecules by tuning Feshbach resonances, removing the atoms from ultracold atom–molecule mixtures in order to produce a pure molecular gas, and using avoided crossings induced by magnetic dipole–dipole and © 2009 by Taylor and Francis Group, LLC
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spin–orbit interactions to prepare molecular samples in specific quantum states. The authors describe the fascinating observation of Stückelberg oscillations in an ultracold gas of molecules and how tuning the scattering length of ultracold atoms via Feshbach resonances can be used to study few- and many-body physics of ultracold systems. In particular, they describe the experimental studies of Efimov trimer states, molecular Bose–Einstein condensates and many-body physics in atomic Fermi gases. A major thrust of recent experimental research has been to produce ultracold molecules in the absolute ground state by combining ultracold atoms. This has been a daunting task. At the end of Chapter 9, the authors describe the results of a successful experiment yielding ultracold molecules in the absolute ground state. The creation and Bose–Einstein condensation of weakly bound diatomic molecules of fermionic atoms is perhaps one of the most important results of cold atom physics. Molecules produced by tuning interatomic interactions using Feshbach resonances are the largest diatomic molecules obtained so far, with a size of the order of thousands of Ångströms. They represent novel composite bosons, which exhibit features of Fermi statistics at short intermolecular distances. While highly excited, these molecules are remarkably stable with respect to collisional relaxation. This is a consequence of the Pauli exclusion principle for identical fermionic atoms. Chapter 10 by Dmitry Petrov, Christophe Salomon, and Georgy Shlyapnikov, entitled “Molecular Regimes in Ultracold Fermi Gases,” introduces theories to describe the physics of molecular regimes in two-component Fermi gases and Fermi–Fermi mixtures. The authors discuss elastic and inelastic collisions of these extended molecules and demonstrate the effects of quantum statistics on interactions of weakly-bound molecules in a molecular Bose–Einstein condensate. For heteronuclear systems, they introduce a model based on the Born–Oppenheimer approximation separating the motion of light and heavy atoms. The model is used for an elegant description of the Efimov states. At the conclusion of the chapter, the authors extend their theory to many-body systems of heavy and light fermions and demonstrate that significant long-range repulsions between weakly-bound molecules may lead to the spontaneous formation of a molecular crystalline phase. Building on the discussion of Chapter 1 and complementing the discussions of Chapters 9 and 10, Chapter 11, “Theory of Ultracold Feshbach Molecules,” written by Thomas Hanna, Hugo Martay and Thorsten Köhler, is a concise description of the microsopic theory of Feshbach molecules. The authors describe the role of atomic hyperfine interactions and the interatomic interactions in determining Feshbach resonances and how Feshbach resonances modify the interaction properties of ultracold atoms. They present the classification of Feshbach resonances and distinguish openchannel-dominated and closed-channel-dominated resonances. As the different types of resonances give rise to different properties of Feshbach molecules, the discussion of this chapter is important for the understanding of how tuning microscopic interaction parameters with magnetic fields modifies the macrosopic properties of an ultracold quantum gas. The creation of low-dimensional quantum gases has opened up exciting possibilities for new research with ultracold atoms and molecules on problems in several different areas of physics. Bose and Fermi gases exhibit unusual properties in two dimensions, and confining ultracold atoms in two-dimensions may result in interesting © 2009 by Taylor and Francis Group, LLC
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decoherence dynamics of quantum gases. Low-dimensional quantum systems may be used as controllable models for a variety of phenomena in condensed-matter physics. For example, polar molecules confined in two dimensions may repel each other at long range, which leads to the formation of self-organizing crystals at ultracold temperatures and to the possibility of designing various spin lattice models. This is the focus of Chapter 12 by Guido Pupillo, Andrea Micheli, Hans-Peter Büchler, and Peter Zoller, entitled “Condensed Matter Physics with Cold Polar Molecules.” The chapter opens with an overview of many-body systems of strongly interacting polar molecules placed in external microwave and dc electric fields. The authors show that the polarization of laser fields, the strength of dc fields, and the number of laser-field couplings can all be used to tune the long-range interactions between polar molecules and generate purely repulsive, step-like or strongly attractive potentials. This can be used to engineer strongly correlated quantum phases in a two-dimensional crystal of molecules. This system can also be used to realize effective lattice models with interparticle interactions described by exotic Hubbard Hamiltonians. Hubbard Hamiltonians are model Hamiltonians designed to describe low-energy physics of interacting fermions or bosons on a lattice. The authors show that internal degrees of freedom in ultracold molecules combined with the tunability of the many-body system of polar molecules in two dimensions may allow researchers to construct a complete toolbox for the simulation of any permutation-symmetric lattice spin model. At the conclusion of the chapter, the authors describe a setup that allows them to engineer many-body systems of ultracold polar molecules in which the two-body interactions are canceled and the dominant interaction is determined by three-body effects. This exotic system may find intriguing applications in condensed-matter physics.
COOLING AND TRAPPING OF PREEXISTING MOLECULES The workhorse of cold-atom physics, laser cooling, is inapplicable to most molecules (if to any at all). Necessity proved to be a mother of invention: a host of techniques, based on new principles independent of laser cooling, have been developed that make it possible to cool preexisting molecules. The swatch of these direct techniques is variegated and attests to the imagination of their inventors. The direct methods are currently being pursued by about 20 research groups worldwide, and are concerned with the development and applications of buffer-gas cooling, Stark or Zeeman deceleration, deceleration by pulsed optical fields, deceleration via collisions in crossed molecular beams, supersonic expansion from a counter-rotating nozzle, or selection of the low-velocity tail of a Maxwell–Boltzmann distribution of molecules in an effusive beam. All direct methods start with relatively hot molecules (200 to 1000 K), usually in the source of a supersonic molecular beam. Whereas the indirect methods are limited to molecules readily formed by photoassociation or magnetoassociation (e.g., diatomic molecules), the direct methods are versatile, applicable to large classes of molecules (e.g., Stark deceleration to all polar molecules; buffergas cooling coupled with magnetic trapping to all paramagnetic molecules) or to any molecules (all other direct techniques). Buffer-gas cooling and Stark deceleration © 2009 by Taylor and Francis Group, LLC
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turned out to be the most effective among the direct techniques; each is described in some detail in a dedicated chapter. Wes Cambell and John Doyle, in Chapter 13, entitled “Cooling, Trap Loading, and Beam Production Using a Cryogenic Helium Buffer Gas,” survey the feats accomplished by thermalizing molecules via elastic collisions with a cryogenic He buffer gas. Buffer-gas cooling makes it possible to attain a temperature of about 0.5 K, which is not ultracold. However, a great advantage of the method (apart from its versatility) is its ability to cool and trap large numbers of molecules. By evaporating a fraction of the trapped molecules, it should be possible to cool the molecular ensemble further down toward the ultracold regime (<1 mK). Nevertheless, so far the initial numbers of trapped molecules and other factors, especially the buffer gas removal, have not made it possible to effectively apply such an evaporative cooling scheme. The number of the magnetically trapped molecules depends critically on the technique of loading the molecules into the buffer gas. Laser ablation can hardly evaporate more than 108 to 1013 atoms or molecules (with a single pulse). Therefore, a considerable effort has recently been put into developing an alternative technique of loading molecules into the cryogenic environment. The new technique is based on the use of a molecular beam made up of the molecules that are to be cooled. The molecular beam is simultaneously employed to transport the molecules into the cryogenic cell. Thus far, it has been possible to thermalize 1012 molecules of the NH radical, obtained by dissociating a pulsed beam of NH3 in an electric discharge. Unlike an conventional supersonic expansion, the authors show that it is possible to precool molecules to temperatures well below their boiling point before they are released through an orifice into a beam. In essence, molecules are cooled inside a cell held at a temperature of about 1 K by high-density helium and come out of an orifice in the cell (into a beam). Thus, molecules come out rotationally cold. If the helium density is high enough, most of the cold molecules inside the cell will exit into the beam with a forward velocity equal to that of the cold helium gas. This has been shown to produce very high fluxes of cold molecules, including cold molecular oxygen injected into a magnetic hexapole guide whose output flux was found to be in excess of 1012 sec−1 . Bas van de Meerakker, Rick Bethlem, and Gerard Meijer in Chapter 14, “Slowing, Trapping and Storing of Polar Molecules by Means of Electric Fields,” describe the techniques of Stark deceleration, dc and ac trapping, and storage of polar molecules using inhomogeneous electric fields. The authors first show that the technique of Stark deceleration (or acceleration) makes it possible not only to arbitrarily vary the velocity of polar molecules but also to select the molecules’ internal state (electronic, vibrational, and rotational) and orientation. The method thus allows for a complete control of molecules. The method makes use of a time-dependent inhomogeneous electric field, by means of which the Stark energy of polar molecules is altered. Stark energy represents the potential energy of molecules in an electric field. Its change is compensated for by a change in the kinetic translational energy of the molecules, as dictated by energy conservation. Depending on whether the Stark energy provided is negative or positive, the field causes either an acceleration or a deceleration along the longitudinal coordinate. In © 2009 by Taylor and Francis Group, LLC
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order to be able to accelerate/decelerate a pulse of molecules with a distribution of coordinates and velocities (spatial and velocity spread), the acceleration/deceleration process must be carried out under the conditions of phase stability. This is discussed in some detail in the chapter, along with the transverse focusing needed to maintain the pulse of molecules on course along the longitudinal coordinate. Sufficiently slow polar molecules in a low-field seeking state can be confined in an electrostatic (dc) trap—a minimum of Stark energy in free three-dimensional space. A special case of an electrostatic trap, also discussed in the chapter, is a storage ring, which provides transverse but not longitudinal confinement for low-field seeking molecules. It is most simply implemented by bending a transverse focuser, such as a hexapole, onto itself. The low-field seekers then have a minimum of potential energy on a circle rather than at a single point. Compared with a trap, a storage ring is capable of confining molecules without the need to bring them to a standstill first. The circling packets of molecules can repeatedly interact, at well-defined times and positions, with electromagnetic fields and/or other atoms or molecules. In order to counteract the spreading of the packet in the ring—and thus to restore the ability to define the timing and position of the stored molecules—a storage ring was constructed consisting of two half-rings separated by a gap. Depending on the switching sequence, the gap can act as a time-varying inhomogeneous electric field (similar to a single stage of a Stark decelerator) that accelerates, decelerates, or just transports or bunches the molecular packet under the conditions of phase stability. The split-ring device thus represents a molecular analog of a synchrotron for charged particles. Bunching ensures a high density of the stored molecules and, in addition, enables injecting multiple packets—either copropagating or counterpropagating—into the ring independent of the packet(s) already stored. Trapping of molecules in high-field seeking states is of particular interest, chiefly because ground-state molecules are always high-field seeking. Since a ground state cannot decay, molecules trapped via their ground state cannot be lost from the trap by two-body relaxation processes, which plague trapping via low-field seeking states in electrostatic or magnetic traps. In the absence of a loss mechanism, ground-state molecules are expected to be amenable to evaporative or sympathetic cooling. We note that relaxation losses have been predicted to be particularly severe for polar molecules in excited rovibrational states. Another reason for developing traps for high-field seekers is that heavier molecules have small rotational constants (and, often, large dipole moments) and so become high-field seeking at relatively small field strengths, thereby precluding the application of large-enough forces required to trap or otherwise manipulate them. Since creating a maximum of a static field in free space is prohibited by a consequence of Maxwell’s equations, known as Earnshaw’s theorem, one has to rely on a trick similar to the one used in a Paul trap for ions: one generates a static field that has the shape of a saddle surface. Such a field focuses molecules in one direction and defocuses them in the perpendicular one. By reversing the polarity of the electrodes that generate the field (i.e., ac switching), the focusing and defocusing directions can be interchanged. The ac switching between the two field configurations causes the molecules to undergo an oscillatory micromotion whose frequency tends to get in step with the switching frequency. Since the force acting on the molecules increases with distance from the saddle point, the amplitude and thus © 2009 by Taylor and Francis Group, LLC
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the kinetic energy of the micromotion increases with distance as well. Because the kinetic energy of the micromotion is positive (as kinetic energy always is), it creates in effect a potential energy well, with a minimum at the trap center. The positive kinetic energy of the micromotion is independent of whether the molecules are in a low- or high-field-seeking state, which makes an ac trap work for either. It is noteworthy that electrostatic traps are about 1 K deep (depending on the molecular species and details of the trap design), and have a volume of typically 1 cm3 . An ac electric trap has a depth of about 1 to 10 mK and a volume of about 10−2 cm3 .
TESTS OF FUNDAMENTAL PHYSICS WITH COLD MOLECULES There is a class of experiments in prospect or progress with cold molecules that could answer questions reaching far beyond the scope of traditional molecular science: these experiments test some of the fundamental symmetries in physics, such as the time-reversal symmetry (T), parity (P), and the Pauli principle. These symmetries are a window into the world of the fundamental forces in nature and thus molecular, table-top experiments that test them are complementary to the high-energy collisional experiments. Particularly promising and interesting is the simultaneous testing of the timereversal symmetry and parity in experiments that search for the permanent electric dipole (EDM) of the electron (and of other elementary particles). A nonzero value of EDM implies the breaking of both T and P. Since the Standard Model predicts an unmeasurably small value for the electron EDM, finding a nonzero EDM would amount to the discovery of physics beyond the Standard Model. Such a discovery would revolutionize physics. In Chapter 15, “Preparation and Manipulation of Molecules for Fundamental Physics Tests,” Michael Tarbutt, Jony Hudson, Ben Sauer, and Ed Hinds show what the search for the EDM of the electron has to do with cold molecules. From the experiments carried out so far it is known that the EDM of the electron does not exceed the value of about 5 × 10−19 D. This is an exceedingly small value, one that would be found if an electron were expanded to the size of the Earth and deformed on the poles by a micron! A dipole moment manifests itself through its Stark effect, that is, by a shift of the energy levels of a system carrying the dipole moment when the system is subject to an electric field. In order for the Stark shift to be detectable, the EDM has to be exposed to as large a field as possible. The strongest fields that one can come by are available inside heavy atoms (where they can reach values of 10 GV/cm, thanks to relativistic effects). Atoms are spherically symmetric, however, and so before any EDM measurement they need to be oriented in the laboratory frame (otherwise the dipole moment would average out). But orienting atoms is difficult, because in doing so one has to rely on their polarizability, which is small. Therefore, it is of great advantage to add to a given heavy atom another atom, and to carry out the Stark effect measurement on the polar molecule that thereby ensues! Polar molecules can be easily oriented. In fact, a Stark decelerator and an electric trap are based on the ability to orient polar molecules easily in the laboratory frame. The use of cold polar molecules further enhances the resolution of the measurement. The experiment of the Hinds group with decelerated YbF molecules already comes close to yielding the © 2009 by Taylor and Francis Group, LLC
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most accurate value of the electron‘s EDM. An increase of the experimental accuracy by a single order of magnitude could lead to a rejection or adoption of some of the alternatives to the Standard Model. Molecules are beginning to play a key role in another area of fundamental interest, namely in testing the temporal and spatial variation of the fundamental constants. Molecular vibration, rotation, hyperfine structure, and other features offer combinations of the fundamental constants that are not available with atoms. Chapter 16, “Variation of the Fundamental Constants as Revealed by Molecules: Astrophysical Observations and Laboratory Experiments,” by Victor Flambaum and Mikhail Kozlov, describes the application of precision molecular spectroscopy to the study of a possible variation of the fundamental constants. The authors show that molecular spectra are mostly sensitive to two dimensionless constants, namely the fine-structure constant and the electron-to-proton mass ratio. The chapter discusses the results which follow from the astrophysical observations of the optical and microwave spectra of molecules, as well as possible laboratory experiments with molecules. Although the accuracy of the laboratory results cannot yet compete with that of the astrophysical observations, there are significantly improved experiments in progress that are likely to reverse this situation soon. The ideas behind these experiments and the preliminary experimental results so far obtained are discussed in detail. The authors show that the sensitivity for the variation of the fine-structure constant is particularly large for narrow, quasidegenerate levels of diatomic molecules. Such levels may come about due to a quasidegeneracy of either hyperfine and rotational levels, or between the fine and vibrational levels within the molecular electronic ground state. The transitions between the quasidegenerate levels correspond to microwave frequencies, which are experimentally accessible, and have narrow linewidths, typically ∼10−2 Hz. The sensitivity of the relative variation (frequencyto-frequency variation ratio) can exceed 105 in such cases.
QUANTUM COMPUTING WITH COLD MOLECULES Already atomic systems such as charged ions and neutral atoms have been shown to be attractive candidates for physical realization of quantum computation, in view of their exceptionally long coherence times and well-developed techniques for cooling and trapping. The main conceptual problems are associated with designing fast and robust multiatom operations for quantum entanglement as well as with scaling these systems to large numbers of qubits. Susanne Yelin, Dave DeMille, and Robin Cote show in Chapter 17, entitled “Quantum Information Processing with Ultracold Polar Molecules,” that ultracold polar molecules are a potentially superior candidate for the realization of quantum bits in a scalable quantum computer. Polar molecules combine the key advantages of neutral atoms and ions while featuring similarly long coherence times. In particular, large ensembles of cold molecules can be trapped and cooled similarly to neutral atoms, but they can then be manipulated individually using electric fields in analogy with ions. The first proposed complete scheme for quantum computing with polar molecules took advantage of the first feature, but not the second. It was based on an ensemble of ultracold polar molecules trapped in a one-dimensional optical lattice, combined © 2009 by Taylor and Francis Group, LLC
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with an inhomogeneous electric field. The requisite entanglement is achieved via the interaction among the dipoles, each of which represents a qubit. Such qubits are individually addressable, thanks to the Stark effect, which is different for each molecule in the inhomogeneous electric field. A subsequent proposal showed that it may be possible to couple polar molecules into a quantum circuit using superconducting wires. The capacitive, electrodynamic coupling to transmission line resonators was proposed in analogy with coupling to Rydberg atoms and Cooper pair boxes. The key feature of molecules is their RF frequency rotational transitions, nicely compatible with microwave circuitry. Coupling individual molecules to microwave strip-lines is advantageous for several reasons. First, it allows for detection of single molecules by remote sensing of transmission line potentials, as well as for efficient quantum-state readout. Second, the molecules can be further cooled using a novel method involving microwave spontaneous emission into on-chip transmission lines. Finally, remotely separated molecules can be coherently coupled to allow for nonlocal operations. Local gating with electrostatic fields can be used to achieve an exceptional degree of addressability. Single-bit manipulations can be accomplished by using local modulated electric fields. The chapter provides a brief introduction to the field of quantum information processing and computation, discusses various platforms for implementing quantum computers, and makes the case for using permanent molecular dipoles as realizations of individually addressable qubits. A look into the future reveals the contours of a superconducting microwave resonator of an optical quantum computer enmeshed with an ensemble of polar molecules.
MUCH OF THE ABOVE WITH COLD MOLECULAR IONS Chapter 18 by Bernhard Roth and Stephan Schiller, entitled “Sympathetically Cooled Molecular Ions: From Principles to First Applications,” provides a superb introduction to the very different techniques used to study cold molecular ions and of their applications. First the authors provide some background on RF ion traps, ion cooling, and cold molecular ion production. The chapter focuses on the interpretation of images of Coulomb clusters formed by fluorescing atomic ions (e.g., alkaline earth positive ions) interacting with nonfluorescing ions (e.g., HD+ or alkaline earth hydride positive ions, formed by reaction of hydrogen molecules with alkaline earth positive ions). The molecular dynamics simulations of Coulomb clusters are compared with the fluorescent images of these same clusters, which allows for a detailed species and cluster-shape identification as well as for an understanding of the heating effects. Sympathetic cooling thus emerges as a general and well-understood method for producing cold molecular ions. Special topics, such as motional resonance coupling and species-selective ion removal follow, before ion-neutral chemical reactions and polyatomic molecule photofragmentation are discussed. The chapter concludes with a dream-come-true spectroscopy of the HD+ molecular ion. It is clear that the field of cold molecular ions represents an area of great opportunity for the pursuits of the ultracold. © 2009 by Taylor and Francis Group, LLC
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PROSPECTS This monograph reviews the past and surveys the present of cold molecule research, whose spirit, scope and promise all point to a bright future indeed. It is evident already that the experimental and theoretical research on cold and ultracold molecules has irreversibly changed atomic, molecular, and optical physics and quantum information science. Its impact on condensed-matter physics, astrophysics and physical chemistry is becoming apparent as well. It is also evident from the material expounded in this volume that the research field of cold molecules has been marching ahead with big strides. What seemed unfeasible less than ten years ago (e.g., the creation of ultracold polar molecules in the absolute ground state by photoassociation, magnetic trapping of dense molecular gases, and the collision experiments with molecules at ultracold temperatures, among others) has now been realized by multiple research groups. The main goal of the cold molecule experiments over the past five years has been the creation of stable and dense ensembles of ultracold molecules. With this goal accomplished, at least for some molecular systems, a part of the experimental effort is now focusing on applications. Ultracold chemistry, molecular Bose–Einstein condensates, and coherent control of ultracold molecular processes are no longer in the realm of theory only. The close interplay between theory and experiment has been instrumental for the success of cold molecule research. As this research is becoming increasingly multidisciplinary, the role of theory is shifting toward identifying new applications of cold and ultracold molecules in a variety of areas of physics and chemistry. In conclusion, we are pleased to say that the properties of ultracold molecules and the prospects for diverse applications of ultracold molecular gases seem to have captured the imagination of the physics and chemistry communities at large. Cold molecules are enjoying a happy life and their ending is nowhere in sight. Roman V. Krems, Canada William C. Stwalley, USA Bretislav Friedrich, Germany
© 2009 by Taylor and Francis Group, LLC
Editors Roman V. Krems is an assistant professor at the University of British Columbia in Vancouver, Canada. He graduated from Moscow State University, Russia, in 1999 and obtained a PhD in physical chemistry from Göteborg University, Sweden, in 2002. He was a Smithsonian Institute predoctoral fellow at the HarvardSmithsonian Center for Astrophysics in 2001–2002 and a postdoctoral fellow at the Harvard-MIT Center for Ultracold Atoms in 2003–2005. His current research focuses on the theoretical study of new methods for the production of ultracold molecules, understanding the effects of external electromagnetic fields on dynamics of molecules at low temperatures and the interaction properties of cold and ultracold molecules, and ultracold chemistry. William C. Stwalley is Board of Trustees Distinguished Professor and Department Head of Physics at the University of Connecticut. He graduated from the California Institute of Technology with a BS in chemistry in 1964 and from Harvard University with a PhD in physical chemistry in 1969. He was a faculty member at the University of Iowa from 1968 to 1993 in chemistry and later in physics as well, serving as director of the Iowa Laser Facility from 1976 to 1993 and George Glockler Professor of Physical Science from 1988–1993. His awards include Fellowship in the American Physical Society, the Optical Society of America, and the American Association for Advancement of Science, an Outstanding Referee award (American Physical Society, APS), the William F. Meggers Award (the Optical Society of America, OSA), and the State of Connecticut Medal of Science (2005). His research has focused on atomic and molecular interactions from collisions, spectroscopy and theory, with particular interest in the interactions of alkali metal (and hydrogen) atoms. Recent work has emphasized such studies at ultracold temperatures using photoassociation and related techniques. Bretislav Friedrich is a research group leader at the Fritz-Haber-Institut der MaxPlanck-Gesellschaft, Berlin, Honorarprofessor at the Technische Universität Berlin, and co-director of the History and Foundations of Quantum Physics Program in Berlin. He graduated from Charles University, Prague, in 1976, and earned his PhD in chemical physics from the Czech Academy of Sciences, in 1981. His awards include an Iberdrola Fellowship (Spain) and an Outstanding Referee award (APS). His experimental and theoretical research at Göttingen (1986–87), Harvard (1987–2003), and Berlin (from 2003) have dealt with interactions of molecules with fields, molecular spectroscopy, molecular collisions, and molecular cooling and trapping. He has taught undergraduate and graduate courses in physical chemistry and molecular physics as well as in the history of science.
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Contributors Naduvalath Balakrishnan Department of Chemistry University of Nevada Las Vegas Las Vegas, Nevada
Dave DeMille Department of Physics Yale University New Haven, Connecticut
Hendrick L. Bethlem Laser Centre Vrije Universiteit Amsterdam, Netherlands
John M. Doyle Department of Physics Harvard University Cambridge, Massachusetts
John L. Bohn JILA University of Colorado Boulder, Colorado
Edward E. Eyler Department of Physics University of Connecticut Storrs, Connecticut
Hans-Peter Büchler Institute for Theoretical Physics III University of Stuttgart Stuttgart, Germany
Francesca Ferlaino Institute of Experimental Physics and Center for Quantum Physics University of Innsbruck Innsbruck, Austria
Wesley C. Campbell Department of Physics Harvard University Cambridge, Massachusetts Robin Côté Department of Physics University of Connecticut Storrs, Connecticut Alexander Dalgarno Institute for Theoretical Atomic, Molecular and Optical Physics Harvard-Smithsonian Center for Astrophysics Cambridge, Massachusetts
and Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences Innsbruck, Austria Victor V. Flambaum School of Physics The University of New South Wales Sydney, Australia Phillip L. Gould Department of Physics University of Connecticut Storrs, Connecticut
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Contributors
Rudolf Grimm Institute of Experimental Physics and Center for Quantum Physics University of Innsbruck Innsbruck, Austria
Steven Knoop Institute of Experimental Physics and Center for Quantum Physics University of Innsbruck Innsbruck, Austria
and
and
Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences Vienna, Austria
Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences Vienna, Austria
Thomas M. Hanna Department of Physics Clarendon Laboratory University of Oxford Oxford, United Kingdom
Thorsten Köhler Department of Physics and Astronomy University College London London, United Kingdom
Edward A. Hinds Centre for Cold Matter The Blackett Laboratory Imperial College London London, United Kingdom Jony J. Hudson Centre for Cold Matter The Blackett Laboratory Imperial College London London, United Kingdom Jeremy M. Hutson Department of Chemistry University of Durham Durham, United Kingdom Paul S. Julienne Joint Quantum Institute National Institute of Standards and Technology and University of Maryland Gaithersburg, Maryland
© 2009 by Taylor and Francis Group, LLC
Mikhail G. Kozlov Petersburg Nuclear Physics Institute Gatchina, Russia Roman V. Krems Department of Chemistry University of British Columbia Vancouver, Canada Eliane Luc-Koenig Laboratoire Aimé Cotton, CNRS Université Paris Sud XI Orsay, France Hugo Martay Department of Physics University of Oxford, Clarendon Laboratory Oxford, United Kingdom Françoise Masnou-Seeuws Laboratoire Aimé Cotton, CNRS Université Paris Sud XI Orsay, France
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Contributors
Gerard Meijer Fritz-Haber-Institut der Max-Planck-Gesellschaft Berlin, Germany Andrea Micheli Institute for Theoretical Physics University of Innsbruck and Institute for Quantum Optics and Quantum Information Innsbruck, Austria Dmitry S. Petrov Laboratoire de Physique Théorique et Modèles Statistiques Université Paris-Sud Orsay, France and Russian Research Center Kurchatov Institute Moscow, Russia Guido Pupillo Institute for Theoretical Physics University of Innsbruck and Institute for Quantum Optics and Quantum Information Innsbruck, Austria Goulven Quéméner Department of Chemistry University of Nevada Las Vegas Las Vegas, Nevada
© 2009 by Taylor and Francis Group, LLC
Bernhard Roth Institute of Experimental Physics Heinrich-Heine-University Düsseldorf Düsseldorf, Germany Christophe Salomon Laboratoire Kastler Brossel Ecole Normale Supérieure Paris, France Ben E. Sauer Centre for Cold Matter The Blackett Laboratory Imperial College London London, United Kingdom Stephan Schiller Institute of Experimental Physics Heinrich-Heine-University Düsseldorf Düsseldorf, Germany Evgeny A. Shapiro Departments of Chemistry and Physics University of British Columbia Vancouver, Canada Moshe Shapiro Departments of Chemistry and Physics University of British Columbia Vancouver, Canada and Department of Chemical Physics The Weizmann Institute Rehovot, Israel
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Georgy V. Shlyapnikov Laboratoire de Physique Théorique et Modèles Statistiques Université Paris-Sud Orsay, France and Van der Waals–Zeeman Institute University of Amsterdam Amsterdam, Netherlands William C. Stwalley Department of Physics University of Connecticut Storrs, Connecticut Michael R. Tarbutt Centre for Cold Matter The Blackett Laboratory Imperial College London London, United Kingdom Timur V. Tscherbul Harvard-MIT Center for Ultracold Atoms and Institute for Theoretical Atomic, Molecular, and Optical Physics
© 2009 by Taylor and Francis Group, LLC
Contributors
Harvard-Smithsonian Center for Astrophysics Cambridge, Massachusetts Sebastiaan Y.T. van de Meerakker Fritz-Haber-Institut der Max-Planck-Gesellschaft Berlin, Germany Susanne F. Yelin Department of Physics University of Connecticut Storrs, Connecticut and Institute for Theoretical Atomic, Molecular and Optical Physics Harvard-Smithsonian Center for Astrophysics Cambridge, Massachusetts Peter Zoller Institute for Theoretical Physics University of Innsbruck and Institute for Quantum Optics and Quantum Information Innsbruck, Austria
Part I Cold Collisions
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Theory of Cold Atomic 1 and Molecular Collisions Jeremy M. Hutson CONTENTS 1.1 1.2
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Collision Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Laboratory and Center of Mass Coordinates . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Cross-Sections and Rate Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Elastic, Inelastic, and Reactive Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Centrifugal Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Structured Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Quantum Collision Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Single-Channel Scattering (for Unstructured Atoms) . . . . . . . . . . . . . . 1.3.1.1 Bound States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1.2 Low-Energy Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Multichannel Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2.1 Atom–Diatom Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2.2 Scattering of Metastable Helium Atoms . . . . . . . . . . . . . . . . . . 1.3.3 Coupled Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3.1 Scattering Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3.2 Numerical Methods for Scattering . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3.3 Decoupling Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3.4 Bound States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3.5 Numerical Methods for Bound States . . . . . . . . . . . . . . . . . . . . . 1.3.4 Quasibound States and Scattering Resonances . . . . . . . . . . . . . . . . . . . . . 1.3.5 Low-Energy Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Collisions in External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6.1 Basis Sets without Total Angular Momentum . . . . . . . . . . . . 1.3.6.2 Spin Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6.3 Zero-Energy Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . 1.4 Reactive Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Langevin Model for Barrierless Reactions. . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 4 4 5 6 7 7 7 7 11 11 13 15 15 16 17 19 19 20 21 21 23 25 26 27 28 28 30 32 34 3
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1.1
Cold Molecules: Theory, Experiment, Applications
INTRODUCTION
An understanding of atomic and molecular interactions and collisions is essential to the study of cold and ultracold molecules. Collisions govern the lifetime of molecules in traps and determine whether proposed cooling schemes will work. Once atoms and molecules are in the ultracold regime, the extent to which their interactions can be controlled depends on a detailed understanding of their collisional properties. The purpose of this chapter is to outline atomic and molecular collision theory and describe the special features that are important to the study of cold molecules.
1.2
CLASSICAL COLLISION THEORY
Studies of cold molecules focus on behavior at temperatures below 1 K and sometimes as low as 1 nK. At such temperatures the de Broglie wavelengths for translational motion become quite large, and at the lower end of the temperature range it becomes meaningless to consider atoms and molecules as classical particles—a quantum-mechanical treatment is essential. Nevertheless, a classical treatment is relevant at higher temperatures and provides many useful concepts, so we will begin with an outline of classical scattering before proceeding to a more detailed treatment of quantum scattering. More detailed treatments are given in various textbooks, such as those of Child [1] and Levine [2].
1.2.1
LABORATORY AND CENTER OF MASS COORDINATES
Consider two atoms or molecules with masses m1 and m2 with positions r1 (t) and r2 (t) as a function of time t. (Bold type indicates vector quantities.) The initial velocities of the particles are given by v1 and v2 . After an elastic or inelastic collision (in which the chemical identities and masses of the collision partners do not change, see Section 1.2.3) the colliding particles have velocities v1 and v2 . These are the laboratory coordinates of the particles; most experiments provide information about collisions in the laboratory frame. For calculations on atomic and molecular collisions, it is almost always convenient to transform to center of mass coordinates. The centre of mass is at R, where R=
m 1
M
r1 +
m 2
M
r2 .
(1.1)
The center of mass moves like a particle with mass M and velocity V, where M = m1 + m2 m m m m 1 2 1 2 v1 + v2 = v1 + v2 . V= M M M M © 2009 by Taylor and Francis Group, LLC
(1.2)
Theory of Cold Atomic and Molecular Collisions
5
The classical kinetic energy T , linear momentum p, and angular momentum L may be written T=
1 1 1 1 m1 v12 + m1 v12 = MV 2 + μv 2 2 2 2 2
p = m1 v1 + m2 v2 = MV
(1.3)
L = m1 (r1 ∧ v1 ) + m2 (r2 ∧ v2 ) = μ(r ∧ v), where μ is the reduced mass, μ = m1 m2 /(m1 + m2 ), and r and v are the relative position and velocity vectors, r = r1 − r2 and v = v1 − v2 . The motion of the particles is governed by the potential energy U. If U is independent of the position of the center of mass, as is the case in field-free space or in the presence of homogeneous fields, the equations of motion (classical or quantum) factorize into separate equations for the center of mass and for the relative motion. The total linear momentum p is conserved in a collision, so the center of mass moves uniformly (constant velocity V) and is unaffected by the collision. The kinetic energy T is conserved in elastic collisions, and L is conserved if the angular momenta of the particles themselves do not change in magnitude or direction. Collision calculations are almost always carried out in the center of mass frame, which moves with the center of mass. The relative motion of the two particles is described as that of a single particle of mass μ subject to an interaction potential V (which is not the same asV in Equation 1.3). For structureless particles V depends only on the interparticle distance r, but more generally it also depends on internal degrees of freedom such as spins and rotations. Once the equations have been solved for the relative motion, they can, if desired, be transformed back into laboratory coordinates.
1.2.2
CROSS-SECTIONS AND RATE COEFFICIENTS
Consider the collision of two particles initially in internal states described by an index i. To simplify notation, it is convenient to use a single index to specify the states of both particles. The angle between the initial and final relative velocities v and v is given by spherical polar coordinates Θ and Φ, where Θ is the deflection angle in the center of mass frame. We start with a well-defined beam of particles with a flux Ii (number of particles per unit area per unit time). After the collision, the flux Ij (number of particles per unit solid angle per unit time) is a function of the deflection angle Θ and is different for each possible set of final internal states j. We define the differential cross-section as dσij Ij = , dω Ii
(1.4)
where ω is an element of solid angle at deflection angle Θ. The corresponding integral cross-section σij is integrated over all possible final directions, so it contains information about the total probability of the transition i → j, 2π π dσij σij = sin Θ dΘ dΦ. (1.5) dω 0 0 © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
Under most circumstances the differential cross-section is independent of Φ in the center of mass frame so that the integration over Φ in Equation 1.5 is equivalent to multiplication by 2π. The integral cross-section should not be confused with the total cross-section, which is usually understood to be summed over all possible final states j; thus it is possible to speak of total differential or total integral cross-sections. The differential cross-section has dimensions of area/solid angle, and the integral cross-section has dimensions of area. Cross-sections are functions of collision energy. However, many experiments are not monoenergetic and involve distributions of energy. The overall rate of a process i → j is given by the rate coefficient kij = vσij , where the extra factor v arises because the collision rate is proportional to v. When there is a distribution of relative velocities f (v) this must be further averaged, kij =
∞
vf (v)σij dv.
(1.6)
0
The velocity distribution function is normalized so that gases it is commonly a Maxwell distribution, f (v) =
μ 2πkB T
3/2
∞
0
f (v) dv = 1. For thermal
−μv 2 4πv exp 2kB T 2
,
(1.7)
but other distributions can be important in cold molecule studies.
1.2.3
ELASTIC, INELASTIC, AND REACTIVE SCATTERING
It is useful to classify the possible outcomes of collisions as elastic, inelastic, and reactive: • An elastic collision is one in which the kinetic energy of relative motion of the two particles does not change, though their individual kinetic energies and velocities may change. • An inelastic collision is one in which energy is transferred between internal and relative kinetic energy, but the chemical constitution of the collision partners is unchanged. • A reactive collision is one in which the products are chemically distinct from the reactants, usually also involving a change in relative kinetic energy. These definitions are not quite universal. Collisions in which the internal state of one or both collision partners changes are sometimes considered inelastic even if the total internal energy is unchanged, so that there is no change in relative kinetic energy. Collisions in which the possibility of atom transfer needs to be taken into account in the dynamics are often considered reactive, even if the final products are indistinguishable from the reactants. © 2009 by Taylor and Francis Group, LLC
Theory of Cold Atomic and Molecular Collisions
1.2.4
7
CENTRIFUGAL BARRIERS
It is usually convenient to consider collisions in spherical polar coordinates that describe the relative positions of the particles. In such a coordinate system the equations of motion for the interparticle distance r are governed by a centrifugally corrected effective potential, VL . For structureless particles where V depends only on r, |L|2 , (1.8) VL (r) = V (r) + 2μr 2 where |L| is the magnitude of the relative angular momentum of the two particles about one another. The interaction potential between two atoms or molecules dies off as r −n at long range, where n ≥ 2 except for collisions between two charged particles. Because the potential is attractive at some or all angles, the effective potential for |L| > 0 usually has a maximum. There is therefore a centrifugal barrier in the effective potential. In classical scattering at kinetic energies below the maximum, the particles cannot penetrate the barrier and do not enter the short-range region. In quantum scattering there is the possibility of tunneling into or through the barrier.
1.2.5
STRUCTURED PARTICLES
In practice, most of the collisions of interest in cold molecule studies involve collision partners with internal structure. Many of the atoms and molecules of interest have unpaired spins, and molecules have vibrational and rotational levels. Even nuclear hyperfine splittings, which are small enough to be neglected in many areas of chemistry, are large enough to be important in cold molecule studies. It is thus crucial to be able to handle collisions between atoms and molecules with internal structure. Although methods based on classical trajectories have had considerable success for inelastic and reactive collisions at room temperature or higher [3], for cold atoms and molecules it is almost always necessary to treat the collisions quantum-mechanically.
1.3
QUANTUM COLLISION THEORY
In quantum mechanics, atomic and molecular collisions may be handled using either time-dependent or time-independent approaches. Time-dependent (wavepacket) methods have recently become very popular for handling molecular collisions at temperatures around room temperature and higher [4,5]. However, at very low collision energies the usual time-dependent methods become hard to implement, both because it takes a very long time to complete a collision and because the absorbing potentials that are usually used at long range tend to cause unphysical reflections. It is thus usual to handle the low-energy collisions that are important in cold molecule studies using time-independent methods.
1.3.1
SINGLE-CHANNEL SCATTERING (FOR UNSTRUCTURED ATOMS)
Most of the processes of interest in cold collision studies involve structured particles, but there are some ideas that are most simply introduced for unstructured particles. © 2009 by Taylor and Francis Group, LLC
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We introduce a time-independent wavefunction Ψ that represents the complete two-particle collision system. For unstructured particles this depends only on r, the relative position vector. We represent the wavefunction in the absence of scattering as a plane wave Ψ0 = eik·r , where k is the wavevector describing the relative motion and its magnitude k is the corresponding wavenumber. The relative kinetic energy Ekin is 2 k 2 /2μ. This corresponds to an incoming flux density Ii = k/μ, which corresponds classically to the relative velocity v for a wave with unit particle density. The total wavefunction, including the effects of scattering, behaves asymptotically (at very large r) as r→∞
Ψ(r) ∼ Ψ0 (r) + f (Θ)
eikr , r
(1.9)
where the complex quantity f (Θ) is the scattering amplitude (which is zero in the absence of any collisions). The scattering is cylindrically symmetric about the initial relative velocity vector so it is convenient to expand the scattering amplitude in Legendre polynomials PL (cos Θ), f (Θ) =
∞
fL PL (cos Θ).
(1.10)
L=0
The incoming plane wave may also be expanded in Legendre polynomials using the identity ∞
eik·r = (2L + 1)iL PL (cos Θ)jL (kr), (1.11) L=0
where the functions jL (r) are spherical Bessel functions [6], which have the asymptotic form r→∞
jL (kr) ∼ (kr)−1 sin(kr − Lπ/2).
(1.12)
The quantum number L represents the orbital angular momentum for end-over-end motion of the two particles around one another. L has nothing to do with any electron orbital angular momentum that is present in the system. For single-channel scattering, different values of L are referred to as different partial waves. The Schrödinger equation for relative motion of two unstructured particles in free space is 2 2 (1.13) − ∇ + V (r) Ψ(r) = EΨ(r). 2μ It is usually convenient to take E = 0 to be the energy of the separated particles, so that V (r) → 0 as r → ∞. Ψ(r) may be expanded as
ψL (r)PL (cos Θ), (1.14) Ψ(r) = r −1 L
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and substituting this into Equation 1.13 gives independent equations for the different values of L, 2 d2 − + VL (r) ψL (r) = EψL (r), (1.15) 2μ dr 2 where the centrifugally corrected potential VL (r) is VL (r) = V (r) +
2 L(L + 1) . 2μr 2
(1.16)
This is simply the quantal form of the classical Equation 1.8. Although the wavefunction of Equation 1.9 is complex, it is sufficient to work with real solutions of Equation 1.15 that are regular at the origin (usually ψL → 0 as r → 0 provided VL (r) 0 at r = 0). Typical solutions are shown in Figure 1.1. They have the long-range form r→∞
ψL (r) ∼ krjL (kr + δL ),
(1.17)
Energy
where jL (kr) is a spherical Bessel function [6] and krjL (kr) (sometimes called a Riccati–Bessel function) is the solution of Equation 1.15 with V (r) = 0, i.e., for free
Scattering states
Centrifugal barrier
Dissociation energy
Bound states
r
FIGURE 1.1 Effective potential for single-channel scattering, showing centrifugal barrier and typical wavefunctions for bound and scattering states.
© 2009 by Taylor and Francis Group, LLC
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motion of noninteracting particles. At very long range, where the centrifugal potential is negligible, the solutions may be written r→∞
ψL (r) ∼ sin(kr − Lπ/2 + δL ).
(1.18)
The quantity δL is the phase shift for partial wave L and is zero in the absence of interaction, i.e., if V (r) is zero. The scattering amplitude may be written in terms of phase shifts as f (Θ) =
∞ 1
(2L + 1)(e2iδL − 1)PL (cos Θ). 2ik
(1.19)
L=0
The differential cross-section is dσ = | f (Θ)|2 dω
(1.20)
and thus involves a double sum over partial waves. Structure in the differential crosssection arises from interference between the contributions of different partial waves. The total elastic cross-section is ∞
4π
(2L + 1) sin2 δL . σ= 2 k
(1.21)
L=0
The equations above refer to scattering of distinguishable particles. For identical bosons (fermions), only even (odd) values of L are allowed and the prefactors are modified. The expressions for cross-sections involve sums over partial waves L that formally run from zero to infinity. In practice, however, for sufficiently high L the centrifugal barrier in the effective potential VL (r) keeps the particles far enough apart that there is no significant scattering. The usual approach is therefore to begin the summation at low L and keep adding contributions until the sum converges, i.e., until several successive L values make negligible contributions. There is an approximate correspondence between the partial wave quantum number L and the classical impact parameter b, which describes a collision in terms of the closest distance that the two particles would approach one another if they were noninteracting, b2 ∼
L(L + 1) . k2
(1.22)
In general terms, a collision can contribute to cross-sections if b is less than the distance outside which the interaction potential is negligible. For heavy-particle collisions at room temperature or above, it is common to need hundreds or even thousands of partial waves to converge the sums. For cold collisions, however, much smaller numbers of partial waves are needed and ultracold cross-sections are often dominated by a single partial wave: L = 0 for distinguishable particles or indistinguishable bosons and L = 1 for indistinguishable fermions. Low partial waves are sometimes referred to using electronic orbital terminology: s-wave, p-wave, d-wave for L = 0, 1, 2, and so on. © 2009 by Taylor and Francis Group, LLC
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Theory of Cold Atomic and Molecular Collisions
1.3.1.1
Bound States
When the interaction potential V (r) is attractive, the Schrödinger equation (1.15) may support bound-state solutions at energies E < 0 (below the dissociation energy) as well as scattering solutions at energies E > 0, as shown in Figure 1.1. In practice, almost all systems of interest support many bound states. The boundary conditions for bound states are different from those for scattering states, and in particular ψL (r) → 0
r→∞
as
or
r → 0.
(1.23)
For deep wells the low-lying bound states are often harmonic-oscillator-like, with a spacing that depends mainly on the curvature of the potential near the minimum. Just below dissociation (at E = 0), however, the bound states are densely packed. Numerical methods for finding bound-state solutions of Equation 1.15 are described in Section 1.3.1.3. However, in conceptual terms a considerable amount may be understood in terms of semiclassical arguments [7]. In semiclassical methods, the Schrödinger equation is expanded semiclassically in powers of . The resulting first-order JWKB (Jeffreys–Wentzel–Kramers–Brillouin) quantization condition gives remarkably accurate results for the vibration-rotation energies EvL of diatomic molecules: ΦL (EvL ) = (v + 21 )π,
(1.24)
where v is the vibrational quantum number and the phase integral ΦL (E) is ΦL (E) =
rmax
rmin
2μ[E − VL (r)] 2
1/2 dr.
(1.25)
The integral is taken between the classical turning points rmin and rmax at energy E, such that VL (rmin ) = VL (rmax ) = E. At energies near the potential minimum, where rmin and rmax are (relatively) close together, ΦL (E) increases relatively slowly with energy and the bound states are far apart. Conversely, near dissociation, where rmin and rmax are far apart, ΦL (E) increases quickly with energy and the bound states are close together. 1.3.1.2
Low-Energy Collisions
Collisions at low energy exhibit special behavior. It is useful to define a k-dependent scattering length aL by − tan δL . (1.26) aL (k) = k The s-wave scattering length a0 (k) (or often just a(k)) is particularly important because the s-wave elastic cross-section is given simply by σ(k) =
© 2009 by Taylor and Francis Group, LLC
4πa2 . 1 + a2 k 2
(1.27)
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Cold Molecules: Theory, Experiment, Applications
The scattering length becomes constant at very low energy, with leading correction terms given by effective range theory [8], 1 a(k) = a(0) + k 2 r0 a(0)2 + O(k 4 ), 2
(1.28)
where r0 is the effective range. The physical interpretation of the zero-energy scattering length a(0) (or often just a) is that, when E = V (r), the wavefunction ψL=0 (r) has zero curvature. As r → ∞, ψL=0 (r) for zero energy is thus a straight line. The extrapolation of this straight line cuts the distance axis at a separation R = a0 , which may be positive or negative. In the special case where there is a bound state exactly at zero energy, the zeroenergy wavefunction at long range is a straight line with zero slope, and the scattering length a0 is infinite. The quantization condition of Equation 1.24 suggests that for L = 0 there will be a zero-energy vibrational level and thus an infinite scattering length when Φ0 (0) = (v + 21 )π. However, this is not quite correct; Gribakin and Flambaum [9] have shown that, if V (r) behaves at long range as V (r) = −Cn /r n , the zero-energy scattering length is given semiclassically by
a0 = a¯ 1 − tan
π π tan Φ0 (0) − , n−2 2(n − 2)
(1.29)
where Φ0 (0) is evaluated at E = 0. The mean scattering length a¯ is
√ 2/(n−2) Γ n−3 n−2 π 2μCn a¯ = cos n−1 n − 2 (n − 2) Γ n−2
(1.30)
and is a function only of n, Cn , and the reduced mass μ. The scattering length given by Equation 1.29 is infinite when Φ 1 1 =v+ + π 2 2(n − 2)
(1.31)
instead of simply v + 21 . Boisseau and colleagues [10,11] have demonstrated that the semiclassical quantization condition of Equation 1.24 breaks down very close to dissociation and have explored corrections to it consistent with Equation 1.29. The zero-energy scattering length is very sensitive to the details of the interaction potential. If we consider multiplying the (attractive) interaction potential V (r) by a factor λ, the typical behavior of a0 for the single-channel case is shown in Figure 1.2. As the potential depth is reduced it supports fewer and fewer bound states, and every time there is a bound state at exactly zero energy there is a√pole in the scattering length. The quantity Φ0 (0) in Equation 1.29 is proportional to λ. If the potential supports nb bound states, then changing it by 2 parts in nb is enough to take the scattering length a0 through a complete cycle (from one pole to the next). A much smaller change in the potential can cause substantial changes in a0 , especially near poles in Figure 1.2. © 2009 by Taylor and Francis Group, LLC
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Theory of Cold Atomic and Molecular Collisions 10
a –a
5
0
–5
–10 0.9
0.95
1
1.05
1.1
l
FIGURE 1.2 The s-wave scattering length as a function of potential scaling factor λ: in this case λ = 1 corresponds to a potential with 100 bound states and the potential behaves at long range as V (r) = −C6 r −6 .
For this reason it is very difficult to obtain reliable scattering lengths from ab initio electronic structure calculations, except for systems with quite shallow wells (such as those for collisions with He). When there is a bound state just below dissociation, the scattering length is given by a=
[2 /(2μEbind )]
(1.32)
where Ebind is the binding energy of the highest level. More generally, for collisions of unstructured particles a0 is a function of the fractional part of the noninteger quantum number at dissociation vD ,
π 1 (1.33) tan vD + 2 π . a0 = a¯ 1 − tan n−2 1.3.1.3
Numerical Methods
Single-channel collision problems are usually handled by propagation methods, whereas bound-state problems can be handled either by propagation or by basis set methods. The general strategy in propagation methods is to define a grid of points between a distance rmin at very short range and a distance rmax at long range. It is useful to divide © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
the integration range into two or more regions. The inner classically forbidden region is the region near the origin where VL (r) > E. Conversely, the classically allowed region is the region where VL (r) < E. The classically allowed region is separated from the inner classically forbidden region by the inner turning point, with VL (r) = E; this is the point at which a classical particle would stop and turn around. For bound-state problems, where VL (r) > E at long range, there is also an outer classically forbidden region at large r where VL (r) > E, which is separated from the classically allowed region by the outer turning point. In general, of course, it is possible for there to be more than one classically allowed region and additional classical turning points. This is particularly common in cold molecule applications when the collision energy is lower than the height of a centrifugal barrier in VL (r). For a scattering calculation, it is sufficient to solve Equation 1.15 numerically by propagating ψL (r) outwards, starting at a point rmin deep in the inner classically forbidden region. In the classically forbidden region, the wavefunction curves away from zero according to Equation 1.15. Thus a solution that starts from zero will increase approximately exponentially until it reaches the inner turning point, then start to oscillate. The usual procedure is to propagate the solution out to a distance rmax and then to match the wavefunction and its derivative to Equation 1.17 to obtain the phase shift δL . This is then repeated for sufficient values of L to obtain convergence of the summations in the expressions for cross-sections. The propagation can be carried out using any of a variety of techniques for solving second-order differential equations, such as the renormalized Numerov method of Johnson [12]. For bound states the problem is a little more difficult, because solutions of Equation 1.15 that satisfy the boundary conditions Equation 1.23 do not exist at all energies. In general, if a solution of the differential equation that satisfies the boundary condition at one end of the range is calculated at an arbitrary energy, it will not satisfy the boundary condition at the other end of the range. Furthermore, propagating a solution of a differential equation into a classically forbidden region is a notoriously unstable procedure, because any exponentially increasing component quickly dominates the (usually desired) exponentially decreasing component, and any information about the exponentially decreasing component becomes lost in the numerical errors. This is not a problem when propagating a solution out of a classically forbidden region, because the exponentially increasing solution is then the one that is wanted. The approach usually adopted is to guess an energy Etrial and propagate a solution ψ+ (r) outwards from rmin to a point rmid in the classically allowed region. A second solution ψ− (r) is then propagated inwards to meet it from a point rmax , far into the outer classically forbidden region. The two solutions are arbitrarily normalized, so their values can always be made to agree at the matching point by renormalizing them. However, for a finite potential VL (r), the derivative ψ (r) of the wavefunction must be continuous, as well as its value ψ(r). The criterion for Etrial to be an eigenvalue is therefore [ψ+ ] (rmid )/ψ+ (rmid ) = [ψ− ] (rmid )/ψ− (rmid ), (1.34) where the superscripts + and − indicate outwards and inwards solutions originating at short and long range, respectively. The matching function [ψ+ ] /ψ+ − [ψ− ] /ψ− is thus a function of energy that is zero when E is an eigenvalue; it would be possible to © 2009 by Taylor and Francis Group, LLC
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15
converge on an eigenvalue simply by using one of the standard numerical procedures for finding a zero of a function, such as the secant method, to find a zero of the difference of the normalized derivatives. However, in the one-dimensional case, it is common to use Numerov propagation with an explicit energy correction formula due to Cooley [13], which gives quadratic convergence in the energy. This algorithm usually converges to ±10−6 cm−1 in less than 10 iterations, so that it offers a very efficient method for finding eigenvalues and eigenfunctions of one-dimensional Schrödinger equations.
1.3.2
MULTICHANNEL SCATTERING
The discussion of quantum scattering so far has concentrated on collisions between unstructured particles with only a single internal state. However, most systems of interest involve structured particles. The Hamiltonian for the colliding pair may generally be written 2 (1.35) − ∇ 2 + Hˆ int (τ) + V (r, τ), 2μ where τ denotes all coordinates except the interparticle distance r and Hˆ int (τ) represents the internal Hamiltonians of the two particles. The intermolecular potential V (r, τ) is now a function of τ as well as r. The wavefunction for a pair of colliding particles may be written in the form
Ψ(r, τ) = r −1 φi (τ)ψi (r), (1.36) i
where the channel functions {φi (τ)} form a basis set in the τ coordinates. The total wavefunction Ψ(r, τ) thus has components in each channel i. For both bound states and scattering, the wavefunction must be regular at the origin, and when V (r) 0 as r → 0 the short-range boundary conditions are ψi (r) → 0
as
r → 0.
(1.37)
The particular sets of channel functions needed are different for each type of scattering problem, and it would be quite impossible to enumerate all the possibilities, so just two relatively simple examples will be given here: atom–diatom scattering and collisions of metastable helium atoms. 1.3.2.1 Atom–Diatom Scattering Consider collisions between a structureless atom A and a closed-shell diatomic molecule BC, which is allowed to rotate but not vibrate. Nuclear spins are neglected. The orientation of the diatomic molecule is described by a unit vector rˆBC with spherical polar coordinates (θBC , φBC ). The corresponding rotational wavefunctions are spherical harmonics Yjmj (ˆrBC ) = Yjmj (θBC , φBC ). The position of the atom with respect to the center of mass of the diatom is given by spherical polar coordinates (r, rˆA ) = (r, θA , φA ) with corresponding spherical harmonics YLmL (ˆrA ). © 2009 by Taylor and Francis Group, LLC
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It is possible to expand the total wavefunction as in Equation 1.36 using simple products of spherical harmonics, φi (τ) = Yjmj (ˆrBC )YLmL (ˆrA ).
(1.38)
In this representation, the index i represents the set of quantum numbers { j, mj , L, mL }. This expansion is suitable under some circumstances, such as calculations on collisions in applied electric and magnetic fields. However, for field-free scattering it is inefficient because it fails to take advantage of the fact that the total angular momentum J is a good quantum number. It is therefore usual to use the Arthurs–Dalgarno total angular momentum representation [14], with basis functions that are eigenfunctions of Jˆ 2 and JˆZ as well as ˆj2 and Lˆ 2 . These are formed as linear combinations of the simple product functions,
JMJ φi (τ) = YjL = jmj LmL |JMJ Yjmj (ˆrBC )YLmL (ˆrA ), (1.39) mj mL
where jmj LmL |JMJ is a Clebsch–Gordan vector coupling coefficient. The index i thus represents the set of quantum numbers { j, L, J, MJ }. The Hamiltonian of the colliding pair is diagonal in the quantum numbers J and MJ (and independent of MJ ). It is therefore adequate to solve equations for each value of J separately, which results in very large savings in computer time. The computational efficiency may be further increased by taking advantage of other symmetries such as total parity. For each value of J, the channel basis set contains a set of functions with values of j and L that can couple together to give a resultant J. There are matrix elements involving the anisotropic part of the interaction potential that are off-diagonal in j and L. A common approach is to include all monomer rotational functions with j limited by a maximum value jmax , and for each j to include all allowed values of L of the desired parity (−1) j+L that satisfy the triangle condition | j − L| ≤ J ≤ j + L. Such calculations are often referred to as close-coupling calculations. The value of jmax required depends on the anisotropy of the interaction potential and it is of course always necessary to check convergence of the results with respect to jmax . The total angular momentum J plays the part of a partial wave quantum number. As for collisions of structureless particles, integral cross-sections involve sums over partial waves and differential cross-sections involve double sums, with interference between different partial waves playing an important role. 1.3.2.2
Scattering of Metastable Helium Atoms
States of individual atoms are usually described by quantum numbers L, S, and Ja for the electronic orbital, spin, and total angular momenta, respectively. However, in scattering and bound-state problems involving pairs of atoms or molecules it is common to use lower-case letters for quantum numbers of individual collision partners and reserve capital letters for quantities that refer to the collision system (or complex) as a whole. Thus, in this subsection we will use l and s for the quantum numbers of a single helium atom and reserve L and S for the end-over-end angular momentum of the atomic pair and the total spin, respectively. © 2009 by Taylor and Francis Group, LLC
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The helium atom has a metastable 3 S1 excited state with quantum numbers a = ja = 1 and a radiative lifetime of 8000 s. To carry out quantum scattering calculations on collisions of two such atoms, we need channel functions to handle the two spin quantum numbers and the mechanical angular momentum L. Once again it would be possible to use simple product functions as channel functions, |s1 ms1 |s2 ms2 |LML ,
(1.40)
but in the absence of external fields it is more efficient to use a total angular momentum representation. There is more than one way to construct such functions from three angular momenta, but the obvious one in this case is to couple s1 and s2 to give a total S and then couple S to L to give a resultant J, |s1 s2 SMS =
s1 ms1 s2 ms2 |SMS |s1 ms1 |s2 ms2 ;
ms1 ms2
|SL JMJ =
SMS LML |JMJ |SMS |LML .
(1.41)
MS ML
Once again, the resulting equations are diagonal in J and independent of MJ . Additional savings may be made by constructing functions that are symmetric with respect to exchange of identical atoms. In the case of metastable 4 He, the functions that are antisymmetric with respect to exchange may be discarded because the atoms are composite bosons.
1.3.3
COUPLED EQUATIONS
Substituting the expansion (1.36) into the total Schrödinger equation and projecting onto a single channel function φj (τ) yields a set of coupled equations,
2 d2 − − E ψ (r) = − Wji (r)ψi (r), j 2μ dr 2
(1.42)
i
where i and j are collective indices that each include all the quantum numbers needed to describe a channel. The coupling matrix W(r) generally contains off-diagonal as well as diagonal terms, Lˆ 2 ∗ Wji (r) = φj (τ) Hint (τ) + V (r, τ) + φi (τ) dτ. (1.43) 2μr 2 The coupling matrix W(r) does not become zero at long range, because different channels correspond to different energies of the separated monomers. Indeed, in some channel basis sets, W(r) may not even be diagonal at long range. However, it is almost always best to transform into a representation in which W(r) is diagonal, r→∞
Wii (r) ∼ wi δij , © 2009 by Taylor and Francis Group, LLC
(1.44)
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Cold Molecules: Theory, Experiment, Applications
before applying scattering boundary conditions. The energies wi , which correspond to levels of (pairs of) free monomers, are often referred to as thresholds because they represent energies at which pairs of monomers can be separated into the states concerned. The channels in a coupled-channel calculation at total energy E are usually divided into open channels, which correspond to energetically accessible monomer levels (E ≥ wi ), and closed channels, which correspond to energetically inaccessible monomer levels (E < wi ). The channel wavenumber ki for channel i is defined through the equation 2 ki2 = |E − wi | = |Ekin,i |, 2μ
(1.45)
where Ekin,i is the kinetic energy for channel i. In quantum scattering, we envisage a situation where there is an incoming wave in a single channel j, corresponding to the incident particles, and outgoing waves in all open channels, corresponding to scattered particles. The corresponding unnormalized wavefunction is asymptotically r→∞ −1
Ψ(r, τ) = r
−1/2 −ikj r+iLj π/2 φj (τ)kj e
+
−1/2 iki r−iLi π/2 Sji φi (τ)ki e
,
i
(1.46) where the sum runs over open channels only. There is a separate solution corresponding to each possible incoming channel, and the solution is characterized at long range by the S-matrix with elements Sji . The S-matrix is an Nopen × Nopen complex symmetric matrix, where Nopen is the number of open channels. It is unitary, that is, SS† = I, where S† indicates the Hermitian conjugate and I is a unit matrix. If the physical problem is factorized into separate sets of coupled equations for different symmetries (such as total angular momentum or parity), there is a separate S-matrix for each symmetry. All properties that correspond to completed collisions, such as elastic and inelastic integral and differential cross-sections, can be written in terms of S-matrices. There are usually several (or many) channels i, with different values of the orbital angular momentum Li , corresponding to each pair of internal states α of the colliding atoms or molecules. We indicate the set of channels corresponding to a particular set of monomer quantum numbers α as i ∈ α. The integral cross-section for molecules to undergo a transition from states α before the collision and to states β after the collision is π
σαβ = 2 gi |δij − Sij |2 , (1.47) kα i∈α j∈β
where the degeneracy factor gi is a function of the channel quantum numbers that depends on the particular channel basis set in use. In practice, the coupled equations usually factorize into sets with different symmetries (such as total angular © 2009 by Taylor and Francis Group, LLC
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Theory of Cold Atomic and Molecular Collisions
momentum), each with its own S-matrix, so that there is an additional sum over symmetries. 1.3.3.1
Scattering Calculations
If the basis set contains N channel functions of a particular symmetry, a single solution of the coupled equations 1.42 is a vector ψ(r) made up of N radial strength functions ψi (r) for i = 1 to N. However, at any energy there are N linearly independent solution vectors that satisfy the boundary conditions (1.37) at short range, and it is usually not possible to select a single one of them that is physically relevant until longrange boundary conditions are applied. In computational terms it is therefore usually necessary to solve for all N solutions simultaneously to obtain an N × N wavefunction matrix Ψ(r) with elements ψij (r). The radial strength functions ψij (r) are in general complex. However, it is almost always possible to transform the problem to work with real functions and to express the complex solutions, when needed, in terms of these. For both open and closed channels, the boundary conditions require that Ψ(r) → 0 at short range. However, at long range the scattering boundary conditions corresponding to real solutions are r→∞
Ψ(r) = J(r) + N(r)K,
(1.48)
where the S-matrix is related to the open-open part K oo of the real symmetric K-matrix by S = (1 + iK oo )−1 (1 − iK oo ).
(1.49)
The matrices J(r) and N(r) are diagonal matrices with elements for open channels given by [15] 1/2
jLj (kj r)
1/2
nLj (kj r),
[J(r)]ij = δij rkj [N(r)]ij = δij rkj
(1.50)
where jL and nL are spherical Bessel functions, and for closed channels given by [J(r)]ij = δij (kj r)−1/2 ILj +1/2 (kj r) [N(r)]ij = δij (kj r)−1/2 KLj +1/2 (kj r),
(1.51)
where IL and KL are modified spherical Bessel functions of the first and third kinds [6]. (Note that the Riccati–Bessel functions used by Johnson [15] are krjL (kr) and krnL (kr) in the notation of Ref. [6].) The use of Bessel functions in place of sines and cosines takes into account the effect of the centrifugal terms in W(r) and makes it possible to apply the boundary conditions at much shorter range. 1.3.3.2
Numerical Methods for Scattering
As in the single-channel case, the coupled-channel equations are usually solved by numerical propagation. One approach is to start a solution matrix Ψ(r) at a distance © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
rmin , far enough into the classically forbidden region (in all channels) to assume that Ψ(rmin ) = 0. The solution is propagated outwards to rmax , which is chosen to be at long enough range that the effects of the potential V (r, τ) are negligible outside rmax . The wavefunction matrix and its derivative are then matched to the boundary conditions Equation 1.48 to obtain the real symmetric matrix K. This is then transformed using Equation 1.49 to obtain the physically meaningful S-matrix. There are many different propagators that have been developed to take account of the special properties of Equation 1.42, and a detailed discussion is beyond the scope of this chapter. Among those commonly used in cold molecule collisions are the renormalized Numerov approach [16,17] and several different log-derivative propagators [15,18,19]. The latter actually propagate the log-derivative matrix, defined as Y(r) =
dΨ [Ψ(r)]−1 , dr
(1.52)
in place of the wavefunction itself. The K matrix can be obtained from the logderivative matrix at rmax without requiring the wavefunction matrix explicitly [15]. The log-derivative propagators have good step-size convergence properties and do not suffer from stability problems arising from the presence of deeply closed channels. The airy-based log-derivative propagator of Alexander and Manolopoulos [19] has particular advantages for cold collisions because it allows very large propagation steps at long range. A variety of program packages are available to construct and solve sets of coupled equations for molecular quantum scattering calculations. These include the MOLSCAT package of Hutson and Green [20] and the HIBRIDON package of Alexander et al. [21]. 1.3.3.3
Decoupling Approximations
The computer time taken to solve a set of coupled equations is generally proportional to N 3 , where N is the number of channel basis functions in the expansion of Equation 1.36. Even for atom–diatom scattering, the size of the basis set required for close-coupling calculations can be prohibitively large, especially for systems with small rotational constants or strongly anisotropic potential energy surfaces. For more complicated systems, such as molecule–molecule scattering or for problems involving open-shell or nonlinear molecules, the problem is substantially worse. There are thus a number of decoupling approximations that factorize the coupled equations into smaller sets that can be solved separately. Approaches that have been popular for collisions at higher energies include the CS (coupled states or centrifugal sudden) approximation and the IOS (infinite-order sudden) approximation. A detailed discussion of these is beyond the scope of this chapter, but there is a good (if old) review by Kouri [22]. Decoupling approximations can save enormous amounts of computer time and can often provide approximate solutions for otherwise intractable scattering problems. However, they necessarily involve approximations, and must be treated with great care for cold molecule interactions. In particular, the CS and IOS approximations in their usual form both approximate the end-over-end angular momentum © 2009 by Taylor and Francis Group, LLC
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Theory of Cold Atomic and Molecular Collisions
operator Lˆ 2 and thus affect centrifugal barriers, which can be extremely important in the ultracold regime. Nevertheless, carefully crafted decoupling approximations can be of great value. 1.3.3.4
Bound States
Provided the interaction potential is sufficiently attractive (which it almost always is), the multichannel Schrödinger equation supports bound states at energies En below the lowest threshold. The corresponding wavefunction for state n is Ψn (r, τ) = r −1
φi (τ)ψin (r).
(1.53)
i
The boundary conditions for bound-state problems are ψin (r) → 0
as
r→0
or
∞
(1.54)
in all channels i. Multichannel bound-state problems are much more complicated than singlechannel problems. Although to a first approximation it may be true that each bound state is supported primarily by a single channel with a well-defined potential well, this is not generally the case. The coupling between channels provided by W(r) mixes and shifts the levels in a nontrivial way that is outside the scope of this chapter but has been explored extensively in the context of the spectroscopy of Van der Waals complexes [23]. 1.3.3.5
Numerical Methods for Bound States
There are two general approaches to the calculation of bound states of multichannel systems: coupled-channel methods and radial basis set methods. Coupled-channel methods [24] operate by direct numerical solution of Equation 1.42, propagating either a wavefunction matrix or a derived quantity such as the log-derivative matrix (Eqution 1.52) across a grid of r values. As in the singlechannel case, the propagation is carried out outwards from a distance rmin at very short range and inwards from a distance rmax at long range to a matching point rmid in the classically allowed region. Solutions that satisfy bound-state boundary conditions (Equation 1.54) exist only for certain values of the energy E, so an extra layer of calculation is needed to locate the energies (eigenvalues). In the multichannel case, the desired wavefunction is a column vector ψn (r) whose elements are the channel wavefunctions ψin (r). In order to start propagating a solution to the coupled equations, it is necessary to know not only the initial values of the functions ψin (r) at rmin and rmax (which are given by the boundary conditions) but also their derivatives (which are not). It is therefore once again necessary to propagate an N × N wavefunction matrix Ψ(r), made up of a complete set of N vectors that spans the space of all possible initial derivatives. The particular wavefunction vector that is continuous (and has a continuous derivative) at the matching point is not identified until after converging on an energy eigenvalue. © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
Because the columns of Ψ(r) span the space of all possible initial derivatives, any wavefunction that satisfies the boundary conditions can be expressed as a linear combination of them. The true wavefunction vector ψn (r) can thus be expressed as ψn (r) = Ψ+ (r)C + , −
−
ψn (r) = Ψ (r)C ,
R ≤ rmid ; R ≥ rmid ,
(1.55)
where Ψ+ (r) and Ψ− (r) are the wavefunction matrices propagated from short and long range and C + and C − are r-independent column vectors that must be found. For an acceptable wavefunction, both ψn and its derivative must match at R = rmid , ψn (rmid ) = Ψ+ (rmid )C + = Ψ− (rmid )C − ; ψn (rmid ) = [Ψ+ ] (rmid )C + = [Ψ− ] (rmid )C − ,
(1.56)
where the primes indicate radial differentiation. Gordon [25] combined these two equations into the single equation + − + − + Ψ Ψ [Ψ ] [Ψ ] C − C − = 0, (1.57) where the matrix on the left-hand side is evaluated at R = rmid . It is a matrix of order 2N × 2N, and is a function of the energy at which the wavefunctions are calculated. A nontrivial solution of Equation 1.57 exists only if the determinant of the matrix on the left is zero, and this is true only if the energy used is an eigenvalue En of the coupled equations. In Gordon’s method, eigenvalues of the coupled equations are located by solving the coupled equations at a series of trial energies E and searching for energies at which the determinant is zero. It is then straightforward to find the transformation vectors C + and C − . Propagating wavefunctions explicitly in the presence of deeply closed channels is notoriously numerically unstable. It is much more satisfactory to use a propagator that is stable in the presence of closed channels, such as one of the log-derivative propagators described above. The multichannel matching condition can be expressed very simply in terms of the log-derivative matrix Y(r) of Equation 1.52. If E is an eigenvalue of the coupled equations, there must exist a wavefunction vector − ψn (rmid ) = ψ+ n (rmid ) = ψn (rmid ) for which − [ψ+ n ] (rmid ) = [ψn ] (rmid ),
(1.58)
Y + (rmid )ψn (rmid ) = Y − (rmid )ψn (rmid ),
(1.59)
[Y + (rmid ) − Y − (rmid )]ψn (rmid ) = 0.
(1.60)
so that
or equivalently
Thus the wavefunction ψn (rmid ) is an eigenvector of the matching matrix Y + (rmid ) − Y − (rmid ) with eigenvalue zero. Eigenvalues En of the coupled equations can be located © 2009 by Taylor and Francis Group, LLC
Theory of Cold Atomic and Molecular Collisions
23
by searching for energies at which the log-derivative matching matrix has a zero eigenvalue, using standard methods for finding zeroes of a function, such as the secant method. The propagator methods described here are implemented in the widely distributed BOUND package [26]. However, most bound-state problems of interest in cold molecule studies, such as those involving open-shell species and applied magnetic fields [27,28] require extensions to the distributed code. A quite different approach for calculating bound states is to expand the radial functions ψin (r) in a basis set of functions {χj (r)}, so that the full wavefunction is expanded
cijn φi (τ)χj (r). (1.61) Ψn (r, τ) = ij
The approach is then simply to calculate the complete Hamiltonian matrix in the product representation and diagonalize it to find the eigenvalues and eigenvectors. This approach is generally preferred for low-lying (deeply bound) states, because in that case quite compact radial basis sets can be used. Methods of this general type have been extensively reviewed in the context of the vibration-rotation spectroscopy of Van der Waals complexes and other molecules that exhibit wide-amplitude motion [29–31]. General-purpose programs are available for closed-shell triatomic and tetraatomic molecules [32,33]. However, these methods present some difficulties for the near-dissociation states that are important for cold molecule studies. The general problem is that quite large basis sets of radial basis functions {χj (r)} are often needed to describe both the short-range and long-range parts of the potential. For N channels and Nr radial basis functions, the full Hamiltonian matrix is of dimension NNr and the time taken to diagonalize it is proportional to N 3 Nr3 . By contrast, the time taken by propagation methods is proportional to N 3 but is linear in the number of propagation steps. There has nevertheless been a considerable amount of work on methods to calculate near-dissociation bound states using product basis sets [34,35].
1.3.4
QUASIBOUND STATES AND SCATTERING RESONANCES
True bound states can exist only at energies below the lowest threshold of the same symmetry as the state concerned. However, quasibound states can exist at energies above threshold, as shown in Figure 1.3. These are (relatively) long-lived states that can be seen in spectroscopy in much the same way as true bound states, but are coupled to a continuum and decay (dissociate) spontaneously. As a quasibound state has a finite lifetime τ, it has a width in energy ΓE rather than a precisely defined eigenvalue, ΓE = /τ.
(1.62)
Quasibound states also produce sharp features in collision properties as a function of energy, and in this context they are usually referred to as scattering resonances. There are two different types of scattering resonance that can occur, as shown schematically in Figure 1.3. A shape resonance corresponds to a state that is confined © 2009 by Taylor and Francis Group, LLC
Energy
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Cold Molecules: Theory, Experiment, Applications
Quasibound states (Feshbach resonances) Centrifugal barrier Quasibound state (shape resonance) Dissociation energy
Bound states
r
FIGURE 1.3 The different types of bound and quasibound states and scattering resonances. For the purpose of illustration only two potential curves are shown, corresponding to the lowest channel and one excited channel.
behind a barrier in a centrifugally corrected potential (Equation 1.16). A shape resonance can exist even in single-channel scattering and decays by tunneling through the barrier. Conversely, a Feshbach resonance [36] (sometimes called a compound state resonance) corresponds to a state that is supported by an attractive well in channels that are energetically closed at the energy of interest. It decays by curve-crossing to the open channel. If there is only one open channel, a resonance (of either type) is characterized by a sharp change in the phase shift δL , which increases sharply by π across the width of the resonance. The phase shift follows a Breit–Wigner form as a function of energy,
ΓE −1 , (1.63) δL (E) = δbg + tan 2(Eres − E) where δbg is a slowly varying background term, Eres is the resonance position, and ΓE is its width (in energy space). In general the parameters δbg , Eres and ΓE are weak functions of energy, but this is often neglected for reasonably narrow resonances. The 1 × 1 S-matrix for single-channel scattering is given by SL (E) = e2iδL (E) .
(1.64)
As δL (E) is real, SL (E) describes a circle of radius 1 in the complex plane across a resonance. © 2009 by Taylor and Francis Group, LLC
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25
When there is more than one open channel, the complex symmetric scattering matrix S has elements Sij . The diagonal S-matrix element in incoming channel i has magnitude |Sii | ≤ 1 and may be written in terms of a complex phase shift δi (E) with a positive imaginary part [37], Sii (E) = e2iδi (E) .
(1.65)
In the multichannel case, the quantity that follows the Breit–Wigner form of Equation 1.63 is the S-matrix eigenphase sum [38,39], which is the sum of the phases obtained from the eigenvalues of the S-matrix. The eigenphases and the eigenphase sum are real, unlike the phases δi obtained from individual diagonal elements, because the S-matrix is unitary, so that all its eigenvalues have modulus 1. Across a resonance, the individual S-matrix elements describe circles in the complex plane [40], Sij (E) = Sbg,ij −
igEi gEj , E − Eres + iΓE /2
(1.66)
where gEi is complex. The radius of the circle in Sij is |gEi gEj |/ΓE , which is usually less than 1. The partial width for channel i is commonly defined as a real quantity, ΓEi = |gEi |2 . For a narrow resonance, the total width is just the sum of the partial widths, ΓE =
ΓEi .
(1.67)
i
The physical interpretation of the partial width is that, when the quasibound state corresponding to the resonance decays, the fractional population produced in product channel i is ΓEi /ΓE .
1.3.5
LOW-ENERGY SCATTERING
At low energies, scattering is commonly dominated by one or a few partial waves and the cross-sections σ(k) may be decomposed into partial wave contributions σL (k) from different incoming partial waves L. The limiting low-energy behavior for different values of L is given by the Wigner threshold laws [41]. For elastic cross-sections, tan δL (k) ∼ k 2L+1
and
σel,L ∼ k 4L ,
(1.68)
where k is the wavenumber in the incoming channel. For a potential that behaves as V (r) = −Cn r −n at long range (with n > 2), there is an additional L-independent term that dominates at high L [42], tan δL (k) ∼ k n−2 © 2009 by Taylor and Francis Group, LLC
and
σel,L ∼ k 2n−6 .
(1.69)
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Cold Molecules: Theory, Experiment, Applications
For inelastic (deexcitation) cross-sections, σinel,L ∼ k 2L−1 .
(1.70)
For the purpose of the Wigner laws, a transition is classed as inelastic if the kinetic energy released is large compared to the centrifugal barrier in the outgoing channels. At low energies, the complex phase shift defined by Equation 1.65 can be expressed in terms of a complex energy-dependent scattering length, a(k) = α(k) − iβ(k) [43], defined by analogy with Equation 1.26 as − tan δi (k) 1 a(k) = = k ik
1 − Sii (k) . 1 + Sii (k)
(1.71)
This expression is to be preferred to the common approximation, a(k) ≈
−Im Sii , ik
(1.72)
because Equation 1.71 remains valid close to resonance when a(k) is large and |1 − Sii | is not close to zero. For an incoming channel with L = 0 (s-wave scattering), the scattering length a(k) becomes constant at limitingly low energy. The elastic and total inelastic cross-sections are exactly [44] σel (k) =
4π|a|2 1 + k 2 |a|2 + 2kβ
(1.73)
and tot σinel (k) =
1.3.6
4πβ . k(1 + k 2 |a|2 + 2kβ)
(1.74)
COLLISIONS IN EXTERNAL FIELDS
Collisions in applied electric and magnetic fields are important for several reasons. Most importantly, applied fields provide ways to control ultracold gases, and for atomic systems this control has already led to the observation of a wide range of new phenomena including molecule formation and transitions to superfluid phases [45]. There is little doubt that applied fields will provide the key to controlling the much richer behavior expected for ultracold molecules. A subsidiary issue is that cold molecules are often manipulated or trapped using applied electric or magnetic fields. Such methods generally rely on the molecules remaining in a particular quantum state, and are disrupted if they undergo transitions to different states. In addition, inelastic collisions that convert internal energy into kinetic energy can eject both collision partners from the trap if the kinetic energy released is greater than the trap depth. The most important effect of an applied electric or magnetic field is that the energy levels of the colliding monomers split and shift by amounts that can be quite large © 2009 by Taylor and Francis Group, LLC
Theory of Cold Atomic and Molecular Collisions
27
compared to the kinetic energy. The shifts in the positions of scattering thresholds have profound effects on collision properties, as will be seen below. Another effect of an applied field is that it breaks the isotropic character of space, so that the total angular momentum is no longer a good quantum number. This presents computational problems because much greater numbers of channels are coupled together. Many problems that are computationally tractable in the absence of fields become intractable when fields are applied. Despite the proliferation of channels and the need for more complicated Hamiltonians, the formal theory of atomic and molecular collisions is not much changed in the presence of fields. Applied fields do reduce symmetry, so that collisions no longer take place in an isotropic environment: total angular momentum is no longer conserved, and differential cross-sections need not be cylindrical symmetrical in the center-of-mass frame. However, in computational terms, it is still necessary to solve sets of coupled differential equations subject to bound-state and scattering boundary conditions. Perhaps the most important effect is that applied fields provide access to resonance phenomena that do not occur in field-free scattering. In particular, they provide access to zero-energy Feshbach resonances, which are at the heart of most approaches to controlling ultracold gases.
1.3.6.1
Basis Sets without Total Angular Momentum
As described in Section 1.3.2 above, field-free scattering calculations are usually carried out in total angular momentum representations. The equations arising from each value of the total angular momentum are independent and may be solved separately. In the presence of an applied field, this symmetry is lost and there is much less advantage in working in a total angular momentum representation. Several cases may be envisaged: • In the presence of a magnetic field with no electric field, the quantities that are conserved are the projection Mtot of the total angular momentum onto the field axis and the total parity [46–48]. • In the presence of an electric field but no magnetic field, Mtot is again preserved but the total parity is lost. Pairs of states with ±|Mtot | are degenerate with one another. States with Mtot = 0 separate into blocks that are even and odd with respect to reflection in a plane containing the field axis. • In the presence of collinear electric and magnetic fields, only Mtot is conserved. • In the presence of electric and magnetic fields that are not collinear, there is no symmetry at all and even Mtot is lost [49]. As in field-free calculations, there are very many basis sets that may be used in different circumstances. A totally uncoupled representation is often convenient because the matrix elements needed are usually quite simple. For example, for collisions of a molecule in a 3 Σ state with an unstructured atom we need basis functions to handle the molecular rotation n, electron spin s, and end-over-end angular momentum © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
L. In this case the totally uncoupled representation is made up of simple product functions φi (τ) = |nmn |sms |LML .
(1.75)
However, other basis functions sometimes describe the real eigenfunctions more compactly and may be useful in arriving at physical interpretations. In this case the functions that describe the diatomic molecule most efficiently at low fields are obtained by coupling n and s to give a resultant j, which does not appear in the basis set of Equation 1.75. 1.3.6.2
Spin Relaxation
Paramagnetic molecules are often confined in magnetic traps, with a magnetic field that is zero at the center of the trap and increases in all directions away from it. Such an arrangement traps molecules in low-field-seeking states, whose energy increases away from the center, but not those in high-field-seeking states. However, because the lowest-energy state in a field is always high-field-seeking, it is always possible for molecules to be transferred from trapped states to untrapped states by inelastic collisions. The rate of this process limits the time for which molecules can be trapped. Calculations on the collisions of cold molecules in fields are discussed in more detail in this book in the chapter by Tscherbul and Krems (Chapter 4). 1.3.6.3
Zero-Energy Feshbach Resonances
Vibrational energy levels are quite closely packed near thresholds. In cold molecule studies, it is common to have several closely spaced thresholds, corresponding to different nuclear hyperfine states or other small energy splittings of the colliding monomers. Because the bound states of the two-body system usually have different Stark and Zeeman effects from the constituent monomers, near-dissociation bound states may often be tuned across thresholds with applied electric and magnetic fields. An example of this is shown in Figure 1.4. At the point where the bound state and the threshold exactly coincide, there is a zero-energy Feshbach resonance. Consider the case of a zero-energy Feshbach resonance as a function of magnetic field B. At constant Ekin , the phase shift follows a form similar to Equation 1.63,
ΓB , (1.76) δ(B) = δbg + tan−1 2(Bres − B) where Bres is the field at which Eres = E = Ethresh + Ekin . An example of this is shown in Figure 1.5. The width ΓB is a signed quantity given by ΓB = ΓE /Δμ, where the magnetic moment difference Δμ is the rate at which the energy Ethresh of the open-channel threshold tunes with respect to the resonance energy, Δμ =
dEres dEthresh − . dB dB
(1.77)
ΓB is thus negative if the bound state tunes upwards through the energy of interest, as in the case in Figure 1.5. © 2009 by Taylor and Francis Group, LLC
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Theory of Cold Atomic and Molecular Collisions
0
(1, 1)–1
(mfin1 mfin2) (–1, 1) (0, 0) (0, 1)
(1, 1)–2
Energy/h (GHz)
–1
(1, 2)–4 (1, 2)–4
(1, 1)
–2
–3
(1, 1)–3 (2, 2)–5 (2, 2)–5 (2, 2)–5
–4 (f1, f2) v ¢ 0
200
400 600 Magnetic field (G)
800
1000
1200
FIGURE 1.4 Tuning of molecular levels (solid lines) and atomic thresholds (dashed lines) for 87 Rb as a function of magnetic field. Feshbach resonances occur at the points marked with 2 filled circles, where a molecular state crosses a threshold. (From Marte, A. et al., Phys. Rev. Lett., 89, 283202, 2002. With permission. Copyright 2002, the American Physical Society.)
When there is only one open channel, the S-matrix element S = e2iδ(B) describes a circle of radius 1 in the complex plane. In the ultracold regime, the background phase shift δbg goes to zero as k → 0 according to Equation 1.26 (with aL (k) replaced by abg , which is finite and becomes constant as k → 0). However, the resonant term still exists. The scattering length passes through a pole when δ(B) = n + corresponding to S = −1. The scattering length follows the formula [51],
π,
a(B) = abg
ΔB 1− . B − Bres
1 2
(1.78)
An example of this is shown in Figure 1.6. The elastic cross-section given by Equation 1.73 thus shows a sharp peak of height 4π/k 2 at resonance. The two widths ΓB and ΔB are related by ΓB = −2abg kΔB .
(1.79)
At limitingly low energy, ΓB is proportional to k while ΔB is constant [52]. For multichannel scattering, the situation is more complicated. As a function of magnetic field, the scattering length passes through a pole only if δ0 (B) passes through © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
10.0
Eigenphase sum/p
9.8
9.6
9.4
9.2
9.0 7168.7553
7168.7550
7168.7556
7168.7559
Magnetic field (G)
FIGURE 1.5 Phase shift for elastic 3 He–NH collisions in the vicinity of an elastic Feshbach resonance at a kinetic energy of 10−6 K. (From González-Martínez, M.L. and Hutson, J.M., Phys. Rev. A, 75, 022702, 2007. With permission. Copyright 2007, the American Physical Society.)
n + 21 π, corresponding to S00 = −1. If there is any inelastic scattering, the radius of the resonant circle in S00 is proportional to k at low energy and this does not occur. The formula followed by the complex scattering length is [53] a(B) = abg +
ares , 2(B − Bres )/Γinel B +i
(1.80)
where ares is a resonant scattering length that characterizes the strength of the resonance. When ares is small, the scattering length shows only small peaks or oscillations across a resonance. Both ares and the background term abg can in general be complex and are independent of kinetic energy at low energy. When ares is complex, the total inelastic cross-section given by Equation 1.74 can show troughs as well as peaks in the vicinity of a resonance.
1.4
REACTIVE SCATTERING
Reactive scattering, in which the products of a collision are chemically different from the reactants, is formally similar to inelastic scattering. However, there are complications that arise from the fact that a basis set that efficiently describes the reactants is usually inefficient to describe the products and vice versa. Even the coordinate system to be used requires some care: for example, Jacobi coordinates © 2009 by Taylor and Francis Group, LLC
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Theory of Cold Atomic and Molecular Collisions 3 × 104 Ekin = 1.0 mK
Re (scattering length) (Å)
2 × 104
1 × 104
0 × 100
–1 × 104
–2 × 104
–3 × 104 7168.7550
7168.7553
7168.7556
7168.7559
Magnetic field (G)
FIGURE 1.6 Scattering length for elastic 3 He–NH collisions in the vicinity of an elastic Feshbach resonance, from calculations at a kinetic energy of 10−6 K. (From GonzálezMartínez, M.L. and Hutson, J.M., Phys. Rev. A, 75, 022702, 2007. With permission. Copyright 2007, the American Physical Society.)
are usually inappropriate because the reduced mass is different in the reactant and product coordinate systems. The approach most commonly used to circumvent these difficulties, at least for triatomic systems, is to express the problem in hyperspherical coordinates, with the relative positions described by a single radial coordinate ρ (the hyperradius) and two hyperangles. The different atomic arrangements simply correspond to different values (or ranges of values) of the hyperangles, and a single basis set is capable of describing all arrangements. The time-independent scattering problem again reduces to a set of coupled differential equations that can be solved by propagation as in the inelastic case. Several variants of hyperspherical coordinates have been used, and a general comparison has been given by Pack and Parker [54]. There are a variety of channel basis sets that have been used by different authors: in some approaches the basis functions are solutions of a fixed-ρ Schrödinger equation that is solved at each value of the hyperradius [54–56], and in others the basis functions represent the reactants and products [57]. At long range (in the hyperradius ρ), the solutions are transformed back into the basis sets appropriate for the reactants or products individually and matched onto Bessel functions to give the scattering S-matrix as described in Section 1.3.3.1. These methods, together with time-dependent (wavepacket) approaches for reactive scattering, have been recently reviewed by Hu and Schatz [58]. © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
The time-independent procedure works well for atom–diatom reactions involving light atoms that proceed on a single potential energy surface. A general-purpose program to solve the time-independent scattering problem in this case is available [57]. However, applying the theory outside this narrow range of systems is not yet routine. Low-energy reactive scattering calculations have been carried out for single-surface systems as heavy as K + K2 [59]. Calculations have also been carried out at higher energy for light systems such as F + H2 [60], where the reaction occurs on several surfaces coupled by spin–orbit interactions. However, heavy-atom systems involving multiple surfaces are not yet tractable, even for triatomic systems. Methods for performing full-dimensional quantum scattering calculations on four-atom and larger systems have been developed [61,62], but are mostly based on time-dependent wavepacket approaches that are not easily applicable to ultracold collisions. There has so far been relatively little work on reactive collisions in the presence of external fields, but Tscherbul and Krems [63] have described the general theory required in the presence of an electric field and have carried out pilot calculations on the LiF + H ↔ Li + FH reactions.
1.4.1
LANGEVIN MODEL FOR BARRIERLESS REACTIONS
There is a considerable class of chemical reactions for which the potential energy surfaces are barrierless with deep wells at short range. These include many ion–molecule reactions as well as atom exchange reactions in atom–molecule and molecule– molecule collisions involving alkali metal dimers. In the ultracold domain where only one or a few partial waves contribute to collision cross-sections, such reactions still require a full quantum treatment. However, for strongly exothermic barrierless reactions at slightly higher temperatures (several mK) it is a reasonable approximation that all collisions that sample the short-range region of the potential surface lead to reaction. This is the basis of the Langevin capture model [64]. The effective potential VL (r) for a partial wave with L > 0, Equation 1.16, is governed at long range by the centrifugal and dispersion terms, VL (R) =
2 L(L + 1) C6 − 6, 2μR2 R
(1.81)
where C6 is the atom–molecule or molecule–molecule dispersion coefficient. There is thus a centrifugal barrier at a distance
RLmax with height
VLmax
=
6μC6 = 2 L(L + 1)
2 L(L + 1) μ
1/4 (1.82)
3/2 (54C6 )−1/2 .
(1.83)
At collision energies below the centrifugal barrier in the incoming channel, the partial cross-sections for each L follow Wigner laws given by Equation 1.70. If there are © 2009 by Taylor and Francis Group, LLC
33
Theory of Cold Atomic and Molecular Collisions
many open channels, however, the inelastic probabilities above the centrifugal barrier come close to their maximum possible value of 1. The cross-sections then vary as E −1 because of the k −2 factor in the expression for the cross-section. At collision energies high enough that many partial waves are involved, the total inelastic cross-section and rate coefficient become 1/3 C6 capture σinel (E) = 3π ; 4E (1.84) 1/3 1/2 1/3 3πC6 E 1/6 2E C6 capture = . kinel (E) = 3π 4E μ 21/6 μ1/2 For atom–molecule and molecule–molecule reactions, the centrifugal barriers for low partial waves are typically a few mK or less. For barrierless reactions, therefore, Langevin behavior sets in at temperatures between 1 and 100 mK. Above this temperature, the details of the short-range potential become unimportant. An example of this is shown in Figure 1.7, which shows inelastic collision rates for Li + Li2 for boson and fermion dimers initially in v = 1 and 2 [65]. The full quantum result approaches the Langevin value at collision energies above about 10 mK.
Rate coefficient (cm3/s–1)
10−9
Bosons: vi = 1 Bosons: vi = 2 Fermions: vi = 1 Fermions: vi = 1
10−10
10−6
Langevin model
10−5
10−4
10−3 Collision energy (K)
10−2
10−1
FIGURE 1.7 Total inelastic rate coefficients for collisions of Li with Li2 (v = 1 and 2, with j = 0 for boson dimers and j = 1 for fermion dimers). The dotted line shows the result of Langevin capture theory. (From Critaš, M.T. et al., Phys. Rev. Lett., 94, 033201, 2005. With permission.)
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Cold Molecules: Theory, Experiment, Applications
REFERENCES 1. Child, M.S., Molecular Collision Theory, Academic Press, London, 1974. 2. Levine, R.D., Molecular Reaction Dynamics, Cambridge University Press, 2005. 3. Aoiz, F.J., Banares, L., and Herrero, V.J., Recent results from quasiclassical trajectory computations of elementary chemical reactions, J. Chem. Soc. Faraday Trans., 94, 2483–2500, 1998. 4. Althorpe, S.C. and Clary, D.C., Quantum scattering calculations on chemical reactions, Annu. Rev. Phys. Chem., 54, 493–529, 2003. 5. Althorpe, S.C., The plane wave packet approach to quantum scattering theory, Int. Rev. Phys. Chem., 23, 219–251, 2004. 6. Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, 9th edn., Dover, New York, 1964. 7. Child, M.S., Semiclassical Mechanics with Molecular Applications, Oxford University Press, Oxford, 1991. 8. Hinckelmann, O. and Spruch, L., Low-energy scattering by long-range potentials, Phys. Rev. A, 3, 642, 1971. 9. Gribakin, G.F. and Flambaum, V.V., Calculation of the scattering length in atomic collisions using the semiclassical approximation, Phys. Rev. A, 48, 546, 1993. 10. Boisseau, C., Audouard, E., and Vigué, J., Quantization of the highest levels in a molecular potential, Europhys. Lett., 41, 349–354, 1998. 11. Boisseau, C., Audouard, E., Vigué, J., and Flambaum, V.V., Analytical correction to the WKB quantization condition for the highest levels in a molecular potential, Eur. Phys. J. D, 12, 199–209, 2000. 12. Johnson, B.R., Renormalized Numerov method applied to calculating bound states of coupled-channel Schrödinger equation, J. Chem. Phys., 69, 4678–4688, 1978. 13. Cooley, J.W., An improved eigenvalue corrector formula for solving the Schrödinger equation for central fields, Math. Comput., 15, 363–374, 1961. 14. Arthurs, A.M. and Dalgarno, A., The theory of scattering by a rigid rotator, Proc. Roy. Soc., Ser. A, 256, 540–551, 1960. 15. Johnson, B.R., Multichannel log-derivative method for scattering calculations, J. Comput. Phys., 13, 445–449, 1973. 16. Johnson, B.R., New numerical methods applied to solving one-dimensional eigenvalue problem, J. Chem. Phys., 67, 4086, 1977. 17. Colavecchia, F.D., Mrugała, F., Parker, G.A., and Pack, R.T, Accurate quantum calculations on three-body collisions in recombination and collision-induced dissociation. II. The smooth-variable discretization-enhanced renormalized Numerov propagator, J. Chem. Phys., 118, 10387–10398, 2003. 18. Manolopoulos, D.E., An improved log-derivative method for inelastic scattering, J. Chem. Phys., 85, 6425–6429, 1986. 19. Alexander, M.H. and Manolopoulos, D.E., A stable linear reference potential algorithm for solution of the quantum close-coupled equations in molecular scattering theory, J. Chem. Phys., 86, 2044, 1987. 20. Hutson, J.M. and Green, S., MOLSCAT computer program, version 14. Distributed by Collaborative Computational Project No. 6, UK Engineering and Physical Sciences Research Council, 1994. 21. Alexander, M.H., Manolopoulos, D.E., Werner, H.-J., and Follmeg, B., HIBRIDON computer program, available at http://www.chem.umd.edu/groups/alexander/hibridon/ hib43/, 1987–2008.
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22. Kouri, D.J., Rotational excitation II: Approximate methods, in Bernstein, R.B., ed., Atom–Molecule Collision Theory: A Guide for the Experimentalist, Plenum Press, New York, 1979, p. 301–358. 23. Hutson, J.M., An introduction to the dynamics of Van der Waals molecules, in Advances in Molecular Vibrations and Collision Dynamics, Vol. 1A, JAI Press, Greenwich, Connecticut, 1991, p. 1–45. 24. Hutson, J.M., Coupled-channel methods for solving the bound-state Schrödinger equation, Comput. Phys. Commun., 84, 1–18, 1994. 25. Gordon, R.G., A new method for constructing wavefunctions for bound states and scattering, J. Chem. Phys., 51, 14, 1969. 26. Hutson, J.M., BOUND computer program, version 5. Distributed by Collaborative Computational Project No. 6, UK Engineering and Physical Sciences Research Council, 1993. 27. González-Martínez, M.L. and Hutson, J.M., Ultracold atom–molecule collisions and bound states in magnetic fields: zero-energy Feshbach resonances in He–NH (3 Σ− ), Phys. Rev. A, 75, 022702, 2007. 28. Hutson, J.M., Tiesinga, E., and Julienne, P.S., Avoided crossings between bound states of ultracold Cesium dimers, Phys. Rev. A, 78, 052703, 2008. 29. Tennyson, J., The calculation of the vibration-rotation energies of triatomic molecules using scattering coordinates, Comput. Phys. Rep., 4, 1–36, 1986. 30. Baˇci´c, Z. and Light, J.C., Theoretical methods for rovibrational states of floppy molecules, Annu. Rev. Phys. Chem., 40, 469–498, 1989. 31. Carrington, T., Methods for calculating vibrational energy levels, Can. J. Chem.–Rev. Can. Chim., 82, 900–914, 2004. 32. Tennyson, J., Kostin, M.A., Barletta, P., Harris, G.J., Polyansky, O.L., Ramanlal, J., and Zobov, N.F., DVR3D: a program suite for the calculation of rotation-vibration spectra of triatomic molecules, Comput. Phys. Commun., 163, 85–116, 2004. 33. Kozin, I.N., Law, M.M., Tennyson, J., and Hutson, J.M., New vibration-rotation code for tetraatomic molecules exhibiting wide-amplitude motion: WAVR4, Comput. Phys. Commun., 163, 117–131, 2004. 34. Tiesinga, E., Williams, C.J., and Julienne, P.S., Photoassociative spectroscopy of highly excited vibrational levels of alkali-metal dimers: Green-function approach for eigenvalue solvers, Phys. Rev. A, 57, 4257–4267, 1998. 35. Mussa, H.Y. and Tennyson, J., Bound and quasi-bound rotation-vibrational states using massively parallel computers, Comput. Phys. Commun., 128, 434–445, 2000. 36. Feshbach, H., Unified theory of nuclear reactions, Ann. Phys., 5, 357–390, 1958. 37. Mott, N.F. and Massey, H.S.W., The Theory of Atomic Collisions, 3rd edn., Clarendon Press, Oxford, 1965, p. 380. 38. Hazi, A.U., Behavior of the eigenphase sum near a resonance, Phys. Rev. A, 19, 920–922, 1979. 39. Ashton, C.J., Child, M.S., and Hutson, J.M., Rotational predissociation of the Ar–HCl Van der Waals complex—close-coupled scattering calculations, J. Chem. Phys., 78, 4025, 1983. 40. Taylor, J.R., Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Wiley, New York, 1972, p. 411–412. 41. Wigner, E.P., On the behavior of cross sections near thresholds, Phys. Rev., 73, 1002–1009, 1948. 42. Sadeghpour, H.R., Bohn, J.L., Cavagnero, M.J., Esry, B.D., Fabrikant, I.I., Macek, J.H., and Rau, A.R.P., Collisions near threshold in atomic and molecular physics, J. Phys. B–At. Mol. Opt. Phys., 33, R93–R140, 2000.
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43. Balakrishnan, N., Kharchenko, V., Forrey, R.C., and Dalgarno, A., Complex scattering lengths in multi-channel atom–molecule collisions, Chem. Phys. Lett., 280, 5–9, 1997. 44. Cvitaš, M.T., Soldán, P., Hutson, J.M., Honvault, P., and Launay, J.M., Interactions and dynamics in Li + Li2 ultracold collisions, J. Chem. Phys., 127, 074302, 2007. 45. Pethick, C.J. and Smith, H., Bose–Einstein Condensation in Dilute Gases, Cambridge University Press, 2002. 46. Volpi, A. and Bohn, J.L., Magnetic-field effects in ultracold molecular collisions, Phys. Rev. A, 65, 052712, 2002. 47. Krems, R.V. and Dalgarno, A., Quantum-mechanical theory of atom–molecule and molecular collisions in a magnetic field: Spin depolarization, J. Chem. Phys., 120, 2296–2307, 2004. 48. Krems, R.V. and Dalgarno, A., Collisions of atoms and molecules in external magnetic fields, in Fundamental World of Quantum Chemistry, Brändas, E.J., and Kryachko, E.S., Eds, Vol. 3, Kluwer Academic, 2004, p. 273–294. 49. Tscherbul, T.V. and Krems, R.V., Controlling electronic spin relaxation of cold molecules with electric fields, Phys. Rev. Lett., 97, 083201, 2006. 50. Marte, A., Volz, T., Schuster, J., Durr, S., Rempe, G., van Kempen, E.G.M., and Verhaar, B.J., Feshbach resonances in rubidium 87: Precision measurement and analysis, Phys. Rev. Lett., 89, 283202, 2002. 51. Moerdijk, A.J., Verhaar, B.J., and Axelsson, A., Resonances in ultracold collisions of Li-6, Li-7, and Na-23, Phys. Rev. A, 51, 4852–4861, 1995. 52. Timmermans, E., Tommasini, P., Hussein, M., and Kerman, A., Feshbach resonances in atomic Bose–Einstein condensates, Phys. Rep., 315, 199–230, 1999. 53. Hutson, J.M., Feshbach resonances in the presence of inelastic scattering: threshold behavior and suppression of poles in scattering lengths, New J. Phys., 9, 152, 2007. Note that there is a typographical error in Equation 22 of this paper: the last term on the right-hand side should read −βres instead of +βres . 54. Pack, R.T, and Parker, G.A., Quantum reactive scattering in 3 dimensions using hyperspherical (APH) coordinates—theory, J. Chem. Phys., 87, 3888–3921, 1987. 55. Schatz, G.C., Quantum reactive scattering using hyperspherical coordinates: results for H + H2 and Cl + HCl, Chem. Phys. Lett., 150, 92–98, 1988. 56. Launay, J.M. and LeDourneuf, M., Hyperspherical close-coupling calculation of integral, cross-sections for the reaction H + H2 → H2 + H, Chem. Phys. Lett., 163, 178, 1989. 57. Skouteris, D., Castillo, J.F., and Manolopoulos, D.E., ABC: a quantum reactive scattering program, Comput. Phys. Commun., 133, 128–135, 2000. 58. Hu, W. and Schatz, G.C., Theories of reactive scattering, J. Chem. Phys., 125, 132301, 2006. 59. Quéméner, G., Honvault, P., Launay, J.M., Soldán, P., Potter, D.E., and Hutson, J.M., Ultracold quantum dynamics: Spin-polarized k + k2 collisions with three identical bosons or fermions, Phys. Rev. A, 71, 032722, 2005. 60. Alexander, M.H., Manolopoulos, D.E., and Werner, H.-J., An investigation of the F + H2 reaction based on a full ab initio description of the open-shell character of the F(2 P) atom, J. Chem. Phys., 113, 11084–11000, 2000. 61. Zhang, D.H. and Light, J.C., Quantum state-to-state reaction probabilities for the H + H2 O → H2 + OH reaction in six dimensions, J. Chem. Phys., 105, 1291–1294, 1996. 62. Meyer, H.D. and Worth, G.A., Quantum molecular dynamics: propagating wavepackets and density operators using the multiconfiguration time-dependent Hartree method, Theor. Chem. Acc., 109, 251–267, 2003.
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63. Tscherbul, T.V. and Krems, R.V., Quantum theory of chemical reactions in the presence of electromagnetic fields, J. Chem. Phys., 129, 034112, 2008. 64. Levine, R.D. and Bernstein, R.B., Molecular Reaction Dynamics and Chemical Reactivity, Oxford University Press, 1987. 65. Cvitaš, M.T., Soldán, P., Hutson, J.M., Honvault, P., and Launay, J.M., Ultracold Li + Li2 collisions: Bosonic and fermionic cases, Phys. Rev. Lett., 94, 033201, 2005.
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Electric Dipoles at 2 Ultralow Temperatures John L. Bohn CONTENTS 2.1 2.2 2.3
General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review of Classical Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum-Mechanical Dipoles in Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Rotating Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Molecules with Lambda-Doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Field due to a Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Example: j = 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Example: j = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Interaction of Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Potential Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Adiabatic Potential Energy Surfaces in Two Dimensions . . . . . . . . . . 2.5.3 Example: j = 1/2 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Adiabatic Potential Energy Curves in One Dimension: Partial Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Asymptotic Form of the Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
39 40 42 42 45 48 51 53 55 57 58 61 62 65 66 67 67
GENERAL REMARKS
Any object with a net positive charge on one end and a net negative charge on the other end possesses an electric dipole moment. In ordinary classical electromagnetism this dipole moment is a vector quantity that can point in any direction, and is subject to electrical forces that are fairly straightforward to formulate mathematically. However, for a quantum mechanical object like an atom or molecule, the strength and orientation of the object’s dipole moment can depend strongly on the object’s quantum mechanical state. This is a subject that becomes relevant in low-temperature molecular samples, where an ensemble of molecules can be prepared in a single internal state, as described in Chapters 5, 9, 13, 14, and 15. In such a case, the mathematical description becomes more elaborate, and indeed the dipole–dipole interaction need not take the classical 39 © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
form given in textbooks. The description of this interaction is the subject of this chapter. We approach this task in three steps: first, we introduce the ideas of how dipoles arise in quantum mechanical objects; second, we present a formalism within which to describe these dipoles; and third, we give examples of the formalism that illustrate some of the basic physics that emerges. The discussion will explore the possible energy states of the dipole, the field generated by the dipole, and the interaction of the dipole with another dipole. We restrict the discussion to a particular “minimal realistic model,” so that the most important physics is incorporated, but the arithmetic is not overwhelming. Although we discuss molecular dipoles in several contexts, our main focus is on polar molecules that possess a Λ doublet in their ground state. These molecules are the most likely, among diatomic molecules at least, to exhibit their dipolar character at moderate laboratory field strengths. Λ-doubled molecules have another peculiar feature, namely, their ground states possess a degeneracy even in an electric field. This means that there is more than one way for such a molecule to align with the field; the two possibilities are characterized by different angular momentum quantum numbers. This degeneracy leads to novel properties of both the orientation of a single molecule’s dipole moment and the interaction between dipoles. In the examples we present, we focus on revealing these novel features. We assume the reader has a good background in undergraduate quantum mechanics and electrostatics. In particular, the ideas of matrix mechanics, Dirac notation, and time-independent perturbation theory are used frequently. In addition, the reader should have a passing familiarity with electric dipoles and their interactions with fields and with each other. Finally, we will draw heavily on the mathematical theory of angular momentum as applied to quantum mechanics in more detail in the classic treatise of Brink and Satchler [1]. When necessary, details of the structure of diatomic molecules have been drawn from Brown and Carrington’s recent authoritative text [2].
2.2
REVIEW OF CLASSICAL DIPOLES
The behavior of a polar molecule is largely determined by its response to electric fields. Classically, an electric dipole appears when a molecule has a little bit of positive charge displaced a distance from a little bit of negative charge. The dipole moment is then a vector quantity that characterizes the direction and magnitude of this displacement: μ =
qξ rξ ,
ξ
where the ξ-th charge qξ is displaced rξ from a particular origin. Because we are interested in forces exerted on molecules, we will take this origin to be the center of mass of the molecule. (Defining μ = 0 would instead identify the center of charge of the molecule—quite a different thing!) By convention, the dipole moment vector points away from the negative charges, and toward the positive charges, inside the molecule. © 2009 by Taylor and Francis Group, LLC
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Electric Dipoles at Ultralow Temperatures
A molecule has many charges in it, and they are distributed in a complex way, as governed by the quantum-mechanical state of the molecule. In general there is much more information about the electrostatic properties of the molecule than is contained in its dipole moment. However, at distances far from the molecule (as compared to the molecule’s size), these details do not matter. The forces that one molecule exerts on another in this limit are strongly dominated by the dipole moments of the two molecules. In this limit, too, the details of the dipole moment’s origin are irrelevant, and we consider the molecule to be a “point dipole,” with a dipole moment characterized by a magnitude μ and a direction μ. ˆ We consider in this chapter only electrically neutral molecules, so that the Coulomb force between molecules is absent. its energy depends on the relative If a dipole μ is immersed in an electric field E, orientation of the field and the dipole, via Eel = −μ · E. This follows simply from the fact that the positive charges will be pulled in the direction of the field, while the negative charges are pulled the other way. Thus a dipole pointing in the same direction as the field is lower in energy than a dipole pointing in exactly the opposite direction. In classical electrostatics, the energy can continuously vary between these two extreme limits. As an object containing charge, a dipole generates an electric field, which is given, r ). For a point as usual, by the gradient of an electrostatic potential, Emolecule = −∇Φ( dipole the potential Φ is given by Φ(r ) =
μ · rˆ , r2
(2.1)
where r = r rˆ denotes the point in space, relative to the dipole, at which the field is to be evaluated [3]. The dot product in Equation 2.1 gives the field Φ a strong angular dependence. For this reason, it is convenient to use spherical coordinates to describe the physics of dipoles, because they explicitly record directions. Setting rˆ = (θ, φ) and μ ˆ = (α, β) in spherical coordinates, the dipole potential becomes Φ=
μ (cos α cos θ + sin α sin θ cos(β − φ)). r2
(2.2)
For the most familiar case of a dipole aligned along the positive z-axis (α = 0), this yields the familiar result Φ = μ cos θ/r 2 . This potential is maximal along the dipole’s axis (θ = 0 or π), and vanishes in the direction perpendicular to the dipole (θ = π/2). From the results above we can evaluate the interaction potential between two dipoles. One of the dipoles generates an electric field, which acts on the other. Taking the scalar product of one dipole moment with the gradient of the dipole potential Equation 2.2 due to the other, we obtain [3] ) = Vd (R
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ˆ μ ˆ 2 − 3(μ 1 · R)( 2 · R) μ 1 · μ , 3 R
(2.3)
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Cold Molecules: Theory, Experiment, Applications
= RRˆ is the relative coordinate of the dipoles. This result is general for any where R orientation of each dipole, and for any relative position of the pair of dipoles. In a special case where both dipoles are aligned along the positive z-axis, and where the vector connecting the centers of mass of the two dipoles makes an angle θ with this axis, the dipole–dipole interaction takes a simpler form: 1 − 3 cos2 θ . (2.4) R3 Note that the angle θ as used here has a different meaning from the one in Equation 2.2. We will use θ in both contexts throughout this chapter, hopefully without causing undue confusion. The form 2.4 of the interaction is useful for illustrating the most basic fact of the dipole–dipole interaction: if the two dipoles line up in a head-to-tail orientation (θ = 0 or π), then Vd < 0 and they attract one another; if they lie side by side (θ = π/2), then Vd > 0 and they repel one another. (This expression ignores a contact potential that must be associated to a point dipole to conserve lines of electric flux [3]. However, real molecules are not point dipoles, and the electrostatic potential differs greatly from this dipolar form at length scales inside the molecule, scales that do not concern us here.) Our main goal in this chapter is to investigate how these classical results change when the dipoles belong to molecules that are governed by quantum mechanics. In Section 2.3 we evaluate the energy of a dipole exposed to an external field. In Section 2.4 we consider the field produced by a quantum-mechanical dipole. In Section 2.5 we address the interaction between two dipolar molecules. Vd (r ) = μ1 μ2
2.3
QUANTUM-MECHANICAL DIPOLES IN FIELDS
Whereas the classical energy of a dipole in a field can take a continuum of values between its minimum and maximum, this is no longer the case for a quantummechanical molecule. In this section we will establish the energy spectrum of a polar molecule in an electric field, building from a set of simple examples. To start, we will define the laboratory z-axis to coincide with the direction of an externally applied electric field, so that E = E zˆ . In this case, the projection of the total angular momentum on the z-axis is a conserved quantity.
2.3.1 ATOMS Our main focus in this chapter will be on electrically polarizable dipolar molecules. But before discussing this in detail, we first consider the simpler case of an electrically polarizable atom, namely, hydrogen. This will introduce both the basic physics ideas, and the angular momentum techniques that we will use. In this case a negatively charged electron separated a distance r from a positively charged proton forms a dipole moment μ = −er . Because dipoles require us to consider directions, it is useful to cast the unit vector rˆ into its spherical components [1]: x √ z y ±i = ∓ 2C1±1 (θφ), = C10 (θφ). r r r © 2009 by Taylor and Francis Group, LLC
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Electric Dipoles at Ultralow Temperatures
Here the C’s are reduced spherical harmonics, related to the usual spherical harmonics by [1] 4π Ckq = Ykq , 2k + 1 and given explicitly for k = 1 by 1 C1±1 (θφ) = ∓ √ sin θe±iφ 2 C10 (θφ) = cos θ.
(2.5)
In general, it is convenient to represent interaction potentials in terms of the functions Ckq (because they do not carry extra factors of 4π), and to use the functions Ykq as wavefunctions in angular degrees of freedom (because they are already properly normalized, by Ylm |Yl m = δll δmm ). Integrals involving the reduced spherical harmonics are conveniently related to the 3-j symbols of angular momentum theory, for example:
d(cos θ) dφCk1 q1 (θφ)Ck2 q2 (θφ)Ck3 q3 (θφ) = 4π
k1 q1
k2 q2
k3 q3
k1 0
k2 0
k3 . 0
The 3-j symbols, in parentheses, are related to the Clebsch–Gordan coefficients. They are widely tabulated and easily computed for applications. In terms of these functions, the Hamiltonian for the atom–field interaction is Hel = −(−er ) · E = ezE = er cos(θ)E = erEC10 (θφ).
(2.6)
The possible energies for a dipole in a field are given by the eigenvalues of the Hamiltonian 2.6. To evaluate these energies in quantum mechanics, we identify the usual basis set of hydrogenic wavefunctions, |nlm, where we ignore spin for this simple illustration: r, θ, φ | nlm = fnl (r)Ylm (θ, φ). The matrix elements between any two hydrogenic states are nlm | −μ · E | n l m = erE
∗ d(cos θ) dφYlm C10 Yl m
= erE(−1)m (2l + 1)(2l + 1) l 1 l l 1 l × . 0 0 0 −m 0 m
(2.7)
Here er = r 2 drfnl (r)rfn l is an effective magnitude of the dipole moment, which can be analytically evaluated for hydrogen [4]. © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
Some important physics is embodied in Equation 2.7. First, the electric field defines an axis of rotational symmetry (here the z-axis). On general grounds, we therefore expect that the projection of the total angular momentum of the molecule onto this axis is a constant. And indeed, this is built into the 3-j symbols: because the sum of all m quantum numbers in a 3-j symbol should add to zero, Equation 2.7 asserts that m = m , and the electric field cannot couple two different m’s together. A second feature embodied in Equation 2.7 is the action of parity. The hydrogenic wavefunctions have a definite parity, that is, they either change sign, or else remain invariant, upon converting from a coordinate system (x, y, z) to a coordinate system (−x, −y, −z). The sign of the parity-changed wavefunction is given by (−1)l . Thus an s-state (l = 0) has even parity, while a p-state (l = 1) has odd parity. For an electric field pointing in a particular direction, the Hamiltonian 2.6 itself has odd parity, and thus serves to change the parity of the atom. For example, it can couple the s and p states to each other, but not to themselves. This is expressed in Equation 2.7 by the first 3-j symbol, whose symmetry properties require that l + 1 + l = even. This seemingly innocuous statement is the fundamental fact of electric dipole moments of atoms and molecules. It says that, for example the 1s ground state of hydrogen, with l = l = 0, does not, by itself, respond to an electric field at all. Rather, it requires an admixture of a p state to develop a dipole moment. (These remarks are not strictly true. The ground state of hydrogen already has a small admixture of odd-parity states, due to the parity-violating part of the electroweak force. This effect is far too small to concern us here, however.) To evaluate the influence of an electric field on hydrogen, therefore, we must consider at least the nearest state of opposite parity, which is the 2p state. These two states are separated in energy by an amount E1s2p . Considering only these two states, and ignoring any spin structure, the atom-plus-field Hamiltonian is represented by a simple 2 × 2 matrix: −E1s2p /2 μE H= . (2.8) μE E1s2p /2 where the dipole matrix element is given by the convenient shorthand μ = 1s, m = √ 0 | ez | 2p, m = 0 = 128 2ea0 /243 [4]. Of course there are many more p states that the 1s state is coupled to. Plus, all states are further complicated by the spin of the electron and (in hydrogen) the nucleus. Matrix elements for all these can be constructed, and the full matrix diagonalized to approximate the energies to any desired degree of accuracy. However, we are interested here in the qualitative features of dipoles, and so limit ourselves to Equation 2.8. The Stark energies are thus given approximately by E± = ± (μE)2 + (E1s2p /2)2 .
(2.9)
This expression illustrates the basic physics of the quantum mechanical dipole. First, there are necessarily two states (or more) involved. One state decreases in energy as the field is turned on, representing the “normal” case where the electron moves to negative z and the electric dipole moment aligns with the field. The other state, © 2009 by Taylor and Francis Group, LLC
Electric Dipoles at Ultralow Temperatures
45
however, increases in energy with increasing field and represents the dipole moment antialigning with the field. Classically, it is of course possible to align the dipole against the field in a state of unstable equilibrium. Similarly, in quantum mechanics this is a legitimate energy eigenstate, and the dipole will remain antialigned with the field in the absence of perturbations. A second observation about the energies of Equation 2.9 is that the energy is a quadratic function of E at low fields, and only becomes linearly proportional to E at higher fields. Thus the permanent dipole moment of the atom, defined by the zero-field limit ∂E− μpermanent ≡ lim , E →0 ∂E vanishes. The atom, in an energy eigenstate in zero field, has no permanent electric dipole moment. This makes sense because, in zero field, the electron’s position is randomly varied about the atom, lying as much on one side of the nucleus as on the opposite side. The transition from quadratic to linear Stark effect is an example of a competition between two opposing tendencies.At low field, the dominant energy scale is the energy splitting E1s2p between opposite parity states. At higher field values, the interaction energy with the electric field becomes stronger, and the dipole is aligned. The value of the field where this transition occurs is found roughly by setting these energies equal to find a “critical” electric field: Ecritical = E1s2p /2μ. For atomic hydrogen, this field is on the order of 109 V/cm. However, at this field it is already a bad approximation to ignore that fact that there are both 2p1/2 and 2p3/2 states, as well as higher-lying p states, and further coupling between p, d, etc., states. We will not pursue this subject further here.
2.3.2
ROTATING MOLECULES
With these basics in mind, we can move on to molecules. We focus here on diatomic, heteronuclear molecules, although the principles are more general. We will consider only electric fields so small that the electrons cannot be polarized in the sense of the previous section; thus we consider only a single electronic state. However, the charge separation between the two atoms produces an electric dipole moment μ in the rotating frame of the molecule. We assume that the molecule is a rigid rotor and we will not consider explicitly the vibrational motion of the molecule, focusing instead solely on the molecular rotation. (More precisely, we consider μ to incorporate an averaging over the vibrational coordinate of the molecule, much as the electron–proton distance r was averaged over for the hydrogen atom in the previous section.) As a mathematical preliminary, we note the following. To deal with molecules, we are required to transform freely between the laboratory reference frame and the bodyfixed frame that rotates with the molecule. The rotation from the lab frame (x, y, z) © 2009 by Taylor and Francis Group, LLC
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to the body frame (x , y , z ) is governed by a set of Euler angles (α, β, γ). The first two angles α = φ, β = θ coincide with the spherical coordinates (θ, φ) of the body frame’s z axis. By convention, we take the positive z direction to be parallel to the dipole moment μ. The third Euler angle γ serves to orient the x axis in a desired orientation within the body frame; it is thus the azimuthal angle of rotation about the molecular axis. Consider a given angular momentum state | jm referred to the lab frame. This state is only a state of good m in the lab frame, in general. In the body frame, which points in some other direction, the same state will be a linear superposition of different m’s, which we denote in the body frame as ω’s to distinguish them. Moreover, this linear superposition will be a function of the Euler angles, with a transformation that is conventionally denoted by the letter D: D(αβγ) | jω =
| jm jm | D(α, β, γ) | jω
m
≡
j
| jmDmω (α, β, γ).
m
This last line defines the Wigner rotation matrices, whose properties are widely j tabulated. For each j, Dmω is a unitary transformation matrix; note that a rotation can only change m-type quantum numbers, not the total angular momentum j. One of the more useful properties of the D matrices, for us, is dα d cos(β) dγ = 8π2
j1 m1
j
j
j
Dm11 ω1 (αβγ)Dm22 ω2 (αβγ)Dm33 ω3 (αβγ) j2 m2
j3 m3
j1 ω1
j2 ω2
j3 . ω3
(2.10)
Because the dipole is aligned along the molecular axis, and because the molecular axis is tilted at an angle β with respect to the field, and because the field defines the z-axis, the dipole moment is defined by its magnitude μ times a unit vector with polar coordinates (β, α). The Hamiltonian for the molecule–field interaction is given by Hel = −μ · E = −μEC10 (βα) = −μEDq0 (αβγ). j∗
(2.11)
For use below, we have taken the liberty of rewriting C10 as a D-function; since the second index of D is zero, this function does not actually depend on γ, so introducing this variable is not as drastic as it seems. To evaluate energies in quantum mechanics we need to choose a basis set and take matrix elements. The Wigner rotation matrices are the quantum-mechanical eigenfunctions of the rigid rotor. With normalization, these wavefunctions are αβγ | nmn λn =
© 2009 by Taylor and Francis Group, LLC
2n + 1 n∗ D . 8π2 mn λn
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47
As we did for hydrogen, we here ignore spin. Thus n is the quantum number of rotation of the atoms about their center of mass, mn is the projection of this angular momentum in the lab frame, and λn is its projection in the body frame. In this basis, the matrix elements of the Stark interaction are computed using Equation 2.10 to yield · E | n mn λ = −μE(−1)mn −λn (2n + 1)(2n + 1) nmn λn | −μ n 1 n n 1 n × . (2.12) −mn 0 mn −λn 0 λn In an important special case, the molecule is in a Σ state, meaning that the electronic angular momentum projection λn = 0. In this case, Equation 2.12 reduces to the same expression as that for hydrogen, apart from the radial integral. This is as it should be: in both objects, there is simply a positive charge at one end and a negative charge at the other. It does not matter if one of these is an electron, rather than an atom. More generally, however, when λn = 0 there will be a complicating effect of lambda-doubling, which we will discuss in the next section. Thus the physics of the rotating dipole is much the same as that of the hydrogen atom. Equation 2.12 also asserts that, for a Σ state with λn = 0, the electric field interaction vanishes unless n and n have opposite parity. For such a state, the parity is related to the parity of n itself. Thus, for the ground state of a 1 Σ molecule with n = 0, the electric field only has an effect by mixing this state with the next rotational state with n = 1. These states are split by an energy Erot = 2Be , where Be is the rotational constant of the molecule. (In zero field, the state with rotational quantum number n has energy Be n(n + 1).) We can formulate a simple 2 × 2 matrix describing this situation, as we did for hydrogen: −Erot /2 −μE H= , (2.13) −μE +Erot /2 where the dipole matrix element is given by the convenient shorthand notation μ = nmn 0 | μq=0 | n mn 0. There is one such matrix for each value of mn . Of course there are many more rotational states that these states are coupled to. Plus, all states would further be complicated by the spins (if any) of the electrons and nuclei. The matrix 2.13 can be diagonalized just as Equation 2.8 was above, and the same physical conclusions apply. Namely, the molecule in a given rotational state has no permanent electric dipole moment, even though there is a separation of charges in the body frame of the molecule. Second, the Stark effect is quadratic for low fields, and linear only at higher fields, with the transition occurring at a “critical field” Ecrit = Erot /2μ.
(2.14)
To take an example, the NH molecule possesses a 3 Σ ground state. For this state, ignoring spin, the critical field is of the order 7 × 106 V/cm. This is far smaller than the field required to polarize electrons in an atom or molecule, but still large for laboratory-strength electric fields. Diatomic molecules with smaller rotational constants, such as LiF, would have correspondingly smaller critical fields. In any event, by the time the critical field is applied, it is already a bad approximation to © 2009 by Taylor and Francis Group, LLC
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ignore coupling to the other rotational states of the molecule, which must be included for an accurate treatment. We do not consider this topic further here.
2.3.3
MOLECULES WITH LAMBDA-DOUBLING
As we have made clear in the previous two subsections, the effect of an electric field on a quantum-mechanical object is to couple states of opposite parity. For a molecule in a Π or Δ state, there are often two such parity states that are much closer together in energy than the rotational spacing. The two states are said to be the components of a “Λ-doublet.” Because they are close together in energy, these two states can then be mixed at much smaller fields than are required to mix rotational levels. The physics underlying the lambda doublet is rather complex, and we refer the reader to the literature for details [2,5]. However, in broad terms, the argument is something like this: a Π state has an electronic angular momentum projection of magnitude 1 about the molecular axis. This angular momentum comes in two projections, for the two senses of rotation about the axis, and these projections are nominally degenerate in energy. The rotation of the molecule, however, can break the degeneracy between these levels, and (it so happens) the resulting nondegenerate eigenfunctions are also eigenfunctions of parity. The main point is that the resulting energy splitting is usually quite small, and these parity states can be mixed in fields much smaller than those required to mix rotational states. To this end, we modify the rigid-rotor wavefunction of the molecule to incorporate the electronic angular momentum: αβγ | jmω =
2j + 1 j∗ Dmω (αβγ). 8π2
(2.15)
Here j is the total (rotation-plus-electronic) angular momentum of the molecule, and m and ω are the projections of j on the laboratory and body-fixed axes, respectively. Using the total j angular momentum, rather than just the molecular rotation n, marks the use of a “Hund’s case a” representation, rather than the Hund’s case b that was implicit in the previous section [2]. In this basis the matrix element of the electric field Hamiltonian 2.11 becomes jmω | −μ · E | j m ω = −μE(−1)
m−ω
(2j
+ 1)(2j
j + 1) −m
1 0
j m
j −ω
1 0
j . (2.16) ω
In Equation 2.16, the 3-j symbols imply conservation laws. The first asserts that m = m is conserved, as we already knew. The second 3-j symbol adds to this the fact that ω = ω . This is the statement that the electric field cannot exert a torque around the axis of the dipole moment itself. Moreover, in the present model we assert that j = j , because the next higher-lying j level is far away in energy, and only weakly mixes with the ground state j. With this approximation, the 3-j symbols have simple © 2009 by Taylor and Francis Group, LLC
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algebraic expressions, and we can simplify the matrix element: jmω | −μ · E | jmω = −μE
mω . j( j + 1)
The physical content of this expression is illustrated in Figure 2.1. Notice that both m and ω can have a sign, and that whether the energy is positive or negative depends on both signs. An essential point is that there is not a unique state representing the dipole aligned with the field. Rather, there are two such states, distinguished by different angular momentum quantum numbers but possessing the same energy. To distinguish these in the following we will refer to the two states in Figures 2.1a and b as molecules of type | a and type | b, respectively. Likewise, for molecules nominally antialigned with the field, we will refer to types | c and | d, corresponding to the two states in Figures 2.1c and d. The existence of these degeneracies will lead to novel phenomena in these kinds of molecules, as we discuss below. As to the lambda doubling, it is, as we have asserted, diagonal in a basis where parity is a good quantum number. In terms of the basis (Equation 2.15), wavefunctions of well-defined parity are given by the linear combinations 1 | jmω ¯ = √ [| jmω ¯ + | jm − ω]. ¯ 2
(a)
m>0
j
(2.17)
(b) w>0 w>0 m<0
j (c)
m>0
j
(d) w<0 w>0 j
m<0
FIGURE 2.1 Energetics of a polar molecule in an electric field. The molecule’s dipole moment points from the negatively charged atom (large circle) to the positively charged atom (small circle) as indicated by the thick arrow. The dashed line indicates the positive direction of the molecules’ body axis, while the vertical arrow represents the direction of the applied electric field. The dipole aligns with the field, on average, if either (i) the angular momentum j aligns with the field and m > 0, ω > 0 (a), or (ii) j aligns against the field and m < 0, ω < 0 (b). Similar remarks apply to dipoles that antialign with the field (c,d).
© 2009 by Taylor and Francis Group, LLC
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Here we define ω ¯ =| ω |, the absolute value of ω. For a given value of m, the linear combinations of ±ω ¯ in Equation 2.17 are distinguished by the parity quantum number = ±1. It is straightforward to show that in the parity basis Equation 2.17 the Λ-doubling is off-diagonal. The net result is that for each value of m, the Hamiltonian for our lambda-doubled molecule can be represented as a 2 × 2 matrix, similar to the ones above: −Q −Δ/2 H= , −Δ/2 Q where Δ is the lambda doubling energy, that is, the energy difference between the two parity states, and |m|ω ¯ Q ≡ μE j( j + 1) is a manifestly positive quantity. This is the Hamiltonian we will treat in the remainder of this chapter. The difference here from the previous subsections is that the basis is now that of Equation 2.15, which diagonalizes the electric field interaction rather than the zero-field Hamiltonian. This change reflects our emphasis on molecules in strong fields where their dipole moments are made manifest. The zero-field Λ-doubling Hamiltonian is considered, for the most part, to be a perturbation. In general, for field interactions that are not infinitely larger than the lambda doublet, the energy eigenstates are superpositions of the strong-field states | jm ± ω. ¯ For each value of m (which, remember, is conserved by the field), the mixing of +ω ¯ and −ω ¯ states is conveniently represented by a mixing angle that we denote by δm : ¯ + sin δm | jm − ω ¯ | jmω ¯ = + = cos δm | jmω ¯ + cos δm | jm − ω. ¯ | jmω ¯ = − = − sin δm | jmω
(2.18)
Explicitly, the mixing angle as a function of field is given by tan δ|m| =
Δ/2 = − tan δ−|m| , Q + Q 1 + η2
in terms of the energy Q and the dimensionless parameter ηm =
Δ . 2Q
Notice that with this definition, δm is positive when m is positive, and δ−m = −δm . The energies of these states are conveniently summarized by the expression Emω ¯
mω ¯ = −μE 1 + η2 , j( j + 1)
m = 0.
This is a very compact way of writing the results that will facilitate writing the expressions below. Notice that the intuition afforded by Figure 2.1 is still intact in © 2009 by Taylor and Francis Group, LLC
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these energies, but by replacing the sign of for the sign of ω. Thus states with m > 0 have negative energy, while states with m < 0 have positive energy. These expressions are somewhat problematic in the limit where the field vanishes, which sets Q = 0. However, this limit exists and it is easily shown that δ| m| = π/4 in this limit, while the energies become −μEsign(m)Δ/2. Likewise, we also have Q = 0 whenever m = 0, for any value of the field strength. Again, we can still write the eigenstates in the form of 2.18, provided that we set δ0 =
π , 4
and understand that the corresponding energies, independent of field, are simply E0ω ¯ = −Δ/2. Like the dipoles considered above, this model has a quadratic Stark effect at low energies, rolling over to linear at electric fields exceeding the critical field given by setting Q = Δ/2. This criterion gives a critical field of Ecrit =
Δj( j + 1) . 2μ| m | ω ¯
(2.19)
To take an example, consider the ground state of OH, which has j = ω ¯ = 3/2, μ = 1.7 Db, and a Λ-doublet splitting of 0.06 cm−1 . In its m = 3/2 ground state, its critical field is ∼1600 V/cm. Again, we have explicitly ignored the spin of the electron. The parity states are therefore easily mixed in fields that are both small enough to easily obtain in the laboratory, and small enough that no second-order coupling to rotational or electronic states needs to be considered. Keeping the relatively small number of molecular states is therefore a reasonable approximation, and highly desirable as it simplifies our discussion. (In OH there is also the ω ¯ = 1/2 state to consider, but it is also far away in energy compared to the Λ-doublet, and we ignore it. It does play a role in the fine structure of OH, however, and this should be included in a quantitative model of OH.)
2.4 THE FIELD DUE TO A DIPOLE Once polarized, each of the Λ-doubled molecules discussed above is itself the source of an electric field. The field due to a molecule is given above by Equation 2.1. In what follows, it is convenient to cast this potential in terms of the spherical tensors defined above: Φ(r ) =
μ · rˆ μ
= 2 (−1)q C1q (αβ)C1−q (θφ). 2 r r q
(2.20)
That this form is correct can be verified by simple substitution, using the definitions of Equation 2.5 and comparing to Equation 2.2. It seems at first unnecessarily complicated to write Equation 2.20 in this way. However, the effort required to do so © 2009 by Taylor and Francis Group, LLC
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will be rewarded when we need to evaluate the potential for quantum-mechanical dipoles below. For a classical dipole, defining the direction (αβ) of its dipole moment would immediately specify the electrostatic potential it generates according to Equation 2.20. However, in quantum mechanics the potential will result from suitably averaging the orientation of μ over the distribution of (αβ), weighted by probabilities that are dictated by the molecule’s wavefunction. To evaluate this, we need to evaluate matrix elements of Equation 2.20 in the basis of energy eigenstates 2.18. This is easily done using Equation 2.10, along with formulas that simplify the 3-j symbols. The result is
jmω | C1q
⎧ ( j + m)( j − m ) ⎪ ⎪ ⎪ , q = +1 − ⎪ ⎨ 2 ω | jm ω = δωω m, q=0 , j( j + 1) ⎪ ⎪ ⎪ ⎪ ⎩+ ( j − m)( j + m ) , q = −1 2
(2.21)
where the quantum numbers m and ω are signed quantities. Also note that this integration is over the molecular degrees of freedom (αβ) in Equation 2.20. This still leaves the angular dependence on (θφ), which characterizes the field in space around the dipole. From expression 2.21 it is clear that this matrix element changes sign upon either (1) reversing the sign of both m and m or (2) changing the sign of ω. Moreover, jmω | C1q | jm − ω = 0, because ω is conserved by the electric field. Using these observations, we can readily compute the matrix elements of Φ in the dressed basis 2.18. Generally they take the form ¯ = μ jmω ¯ | C1q | jm ω ¯ (−1)q jmω ¯ | Φ(r ) | jm ω
C1−q (θφ) . r2
(2.22)
The matrix elements in front of Equation 2.22 represent the quantum-mechanical manifestation of the dipole’s orientation. These matrix elements follow from the above definition of the eigenstates (Equation 2.18). Explicitly, jmω, ¯ | C1q | jm ω, ¯ = cos(δm + δm ) jmω ¯ | C1q | jm ω ¯ jmω, ¯ −| C1q | jm ω, ¯ + = jmω, ¯ + | C1q | jm ω, ¯ −
(2.23)
= − sin(δm + δm ) jmω ¯ | C1q | jm ω. ¯ In all these expressions, the value of q is set by angular momentum conservation to q = m − m . This description, while complete within our model, nevertheless remains somewhat opaque. Let us therefore specialize it to the case of a particular energy eigenstate | jmω. ¯ In this state, the averaged electrostatic potential of the dipole is cos θ mω . cos 2δm Φ(r ) ≡ jmω ¯ | Φ(r ) | jmω ¯ = μ j( j + 1) r2 © 2009 by Taylor and Francis Group, LLC
(2.24)
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53
Here the factor in parentheses is a quantum-mechanical correction to the magnitude of the dipole moment. The factor cos 2δm expresses the degree of polarization: in a strong field, δm = 0 and the dipole is at maximum strength, whereas in zero field δm = π/4 and the dipole vanishes. Notice that for a given value of m, the potential generated by the states with = ± differ by a sign. this is appropriate, because these states correspond to dipoles pointing in opposite directions (Figure 2.1). The off-diagonal matrix elements in Equation 2.22 are also important, for two reasons. First, it may be desirable to create superpositions of different energy eigenstates, and computing these matrix elements requires the off-diagonal elements in Equation 2.22, as we will see shortly. Second, when two dipoles interact with each other, one will experience the electric field due to the other, and this field need not lie parallel to the z-axis. Hence, the m quantum number of an individual dipole is no longer conserved, and elements of Equation 2.22 with q = 0 are required.
2.4.1
EXAMPLE: j = 1/2
To illustrate these abstract points, we consider here the simplest molecular state with a Λ doublet: a molecule with j = 1/2, which has ω ¯ = 1/2, and consists of four internal states in our model. Based on the discussion above, we tabulate the matrix elements between these states in Table 2.1. Because we have j = 1/2, we suppress the index j in this section. For concreteness, we focus on a type | a molecule, as defined in Figure 2.1. This molecule aligns with the field and produces an electrostatic potential (Equation 2.24). However, if the molecule is prepared in a state that is a superposition of this state with another, a different electrostatic potential can result. We first note that combining | a with | b produces nothing new, because both states generate the same potential. An alternative superposition combines state | a with state | c. In this case the two states have the same value of m, but are nevertheless nondegenerate. We define iω0 t 1 −iω0 t 1 | ψac = Ae ω, ¯ + + Be ω, ¯ − , 2 2 for arbitrary complex numbers A and B with | A |2 +| B |2 = 1. Because the two states are nondegenerate, it is necessary to include the explicit time-dependent phase factors, where ω0 = | Emω ¯ |/, and 2ω0 is the energy difference between the states. As usual in quantum mechanics, these phases will beat against one another to make the observables time dependent. Now some algebra identifies the mean value of the electrostatic potential, averaged over state | ψac , as μ | A |2 −| B |2 cos 2δ1/2 ψac | Φ(r ) | ψac = 3 cos θ − 2| AB | sin 2δ1/2 cos(2ω0 t − δ) . (2.25) r2 This potential has the usual cos θ angular dependence, meaning that the dipole remains aligned along the field’s axis. However, the magnitude, and even the sign, of the dipole © 2009 by Taylor and Francis Group, LLC
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TABLE 2.1 ¯ | C1q | jm ω ¯ for a j = 1/2 Molecule Matrix Elements jm ω
1 ω+ ¯ 2 1 ω− ¯ 2 1 ¯ − ω+ 2 1 − ω− ¯ 2
1 ω+ ¯ 2
1 ω− ¯ 2
1 cos 2δ1/2 3
1 − sin 2δ1/2 3
1 − sin 2δ1/2 3 √ 2 3
1 − cos 2δ1/2 3
0
2 3
0
1 − cos 2δ1/2 3
1 − sin 2δ1/2 3
1 − sin 2δ1/2 3
1 cos 2δ1/2 3
0
√ −
2 3
1 − ω+ ¯ 2
1 − ω− ¯ 2
√ −
2 3
0 √
To obtain matrix elements of the electrostatic potential Φ(r ), these matrix elements should be multiplied by (−1)q C1−q (θφ)μ/r 2 , where q = m − m .
change over time. The first term in square brackets in Equation 2.25 gives a constant, dc component to the dipole moment, which depends on the population imbalance | A |2 −| B |2 between the two states. The second term adds to this an oscillating component with angular frequency 2ω0 . The leftover phase δ is an irrelevant offset, and comes from the phase of A∗ B, that is, the relative phase of the two components at time t = 0. It is therefore possible to construct a superposition of states of the dipole, such that the effective dipole moment of the molecule bobs up and down in time. The amount that the dipole bobs, relative to the constant component, can be controlled by the relative population in the two states. Moreover, the degree of polarization of the molecule plays a significant role. For a fully polarized molecule, when δ1/2 = 0, only the dc portion of the dipole persists, although even it can vanish if there is equal population in the two states, “dipole up” and “dipole down.” As another example, we consider the superposition of | a with | d. Now the two states have different values of m as well as different energies: iω0 t 1 −iω0 t 1 ¯ + + Be ¯ + . | ψad = Ae 2 ω, − 2 ω, The dipole potential this superposition generates is cos θ μ ψad | Φ(r ) | ψad = cos 2δ1/2 | A |2 −| B |2 3 r2 2μ sin θ + . Re A∗ Be−i(φ+2ω0 t) 3 r2 This expression can be put in a useful and interesting form if we parametrize the coefficients A and B as α α A = cos e−iβ/2 B = sin eiβ/2 . 2 2 © 2009 by Taylor and Francis Group, LLC
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This way of writing A and B seems arbitrary, but it is not. It is the same parametrization that is used in constructing the Bloch sphere, which is a powerful tool in the analysis of any two-level system [6]. This parametrization leads to the following expression for the potential: ψad | Φ(r ) | ψad =
1μ cos 2δ cos α cos θ + sin α sin θ cos((β − 2ω t) − φ) . 1/2 0 3 r2 (2.26)
The interpretation of this result is clear upon comparing it to the classical expression 2.2. First consider that the molecule is perfectly polarized, so that cos 2δ1/2 = 1. Then Equation 2.26 represents the potential due to a dipole whose polar coordinates are (α, β − ωt). That is, this dipole makes (on average) an angle α with respect to the field, and it precesses about the field with an angular frequency 2ω0 . Interestingly, even in this strong field limit where the field nominally aligns the dipole along z, quantum mechanics allows the dipole to point in quite a different direction. As the field relaxes, the z-component reduces, but this dipole still has a component precessing about the field.
2.4.2
EXAMPLE: j = 1
We also consider a molecule with spin j = 1. Here there are in principle three mixing angles, δ1 , δ0 , and δ−1 . However, as noted above we have δ−1 = −δ1 and δ0 = π/4, so that the entire electric field dependence of these matrix elements is incorporated in the single parameter δ1 . In this notation, the matrix elements of the electrostatic potential for a j = 1 molecule are given in Table 2.2. Similar remarks apply to the spin-1 case as applied to the spin-1/2 case. If the molecule is in an eigenstate, say | +1ω, ¯ −, then the expectation value of the dipole points along the field axis, and its distribution has the usual cos θ dependence. In an eigenstate with m = 0, however, the expectation value of the dipole vanishes altogether. As before, the molecule can also be in a superposition state. No matter how complicated this superposition is, the expectation value of the dipole must instantaneously point in some direction, since the only available angular dependence resides in the C1q functions, which yield only dipoles. In other words, no superposition can generate the field pattern of a quadrupole moment, for example. Where this dipole points, and how its orientation evolves with time, however, can be nontrivial. For example, a superposition of | +1ω, ¯ + and | +1ω, ¯ − can bob up and down, just like the analogous superposition for j = 1/2. However, for j = 1 molecules additional superpositions are possible. For example, consider the combination | ψ3 = Aeiω0 t | +1ω, ¯ + + BeiωΔ t | 0ω, ¯ + + Ceiω0 t | −1ω, ¯ −, where ωΔ = Δ/2 is a shorthand notation for half the lambda doubling energy. Let us further assume for convenience that A, B, and C are all real. Then the expectation © 2009 by Taylor and Francis Group, LLC
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TABLE 2.2
¯ | C1q | jm ω ¯ for a j = 1 Molecule Matrix Elements of jmω ¯ | +1ω−
¯ | 0ω+
¯ | 0ω−
¯ | −1ω+
¯ | −1ω−
+1ω+ ¯ |
1 cos 2δ1 2
1 − sin 2δ1 2
1 − cos(δ1 + π/4) 2
1 sin(δ1 + π/4) 2
0
0
+1ω− ¯ |
1 − sin 2δ1 2
1 − cos 2δ1 2
1 sin(δ1 + π/4) 2
1 cos(δ1 + π/4) 2
0
0
0ω+ ¯ |
1 cos(δ1 + π/4) 2
1 − sin(δ1 + π/4) 2
0
0
1 − cos(−δ1 + π/4) 2
1 sin(−δ1 + π/4) 2
0ω− ¯ |
1 − sin(δ1 + π/4) 2
1 − cos(δ1 + π/4) 2
0
0
1 sin(−δ1 + π/4) 2
1 cos(−δ1 + π/4) 2
0
0
1 cos(−δ1 + π/4) 2
1 − sin(−δ1 + π/4) 2
1 − cos 2δ1 2
1 − sin 2δ1 2
−1ω+ ¯ |
1 1 1 1 − sin(−δ1 + π/4) − cos(−δ1 + π/4) − sin 2δ1 cos 2δ1 2 2 2 2 q 2 To obtain the matrix elements of the electrostatic potential Φ(r ), these matrix elements should be multiplied by (−1) C1−q (θφ)μ/r , where q = m − m . −1ω− ¯ |
0
© 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
¯ | +1ω+
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Electric Dipoles at Ultralow Temperatures
value of the electrostatic potential is ψ3 | Φ(r ) | ψ3 =
cos θ μ μ sin θ cos 2δ1 A2 + C 2 + √ cos(δ1 + π/4)B 2 2 r2 r 2 × [A cos((ωΔ − ω0 )t − φ) − C cos(−(ωΔ − ω0 )t − φ)].
By analogy with remarks in the previous section, this represents a dipole with a constant component along the z-axis, which depends on both the strength of the field and on | A |2 + | C |2 , the total population in the ±m states. It also has a component in the (x–y)-plane (given by the sin θ dependence), orthogonal to the field’s direction. In the case where C = 0, this component would precess around the field axis with a frequency ωΔ − ω0 , in a clockwise direction as viewed from the +z direction. On the other hand, if A = 0, this component would rotate at this frequency but in a counterclockwise direction. If both components are present and A = C, then the result will be, not a rotation, but an oscillation of this component from, say, +x to −x, in much the same way that linearly polarized light in a superposition of left- and right-circularly polarized components. More generally, if A = C, then the tip of the dipole moment will trace out an elliptical path. However, in the limit of zero field, ω0 reduces to ωΔ and these time-dependent effects go away.
2.5
INTERACTION OF DIPOLES
Having thus carefully treated individual dipoles and their quantum-mechanical matrix elements, we are now in a position to do the same for the dipole–dipole interaction between two molecules. This interaction depends on the orientation of each dipole, μ 1 . This interaction has the form (Equation 2.3): and μ 2 , and on their relative location, R ˆ μ ˆ 2 − 3(μ μ 1 · μ 1 · R)( 2 · R) R3 √ 6μ1 μ2
=− (−1)q [μ1 ⊗ μ2 ]2q C2−q (θφ). R3 q
) = Vd (R
(2.27)
In going from the first line to the second, we assume that the intermolecular axis = (R, θ, φ). The makes an angle θ with respect to the laboratory z-axis, so that R angles θ and φ thus stand for something slightly different than in the previous section. The second line in Equation 2.27 also rewrites the interaction in a compact tensor notation that is useful for the calculations we are about to do. Here √
1 1 q 2 [μ1 ⊗ μ2 ]2q = 5 C1q1 (β1 α1 )C1q2 (β2 α2 ) (−1) q −q1 −q2 q1 q2
denotes the second-rank tensor composed of the two first-rank tensors (i.e., vectors) C1q1 (β1 α1 ) and C1q2 (β2 α2 ) that give the orientation of the molecular axes [1]. Equation 2.27 highlights the important point that the orientations of the dipoles are © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
intimately tied to the relative motion of the dipoles: if a molecule changes its internal state and sheds angular momentum, that angular momentum may appear in the orbital motion of the molecules around each other.
2.5.1
POTENTIAL MATRIX ELEMENTS
Equation 2.27 is a perfectly reasonable way of writing the classical dipole–dipole interaction. Quantum mechanically, however, we are interested in molecules that are in particular quantum states | jmω, ¯ , rather than molecules whose dipoles point in particular directions (α, β). We must therefore construct matrix elements of the interaction potential 2.27 in the basis we have described in Section 2.3.3. Writing the interaction in the form above has the advantage that each term in the sum factors into three pieces: one depending on the coordinates of molecule 1, another depending on the coordinates of molecule 2, and a third depending on the relative coordinates (θ, φ). This makes it easier to evaluate the Hamiltonian in a given basis. For two molecules we consider the basis functions α1 β1 γ1 | jm1 ω, ¯ 1 α2 β2 γ2 | jm2 ω, ¯ 2 ,
(2.28)
as defined above. In this basis, matrix elements of the interaction become jm1 ω, ¯ 1 ; jm2 ω, ¯ 2 | Vd (θφ) | jm1 ω, ¯ 1 ; jm2 ω, ¯ 2 √ 2 1 1 = − 30μ1 μ2 q −q1 −q2 × jm1 ω, ¯ 1 | C1q1 | jm1 ω, ¯ 1 jm2 ω, ¯ 2 | C1q2 | jm2 ω, ¯ 2
C2−q (θφ) (2.29) R3
¯ are evaluated in Equawhere matrix elements of the form jmω, ¯ | C1q | jm ω, tion 2.23. Conservation of angular momentum projection constrains the values of the summation indices, so that q1 = m1 − m1 , q2 = m2 − m2 , and q = q1 + q2 = (m1 + m2 ) − (m1 + m2 ). To make this model concrete, we report here the values of the second-rank reduced spherical harmonics [1]: 1 (3 cos2 θ − 1) 2 1/2 3 =∓ cos θ sin θe±iφ 2 1/2 3 = sin2 θe±2iφ . 8
C20 = C2±1 C2±2
We also tabulate the relevant 3-j symbols in Table 2.3. Viewed roughly as a collision process, we can think of two molecules approaching each other with angular momenta m1 and m2 , scattering, and departing with angular © 2009 by Taylor and Francis Group, LLC
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Electric Dipoles at Ultralow Temperatures
TABLE 2.3 The 3-j Symbols Needed to Construct the Matrix Elements in Equation 2.29
q
q1
q2
0 0 1 2
0 1 1 1
0 −1 0 1
2 q
1 1 −q1 −q2 √ 2/15 √ 1/ 30 √ −1/ 10 √ 1/ 5
Source: Brink, D.M. and Satchler, G.R., Angular Momentum, 3rd ed., Oxford University Press, 1993. Note that these symbols remain invariant under interchanging the indices q1 and q2 , as well as under simultaneously changing the signs of q1 , q2 , and q.
momenta m1 and m2 , in which case q is the angular momentum transferred to the relative angular momentum of the pair of molecules. Remarkably, apart from a numerical factor that can be easily calculated, the part of the quantum-mechanical dipole–dipole interaction corresponding to angular momentum transfer q has an angular dependence given simply by the multipole term C2−q . Suppose that the molecules, when far apart, are in the well-defined states of 2.28. Then the diagonal matrix element of the dipole–dipole potential evaluates to (1 − 3 cos2 θ) m1 1 ω ¯ m 2 2 ω ¯ μ2 cos 2δm1 cos 2δm2 . (2.30) μ1 j( j + 1) j( j + 1) R3 This has exactly the form of the interaction for classical, polarized dipoles, as in Equation 2.4. The difference is that each dipole μ1 , μ2 , is replaced by a quantumcorrected version (in large parentheses). It is no coincidence that this is the same quantum-corrected dipole moment that appeared in expression 2.24 for the field due to a single dipole. When both dipoles are aligned with the field, we have m1 1 > 0 and m2 2 > 0 (e.g., both molecules are of type | a), and the interaction has the angular dependence ∝ (1 − 3 cos2 θ). On the other hand, when one dipole is aligned with the field and the other is against (e.g., one molecule is of type | a and the other is of type | c), then the opposite sign occurs – just as we would expect from classical intuition. More generally, at finite electric field, or at finite values of R, the molecules do not remain in the separated-molecule eigenstates 2.28, because they exert torques on one another. The interaction among several different internal molecular states makes the scattering of two molecules a “multichannel problem,” the formulation and solution of which is described in Chapters 5, 9, 13, 14, and 15. However, a good way to visualize the action of the dipole–dipole potential on the molecules is to construct an adiabatic , the relative surface. To do so, we diagonalize the interaction at a fixed value of R location of the two molecules. Before doing this, we must consider the quantum statistics of the molecules. If the two molecules under consideration are identical bosons or identical fermions (with © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
μ1 = μ2 = μ), then the total two-molecule wavefunction must account for this fact. This total wavefunction is α1 β1 γ1 | jm1 ω ¯ 1 α2 β2 γ2 | jm2 ω ¯ 2 Fjω;m ¯ 1 1 m2 2 (R, θ, φ). This wavefunction is either symmetric or antisymmetric under the exchange of the two particles, which is accomplished by swapping the internal states of the molecules, while simultaneously exchanging their center-of-mass coordinates, that to −R : is, by mapping R (α1 β1 γ1 ) ↔ (α2 β2 γ2 ) R→R
(2.31)
θ→π−θ φ → π + φ.
For the molecule’s internal coordinates, a wavefunction with definite exchange symmetry is given by α1 β1 γ1 | jm1 ω ¯ 1 α2 β2 γ2 | jm2 ω ¯ 2 s =
1
2(1 + δm1 m2 δ1 2 ) × α1 β1 γ1 | jm1 ω ¯ 1 α2 β2 γ2 | jm2 ω ¯ 2
+ sα2 β2 γ2 | jm1 ω ¯ 1 α1 β1 γ1 | jm2 ω ¯ 2 . (2.32)
The new index s = ±1 denotes whether the combination 2.32 is even or odd under the → −R interchange. If s = +1, then F must be symmetric under the transformation R for bosons, and odd under this transformation for identical fermions. If s = −1, the reverse must hold. We now have the tools required to consider the form of the dipole–dipole interaction beyond the “pure” dipolar form 2.30. The details of this analysis will depend on the Schrödinger equation to be solved. The Schrödinger equation reads
2 2 − ∇ + Vd + HS Ψ = EΨ. 2mr
Here HS stands for the threshold Hamiltonian that includes Λ-doubling and electric field interactions, and is assumed to be diagonal in the basis 2.32; and mr is the reduced mass of the pair of molecules. In the usual way, we expand the total wavefunction Ψ as 1
Ψ(R, θ, φ) = Fi (R, θ, φ) | i , R i
where the index i stands for the collective set of quantum numbers { jω; ¯ m1 1 m2 2 s}. © 2009 by Taylor and Francis Group, LLC
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Electric Dipoles at Ultralow Temperatures
Inserting this expansion into the Schrödinger equation and projecting onto the ket i | leads to the following set of coupled equations: 2 ∂ ∂2 ∂ 1 1 ∂2 − Fi sin θ + + 2 2mr ∂R2 ∂θ R sin θ ∂θ R2 sin2 θ ∂φ2
+ i | Vd | i Fi + i | HS | iFi = EFi . i
If we keep N channels i, then this represents a set of N coupled differential equations. We can, in principle, solve these subject to physical boundary conditions for any bound or scattering problem at hand. For visualization, however, we will find it convenient to reduce these equations to fewer than three independent variables. We carry out this task in the following subsections.
2.5.2 ADIABATIC POTENTIAL ENERGY SURFACES IN TWO DIMENSIONS Applying an electric field in the zˆ -direction establishes zˆ as an axis of cylindrical symmetry for the two-body interaction. The angle φ determines the relative orientation of the two molecules about this axis, thus the interaction cannot depend on this angle. To handle this, we include an additional factor in our basis set, 1 | ml = √ exp(iml φ). 2π We then expand the total wavefunction as ΨMtot (R, θ, φ) =
1
F (R, θ)| ml | i . R i ml i ml
In each term of this expression the quantum numbers must satisfy the conservation requirement for fixed total angular momentum projection, Mtot = m1 + m2 + ml . In addition, applying exchange symmetry to each term requires that Fi,ml (R, π − θ) = s(−1)ml Fi,ml (R, θ) for bosons, and s(−1)ml +1 Fi,ml (R, θ) for fermions. Inserting this expansion into the Schrödinger equation yields a slightly different set of coupled equations: 2 − 2mr +
∂2 1 ∂ ∂ + sin θ Fi,ml ∂θ ∂R2 R2 sin2 θ ∂θ
2 ml2 2mr
R2 sin2 θ
Fiml +
i ml
i | Vd2D | i Fi m + i | HS | iFiml = EFiml . (2.33) l
This substitution has the effect of replacing the differential form of the azimuthal kinetic energy, ∝ ∂ 2 /∂φ2 , by an effective centrifugal potential ∝ ml2 /R2 sin2 θ. In © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
addition, the matrix elements of the dipolar potential Vd2D are slightly different from those of Vd . Recall that the (θ, φ)-dependent part of the matrix element 2.29 is proportional to C2−q (θ, φ), which we will write explicitly as C2−q (θφ) ≡ C2−q (θ) exp(−iqφ). This equation explicitly defines a new function C2−q (θ) that is a function of θ alone, and that is proportional to an associated Legendre polynomial. The matrix element of the potential now includes the following integral:
1 1 dφ √ e−iml φ C2−q (θ)e−iqφ √ eiml φ 2π 2π C2−q (θ) = dφei(Mtot −Mtot )φ 2π
ml | C2−q | ml =
C2−q (θ), = δMtot Mtot
which establishes the conservation of the projection of total angular momentum by the dipole–dipole interaction. Therefore, matrix elements of Vd2D in this representation are identical to those in of Vd in Equation 2.29 except that the factor exp(−iqφ) is . replaced by δMtot Mtot With these matrix elements in hand, we can construct solutions to the coupled differential equations 2.33. However, to understand the character of the potential surface, it is useful to construct adiabatic potential energy surfaces. This means that, for a fixed relative position of the molecules (R, θ), we find the energy spectrum of Equations 2.33 by diagonalizing the Hamiltonian Vc2D + Vd2D + HS , where Vc2D is a short-hand notation for the centrifugal potential discussed above. This approximation is common throughout atomic and molecular physics, and amounts to defining a single surface that comes as close as possible to representing what is, ultimately, multichannel dynamics.
2.5.3
EXAMPLE: j = 1/2 MOLECULES
Analytic results for the adiabatic surfaces are rather difficult to obtain. Consider the simplest realization of our model, a molecule with spin j = 1/2. In this case each molecule has four internal states (two values of m and two values of ), so that the two-molecule basis comprises 16 elements. Dividing these according to exchange symmetry of the molecules’ internal coordinates, there are ten channels within the manifold of s = +1 channels, and six within the s = −1 manifold. These are the cases we will discuss in the following, although the same qualitative features also appear in higher-j molecules. As the simplest illustration of the influence of internal structure on the dipolar interaction, we will focus on the lowest-energy adiabatic surface, and show how it differs from the “pure dipolar” result (Equation 2.30) as the molecules approach one another. The physics underlying this difference arises from the fact that the dipole–dipole interaction becomes stronger as the molecules get closer together, and at some point this © 2009 by Taylor and Francis Group, LLC
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Electric Dipoles at Ultralow Temperatures
interaction is stronger than the action of the external field that holds their orientation fixed in the lab. The intermolecular distance at which this happens can be approximately calculated by setting the two interactions equal, μ2 /R03 = (μE)2 + (Δ/2)2 , yielding a characteristic distance R0 =
μ2 (μE)2 + (Δ/2)2
1/3 .
When R R0 , the electric field interaction is dominant, the dipoles are aligned, and the interaction is given by Equation 2.30. When R becomes comparable to, or less than, R0 , then the dipoles tend to align in a head-to-tail orientation to minimize their energy, regardless of their relative location. Before proceeding, it is instructive to point out how large the scale R0 can be for realistic molecules. For the OH molecule considered above, with μ = 1.7 D and Δ = 0.06 cm−1 , the molecule can be polarized in a field of E ≈ 1600 V/cm. At this field, the characteristic radius is approximately R0 ≈ 120a0 (where a0 = 0.053 nm is the Bohr radius), far larger than the scale of the molecules themselves. Therefore, while the dipole–dipole interaction is by far the largest interaction energy at large R, over a significant range of R this potential does not take the usual dipolar form. To take an even more extreme case, the molecule NiH has a ground state of 2 Δ symmetry with j = 5/2 [2]. Because it is a Δ, rather than a Π, state, its Λ doublet is far smaller, probably on the order of ∼10−5 cm−1 . This translates into a critical field of E ≈ 0.5 V/cm, and a characteristic radius at this field R0 ≈ 2000a0 ≈ 0.09 μm. This length is approaching a nonnegligible fraction of the interparticle distance in a Bose–Einstein condensed sample of such molecules (assuming a density of 1014 cm−3 , this spacing is of order 0.2 μm). Deviations from the simple dipolar behavior may thus influence the macroscopic properties of a quantum degenerate dipolar gas (see Chapter 12). As an example, we present in Figure 2.2 sections of the lowest-energy adiabatic potential energy surfaces for a fictitious j = 1/2 molecule whose mass, dipole moment, and Λ doublet are equal to those of OH. These were calculated in a strongfield limit with E = 104 V/cm. Each row corresponds to a particular intermolecular separation, which is compared to the characteristic radius R0 . However, as noted above, there are two possible ways for the molecule to have its lowest energy, as illustrated by parts (a) and (b) of Figure 2.1. Interestingly, it turns out that these give rise to rather different adiabatic surfaces. To illustrate this, we show in the left column of Figure 2.2 the surface fora pair of type | a molecules, which corresponds at infinitely large R to the channel 21 +, 21 +; s = 1 ; and in the right column we show combinations of one type | a and one type | b molecule. In the latter case, there are two possible symmetries corresponding to s = ±1, both of which are shown. Finally, for comparison, the unperturbed “pure dipole” result is shown in all panels as a dotted line. Consider first two molecules of type | a (left column of Figure 2.2). For large distances R > R0 (Figure 2.2a), the adiabatic potential deviates only slightly from the pure-state result, reducing the repulsion at θ = π/2. When R approaches the © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications |aaÒ
(a)
|ab, sÒ
(d)
Energy (K)
0.02 0.00 1.5R0 –0.02
Energy (K)
(b) 0.08
(e)
0.00
R0 s = +1 s = –1
–0.08
(f )
Energy (K)
(c) 0.0
0.5R0 –0.5 –1.0 0
p/2 q
p
0
p/2 q
p
FIGURE 2.2 Angular dependence of adiabatic potential energies for various combinations of molecules at different interparticle spacings R, which are indicated on the right side. Dotted lines: diagonal matrix element of the interaction, assuming both molecules remain strongly aligned with the electric field. Solid and dashed lines: adiabatic surfaces. These surfaces are based on the j = 1/2 model discussed in the text, using μ = 1.68 D, Δ = 0.056 cm−1 , mr = 8.5 amu, and E = 104 V/cm, yielding R0 ∼ 70a0 . The left-hand column presents results for two molecules of type | a, as labeled in Figure 2.1; in the right column are results for one molecule of type | a and one of type | b, which necessitates specifying an exchange symmetry s.
characteristic radius R0 (Figure 2.2b), the effect of mixing in the higher-energy channels becomes apparent. Finally, when R < R0 (Figure 2.2c), the mixing is even more significant. In this case the dipole–dipole interaction is the dominant energy, with the threshold energies serving as a small perturbation. As a consequence, the quan tum numbers 21 +, 21 +; s = 1 can no longer identify the channel. It is beyond the scope of this chapter to discuss the corresponding eigenstates in detail. Nevertheless, we find that for R < R0 the channel | aa (repulsive at θ = π/2) is strongly mixed with the channel | ac (attractive at θ = π/2). The combination is just sufficient that the two channels nearly cancel out one another’s θ-dependence. At the ends of the range, however, the adiabatic curve is contaminated by a small amount of channels containing centrifugal energy ∝ 1/ sin2 θ. The right column of Figure 2.2 shows adiabatic curves for the mixed channels, one molecule of type | a and one of type | b. In this case there are two possible signs of s; © 2009 by Taylor and Francis Group, LLC
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Electric Dipoles at Ultralow Temperatures
these are distinguished by using solid lines for channel 21 +, − 21 −; s = 1 and dashed lines for channel 21 +, − 21 −; s = −1 . Strikingly, these surfaces are different both from one another, and from the surfaces in the left column of the figure. Ultimately this arises from different kinds of channel couplings in the potentials (Equation 2.29). Note that, while type | a and type | b molecules are identical in their interaction energy with the electric field, they still represent different angular momentum states. Nevertheless, molecules in these channels still closely reflect the pure dipolar potential at large R, and become nearly θ-independent for small R. A further important point is that the potentials described here represent large energies as compared to the mK or μK translational kinetic energies of cold molecules, and will therefore significantly influence their dynamics. Further, the potentials depend strongly on the value of the electric field of the environment, both through the direct effect of polarization on the magnitude of the dipole moments, and through the influence of the field on the characteristic radius R0 . It is this sensitivity to field that opens the possibility of control over interactions in an ultracold dipolar gas (see Chapter 12). Although we have limited the discussion here to the lowest adiabatic state, interesting phenomena are also expected to arise due to avoided crossings in excited states. Notable is a collection of long-range quasibound states, whose intermolecular spacing is roughly centered around R0 [7]. Such states could conceivably be used to associate pairs of molecules into well-characterized transient states, furthering the possibilities of control of molecular interactions.
2.5.4 ADIABATIC POTENTIAL ENERGY CURVES IN ONE DIMENSION: PARTIAL WAVES For many scattering applications, it is not necessarily convenient to express the dipole– dipole interaction as a surface (more properly, a set of surfaces) in the variables (R, θ, φ) describing the relative position and orientation of the molecules. Rather, it is useful to expand the relative angular coordinates (θ, φ) in a basis as well. To do this, the basis set 2.28 is augmented by spherical harmonics describing the relative orientation of the molecules, to become ¯ 1 α2 β2 γ2 | jm2 ω ¯ 2 θφ | lml , α1 β1 γ1 | jm1 ω with
θφ | lml = Ylml (θφ) =
2l + 1 Clml (θφ). 4π
The total wavefunction is therefore described by the superposition ΨMtot (R, θ, φ) =
1
F Y (θφ) | i , R i ,l ,ml l ml i ,l ,ml
which represents a conventional expansion into partial waves. The wavefunction is, as above, restricted by conservation of angular momentum to require Mtot = m1 + © 2009 by Taylor and Francis Group, LLC
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m1 + ml to have a constant value. Moreover, the effect of the symmetry operation 2.31 on | lml is to introduce a phase factor (−1)l . Therefore, the wavefunction is restricted to s(−1)l = 1 for bosons, and s(−1)l = −1 for fermions. The effect of this extra basis function is to replace the C2−q factor in Equation 2.29 by its matrix element C2−q → lml | C2−q | l ml =
l (2l + 1)(2l + 1)(−1)ml 0
2 0
l 0
l −ml
2 −q
l . ml
(2.34)
From this expression it is seen that the angular momentum q, lost to the internal degrees of freedom of the molecules, appears as the change in their relative orbital angular momentum, that is, ml = ml + q. From here, the effects of field dressing are exactly as treated above. The quantum number l, as is usual in quantum mechanics when treated in spherical coordinates, represents the orbital angular momentum of the pair of molecules about their center of mass. Following the usual treatment, this leads to a set of coupled radial Schrödinger Equations for the relative motion of the molecules: −
2 d2 Filml 2 l(l + 1) + Filml + i | Vd1D | i Fi l m + i | HS | iFilml = EFilml . 2 2 l 2mr dR 2mr R i l ml
(2.35) The second term in this expression represents a centrifugal potential ∝1/R2 , which is present for all partial waves l > 0.
2.5.5 ASYMPTOTIC FORM OF THE INTERACTION Casting the Schrödinger equation as an expansion in partial waves, and the interaction as a set of curves in R, rather than as a surface in (R, θ, φ), allows us to explore more readily the long-range behavior of the dipole–dipole interaction. The first 3-j symbol in Equation 2.34 vanishes unless l + 2 + l is an odd number, meaning that even (odd) partial waves are coupled only to even (odd) partial waves. Moreover, the values of l and l can differ by at most two. Thus the dipolar interaction can change the orbital angular momentum state from l = 2 to l = 4, for instance, but not to l = 6. Finally, the interaction vanishes altogether for l = l = 0, meaning that the dipole–dipole interaction nominally vanishes in the s-wave channel. Because all other channels have higher energy, due to their centrifugal potentials, it appears that the lowest adiabatic curve is trivially equal to zero. This is not the case, however, because the s-wave channel is coupled to a nearby d-wave channel with l = 2. Ignoring higher partial waves, the Hamiltonian corresponding to a particular channel i at long range has the form
0 A20 /R3
© 2009 by Taylor and Francis Group, LLC
A02 /R3 . 32 /mr R2 + A22 /R3
Electric Dipoles at Ultralow Temperatures
67
Here A02 = A20 and A22 are coupling coefficients that follow from the expressions derived above. Note that all these coefficients are functions of the electric field E. Now, the comparison of dipolar and centrifugal energies defines another typical length scale for the interaction, namely, the one where μ2 /R3 = 2 /mr R2 , defining a “dipole radius” RD = μ2 mr /2 . (More properly, one could define an electric-field-dependent radius by substituting A02 for μ2 .) For OH, this length is ∼6800a0 , while for NiH it is ∼9000a0 . When the molecules are far apart, R > RD , the dipolar interaction is a perturbation. The size of this perturbation on the s-wave interaction is found through second-order perturbation theory to be A202 mr 1 (A02 /R3 )2 − 2 ∼− . 3 /mr R2 32 R4 Therefore, at very large intermolecular distances, the effective potential, as described by the lowest adiabatic curve, carries an attractive 1/R4 dependence on R, rather than the nominal 1/R3 dependence. At closer range, when R < RD , the 1/R3 terms dominate the 1/R2 centrifugal interaction, and the potential reduces again to the expected 1/R3 dependence on R.
ACKNOWLEDGMENTS Many fruitful discussions of molecular dipoles over the years are acknowledged, notably those with Aleksandr Avdeenkov, Doerte Blume, Daniele Bortolotti, Jeremy Hutson, and Chris Ticknor. This work was supported by the National Science Foundation.
REFERENCES 1. Brink, D.M. and Satchler, G.R., Angular Momentum, 3rd ed., Oxford University Press, 1993. 2. Brown, J. and Carrington, A., Rotational Spectroscopy of Diatomic Molecules, Cambridge University Press, 2003. 3. Jackson, J.D., Classical Electrodynamics, 2nd ed., Wiley, New York, 1975. 4. Bethe, H.A. and Salpeter, E.E., Quantum Mechanics of One- and Two-Electron Atoms, Plenum Press, 1977. 5. Hougen, J.T., The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic Molecules, NBS Monograph 115, available at http://physics.nist.gov/Pubs/Mono115/ 6. Allen, L. and Eberly, J.H., Optical Resonance and Two-Level Atoms, Wiley, New York, 1975. 7. Avdeenkov, A.V., Bortolotti, D.C.E., and Bohn, J.L., Field-linked states of ultracold polar molecules, Phys. Rev. A, 69, 012710, 2004.
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Inelastic Collisions and 3 Chemical Reactions of Molecules at Ultracold Temperatures Goulven Quéméner, Naduvalath Balakrishnan, and Alexander Dalgarno CONTENTS 3.1 3.2
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inelastic Atom–Molecule Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Vibrational and Rotational Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.1 Collisions at Cold and Ultracold Temperatures . . . . . . . . . . . 3.2.1.2 Shape Resonances in Molecular Collisions . . . . . . . . . . . . . . . 3.2.1.3 Feshbach Resonances in Molecular Collisions . . . . . . . . . . . 3.2.2 Quasiresonant Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Atom–Molecular Ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Chemical Reactions at Ultracold Temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Tunneling-Dominated Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.1 Reactions at Zero Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.2 Feshbach Resonances in Reactive Scattering . . . . . . . . . . . . . 3.3.2 Barrierless Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.1 Collision Systems of Three Alkali Metal Atoms . . . . . . . . . . 3.3.2.2 Role of PES in Determining Ultracold Reactions . . . . . . . . 3.3.2.3 Relaxation of Vibrationally Excited Alkali Metal Dimers 3.3.2.4 Reactions of Heteronuclear and Isotopically Substituted Alkali-Metal Dimer Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Inelastic Molecule–Molecule Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Molecules in the Ground Vibrational State . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Vibrationally Inelastic Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70 71 71 71 79 80 82 83 84 86 87 92 95 95 101 104 106 108 108 112 115 116 117
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Cold Molecules: Theory, Experiment, Applications
INTRODUCTION
The development of techniques for cooling and trapping of a wide variety of atomic and molecular species in recent years has created exciting opportunities for probing and controlling atomic and molecular encounters with unprecedented precision [1]. While many of the initial studies of cold atoms and molecules were centered on the creation of dense samples of cold and ultracold matter, more recent work has focused on the manipulation and control of intermolecular interactions, with the ultimate aim of achieving quantum control of atomic and molecular collisions [2]. Although the ideas of quantum control of chemical reactions were proposed many years ago, the ability to create ultracold molecules in specific quantum states has given further stimulus to this field. Its development requires that molecular properties and collisional behavior be well understood at cold and ultracold temperatures where the dynamics of molecules are dramatically different compared to collisions at elevated temperatures. Over the last ten years significant progress has been achieved both in theoretical and experimental works. The experimental methods such as photoassociation spectroscopy, magnetic tuning of Feshbach resonances, buffer-gas cooling, and Stark deceleration [3–6] have been developed and applied to a variety of molecular systems. Novel methods to study ultracold chemical reactions involving ion–molecule systems in a linear Paul trap have been proposed [7]. External control of chemical reactions using electric and magnetic fields is another area of active interest [8]. The aim of this chapter is to provide an overview of recent progress in characterizing molecular processes and chemical reactions at cold and ultracold temperatures, with particular emphasis on theoretical developments in quantum-dynamics simulations of atom–molecule collision systems over the last ten years. In contrast to scattering at thermal energies, ultracold collisions offer fascinating and unique opportunities to study molecular encounters in the extreme quantum regime where the entire collision can be dominated by a single partial wave. One of the main motivations of current experimental efforts to create dense samples of ultracold molecules is to study the possibility of chemical reactions at temperatures close to absolute zero. While Wigner’s law [9,10] predicts that rate coefficients of exothermic processes are finite in the zero-energy limit, it does not say if the rate coefficient will be large enough for reactions to be observable in an experiment. Nor does it say anything about how the rate coefficients at zero temperature depend on the interaction potential. Most chemical reactions between neutral atoms and molecules involve an energy barrier and it is not clear if chemical reactions between them generally occur with measurable rates at ultracold temperatures. Calculations for the F + H2 reaction, which proceeds by tunneling at low energies, have shown that the reaction may occur with a significant rate at ultracold temperatures [11]. There is also experimental and theoretical evidence that chemical reactions involving heavy-atom tunneling of carbon [12] and fluorine [13] atoms can occur with significant rate coefficients at low temperatures. Photoassociation experiments involving alkali-metal systems have stimulated considerable interest in chemical reactivity in alkali-metal dimer–alkali-metal atom collisions at ultracold temperatures [14]. Rearrangement collisions in identical particle alkali-metal trimer systems occur without energy barrier, and recent studies have
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indicated that chemical reactions in alkali-metal atom–alkali-metal dimer collisions may be very fast at ultracold temperatures. Unlike tunneling-dominated reactions, the limiting values of the rate coefficients for alkali-metal systems are less sensitive to vibrational excitation of the dimer. In this chapter, we give an overview of recent studies of ultracold atom–molecule collisions, focusing on nonreactive and reactive systems and the effect of vibrational excitation of the molecule on the collisional outcome. We will discuss both tunneling-dominated and barrierless reactions and examine recent efforts in extending these studies to ionic systems as well as molecule–molecule systems. We consider mostly the novel aspects of collisional dynamics of atom–diatom systems at cold and ultracold temperatures with illustrative results for specific systems. For more comprehensive discussion of cold and ultracold collisions including reactive and nonreactive processes and the effect of external fields we refer the reader to several review articles [6,8,13–15] that have appeared in the last few years. For details of the theoretical formalisms we refer to the chapters by Hutson and by Tscherbul and Krems.
3.2
INELASTIC ATOM–MOLECULE COLLISIONS
Theoretical studies of ultracold molecules received intense interest after the success of ultracold atom photoassociation and buffer-gas-cooling experiments, which demonstrated that a wide array of molecular systems with thermal and nonthermal vibrational energy distributions can be created in ultracold traps. The collisional loss of trapped molecules is an important issue in these experiments. Photoassociation produces molecules in highly excited vibrational levels. Whether the excited molecules decay by vibrational quenching or through chemical reaction is an intriguing question. Although an extensive literature exists on the collisional relaxation of vibrationally excited molecules at elevated temperatures not much was known about the magnitude of the relaxation rate coefficients at temperatures lower than 1 K. A few earlier reports [16–18] on atom–diatom collisions in the Wigner threshold regime had been published, but a detailed investigation of the dependence of the relaxation rate coefficients on the internal energy of molecules and their sensitivity to details of the interaction potential has not been carried out. Here, we give a brief account of recent quantum-dynamics calculations of vibrational and rotational energy transfer in atom–diatom collisions at cold and ultracold temperatures. We focus on a few representative systems to illustrate the main features of energy transfer in nonreactive atom–molecule collisions at ultracold temperatures, and we show how the corresponding rate coefficients are influenced by rotational or vibrational excitation of the molecule.
3.2.1 VIBRATIONAL AND ROTATIONAL RELAXATION 3.2.1.1
Collisions at Cold and Ultracold Temperatures
As discussed in the chapter by Hutson, at very low energies, scattering is dominated by s-waves, and the scattering cross-section can be expressed in terms of a single parameter called the scattering length. For single-channel scattering where only elastic © 2009 by Taylor and Francis Group, LLC
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scattering is possible, the scattering length is a real quantity and the magnitude of the cross-section in the s-wave limit is given by σ = 4πa2 , where a is the scattering length. For multichannel scattering, as in vibrationally or rotationally inelastic collisions of molecules, the scattering length is a complex number and is denoted as avj = αvj − iβvj where v and j are, respectively, the initial vibrational and rotational quantum numbers of the molecule [10,19]. The limiting value of the elastic cross-section in el = 4π|a |2 = 4π(α2 + β2 ). The the presence of inelastic scattering is given by σvj vj vj vj total inelastic quenching cross-section from a given initial rovibrational level of the molecule is related to the imaginary part of the scattering length through the relation in = 4πβ /k where k is the wavevector in the incident channel. The quenching σvj vj vj vj in = rate coefficient becomes constant at ultralow temperatures and is given by kvj 4πβvj /μ at zero temperature, where μ is the reduced mass of the collisional system. Thus, the rovibrational relaxation rate coefficients attain finite values for different initial vibrational and rotational levels of the molecule. The dependence of the rate coefficients on v and j has been an important issue in cold molecule research because exothermic vibrational and rotational relaxation collisions are a major pathway for trap loss in cooling and trapping experiments. Initial studies of rotational and vibrational relaxation of atom–molecule systems at cold and ultracold temperatures have mostly focused on van der Waals systems such as He–H2 [19–22], He–CO [23,24], and He–O2 [25]. Owing to the importance of some of these systems in astrophysical environments, extensive calculations of low-temperature behavior of rate coefficients have been performed for collisions of H2 [19] and CO [23,24] with both 3 He and 4 He. For both systems reasonably accurate intermolecular potentials have been reported. The initial calculations on the He–H2 system employed the potential energy surface (PES) of Muchnick and Russek (MR) [26]. For He–H2 , vibrational excitation of the H2 molecule has a dramatic effect on the zero-temperature quenching rate coefficients. As illustrated in Figure 3.1, the vibrational quenching rate coefficients increase by about three orders of magnitude between v = 1 and v = 10 of the H2 molecule [19]. The quenching rate coefficients exhibit a minimum at around 10 K, which roughly corresponds to the depth of the van der Waals interaction potential. This behavior appears to be a characteristic of vibrational quenching rate coefficients. For incident energies lower than the well depth the rate coefficient exhibits a minimum and with subsequent decrease in temperature the rate coefficient begins to increase before attaining the Wigner limit. For systems with deeper van der Waals wells the minimum is shifted to higher temperatures. Measurements of vibrational relaxation rate coefficients for the H2 −CO system have confirmed this behavior [27]. Balakrishnan, Forrey, and Dalgarno [28] investigated vibrational relaxation of H2 in collisions with H atoms for vibrational quantum numbers v = 1 to 12 of the H2 molecule. They adopted a nonreactive scattering formalism and neglected the rotational motion of the H2 molecule. The calculations showed that vibrational relaxation rate coefficients are strongly dependent on the initial vibrational level of the H2 molecule. The relaxation rate coefficients were found to increase by about seven orders of magnitude between vibrational levels v = 1 and v = 12. The dramatic variation in the rate coefficients with increase in vibrational excitation was explained in terms of the matrix elements of the interaction potential between © 2009 by Taylor and Francis Group, LLC
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Inelastic Collisions and Chemical Reactions of Molecules
Rate coefficient (cm3/molecule/sec)
10−12
10−14
10−16
u = 10 u=9 u=8 u=7 u=6 u=5 u=4 u=3 u=2 u=1
10−18
10−20 −6 10
10−4
10−2 100 Temperature (K)
102
104
FIGURE 3.1 Rate coefficients for the quenching of H2 (v, j = 0) by collisions with He atoms as functions of the temperature for v = 1 to 10 of the H2 molecule. (From Balakrishnan, N. et al., Phys. Rev. Lett., 80, 3224, 1998. With permission.)
the vibrational wavefunctions as functions of the atom–molecule center-of-mass separation. Vibrational relaxation rate coefficients in atom–molecule systems are often influenced by van der Waals complexes formed during the collision process. Decay of these complexes leads to resonances in the energy dependence of the relaxation crosssections (see Figure 3.2 for Ar–D2 collisions). Applying effective range theory to describe ultracold collisions of the He–H2 system, Balakrishnan and colleagues [19] and Forrey and colleagues [20] demonstrated that vibrational predissociation lifetimes of resonances that lie close to the energy threshold can be derived accurately from the value of the zero-temperature quenching rate coefficient. This formalism was extended to describe vibrational relaxation of trapped molecules and it was shown that the vibrational relaxation rate is controlled by the most weakly bound state of the van der Waals complex [22]. In a related work Dashevskaya and colleagues [29] have shown that vibrational quenching of H2 (v = 1, j = 0) at low temperatures can be described using a two-channel approximation within the quasiclassical method provided appropriate parameters are employed in the calculations. Subsequently, Côté and colleagues [30] generalized this method to predict vibrational relaxation lifetimes of atom–diatom van der Waals complexes with energies near the dissociation threshold. © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications 10–4
j¢ = 2 j¢ = 4 j¢ = 6 j¢ = 8
Cross-section (10–16 cm2)
10–6
10–8
10–10
10–12 –8 10
10–6
10–4
10–2
Kinetic energy
100
102
104
(cm–1)
FIGURE 3.2 Cross-sections for quenching of the v = 1, j = 0 level of D2 in collisions with Ar atoms resolved into the different rotational levels j in v = 0 as functions of the incident kinetic energy. (From Uudus, N. et al., J. Chem. Phys., 122, 024304, 2005. With permission.)
Unlike in thermal energy collisions, the presence of a weakly bound state in the vicinity of a channel threshold can dramatically influence the cross-sections in the ultracold regime. This is illustrated in Figure 3.2 where the cross-sections for vibrational relaxation of D2 (v = 1, j = 0) in collisions with Ar atoms [31] are presented over an energy range of 10−8 to 103 cm−1 . The cross-sections exhibit a curvature characteristic of a resonant enhancement in the energy range 10−5 to 10−3 cm−1 . Such enhancement of the cross-section can occur when the interaction potential supports a virtual state or a very weakly bound state near the channel threshold leading to a zero-energy resonance. The virtual state is characterized by a large negative scattering length while the bound state is characterized by a large positive scattering length. For the present case the real part of the scattering length is large and positive (α10 = 97.0 Å) and the resonance occurs from the decay of a loosely bound van der Waals complex supported by the entrance channel potential. For energies below 10−2 cm−1 the cross-section is dominated by s-wave scattering in the incident channel, and the zero-energy resonance arises from s-wave scattering in the entrance channel. The resonant enhancement is more clearly seen in the plot of the reaction probability as a function of the kinetic energy, shown in Figure 3.3. The probability peaks at an energy of 6.0 × 10−4 cm−1 , which roughly corresponds to the binding © 2009 by Taylor and Francis Group, LLC
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Inelastic Collisions and Chemical Reactions of Molecules 1e–11
8e–12
Probability
6e–12
4e–12
2e–12
0e+00 10–6
10–5
10–3
10–4
Incident kinetic energy
10–2
(cm–1)
FIGURE 3.3 Total probability of quenching of the v = 1, j = 0 level of D2 in collisions with Ar atoms as a function of the incident kinetic energy. The peak value of the probability corresponds to a zero energy resonance. (From Uudus, N. et al., J. Chem. Phys., 122, 024304, 2005. With permission.)
energy of the quasibound state. The resonance appears in the scattering calculations at energies above the threshold due to its close proximity to the channel threshold. The binding energy of the quasibound state can be estimated using the scattering length approximation [10,31]. The magnitude of the binding energy is given by |Eb | = 2 cos 2γ10 /(2μ|a10 |2 ) where μ is the reduced mass of the Ar–D2 system, a10 = α10 − iβ10 is the scattering length for the v = 1, j = 0 level, and γ10 = tan−1 (β10 /α10 ). This yields a value of |Eb | = 4.9 × 10−4 cm−1 , in reasonable agreement with the exact value derived from scattering calculations. A more accurate value of the binding energy can be obtained using the effective range formula given by Forrey and colleagues [20]: α10 r0 2 2α10 r0 |Eb | = 1− − 1− |a10 |2 |a10 |2 μr02 where r0 is the effective range of the potential, which may be evaluated by fitting the low-energy behavior of the phase shift for the elastic channel to the standard 2 /2. For Ar–D (v = 1, j = 0) effective range formula, k10 cot δ10 = −1/α10 + r0 k10 2 collisions, the effective range formula yields r0 = 16.32 Å. The resonance position © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
calculated using the effective range approximation is |Eb | = 5.95 × 10−4 cm−1 , in excellent agreement with the value of 6.0 × 10−4 cm−1 obtained from the scattering calculations. It is generally very difficult to accurately evaluate energies of such weakly bound states using standard bound state codes and the effective range formula provides a convenient and reliable method to calculate binding energies of weakly bound states that lead to zero-energy resonances. One of the challenging aspects of cold and ultracold collisions is the sensitivity to details of the interaction potential. Even the best available methods for the electronic structure calculations of PESs result in errors much larger than the collision energies in the cold and ultracold regime, and the dynamics calculations are often sensitive to small changes in the interaction potential. To explore the sensitivity of cold and ultracold collisions to details of the interaction potential, Lee and colleagues [32] performed a comparative study of the ultracold collision dynamics of the He–H2 system using the MR potential and a more recent ab initio potential developed by Boothroyd, Martin, and Peterson (BMP) [33]. The BMP potential was considered to be an improvement, approaching chemical accuracy, over all conformations compared to the MR potential. However, significant differences were observed for vibrational relaxation of the v = 1, j = 0 state of the H2 molecule in collisions with He computed using the two surfaces. The limiting value of the quenching rate coefficient on the BMP surface was found to be about three orders of magnitude larger than that of the MR surface. The difference was attributed to the more anisotropic nature of the BMP surface leading to larger values of the off-diagonal elements responsible for driving vibrational transitions. Indeed, it was found that the vibrational quenching of the v = 1, j = 0 level was dominated by the transition to the v = 0, j = 8 level, which is driven by the high-order anisotropic terms of the interaction potential. To explore the behavior of inelastic collisions involving polar molecules at ultracold temperatures, Balakrishnan, Forrey and Dalgarno [23] investigated vibrational and rotational relaxation of CO in 4 He–CO collisions. The dynamics of the He–CO system was found to exhibit significantly different features at low temperatures compared to the He–H2 system. Quantum scattering calculations of 4 He and 3 He collisions with the CO molecule revealed that the larger reduced mass and the deeper van der Waals interaction potential of the He–CO system give rise to a number of shape resonances in the energy dependence of vibrational relaxation cross-sections [23,24]. The effect of shape resonances on low-temperature vibrational relaxation rate coefficients will be discussed in the next subsection. The computed values of vibrational relaxation rate coefficients for both 3 He and 4 He collisions with CO(v = 1) have been found to be in good agreement with the experimental data of Reid and colleagues [34] in the temperature range 35 to 100 K. Calculations of vibrational relaxation rate coefficients for the He–CO system in the temperature range 35 to 1500 K have also been reported by Krems [35]. He has shown that inclusion of the centrifugal distortion of the vibrational wavefunction enhances the relaxation process, and that the quenching rate coefficients are sensitive to high-order anisotropic terms in the angular expansion of the interaction potential [36]. Bodo, Gianturco, and Dalgarno [37] have extended the work of Balakrishnan and colleagues [23] to study the vibrational relaxation of excited CO(v = 2, j = 0, 1) molecules in collision with 4 He atoms at ultralow energies. They found that © 2009 by Taylor and Francis Group, LLC
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vibrational quenching of CO(v = 2, j = 0, 1) in collisions with 4 He is dominated by the v = 2 → v = 1 transition. The cross-sections for the v = 2 → v = 0 transition were found to be about four orders of magnitude smaller than the single quantum transitions for both j = 0 and j = 1 initial rotational levels. Ultracold vibrational relaxation of a number of other molecules in collisions with He atoms has been reported by a number of other investigators in recent years. Stoecklin, Voronin, and Rayez [38] reported the vibrational relaxation of F2 in collisions with 3 He atoms. In this study, they constructed the PES of He–F2 using ab initio points obtained by high-level molecular electronic-structure calculations, and reported the cross-sections for elastic scattering and inelastic relaxation of F2 (v = 0, 1, j = 0) for collision energies in the range 10−6 to 2000 cm−1 . A similar study has been reported by the same authors for the 3 He + HF(v = 0, 1, j = 0, 1) system [39]. The vibrational quenching cross-sections were found to be very small compared to pure rotational quenching, in agreement with the results for the He–CO system. This is due to the weak dependence of the He–HF PES on the HF internuclear distance and the strong anisotropy of the interaction potential. Bodo and Gianturco [40] presented a comparative study of vibrational relaxation of CO(v = 1, 2, j = 0), HF(v = 1, 2, j = 0), and LiH(v = 1, 2, j = 0) in collisions with 3 He and 4 He atoms in the Wigner regime. The quenching rate coefficients were found to depend strongly on the collision partner. They reported rate coefficients in the range 10−21 to 10−19 cm3 /sec for CO, 10−16 to 10−15 cm3 /sec for HF, and 10−14 to 10−11 cm3 /sec for LiH. The differences were attributed to the features of the intermolecular forces between the diatomic molecules and the He atoms. The interaction potential of the He–CO system is almost isotropic and is characterized by small vibrational couplings elements. The He–HF system is more anisotropic and the couplings between vibrational states are more significant. The interaction potential of the He–LiH system is very anisotropic and it exhibits strong vibrational couplings. There is considerable ongoing experimental interest in cooling and trapping NH [41,42] and OH [43,44] molecules using the buffer gas cooling and Stark deceleration methods described in the chapters by Doyle and by Meijer. Krems and colleagues [45] and Cybulski and colleagues [46] reported cross-sections and rate coefficients for elastic scattering and Zeeman relaxation in 3 He–NH collisions from ultralow energies to 10 cm−1 . The calculations were performed using the rigid rotor approximation and an accurate He–NH PES. It was demonstrated that the elastic scattering of NH molecules with He atoms in weak magnetic fields is at least five orders of magnitude faster than the Zeeman relaxation, which suggests that the NH molecule is a good candidate for buffer-gas cooling. In a related study González–Sánchez and colleagues [47] examined rotational relaxation and spin-flipping in collisions of OH with He atoms at ultralow energies. They found that the rotational relaxation processes dominate the elastic process as the collision energy is decreased to zero. While theoretical prediction of rate coefficients for vibrational and rotational relaxation in a number of atom–diatom systems has been made, comparable experimental results are not available for the majority of these systems. The first measurements of vibrational relaxation of molecules at temperatures below 1 K were reported by Weinstein and colleagues [48]. In their study, CaH molecules slowed down by elastic collisions with 3 He buffer gas atoms were trapped in an inhomogeneous magnetic © 2009 by Taylor and Francis Group, LLC
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field. An upper bound of the rate coefficients for spin-flipping transitions in CaH as well as vibrational relaxation of CaH molecules in the v = 1 vibrational level in collisions with 3 He atoms were estimated at a temperature of about 500 mK. Balakrishnan and colleagues [49] presented a theoretical analysis of the vibrational relaxation of CaH in collisions with 3 He atoms based on quantum close-coupling calculations and an ab initio PES for the He−CaH system developed by Groenenboom and Balakrishnan [50]. In a related study, Krems and colleagues [51] reported cross-sections for spin-flipping transitions in CaH induced by collisions with 3 He and obtained results in close agreement with the experimentally derived values of Weinstein and colleagues [48]. Krems and colleagues demonstrated that at low energies, spin-flipping transitions in the N = 0 rotational level of 2 Σ molecules induced by structureless atoms occur through coupling to the rotationally excited N > 0 levels and that the corresponding rate coefficients are determined by the spin–rotation interaction with the transiently rotationally excited molecule. Table 3.1 provides a compilation of zero-temperature quenching rate coefficients for vibrational and rotational relaxation in a number of atom–diatom systems.
TABLE 3.1 Zero-Temperature Inelastic Rate Coefficients for Different Atom–Molecule Systems System
Initial (v, j )
kT =0 (cm3 /sec)
Reference
H + H2
(v = 1, j = 0)
1.0 × 10−17
[28]
(v = 1, j = 0) (v = 10, j = 0)
3 × 10−17 3.6 × 10−14
[19] [19]
4 He + CO
(v = 1, j = 0) (v = 1, j = 1)
6.5 × 10−21 9.0 × 10−19
[23] [23]
3 He + CO
(v = 1, j = 0) (v = 2, j = 0) (v = 1, j = 0) (v = 2, j = 0)
1.3 × 10−19 2.1 × 10−19 5.3 × 10−21 1.3 × 10−20
[40] [40] [40] [40]
3 He + CaH
(v = 0, j = 1) (v = 1, j = 0)
3.5 × 10−12 2.6 × 10−17
[49] [49]
3 He + HF
(v = 1, j = 0) (v = 2, j = 0) (v = 1, j = 0) (v = 2, j = 0)
3.1 × 10−16 2.6 × 10−15 8.1 × 10−16 6.5 × 10−15
[40] [40] [40] [40]
(v = 1, j = 0) (v = 2, j = 0) (v = 1, j = 0) (v = 2, j = 0)
9.0 × 10−14 3.6 × 10−12 3.8 × 10−13 1.5 × 10−11
[40] [40] [40] [40]
3 He + H
2
4 He + CO
4 He + HF
3 He + LiH 4 He + LiH
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3.2.1.2
Shape Resonances in Molecular Collisions
At energies above the onset of the s-wave regime, cross-sections will be dominated by contributions from nonzero angular momentum partial waves. If the interaction potential includes an attractive part, the effective potentials for nonzero angular momentum partial waves may possess centrifugal barriers that introduce shape resonances in the collision energy dependence of the cross-section. This is illustrated in Figure 3.4 for the vibrational relaxation of CO(v = 1, j = 0) in collisions with 4 He atoms. The sharp features in the energy dependence of the cross-section for energies between 0.1 and 10.0 cm−1 arise from shape resonances supported by the van der Waals interaction potential between He and the CO molecule. As shown in Figure 3.5, when integrated over the velocity distribution of the colliding species, the shape resonances lead to significant enhancement of the vibrational relaxation rate coefficient for temperatures between 0.1 and 10.0 K. Similar results have been found for vibrational relaxation of CO [24], O2 [25], and CaH [49] in collisions with 3 He atoms. The sharp features in the energy dependence of the vibrational relaxation cross-sections for the Ar + D2 sytem shown in Figure 3.2 also arises from shape resonances supported by the Ar–D2 van der Waals potential. The effect is generally more pronounced for systems composed of heavier diatomic molecules and
10−5
Cross-section (10−16 cm2)
10−6
10−7
10−8
10−9 10−5
10−3
10−1 Energy (cm−1)
101
103
FIGURE 3.4 Cross-section for the quenching of the v = 1, j = 0 level of CO in collisions with 4 He as a function of the incident kinetic energy. (From Balakrishnan, N. et al., J. Chem. Phys., 113, 621, 2000. With permission.)
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Rate coefficient (cm3/sec)
u = 1, j = 1
10−19
10−20 u = 1, j = 0
10−21 10−5
10−3
10−1
101
103
T (K)
FIGURE 3.5 Rate coefficients for the quenching of CO(v = 1, j = 0, 1) by collisions with 4 He as functions of the temperature. (From Balakrishnan, N. et al., J. Chem. Phys., 113, 621, 2000. With permission.)
interaction potentials with deep van der Waals wells for which the density of states will be much higher, leading to rich resonance structures in the cross-sections. 3.2.1.3
Feshbach Resonances in Molecular Collisions
Feshbach resonances occur in multichannel scattering in which an unbound (continuum) channel is coupled to a bound state of another channel. If the energy of the interacting system in the unbound channel lies close to that of the bound state and the coupling between the two channels is strong, the cross-section may change dramatically in the vicinity of the resonance. In the Feshbach resonance method for producing ultracold molecules, an external magnetic field is used to tune the energy of the bound pair to that of the separated atoms. In atom–diatom systems, the bound state may correspond to a quasibound state of the atom–diatom van der Waals complex. Channel potentials corresponding to different initial vibrational and rotational levels of the diatom may induce Feshbach resonances. For the He–CO system Feshbach resonances were found to occur near channel thresholds corresponding to the j = 1 rotational level in the v = 0 and v = 1 vibrational levels. Figure 3.6 shows the Feshbach resonance in the elastic scattering cross-sections in the v = 1, j = 0 channel in the vicinity of the v = 1, j = 1 level. The presence of the Feshbach resonance close © 2009 by Taylor and Francis Group, LLC
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Cross-section (10−16 cm2)
260
240
220
200 3.8080
3.8090
3.8100
3.8110
Energy (cm−1)
FIGURE 3.6 Feshbach resonance in the elastic scattering cross-section of CO(v = 1, j = 0) by 4 He atoms. The resonance occurs just below the opening of the v = 1, j = 1 level shown by the vertical line. The energy is relative to the v = 1, j = 0 level of the CO molecule. (From Balakrishnan, N. et al., J. Chem. Phys., 113, 621, 2000. With permission.)
to the opening of the j = 1 level has a dramatic effect on the vibrational quenching cross-sections from the v = 1, j = 1 level of the CO molecule. Since the Feshbach resonance occurs so close to the threshold of the v = 1, j = 1 channel, its effect on scattering in the v = 1, j = 1 level is similar to that of the zero-energy resonance discussed previously for the Ar + D2 system. This is illustrated in Figure 3.5 (see also Table 3.1) where we compare the rate coefficients for vibrational relaxation from the v = 1, j = 0 and v = 1, j = 1 levels of the CO molecule. The zero-temperature limiting value of the quenching rate coefficient of the v = 1, j = 1 level is about two orders of magnitude larger than for the v = 1, j = 0 level. Similar Feshbach resonances have also been shown to occur in the vibrational and rotational predissociation of He–H2 van der Waals complexes [20]. Forrey and colleagues [20] has successfully used the effective range theory to predict the predissociation lifetimes of these resonances. The Feshbach resonances can be used as a very sensitive probe for the interaction potential and also to selectively break or make bonds in chemical reactions. The coupling between the bound and unbound states can be modified by applying an external electric or magnetic field, and this provides an important mechanism for creating or eliminating Feshbach resonances and thereby controlling the collisional © 2009 by Taylor and Francis Group, LLC
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outcome. Krems have shown that weakly bound van der Waals complexes can be dissociated by tuning a Feshbach resonance using an external magnetic field [52]. In this case the dissociation occurs through coupling between Zeeman levels of the bound and unbound channels and the magnitude of the coupling is varied by changing the external magnetic field.
3.2.2
QUASIRESONANT TRANSITIONS
Although the properties of cold and ultracold collisions are quite different from scattering at thermal energies and quantum effects dominate at low temperatures, a remarkable correlation between classical and quantum-dynamics has been discovered in the relaxation of rovibrationally excited diatomic molecules. Experiments performed nearly two decades ago [53,54] showed that collisions of rotationally excited diatomic molecules with atoms may result in very efficient internal energy transfer between specific rotational and vibrational degrees of freedom. The energy transfer becomes highly efficient when the collision time is longer than the rotational period of the molecule. This effect has since been termed “quasiresonant rotation–vibration energy transfer.” The experimental results revealed that the quasiresonant (QR) transitions satisfy the propensity rule Δj = −4Δv or Δj = −2Δv where Δv = vf − vi and Δj = jf − ji [53,54]. This inelastic channel dominates over all other rovibrational transitions. The QR transfer is generally insensitive to details of the interaction potential. Rather, the QR process involves conservation of the action I = nv v + nj j, where nv and nj are small integers. Forrey and colleagues [21] found that the QR transitions also occur in cold and ultracold collisions of rotationally excited diatomic molecules with atoms and that the process is largely insensitive to the details of the interaction potential even in the ultracold regime. The Δj = −2Δv QR transition in He + H2 collisions [55] is illustrated in Figure 3.7 where the zerotemperature vibrational and rotational transition rate coefficients for different initial vibrational levels of the H2 molecule are plotted as functions of the initial rotational level. For initial rotational levels greater than 12 the QR transition becomes the dominant energy transfer mechanism compared to pure rotational quenching. The gap at j = 22 occurs because the Δj = −2Δv transition is energetically not accessible for this initial state at zero temperature. Forrey and colleagues [21] found that the QR process is even more dominant at low temperatures than at thermal energies. Remarkably, classical trajectory calculations [21] were successful in correctly predicting the correlation between Δj and Δv at ultracold temperatures even though the changes in v and j were fractional. Extensive studies of QR energy transfer in cold and ultracold temperatures have been reported by Forrey and colleagues [56–59]. Ruiz and Heller [60] have recently published a review paper providing a detailed analysis of QR phenomenon using semiclassical techniques. McCaffery and colleagues [61–63] have also reported a number of quasiclassical trajectory calculations of the QR process in thermal energy collisions and successfully interpreted a large body of experimental data based on the QR phenomenon and simple parametric models. © 2009 by Taylor and Francis Group, LLC
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Inelastic Collisions and Chemical Reactions of Molecules 10−10 Dv = 0, Dj = −2 10−12
Rate coefficient (cm3/sec)
Dv = +1, Dj = −2
10−14 u=0 u=1
10−16
u=2 u=3 u=4
10−18
10−20
Dv = −1, Dj = +2
0
4
8
12
16
20
24
28
j
FIGURE 3.7 Zero-temperature rate coefficients for 4 He + H2 (v, j) collisions as functions of the initial vibrational and rotational quantum numbers. (From Forrey, R.C. et al., Phys. Rev. A, 64, 022706, 2001. With permission.)
3.2.3 ATOM–MOLECULAR ION COLLISIONS Dynamics of ionic systems are different from collisions of neutral species. The shortrange part of the interaction potential for ionic systems is usually more anisotropic and the long-range part has an attractive component that is determined by the polarizability of the atom and it vanishes as 1/R4 where R is the atom–molecule center-of-mass distance [64]. Due to the strong polarizability term, the interaction potential for ionic systems extends to longer range compared to neutral atom–molecule systems. Therefore, it is important to understand the effect of both the short- and long-range part of the interaction potential on the scattering dynamics. For this purpose, several studies have focused on ultracold collisions between molecular ions and neutral atoms as well as neutral molecules and atomic ions. Bodo and colleagues [64] investigated rotational quenching in Ne+ 2 + Ne and + He2 + He collisions at ultralow energies. They found that the Wigner regime begins at a collision energy of 10−4 cm−1 for the He system and 10−6 cm−1 for the Ne system. In general, the s-wave Wigner regime was found to occur at lower energies for ionic systems compared to neutral species. For example, in He + H2 [19] and He + O2 [25] collisions the Wigner regime begins at collision energies of about 10−2 cm−1 . The differences are attributed to the long range of the ion–neutral interaction potential, © 2009 by Taylor and Francis Group, LLC
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which enhances contributions from higher partial waves. The differences between the He and Ne systems can be attributed to the mass difference and to the strength of the long-range interaction potentials. For the heavier Ne system the long-range part is more attractive, which increases the contribution of higher-order partial waves at ultralow energies. The magnitude of the zero-energy rate coefficients for rotational quenching in these molecular ions is on the order of 10−9 cm3 /sec, which is considerably larger than for collisions involving neutral species. Stoecklin and colleagues [65] have performed a comparative study of collisions of 3 4 N+ 2 (v = 1, j = 0) molecular ions with neutral He or He atoms, and the scattering of neutral N2 (v = 1, j = 0) molecules in collisions with 3 He or 4 He atoms. The vibrational quenching cross-sections for these systems are presented in Figure 3.8. While the behavior of the quenching cross-sections of neutral N2 molecules with 3 He and 4 He atoms was found to be similar, they observed a striking difference 3 4 between the quenching cross-sections of N+ 2 in collisions with He and He atoms. In particular, the resonance positions in the quenching cross-sections were significantly 3 shifted and the number of resonances was different. The N+ 2 + He system has a −2 −1 shape resonance at a collision energy of 10 cm , which is absent for the N+ 2 + 4 He system. Furthermore, the zero-energy quenching rate coefficient is an order of 3 4 magnitude larger for collisions of N+ 2 (v = 1, j = 0) with He than with He, whereas 3 it is comparable for collisions of N2 (v = 1, j = 0) with both He and 4 He. The 3 differences may be due to a virtual state in the N+ 2 (v = 1, j = 0) + He collision system. More recently, Guillon and colleagues [66] studied the effect of spin-rotation interaction on vibrational and rotational quenching for the He–N+ 2 system. They found that the vibrational quenching is not modified by the spin–rotation coupling, while rotational transitions are sensitive to the fine-structure interactions. Table 3.2 provides a compilation of zero-temperature quenching rate coefficients for vibrational and rotational relaxation in several ion atom–molecule systems.
3.3
CHEMICAL REACTIONS AT ULTRACOLD TEMPERATURES
The last three decades have seen impressive progress in the experimental and theoretical descriptions of chemical reactions between atoms and small molecular systems. While fully state-resolved experiments have been performed for a large number of collision systems, accurate quantum dynamics calculations have been restricted to systems involving light atoms such as H + H2 , F + H2 , Cl + H2 , C + H2 , N + H2 , O + H2 and some of their isotopic analogs [67,68]. Most experimental studies of small molecular systems focused on reactions at thermal or elevated collision energies, although some recent measurements have been extended to temperatures as low as 10 K for some astrophysically relevant systems [69]. Due to the possibility of achieving coherent chemistry there has been substantial interest in understanding the behavior of chemical reactions at cold and ultracold temperatures. At these temperatures perturbations introduced by external electric and magnetic fields are significant compared to the collision energies involved and external fields can be employed to control and manipulate the reaction outcome. Over the last seven years a number of studies of ultracold atom–diatom chemical reactions have been reported both for © 2009 by Taylor and Francis Group, LLC
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(a)
103 Isotope 3 Isotope 4
102
s (10–16 cm2 )
101
100
10–1
10–2
10–3
10–4 –8 10
10–6
10–4
10–2 E
(b)
100
102
(cm–1)
10–2
Isotope 3 Isotope 4
s (10–16 cm2)
10–3
10–4
10–5
10–6
10–7
10–6
10–4
10–2
100
102
–1
E (cm ) + 3 FIGURE 3.8 Vibrational quenching cross-sections for N+ 2 (v = 1, j = 0) + He and N2 (v = 1, j = 0) + 4 He (a) and N2 (v = 1, j = 0) + 3 He and N2 (v = 1, j = 0) + 4 He (b) systems. (From Stoecklin, T. and Voronin, A., Phys. Rev. A, 72, 042714, 2005. With permission.)
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TABLE 3.2 Zero-Temperature Inelastic Rate Coefficients for Different Ion Atom–Molecule Systems Initial (v, j )
kT =0 (cm3 /sec)
Reference
Ne+ 2 + Ne
(v = 0, j = 2) (v = 0, j = 4) (v = 0, j = 6)
3.3 × 10−10 7.3 × 10−10 7.0 × 10−10
[64] [64] [64]
He+ 2 + He
(v = 0, j = 2) (v = 0, j = 4) (v = 0, j = 6)
6.7 × 10−10 8.4 × 10−10 1.2 × 10−9
(64] [64] [64]
3 N+ 2 + He
(v = 1, j = 0)
4 × 10−14
[65]
4 N+ 2 + He
(v = 1, j = 0)
3 × 10−15
[65]
System
reactions with and without an energy barrier. Chemical reactions at low temperatures often behave quite differently from reactions at elevated temperatures. In particular, the weak van der Waals interaction potential, which does not play any significant role at high temperatures, may have a dramatic effect on the outcome of reactions at low temperatures.
3.3.1 TUNNELING-DOMINATED REACTIONS In the course of the last 10 to 15 years there has been much interest in understanding the role of resonances in chemical reactions that involve an energy barrier for which the reactivity is primarily driven by tunneling at low temperatures. Recent studies on F + H2 /HD/D2 [11,70–80], Cl + HD [81,82], H + HCl/DCl [83], Li + HF/LiF + H [84,85], O + H2 [86,87], and F + HCl/DCl [88] reactions have demonstrated that decay of quasibound states of the van der Waals interaction potential in the entrance and exit channels may give rise to narrow Feshbach resonances in the cross-sections. The reactions of F with H2 and HD have been the subject of numerous quantum scattering calculations over the last two decades. The two reactions have emerged as benchmark systems for experimental and theoretical studies of resonances in chemical reactions. A detailed analysis of the low-energy resonance features in the F + H2 reaction was presented by Castillo and colleagues in 1996 [70]. They demonstrated that several resonances that appear in the energy dependence of the cumulative reaction probability for the F + H2 reaction arise due to the van der Waals interaction potential in the product HF + H channel. In particular, the origin of the resonances has been attributed to the van der Waals potential associated with the HF(v = 3, j = 0 − 3) channels. A large number of experimental and theoretical papers has appeared that examined various aspects of these resonance features [11,70–72,75–80]. Among these studies, the work of Takayanagi and Kurosaki [71] deserves special attention. They showed that for F + H2 , F + HD, and F + D2 reactions, reactive scattering © 2009 by Taylor and Francis Group, LLC
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resonances occur due to the decay of quasibound states of the van der Waals potential in the entrance channel of the reaction. These Feshbach resonances are associated with the decay of quasibound states of adiabatic potentials corresponding to F· · · H2 (v = 0, j = 0, 1), F· · · HD(v = 0, j = 0 − 2), and F· · · D2 (v = 0, j = 0 − 2) complexes, obtained by diagonalizing the J = 0 Hamiltonian in a basis set of the asymptotic rovibrational states of the reactant molecules. 3.3.1.1
Reactions at Zero Temperature
Balakrishnan and Dalgarno [11] showed that quasibound states of the F + H2 van der Waals complex have a dramatic effect on the reactivity in the Wigner threshold regime. They found that the F + H2 (v = 0, j = 0) reaction has a rate coefficient of 1.25 × 10−12 cm3 /sec in the zero-temperature limit. Their calculation was based on the widely used PES of Stark and Werner [89] for the F + H2 system. The relatively large value of the zero-temperature rate coefficient is due to the presence of a small narrow energy barrier for the reaction so that tunneling of the H atom is efficient. The vibrational level resolved cross-sections for the F + H2 (v = 0, j = 0) reaction are shown in Figure 3.9 for the total angular momentum quantum number J = 0.
102
Cross-section (10−16 cm2)
100
u¢ = 2
10−2
u¢ = 1
u¢ = 0
10−4
10−6 10−7
10−6
10−5
10−4 10−3 Kinetic energy (eV)
10−2
10−1
FIGURE 3.9 J = 0 cross-sections for F + H2 (v = 0, j = 0) → HF(v ) + H reaction for v = 0 − 2 as functions of the incident kinetic energy. (From Balakrishnan, N. and Dalgarno,A., Chem. Phys. Lett., 341, 652, 2001. With permission.)
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The v = 2 level is the dominant channel at low energies in agreement with the behavior at higher energies. Subsequent calculations showed that at low energies the F + HD reaction is dominated by the formation of the HF product with an HF/DF branching ratio of about 5.5 [74]. The formation of the DF product is suppressed because tunneling of the heavier D atom is less efficient. In earlier quantum calculations, Baer and colleagues [90, 91] reported HF/DF branching ratios ranging from 1.5 at 450 K to 6.0 at 100 K. Their low-temperature value is in close agreement with the zero-temperature limit obtained by Balakrishnan and Dalgarno [74]. Figure 3.10 compares the J = 0 cross-sections for the F + H2 (v = 0, j = 0) and F + HD(v = 0, j = 0) reactions over a kinetic energy range of 10−7 to 1.0 eV. In the Wigner threshold regime, the reactivity of the F + H2 system is an order of magnitude greater than that of the F + HD reaction. A rigorous analysis of the scattering resonances in the F + HD reaction was recently presented by De Fazio and colleagues [80], who provided a detailed characterization of the resonances supported by the entrance and exit channels of the van der Waals potential and discussed the effect of higher total angular momenta on the position and lifetime of the resonances. Stereodynamical aspects of the F + H2 collision and the effect of polarization of the H2 molecule on the outcome of the reaction at low energies were recently 102
Cross-section (10−16 cm2)
101
100
10−1
10−2
10−3
10−4 10−7
10−6
10−5
10−4 10−3 Kinetic energy (eV)
10−2
10−1
FIGURE 3.10 Comparison of J = 0 cross-sections for F + HD(v = 0, j = 0) → HF + D (solid curve), F + HD(v = 0, j = 0) → DF + H (short-dashed curve) and F + H2 (v = 0, j = 0) → HF + H (long-dashed curve) reactions as functions of the incident kinetic energy. (From Balakrishnan, N. and Dalgarno, A., J. Chem. Phys. A, 107, 7101, 2003. With permission.)
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explored by Aldegunde and colleagues [77]. They argued that a reactant polarization scheme can be exploited to control state-to-state dynamics of the reaction. To explore the role of tunneling in chemical reactions at cold and ultracold temperatures Bodo, Gianturco, and Dalgarno [73] investigated the dynamics of the F + D2 system at low and ultralow energies. They found that compared to the F + H2 reaction, the reactivity of F + D2 is significantly suppressed in the Wigner regime with an HF/DF ratio of about 100. This is illustrated in Figure 3.11 where the J = 0 cumulative
Cumulative reaction probabilities
(a) 0.02
0.015
H2
0.01
D2 (×10)
0.005
0
0
0.0002
0.0004 0.0006 Collision energy (eV)
0.0008
0.001
10–3
10–2
(b) 103
Reaction cross-section (Å2)
102 101 100 10–1 10–2 10–3 10–4 –8 10
10–7
10–6 10–5 10–4 Collision energy (eV)
FIGURE 3.11 Comparison of the J = 0 cumulative reaction probabilities (a) and crosssections (b) of F + H2 and F + D2 reactions as functions of the incident kinetic energy. (From Bodo, E. et al., J. Phys. B: At. Mol. Opt. Phys., 37, 3641, 2004. With permission.)
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reaction probability and cross-sections for the F + H2 and F + D2 reactions are plotted as functions of the incident kinetic energy from the Wigner limit to 0.01 eV. The dramatic suppression of the F + D2 reaction in the Wigner regime cannot be explained based on tunneling alone. A closer examination of the reaction probabilities for the F + H2 and F + D2 reaction revealed that an unusual enhancement of the reactivity occurs for the F + H2 reaction at about 3 × 10−5 eV [75] (see Figure 3.11a). This feature was attributed to the presence of a virtual state. The enhancement of the limiting value of the rate coefficient for the F + H2 reaction was also ascribed to the virtual state. Evidence for the virtual state is the presence of a Ramsauer–Townsend minimum in the elastic cross-section at an energy of about 3 × 10−5 eV and a negative value of the real part of the scattering length for the F + H2 reaction [75]. To explore how isotope substitution modifies reactivity at low temperatures Bodo and colleagues [75] artificially varied the mass of the hydrogen atoms in the calculation for the F + H2 reaction from 0.5 to 1.5 amu. As illustrated in Figure 3.12, for an H atom mass of 1.12 amu the virtual state induces a zero energy resonance at which the real part of the scattering length diverged to infinity. For the same value of the H atom mass, the zero-temperature rate coefficient of the reaction attains a value of 1.0 × 10−9 cm3 /sec, which is about three orders of magnitude larger than that of the F + H2 reaction in the Wigner limit. The variation of the scattering length of the F + H2 system as a function of the mass of the pseudo-hydrogen atom is similar to the variation of scattering length as a function of the magnetic field in the vicinity of a Feshbach resonance. Another example of a tunneling-dominated reaction is the Li + HF → LiF + H reaction. At cold and ultracold temperatures the reaction occurs by tunneling of the relatively heavy fluorine atom. The LiH + F channel is energetically not accessible at low energies and tunneling of the H atom is not involved in this reaction at low temperatures. Due to strong electric dipole forces exerted by the HF molecule, the van der Waals interaction potential of the LiHF system is deeper compared to the F + H2 system. The Li· · · HF van der Waals potential well is about 0.24 eV (1936.0 cm−1 ) and the H· · · LiF potential well is about 0.07 eV (565.0 cm−1 ). Quantum scattering calculations for Li + HF and LiF + H reactions by Weck and Balakrishnan [84,85] with the PES of Aguado and colleagues [92] have shown that for incident energies below 10−3 eV the reaction cross-section exhibits a large number of resonances. The energy dependence of the J = 0 cross-section for Li + HF(v = 0, j = 0) collisions is shown in Figure 3.13. Detailed bound-state calculations of the LiHF van der Waals complexes revealed that for the Li + HF(v = 0, j = 0) reaction the resonances correspond to the decay of Li· · · HF(v = 0, j = 1 − 4) van der Waals complexes. Calculations with vibrationally excited HF molecules showed that the reaction becomes about 600 times more efficient in the Wigner regime when HF is excited to the v = 1 vibrational level. As seen in Figure 3.13, a unique feature of the Li + HF(v = 0, j = 0) reaction is the presence of a strong peak at 5.0 × 10−4 eV at which the reaction cross-section is about six orders of magnitude larger than the background cross-section. The results for both Li + HF [84] and LiF + H [85] reactions with thermal and nonthermal vibrational excitation suggest that heavy-atom tunneling may play an important role in chemical reactions at cold and ultracold temperatures. An experimental study [12] of an organic ring expansion reaction at 8 K has © 2009 by Taylor and Francis Group, LLC
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Inelastic Collisions and Chemical Reactions of Molecules (a)
150 100
a (Å)
50 0 –50 –100 –150 (b) 10–8
R(T = 0) (cm3/sec)
10–9 10–10 10–11 10–12 10–13 10–14
|Re(Ep)| (meV)
(c)
100 10–1 10–2 10–3 10–4 10–5
0.6
0.8
1
1.2 1.4 1.6 Mass of both H atoms
1.8
2
FIGURE 3.12 Real part of the scattering length (a), zero-temperature limiting values of the rate coefficient (b), and positions of quasibound states (c) of the F· · · H2 complex as functions of the mass of a pseudo hydrogen atom. (From Bodo, E. et al., J. Phys. B: At. Mol. Opt. Phys., 37, 3641, 2004. With permission.)
shown that the reaction occurs almost exclusively by carbon tunneling. The tunneling contribution was found to be orders of magnitude greater than over the barrier contribution. In a recent work, Tscherbul and Krems [93] have explored the Li + HF and LiF + H reactions in the presence of an external electric field. They have shown that, for temperatures below 1 K, the reaction probability can be significantly influenced by electric fields. © 2009 by Taylor and Francis Group, LLC
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Cross-section (10−16 cm2)
Cross-section (10−16 cm2)
1 0.1
u=1
0.01 0.001 0.0001 1e–05 1e–06 1 u=0
0.1 0.01 0.001 0.0001 1e–05 1e–06 1e–07 1e–07
1e–06
1e–05
0.0001 0.001 Kinetic energy (eV)
0.01
0.1
FIGURE 3.13 Cross-sections for LiF formation (solid curve) and nonreactive scattering (dashed curve) in Li + HF(v, j = 0) collisions as functions of the incident kinetic energy: results for v = 0 (a); results for v = 1 (b). (From Weck, P.F. and Balakrishnan, N., J. Chem. Phys., 122, 154309, 2005. With permission.)
3.3.1.2
Feshbach Resonances in Reactive Scattering
Van der Waals complexes formed during collisions can either undergo vibrational predissociation or vibrational prereaction leading to sharp features in the energy dependence of the cross-sections. The term “prereaction” refers to the process in which a rotationally or vibrationally excited van der Waals complex decays through chemical reaction rather than by rotational or vibrational predissociation. For reactions with energy barriers the chemical reaction pathway may involve tunneling. The reactions of Cl with H2 and HD are dominated by tunneling at low temperatures [81–83]. Compared to the F + H2 and F + HD reactions, the energy barrier for the Cl + H2 reaction is much larger and the reactivity at low energies is significantly suppressed. Nearly a decade ago, Skouteris and colleagues [81] showed that the van der Waals interaction potential between Cl and HD determines the reaction outcome, despite the fact that the depth of the van der Waals interaction potential for the Cl· · · HD system is less than one-tenth of the height of the reaction barrier. Quantum scattering calculations of the Cl + HD reaction on PESs without the van der Waals interaction potential predict nearly equal probabilities for HCl and DCl products [81]. However, if the potential surface includes the van der Waals interaction a strong preference for the DCl product occurs at thermal energies, in agreement with experimental results. The effect of these weakly bound states on the reactivity at cold and ultracold temperatures has recently been explored by Balakrishnan [82]. © 2009 by Taylor and Francis Group, LLC
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The cross-sections for HCl and DCl formation and nonreactive rovibrational transitions in Cl + HD(v = 1, j = 0) collisions for total angular momentum quantum number J = 0 are shown in Figure 3.14 as functions of the total energy [82]. The sharp features in the cross-sections correspond to Feshbach resonances arising from the decay of quasibound van der Waals complexes formed in the entrance channel of the reaction. The quasibound states can be identified by examining bound states of adiabatic potentials correlating with the v = 1, j = 0 and v = 1, j = 1 levels of the HD molecule. They are displayed in Figure 3.15 as functions of the atom–molecule separation. The adiabatic potential curves are computed by diagonalizing the diabatic potential energy matrix obtained in a basis set of rovibrational levels of the HD molecule at each value of the atom–molecule separation. The Feshbach resonances labeled B, C, D, and E in Figure 3.14 result from the decay of the corresponding quasibound states shown in Figure 3.15. The cross-sections in Figure 3.14 do not show a peak corresponding to the metastable state A. This is because the state A is too deeply bound and it is not accessible through scattering in the v = 1, j = 0 channel. Figure 3.14 also shows that the quasibound states preferentially undergo prereaction than predissociation. This
102
101
Cross-section (10−16 cm2)
100
10–1
Cl + HD – nonreactive B
C
D
10–2
E
HCl + D – reactive 10–3
10–4
DCl + H – reactive 10–5
10–6 0.685
0.69
0.695
0.7
0.705
Total energy (eV)
FIGURE 3.14 Reactive and nonreactive scattering cross-sections for Cl + HD(v = 1, j = 0) reaction as functions of the incident kinetic energy. (From Balakrishnan, N. J. Chem. Phys., 121, 5563, 2004. With permission.)
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0.7 E
u = 1, j = 1
D C 0.69
B
Energy (eV)
u = 1, j = 0
A 0.68
0.67
5
10
15
20
R (au)
FIGURE 3.15 Adiabatic potential energy curves of the Cl + HD system correlating with the HD(v = 1, j = 0) and HD(v = 1, j = 1) levels as functions of the atom–molecule separation. Quasibound levels responsible for the resonances in Figure 3.14 are labeled by B, C, D, and E. (From Balakrishnan, N., J. Chem. Phys., 121, 5563, 2004. With permission.)
is due to the enhanced coupling of the resonance states with the reactive channels compared to those with the nonreactive channels. The wavefunctions of quasibound states B and E are shown in Figure 3.16 as functions of the atom–molecule separation. Although the wavefunction of the weakly bound state E extends far beyond the transition state region of the reaction, it preferentially undergoes prereaction rather than predissociation. Thus, regions of the interaction potential far away from the transition state region may have a significant effect on reactivity, especially when part of the wavefunction in that region is sampled by a resonance state that is coupled to the reactive channel. Figure 3.14 also demonstrates that because the reaction is dominated by tunneling at low temperatures, the formation of the DCl + H product, which involves the tunneling of the D atom, is severely suppressed in the threshold regime. This is more clearly illustrated in Figure 4 of Ref. [82] where the reaction cross-section is plotted as a function of the incident kinetic energy. Table 3.3 provides a compilation of zero-temperature quenching rate coefficients for a number of atom–diatom chemical reactions, which are dominated by tunneling at low energies. © 2009 by Taylor and Francis Group, LLC
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0.02
B E Adiabatic potential
0
0
–0.01
–0.01
–0.02
0.01 Potential energy (eV)
Wavefunction (arb. units)
0.01
0
5
10
15
20
25
30
–0.02 35
R (au)
FIGURE 3.16 Adiabatic potential and the wavefunctions of the quasibound levels B and E shown in Figure 3.15 as functions of the atom–molecule separation. Amplitudes of the wavefunctions have been reduced by a factor of 10 for the convenience of plotting. (From Balakrishnan, N., J. Chem. Phys., 121, 5563, 2004. With permission.)
3.3.2 3.3.2.1
BARRIERLESS REACTIONS Collision Systems of Three Alkali Metal Atoms
As discussed by Hutson and Soldán in two recent reviews [6,14], progress on the production of ultracold molecules and the creation of molecular Bose–Einstein condensates of alkali-metal systems have motivated theoretical studies on ultracold atom–dimer alkali-metal collisions. Here, we give an overview of recent calculations for spin-polarized triatomic alkali-metal systems, Li + Li2 [94–97], Na + Na2 [98,99], and K + K2 [100]. These results have all been obtained using the reactive scattering code written by Launay and Le Dourneuf [101], based on a time-independent quantum formalism. Refs. [94,95,97] and [100] describe the details of the PESs and dynamics calculations. The quantum dynamics studies of Refs. [98] and [99] have been performed using the Na3 PES reported by Higgins and colleagues [102], while the results of Ref. [96] have been obtained using the Li3 PES calculated by Colavecchia and colleagues [103]. Another PES for Li3 has been constructed by Brue and colleagues [104]. © 2009 by Taylor and Francis Group, LLC
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TABLE 3.3 Zero-Temperature Quenching Rate Coefficients for Tunneling Dominated Reactions (the Final Arrangement has also been Specified) Initial (v, j )
kT =0 (cm3 /sec)
Reference
F + D2
(v = 0, j = 0) (v = 0, j = 0) (v = 0, j = 0) (v = 0, j = 0)
1.3 × 10−12 (H + HF) 2.8 × 10−14 (D + HF) 0.5 × 10−14 (H + DF) 2.1 × 10−14 (D + DF)
[11] [74] [74] [73]
F + HCl F + HCl
(v = 0, j = 0) (v = 1, j = 0)
F + HCl
(v = 2, j = 0)
F + DCl
(v = 1, j = 0)
1.2 × 10−17 4.0 × 10−15 5.3 × 10−15 5.2 × 10−13 3.3 × 10−13 4.4 × 10−21
(Cl + HF) (Cl + HF) (F + HCl) (Cl + HF) (F + HCl) (Cl + DF)
[88] [88] [88] [88] [88] [88]
H + HCl
(v = 0, j = 0) (v = 1, j = 0) (v = 2, j = 0) (v = 1, j = 0) (v = 2, j = 0)
2.4 × 10−19 (Cl + H2 ) 7.2 × 10−14 (Cl + H2 ) 1.9 × 10−11 (Cl + H2 ) 7.8 × 10−15 (Cl + HD) 1.7 × 10−12 (Cl + HD)
[83] [83] [83] [83] [83]
Cl + HD
(v = 1, j = 0)
1.7 × 10−13 (D + HCl) 7.1 × 10−16 (H + DCl) 7.8 × 10−14 (Cl + HD)
[82] [82] [82]
Li + HF
(v = 0, j = 0) (v = 1, j = 0) (v = 1, j = 0)
4.5 × 10−20 2.8 × 10−17 3.8 × 10−15 1.6 × 10−14 1.7 × 10−14 2.7 × 10−13
[84] [84] [85] [85] [85] [85]
System F + H2 F + HD
H + DCl
H + LiF
(v = 2, j = 0)
(H + LiF) (H + LiF) (Li + HF) (H + LiF) (Li + HF) (H + LiF)
Quantum scattering calculations show that ultracold reactions of alkali metal atoms with alkali metal dimers are much more efficient than tunneling-driven processes. This can be explained based on the following considerations. First, the indistinguishability of the atoms may play a role. For a homonuclear triatomic system, the three possible arrangement channels are the same. Also, the presence of identical two-body potentials in all three arrangement channels may enhance the depth of the three-body interaction potential. Consider the two-body term (additive term) of a homonuclear triatomic system at equilateral geometries. Because the distances between the three identical atoms are the same, the same diatomic potential term is added three times to yield the two-body terms of the triatomic system. If the diatomic potential is deep or repulsive, the two-body term will be three times deeper or repulsive at equilateral geometries. In contrast, for triatomic systems with distinguishable atoms, the diatomic © 2009 by Taylor and Francis Group, LLC
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pairs are not the same. For example, for the Li + HF system, the three diatomic fragments are distinctly different. They have completely different electronic structures and properties such as minima, equilibrium distances, turning points, and the nature and range of the interaction. Thus, at an equilateral configuration for which the three diatomic distances are the same but the diatomic potential energies are different (one may be attractive while the other two may be repulsive), the overall two-body term can be weaker than the individual two-body interaction potential. This could lead to a triatomic interaction region that is less strong than that of a homonuclear triatomic system. Second, the topology of the PES plays a significant role. For all alkali-metal trimer systems, the minimum energy configurations arise at equilateral and collinear geometries as shown by Soldán and colleagues [105]. Furthermore, the surface is barrierless so that all atom–dimer alkali-metal collisional approaches are energetically possible. In contrast, most of the nonalkali systems discussed above are characterized by a collinear (or bend) atom–diatom approach, and dominated by a repulsive barrier in the triatomic transition-state region. The particular topology of the PES arises from the three-body interaction potential. Also, the couplings between different electronic surfaces of the triatomic system give rise to conical intersections, which create a repulsive barrier in the transition-state region. If the energy barrier is high and its width large, tunneling will be very inefficient, leading to very small values for the reaction rate coefficients in the Wigner regime. Although reactivity can be enhanced when resonances are present, the background scattering in reactive cross-sections is generally quite small. In contrast, for almost all alkali-metal trimer systems, the conical intersections arise at small interatomic distances where the PES is sufficiently repulsive that it plays no significant role at ultralow energy and the reactivity is not influenced by such repulsive barriers. Third, the density of states of the system plays an important role. If the density of states is small as for light systems, the typical energy spacing is rather large. For heavy systems such as alkali-metal systems, the density of states is very large and the narrow energy spacing can lead to very strong couplings, especially in the vicinity of avoided crossings. The strong couplings will lead to efficient energy transfer between different quantum states. This may also explain the relatively weak dependence of the vibrational quenching rate coefficients on the initial vibrational state of the molecule in atom–dimer alkali-metal collisions. Figure 3.17 provides a comparison between elastic and quenching rate coefficients for 39 K + 39 K2 (v = 1, j = 0) [100] collisions at energies ranging from 10−9 K to 10−2 K. The results include contributions from the total angular momentum quantum numbers J = 0 to 5. For low vibrational states of the K2 dimer, quenching processes are more efficient than elastic scattering at ultracold energies. For example, at a temperature of 10−9 K, the quenching rate coefficient is about 10−10 cm3 /sec compared to 10−13 cm3 /sec for elastic scattering. The quenching processes lead to collisional relaxation of the molecules to lower rovibrational states resulting in trap loss. Similar results have been obtained for Na + Na2 [98,99] and Li + Li2 [94,96,97] collisions. The zero-energy quenching rate coefficients for these systems are found to be on the order of 10−11 to 10−10 cm3 /sec. The quenching processes are more efficient than the elastic collisions at ultracold temperatures. Thus, quenching of © 2009 by Taylor and Francis Group, LLC
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Elastic rate coefficient (cm3/sec)
Total 10–10
10–11
J = (0–5) 10–12
10–13 –9 10
10–8
10–7
10–6 10–5 Collision energy (K)
10–4
10–3
10–2
10–4
10–3
10–2
Quenching rate coefficient (cm3/sec)
(b) Total Langevin
10–10
J = (0–5)
10–11 –9 10
10–8
10–7
10–6 10–5 Collision energy (K)
FIGURE 3.17 Elastic (a) and quenching (b) rate coefficients vs. the collision energy for 39 K + 39 K (v = 1, j = 0) scattering. The rate coefficient from a capture model is also shown 2 for the quenching processes. (From Quéméner, G. et al., Phys. Rev. A, 71, 032722, 2005. With permission.)
vibrationally excited alkali-metal dimers in collisions with alkali-metal atoms occurs with significant rates at ultracold temperatures. The typical magnitude of these rate coefficients from the scattering calculations is in reasonable agreement with experimental results. In a recent experiment, Staanum and colleagues [106] reported a value of 9.8 × 10−11 cm3 /sec for the relaxation of low-lying vibrational levels of Cs2 © 2009 by Taylor and Francis Group, LLC
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in collisions with Cs atoms at a temperature of 60 × 10−6 K. In a separate experiment, Zahzam and colleagues [107] determined a quenching rate coefficient of 2.6 × 10−11 cm3 /sec for the same system at a temperature of 40 × 10−6 K. Wynar and colleagues [108] estimated inelastic rate coefficients of 8 × 10−11 cm3 /sec for Rb + Rb2 collisions. Mukaiyama and colleagues [109] reported an inelastic rate coefficient of 5.5 × 10−11 cm3 /sec for collisions of Na atoms with Na2 molecules created by the Feshbach resonance method, while Syassen and colleagues [110] obtained a value of 2 × 10−10 cm3 /sec for collisions of Rb atoms with Rb2 molecules. At large atom–diatom separations R, the interaction potential can be approximated by an effective potential, composed of a repulsive centrifugal term and the long-range interaction potential. For alkali-metal trimer systems, the atom–diatom long-range potential is a van der Waals interaction potential and behaves as −C6 /R6 . The classical capture model (also known as the Langevin model) has been shown to work quite well for these systems [100] at certain energy regimes. The Langevin model is based on the assumption that if the total energy of the system can overcome the effective barrier, then the elastic probability is zero and the quenching probability is one. Otherwise, the elastic probability is one and the quenching probability is zero. Classically, the barrier prevents the atom and the molecule from accessing the region of strong coupling with the other channels. Results obtained using the Langevin model for the K + K2 system are shown in Figure 3.17 along with the quantum results. As Figure 3.17 illustrates, three different regions can be distinguished for the K + K2 system. The first region, for collision energies below 10−6 K, corresponds to the Wigner regime where the threshold laws apply. In this regime, the quenching rate coefficient tends to a constant while the elastic component approaches zero as the square root of the collision energy. A quantum description is required and classical models cannot describe the dynamics in this regime. The second region, above a collision energy of 10−3 K, corresponds to the Langevin regime. Here, the difference between the full quantum calculation and the classical model is within 10%. Similar results have been found for 7 Li + 7 Li2 (v = 1, 2, j = 0) and 6 Li + 6 Li2 (v = 1, 2, j = 1) collisions by Cvitaš and colleagues [94], who showed that in the Langevin regime the rate coefficients become independent of the vibrational level of the molecule. The third region is intermediate between the Wigner and Langevin regimes, which typically corresponds to a quantum calculation with two or three partial waves. As shown in Figure 3.17b the height of each J-resolved effective barrier depicted by vertical lines for J = 1 to 5 corresponds approximately to the maximum of each J-resolved quenching rate coefficient. This indicates that the quenching process becomes significant when the barrier height is energetically overcome. Whenever the elastic and quenching rate coefficients have comparable values, the Langevin regime is reached, as shown in Figure 3.17. Since the quenching probability is almost equal to one and the elastic probability is almost equal to zero, elastic and quenching transition matrix elements are both equal to one and the cross-sections and rate coefficients for the two processes become similar. The above analysis shows that at energies above the onset of the Wigner regime the Langevin model can correctly describe the dynamics for barrierless atom–dimer alkali-metal collisions because of the strong inelastic couplings. However, the © 2009 by Taylor and Francis Group, LLC
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classical model is not suitable for tunneling-dominated reactions. Accurate quantum dynamics calculations are computationally demanding for heavier systems such as Cs + Cs2 , Rb + Rb2 , Rb + RbCs, or Cs + RbCs and the classical model may be used to qualitatively describe the dynamics of these systems at energies above the s-wave regime. The temperature dependence of the rate coefficients predicted by the Langevin model is given by the simple formula in atomic units: kLang (T ) = π
8kB T πμ
1/2
2C6 kB T
1/3 Γ
2 3
kLang (10–10 cm3/sec)
where μ is the reduced mass of the atom–diatom collisional system and C6 the dominant atom–diatom long-range coefficient. In Figure 3.18 we show the rate coefficients predicted by the Langevin model as functions of the temperature for different atom–diatom and diatom–diatom collisions involving Rb or Cs atoms. For Rb–RbCs, Cs–RbCs, and RbCs–RbCs collisions we used the corresponding C6 coefficients calculated by Hudson and colleagues [111]. In the absence of similar data for Cs + Cs2 and Rb + Rb2 collisions, we approximated the atom–dimer C6 coefficient by multiplying the corresponding atom–atom value by a factor of two. For Cs + Cs2 collisions we used the C6 coefficient for Cs–Cs interaction calculated by Amiot and colleagues [112] and for Rb + Rb2 collisions we adopted the C6 coefficient for Rb–Rb interaction reported by Derevianko and colleagues [113]. As shown in Figure 3.18 the Langevin model predicts rate coefficients within an order of
1 Rb + Rb2 Rb + Rb2, Wynar et al. (exp, 2000) Cs + Cs2 Cs + Cs2, Staanum et al. (exp, 2006) Cs + Cs2, Zahzam et al. (exp, 2006) Rb + RbCs Rb + RbCs, Hudson et al. (exp, 2008) Cs + RbCs Cs + RbCs, Hudson et al. (exp, 2008) RbCs + RbCs
0.1 0
100
200
300
400
500
600
T (mK)
FIGURE 3.18 Rate coefficients as functions of the temperature predicted by the Langevin model compared with experimental data for different collisional processes involving Rb or Cs atoms. The curves correspond to the Langevin results and the symbols denote the experimental results.
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magnitude of the experimental values for the different systems. The predicted results for the Cs + Cs2 , Rb + RbCs, and Cs + RbCs collisions agree with the corresponding experimental data of Refs. [106,107], and [111] within the reported error bars. For Rb + Rb2 collisions the Langevin model predicts results in very close agreement with the experimental result of Wynar and colleagues [108]. Thus, the Langevin model appears to be valid for describing collisional properties of these systems at μK temperatures. The model also confirms that cold and ultracold collisions of alkali-metal atoms and dimers are essentially characterized by the leading term in the long-range part of the interaction potential. 3.3.2.2
Role of PES in Determining Ultracold Reactions
Potential energy surfaces are the key ingredients that enter in quantum dynamics calculations. While the two-body terms are generally well known and accurate, the three-body terms are more difficult to compute with high precision because they are nonadditive and involve correlations between the three atoms. As a consequence, quantum dynamics calculations may suffer from the quality and degree of accuracy of the three-body terms. The sensitivity of the ultracold collision cross-sections to the details of the PES has been investigated for atom–dimer alkali-metal systems by Quéméner and colleagues [99] for Na + Na2 (v = 1 − 3, j = 0) and by Cvitaš and colleagues [97] for Li + Li2 (v = 0 − 3, j = 0). In these studies, a linear scaling factor λ has been included to tune the three-body interaction term in the PES and cross-sections were calculated as a function of λ. Other studies have compared the dynamics with and without the three-body terms for Na + Na2 (v = 1, j = 0) [98] and for Li + Li2 (v = 0 − 10, j = 0) [96] collisions. The dependence of the total quenching cross-sections on the three-body term for Na + Na2 (v = 1 − 3, j = 0) scattering [99] at a collision energy of 10−9 K is shown in Figure 3.19. The contribution of each final vibrational level is also plotted. For the v = 1 vibrational level, the cross-sections are very sensitive to the details of the three-body potential. A change of 1% in λ leads to a significant change of 75% in the cross-sections. At the minimum of the Na3 potential, a change of 1% corresponds approximately to 10 K. At present, ab initio calculations of PESs cannot be done with such accuracy. Thus it appears that it is difficult to get an accuracy better than two orders of magnitude in the cross-sections for Na + Na2 (v = 1, j = 0) collisions. However, for v = 2 and 3, the total cross-sections show a weaker dependence on the three-body term. The state-resolved cross-sections do not show strong dependence on the three-body term, compared to collisions of molecules in v = 1. Figure 3.20 shows the dependence of the state-to-state cross-sections on the three body term for the reaction Na + Na2 (v = 3, j = 0) → Na + Na2 (vf = 2, 1, 0, jf ) at a collision energy of 10−9 K. The oscillations in the cross-sections, when λ is modified, are due to Feshbach resonances which arise when a triatomic quasibound state (or a virtual state) crosses the energy threshold as the strength of the interaction potential is decreased (increased). When such a Feshbach resonance occurs, Cvitaš and colleagues [97] have argued, strong resonant peaks appear in the cross-sections if inelastic couplings are weak. In contrast, weak oscillations of one order of magnitude at best appear when inelastic couplings are strong, as in alkali-metal trimer © 2009 by Taylor and Francis Group, LLC
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10–9
10–10
Quenching cross-section (cm2)
10–8 Na + Na2 (u = 2, j = 0)
Total uf = 1 uf = 0
10–9
10–10 10–8 Total uf = 0
Na + Na2 (u = 3, j = 0)
uf = 1 uf = 2
10–9
10–10 0.98
0.99
1
1.01
1.02
l
FIGURE 3.19 Dependence of the quenching cross-sections on the three-body term at a collision energy of 10−9 K for Na + Na2 (v = 1 − 3, j = 0). (From Quéméner, G. et al., Eur. Phys. J. D, 30, 201, 2004. With permission.)
systems. A generalization of this effect has been discussed by Hutson [114]. Larger modifications of the three-body term affect the cross-sections significantly. This can also be seen in Figure 3.21 for Li + Li2 collisions [96] in which the three-body term is excluded from the dynamics calculation. The state-to-state cross-sections exhibit a stronger dependence on the three-body term. © 2009 by Taylor and Francis Group, LLC
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(a) u f = 2
8e–11 6e–11 1.02
4e–11
1.01
2e–11 0
1.00 l
2 on (cm ) Cross-secti
1e–10
16
12
0.99
8
4
jf
0
0.98
(b) u f = 1
2 on (cm )
1e–10
Cross-secti
8e–11 6e–11 1.02
4e–11
1.01
2e–11
l
1.00 0 24 20
16 12 jf
0.99 8
4
0
0.98
(c) u f = 0
Cross-secti
2 on (cm )
1e–10 8e–11 6e–11 4e–11
1.02 1.01
2e–11
l
1.00 0
28 24 20 16 12 8 jf
0.99 4
0
0.98
FIGURE 3.20 Variation of state-to-state cross-sections at a collision energy of 10−9 K for Na + Na2 (v = 3, j = 0) → Na + Na2 (vf = 2, 1, 0, jf ) as a function of the parameter λ. See text for details. (From Quéméner, G. et al., Eur. Phys. J. D, 30, 201, 2004.) © 2009 by Taylor and Francis Group, LLC
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+ 7Li2 (u, j = 0)
10–9 Full calculation V3 = 0
Rate coefficient (cm3/sec)
10–10 Quenching 10–11
10–12 Elastic 10–13
10–14
0
1
2
3
4
5
6
7
8
9
10
Initial vibrational number, u
FIGURE 3.21 Dependence of the elastic and quenching rate coefficients on the vibrational excitation of the molecule for 7 Li + 7 Li2 (v = 0 − 10, j = 0) scattering at a collision energy of 10−9 K. The bold line corresponds to the full calculation and the thin line corresponds to the calculation without the three-body term of the PES. (From Quéméner, G. et al., Phys. Rev. A, 75, 050701(R), 2007. With permission.)
3.3.2.3
Relaxation of Vibrationally Excited Alkali Metal Dimers
Ultracold diatomic molecules produced in photoassociation or Feshbach resonance methods are usually created in excited vibrational states. Theoretical studies involving highly vibrationally excited molecules are challenging due to the large number of energetically open reaction channels present in the quantum calculation. This puts severe restriction on the calculations for vibrationally excited molecules at low temperatures. Ultracold quantum dynamics calculations for collisions of highly vibrationally excited molecules have been reported in 2007 by Quéméner and colleagues [96] for the Li + Li2 system. A three-atom problem is generally described by two different kinds of asymptotic channels. The first kind are the single-continuum states (SCSs). They correspond to configurations where the triatomic system dissociates asymptotically into an atom and a diatomic molecule. The second kind are the double-continuum states (DCSs). They correspond to configurations where the triatomic system dissociates asymptotically into three separated atoms. Because highly vibrationally excited molecular states lie close to and below the triatomic dissociation limit, they are also coupled with the DCSs, which lie above the dissociation limit. As a consequence, DCSs have to be included in quantum simulations of atom–diatom systems involving © 2009 by Taylor and Francis Group, LLC
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highly vibrationally excited diatomic molecules [96]. This dramatically increases the size and complexity of the quantum dynamics problem. Figure 3.21 shows the dependence of the rate coefficients for 7 Li + 7 Li2 (v = 0 − 10, j = 0) collisions on the vibrational quantum number at a collision energy of 10−9 K [96]. Quenching processes are more efficient than the elastic scattering for both high and low vibrational levels. Similar results have been found for 6 Li + 6 Li (v = 0 − 9, j = 1) system composed of fermionic atoms [96]. Quenching 2 rate coefficients show a slight decrease when the molecule is in its highest vibrational state. This is because the overlap of the wavefunctions of highly excited diatomic molecules with low-lying vibrational levels is very small, leading to small values for the interaction potential coupling matrix elements between the initial and final states [115]. These results do not directly apply to ultracold molecules composed of fermionic atoms created near a Feshbach resonance. In these experiments, the quenching processes are suppressed because the atom–atom scattering length is tuned to large and positive values, leading to an efficient Pauli blocking mechanism, as explained by Petrov and colleagues [116]. In the theoretical study of Li + Li2 collisions, the atom–atom scattering length is small and negative, and no suppression of the quenching processes is found for the molecule in the last vibrational state. Thus, the sign and magnitude of the Li–Li scattering length play a crucial role in the mechanism that suppresses quenching collisions. The quenching rates displayed in Figure 3.21 show an irregular dependence on the vibrational state of the molecule. This has already been seen for H + H2 collisions [28,117]. In contrast, experimental measurements of quenching rate coefficients for Cs + Cs2 collisions [106,107] do not show any dependence on the vibrational state of the molecule. The differences between these systems can be explained based on the following considerations. First, the theoretical study applies to spin-polarized atom–dimer alkali-metal systems, whereas it is not the case for the experiment. A full theoretical treatment should involve the electronic and nuclear spins of the alkali metal atoms as well as couplings between electronic surfaces of different spins. This is beyond the scope of quantumdynamics calculations at present and will involve significant new code development and massive computational efforts. Second, the dynamics of the two systems are different. The lithium system is lighter and has a more attractive three-body term than the cesium system [105]. The triatomic adiabatic potential energy curves are well separated for a light system such as Li3 and very dense for a heavy system such as Cs3 . The density of states strongly influences the nature of vibrational relaxation. The effect of the density of states has been illustrated in calculations for the Li + Li2 system, which exclude the three-body term. Removing the three-body term makes the energy levels more sparse. This results in a more regular and monotonic dependence of the rate coefficients on the vibrational levels v = 3 to 9 as illustrated in Figure 3.21. This conclusion is in agreement with a previous work of Bodo and colleagues [117] for the H + H2 system. When they increased the density of states of the triatomic system, they found no significant dependence on the vibrational states of the molecule. However, for v = 10, they obtained the same results with and without the three-body term. The three-body term thus appears to be less significant for collisions of molecules in high vibrational levels. Since the © 2009 by Taylor and Francis Group, LLC
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three-body term vanishes at large separations it is only important in the short-range interaction region while high vibrational states involve spatially extended molecules and sample the long-range part of the interaction potential. Therefore, three-body terms, which are the most difficult interaction energy terms to compute numerically for triatomic systems, may be neglected to a first approximation in dynamics of highly vibrationally excited molecules. In the experiments, rate coefficients have been measured for temperatures of 40 × 10−6 K [107] and 60 × 10−6 K [106] for Cs + Cs2 , while the theoretical rate coefficients for Li + Li2 were reported for a temperature of 10−9 K, which corresponds to the Wigner threshold regime. Thus, it is likely that the experimental measurements did not probe the limiting values of the rate coefficients in the Wigner regime. 3.3.2.4
Reactions of Heteronuclear and Isotopically Substituted Alkali-Metal Dimer Systems
Reactive collisions involving heteronuclear molecules are currently of great interest. A major goal of recent experiments has been to produce heteronuclear alkali-metal dimers in their electronic ground state. Examples include RbCs [111,118,119], NaCs [120,121], KRb [122,123], LiCs [124], and mixed isotopes 6 Li7 Li [125]. Quantum dynamics calculations for heteronuclear systems are more difficult. They are more interesting from a chemistry perspective because the reactive channels in collisions of heteronuclear molecules can be distinguished from inelastic channels. Cvitaš and colleagues [95,97] have explored the quantum dynamics of 7 Li + 6 Li7 Li(v = 0, j = 0), 7 Li + 6 Li (v = 0, j = 1), 6 Li + 7 Li (v = 0, j = 0), 2 2 and 6 Li + 6 Li7 Li(v = 0, j = 0) collisions. The J = 0 cross-sections for elastic scattering and chemical reactions in collisions of 7 Li with 6 Li7 Li(v = 0, j = 0) are presented in Figure 3.22a for a wide range of collision energies. The cross-sections are believed to be converged for energies up to 10−4 K. The reactive process, which leads to 6 Li + 7 Li2 (v = 0, j = 0) dominates over the elastic scattering at ultralow energies. However, the reactive process is less efficient than the vibrational relaxation process in homonuclear alkali systems discussed above. For instance, at a collision energy of 10−9 K, the ratio of the cross-sections for reactive and elastic collisions presented in Figure 3.22 is larger for the homonuclear system compared to the heteronuclear system. Cvitaš and colleagues attributed the smaller ratio to the presence of only one open channel in the heteronuclear reaction. The J = 1 elastic and reactive cross-sections for 7 Li + 6 Li2 (v = 0, j = 1) collisions are presented in Figure 3.22b. The reactive process leading to the formation of 6 Li7 Li molecules is slightly more efficient than elastic scattering in the Wigner regime. The other possible collision processes are 6 Li + 6 Li7 Li(v = 0, j = 0) and 6 Li + 7 Li (v = 0, j = 0). However, only elastic scattering occurs in these systems at 2 ultralow collisions energies. These results provide important implications for the experiments on the production of 6 Li7 Li in an ultracold mixture of 6 Li and 7 Li atoms. Cvitaš and colleagues proposed removing quickly the 7 Li atomic gas after the formation of the 6 Li7 Li molecules © 2009 by Taylor and Francis Group, LLC
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Inelastic Collisions and Chemical Reactions of Molecules (a) 10−10 Elastic: uf = 0, jf = 0 Inelastic: uf = 0, jf = 1
10−11
Reactive: uf = 0, jf = 0 Reactive: uf = 0, jf = 2
Cross-section (cm2)
10−12 10−13 10−14 10−12
10−15
10−13 10−14
10−16
10−15 10−16
10−17 10−18 −9 10
10−17
0
10−8
0.2
10−7
0.4
10−6
0.6
10−5
0.8
10−4
1
10−3
10−2
10−1
100
Collision energy (K)
(b) 10−8 Elastic: uf = 0, jf = 0 Reactive: uf = 0, jf = 0 Reactive: uf = 0, jf = 1 Reactive: uf = 0, jf = 2 Reactive: uf = 0 (total)
10−9 10−10
s (cm2)
10−11 10−12 10−13 10−14 10−15 10−16 10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
E(K)
FIGURE 3.22 Elastic and reactive s-wave cross-sections for 7 Li + 6 Li7 Li(v = 0, j = 0) (a) and 7 Li + 6 Li2 (v = 0, j = 1) (b). (From Cvitaš, M.T. et al., Phys. Rev. Lett., 94, 200402, 2005. With permission.)
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in their ground state to prevent the destructive reactive processes. But keeping the 6 Li atomic gas is recommended for sympathetic cooling of the 6 Li7 Li molecules because only elastic collisions are possible. Removing the 6 Li atoms will leave the fermionic heteronuclear 6 Li7 Li molecules in the trap. In the absence of 6 Li atoms, evaporative cooling by collisions between fermionic 6 Li7 Li molecules will not be effective because s-wave collisions of identical fermionic dimers will be suppressed due to the Pauli exclusion principle. Table 3.4 provides a compilation of zero-temperature quenching rate coefficients for different alkali-metal trimer systems.
3.4
INELASTIC MOLECULE–MOLECULE COLLISIONS
Many of the studies of cold and ultracold molecules have focused on reactive and nonreactive scattering in atom–molecule collisions. At high densities of trapped molecules, molecule–molecule collisions need to be considered. The presence of rotational and vibrational degrees of freedom in both collision partners make molecule– molecule systems especially interesting. However, quantum dynamics calculations of molecule–molecule collisions are significantly more challenging. Most dynamical calculations of molecule–molecule scattering have relied on the rigid rotor approximation, and some recent studies have adopted the coupled-states approximation. Calculations performed at high collision energies have employed more approximate methods based on semiclassical techniques. Here, we present a brief account of recent dynamical calculations for the H2 + H2 system and discuss some ongoing work on full-dimensional quantum calculations of rovibrational transitions in H2 –H2 collisions. A brief discussion of hyperfine transitions in molecule–molecule systems is also provided for the illustrative examples of the O2 + O2 and OH + OH/OD + OD systems.
3.4.1
MOLECULES IN THE GROUND VIBRATIONAL STATE
The H2 + H2 system is the simplest neutral tetra-atomic system and it serves as a prototype for describing collisions between diatomic molecules. Although there have been a number of experimental and theoretical studies over the past several years on the H2 + H2 system (see [126] and references therein), only a few studies have explored the collision dynamics in the cold and ultracold regime. Forrey [127] presented a study of rotational transitions in H2 (v = 0, j = 2) + H2 (v = 0, j = 2) collisions at cold and ultracold collision energies using a rigid rotor model. He calculated the real and imaginary components of the complex scattering length for H2 (v = 0, j = 2, 4, 6, 8) + H2 (v = 0, j = j) collisions and showed that the imaginary parts decrease with increasing j, j . The imaginary parts of the scattering lengths are found to be small compared to the real parts, leading to small inelastic cross-sections. Maté and colleagues [128] reported an experimental study of the rate coefficient for the H2 (v = 0, j = 0) + H2 (v = 0, j = 0) → H2 (vf = 0, jf = 0) + H2 (vf = 0, jf = 2) transition at temperatures between 2 and 110 K. They found good agreement between the experimental results and quantum-dynamics calculations based on the rigid rotor model using a PES reported by Diep and Johnson © 2009 by Taylor and Francis Group, LLC
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TABLE 3.4 Zero-Temperature Quenching Rate Coefficients for Different Atom–Dimer Alkali Metal Systems Initial (v, j )
kT =0 (cm3 /sec)
Reference
39 K + 39 K 2 40 K + 40 K 2 41 K + 41 K 2
(v = 1, j = 0) (v = 1, j = 1) (v = 1, j = 0)
1.1 × 10−10 8.0 × 10−11 9.8 × 10−11
[100] [100] [100]
7 Li + 7 Li
(v = 1, j = 0) (v = 2, j = 0) (v = 3, j = 0) (v = 4, j = 0) (v = 5, j = 0) (v = 6, j = 0) (v = 7, j = 0) (v = 8, j = 0) (v = 9, j = 0) (v = 10, j = 0) (v = 1, j = 1) (v = 2, j = 1) (v = 3, j = 1)
2.1 × 10−11 1.5 × 10−11 4.4 × 10−11 3.0 × 10−11 1.2 × 10−10 8.9 × 10−11 3.3 × 10−10 1.6 × 10−10 2.9 × 10−10 2.4 × 10−11 3.3 × 10−11 2.0 × 10−11 5.1 × 10−11
[96] [96] [96] [96] [96] [96] [96] [96] [96] [96] [96] [96] [96]
(v = 1, (v = 2, (v = 1, (v = 2,
j = 0) j = 0) j = 1) j = 1)
5.6 × 10−10 9 × 10−11 2.8 × 10−10 4 × 10−10
[94] [94] [94] [94]
(v = 1, j = 0) (v = 2, j = 0) (v = 3, j = 0)
2.9 × 10−10 1.1 × 10−10 6.1 × 10−11
[99] [99] [99]
(v = 0, (v = 1, (v = 2, (v = 3, (v = 0, (v = 1, (v = 2, (v = 3, (v = 1, (v = 2, (v = 3, (v = 1, (v = 2, (v = 3,
4.1 × 10−12 2.1 × 10−10 4.4 × 10−10 4.0 × 10−10 4.4 × 10−11 5.2 × 10−10 2.6 × 10−10 3.0 × 10−10 2.6 × 10−10 3.5 × 10−10 4.4 × 10−10 2.8 × 10−10 5.3 × 10−10 4.6 × 10−10
[97] [97] [97] [97] [97] [97] [97] [97] [97] [97] [97] [97] [97] [97]
System
2
6 Li + 6 Li 2
7 Li + 7 Li
2
6 Li + 6 Li
2
23 Na + 23 Na
2
7 Li + 6 Li7 Li
7 Li + 6 Li
2
6 Li + 6 Li7 Li
6 Li + 7 Li
2
j = 0) j = 0) j = 0) j = 0) j = 1) j = 1) j = 1) j = 1) j = 0) j = 0) j = 0) j = 0) j = 0) j = 0)
(DJ) [129]. Montero and colleagues [130] have investigated cold inelastic collisions of n-H2 molecules in the ground vibrational state, using a 3:1 gas mixture of orthoand para-hydrogen. They obtained good agreement between the experimental data and the theoretical calculations based on the DJ PES. © 2009 by Taylor and Francis Group, LLC
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Using a quantum formalism based on the rigid rotor model, Lee and colleagues [126] have recently presented a comparative analysis of cross-sections for rotationally inelastic collisions between H2 molecules at low and ultralow energies. The elastic cross-sections for H2 (v = 0, j = 0) + H2 (v = 0, j = 0) collisions obtained in this study are presented in Figure 3.23 for two different PESs. The limiting value of the elastic scattering cross-section in the ultralow energy regime is 1.91 × 10−13 cm2 with the PES of Boothroyd, Martin, Keogh, and Peterson (BMKP) [131] and 1.74 × 10−13 cm2 with the DJ PES [129]. For low collision energies the dynamics is sensitive to higher-order anisotropic terms in the angular expansion of the interaction potential. A diatom–diatom scattering length of 5.88 Å was obtained for the DJ PES and 6.16 Å for the BMKP PES. Cross-sections for the quenching of rotationally excited H2 molecules in H2 (v = 0, j = 2) + H2 (v = 0, j = 0) and H2 (v = 0, j = 2) + H2 (v = 0, j = 2) collisions were presented at low and ultralow energies. Quantum calculations of rotational relaxation of CO in cold and ultracold collisions with H2 have recently been performed by Yang and colleagues [132,133]. They reported quenching rate coefficients for j = 1 to 3 of the CO molecule in collisions with both ortho- and para-H2 [132]. Due to the relatively deep van der Waals interaction potential for the H2 –CO system the cross-sections exhibit a number of narrow resonances for collision energies between 1.0 and 40.0 cm−1 . The signatures of these resonances are present in the temperature dependence of the rate coefficient, which shows broad oscillatory features in the temperature range of 10−2 to 50 K [132].
5 ¥ 103 This work: DJ PES
Elastic cross-section (10–16 cm2)
This work: BMKP PES 4 ¥ 103
3 ¥ 103
2 ¥ 103
1 ¥ 103
0 10–9
10–8
10–7
10–6
10–5 10–4 10–3 Collision energy (eV)
10–2
10–1
100
101
FIGURE 3.23 Elastic cross-section of H2 (v = 0, j = 0) + H2 (v = 0, j = 0) as a function of the collision energy. (From Lee, T.G. et al., J. Chem. Phys., 125, 114302, 2006. With permission.) © 2009 by Taylor and Francis Group, LLC
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Bohn and coworkers have reported extensive calculations of hyperfine transitions in ultracold molecule–molecule collisions. Avdeenkov and Bohn studied ultracold collisions between O2 molecules [134]. They used a quantum-mechanical formalism based on the rigid rotor model and included the electronic spin structure of the O2 molecules. Couplings between the rotational angular momentum and the electronic spin of the molecules lead to rotational fine structure. Avdeenkov and Bohn discussed the elastic and inelastic loss processes in 17 O2 + 17 O2 and 16 O2 + 16 O2 collisions. They found that for collision energies below 10−2 K, elastic collision cross-sections are larger than inelastic spin-flipping transitions. Based on relative magnitudes of elastic and inelastic spin-flipping cross-sections they concluded that 17 O molecules would be good candidates for evaporative cooling. In contrast, for 2 16 O + 16 O collisions, inelastic processes are more efficient than elastic collisions 2 2 so that 16 O2 molecules are prone to collisional trap loss. Avdeenkov and Bohn also studied ultracold collisions between OH [135,136] and OD radicals [137] in the presence of an applied electric field. They showed that elastic scattering is more efficient than inelastic processes for ultracold collisions between fermionic OD molecules [137], inhibiting state-changing collisions. The energy dependence of elastic and inelastic cross-sections for OH + OH and OD + OD collisions is illustrated in Figure 3.24 for an applied electric field of ε = 100 V/cm. While the elastic cross-sections approach finite values in the Wigner regime for the bosonic system of OH molecules and the fermionic system of OD molecules, the
10–11
s (cm2)
10–12
10–13
10–14
10–8
10–6
10–4 Energy (K)
10–2
100
FIGURE 3.24 Elastic and inelastic cross-sections for OD + OD (thick gray line) and OH + OH (thin black line) collisions for an applied electric field of ε = 100 V/cm. Solid and dashed lines refer to elastic and inelastic cross-sections, respectively. (From Avdeenkov, A.V. and Bohn, J.L., Phys. Rev. A, 71, 022706, 2005. With permission.)
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inelastic cross-sections exhibit a totally different behavior. At ultralow energies when molecules interact with an electric field, s-wave scattering yields an inelastic cross−1/2 section diverging as Ecoll for bosonic systems, while p-wave scattering yields an 1/2 inelastic cross-section vanishing as Ecoll for fermionic systems. Thus, the inelastic processes are suppressed for the fermionic system. The differences are attributed to the bosonic/fermionic character of the molecules and to the applied electric field. In the absence of an applied electric field, the elastic cross-section of the fermionic 2 , faster than the inelastic cross-section, which decreases system would decrease as Ecoll 1/2 as Ecoll . Ticknor and Bohn [138] subsequently studied OH–OH collisions in the presence of a magnetic field. They showed that magnetic fields of several thousand gauss reduce inelastic collisions by about two orders of magnitude. Based on these results, they concluded that magnetic trapping may be favorable for OH molecules.
3.4.2 VIBRATIONALLY INELASTIC TRANSITIONS The theoretical studies presented above for molecule–molecule collisions refer to rigid rotor molecules. Pogrebnya and Clary [139,140] have investigated vibrational relaxation in collisions of hydrogen molecules using a full-dimensional quantumdynamics formalism with an angular momentum decoupling approximation in the body fixed frame. They studied H2 (v = 1, j) + H2 (v = 0, j ) collisions involving both para–para and ortho–ortho combinations for collision energies of 1 meV (11.6 K) to 1 eV (11,604 K). Time-dependent quantum-mechanical calculations based on the multiconfiguration time-dependent Hartree approach have been recently applied to study full-dimensional quantum dynamics of the H2 –H2 system [141,142]. These methods are suitable at high collision energies and have not been applied to cold and ultracold collisions. Very recently, a vibrational energy transfer mechanism for ultracold collisions between para-hydrogen molecules has been explored by Quéméner and colleagues [143] using a full-dimensional quantum theoretical formalism implemented in a new code written by Krems [144]. The quantum scattering calculations are based on the theory described by Arthurs and Dalgarno [145], Takayanagi [146], Green [147], and Alexander and DePristo [148]. The H2 molecules were initially in different quantum states characterized by the vibrational quantum number v and the rotational angular momentum j. A combination of two rovibrational states of H2 was referred to as a combined molecular state (CMS). A CMS, denoted as (vjv j ), represents a unique quantum state of the diatom–diatom system before or after a collision. The cross-sections for H2 (v = 1, j = 0) + H2 (v = 0, j = 0), H2 (v = 1, j = 2) + H2 (v = 0, j = 0), and H2 (v = 1, j = 0) + H2 (v = 0, j = 2) collisions are presented in Figure 3.25a. Using the CMS notation, this corresponds, respectively, to initial CMSs (1000), (1200), and (1002). The elastic cross-sections are almost independent of the different initial rovibrational states of the molecules, but the inelastic cross-sections are strongly dependent on the initial rotational and vibrational levels of the H2 molecules. At 10−6 K, the inelastic relaxation of H2 (v = 1, j = 0) is almost six orders of magnitude more efficient in collisions of H2 (v = 0, j = 2) than in collisions with H2 (v = 0, j = 0) and two orders of magnitude more efficient than in collisions of H2 (v = 1, j = 2) with H2 (v = 0, j = 0). At 25.45 K, which © 2009 by Taylor and Francis Group, LLC
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(a)
H2(u = 1, j) + H2(u = 0, j ¢)
104 103
Elastic
Cross-section (10–16 cm2)
102
Inelastic
101 100
j = 0, j¢ = 0 j = 2, j¢ = 0 j = 0, j¢ = 2
10–1
Inelastic
10–2 10–3 Inelastic
10–4 10–5 10–6
10–4
10–3 10–2 10–1 Collision energy (K)
100
101
102
02 10
H2(u = 1, j = 0) + H2(u = 0, j = 0)
12
103
00
H2(u = 1, j = 0) + H2(u = 0, j = 2) H2(u = 1, j = 2) + H2(u = 0, j = 0)
10
104
00
Ecoll = 10–6 K
105
102 101
10–3
10–5
02 08
06 04
06 06 00
00
00
10–4
04 00
02
02
10–2
04 02
04
02
04
10–1
08
00
100
02 00
Cross-section (10–16 cm2)
(b)
10–5
10–6 Final combined molecular state
FIGURE 3.25 Elastic and inelastic cross-sections for the collisions H2 (v = 1, j = 0) + H2 (v = 0, j = 0), H2 (v = 1, j = 2) + H2 (v = 0, j = 0), and H2 (v = 1, j = 0) + H2 (v = 0, j = 2): total cross-sections (a); state-to-state cross-sections at 10−6 K (b). (From Quéméner, G. et al., Phys. Rev. A, 77, 030704 (R), 2008. With permission.)
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corresponds to the energy difference between the CMSs (1002) and (1200), the inelastic cross-sections become comparable for H2 (v = 1, j = 2) + H2 (v = 0, j = 0) and H2 (v = 1, j = 0) + H2 (v = 0, j = 2) collisions. The inelastic scattering depends on the type and the combination of rovibrational levels involved in the collision, whether it involves ground-state molecules, H2 (v = 0, j = 0), vibrationally excited molecules, H2 (v = 1, j = 0), rotationally excited molecules H2 (v = 0, j = 2), or rovibrationally excited molecules, H2 (v = 1, j = 2). H2 molecules are weakly interacting and characterized by shallow van der Waals interaction at large separations and the computed rate coefficients are not representative of strongly interacting alkali-metal dimer systems. For example, Mukaiyama and colleagues [109] reported an inelastic rate coefficient of 5.1 × 10−11 cm3 /sec for collisions between two weakly bound Na2 molecules created by the Feshbach resonance method. In a similar study Syassen and colleagues [110] reported a rate coefficient of 3 × 10−10 cm3 /sec for collisions between two Rb2 molecules. Zahzam and colleagues [107] estimated a rate coefficient of about 10−11 cm3 /sec for collisions between Cs2 molecules. Ferlaino and colleagues [149] measured a rate coefficient of 9 × 10−11 cm3 /sec for collisions between Cs2 molecules in the nonhalo regime. These last authors also measured the variation of the inelastic rate coefficient as a function of the atom–atom scattering length for collisions between tunable halo dimers of Cs2 . The large values of the inelastic rate coefficients for alkali-metal dimer systems are attributed to strong inelastic couplings and deeper potential energy wells. The state-to-state cross-sections for three rovibrational combinations of the H2 –H2 system are presented in Figure 3.25b. The magnitude of the inelastic cross-sections depends on the propensity of the diatom–diatom system to conserve the internal energy and the total rotational angular momentum of the colliding molecules [143]. The final state-to-state distribution in H2 (v = 1, j = 0) + H2 (v = 0, j = 0) collisions shows that there is no preferential population of rotational quantum number of either of the colliding molecules. The conservation of the total rotational angular momentum would entail a large change in the internal energy of the molecules. Thus, the purely vibrational transition (1000) → (0000) is not efficient because the energy gap is large. On the other hand, (near) conservation of the internal energy requires a large change in the total rotational angular momentum of the colliding molecules and the transition (1000) → (0800) is not dominant either. However, the state-to-state cross-sections for H2 (v = 1, j = 2) + H2 (v = 0, j = 0) collisions indicate that the transition (1200) → (1000) is more efficient than all the other transitions combined. For this transition, the total rotational angular momentum change and the internal energy transfer are both minimized, leading to a more efficient energy transfer process. The state-to-state cross-sections for H2 (v = 1, j = 0) + H2 (v = 0, j = 2) collisions present an interesting scenario in which the transition (1002) → (1200) is highly efficient and selective. In this case, the total rotational angular momentum is conserved and the internal energy is almost unchanged [the energy gap between (1200) and (1002) is only 25.45 K]. This creates a favorable situation leading to a near-resonant energy transfer process. This particular mechanism cannot occur in atom–diatom systems because simultaneous conservation of rotational angular momentum and internal energy of the molecule cannot occur. The mechanism is reminiscent of quasiresonant energy transfer in collisions of rotationally excited diatomic molecules © 2009 by Taylor and Francis Group, LLC
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TABLE 3.5 Zero-Temperature Inelastic Rate Coefficients for Different Molecule–Molecule Systems System
Initial (v, j , v , j )
kT =0 (cm3 /sec)
Reference
H2 + H2
(v = 1, j = 0, v = 0, j = 0) (v = 2, j = 0, v = 0, j = 0) (v = 3, j = 0, v = 0, j = 0) (v = 4, j = 0, v = 0, j = 0) (v = 5, j = 0, v = 0, j = 0) (v = 6, j = 0, v = 0, j = 0) (v = 1, j = 0, v = 1, j = 0) (v = 2, j = 0, v = 1, j = 0) (v = 3, j = 0, v = 1, j = 0) (v = 4, j = 0, v = 1, j = 0) (v = 2, j = 0, v = 2, j = 0)
8.9 × 10−18 3.9 × 10−17 1.2 × 10−16 1.8 × 10−16 6.9 × 10−16 2.9 × 10−15 2.7 × 10−16 1.3 × 10−14 1.6 × 10−14 9.0 × 10−15 6.1 × 10−16
[143] [143] [143] [143] [143] [143] [143] [143] [143] [143] [143]
H2 + H2
(v = 0, j = 2, v = 0, j = 0) (v = 1, j = 2, v = 0, j = 0) (v = 2, j = 2, v = 0, j = 0) (v = 3, j = 2, v = 0, j = 0) (v = 4, j = 2, v = 0, j = 0) (v = 1, j = 2, v = 1, j = 0) (v = 2, j = 2, v = 1, j = 0)
3.9 × 10−14 3.7 × 10−14 8.7 × 10−14 1.5 × 10−13 2.5 × 10−13 8.8 × 10−14 6.1 × 10−14
[143] [143] [143] [143] [143] [143] [143]
H2 + H2
(v (v (v (v (v
= 2) = 2) = 2) = 2) = 2)
5.6 × 10−12 3.4 × 10−12 2.7 × 10−12 2.1 × 10−12 5.2 × 10−12
[143] [143] [143] [143] [143]
H2 + CO
(v = 0, j = 0, v = 0, j = 1) (v = 0, j = 0, v = 0, j = 2) (v = 0, j = 0, v = 0, j = 3)
2.0 × 10−12 3.0 × 10−11 1.2 × 10−10
[132] [132] [132]
H2 + CO
(v = 0, j = 1, v = 0, j = 1) (v = 0, j = 1, v = 0, j = 2) (v = 0, j = 1, v = 0, j = 3)
1.2 × 10−11 4.0 × 10−11 8.5 × 10−11
[132] [132] [132]
= 1, = 2, = 3, = 4, = 2,
j j j j j
= 0, v = 0, v = 0, v = 0, v = 0, v
= 0, j = 0, j = 0, j = 0, j = 1, j
with atoms discussed previously [21,53,54,60], but with a purely quantum origin. The near-resonant process may be an important mechanism for collisional energy transfer in ultracold molecules formed by photoassociation of ultracold atoms and for chemical reactions producing identical molecules. Table 3.5 provides a compilation of zero-temperature rate coefficients for rotational and vibrational quenching in molecule–molecule systems.
3.5
SUMMARY AND OUTLOOK
In this chapter we have given an overview of recent theoretical studies of atom– molecule and molecule–molecule collisions at cold and ultracold temperatures. © 2009 by Taylor and Francis Group, LLC
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Although such systems have been extensively studied at higher collision energies over the last few decades, the new experimental breakthroughs in creating dense samples of cold and ultracold molecules have provided unprecedented opportunities to explore elastic, inelastic, and reactive collisions at temperatures close to absolute zero. These studies have revealed unique aspects of molecular collisions and energy transfer mechanisms that are otherwise not evident in thermal energy collisions. The long duration of collisions combined with large de Broglie wavelengths at cold and ultracold temperatures leads to interesting quantum effects. Calculations have shown that reactions with insurmountable energy barriers may still occur at temperatures close to absolute zero, and in certain cases, with appreciable rate coefficients. Such tunneling-dominated reactions have been the topic of many recent investigations and may soon be amenable to experimental investigation in the cold and ultracold regime. That the rates of these reactions can be enhanced by vibrational excitation of the molecule is an interesting scenario for experimental studies of ultracold chemical reactions. The studies of alkali-metal homonuclear and heteronuclear trimer systems have stimulated considerable experimental interest in investigating chemical reactivity at ultracold temperatures. The challenge for theory is to describe collisions involving highly vibrationally excited molecules. Recent theoretical calculations have indicated that the three-body interaction potential can be neglected for highly vibrationally excited molecules, offering significant savings in computational effort. Even so, heavier alkali-metal trimer systems pose a daunting computational challenge. The quantitative description of ultracold molecule–molecule collisions is another challenging topic. The recent progress on the H2 –H2 system will be difficult to implement for heavier systems due to the large number of rovibrational levels of the molecules. The study of H2 –H2 collisions has shown that, for certain combinations of rovibrational levels, the energy transfer may occur to specific final rovibrational states. In such cases, the calculations can use a much smaller basis set without compromising the accuracy. Currently there is substantial interest in controlling the collisional outcome using external electric and magnetic fields. While the idea of coherent control of molecular collisions and chemical reactivity has existed for a long time and some important progress has been achieved, the possibility of creating coherent and dense samples of molecules in specific quantum states has given further impetus to the field of controlled chemistry. We expect that the coming years will see a far greater activity in this direction driven by cold and ultracold molecules and also by the possibility of controlling chemical reactivity using external electric and magnetic fields. Electronically nonadiabatic effects in ultracold collisions is a largely unexplored area, which can be expected to attract much attention.
ACKNOWLEDGMENTS This work was supported by NSF grants PHY-0555565 (N.B.), AST-0607524 (N.B.), and by the Chemical Science, Geoscience and Bioscience Division of the Office of Basic Energy Science, Office of Science, U.S. Department of Energy (A.D.). © 2009 by Taylor and Francis Group, LLC
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spectroscopy, photoassociative molecule formation, and trapping of ultracold 39 K85 Rb, Eur. Phys. J. D, 31, 165, 2004. Kraft, S.D., Staanum, P., Lange, J., Vogel, L., Wester, R., and Weidemüller, M., Formation of ultracold LiCs molecules, J. Phys. B: At. Mol. Opt. Phys., 39, S993, 2006. Schlöder, U., Silber, C., and Zimmermann, C., Photoassociation of heteronuclear lithium, Appl. Phys. B: Lasers Opt., 73, 801, 2001. Lee, T.-G., Balakrishnan, N., Forrey, R.C., Stancil, P.C., Schultz, D.R., and Ferland, G.J., State-to-state rotational transitions in H2 + H2 collisions at low temperatures, J. Chem. Phys., 125, 114302, 2006. Forrey, R.C., Cooling and trapping of molecules in highly excited rotational states, Phys. Rev. A, 63, 051403(R), 2001. Maté, B., Thibault, F., Tejeda, G., Fernández, J.M., and Montero, S., Inelastic collisions in para-H2 : Translation-rotation state-to-state rate coefficients and cross sections at low temperature and energy, J. Chem. Phys., 122, 064313, 2005. Diep, P., and Johnson, J.K., An accurate H2 –H2 interaction potential from first principles, J. Chem. Phys., 112, 4465, 2000. Montero, S., Thibault, F., Tejeda, G., and Fernández, J.M., Rotranslational stateto-state rates and spectral representation of inelastic collisions in low-temperature molecular hydrogen, J. Chem. Phys,. 125, 124301, 2006. Boothroyd, A.I., Martin, P.G., Keogh, W.J., and Peterson, M.J., An accurate analytic H4 potential energy surface, J. Chem. Phys., 116, 666, 2002. Yang, B., Stancil, P.C., Balakrishnan, N., and Forrey, R.C., Quenching of rotationally excited CO by collisions with H2 , J. Chem. Phys., 124, 104304, 2006. Yang, B., Perera, H., Balakrishnan, N., Forrey, R.C., and Stancil, P.C., Quenching of rotationally excited CO in cold and ultracold collisions with H, He and H2 , J. Phys. B: At. Mol. Opt. Phys., 39, S1229, 2006. Avdeenkov, A.V. and Bohn, J.L., Ultracold collisions of oxygen molecules, Phys. Rev. A, 64, 052703, 2001. Avdeenkov, A.V. and Bohn, J.L., Collisional dynamics of ultracold OH molecules in an electrostatic field, Phys. Rev. A, 66, 052718, 2002. Avdeenkov, A.V. and Bohn, J.L., Linking ultracold polar molecules, Phys. Rev. Lett., 90, 043006, 2003. Avdeenkov, A.V. and Bohn, J.L., Ultracold collisions of fermionic OD radicals, Phys. Rev. A, 71, 022706, 2005. Ticknor, C. and Bohn, J.L., Influence of magnetic fields on cold collisions of polar molecules, Phys. Rev. A, 71, 022709, 2005. Pogrebnya, S.K. and Clary, D.C., A full-dimensional quantum dynamical study of vibrational relaxation in H2 + H2 , Chem. Phys. Lett., 363, 523, 2002. Pogrebnya, S.K., Mandy, M.E., and Clary, D.C., Vibrational relaxation in H2 + H2 : full-dimensional quantum dynamical study, Int. J. Mass. Spectrom., 223–224, 335, 2003. Panda, A.N., Otto, F., Gatti, F., and Meyer, H.-D., Rovibrational energy transfer in ortho-H2 + para-H2 collisions, J. Chem. Phys., 127, 114310, 2007. Otto, F., Gatti, F., and Meyer, H.-D., Rotational excitations in para-H2 + para-H2 collisions: Full- and reduced-dimensional quantum wave packet studies comparing different potential energy surfaces, J. Chem. Phys., 128, 064305, 2008. Quéméner, G., Balakrishnan, N., and Krems, R.V., Vibrational energy transfer in ultracold molecule–molecule collisions, Phys. Rev. A, 77, 030704(R), 2008. Krems, R.V., TwoBC—quantum scattering program, University of British Columbia, Vancouver, Canada, 2006.
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145. Arthurs, A.M. and Dalgarno, A., The theory of scattering by a rigid rotator, Proc. Roy. Soc. A, 256, 540, 1960. 146. Takayanagi, K., The production of rotational and vibrational transitions in encounters between molecules, Adv. At. Mol. Phys., 1, 149, 1965. 147. Green, S., Rotational excitation in H2 –H2 collisions: Close-coupling calculations, J. Chem. Phys., 62, 2271, 1975. 148. Alexander, M.H. and DePristo, A.E., Symmetry considerations in the quantum treatment of collisions between two diatomic molecules, J. Chem. Phys., 66, 2166, 1977. 149. Ferlaino, F., Knoop, S., Mark, M., Berninger, M., Schöbel, H., Nägerl, H.-C., and Grimm, R., Collisions between tunable halo dimers: exploring an elementary four-body process with identical bosons, Phys. Rev. Lett., 101, 023201, 2008.
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Effects of External 4 Electromagnetic Fields on Collisions of Molecules at Low Temperatures Timur V. Tscherbul and Roman V. Krems CONTENTS 4.1 4.2
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collisions in Magnetic Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Zeeman Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Tunable Shape Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Tunable Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Collisions in Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Stark Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Scattering of Molecular Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Electric-Field-Induced Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Collisions in Superimposed Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Effects of Electric Fields on Magnetic Feshbach Resonances . . . . . 4.4.2 Collisions near Tunable Avoided Crossings . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Effects of Field Orientations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Differential Scattering in Electromagnetic Fields . . . . . . . . . . . . . . . . . . 4.5 Collisions in Restricted Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Cold Controlled Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
125 126 127 131 134 134 134 136 137 138 140 141 144 150 156 162 164 164
INTRODUCTION
As discussed in Chapters 9 through 17 of this book, diatomic molecules containing unpaired electrons have recently become of increased interest to researchers of lowtemperature gases, condensed-matter physics, precision spectroscopy, and quantum 125 © 2009 by Taylor and Francis Group, LLC
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computation. The interaction of open-shell molecules with magnetic fields allows for magnetic trapping and thermal isolation of molecular ensembles at cold and ultracold temperatures (Chapter 13), novel methods to study fundamental symmetries of nature (Chapters 15 and 16) and mechanisms to control molecular collisions externally. These and other applications of open-shell molecules stimulated the design of laboratory superconducting magnets [1], which can now generate magnetic fields of up to 6 T. The experimental work on Stark deceleration of molecular beams described in Chapter 14 stimulated the development of techniques for the generation of tunable dc electric fields up to 200 kV/cm. As described in Chapter 12, dc electric fields can be used in combination with microwave laser radiation to engineer long-range interaction potentials between ultracold polar molecules confined by an optical lattice. Electric fields can also be used to confine ultracold molecules in electrostatic traps [2] and to generate cold molecules with tunable velocities [3]. Interactions with external electromagnetic fields thus play an important role in most studies of cold and ultracold molecules, both theoretical and experimental. The perturbations exerted by external electromagnetic fields on molecular energy levels are often larger than the kinetic energy of molecules at temperatures below 1 K. Collisions of molecules in a cold gas may therefore be significantly affected by the presence of external fields. The purpose of this chapter is to discuss the effects of external fields on dynamics of molecular collisions at cold and ultracold temperatures and outline the prospects for new discoveries in the research of molecule–field interactions at low temperatures. The experimental work on collision dynamics of low-temperature molecules in external fields may lead to the development of the research field of cold controlled chemistry [4] and we will particularly focus the discussion on mechanisms for external field control of intermolecular interactions. Most of the results presented are based on rigorous quantum-mechanical calculations. The quantum theory of molecular collisions in the presence of external fields is described in Chapter 1.
4.2
COLLISIONS IN MAGNETIC TRAPS
The development of the experimental techniques for confining cold molecules in magnetic [5], electrostatic [2], or laser-field traps [6] has opened up exciting possibilities for new research in molecular physics. For example, confining molecules in external field traps allows for spectroscopy measurements with enhanced interrogation time [7]. This has been used for measuring radiative lifetimes of molecular energy levels with unprecedented precision [8]. External field traps provide thermal isolation, which is necessary for cooling molecules to ultracold temperatures. Trapping fields modify the symmetry of intermolecular interactions and molecules in traps may exhibit interesting dynamics, not observable with thermal gases. Chapter 13 provides a detailed description of magnetic trapping experiments. Magnetic traps have been designed to confine and thermally isolate large ensembles of paramagnetic molecules (∼1013 particles) in a wide range of temperatures (∼1 μK to 700 mK). This makes magnetic traps a particularly useful instrument for experimental studies of molecular collisions at cold and ultracold temperatures. The versatility of magnetic traps is, however, limited by collision-induced Zeeman relaxation. © 2009 by Taylor and Francis Group, LLC
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4.2.1
127
ZEEMAN RELAXATION
The rotational energy levels of open-shell molecules placed in a magnetic field split into manifolds of Zeeman sublevels. Figure 4.1 shows the Zeeman energy sublevels of a CaD molecule in the ground electronic state 2 Σ as functions of an applied magnetic field. CaH was the first molecule thermally isolated in a magnetic trap [5]. Magnetic trapping selects molecules in the state with the lowest-energy rotational angular momentum N = 0 and the electron spin projection aligned along the magnetic field axis. This state is shown by the dashed line in Figure 4.1. Zeeman energy levels with the positive derivative with respect to the magnetic field are often called “lowfield-seeking (LFS) states,” and Zeeman levels with the negative derivative are referred to as “high-field-seeking (HFS) states.” LFS states are collisionally unstable and may decay to the lowest-energy HFS state. This process leads to trap loss and heating of molecular samples in the buffer-gas-cooling experiments described in Chapter 13. Collision-induced Zeeman relaxation of magnetically trapped molecules has therefore been studied by several authors, both theoretically [9–16] and experimentally [18,19]. Translational energy thermalization of molecules in buffer-gas-cooling experiments is mediated by elastic collisions with helium atoms. In order to estimate the efficiency of buffer-gas-cooling experiments, it is necessary to understand the mechanism of Zeeman relaxation and evaluate the cross-sections for elastic scattering and 20
15
Energy (cm–1)
N=2
10
5
N=1
0 N=0 0
1
2
3 4 5 Magnetic field (T)
6
7
FIGURE 4.1 Zeeman energy levels of the CaD(2 Σ) molecule. The dashed line indicates the energy of the molecule confined in a magnetic trap.
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Zeeman relaxation in collisions of molecules with He atoms in the cold temperature regime (T ∼ 0.5 to 1 K). Figure 4.2 presents the ratio of cross-sections for elastic scattering and Zeeman relaxation in collisions of NH molecules in the ground rotational state with 3 He atoms as a function of the collision energy and the magnetic field strength. NH molecules are currently being studied in the buffer-gas-cooling experiments [19]. The ground electronic state of NH is 3 Σ and the structure of molecular energy levels for NH in a magnetic field is slightly more complicated than the illustration in Figure 4.1. The ground rotational state of the NH molecule is characterized by the total angular momentum value J = 1. The total angular momentum includes the electron spin and the rotational angular momentum of the molecule. The J = 1 state gives rise to three Zeeman sublevels corresponding to the projections MJ = 0, −1 and +1 of J on the magnetic field axis. The MJ = +1 sublevel is the LFS state and the buffer-gas-cooling experiments confine molecules in this state in a magnetic trap. Zeeman relaxation involves transitions from the MJ = +1 state to the MJ = 0 and MJ = −1 states. The results presented in Figure 4.2 represent the sum of the two transitions. Figure 4.2 illustrates that the probability of Zeman relaxation in NH–He collisions is very small and extremely sensitive to the magnitude of an external magnetic field at low collision energies. The efficiency of Zeeman relaxation in collisions of molecules in the Σ electronic states is determined by the magnitude of the spin–rotation and spin–spin interactions giving rise to the molecular fine-structure [13]. The results of Figure 4.2 indicate that the fine-structure interactions in the NH molecule are weak.
1010 B=0 Elastic-to-inelastic ratio
B = 0.01 T 108
B=1T
106 B=2T
B=3T 104 10–4
10–3
10–2
10–1
100
101
Collision energy (cm–1)
FIGURE 4.2 Ratio of cross-sections for elastic scattering and Zeeman relaxation in collisions of rotationally ground-state NH(3 Σ) molecules with 3 He atoms (1 K = 0.695 cm−1 ). The curves are labeled by the magnetic field magnitude. (Adapted from Cybulski, H. et al., J. Chem. Phys., 122, 094307, 2005. With permission.)
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It should be noted that the effects of the spin–rotation and spin–spin interactions in most other stable radicals in the Σ electronic states are much more significant and Zeeman relaxation in collisions of other diatomic molecules with He atoms should be more efficient. The He atom is a weak perturber and the interaction of NH molecules with He atoms is weak. Angular momentum transfer in molecular collisions is induced by the anisotropy of the interaction potential between the collision partners. Therefore, Zeeman relaxation in collisions of molecules with other atoms and in molecule– molecule collisions should be expected to be more efficient. Figure 4.2 also demonstrates that the relative probability of Zeeman relaxation is significantly enhanced in a narrow interval of collision energies around 1 K, which indicates the presence of a scattering resonance. The influence of scattering resonances on inelastic collisions has been discussed in Chapter 3 and will be further examined in the following sections. Volpi and Bohn were the first to discover that Zeeman relaxation in collisions of ultracold molecules is extremely sensitive to the magnitude of an external magnetic field [12]. When molecules react, the rotational motion of the collision complex gives rise to a centrifugal force that may suppress collisions at low temperatures. As explained in Chapter 1, the total scattering wavefunction of the collision complex can be decomposed into contributions from different angular momenta of the rotational motion called partial waves. Collisions at ultracold temperatures (<1 mK) are entirely determined by a single partial-wave scattering. For collisions of bosons or distinguishable fermions, this is a state of zero orbital angular momentum. Due to conservation of the total angular momentum projection on the external field axis, collision-induced Zeeman relaxation must be accompanied by changes of the orbital angular momentum for the rotation of the collision complex. If the initial collision channel is characterized by zero orbital angular momentum, the rotational motion of the collision complex must be accelerated as a result of Zeeman relaxation. This gives rise to long-range centrifugal barriers in the outgoing collision channels. The magnitude of the external field determines the splitting between the Zeeman levels and consequently the amount of the kinetic energy released as a result of the inelastic transitions. If the kinetic energy is larger than the maximum of the long-range centrifugal barriers, the inelastic transitions are unconstrained and efficient. At low external fields, however, the centrifugal barriers in the outgoing channels suppress inelastic scattering. Figure 4.3 illustrates this mechanism and Figure 4.4 presents the zero temperature rate constant for Zeeman relaxation in collisions of rotationally ground-state NH(3 Σ) molecules with 3 He atoms as a function of the magnetic field. Ultracold ensembles of atoms are usually created by evaporative cooling in a magnetic trap. The evaporative cooling relies on repetitive rethermalization of the translational energy in collisions of trapped atoms with each other. The evaporative cooling of molecules to ultracold temperatures will rely on translational energy thermalization in elastic molecule–molecule collisions. Zeeman relaxation in molecule–molecule collisions may be significantly more efficient than Zeeman transitions in atom–molecule collisions discussed above. Accurate quantum calculations of cross-sections for molecule–molecule collisions in the presence of a magnetic field are, however, very complicated and computationally demanding. At present, there are no reliable data on the rates of Zeeman relaxation in molecule–molecule collisions at magnetic fields below 1 T and collision energies below 1 K. © 2009 by Taylor and Francis Group, LLC
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Low field
High field
FIGURE 4.3 External field suppression of the role of centrifugal barriers in outgoing reaction channels. Incoming channels are shown by full curves, outgoing channels by dashed curves. An applied field separates the energies of the initial and final channels and increases the kinetic energy in the outgoing channels. (Adapted from Krems, R.V., Int. Rev. Phys. Chem., 24, 99, 2005. With permission.)
The mechanism of Zeeman relaxation in collisions of molecules in electronic states with nonzero electronic orbital angular momenta is different from that in collisions of Σ-state molecules. The response of non-Σ-state molecules to a magnetic field is determined by both the electron spin and the orbital angular momentum of the electrons in the open electronic shell. The orbital motion of the electrons induces electronic anisotropy, which gives rise to multiple adiabatic interaction potentials between the collision partners [20]. Consider, for example, the collision system of a hydrogen atom in an excited P state and a structureless atom, such as He. The interaction between the atoms can be described by an effective potential as a function of the interatomic distance and an angle between the direction of the electronic P-orbital and the interatomic separation line. The angular dependence of this potential is the electronic anisotropy. An alternative description of the interatomic interaction can be © 2009 by Taylor and Francis Group, LLC
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Rate constant (cm3/sec)
10–17
10–18
10–19
10–20
10–21 0
0.2
0.4 0.6 Magnetic field (T)
0.8
1
FIGURE 4.4 Zero-temperature rate constant for Zeeman relaxation in collisions of rotationally ground-state NH(3 Σ) molecules in the maximally stretched spin level with 3 He atoms. Such field dependence is typical for Zeeman or Stark relaxation in ultracold collisions of atoms and molecules without hyperfine interaction. The variation of the relaxation rates with the field is stronger and extends to larger field values for systems with smaller reduced mass. (Adapted from Krems, R.V., Int. Rev. Phys. Chem., 24, 99, 2005. With permission.)
based on the Born–Oppenheimer potentials of the two-atom system. The interaction between a P-state atom and a closed-shell atom gives rise to two adiabatic potentials of Σ and Π symmetries. The energy separation between the interaction potentials is determined by the electronic anisotropy. The energy separation between the adiabatic electronic potentials of the collision systems involving molecules in non-Σ states is usually much larger than the kinetic energy of cold collisions. The electronic anisotropy couples directly the Zeeman sublevels of the open-shell molecules and Zeeman relaxation in collisions of such molecules is therefore very efficient [21]. It is unlikely that molecules in other than Σ electronic states can be evaporatively cooled in a magnetic trap, unless the cooling process can be carried out in extremely shallow magnetic traps and the Zeeman relaxation is suppressed due to the centrifugal barrier effect discussed above.
4.2.2 TUNABLE SHAPE RESONANCES Figure 4.5 presents the cross-section for the |MS = 1 → |MS = −1 transition in collisions of NH molecules in the ground rotational state with He atoms as a function of the collision energy and the magnetic field strength. The cross-section for elastic collisions of NH with He at zero magnetic field shows a pronounced peak at the collision energy of about 1 K [15]. This enhancement is due to a shape resonance (cf., Chapter 1). The cross-section for the Zeeman relaxation is similarly enhanced by the resonance. Figure 4.5 shows that the cross-section for the |MS = 1 → |MS = −1 transition exhibits two peaks. The location of the higher-energy peak is independent © 2009 by Taylor and Francis Group, LLC
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–6
0.1 5 0.2 Ma g ne 5 0 tic fiel .35 d( T) 0.45
–6 –8 –10 –12
1.6
1.4 1 . Colli 2 1 0.8 sion energ 0.6 0.4 y (K) 0.2
–8 –9 –10 –11 –12
0.0 5
Cross-section
–7
5 6
Ratio kND/kNH
4 5 3 4 2 3 0.4
0.8
1.2
0.4
0.8
1.2
Temperature (K)
FIGURE 4.5 The logarithm of the cross-section for the |MS = 1 → |MS = −1 transition in collisions of NH molecules with 3 He atoms as a function of the magnetic field and collision energy. (From Campbell, W.C. et al.., Phys. Rev. Lett., 102, 013003, 2009. With permission.)
of the magnetic field, whereas the lower-energy peak shifts toward lower collision energies as the magnetic field increases. This demonstrates that the first peak is due to trapping behind the centrifugal barrier in the incoming collision channel corresponding to the MS = +1 state. The second peak occurs due to the centrifugal barrier in the outgoing collision channel corresponding to the MS = −1 state. As the separation between the Zeeman states increases with the magnetic field magnitude, the effective kinetic energy in the outgoing collision channel increases and the resonance position in the MS = −1 state approaches the collision threshold of the MS = +1 state. An indirect effect of these shape resonances on the Zeeman relaxation cross-sections in magnetically trapped NH molecules has been recently observed in the experiments of Campbell and colleagues [19]. © 2009 by Taylor and Francis Group, LLC
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The number of shape resonances in atom–molecule scattering typically increases with the mass of the colliding partners and the strength of the interaction between the collision partners. If the molecule has a large dipole moment and a small rotational constant, the number and positions of shape resonances may also be sensitive to an external electric field. Figure 4.6 presents the cross-sections for inelastic Zeeman and hyperfine relaxation in collisions of YbF(2 Σ) molecules with He atoms at a collision energy of 0.1 K. As the figure shows, the scattering resonances enhance the probability of inelastic scattering by several orders of magnitude. As the temperature of a molecular ensemble is reduced, the Maxwell–Boltzmann distribution of molecular speeds narrows down. Energy-resolved scattering resonances may therefore have a dramatic effect on elastic and inelastic collision rates in a cold gas [14]. For example, a single shape resonance in collisions of CaH molecules with He atoms at the collision energy
Cross-section (Å2)
(a) E = 0 kV/cm
10–2
10–4
Cross-section (Å2)
(b) 102 E = 10 kV/cm 100
10–2
10–4
Cross-section (Å2)
(c)
100
E = 20 kV/cm
10–2
10–4 0
0.2
0.4 0.6 Magnetic field (T)
0.8
1
FIGURE 4.6 Magnetic field dependence of the Zeeman (full line) and hyperfine relaxation (dashed line) cross-sections at zero electric field: (a) E = 0 kV/cm; (b) E = 10 kV/cm; (c) E = 20 kV/cm. The symbols in the upper panel are the results of the calculations without the spin–rotation interaction. The collision energy is 0.1 K. (Adapted from Tscherbul, T.V. et al., Phys. Rev. A, 75, 033416, 2007. With permission.)
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0.02 cm−1 enhances the rate for collisionally induced spin relaxation at 0.4 K by three orders of magnitude [14]. The results presented in this section therefore suggest that collision rates in cold gases may be tuned by an external field inducing scattering resonances. The effects of electric fields as well as the effects of combined electric and magnetic fields on cold and ultracold collisions of molecules will be discussed in more detail in the following sections.
4.2.3 TUNABLE FESHBACH RESONANCES Gonzalez-Martinez and Hutson presented a detailed study of scattering Feshbach resonances in collisions of NH(3 Σ) molecules with He atoms in a magnetic field [16]. In order to analyze the energy and magnetic field dependence of cross-sections for NH–He collisions in the vicinity of Feshbach resonances, they developed a package of programs to compute the energies and wavefunctions of the bound states of the collision complex. Zero-energy Feshbach resonances occur when the energy of the threshold becomes degenerate with the energy of a bound state of the collision complex. Feshbach resonances in collisions of molecules may be tuned by varying the magnetic field as in the experiments with ultracold atoms described in Chapters 9 through 11. The properties of magnetic Feshbach resonances in collisions involving molecules may, however, be different from those in collisions of alkali metal atoms. For example, Gonzalez-Martinez and Hutson found that Feshbach resonances in collisions of molecules in metastable excited states may be significantly suppressed due to efficient inelastic processes. The resonance peaks in the elastic cross-sections for collisions of molecules in excited states were found to be much smaller than in collisions of atoms or molecules in the absolute ground state (see Chapter 1 and Ref. [17] for more details).
4.3 4.3.1
COLLISIONS IN ELECTRIC FIELDS STARK RELAXATION
The rotational energy levels of a polar molecule placed in an electric field split into manifolds of Stark sublevels. Figure 4.7, for example, shows the effect of electric fields on the rotational structure of the OH molecule in the ground state. The Stark effect can be used to confine molecules in the LFS states in an electrostatic trap. In order to understand the stability of molecular ensembles in electrostatic traps, Bohn and colleagues carried out extensive quantum calculations of collision rates for ultracold molecules in electric fields [22–26]. In general, the rate constants for Stark relaxation at ultracold collision energies were found to be quite large, which precludes the possibility of evaporative cooling of molecules in electrostatic traps. Both Zeeman and Stark relaxation in collisions of molecules are determined by the anisotropy of intermolecular interactions. The mechanisms of Zeeman and Stark transitions in collisions of Σ-state molecules are, however, different. For example, Zeeman relaxation in collisions of CaH molecules with He atoms is induced by an interplay of the atom–molecule interaction anisotropy coupling different rotational states of the molecules and the spin–rotation interaction coupling the rotational © 2009 by Taylor and Francis Group, LLC
Effects of External Electromagnetic Fields on Collisions of Molecules (a)
0.090 f
0.088 Energy (K)
135
0.086 0.084 0.082 0.080 200
0
400
600
800
1000
800
1000
Electric field (V/cm) (b) 0.004
Energy (K)
0.002 0.000 –0.002
e
–0.004 –0.006 0
200
400
600
Electric field (V/cm)
FIGURE 4.7 The Stark effect in ground-state OH molecules. (a) The states that have odd parity in zero electric field; (b) those of even parity. In zero field the f states and the e states are separated by the Λ-doublet energy. (Adapted from Ticknor, C., and Bohn, J.L., Phys. Rev. A, 71, 022709, 2005. With permission.)
motion of the molecules with the electron spin states. The Zeeman transitions in 2 Σ molecules cannot occur in the absence of the spin–rotation interaction, even if the atom–molecule interaction anisotropy is very large. In contrast, the Stark energy levels of a polar molecule are directly coupled by the anisotropy of the interaction between the molecule and its collision partner. The anisotropy of atom–molecule and molecule–molecule interactions is usually quite large, which leads to significant rates for Stark relaxation in cold collisions of molecules. As in the case of Zeeman relaxation, Stark transitions in s-wave collisions of ultracold molecules must be accompanied by angular momentum transfer giving rise to long-range angular momentum barriers in the outgoing scattering channels. The angular momentum effect illustrated by Figure 4.3 will therefore similarly suppress both Stark and Zeeman transitions. The suppression should be more effective for light molecules and systems with weak long-range interactions characterized by larger centrifugal barriers in the states of nonzero partial waves. Whether the suppression of © 2009 by Taylor and Francis Group, LLC
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Stark relaxation in collisions of ultracold polar molecules in shallow electrostatic traps will be significant enough to allow for evaporative cooling of molecular ensembles remains to be determined.
4.3.2
SCATTERING OF MOLECULAR DIPOLES
Ultracold collisions of molecules are predominantly determined by the long-range part of the interaction potential between the collision partners. As discussed in Chapter 2, the long-range interaction between polar molecules placed in an electric field is dominated by the dipole–dipole forces, which can be tuned by varying the external electric fields. Figure 4.8 illustrates a characteristic dependence of rate (a)
K (cm3/sec)
10–9
Kel
10–10 Kinel 10–11
10–12 101
102
103 104 Electric field (V/cm)
105
(b) 10–8
K (cm3/sec)
E0 10–9
10–10
10–11 101
Kel
Kinel 102
103 104 Electric field (V/cm)
105
FIGURE 4.8 Rate constants vs. electric field for OH–OH collisions with molecules initially in their |F = 2MF = 2, = − state (the gray line of Figure 4.7). Shown are the collision energies 100 mK (a) and 1 mK (b). Solid lines denote elastic-scattering rates, while dashed lines denote rates for inelastic collisions, in which one or both molecules change their internal state. These rate constants exhibit characteristic oscillations in field when the field exceeds a critical field of about 1000 V/cm. (Adapted from Avdeenkov, A.V. and Bohn, J.L., Phys. Rev. A, 66, 052718, 2002. With permission.)
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constants for elastic scattering and inelastic collisions on the electric field magnitude. The oscillations arise due to bound states of the two-molecule complex supported by the long-range dipole–dipole interactions. At the same time, the calculations of Avdeenkov and Bohn [22–24] showed that the dipole–dipole interaction may lead to long-range minima and barriers in the molecule–molecule interaction potential. The shapes of these long-range minima and barriers depend on the magnitude of an external electric field that polarizes the interacting molecules. The dipole–dipole interaction can thus be used to suppress molecular collisions at ultracold temperatures. Chapter 12 presents a detailed discussion of how the dipole–dipole interaction can be used to engineer repulsive interactions between ultracold polar molecules in external field traps. The dipole–dipole interaction between molecules is determined by the properties of the molecules, such as the rotational constant and the permanent dipole moment, as well as the strength and direction of the external electric field. Ticknor has recently shown that collision cross-sections of ultracold molecules in the absolute ground state have a universal dependence on the dipole–dipole interaction in the limit of high electric fields [27]. In particular, he found that at high electric fields—the regime of strong dipolar scattering—the elastic scattering cross-sections are uniquely determined by the mass of the colliding particles, the induced dipole moments and the collision energy, and are independent of the details of the interaction potentials at short range. The analysis is based on rescaling the Schrödinger equation written in a multichannel form as follows:
Cll d2 l(l + 1) − + ξ ψl = − ψl , (4.1) 2 2 dy y y3 l
where l is the index of the partial wave expansion introduced in Chapter 1, y = R/D, R is the intermolecular distance, D is the length scale and Cll denotes the matrix elements of the dipole–dipole interaction coupling different partial wave states of the colliding molecules, and the short-range interaction potential has been neglected. Equation 4.1 depends on a single parameter ξ. The regime of strong dipolar scattering corresponds to large values of ξ. Figure 4.9 shows that the T -matrix elements for collisions of different molecules converge to the same line as a function of ξ at values of ξ > 100. This suggests that the collision cross-sections of molecular dipoles in the presence of high electric fields can be evaluated for any collision system based on the results of a quantum calculation for a single model system.
4.3.3
ELECTRIC-FIELD-INDUCED RESONANCES
Electric fields can be used to tune shape and Feshbach scattering resonances through a variety of mechanisms. The Stark effect may lead to tunable shape resonances due to trapping behind the centrifugal barriers in the outgoing Stark states. Such resonances would be analogous to the shape resonance in the outgoing Zeeman state illustrated by Figure 4.5. As mentioned above, however, the couplings between the Stark states are usually much larger than the matrix elements coupling different Zeeman states in Σ-state molecules. The resonances in the lower-energy Stark states may therefore © 2009 by Taylor and Francis Group, LLC
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NaCs
102
T
101
x1/2
CsRb
4
100
1010
Rydberg
RbK 10–1
109 1019
10–2 –2 10
10–1
100
101
102 x
103
1020
104
1021
105
106
FIGURE 4.9 Collision T -matrix elements for scattering of molecules in the absolute ground state as functions of ξ. (Adapted from Ticknor, C., Phys. Rev. Lett., 100, 133202, 2008. With permission.)
have a significantly different effect on the dynamics of molecular collisions than the resonances in the lower-energy Zeeman states. For example, the field-dependent resonance of Figure 4.5 does not modify the cross-section for elastic scattering of molecules in the LFS Zeeman state. Similar resonances in the HFS Stark states may influence the scattering wavefunction in the elastic channel of LFS molecules. Electric fields shift bound energy levels of collision complexes involving polar molecules with respect to the collision threshold and may lead to diagrams of molecular energy levels similar to those discussed in Chapter 1. The Stark effect may therefore give rise to zeroenergy Feshbach resonances, similar to magnetic-field-induced Feshbach resonances in collisions of paramagnetic molecules. Unlike magnetic fields, electric fields couple different partial waves of the collision complex and states of different parity. These couplings may have a dramatic effect on differential scattering of polar molecules (see Section 4.4 below) and modify shape scattering resonances. Figure 4.10 demonstrates the effect of electric fields on the probabilities of the chemical reaction of LiF molecules with H atoms and the crosssections for vibrational relaxation of LiF molecules in the first vibrationally excited state in collisions with H atoms at low collision energies. The presence of electric fields apparently results in couplings that shift the position of the shape resonance, which suppresses the cross-sections for the chemical reaction and vibrationally inelastic processes at collision energies near 1 K [28].
4.4
COLLISIONS IN SUPERIMPOSED ELECTRIC AND MAGNETIC FIELDS
Superimposed electric and magnetic fields can be used to control microscopic interactions between atoms and molecules in a wider range of field parameters than © 2009 by Taylor and Francis Group, LLC
Effects of External Electromagnetic Fields on Collisions of Molecules Chemical reaction
(a)
Cross-section (Å2)
139
10–2
10–4
10–6 10–3 (b)
10–2
10–1 Vibrational relaxation
102
No electric field E = 200 kV/cm E = 50 kV/cm E = 150 kV/cm
101 Cross-section (Å2)
100
100 10–1 10–2 10–3 10–3
10–2
10–1 Collision energy (cm–1)
100
FIGURE 4.10 (a) Total cross-sections for the chemical reaction of LiF(v = 1, N = 0) molecules with H atoms as functions of the collision energy at zero electric field (circles) and electric fields of 100 kV/cm (triangles), 150 kV/cm (squares), and 200 kV/cm (diamonds). (b) Cross-sections for nonreactive vibrational relaxation of LiF(v = 1, N = 0) molecules in collisions with H atoms. (Adapted from Tscherbul, T.V. and Krems, R.V., J. Chem. Phys., 129, 034112, 2008. With permission.)
may be possible with a single field. For example, electric fields may shift the positions of magnetic Feshbach resonances or broaden the resonances. The following sections discuss the effects of combined electric and magnetic fields on collision dynamics of atoms and molecules at cold and ultracold temperatures. In particular, Section 4.4.1 describes how electric fields may modify magnetic Feshbach resonances or induce new resonances in ultracold collisions by coupling scattering states with different orbital angular momenta. Experimental studies of cold molecules in external field traps may be particularly interesting as a novel approach to explore the effects of combined electric and magnetic fields on molecular collisions. When molecules are trapped, their magnetic or electric dipole moments are aligned by the confining field © 2009 by Taylor and Francis Group, LLC
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(cf. Chapter 13). The alignment of molecular dipole moments restricts the symmetry of the interaction potential between the collision partners. Superimposed external fields can then be used to induce reorientation of trapped molecules, leading to nonadiabatic transitions between different electronic states. This may modify the mechanisms of inelastic scattering and chemical reactions at low temperatures and result in enhancement or suppression of the collision rates. Section 4.4.2 demonstrates that a combination of electric and magnetic fields can be used to create magnetically aligned molecules with the orientation of the magnetic dipole moments that is extremely sensitive to collisional perturbations. Section 4.4.3 presents a few examples of molecular collision processes that can be sensitive not only to the magnitude but also to the relative orientation of superimposed electric and magnetic fields. Section 4.4.4 describes the effects of electromagnetic fields on angle-resolved differential scattering of cold polar molecules.
4.4.1
EFFECTS OF ELECTRIC FIELDS ON MAGNETIC FESHBACH RESONANCES
The duration of an ultracold collision is very long. The scattering dynamics of ultracold atoms and molecules is therefore often sensitive to weak interactions with external fields. The results of Refs. [29] and [30], for example, show that collisions of ultracold atoms can be controlled by dc electric fields. When two different atoms collide, they form a heteronuclear collision complex with an instantaneous dipole moment that can interact with external electric fields. The dipole moment function of the collision complex is typically peaked around the equilibrium distance of the diatomic molecule in the vibrationally ground state and quickly decreases as the atoms separate. Only a small part of the scattering wavefunction samples the interatomic distances, where the dipole moment function is significant and the oscillatory structure of the scattering wavefunction diminishes the interaction of the collision complex with external electric fields. Collisions of atoms are therefore usually insensitive to dc electric fields of moderate strength (<200 kV/cm). At the same time, the interaction with an electric field couples states of different orbital angular momenta. The zero angular momentum s-wave motion of ultracold atoms is coupled to an excited p-wave scattering state, in which the colliding atoms rotate about each other with the angular momentum l = 1 au. The probability density of the p-wave scattering wavefunction at small interatomic separations is very small and the coupling between the s- and p-collision states is suppressed. However, the interaction between the s-wave and p-wave scattering states is dramatically enhanced near scattering resonances. Figure 4.11 illustrates this phenomenon with plots of the scattering cross-sections as functions of the reduced mass of the two-atom collision system. Increasing the reduced mass increases the number of bound states supported by the interaction potential and a resonance occurs when a new bound state appears at the collision threshold. The peaks in the elastic s-wave cross-section correspond to zero-energy s-wave resonances; the peaks in the elastic p-wave cross-section correspond to zeroenergy p-wave resonances. The s → p cross-section is enhanced at both the s-wave and p-wave resonances. This indicates that the s → p coupling and the interaction of the collision complex with electric fields can be enhanced either by an s-wave or by a p-wave scattering resonance at zero-energy. The resonance enhancement of the © 2009 by Taylor and Francis Group, LLC
Effects of External Electromagnetic Fields on Collisions of Molecules
141
1010
Cross-section (au)
105 100 10–5 10–10 10–15 5.8
6
6.2 6.4 6.6 Reduced mass (cu)
6.8
FIGURE 4.11 Variation of the cross-sections for elastic s-wave scattering (upper full curve), elastic p-wave scattering (lower full curve) at zero electric field, and the s → p transition (dashed curve) at an electric field magnitude of 100 kV/cm as a function of the reduced mass at the collision energy 10−11 cm−1 . Increasing the reduced mass brings more bound states in. The peaks in the elastic s-wave cross-section correspond to zero-energy s-wave resonances; the peaks in the elastic p-wave cross-section correspond to zero-energy p-wave resonances. The s → p cross-section is enhanced at both the s-wave and p-wave resonances.
s → p coupling changes the cross-section for the s → p transition by 4 to 6 orders of magnitude. Figure 4.12 illustrates the dependence of the cross-sections for elastic scattering in ultracold collisions of two atoms near a p-wave or s-wave resonance of Figure 4.11 on the electric field magnitude. Interactions between molecules are generally characterized by significant dipole moment functions. For example, Figure 4.13 demonstrates that the dipole moment function of the RbCs–RbCs collision complex can be as large as 6 D, which is comparable with the magnitude of the dipole moment function for the LiCs molecule used for the calculations presented in Figures 4.11 and 4.12. The mechanism of electric field control described above may therefore be used to manipulate ultracold collisions and chemical reactions of molecules.
4.4.2
COLLISIONS NEAR TUNABLE AVOIDED CROSSINGS
The analysis of Ref. [13] and the quantum calculations of Ref. [14] demonstrate that the Zeeman transitions in collisions of 2 Σ molecules in the ground rotational state are induced by the spin–rotation interaction in the rotationally excited states and a coupling between the rotationally ground and excited states of the molecules. As the magnetic field increases, the energy of the lowest LFS state of a 2 Σ molecule approaches the energy of HFS states corresponding to the rotationally excited states (see Figure 4.1). For example, the energy of the spin-up Zeeman state of N = 0 of CaD becomes degenerate with the energy of the spin-down Zeeman state of N = 1 at the magnetic field 4.65 T. The LFS state of N = 0 and the HFS states of N = 1 have © 2009 by Taylor and Francis Group, LLC
142
(a)
s-wave elastic scattering cross-section (au)
Cold Molecules: Theory, Experiment, Applications 107
Variation near p-wave resonance
106
105
(b)
s-wave elastic scattering cross-section (au)
20
40
60
80
100
80
100
1013
Variation near s-wave resonance
1012
1011
1010
109 20
40
60 Electric field (kV/cm)
FIGURE 4.12 Variation of elastic s-wave scattering cross-section at zero collision energy with the electric field in the presence of a p-wave resonance (a) and s-wave resonance (b) near threshold calculated using a single-channel model calculation with the resonances induced as shown in Figure 4.11.
different symmetries. Therefore, the variation of the magnetic field in the circled area of Figure 4.1 does not affect collision dynamics of trapped CaD molecules [31]. An external electric field may, however, couple states with different symmetries. In the presence of an electric field, the crossing in the circled area of Figure 4.2 becomes an avoided crossing and the collision dynamics of CaD in the LFS Zeeman state becomes very sensitive to both the magnetic and electric fields. To illustrate this effect, Figure 4.14 shows the cross-section for magnetic spin reorientation in collisions of CaD molecules with He atoms computed as a function of electric and magnetic fields at a collision energy of 0.5 K. In the absence of electric fields or at magnetic fields far detuned from the circled area of Figure 4.1, collisions with He cannot significantly change the orientation of the electron spin of the © 2009 by Taylor and Francis Group, LLC
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Effects of External Electromagnetic Fields on Collisions of Molecules 0
Dipole moment (D)
–1 –2 –3 –4 –5 –6 –7
0
10
20 30 RbCs–RbCs distance (au)
40
50
FIGURE 4.13 Dipole moment functions of the RbCs–RbCs molecular complex calculated using a MRCISD quantum chemistry method with the Stuttgart 1997 effective core potentials. Full line, z-component; dashed line, y-component. The results are for the T -shaped configuration of the complex with Cs atom pointing toward the molecule. (The calculation is a courtesy of Dr. Jacek Klos of the University of Maryland.)
molecule. At certain combinations of electric and magnetic fields, however, the spinup state of CaD becomes very unstable and even weak collisional perturbations result in spin reorientation. The efficiency of collisional spin reorientation of 2 Σ molecules in the rotationally ground state may thus be controlled with superimposed electric and magnetic fields. This mechanism can be used to induce transitions between different adiabatic states in reactions of open-shell molecules with open-shell atoms [32] and manipulate the efficiency of spin-forbidden chemical reactions in a magnetic trap. Consider, for example, a chemical reaction between a 2 Σ diatomic molecule in the rotationally ground state (e.g., CaH) and an atom with one unpaired electron (e.g., Na) in a 2 S electronic state in a magnetic trap. The interaction between a 2 Σ molecule and a 2 S atom gives rise to two electronic states corresponding to the total spin values of the reactive complex S = 1 and S = 0. The configuration of the trapping field ensures that the magnetic moments of both the atom and the molecule are coaligned [5]. Therefore, if the atom and the molecule are both confined in a magnetic trap, they are initially in the state with the total spin S = 1. The interactions between 2 S atoms and 2 Σ molecules in the triplet spin state are typically characterized by strongly repulsive exchange forces, leading to significant reaction barriers. The interactions in the singlet spin state S = 0 are usually strongly attractive, leading to short-range minima and insertion chemical reactions. Figure 4.15 presents a schematic illustration of these interactions. In the absence of nonadiabatic interactions between the S = 0 and S = 1 states, chemical reactions of atoms and molecules with aligned magnetic moments should therefore be much slower than reactions in the singlet spin state. The nonadiabatic coupling between the different electronic states of the A(2 S)–BC(2 Σ) © 2009 by Taylor and Francis Group, LLC
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Lg(cross-section):
–2 –1
0
1
2
4.85
Magnetic field (T)
4.8
4.75
4.7
4.65
4.6
0
10
20 30 Electric field (kV/cm)
40
FIGURE 4.14 Decimal logarithm of the cross-section for spin relaxation in collisions of CaD(2 Σ) molecules in the rotationally ground state with He atoms as a function of electric and magnetic fields. The fields are parallel. The collision energy is 0.5 K. The cross-section increases exponentially near the avoided crossings. (Adapted from Abrahamsson, E. et al., J. Chem. Phys., 127, 044302, 2007, American Institute of Physics. With permission.)
collision complex may be induced by the spin–rotation interaction responsible for spin reorientation in CaH–He collisions described above [32]. As Figure 4.14 clearly demonstrates, the spin–rotation interaction can be effectively manipulated with a combination of external electric and magnetic fields.
4.4.3
EFFECTS OF FIELD ORIENTATIONS
The energy levels of a polar paramagnetic molecule in combined electric and magnetic fields depend not only on the magnitude of the applied fields but also on their relative orientation. Crossed electric and magnetic fields break the symmetry of the scattering problem and couple different total angular momentum projections, which adds to the complexity of the problem. Even simple systems such as the hydrogen atom may exhibit very complicated behavior when placed in crossed electric and magnetic fields [35]. The effects of crossed fields on the spectrum of a 2 Σ molecule can be described by the following effective Hamiltonian [32–34] H = Be N 2 + γN · S + 2μ0 B · S − E · d, © 2009 by Taylor and Francis Group, LLC
(4.2)
Effects of External Electromagnetic Fields on Collisions of Molecules
145
A + BC B + AC
FIGURE 4.15 Schematic illustration of minimum energy profiles for an A(2 S) + BC(2 Σ) chemical reaction in the singlet-spin (lower curve) and triplet-spin (upper curve) electronic states. Electric fields may induce nonadiabatic transitions between the different spin states and modify the reaction mechanism.
where Be is the rotational constant of the molecule, γ is the spin–rotation interaction constant, N is the rotational angular momentum, S is the electron spin, E is the electric field vector, and d is the permanent dipole moment of the molecule. If the laboratory z-axis is chosen to coincide with the direction of the magnetic field vector B, the interaction with magnetic fields becomes 2μ0 BSz , where Sz is the z-component of the electron spin and μ0 is the Bohr magneton. The last term in Equation 4.2 can be written as −Ed cos θ, where θ is the angle between the electric field vector and molecular axis. The spherical harmonic addition theorem [36] yields −Ed cos θ = −Ed
4π ∗ ˆ ˆ Y (d)Y1q (E), 3 q 1q
(4.3)
where the spherical harmonics Y1q depend on the orientation of the unit vectors dˆ and Eˆ in the laboratory frame (relative to the direction of the magnetic field). In order to obtain the energy levels of the molecule, consider the matrix of the Hamiltonian of Equation 4.2 in the fully uncoupled space-fixed basis [13,36] |NMN |SMS ,
(4.4)
where |NMN are the rotational wavefunctions and |SMS are the spin wavefunctions of the molecule. The total angular momentum projection is given by MJ = MN + MS . The matrix elements of the rotational angular momentum, the spin– rotation interaction, and the magnetic field are presented in Refs. [13] and [36]. The matrix element of the interaction with electric fields (Equation 4.3) can be © 2009 by Taylor and Francis Group, LLC
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written as SMS |NMN | − Ed cos θ|SMS |N MN = −δMS M Ed S
×
4π 3
∗ ˆ ˆ (4.5) NMN |Y1q (d)|N MN Y1q (E). q
Evaluating the integral over three spherical harmonics [36], we can write Equation 4.5 in the form [32] SMS |NMN | − Ed cos θ|SMS |N MN 1/2 4π N = −δMS M Ed(−)MN [(2N + 1)(2N + 1)]1/2 S 0 3 N 1 N × Y1,MN −MN (χ, 0), −MN MN − MN MN
1 0
N 0
(4.6)
where we have assumed that the electric field is oriented at the angle χ with respect to the magnetic field in the xz-plane, that is, Eˆ = (χ, 0). (Note that there are typographical errors in Ref. [32]: (i) the right-hand side of Equation (7) should be multiplied by (4π/3)1/2 ; (ii) the matrix elements of the interaction with √ electric fields √ between different MN in Equations 13 and 14 should be divided by 6 instead of 3; (iii) the basis state |11 21 in Equation 13 should read |1 − 1 21 ). Equation 4.6 gives the most general form of the electric-field-induced interaction in polar molecules. For example, substituting χ = 0, we obtain the important particular case of the interaction with parallel fields [32,34] SMS |NMN | − Ed cos θ|SMS |N MN || = −δMS M δMN MN Ed(−)MN [(2N + 1)(2N + 1)]1/2 S N 1 N N 1 N × . 0 0 0 −MN 0 MN
(4.7)
This expression shows that there is no coupling between the states with different MN in parallel fields. This is similar to the selection rule ΔMN = 0 for the photoexcitation of the molecule with linearly polarized laser light [36]. For the discussion below, it is also instructive to consider the case of perpendicular electric and magnetic fields. Setting χ = π/2 in Equation 4.6 yields SMS |NMN | − Ed cos θ|SMS |N MN ⊥ 1 = ±δMS M δMN ,MN ∓1 Ed(−)MN √ [(2N + 1)(2N + 1)]1/2 S 2 N 1 N N 1 N . × 0 0 0 −MN MN − MN MN
© 2009 by Taylor and Francis Group, LLC
(4.8)
Effects of External Electromagnetic Fields on Collisions of Molecules
147
This equation demonstrates that only the states with ΔMN = ±1 are coupled in perpendicular electric and magnetic fields. The coupling matrix elements given by Equation 4.8 thus have the same selection rules as the interaction of the molecule with a circularly polarized laser light [36]. Abrahamsson, Tscherbul, and Krems [32] computed the energy levels of the CaD(2 Σ) molecule as functions of the angle χ by numerical diagonalization of Hamiltonian 4.2 using Equation 4.6. Their results indicated that the dependence of the Stark shifts on χ is most significant near the avoided crossings discussed in Section 4.4.2. The dependence of molecular energy levels on the electric and magnetic fields near the avoided crossings can be evaluated analytically for both parallel and perpendicular orientations of the fields [32]. Consider first the case of parallel fields. The lowest magnetically trappable state of CaD is |N = 0, MN = 0, MS = 21 . Electric fields couple this state with the rotationally excited state |N = 1, MN = 0, MS = 21 , which is in turn coupled with the spin-down state |N = 1, MN = 1, MS = − 21 via the spin–rotation interaction. The matrix representation of Hamiltonian 4.2 in the basis of the three states is ⎛
|0 0 21
⎜ ⎜ μ0 B/2 ⎜ ⎜ ⎝ 0 √ −Ed/ 3
|1 1 − 21 0 2Be − μ0 B/2 − γ/2 √ γ/ 2
|1 0 21 √ −Ed/ 3 √ γ/ 2 2Be + μ0 B/2
⎞ ⎟ ⎟ |0 0 1 2 ⎟ ⎟ |1 1 − 1 , 2 ⎠ |1 0 21
(4.9)
where we have used a shorthand notation |NMN MS for basis states 4.4. The energy of the spin-up state |00 21 increases and that of the spin-down state |11 − 21 decreases with increasing the magnetic field. At the crossing point defined by B = μ10 (2Be − γ/2), the matrix 4.9 can be diagonalized analytically to yield γ2 E 2 d 2 1/2 1 = μ0 B/2, 2,3 = Be + μ0 B/2 ± Be 1 + 2 + . (4.10) 2Be 3Be2 Equation 4.10 shows that electric fields induce an avoided crossing between the degenerate Zeeman levels lifting their degeneracy by Δ = 1 − 3 = Be [ 1 + γ2 /2Be2 + E 2 d 2 /3Be2 − 1]. If the interaction with electric fields is small compared to the rotational constant of the molecule, the expansion of the square root suggests quadratic dependence of the splitting on the electric field strength Δ ∼ Be (γ2 /2Be2 + d 2 E 2 /3Be2 ). Note that in the absence of electric fields, the crossing is real and occurs at a slightly different value of the magnetic field because the off-diagonal matrix elements of the spin–rotation interaction shift the energy of the rotationally excited levels. The crossing also becomes real if the spin–rotation interaction is omitted; the quasidegenerate states |00 21 and |11 − 21 would then again remain uncoupled. It is thus a combination of the spin–rotation interaction and the electric field that leads to the avoided crossing of the ground and the first excited rotational levels [34]. In the case of perpendicular fields, Equation 4.8 shows that the initial spin-up state |00 21 is coupled with the two rotationally excited states |11 21 and |1 − 1 21 . © 2009 by Taylor and Francis Group, LLC
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The second of these states is coupled by the spin–rotation interaction to the state |10 − 21 . The matrix of the Hamiltonian in the basis of the three states can be written as ⎞ ⎛ |1 0 − 21 |1 −1 21 |0 0 21 ⎟ ⎜ √ ⎟ |0 0 1 ⎜ μ0 B/2 0 −Ed/ 6 2 ⎟ ⎜ (4.11) √ ⎟ |1 0 − 1 . ⎜ 2 ⎠ ⎝ 0 2Be − μ0 B/2 γ/ 2 1 √ √ |1 −1 2 −Ed/ 6 γ/ 2 2Be + μ0 B/2 − γ/2 The crossing between the ground and the first excited rotational states now occurs at B = μ20 Be . For this particular value of the magnetic field, the matrix 4.11 can be diagonalized analytically to yield 1 = μ0 B/2,
γ2 E 2 d 2 1/2 2,3 = Beγ + μ0 B/2 ± Beγ 1 + 2 + , 2 2Beγ 6Beγ
(4.12)
where we have introduced the notation Beγ = Be − γ/4. Equations 4.10 and 4.12 show that the dependence of the Zeeman levels on the electric field is similar for crossed and parallel electric and magnetic fields. However, the splitting between the levels is smaller in perpendicular fields because the√electric field-dependent offdiagonal matrix element in Equation 4.11 is a factor 1/ 2 smaller. If the angle between the electric and magnetic fields is not equal to zero or π/2, five basis states are required to represent the Hamiltonian matrix [32]. The energy levels can be computed numerically by diagonalizing the Hamiltonian matrix including all relevant coupling terms. Figure 4.16 shows the energy levels of CaD at a constant electric field of 20 kV/cm as functions of the magnetic field at selected values of the angle between the fields. For parallel fields, there is only one avoided crossing between the |00 21 and |11 − 21 levels, as discussed above. Another crossing can be induced by rotating the fields, as illustrated in Figure 4.16b for χ = π/4. As the angle χ increases, the two avoided crossings approach each other and merge at χ = π/2. The crossing in perpendicular fields is shifted to higher magnetic fields by γ/2μ0 , as demonstrated by Equations 4.9 and 4.11. Equations 4.9 and 4.11 establish that avoided crossings can occur between nearly degenerate Zeeman states of opposite parity. The near-degeneracy can also be achieved by varying the electric field at a fixed value of the magnetic field. Figure 4.17 demonstrates that different Stark levels exhibit avoided crossings at different electric fields just like the Zeeman levels do in Figure 4.16. The results shown in Figures 4.16 and 4.17 suggest that the positions of the crossings can be controlled not only by varying the strength of the magnetic and electric fields, but also by changing the relative orientation of the fields. In addition, Figures 4.16 and 4.17 demonstrate that by rotating the fields it is possible to transform real crossings into avoided crossings. As alluded to earlier, the spin–rotation interaction couples different spin states and induces spin relaxation in collisions of ground-state 2 Σ molecules with He atoms. Figure 4.17 shows that by increasing the electric field, it is possible to adiabatically transfer the initial pure spin state into a mixture of different spin states, thereby © 2009 by Taylor and Francis Group, LLC
149
(a)
2.14
Energy (cm–1)
Effects of External Electromagnetic Fields on Collisions of Molecules
2.12 c=0 2.1
(b)
2.14
Energy (cm–1)
2.08
2.12 c = p/4 2.1
(c)
2.14
Energy (cm–1)
2.08
2.12 c = p/2 2.1
2.08 4.62
4.64
4.66 4.68 4.7 Magnetic field (T)
4.72
4.74
FIGURE 4.16 Energy levels of the CaD molecule as functions of the magnetic field near the avoided crossing of the N = 0 and N = 1 levels (encircled in Figure 4.1). The angle between the electric and magnetic fields is 0 (a), π/4 (b), and π/2 (c). The initial LFS Zeeman state is shown by the dashed line. The electric field is 20 kV/cm.
inducing spin relaxation. Varying the angle between the fields can therefore be used to control spin-changing transitions in molecular collisions. Figure 4.18 shows the cross-sections for spin relaxation in collisions of CaD molecules with He atoms for the parallel and perpendicular orientations of the fields [32]. The cross-sections increase exponentially at the electric field of 22 kV/cm for parallel fields and 24 kV/cm for perpendicular fields. The positions of the resonances in Figure 4.18 match the avoided crossings in Figure 4.17. The results shown in Figure 4.2 demonstrate that the resonances induced by avoided crossings shift to smaller electric fields with increasing the angle between the electric and magnetic fields. The peak value of the inelastic cross-section shown in Figure 4.18 can be as high as 100 Å2 , comparable with the cross-section for elastic scattering. © 2009 by Taylor and Francis Group, LLC
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Energy (cm–1)
2.12
c=0
Off 2.1
On
2.08 (b) 2.12 Energy (cm–1)
c = p/4
On
2.1
On
2.08 (c) 2.12 Energy (cm–1)
On
c = p/2
2.1
2.08 19
20
21 22 23 Electric field (kV/cm)
24
25
FIGURE 4.17 Electric field dependence of the energy levels of CaD(2 Σ) at different values of the angle χ between the electric and magnetic fields. The magnetic field is 4.7 T. (Adapted from Abrahamsson, E. et al., J. Chem. Phys., 127, 044302, 2007. With permission.)
4.4.4
DIFFERENTIAL SCATTERING IN ELECTROMAGNETIC FIELDS
The development of the experimental techniques for the production of slow molecular beams described in Chapter 14 offers the unique possibility of studying state and angle-resolved differential scattering of molecules in the presence of external electromagnetic fields. Molecular beam experiments with Stark decelerated and guided molecular beams can be designed to probe fully state-resolved differential crosssections (DCSs), which contain detailed information about the collision process. This © 2009 by Taylor and Francis Group, LLC
151
Effects of External Electromagnetic Fields on Collisions of Molecules 200
Cross-section (Å2)
150
c = p/2
c=0
100
50
0 22
23 Electric field (kV/cm)
24
25
FIGURE 4.18 Cross-sections for spin relaxation as functions of the electric field at the angles χ = 0 and χ = π/2 between the electric and magnetic fields. The magnetic field is 4.7 T. (Adapted from Abrahamsson, E. et al., J. Chem. Phys., 127, 044302, 2007. With permission.)
information can be used to elucidate the reaction mechanisms [41,42], fine-tune the interaction potential between molecules [43], and identify reactive scattering resonances [42]. Unlike the integral cross-sections and rate constants, the DCS is a coherent superposition of different partial wave contributions. The interference between the different scattering states gives rise to an oscillatory structure of the DCS as a function of the scattering angle, which is very sensitive to fine details of the interaction potential. This suggests that the angular dependence of the DCS at low collision energies may be particularly sensitive to the potential energy surface and the strength of an applied field. Alternatively, the DCS can be viewed as a diffraction pattern that arises when matter waves scatter off the impenetrable target. Combining the optical analogy with the energy-sudden approximation, Lemeshko and Friedrich [39] derived an approximate expression for the scattering amplitude based on the Fraunhofer model of diffraction. The differential cross-sections for NO(2 Π)–Ar collisions calculated using the Fraunhofer model at a collision energy of 442 cm−1 were found to be in a good qualitative agreement with previous close-coupling calculations [40]. At variance with previous theoretical work, Lemeshko and Friedrich considered the more general case where the initial collision flux is not parallel to the quantization axis defined by the electric field. Rotating the direction of the incoming collision flux introduces additional Bessel functions in the expansion of the scattering wavefunction, leading to noticeable changes in the integral and differential cross-sections [39]. © 2009 by Taylor and Francis Group, LLC
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The differential cross-section for the transition between the field-dressed states α and α can be written as [13,44] dσα→α 1 ˆ 2 = 2 |qα→α (Rˆ i , R)| ˆ ˆ kα dRi dR
(4.13)
where ˆ = 2π qα→α (Rˆ i , R)
l,ml l ,ml
∗ ˆ αm ;α l m il−l Ylm (Rˆ i )Yl m (R)T l l l
l
(4.14)
is the scattering amplitude parametrized by the direction of the incoming (Rˆ i ) and ˆ collision fluxes [13], and Ylml are the spherical harmonics. The T outgoing (R) matrix elements Tαml ;α m in Equation 4.14 can be obtained from exact quantum l scattering calculations as described in Chapter 1. The fully averaged integral crosssection considered previously can be obtained by averaging the differential crosssection over Rˆ i and integrating over Rˆ as follows: dσα→α 1 σα→α = dRˆ i dRˆ . (4.15) 4π dRˆ i dRˆ In crossed molecular beam scattering experiments, the direction of approach of the reactants is fixed by the configuration of the colliding beams [41]. Since by convention the incoming flux vector is directed along the space-fixed z-axis (and parallel to the field vectors E and B), we have Rˆ i = 0, and the general expression for the scattering amplitude (4.14) reduces to
ˆ = π1/2 ˆ α0;α m . qα→α (R) i− (2 + 1)1/2 Y m (R)T (4.16) ,m
Subsituting the result (4.16) into Equation 4.13, we obtain the differential crosssection corresponding to Rˆ i = 0: 2 dσα→α π − 1/2 ˆ = 2 i (2 + 1) Y m (R)Tα0;α m . ˆ k dR α
(4.17)
,m
Note that the projection m in Equations 4.16 and 4.17 vanishes because we require that the wavevector of the incident plane wave be parallel to the space-fixed quantization axis. The same choice is adopted in the conventional theory of atom–molecule collisions in the absence of fields [45]. Generally, the initial collision flux may not be parallel to the quantization axis (defined by the direction of the external field), and the differential cross-section Equation 4.13 must depend on both the orientation of the incoming flux and the direction of the outgoing flux. In this section, we restrict the discussion to the particular case of the DCS evaluated at fixed Rˆ i = 0 (Equation 4.17). The differential cross-sections given by Equation 4.17 are functions of the scattering angle θ defined by Rˆ = (θ, φ) at φ = 0 [44]. Figure 4.19 shows the DCS for © 2009 by Taylor and Francis Group, LLC
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Ec = 10–9 cm–1
1 ¥ 10–3
Ec = 0.01 cm–1
Differential cross-section (Å2 / sr)
Ec = 0.5 cm–1 ×50
8 ¥ 10–4
6 ¥ 10–4
4 ¥ 10–4
×104
2 ¥ 10–4
0
0
50
100 Scattering angle, θ (deg)
150
200
FIGURE 4.19 Spin relaxation DCS as functions of the scattering angle at a collision energy of 10−9 cm−1 (dash-dotted line), 10−2 cm−1 (dashed line), and 0.5 cm−1 (full line). The collision energies chosen correspond to the transitions (i), (ii), and (iii) discussed in the text. (Adapted from Tscherbul, T.V., J. Chem. Phys., 128, 244305, 2008. With permission.)
spin relaxation at a magnetic field of 0.5 T at three representative collision energies, each corresponding to the dominant partial wave transition at a given energy: (i) l = 0 → l = 2 (Ec = 10−9 cm−1 ), (ii) l = 1 → l = 1 (Ec = 10−2 cm−1 ), and (iii) l = 3 → l = 3 (Ec = 0.5 cm−1 ). Even at the lowest collision energy, the crosssections display a pronounced angular dependence due to the anisotropy of the scattering amplitude for the transitions (i) to (iii). In constrast, the elastic DCS (not shown) is determined by the l = 0 → l = 0 transition and is therefore independent of the scattering angle θ in the limit of low collision energy. The angular profiles of the DCS shown in Figure 4.19 can be explained using the definition of Equation 4.17. Krems and Dalgarno [13] showed that spin relaxation transitions must be accompanied by the change of the orbital angular momentum projection ml → ml = ml + 1. As a consequence, the sum in Equation 4.17 reduces to the single term |Tα00,α 21 Y21 (cos θ, 0)|2 so that the DCS for transition (i) can be written as dσl=0→l =2 /dθ ∝ sin2 2θ. Figure 4.19 demonstrates that the calculated DCS at a collision energy of 10−9 cm−1 does assume a similar bimodal dependence, © 2009 by Taylor and Francis Group, LLC
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although the two peaks have different magnitudes because of the admixture of the = 1 partial wave. Using the same arguments, it is easy to show that the angular dependence of the DCS for transition (ii) has the form dσl=1→l =1 /dθ ∝ sin2 θ, which is shown in Figure 4.19 as a broad maximum at a collision energy of 0.01 cm−1 . As demonstrated in Refs. [14] and [44], the DCS for transition (iii) is greatly enhanced by the presence of an l = 3 shape resonance in the incoming collision channel [14]. In the vicinity of the resonance, the spin relaxation DCS is proportional to |Y31 (θ, 0)|2 , or equiva2 . The DCS given by this expression has lently dσl=3→l =3 /dθ ∝ sin2 θ[4 − 5 sin2 θ]√ −1 two nontrivial nodes at θ1,2 = sin (±2/ 5). The numerical values of θ1 = 63◦ and θ2 = 117◦ are in good agreement with the minima of the calculated DCS for Ec = 0.5 cm−1 shown in Figure 4.19. These simple expressions show that the angular dependence of DCSs for spin relaxation at low temperatures is completely determined by a single partial wave contribution. This is also the case in the vicinity of a shape resonance. Therefore, experimental measurements of spin relaxation DCSs may provide detailed information about the individual partial wave components and shape resonances in cold collisions. Equation 4.17 shows that the angular dependence of the DCS is determined by the relative weights of the T -matrix elements between the states with different l. Because the magnetic fields do not change the partial wave contributions [44], the angular dependence of the spin relaxation DCS at collision energies of 0.01 and 0.5 cm−1 is not very sensitive to the magnetic field. As shown in Ref. [31], the absolute magnitude of the integral cross-sections in the multiple partial wave regime is also a slowly varying function of the magnetic field. On the contrary, the partial wave structure of spin relaxation cross-sections is strongly modified by electric fields [44]. Figure 4.20 is an energy–angle plot of the spin relaxation DCS at a magnetic field of 0.5 T and different electric fields. The DCS at zero electric field shows a triple peak corresponding to the l = 3 shape resonance [14]. Electric fields couple different rotational states and partial waves, which complicates the angular dependence of the DCS. For example, the DCS in the collision energy range of 0.01 to 1 cm−1 is a superposition of four different partial wave transitions [44]. Figure 4.20 reveals that electric fields tend to shift angular distributions in the backward direction. At the lowest collision energies, the DCS in the presence of an electric field shows a small forward peak due to the contribution from the l = 0 → l = 2 transition. The dynamics of spin relaxation changes dramatically in the presence of DC electric fields. The electric field-induced coupling between the ground and the excited rotational states leads to indirect couplings between different partial waves [44] and the enhancement of the forbidden l = 0 → l = 2 transition [13] at ultralow temperatures. This, in turn, leads to the increase of the spin relaxation cross-sections by many orders of magnitude [31,44]. We have found that the angular dependence of spin relaxation DCSs at low energies can be well approximated by a single spherical harmonic of the form |Yl m (θ, 0)|2 , where index l is determined by the dominant partial l wave at a given collision energy, and the projection ml is fixed by the conservation of the total angular momentum projection, that is, ml = 1 [13]. In particular, Figure 4.20
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Effects of External Electromagnetic Fields on Collisions of Molecules (a)
DC
05
.00
S0
0
0.2
Co lli
0.4
0
sio
15
n en
0.6
gy er
0 10 gle an g n i 50 catter S
0.8 1
(b)
004
0.0 S DC
002
0.0
0
0.2
Co lli
0.4
0
sio
15
n en
0.6
gy er
0 10 gle an g in r e 0 t 5 t S ca
0.8 1
FIGURE 4.20 Spin relaxation DCSs (in units of Å2 /sr) vs. scattering angle (in degrees) and collision energy (in cm−1 ) at zero electric field (a) and E = 100 kV/cm (b). The magnetic field is 0.5 T. (Adapted from Tscherbul, T.V., J. Chem. Phys., 128, 244305, 2008. With permission.)
shows that near the l = 3 shape resonance, the DCS assumes a characteristic form with three distinct maxima given by |Y31 (θ, 0)|2 . The angular dependence of the DCS is unaffected by magnetic fields. In contrast, electric fields induce couplings between the states with different l, and shift the angular distribution of the scattered molecules in the backward direction as shown in Figure 4.20. © 2009 by Taylor and Francis Group, LLC
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The results presented in Figure 4.20 might be useful for the interpretation of crossed beam scattering experiments with cold molecules. Such experiments probe angular distributions of scattered molecules in the laboratory frame rather than the averaged integral cross-sections. The results shown in Figure 4.20 suggest that the dependence of the DCS on the scattering angle can be modified by electric fields of up to 100 kV/cm. The results obtained in this section are applicable to all 2 Σ molecules, including CaF andYbF. The angular dependence of spin relaxation in these molecules can in principle be observed using slow molecular beams described in Chapter 14. The DCS is a rich source of information about atomic and molecular interactions as it allows one to unfold the partial wave averaging and analyze each l-contribution separately [43]. In addition, spin relaxation cross-sections are extremely sensitive to the atom–molecule interaction potential [19]. Therefore, we expect the spin relaxation DCS to be a sensitive probe of the anisotropic molecular interactions. As shown in Section 4.2, electric fields enhance the l → l ± 1 transitions, and thus the sensitivity of the cross-sections to the anisotropic part of the interaction potential. Figure 4.20 illustrates that spin relaxation DCSs assume a specific θ dependence near a shape resonance, which can be used for detection and assignment of shape resonances in cold collisions. The kinematic cooling technique [46] relies on collisions of thermal molecular beams to stop the molecules with translational energies satisfying certain kinematic constraints. For the kinematic cooling to be efficient, it is essential that a large number of molecules be scattered in a specific direction, which is dictated by the initial configuration and velocities of the colliding beams [46]. Figure 4.20 shows that electric fields shift angular distributions of inelastically scattered CaD molecules from sideways-peaked to backward-peaked. Applying an external dc electric field may therefore enhance the yield of cold molecules by focusing their angular distributions in preferred direction.
4.5
COLLISIONS IN RESTRICTED GEOMETRIES
As described in Chapter 9, counterpropagating laser beams create standing waves that can be used to trap ultracold atoms and generate an optical lattice [47]. Photoassociation of ultracold atoms on an optical lattice will produce a lattice of molecules suspended in three dimensions by laser fields, which can be used to study collisions of individual molecules. Optical lattices can also be used to confine the motion of ultracold molecules in low dimensions, which may alter dynamics of molecular collisions and allow for the study of intermolecular interactions in confined geometries [48–50]. The development of experimental techniques for confining ultracold molecules in cigar-shaped (one-dimensional) or pancake-shaped (two-dimensional) optical traps [51] may therefore open up new possibilities to study controlled molecular interactions. For example, Petrov and Shlyapnikov demonstrated that elastic collisions of ultracold atoms in the presence of strong harmonic confinement are modified by the trapping potential [49]. To examine the effects of external confinement on inelastic and reactive collisions of molecules trapped in a pancake configuration on an optical lattice, let us first consider the (unrealistic) limit of extremely tight confinement. The molecules are then restricted to move in two dimensions. © 2009 by Taylor and Francis Group, LLC
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As explained by Sadeghpour and colleagues [48], the energy dependence of elastic and reactive cross-sections depends on the dimensionality of the collision system. Wigner showed that the energy dependence of the scattering cross-section for unconstrained three-dimensional collisions near threshold is determined by the values of the orbital angular momentum of the collision complex before and after the collision [52]. In particular, he demonstrated that the elastic (internal energy conserving) scattering cross-section varies near threshold as v 2l+2l , where v is the collision velocity, l is the value of the orbital angular momentum before the collision and l is the orbital angular momentum of the outgoing scattering wave. He also showed that the cross-sections for collisional energy relaxation (inelastic scattering) vary near threshold as v (2l−1) . The quantum number l is not defined for collisions in two dimensions. The relative motion of two particles confined in a plane is described by the following Hamiltonian (in atomic units) H=−
l 2 (φ) 1 d d ρ + z 2 + Has + V , 2μρ dρ dρ 2μρ
(4.18)
where ρ is the center-of-mass separation and μ is the reduced mass of the colliding particles, l z is the operator describing the rotation of the collision complex about the quantization axis, φ is the angle specifying the orientation of the vector ρ in the space fixed coordinate system, Has is the asymptotic Hamiltonian describing the separated atoms or molecules, and V is the interparticle interaction potential. The quantization axis is directed along the normal to the confinement plane. The total wavefunction can be expanded in a basis of product wavefunctions Rsm (ρ)Θm (φ)ψs , where ψs represent the eigenfunctions of Has and Θm (φ) = √ 1 eimφ are the eigenfunctions of the l z operator labeled by the quantum number (2π) m. The eigenvalues s of the Hamiltonian Has correspond to the asymptotic energies of the interacting particles. The collision channels are specified by the quantum numbers s and m. The collision channels with the same value of s are degenerate. The interaction potential between atoms or molecules may induce transitions between different collision channels as the atoms or molecules collide. The collision process is described by the coupled differential equations of the form
1 d d m2 2 ρ − 2 + ks Rsm (ρ) = 2μ Vsm;s m Rs m (ρ), ρ dρ dρ ρ
(4.19)
sm
where ks2 = 2μ(E − s ), E is the total energy of the system, and Vsm;s m denote the matrix elements of the interaction potential. The interaction potential between neutral atoms or molecules vanishes faster than ∝ 1/ρ3 as ρ → ∞. If this condition is satisfied, the interaction potential can be neglected at large ρ and the solutions to Equation 4.19 can be written as superpositions of Hankel functions of the first (1) (2) Hm (ks ρ) and second Hm (ks ρ) kind. The solutions corresponding to the incoming part Ism (ks ρ) and the outgoing part Esm (ks ρ) of the collision wavefunction in channel © 2009 by Taylor and Francis Group, LLC
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(sm) can be defined as Ism (ks ρ) = Esm (ks ρ) =
π πμ (−i)m e−i 4 Hm(2) (ks ρ)Θm (φ)ψs 2
πμ m i π (1) (i) e 4 Hm (ks ρ)Θm (φ)ψs . 2
(4.20)
If the collision flux is prepared in a single quantum channel (sm), the collision wavefunction has the form Ism −
Usm;s m Es m ,
(4.21)
s m
which defines the collision U matrix [52]. The integral cross-sections for elastic and inelastic scattering are given in terms of the matrix elements of U by σsm;s m =
2 π 1 (−1)m+1 ei 2 Usm;s m − δs,s δm,m . ks
(4.22)
To obtain this expression, we assumed that the incident collision flux is represented by a plane wave propagating in the x-direction and used the Jacobi–Anger expansion of a plane wave to express it in terms of the incident and outgoing scattering waves (Equation 4.20):
ks μ
− 1 2
eiks x ψs =
π −1 π (−1)m ei 4 ks 2 Ism (ks ρ) + (−1)m e−i 2 Esm (ks ρ) . m
(4.23) The formulation of the collision problem as described above allows us to determine the dependence of the two-dimensional scattering cross-sections on the collision energy in the limit of small collision velocities using the formalism of Wigner [52] directly. Wigner showed that the U-matrix entering Equation 4.21 can be represented as U = ω 1 + ij(q − R)−1 j ω, (4.24) where R is the Wigner’s R-matrix, q is a matrix of the logarithmic derivatives of Esm from Equation 4.20, qsm = Esm /esm , esm = (1/2μ)dEsm /dρ, ω is a unitary diagonal matrix defined by esm = |esm |ω∗sm ,
(4.25)
and jsm is a diagonal matrix defined by 2 ∗ jsm = i(qsm − qsm ).
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(4.26)
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The threshold dependence of the U-matrix on the collision energy can thus be found analytically by analyzing the form of the scattering waves (Equation 4.20) at small collision velocities. The value of m = 0 corresponds to s-wave scattering in two dimensions. In the (1) limit of small argument η = ks ρ, the Hankel function H0 approaches the following function of η ' η2 2i & η (1) + ln +γ . (4.27) H0 (η) → 1 − 4 π 2 The derivative of the Hankel function with respect to η is
η 2i (1) H0 (η) → − + 2 πη
(4.28)
2 ∝ const. and Using these expressions and Equations 4.25 and 4.26, we find that js0 −1 the matrix elements (q − R) sm;sm with one or both of m and m equal to zero vanish as 1/ ln(η) as η → 0. Using Equations 4.22 and 4.21, the elastic scattering cross-section can be written as 1 2 σs0,s0 = |1 + iω2s0 + iω2s0 js0 ((q − R)−1 )s0,s0 |2 . (4.29) ks
The term (1 + iω2s0 ) is proportional to ks2 so it can be omitted at small ks by comparison with the third term and we find that the energy dependence of the elastic cross-section for s-wave scattering is given by σs0,s0 ∝
1 ks ln2 ks
.
(4.30)
For collision processes that change the projection of the orbital angular momentum (s0 → sm), the cross-section can be written as [52] σsm;sm =
1 | jsm jsm ((q − R)−1 )sm;sm |2 . ks
(4.31)
To determine the threshold behavior of the cross-sections for scattering in collision channels with |m| > 0, we express the Hankel functions in Equation 4.20 in terms of the Bessel and Neuman functions. Using the asymptotic expansions of the Bessel 2 ∝ k 2|m| as k → 0. This yields the following and Neuman functions, we find that jsm s s energy dependence of the cross-section near threshold: σs0,sm ∝ ks2|m|−1
1 2
ln ks
.
(4.32)
Processes described by Equation 4.32 are particularly important for inelastic collisions and angular momentum depolarization of molecules trapped in two dimensions at ultracold temperatures. © 2009 by Taylor and Francis Group, LLC
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It is also necessary to consider transitions between states of nonzero angular momentum m. Such transitions determine collision properties of fermionic atoms and 2|m| 2 molecules confined in two dimensions. Using the above result jsm ∝ ks and noting −1 that (q − R) sm;sm ∝ const. as ks → 0, we find the following energy dependence of the scattering cross-sections
σsm;sm ∝ ks2|m|+2|m |−1 .
(4.33)
When collisions release energy, the energy dependence of the scattering crosssections near threshold does not depend on the angular momentum in the final collision channel. For example, the cross-section for inelastic energy transfer in (2l−1) three-dimensional collisions is proportional to ks [52]. For reactive or inelastic collisions changing the quantum number s, Equation 4.31 transforms into σsm;s m =
2 1 jsm js m ((q − R)−1 )sm;s m . ks
(4.34)
According to Wigner [52], the off-diagonal matrix elements ((q − R)−1 )sm;s m are then independent of energy at small collision energies if m = 0. For m = 0, we obtain ((q − R)−1 )sm;s m ∝
1 ln ks
(4.35)
so the energy dependence of the inelastic s-wave scattering cross-section is the same as that of the elastic cross-section given by Equation 4.30: σs0,s m ∝
1 ks ln2 ks
.
(4.36)
When |m| > 0, the energy dependence of the scattering amplitude is determined 2 ∝ k 2|m| so the inelastic scattering cross-section for collisions with by the term jsm s angular momentum m is given by σsm;s m ∝ ks2|m|−1 .
(4.37)
The analysis above can be generalized to describe collisions of molecules on an optical lattice with harmonic confinement in one dimension. The Hamiltonian of the collision system can be written as H=−
1 Δ + V + Has + Vz , 2μ
(4.38)
where the confining potential Vz = az2 acts only on the colliding particles in the initial state s and can be ignored at short range [49]. The collision dynamics at finite interatomic distances is then described by a system of coupled differential equations in three-dimensional Jacobi coordinates presented, for example, in Ref. [29]. When © 2009 by Taylor and Francis Group, LLC
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the atoms or molecules react to produce other particles or change their internal state s, their translational energy becomes much larger than Vz so they are no longer confined. At infinite interparticle separation, the interaction potential V vanishes, and Equation 4.38 can be written as a sum of Equation 4.18 and a Hamiltonian describing the motion along the z-coordinate. The total wavefunction should now be expanded in a basis of product states Fs,α (x, y, z)ψs . For the initial channel s, the function Fs,α (x, y, z) can be represented as Rsm (ρ)Θm (φ)χ(z), where χ(z) describes the harmonic motion of the particles along z, and Rsm (ρ) can be expressed as a superposition of the functions defined in Equation 4.20. For all other channels s = s, the functions Fs ,α (x, y, z) = ˆ where R is the center-of-mass separation between the colliding partiFs lm (R)Ylm (R), cles and Ylm are spherical harmonics. The functions Fs lml can be written as superpositions of spherical Hankel functions. They are properly normalized and correspond to Wigner’s functions Es ,lm [52]. For elastic scattering, we have to consider only the initial channel and Equation 4.22 applies, leading to the result of Equation 4.30. For reactive scattering, Equation 4.31 must be modified to include js lm and (q − R)−1 )sm;s lm expressed in terms of the functions Fs lm at infinite interatomic separation. This modification, however, does not change the energy dependence of the cross-section (4.37) determined by jsm and the leading term in the expansion of (q − R)−1 )sm;s lm from Equation 4.20. The results of Equations 4.30, 4.37, and 4.33 thus apply to scattering in quasi-two-dimensional geometry accompanied with loss of confinement. Table 4.1 summarizes the comparison of the energy dependence of the elastic and inelastic cross-sections on the collision velocity for unconstrained collisions in three dimensions and collisions in the presence of strong confinement. Petrov and Shlyapnikov term the regime described here as “quasi-2D” [49]. Although Table 4.1 does not provide information on the absolute values of the cross-sections, it shows that inelastic collisions of molecules in quasi-two-dimensional scattering at small energies must be suppressed. External confinement also changes the symmetry of long-range intermolecular interactions. For example, the dipole–dipole interaction averaged over the scattering wavefunction of polar molecules vanishes in the limit of ultracold s-wave scattering in three dimensions (see Chapter 2), but remains significant
TABLE 4.1 Dependence of the Scattering Cross-Sections on the Collision Velocity v Near Threshold Elastic Collisions s-wave s-wave to non-s-wave Non-s-wave to non-s-wave Inelastic Collisions s-wave relaxation Non-s-wave relaxation
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Three-Dimensional
Quasi-Two-Dimensional
σ = const.
1 v ln2 v σ ∝ v 2|m|−1 12 ln v |−1 2|m|+2|m σ∝v
σ ∝ v 2l
σ ∝ v 2l+2l σ ∝ 1/v σ ∝ v 2l−1
σ∝
1 v ln2 v σ ∝ v 2|m|−1
σ∝
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in two dimensions. Measurements of molecular collisions in confined geometries may thus provide a sensitive probe of long-range intermolecular interactions and quantum phenomena in collision physics. Confining molecules in low dimensions may be a practical tool for increasing the stability of ultracold molecular gases. The symmetry of the collision problem can be completely destroyed by a combination of confining laser fields and external static electromagnetic fields [50]. Measurements of chemical reactions in confined geometries may thus be a novel approach to study stereodynamics and differential scattering at ultracold temperatures.
4.6
COLD CONTROLLED CHEMISTRY
Since early experiments on chemical reaction dynamics [53], the efforts of many researchers have been to achieve external control over chemical reactions. External field control of molecular collisions in the gas phase, however, remains a significant challenge. External control of bimolecular reactions is complicated by thermal motion of molecules that randomizes molecular encounters and diminishes the effects of external fields on molecular collisions. Thermal motion precludes coherent control of molecular collisions [54]. The effects of the thermal motion can be reduced by cooling molecular gases to low temperatures. Electromagnetic fields may influence molecular collisions significantly only when the translational energy of the molecules is smaller than the perturbations due to interactions with external fields. Static magnetic and electric fields (up to 5 T and 200 kV/cm, respectively) as well as off-resonant laser fields readily available in the laboratory shift molecular energy levels by up to a few Kelvin so external field control of gas-phase molecular dynamics may be most easily achieved at temperatures near or less than 1 K. The experimental work on the creation of dense ensembles of cold molecules and the study of intermolecular interactions in the presence of electromagnetic fields may therefore lead to the development of a new research field of cold controlled chemistry. The research of chemistry at cold and ultracold temperatures may result in both practical and fundamental applications. When molecules are cooled to low temperatures, inelastic and reactive collisions become extremely state-selective and propensities for populating specific rovibrational states are enhanced (see Chapter 3). This could be used for an efficient production of atoms or molecules with inverted populations of internal energy levels and the development of new atomic or molecular lasers. Bose-stimulated photodissociation may be used for controlled preparation of entangled pairs of molecules [55]. Entanglement of spatially separated molecules is necessary for the study of quantum information transfer and the development of quantum computing schemes based on atomic and molecular systems. The production of entangled molecules may also be used for the realization of coherent control of bi-molecular chemical reactions [54]. Chemical reactions in a cold buffer gas can provide a rich source for slow molecular beams [56]. Slow molecular beams may find a lot of applications in chemistry research, ranging from high-precision spectroscopy, to novel scattering experiments, to studies of collective dynamics of strongly interacting systems at low temperatures. Studies of low-temperature chemical reactions in the presence of external fields may elucidate several fundamental questions of modern chemical physics. As © 2009 by Taylor and Francis Group, LLC
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demonstrated in this chapter, electric fields may induce avoided crossings between states otherwise uncoupled. Measurements of cross-sections for molecular collisions and chemical reactions as functions of the electric field magnitude may provide information on the role of avoided crossings and the associated geometric phase in chemical dynamics of molecules. Measurements of chemical reactions in electric and magnetic traps may be used as a new approach to probe the effects of molecular alignment and symmetry of intermolecular interactions on molecular collisions. Measurements of chemical reactions at cold temperatures, where molecular collisions are determined by a few partial waves, may elucidate the effects of interference between different scattering states and orbiting shape resonances on chemical dynamics. Molecules cooled to low temperatures can be confined in unusual geometries [57,58], which may be used to study the effects of complex gage potentials on molecular structure and collisions. Studies of ultracold molecules in confined geometries may also result in many fundamental applications. The energy dependence of cross-sections for molecular collisions in confined and quasiconfined geometries is different from Wigner’s threshold laws in three dimensions [48–50]. Chemical reactions and inelastic collisions of molecules under external confinement must therefore be modified. External confinement also changes the symmetry of long-range intermolecular interactions. For example, the dipole–dipole interaction averaged over the scattering wavefunction of polar molecules vanishes in the limit of ultracold s-wave scattering in three dimensions, but remains significant in two dimensions. Measurements of molecular collisions in confined geometries may thus provide a sensitive probe of long-range intermolecular interactions and quantum phenomena in collision physics. Confining molecules in low dimensions may be a practical tool for increasing the stability of ultracold molecular gases. The symmetry of the collision problem can be completely destroyed by a combination of confining laser fields and external static electromagnetic fields [50]. Measurements of chemical reactions in confined geometries may thus be a novel approach to study stereodynamics and differential scattering at ultracold temperatures. As demonstrated in this chapter, electric and magnetic fields can be used to tune fine interactions responsible for intermolecular energy transfer at low temperatures. Studies of molecular collisions in external fields may therefore be used to probe the role of fine interactions such as the spin–rotation interaction or the second-order spin–orbit coupling in determining chemical reactivity of cold molecules. Using electromagnetic fields to induce scattering Feshbach resonances in molecular collisions at ultracold temperatures may be an important tool for studying energy transfer between molecules as well as the internal energy redistribution in polyatomic molecules. Feshbach resonances enhance the interaction time of the reactants. Dynamics of energy transfer and chemical reactions in the presence of scattering resonances is therefore dramatically modified. Tuning scattering resonances in collisions involving polyatomic molecules may be a novel approach to elucidating the role of ergodicty and multiple encounters in molecular energy transfer. Resonances may also affect the collective properties of ultracold gases such as diffusion. Measurements of dynamics of ultracold gases in the presence of external electromagnetic fields may therefore lead to the observation of new phenomena and the development of new theories for strongly interacting molecular gases. © 2009 by Taylor and Francis Group, LLC
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ACKNOWLEDGMENT We thank Alisdair Wallis for verifying some of our equations.
REFERENCES 1. Harris, J.G.E., Campbell, W.C., Egorov, D., Maxwell, S.E., Michniak, R.A., Nguyen, S.V., van Buuren, L.D., and Doyle, J.M., Deep superconducting magnetic traps for neutral atoms and molecules, Rev. Sci. Instrum., 75, 14, 2004. 2. van Veldhoven, J., Bethlem, H.L., Schnell, M., and Meijer, G., Versatile electrostatic trap, Phys. Rev. A, 73, 063408, 2006. 3. Heiner, C.E., Bethlem, H.L., and Meijer, G., Molecular beams with a tunable velocity, Phys. Chem. Chem. Phys., 8, 2666, 2006. 4. Krems, R.V., Cold controlled chemistry, Phys. Chem. Chem. Phys., 10, 4079, 2008. 5. Weinstein, J.D., deCarvalho, R., Guillet, T., Friedrich, B., and Doyle, J.M., Magnetic trapping of calcium monohydride molecules at millikelvin temperatures, Nature, 395, 148, 1998. 6. Grimm, R., Weidemüller, M., and Ovchinnikov, Y.B., Optical dipole traps for neutral atoms, Adv. At. Mol. Opt. Phys., 42, 95, 2000. 7. Doyle, J., Friedrich, B., Krems, R.V., and Masnou-Seeuws, F., Quo vadis, cold molecules?, Eur. Phys. J. D, 31, 149, 2004. 8. van de Meerakker, S.Y.T., Vanhaecke, N., van der Loo, M.P.J., Groenenboom, G.C., and Gerard Meijer, Direct measurement of the radiative lifetime of vibrationally excited OH radicals, Phys. Rev. Lett., 95, 013003, 2005. 9. Bohn, J.L., Molecular spin relaxation in cold atom–molecule scattering, Phys. Rev. A, 61, 040702, 2000. 10. Bohn, J.L., Cold collisions of O2 with helium, Phys. Rev. A, 62, 032701, 2000. 11. Avdeenkov, A.V. and Bohn, J.L., Ultracold collisions of oxygen molecules, Phys. Rev. A, 64, 053602, 2001. 12. Volpi, A. and Bohn, J.L., Magnetic-field effects in ultracold molecular collisions, Phys. Rev. A, 65, 052712, 2002. 13. Krems, R.V. and Dalgarno, A., Quantum mechanical theory of atom–molecule and molecular collisions in a magnetic field: Spin depolarization, J. Chem. Phys., 120, 2296, 2004. 14. Krems, R.V., Dalgarno, A., Balakrishan, N. and Groenenboom, G.C., Spin-flipping transitions in doublet-sigma molecules induced by collisions with structureless atoms, Phys. Rev. A, 67, 060703, 2003. 15. Krems, R.V., Sadeghpour, H.R., Dalgarno, A., Zgid, D., Klos, J., and Chalasinski, G., Low-temperature collisions of NH(X 3 Σ) molecules with He atoms in a magnetic field: An ab initio study, Phys. Rev. A, 68, 051401(R), 2003. 16. Gonzalez-Martinez, M.L. and Hutson, J.M., Ultracold atom–molecule collisions and bound states in magnetic fields: tuning zero-energy Feshbach resonances in He + NH(3 Σ), Phys. Rev. A, 75, 022702, 2007. 17. Hutson, J.M., Feshbach resonances in ultracold atomic and molecular collisions: threshold behaviour and suppression of poles in scattering lengths, New. J. Phys., 9, 152, 2007. 18. Maussang, K., Egorov, D., Helton, J.S., Nguyen, S.V., and Doyle, J.M., Zeeman relaxation of CaF in low temperature collisions with helium, Phys. Rev. Lett., 94, 123002, 2004.
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19. Campbell, W.C., Tscherbul, T.V., Lu, H.-I., Tsikata, E., Krems, R.V., and Doyle, J.M., Mechanism of collisional spin relaxation in 3 Σ molecules, Phys. Rev. Lett., 102, 013003, 2009. 20. Krems, R.V., Groenenboom, G.C., and Dalgarno, A., Electronic interaction anisotropy between atoms in arbitrary angular momentum states, J. Phys. Chem. A, 108, 8941, 2004. 21. Ticknor, C. and Bohn, J.L., Influence of magnetic fields on cold collisions of polar molecules, Phys. Rev. A, 71, 022709, 2005. 22. Avdeenkov, A.V. and Bohn, J.L., Collisional dynamics of ultracold OH molecules in an electrostatic field, Phys. Rev. A, 66, 052718, 2002. 23. Avdeenkov, A.V. and Bohn, J.L., Linking ultracold polar molecules, Phys. Rev. Lett., 90, 043006, 2003. 24. Avdeenkov, A.V., Bortolotti, D.C.E., and Bohn, J.L., Field-linked states of ultracold polar molecules, Phys. Rev. A, 69, 012710, 2004. 25. Ticknor, C. and Bohn, J.L., Long-range scattering resonances in strong-field-seeking states of polar molecules, Phys. Rev. A, 72, 032717, 2005. 26. Avdeenkov, A.V., Kajita, M., and Bohn, J.L., Suppression of inelastic collisions of polar 1 Σ state molecules in an electrostatic field, Phys. Rev. A, 73, 022707, 2006. 27. Ticknor, C., Collisional control of ground state polar molecules and universal dipolar scattering, Phys. Rev. Lett., 100, 133202, 2008. 28. Tscherbul, T.V. and Krems, R.V., Quantum theory of chemical reactions in the presence of electromagnetic fields, J. Chem. Phys., 129, 034112, 2008. 29. Li, Z. and Krems, R.V., Electric-field-induced Feshbach resonances in ultracold alkali metal mixtures, Phys. Rev. A, 75, 032709, 2007. 30. Krems, R.V., Controlling collisions of ultracold atoms with dc electric fields, Phys. Rev. Lett., 96, 123202, 2006. 31. Tscherbul, T.V. and Krems, R.V., Controlling electronic spin relaxation of cold molecules with electric fields, Phys. Rev. Lett., 97, 083201, 2006. 32. Abrahamsson, E., Tscherbul, T.V., and Krems, R.V., Inelastic collisions of cold polar molecules in non-parallel electric and magnetic fields, J. Chem. Phys., 127, 044302, 2007. 33. Brown, J.M. and Carrington, A., Rotational Spectroscopy of Diatomic Molecules, Cambridge University Press, 2003. 34. Friedrich, B. and Herschbach, D., Steric proficiency of polar 2 Σ molecules in congruent electric and magnetic fields, Phys. Chem. Chem. Phys., 2, 419, 2000. 35. Wiebusch, G., Main, J., Krüger, K., Rottke, H., Holle, A., and Welge, K.H., Hydrogen atom in crossed magnetic and electric fields, Phys. Rev. Lett., 62, 2821, 1989. 36. Zare, R.N., Angular Momentum, Wiley, New York, 1988. 37. Svanberg, S., Atomic and Molecular Spectroscopy: Basic Aspects and Practical Applications, Sect. 7.1.5, Springer, 2004. 38. Mark, M., Kraemer, T., Waldburger, P., Herbig, J., Chin, C., Nägerl, H.-C., and Grimm, R., Stückelberg interferometry with ultracold molecules, Phys. Rev. Lett., 99, 113201, 2007. 39. Lemeshko, M. and Friedrich, B., An analytic model of rotationally inelastic collisions of polar molecules in electric fields, J. Chem. Phys., 129, 024301, 2008. 40. de Lange, M.J.L., Drabbels, M., Griffiths, P.T., Bulthuis, J., Stolte, S. and Snijders, J.G., Steric asymmetry in state-resolved NO–Ar collisions, Chem. Phys. Lett., 313, 491, 1999. 41. Bernstein, R.B., Chemical Dynamics via Molecular Beam and Laser Techniques, Clarendon Press, Oxford, 1982.
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42. Skodje, R.T., Skouteris, D., Manolopoulos, D.E., Lee, S.-H., Dong, F., and Liu, K., Resonance-mediated chemical reaction: F + HD → HF + D, Phys. Rev. Lett., 85, 1206, 2000. 43. Buck, U., Huisken, F., Schleusener, J., and Pauly, H., Differential cross sections for the excitation of single rotational quantum transitions: HD + Ne, Phys. Rev. Lett., 38, 680, 1977. 44. Tscherbul, T.V., Differential scattering of cold molecules in superimposed electric and magnetic fields, J. Chem. Phys., 128, 244305, 2008. 45. Balint-Kurti, G.G., The theory of rotationally inelastic molecular collisions, in International review of Science, Ser. II, Vol. 1, Eds. Buckingham, A.D. and Coulson, C.A., Butterworths, 1975, p. 286. 46. Elioff, M.S., Vaneltini, J.J., and Chandler, D.W., Formation of NO( j = 7.5) molecules with sub-kelvin translational energy via molecular beam collisions with argon using the technique of molecular cooling by inelastic collisional energy-transfer, Eur. Phys. J. D, 31, 385, 2004. 47. Bloch, I. and Greiner, M., Exploring quantum matter with ultracold atoms in optical lattices, Adv. At. Mol. Phys. 52, 1, 2005. 48. Sadeghpour, H.R., Bohn, J.L., Cavagnero, M.J., Esry, B.D., Fabrikant, I.I., Macek, J.H., and Rau, A.R.P., Threhsold phenomena in atomic and molecular physics, J. Phys. B: At. Mol. Opt. Phys., 33, R93, 2000. 49. Petrov, D.S. and Shlyapnikov, G.V., Interatomic collisions in a tightly confined Bose gas, Phys. Rev. A, 64, 012706, 2001. 50. Li, Z., Alyabyshev, S.V., and Krems, R.V., Ultracold collisions in two dimensions, Phys. Rev. Lett., 100, 073202, 2008. 51. Bloch, I., Ultracold quantum gases in optical lattices, Nature Physics, 1, 23, 2005. 52. Wigner, E.P., On the behavior of cross sections near thresholds, Phys. Rev., 73, 1002, 1948. 53. Tailor, E.H. and Datz, S., Study of chemical reaction mechanisms with molecular beams: The reaction of K with HBr, J. Chem. Phys., 23, 1711, 1955. 54. Shapiro, M. and Brumer, P., Principles of the Quantum Control of Molecular Processes, Wiley Inter-Science, 2003. 55. Moore, M.G. and Vardi, A., Bose-enhanced chemistry: amplification of selectivity in the dissociation of molecular Bose-Einstein condensates, Phys. Rev. Lett., 88, 160402, 2002. 56. Maxwell, S.E., Brahms, N., deCarvalho, R., Glenn, D.R., Helton, J.S., Nguyen, S.V., Patterson, D., Petricka, J., DeMille, D., and Doyle, J.M., High-flux beam source for cold, slow atoms or molecules, Phys. Rev. Lett., 95, 173201, 2005. 57. Patterson, D. and Doyle, J.M., Bright, guided molecular beam with hydrodynamic enhancement, J. Chem. Phys., 126, 154307, 2007. 58. Heiner, C.E., Carty, D., Meijer, G., and Bethlem, H.L., A molecular synchrotron, Nature Phys., 3, 115, 2007.
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Part II Photoassociation
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Ultracold Molecule 5 Formation by Photoassociation William C. Stwalley, Phillip L. Gould, and Edward E. Eyler CONTENTS 5.1
Ultracold Molecule Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Cold and Ultracold Molecule Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 The Process of Photoassociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Photoassociation of Ultracold Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3.1 Like Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3.2 Unlike Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3.3 Involving Molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3.4 In a Quantum Degenerate Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3.5 In an Electromagnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3.6 In an Optical Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Characterization and State-to-State Transfer of Ultracold Alkali Dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Levels Near Dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Strongly Bound X State Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2.1 Production by Photoassociation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2.2 Enhancement in Double-Minimum Potentials . . . . . . . . . . . . 5.2.2.3 Enhancement by Resonant Coupling of Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2.4 Stimulated Raman Transfer to Deeply Bound Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Further Connections and Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Ultracold Molecular Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Tests of Fundamental Physical Constants and Symmetries . . . . . . . . 5.3.3 Quantum Computing with Ultracold Polar Molecules . . . . . . . . . . . . . 5.3.4 Ultracold Collisions and Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Final Speculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
170 170 170 172 172 182 188 189 192 193 194 194 201 201 202 204 206 208 208 209 210 210 211 211 211
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Cold Molecules: Theory, Experiment, Applications
ULTRACOLD MOLECULE FORMATION COLD AND ULTRACOLD MOLECULE FORMATION
There are two basic approaches to forming ensembles of molecules with cold and ultracold translational temperatures (below 1 K and 1 mK, respectively): (1) cooling warmer molecules directly and (2) cooling ensembles of atoms to cold and ultracold translational temperature, followed by association of these atoms into molecules without significant heating. The former approach includes many specific strategies, such as helium buffer gas cooling (more generally “sympathetic cooling”) as described in Chapter 13 by Campbell and Doyle, and electric dipole deceleration (more generally magnetic dipole and polarizability deceleration) as described in Chapter 14 by van de Meerakker, Bethlem, and Meijer. Other possible strategies are discussed there and elsewhere [1,2]. Cooling of ensembles of atoms to ultracold temperatures, for example, hundreds of μK in a magneto-optical trap (MOT), is now well known, and will not be discussed here. There are a number of approaches for associating atoms into molecules (without significant heating). The oldest and simplest approach (Section 5.1.3.1) is ordinary (one color) photoassociation (PA), that is, free→bound absorption red-detuned from an atomic resonance, followed by bound→bound or bound→free radiative emission, the former producing a bound molecule of two ground-state atoms. Alternatively, one can use two-color PA, especially with short duration pulsed lasers as described in Chapters 7 and 8 (“PUMP-DUMP” and “adiabatic Raman,” respectively). It is often useful to think of two-color PA as an “optical Feshbach resonance,” where the frequency difference between two lasers can be readily tuned. Even diatomic molecules have a complex structure of rovibrational hyperfine levels just below (see Chapter 6 by Julienne) and just above (see Chapter 1 by Hutson) a given asymptote. With high-resolution cw lasers, the intensities are usually low enough that the near dissociation structure is intensity independent. Other “non-optical” Feshbach resonances can be tuned with a magnetic field (or an electric field or both). When a Feshbach resonance is tuned close to an appropriate asymptote, the probability of magnetoassociation (MA) (or electroassociation) becomes significant. This magnetic tuning has led to the first “molecular BoseEinstein Condensate (BEC)” of bosonic “Feshbach molecules” formed from two fermionic atoms (e.g., 6 Li2 , 40 K2 ), although the molecules are extremely weakly bound (in states that are the negative energy correlates of the Feshbach resonances). Chapter 4 deals with these Feshbach resonances in collisions, while Chapters 9 to 11 deal more specifically with MA and related few-body physics issues, such as the socalled BEC–BCS (Bardeen-Coofer-Schrieffer) crossover. Recently it has been shown that PA rates can be enhanced and depressed by many orders of magnitude by tuning a magnetic field in the vicinity of a Feshbach resonance threshold [3,4].
5.1.2 THE PROCESS OF PHOTOASSOCIATION The interaction of an ensemble of atoms with light of specific frequencies has long been of interest in physics and astronomy. The fact that atoms such as H, He, Ne, Ar, and Hg show line spectra in both absorption and emission was pivotal in the © 2009 by Taylor and Francis Group, LLC
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development of quantum mechanics. At high atomic densities, the lines develop finite frequency widths and complex shapes associated in part with interatomic interactions, especially in the wings of the spectral lines. In general, an atomic line shape (e.g., Li 2s → 2p) involves all potential energy curves associated with the interaction of two ground state Li(2s) atoms and all potential energy curves associated with the interaction of one excited state Li(2p) atom with one Li(2s) atom, where the small spin–orbit interaction of the Li(2p) atom is neglected for simplicity. Those potential 3 + curves (shown in Figure 5.1) correspond to the X 1 Σ+ g and a Σu states (for two 1 + 1 + 1 1 3 + 3 3 3 Li(2s) atoms) and the 2 Σg , A Σu , 1 Πg , B Πu , 1 Σg , 2 Σ+ u , 1 Πg , and b Πu states (for one Li(2p) atom plus one Li(2s) atom). However, because of the electric dipole selection rules for transitions between these states (ΔΛ = 0, ±1; ΔS = 0; u ↔ 1 1 + g), only the A 1 Σ+ u and B Πu states are involved in transitions to and from the X Σg 3 state, and only the 1 3 Σ+ g and 1 Πg states are involved in the transitions to and from 3 + the a Σu state. Thus, four different pairs of electronic states contribute to absorption line broadening and line shapes. In addition, the colliding atoms can absorb light and form a bound state of the various excited states or they can reach a continuum (or “free”) state, which will fall apart to Li(2p) + Li(2s). These processes are called free→bound and free→free absorption, respectively. An alternative name for free→bound absorption is photoassociation (PA) and it commonly occurs on the low-frequency (“red”) side of a line shape. In the Li(2s → 2p) “red” wing, the dominant triplet contribution is absorption 3 + from the a 3 Σ+ u state to the 1 Σg state, as illustrated in Figure 5.1 [5]. A hot thermal ensemble of Li atoms will also typically include some Li2 molecules (e.g., in thermodynamic equilibrium at 1 atm and 2033 K [5], the Li2 pressure is 0.01 atm). Most of these molecules are in the strongly bound X 1 Σ+ g state, but some 3 + are in the weakly bound a Σu state as well. Molecules in these two states can likewise absorb light to form bound excited molecules (bound→bound absorption) or free excited molecules (which dissociate to Li(2p) + Li(2s); bound→free absorption). An alternative name for bound→free absorption is photodissociation. Similar considerations apply to line shapes of more highly excited atoms, such as the K(4s → 4d5/2 ) dipole-forbidden transition. Here the very weak atomic line is at 27,397.01 cm−1 , but the stronger PA from colliding a 3 Σ+ u atoms extends far to the red, with a prominent peak at 17,400 cm−1 , as reviewed in Ref. [6]. This is because the electric dipole transition moment for the 2 3 Πg ← a 3 Σ+ u transition, which vanishes for isolated atoms, increases rapidly as the molecular internuclear distance R decreases. This 2 3 Πg ← a 3 Σ+ u photoassociation is shown in Figure 5.2 from Ref. [6], where two experiments and a theoretical model are seen to be in reasonable agreement. Note, however, that the model involved thousands of 2 3 Πg state rovibrational levels (v ≤ 50, J ≤ 300), an 800 K Boltzmann distribution for a 3 Σ+ u colliding atoms, and an electric dipole transition moment function (which vanishes as R → ∞). It is clear because of the relatively simple shapes of the spectra in Figure 5.2 that there is little information content in these spectra, principally because of the averaging over a broad thermal distribution of collisional energies and angular momenta. We shall see in the following sections how this drastically changes when the colliding atoms are ultracold, typically a million or more times colder!
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21Su+
23Sg+
2 ¥ 104 13Pg
E (cm–1)
2s + 2p
13Sg+
B1Pu
1 ¥ 104
A1Su+ 13Su+
2s + 2s
0 X1Sg+
0
5
10 R (Å)
FIGURE 5.1 Li2 potential curves used for modeling lithium absorption spectra near the Li(2s → 2p) resonance lines at high pressure (1 atm) and temperature (2033 K). The vertical lines indicate selected frequencies at which photoassociation from colliding atoms to boundstate Li2 1 3 Σ+ g molecules occurs (the dominant triplet contribution to the line shape on the low-frequency “red” side of resonance). (From Erdman, P.S. et al., J. Quant. Spectrosc. Radiat. Transf., 88, 447, 2004. With permission.)
Similar considerations also apply to emission line broadening and line shapes, but will not be discussed here.
5.1.3 5.1.3.1
PHOTOASSOCIATION OF ULTRACOLD ATOMS Like Atoms
Ultracold atoms (T < 1 mK) have remarkably different PA spectra (Figure 5.3, from [7]) from the much higher temperature PA spectra discussed in Section 5.1.2. Now the Boltzmann distribution of thermal collision energies is very sharp, with typical temperatures of hundreds of μK. A 300 μK temperature corresponds to a 6 MHz energy spread (0.0002 cm−1 ) and such narrow widths can be observed at low laser intensity. However, the collisional angular momenta capable of reaching small (<100 Å) internuclear distances are also greatly reduced. As shown in Figure 5.4 for 39 K, the J = 3 © 2009 by Taylor and Francis Group, LLC
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Intensity (arb. units)
0.8
0.6
0.4
0.2
0 5670
5730
5790
5850
l (A)
FIGURE 5.2 Absorption observations (•, ) of the K2 diffuse band at 800 K corresponding to 2 3 Πg ← a 3 Σ+ u photoassociation, compared to a quantum-mechanical simulation, (solid line). (From Luh, W.T. et al., Chem. Phys. Lett., 144, 221, 1988. With permission.)
( f -wave) centrifugal barrier is nearly 4 mK high and those for J > 3 are much higher, preventing PA inside 50 Å except for s-, p-, and d-waves (J = 0, 1 and 2). A d-wave collision absorbing a single photon into an Ω = 0 state can produce molecular J values up to 3, while a d-wave collision absorbing into an Ω = 1 state can produce molecular J values up to 4 [8], as shown in Figure 5.5. The potential of photoassociation of ultracold atoms was first pointed out in a landmark paper by Thorsheim, Weiner, and Julienne [9]. Their calculations used a relatively hot temperature of 10 mK. In 1993, two landmark experimental papers [10,11] demonstrated ultracold PA in Na2 and Rb2 , respectively, at significantly lower temperatures. There are many techniques for observing PA, as listed in Table 5.1. Figure 5.3 is an example of trap-loss detection as used for the initial Rb2 PA result [11], while Figure 5.5 and the initial Na2 PA result are examples of direct molecular ionization. A third technique that only works for predissociative levels is fragmentation spectroscopy [12], shown in Figure 5.6. A fourth technique is direct detection of 3 + the ultracold molecules formed by PA followed by decay into the X 1 Σ+ g and a Σu states correlating to two ground-state atoms. This technique is of particular interest for this book. We shall see in Section 5.1.3.2 that this last technique is also the method of choice for detecting heteronuclear PA used to form ultracold heteronuclear molecules. © 2009 by Taylor and Francis Group, LLC
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Fluorescence
u=9
0–g
10
11
12
13
14
15
126
127
128
129
130
16
17
18 19 20 21 22 23
0u+
u = 124 1g
125
13,039.9
13,040.5
135
13,041.1
13,041.7
13,042.3
Laser frequency (cm–1)
FIGURE 5.3 Trap-loss spectrum near the 4p3/2 + 4s1/2 asymptote of 39 K2 , which clearly shows the 0g− , 0u+ , and 1g vibrational levels. The K(4s 2 S → 4p 2 P3/2 ) transition is at 13,042.876 cm−1 . Each trap-loss peak has rotational structure at high resolution. Vibrational quantum numbers have been assigned only for the 0g− and 1g states. (From Stwalley, W.C. and Wang, H., J. Mol. Spectrosc., 195, 194, 1999. With permission.)
4 3
E (×10–3 cm–1)
2 T = 1 mK
1 0
K (42 S1/2) + K (42 S1/2)
–1 J=3 J=2 J=1 J=0
–2 –3 –4 20
40
60
80
100 R (Å)
120
140
160
180
FIGURE 5.4 Long-range potential (J = 0) and J = 1, 2, and 3 effective potentials UJ = V + {2 [J(J + 1) − Ω2 ]/2μR2 )} for two colliding 39 K ground-state atoms. Even at an energy of 3 kT (∼1 mK), only the s-, p-, and d-waves reach distances inside 100 Å. (From Stwalley, W.C. and Wang, H., J. Mol. Spectrosc., 195, 194, 1999. With permission.)
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2 J=1
4 3
2
u = 100
Ion signal
J=1
3 4
2
J=1
0.0
0.5
1.0
u = 108
3 4
1.5
2.0 2.5 Frequency (GHz)
3.0
3.5
4.0
FIGURE 5.5 Examples of K2 high-resolution PA spectra detected by resonant molecular ionization: 1g (4p 2 P3/2 ) levels v = 93, 100, and 108, bound by 25.6, 16.8, and 9.9 cm−1 , respectively. (From Pichler, M. et al., J. Chem. Phys., 118, 7837, 2003. With permission.)
There is a massive literature on PA of ultracold atoms (primarily like atoms) recently and admirably reviewed by Jones and colleagues [13]. Early work emphasizing Na2 and Rb2 PA is reviewed in Refs. [14,15]. The well-studied example of K2 is treated in considerable detail in Ref. [7], while Cs2 is the focus of the review of Masnou-Seeuws and Pillet [16]. The field of cold and ultracold collisions (not just PA) is admirably reviewed in Ref. [17]. We can only touch on a few topics here. © 2009 by Taylor and Francis Group, LLC
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TABLE 5.1 Detection Techniques for Photoassociation (PA) of Ultracold Alkali Metal Atoms M Type
Equation
I. Trap loss (decrease in atomic fluorescence) II. Molecular ionization
III. Fragmentation IV. X, a state ionization
Comments
(
)
“Cold” atoms (no loss) “Hot” atoms (loss)
M2 * → M + M + hν
3 + X 1 Σ+ g or a Σu (loss) Autoionization or two-photon ionization
M2 * → M2 + hν M2 * + (1 or 2) hν − → M+ 2 +e + → M + M + e− M2 * → M + M* M* + 2hν → M+ + e− M2 * → M2 + hν − M2 + (2 or 3)hν → M+ 2 +e
REMPIa REMPI
a REMPI = Resonance-enhanced multiphoton ionization, where usually two or three photons are
involved. Two-color rather than one-color ionization is also frequently used, and ions can be detected with or without mass resolution.
u = 112
1g
114
116
118
120
122
42 S1/2 + 42 P3/2
124 126
Trap-loss spectrum
0u+
Fragmentation spectrum
13,035.1
13,037.1
13,041.1
13,039.1
13,043.1
–1)
Laser frequency (cm
FIGURE 5.6 High-resolution trap-loss spectrum and fragmentation spectrum for PA of 39 K in the region within 8 cm−1 of the 4s + 4p3/2 asymptote. (From Stwalley, W.C. and Wang, H., J. Mol. Spectrosc., 195, 194, 1999. With permission.)
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Ultracold Molecule Formation by Photoassociation
First, although many nonalkali metal atoms have been photoassociated at ultracold temperatures (e.g., H, He(2 3 S), Ca, Sr,Yb), we are not aware of any examples of direct detection of ultracold nonalkali metal molecules formed by PA. Thus we focus on alkali atoms and molecules in this chapter. Second, the overwhelming majority of one-color PA studies are at the two nmin s + nmin pJ alkali metal asymptotes (nmin = 2, 3, 4, 5, and 6 for Li, Na, K, Rb, and Cs; J = 1/2, 3/2). One fascinating study has been carried out by M. Pichler and colleagues at the Cs(6s) + Cs(7pJ ) asymptotes [18,19]. Although the spectra are weak and the data are limited and accurate theory is not yet available, it is clear that there are hyperfine–split PA resonances on the blue as well as the red side of the resonance lines. The blue features are thought to be due to quasibound states associated with potential barriers that are expected to occur at these asymptotes. Of course, many states observed between the nmin s + nmin p3/2 asymptote and the nmin s + nmin p1/2 asymptote can predissociate and are thus quasibound. An example is the predissociation of the 0u+ (p3/2 ) state of K2 [20] shown in Figure 5.6, detected by the fragmentation technique [12]. Thus far we have considered only one-color PA (usually one photon as well, although in the initial Na2 PA work [10], a second photon of the same color ionized the photoassociated molecule). However, two-color two-photon PA is a powerful extension of the one-color technique. There are two varieties as shown in Figure 5.7: the “ladder” and the “lambda.” The ladder technique could also be called pump–probe Wg or u(u, J )
Probe
Wu¢ or g(u¢, J ¢)
Wu¢ or g(u¢, J ¢)
Pump
Pump
W≤g or u(k≤, J ≤)
W≤g or u(k≤, J ≤)
“Ladder”
Dump
Wg or u(u, J) “Lambda”
FIGURE 5.7 The two schemes for free-bound-bound two-color two-photon PA (optical– optical double resonance): the “ladder” and the “lambda,” both starting in a continuum state Ωg/u (k , J ). (From Stwalley, W.C. and Wang, H., J. Mol. Spectrosc., 195, 194, 1999. With permission.)
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optical–optical double resonance (PPOODR), and the lambda technique could also be called pump–dump optical–optical double resonance (PDOODR), or a stimulated Raman process. The lambda approach is the focus of Chapters 6, 7, and 8, because formation of ultracold X- and a-state molecules is the goal. The critical difference compared to ordinary OODR is that the initial state is a continuum state (albeit with an ultracold kinetic energy). An impressive example of ladder two-color PA is given in Ref. [21], where Na2 doubly-excited autoionizing states in a potential curve slightly outside the Na+ 2 potential energy curve were formed and mapped out, as schematically illustrated in Figure 5.8. Another interesting example is presented in Ref. [23], where K2 highly excited weakly bound long-range states near the (4s + 6s, 7s, 4d, 5d, and 6d) asymptotes [22] were formed (Figure 5.9) and where autoionization of some but not all levels above the K+ 2 (v = 0, J = 0) level was observed. A final example of particular relevance to this book is based on a Band–Julienne proposal [24] for ladder two-color PA. This scheme has been used with K2 to produce ultracold Na+2
P3/2 + P3/2
V2 (PI)
1g Ion yield
0g–
S1/2 + P3/2
V1 (PA)
S1/2 + S1/2
FIGURE 5.8 Schematic diagram of a ladder two-color PA experiment in Na2 . The PA laser (red-detuned) is fixed on a rovibrational level of either the singly-excited 1g or the 0g− potential, while the photoionization (PI) laser (blue-detuned) is scanned through the doubly-excited 3p3/2 + 3p3/2 asymptote. Ions produced by autoionization below this asymptote are due to excitation of doubly-excited states. (From Amelink, A. et al., Phys. Rev. A, 61, 042707, 2000. With permission.)
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Ultracold Molecule Formation by Photoassociation
X 1 Σ+ g (v = 0) molecules [25], as discussed in Section 5.2.2.1. Other means of forming X state v = 0 ultracold molecules are discussed later in this section and in Section 5.2.2. The lambda two-color PA approach has been used extensively to study the complex molecular structure in the region near dissociation to two ground state atoms, for example, in Li2 [27–29], Rb2 [26] and K2 [30], the topic focused on in Chapter 6. An example from Ref. [26] is shown in Figure 5.10, where the “Raman”
62S1/2
1u(4s + 6s) u* +1
u* +2 u* +3
u*
u* +4
Ion signal (arb. units)
52D5/2 52D3/2 1u(4s + 5d)
u* +2 u* +1 u* +3
u*
1u or 0u– (4s + 6d) u* +2
62D5/2 62D3/2
u* +1 u*
0.0
u* +3
1.0
3.0 2.0 Relative laser frequency (cm–1)
4.0
5.0
FIGURE 5.9 Two-color ladder-type optical–optical double-resonance PA spectra via the 0g− (4p3/2 ) v = 0, J = 2 level. The panels show the spectra just below the 4s + 6s asymptote (top), the 4s + 5d asymptote (middle) and the 4s + 6d asymptote (bottom). Comparison with theoretical calculations suggests that the absolute vibrational quantum numbering is as shown, with v ∗ = 0 [22]. (From Normand, B. and Stwalley, W.C., J. Chem. Phys., 121, 285, 2004. With permission.)
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frequency difference ν2 − ν1 directly determines the binding energies of 12 levels within 20 GHz of the lowest-energy hyperfine asymptote for two ground-state 85 Rb atoms. More generally, the lambda two-color PA approach can provide the first step toward forming X (v = 0, J = 0) homonuclear molecules. Such molecules (and possibly X (v = 0, J = 1) homonuclear molecules in the absence of atoms as well) at ultracold temperatures should be very resistant to collisional relaxation and thus should readily Bose–Einstein condense (they are all automatically composite bosons). Note that the near-degeneracy of a larger number of |I, mI nuclear spin states
52S1/2 + 52P1/2
12,580
12,570 0g–
12,560
n2
n1
ns
V (R) (cm–1)
52S1/2 + 52S1/2 0
–0.5
F=6 F=5 F=4
–1 20.0
30.0
40.0
50.0
R(a0)
FIGURE 5.10 Two-color lambda-type photoassociation spectroscopy of 85 Rb2 . Colliding, trapped ultracold 85 Rb atoms are irradiated by lasers of frequencies ν1 and ν2 . Spontaneous emission from the excited level at frequency νs leads to loss of the atoms from the trap. Optical–optical double-resonance signals (free-bound-bound) are observed when the frequency difference ν2 − ν1 coincides with the binding energy of a lower-state vibrational level. (From Tsai, C.C. et al., Phys. Rev. Lett., 79, 1245, 1997. With permission.)
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Ultracold Molecule Formation by Photoassociation
181
may be a significant complication in achieving BEC, as discussed in Ref. [1] and Section 5.1.3.4 below. However, it is theoretically predicted and experimentally established that at small detunings, one-color PA followed by spontaneous emission forms X and a state ultracold molecules in very high vibrational levels, as discussed in Section 5.2.1. For −1 of disexample, in 85 Rb2 , the X 1 Σ+ g state levels v = 111–117 (within 13 cm sociation) are observed [33]. Even higher levels are also predicted to form, but are quickly photodissociated in the experiment. Virtually no population goes to v = 0 −1 and in 39 K2 , X 1 Σ+ g , the v = 35 and 36 levels (bound by ∼1900 cm ) are the lowest levels with a detectable population [31]. It is possible, however, that a stimulated Raman transfer (free → bound → bound) from the X 1 Σ+ g continuum to a level like v = 36 in K2 could be efficient; a second stimulated Raman transfer from K2 v = 36 to v = 0 (bound → bound → bound) could then be nearly 100% efficient as noted long ago [1] (Figure 5.11). Further discussion of stimulated Raman transfer occurs in Sections 5.1.3.4 (quantum degenerate gases), 5.1.3.5 (external fields and Feshbach resonances), and 5.2.2.4 (Raman transfer to deeply-bound levels), as well as in Chapters 7 and 8 on ultrafast PA. It is important to realize that there is an extensive theoretical and spectroscopic foundation for assigning and understanding PA spectra of the homonuclear alkali dimers [7,13,34]. The long-range potentials, including their interactions (“Movre– Pichler” analysis), are accurately known; the example of K2 is shown in Figure 5.12. High-quality ab initio calculations of ground and excited states give potential energy curves at shorter internuclear distances. The correlations of the short-range and longrange states is shown for K2 in Figure 5.13. These ab initio calculations also provide dipole moments and transition dipole moments for all alkali metal dimers [13,35,36]. Solutions of the radial Schrödinger equation using these potentials (or experimentally based potentials such as RKR, IPA) provide the rovibrational energies, Ev,J , and radial wavefunctions ψv,J (R) for both bound and free (continuum) states. Combining initial and final radial wavefunctions and energies, one can obtain Franck–Condon factors, |v , J |v , J |2 , for estimating relative intensities of all absorption and emission processes, where v , J label the upper electronic state and v , J label the lower electronic state. If the transition dipole moment function between the two electronic states is available, the absolute values of Einstein A and B coefficients (for spontaneous emission, and for stimulated emission and absorption, respectively) can be accurately calculated. Other processes (e.g., predissociation of PA-excited levels [8]) can also be calculated given the appropriate matrix elements between radial wavefunctions. Normally it is assumed that emission from the states just below the nmin 2 2S 1/2 + nmin P1/2,3/2 asymptotes leads directly and exclusively to molecules in the 1 + X Σg state or the a 3 Σ+ u state, depending on whether the upper state is of u or g symmetry. Recently, this has been questioned for Rb2 [33] and Cs2 [37]. For example, spontaneous emission from the 1g ∼ 1 1 Πg state to 0u+ ∼ A 1 Σ+ u can then lead 1 Σ+ emission to low-v levels of the X state (rather than the a state). → X to A 1 Σ+ u g Moreover, optical pumping from all low-v levels except v = 0 can lead to an accumulation of population in the v = 0 level [37,38], with 70% of a population of Cs2 molecules ending up in the v = 0 level in the former reference. © 2009 by Taylor and Francis Group, LLC
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20
Energy (cm–1 ¥ 103)
u ª191
K(4s) + K(4p)
A 1Su+
15 u =18
PA
10
STIRAP
X 1Sg+
5
K(4s) + K(4s)
u = 36 0
0
10
20
30
40
R (Å) FIGURE 5.11 A combination of the demonstrated formation of X 1 Σ+ g (v = 36) transla39 tionally ultracold K2 molecules [31] with the well-documented stimulated Raman adiabatic passage (STIRAP) technique [32] could produce small numbers of translationally ultracold 39 X 1 Σ+ g (v = 0) K2 molecules. (From Bahns, J.T. et al., Adv. At. Mol. Opt. Phys., 42, 171, 2000. With permission.)
It is also normally assumed that highly detuned (say >100 cm−1 ) levels are difficult to form by one-color PA. However, the earliest example of Rb2 PA showed this not to be the case [11]; PA was observed for detunings up to 953 cm−1 . Another recent example is LiCs (see Section 5.1.3.2) where the B 1 Π v = 4 level (with classical turning points at ∼3.5 and 5 Å) was formed by PA. This level subsequently radiatively decays to levels as low as v = 0 in the X state [39]. It has been predicted that PA 1 + into the 3 1 Σ+ u state of Cs2 would produce X Σg (v = 0) molecules with ∼8% efficiency [40]. In the future, significant effort might be saved by identifying states at large red detunings that could efficiently directly populate the v = 0 level of the X state, particularly when enhanced by the potential barrier of a double-minimum potential or by resonant coupling, as discussed in Sections 5.2.2.2 and 5.2.2.3, respectively. 5.1.3.2
Unlike Atoms
It has long been recognized that colliding pairs of different atoms could be photoassociated to form polar heteronuclear molecules [41]. However, it is only © 2009 by Taylor and Francis Group, LLC
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Ultracold Molecule Formation by Photoassociation
300
39K 2
Hund’s case (c) potentials
100 0–g 0u–
2g
1g 1u 0–g
E (wavenumber)
200
1u 42S1/2 + 42P3/2
0+g 1g
0
1u
2u
42S1/2 + 42P1/2
0u+
–100 0–g
0u–
0u+
1g
–200
–300 10
20
30
40
50
R (Å)
FIGURE 5.12 The 16 adiabatic Hund’s case (c) potential energy curves of K2 dissociating to the 4s1/2 + 4p3/2 and 4s1/2 + 4p1/2 asymptotes, based on the C3 , C6 , and C8 values of Ref. [7]. The solid curves are the seven observed states that support bound states and are optically allowed from the 4s1/2 + 4s1/2 asymptote. (From Stwalley, W.C. and Wang, H., J. Mol. Spectrosc., 195, 194, 1999. With permission.)
recently [42] that the first such ultracold polar molecule, RbCs, was formed by PA, and still there are only four polar molecules formed by PA using published ultracold PA spectra: RbCs [42,43], KRb [44,45], NaCs [46], and LiCs [39]. PA spectra were recently published for YbRb, but the corresponding ground-state molecules have not yet been detected [47]. We will emphasize the example of KRb here because it is the most completely studied. © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications 0 +g 1g 0 –u 2 3S +u
2g 1
1 2Pg
+ g
3S + S u +2C3/R3
1g 1u
1 1Pg 1P
0 –g
3P g
u
B 1Pu
42S1/2 + 42P3/2
2u
+C3/R3
1g 2 1S +g
1
3
Pg
0 +u
Pu
–C3/R3
1 3S +g
0 +g 1S + u
A 1S +u
3S + g
–2C3
1u
/R3
0 u–
b 3Pu
0 –g
42S1/2 + 42P1/2
1 +g 0 u+ Hund’s case (c)
Hund’s case (a) a 3S u+ X1S g+
1S + g
3S + u
0 +g 0 u– 1 u
42S1/2 + 42S1/2
–C6/R6 I
RLR
II
RFS
III
IV
R
FIGURE 5.13 Correlation diagram for electronic states leading to the three lowest asymptotes of K2 . States at short internuclear distances are described by Hund’s case (a) or (b) quantum numbers. As R increases, the Ω components of the short-distance states split into a variety of Hund’s case (c) states, especially as the spin–orbit splitting of the 4p state exceeds the magnitude of the interaction potential. (From Stwalley, W.C. and Wang, H., J. Mol. Spectrosc., 195, 194, 1999. With permission.)
The potential energy curves of KRb at short (Figure 5.14), long (Figure 5.15), and intermediate internuclear distance (Figure 5.16) provide a detailed understanding of eight different attractive Hund’s case (c) electronic states excited in PA. At long range, there is a lower triad (2(1), 2(0+ ), and 2(0− )) dissociating to the K(4s) + Rb(5p1/2 ) asymptote, and a second triad (3(1), 3(0+ ) and 3(0− )) and a dyad (4(1) and 1(2)) dissociating to the K(4s) + Rb(5p3/2 ) asymptote, also shown in Figure 5.15. © 2009 by Taylor and Francis Group, LLC
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33S+ 15
KRb
23P
4P + 5S 21P
11P
4S + 5P
31S+
Energy (¥ 103 cm–1)
10 23S+ 21S+
5
KRb
13P
a3S+ 4S + 5P
0
X1S+ –5
5
15
10
20
25
R(a0)
FIGURE 5.14 Potential energy curves (in 103 cm−1 vs. R in a0 ) of the KRb molecule based on the high quality ab initio calculations of Rousseau and colleagues [48]. (From Wang, D. et al., Eur. Phys. J. D, 31, 165, 2004. With permission.)
A PA segment shown in Figure 5.17 shows assignments of seven of the eight states, which were observed as expected (using KRb+ ion detection). As R is decreased, there are many crossings of the electronic states shown in Figure 5.16 and also in Figure 5.18 (a correlation diagram similar to Figure 5.13 for K2 ). Figure 5.16 is essentially a “road map” for predissociation of levels between the 4s1/2 + 5p1/2, 3/2 asymptotes, such as the known predissociation of the 1 1 Π ∼ 4(1) and 2 1 Π ∼ 5(1) states [49]. For example, the 4(1) state mixes asymptotically with the 1(2), while the 1(2) state mixes with the other components of the 2 3 Π state near 13a0 , which then mix with the 2 3 Σ+ state near 12a0 , finally predissociating via the 2 3 Σ+ state to the 4s1/2 + 5p1/2 asymptote. The 5(1) case is even more complex, probably beginning with its mixing with the 4(1) state itself. These complex multichannel predissociations have not yet been tackled theoretically, unlike simple two-channel “curve-crossing” predissociations (e.g., predissociation of the 1g state of K2 by the 0g+ state [8]). © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications 13.2 1
2
0+
0– 1
Energy (×103 cm–1)
13.0 0+
12.8
2
0–
K(42P1/2) + Rb(52S1/2)
1
1 K(42S1/2) + Rb(52P3/2)
1 12.6
K(42P3/2) + Rb(52S1/2)
0–
0+ K(42S1/2) + Rb(52P1/2)
0–
1 0+
20
30
40 R (a0)
50
60
FIGURE 5.15 The 16 excited long-range Hund’s case (c) potential energy curves of KRb at the K(4p1/2, 3/2 ) + Rb(5s1/2 ) and K(4s1/2 ) + Rb(5p1/2, 3/2 ) asymptotes. Note that the potentials at the two lower asymptotes are attractive while those at the two upper asymptotes are repulsive. (From Wang, D. et al. Eur. Phys. J. D, 31, 165, 2004. With permission.)
A summary of the differences between like-atom and unlike-atom PA is given in Table 5.2. The dipole moment functions of the heteronuclear alkali metal dimer X and a states have recently been calculated accurately and reviewed [36]. Unlike homonuclear dimers, which can emit only quadrupole moment radiation with extremely long lifetimes (at least days), heteronuclear dimers can radiate. In KRb, which has a relatively small dipole moment, all vibrational levels have at least a 103 sec radiative lifetime, the levels near dissociation being far longer lived [50]. For the most short-lived heteronuclear alkali dimer (probably LiCs), the simple scaling of dipole moment squared times vibrational frequency cubed [50] suggests a lifetime of at least 0.4 s, using the recent dipole moment calculations of Ref. [36]. The dipole moment expectation values in various X 1 Σ+ vibrational levels vary greatly as well, being much smaller near dissociation. Some examples of these important dipole properties from Ref. [50] are listed in Table 5.3. Table 5.3 shows the wide variation in dipole properties in the X 1 Σ+ state of KRb, with the (molecule-fixed) dipole moment expectation values decreasing by almost 104 between v = 98 and v = 0. However, the levels observed in ultracold molecule formation by PA (v = 86–92) have dipole moments only one order of magnitude smaller than v = 0. It is interesting that v = 1 is longer lived than most levels except very near dissociation. The levels to which the strongest emission takes place are the © 2009 by Taylor and Francis Group, LLC
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Ultracold Molecule Formation by Photoassociation 14 33S+
23P
J
Energy (×103 cm–1)
13
K
Rb
3/2 4pJ + 5s1/2 21P
4(0+)
1/2
1(2)
3/2 4s1/2 + 5pJ 1/2
4(1) 3(1)
12
33(0+), 11P
3(0–)
2(0–), 2(1) 2(0
+)
11 PA
23S+
31S+ 13P 10 10
21S+ 15
20
25
R (a0)
FIGURE 5.16 Potential energy curves at intermediate distances showing the correlations between the potential curves of the short-range states in Figure 5.14 (2S+1 Λ± ) and those of the long-range states in Figure 5.15 (Ω± ). Also shown is the approximate minimum PA distance (∼23a0 ) for the lower triad corresponding to a level detuned 95 cm−1 below the K(4s1/2 ) + Rb(5p1/2 ) asymptote. (From Wang, D. et al., Eur. Phys. J. D, 31, 165, 2004. With permission.)
next-lower levels only for low v. For v = 86 and 92, the largest Einstein A coefficients are to v = 76 and 78! An important distinction between homonuclear and heteronuclear molecules is the form of the long-range interaction between the ground-state atom and its excited partner: C3 /R3 (dipole–dipole) vs. C6 /R6 (van der Waals). This has important consequences for both the free-bound absorption and the subsequent bound-bound emission [35], and thus the relative merits of trap-loss and REMPI detection, as discussed in Section 5.2.1. Regarding REMPI detection, it is also worth noting that the alkali metal diffuse bands (both heteronuclear and homonuclear) [52] are often at wavelengths for which this process is efficient. Many of these bands originate (in absorption) in the shallow a 3 Σ+ (u) states where 3/4 of the collisions of ground-state atoms occur. In addition, it is worth noting that (unlike homonuclear dimers) all states at the 4s1/2 + 5p1/2, 3/2 asymptotes have optically allowed transitions to the a 3 Σ+ state. Moreover, at small detunings, this emission is either exclusively or predominantly to the a 3 Σ+ state, as indicated in Table 5.4. This is in contrast to homonuclear dimers where u/g symmetry causes upper g states to emit only to the a 3 Σ+ u state while upper u states emit only to the X 1 Σ+ state. g © 2009 by Taylor and Francis Group, LLC
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10.4
8.5 2(1)
2(0–)
4(1)
7.1 5.6
6.7
3(0–)
2(0+)
15.7
2(0–)
3(0+)
39K 85Rb+
signal (arb. units)
12.4
3(1) 10.9
12,546
12,548 PA frequency (cm–1)
FIGURE 5.17 PA spectrum for 39 K85 Rb showing eight vibrational bands (from seven electronic states) in the region 12,545.3–12,549.8 cm−1 . Preliminary state numbers and Ω(±) symmetry values are given for each band. It should be noted that many levels are perturbed and thus not well represented by a single vibronic state. Also given above the symmetry is 103 Bv in cm−1 , with smaller Bv (rotational constant) corresponding to a longer-range level, that is, one with a larger outer turning point. All eight state symmetries (including Ω = 2, not occurring in this region) can be reached by dipole-allowed transitions from colliding ground state atoms (Ω = 0± , 1). (From Wang, D. et al., Eur. Phys. J. D, 31, 165, 2004. With permission.)
Finally, as noted above, we believe that predissociation will be more common in the p3/2 − p1/2 gap in the heteronuclear case than in the homonuclear case. Not only is the u/g symmetry gone in the heteronuclear case, but there are more curve crossings and heterogeneous (ΔΩ = ±1) as well as homogeneous (ΔΩ = 0) couplings. The KRb case also puts the various states very close together (see Figure 5.16) because of the near degeneracy (within 800 cm−1 ) of the 4s1/2 + 5p1/2,3/2 and 4p1/2, 3/2 + 5s1/2 asymptotes. Unlike the homonuclear case, there apparently has been no two-color PA, either of the “ladder” or “lambda” variety, in heteronuclear molecules. One reason may be the development of techniques for spectroscopy of the ultracold molecules themselves, described in Section 5.2. 5.1.3.3
Involving Molecules
It is clear that atom–molecule and molecule–molecule colliding pairs could be photoassociated to form polyatomic molecules. However, we are not aware of any results © 2009 by Taylor and Francis Group, LLC
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Ultracold Molecule Formation by Photoassociation Short range
Long range
Asymptotes
5(0+) 5(0–) 7(1) 2(2) 6(1)
3 3S+ [3] 2 3P [6] 2 1P [2] 1 1P [2] 3 1S+ [1] 2 1S+ [3] 2 1S+ [1] 1 3P+ [6]
4p 2P3/2 + 5s 2S1/2 [8]
5(1) 4(0–) 4(0+) 1(2) 4(1) 3(0+) 3(1) 3(0–) 2(0–) 2(1) 2(0+)
[Total short distance degeneracy 24]
4p 2P1/2 + 5s 2S1/2 [4] Dyad 4s 2S1/2 + 5p 2P3/2 [8] Upper triad Lower triad 4s 2S1/2 + 5p 2P1/2 [4] [Total asymptotic degeneracy 24]
Hund’s case (c)
Hund’s case (a) or (b)
1(0–) 1(1) 1(0+)
a 3S+ X 1S+
4s 2S1/2 + 5s 2S1/2
FIGURE 5.18 Correlation diagram for the 4s + 5s and 4s + 5p1/2, 3/2 asymptotes of KRb.
at this time, primarily because of the lack of dense samples of ultracold molecules until very recently. For a cold molecule like para-H2 in a single quantum state, such PA should be relatively strong. The particular case of the PA of Cs + para-H2 just below the Cs 7p1/2 + para-H2 and the Cs 7p3/2 + para-H2 asymptotes has been proposed for study [53], because the reactions Cs(7p1/2,3/2 ) + H2 → CsH + H
(5.1)
are well known at higher temperatures. The competition between reaction and radiative decay of the excited Cs–H2 complex at low temperatures may be quite interesting. 5.1.3.4
In a Quantum Degenerate Gas
This section and the next two are very much intertwined, particularly as techniques based on magnetically tunable Feshbach resonances and on a wide variety of optical lattices are increasingly used in studies of quantum degenerate gases. Our goal here © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
TABLE 5.2 Comparison of Formation of Homonuclear and Heteronuclear Alkali Metal Diatomic Molecules in Bound Excited States by PA Homonuclear
Heteronuclear
No dipole No IR emission Strong excited-state LR (C3 /R3 ) Strong PA Weak bound → bound emission Comparable X and a molecule formation REMPI of X and a states efficient REMPI at diffuse bands works well Trap-loss detection works well u, g symmetry (ignoring hyperfine coupling) Emission either to X or a Occasional predissociation in p3/2 − p1/2 gap
Small to large dipole Slow IR emission (τ > 0.1 s) Weak excited-state LR (C6 /R6 ) Weak PA Strong bound → bound emission Comparable X and a molecule formation REMPI of X and a states efficient REMPI at diffuse bands works well Trap-loss detection is difficult No u, g symmetry Always emission to a, sometimes to X Frequent predissociation in p3/2 − p1/2 gap
LR, long-range potential; IR, infrared; REMPI, resonance-enhanced multiphoton ionization.
is to somewhat separately introduce a few relevant topics related to each of these intertwined subjects. The first quantum degenerate gaseous systems were ensembles of bosonic alkali metal atoms in specific quantum states (Bose–Einstein condensates, BECs). With
TABLE 5.3 Examples of Dipole Properties for Vibrational Levels v (with J = 0) in the X 1 Σ+ Ground State of 39 K85 Rb v 0 1 4 12 40 60 80 86a 92b 98c
Binding Energy v (cm−1 )
Dipole Moment Expectation Value μv,0 (eÅ)
−4179.5524 −4104.1648 −3880.7997 −3306.1028 −1561.8206 −635.5175 −95.8234 −29.7656 −4.0526 −0.0036
0.136 0.136 0.136 0.134 0.117 0.088 0.036 0.018 0.0042 0.000017
a Lowest level observed in PA in Ref. [51]. b Highest level observed in PA in Ref. [51]. c Last level predicted [50]. d The most likely Δv in emission [50].
© 2009 by Taylor and Francis Group, LLC
Radiative Lifetime τv,0 (103 sec)
Δvd
— 142.9 32.11 7.935 1.265 1.013 2.044 4.048 15.67 1719
— −1 −1 −2 −2 −3 −7 −10 −14 −20
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TABLE 5.4 Asymptotic (∼50a0 ) Relative Singlet (→1(0+ ) ∼ X 1 Σ+ ) and Triplet (→1(0− ) and 1(1) ∼ a 3 Σ+ ) Spontaneous Emission from Various Hund’s Case (c) States Correlating with the K(4s) + Rb(5pJ ) Asymptotes Fraction Asymptote
Upper State
Singlet
Triplet
4s + 5p1/2
2(0+ )
0.18 0.00 0.22 ∼0.25a 0.00 ∼0.25a ∼0.25a 0.00
0.82 1.00 0.78 ∼0.75a 1.00 ∼0.75a ∼0.75a 1.00
4s + 5p3/2
2(0− ) 2(1) 3(0+ ) 3(0− ) 3(1) 4(1) 1(2)
a Statistical; transition moments not available.
typical BEC temperatures of 100 nK and typical atom densities of 1014 atoms/cm3 , compared to MOT values of 100 μK and 1011 atoms/cm3 , PA becomes much faster and linewidths much narrower. For example, the landmark work of Wynar and colleagues [54] found two-color PA linewidths as small as 1.5 kHz vs. a typical 10–100 MHz in MOTs. The narrow linewidths have allowed study of atom– condensate and molecule–condensate mean field effects [54] and of small spectral light shifts [55,56]. The very high PA rate was first observed in 23 Na [57], where it was also established that there was a factor of two reduction in the PA rate in a BEC as predicted theoretically [13]. Recently, the long-sought saturation of this very high rate has been observed in 7 Li [56]. The various issues and theoretical predictions (e.g., Refs. [58–60]) are nicely reviewed in Ref. [13]. Several years ago it was suggested that it might be possible to coherently oscillate back and forth between an atomic BEC and a “molecule” BEC [61,62]. It should be noted that the “molecules” were to be formed in a very weakly bound metastable level (e.g., bound by only 636 MHz in Ref. [54]), not in v = 0, J = 0 of the electronic ground state. Further work [58,59,63,64] has not fully resolved the issue. Quantum degenerate atomic Fermi gases should also readily photoassociate. We are not aware of much work in this area so far, although there is an interesting theoretical paper on two-color spectroscopy of fermions in the BEC–BCS crossover region [65]. In contrast, magnetoassociation (MA) of Fermi gases near Feshbach resonances (next section and Chapters 9 to 11) has been extensively studied, and it is now understood that PA can be considerably enhanced near these same Feshbach resonances [3,4]. © 2009 by Taylor and Francis Group, LLC
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Finally, we note the interesting suggestion in Ref. [13] regarding the possibility of PA directly into the electronic ground state with microwave or even radio frequency (RF) photons, given the low temperature and high density of quantum degenerate gases. Molecule formation with RF (or even lower frequencies) has been utilized [66–68], and in heteronuclear systems with electric dipole moments, PA with microwaves [69] or infrared [70] photons has been proposed. Radio frequencies have also been used (e.g., by the Innsbruck group) for bound → bound and bound → free spectroscopy of Li2 and Cs2 levels very near dissociation [71–75]. 5.1.3.5
In an Electromagnetic Field
Because studies of ultracold atoms and molecules usually involve atomic and molecular traps and manipulations with electromagnetic fields, the impact of these external fields on atomic and molecular energy levels, line widths, scattering properties, and so on must usually be considered. At weak field strengths, there are well-understood Stark and Zeeman shifts. Far more interesting are the magnetically tunable Feshbach resonances that occur at weak to moderate magnetic fields, which allow precise control of interatomic interactions by static magnetic fields [76–78] (see also Chapters 1, 4, 6, 9, 10, and 11). Such resonances are an elegant and powerful tool for manipulation of ultracold gases. As the magnetic field is tuned toward a resonance from lower field, the resonance energy above the collisional asymptote drops toward zero. As this occurs, the scattering length a diverges to −∞. When a new bound state appears it switches sign to +∞, and then decreases as the field is increased further. The case of 85 Rb is particularly interesting, because the zero-field scattering length is negative, and hence a large Bose–Einstein condensate of 85 Rb is unstable [79]. However, by tuning to fields above ∼155 g (as shown in Figure 5.19), the scattering the scattering length becomes positive and a stable BEC can be created [79]. A note on terminology: the Feshbach resonance threshold (the magnetic field where the resonance energy is at the collisional asymptote) is often simply referred to as the “Feshbach resonance.” The negative-energy bound state of a molecule below the threshold but obviously correlated with a specific Feshbach resonance is sometimes referred to as a “Feshbach molecule.” There are other effects of the rapid change in scattering length with magnetic field. The elastic cross-section (8πa2 for identical particles) becomes extremely large near resonance, and goes to zero when a = 0 [80], as do the inelastic cross-sections. It has also recently been noted that Feshbach resonances near threshold can enhance the rate of PA by several orders of magnitude [3,4]. Magnetic Feshbach resonances may be conveniently detected using PA as in Ref. [81]. Such resonances have been observed by a variety of techniques in a large number of homonuclear and heteronuclear systems as discussed in Chapters 9 to 11. Bose–Einstein condensation of pairs of fermions magnetoassociated to form 6 Li2 or 40 K “Feshbach molecules” was achieved in 2003 [82–85]. We also note the very 2 interesting crossover between such BECs and the so-called BCS quantum gases, where there are no two-body bound molecules but rather many-body bound Cooper pairs [86]. Finally, we point out that the transfer from a Feshbach molecule to a bound © 2009 by Taylor and Francis Group, LLC
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6000
–4000
4000
145
160
175
2000
0 157
159
161 Magnetic field (G)
163
165
FIGURE 5.19 Scattering length data (a0 ) as a function of the magnetic field. The solid line illustrates the expected shape of the Feshbach resonance, using a peak position and resonance width consistent with previous measurements. For reference, the shape of the full resonance has been included in the inset. (From Cornish, S.L. et al., Phys. Rev. Lett., 85, 1795, 2000. With permission.)
lower level is a lambda-type bound-bound Raman transition and not PA. These Raman processes are discussed in Section 5.2.2.4. Feshbach resonances can in principle also be observed by tuning a static electric field [87] or an optical (or RF, microwave, etc.) field [88,89] or a combination of both [87,90]. The electric field dependence of PA for polar molecules can now be accurately modeled [91,92]. 5.1.3.6
In an Optical Lattice
Optical lattices occupied by atoms and/or molecules represent a new and rapidly expanding frontier of “artificial” solid-state physics, in which the possibilities for lattice structure and site species are each enormous, perhaps a “quantum wonderland” [93–96]. The optical lattice (in 1D, 2D or 3D or “quasi-1D” or “quasi-2D”) is formed by standing waves of laser beams. Polarizable atoms or molecules experience a periodic potential with minima at the antinodes of the standing wave, separated by half of the laser wavelength. The first demonstration of an optical lattice was in 3D with Na atoms [97]. The barriers between periodic potential minima can be tuned with the laser intensity, from large barriers that produce strong localization (“crystallization”) to small barriers that allow hopping between sites (“melting” and even “superfluidity”). In principle, one could make lattices using other types of atomic and molecular traps (e.g., magnetic traps), but we will not discuss this topic here. © 2009 by Taylor and Francis Group, LLC
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It is possible to create an atomic Mott insulator phase in an optical lattice, in which sites in a given region are occupied by a specific integer number of atoms [98]. A lattice with two atoms per site is ideal for making diatomic molecules by PA (or MA) and protecting them from collisions; in a sense, PA is then converted from a free → bound process to a bound → bound process, because the two atoms are confined in the lattice site. Homonuclear and heteronuclear diatomic molecule formation in this and other ways has been proposed ([99] and [100], respectively) and more recently observed ([101–104] and [67,105], respectively). Similarly, sites with three atoms/site may be ideal for two-step PA to form trimers, by first making a dimer and then using a second PA step to make a trimer from the remaining atom and the dimer. Tetramers, pentamers, and so forth might be built up in more heavily loaded lattices. Also of interest is the formation of repulsively “bound” pairs of atoms in an optical lattice [106], made possible by the lack of dissipation (phonons) in an optical lattice.
5.2 5.2.1
CHARACTERIZATION AND STATE-TO-STATE TRANSFER OF ULTRACOLD ALKALI DIMERS LEVELS NEAR DISSOCIATION
Photoassociation to an excited electronic state is the first step in ultracold molecule formation. It must be followed by emission to either the lowest singlet (X) electronic state or lowest triplet (a) state in order to form molecules that are stable against radiative decay. In general, the photoassociation process favors high vibrational levels near the excited-state dissociation limit. This is due to the better overlap of these levels with the initial continuum wavefunction of the colliding atoms. These excited states have their outer turning points at large R, so when they decay to the X or a states they tend to populate high vibrational levels. Starting with the first report of ultracold molecule formation by PA [107], detection of the X- or a-state molecules has usually used REMPI with a pulsed laser, followed by time-of-flight (TOF) mass analysis in order to distinguish dimer ions from atomic ions. This detection method can be both efficient and state selective. By scanning the frequency of the ionizing laser, the distribution of population among the various vibrational levels can be mapped out. This state-selective detection is important to many of the applications of ultracold molecules, such as state transfer and ultracold chemistry. Of course, for it to work, the bandwidth of the pulsed laser must be less than the spacing between vibrational levels. In our work in Rb2 and KRb, a pulsed dye laser bandwidth of 0.2 cm−1 is sufficient to resolve the vibrational levels typically populated. However, it should be pointed out that in initial measurements, when searching for both PA lines and detection lines, a bandwidth broad enough to ionize a range of levels can actually be a benefit. In the following, we briefly describe our results for the production and detection of high-lying X- and a-state molecules in both KRb and Rb2 , focusing on the case of KRb(X). As mentioned earlier (Section 5.1.3.2), one of the main differences between heteronuclear KRb and homonuclear Rb2 involves the long-range excited-state potentials. In KRb, the excited-state potentials behave as C6 /R6 at long range, while in Rb2 , © 2009 by Taylor and Francis Group, LLC
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the long-range excited-state potential is usually C3 /R3 . For a given PA detuning, the PA will take place at a smaller R in KRb than in Rb2 . Because the ground-state potentials for KRb and Rb2 are similar, both scaling as C6 /R6 at long range, the Franck–Condon factors for decay favor more deeply-bound ground-state vibrational levels in KRb than in Rb2 . In KRb, the use of ionization detection is particularly important because observing PA-induced trap loss is much more difficult than in a homonuclear system such as Rb2 . This is again due to the character of the long-range excited-state potentials. The C6 /R6 potential for KRb means that PA takes place at relatively short range where there are fewer atom pairs available. However, when PA does occur, the probability for decay to a bound level of the ground-state potential, as opposed to dissociation back into the continuum, is much greater for KRb than for Rb2 . So both systems tend to produce similar numbers of molecules, but the PA step is more favorable in Rb2 , while the decay step to produce molecules is more efficient in KRb. Another issue in trap-loss detection is the competition between homonuclear and heteronuclear trap loss in a given experiment, making assignments quite difficult. For example, when searching for KRb trap loss just below the 4s1/2 + 5p1/2 asymptote, stronger trap loss is simultaneously observed from just below the Rb2 5s1/2 + 5p1/2 asymptote. However, for detection of KRb+ ions by REMPI from the X and/or a states of KRb, there is no competition from ions due to Rb2 PA (the Rb+ and Rb+ 2 ions are easily distinguishable by their differing TOFs). Thus we believe that REMPI TOF mass spectrometer detection of X- and a-state heteronuclear molecules is normally preferable to trap-loss spectroscopy. The KRb experiments take place in a dual-species vapor-cell MOT [45]. Diode lasers at 767 and 780 nm are used to cool and trap 39 K and 85 Rb atoms, respectively, in the dark-spot MOT configuration. Atomic densities of 3 × 1010 cm−3 for 39 K and 1 × 1011 cm−3 for 85 Rb are realized, with corresponding temperatures of 300 and 100 μK. Photoassociation is induced by illuminating the overlapping cold atomic clouds with light from a tunable cw titanium-sapphire laser (Coherent 89929). This laser typically provides 500 mW with a linewidth of ∼1 MHz and its output is focused down to approximately match the size (∼300 μm diameter) of the cold atom clouds. For ionization detection of the resulting X- or a-state molecules, we use a pulsed dye laser (Continuum ND6000), pumped by the frequency-doubled output of a Nd:YAG laser operating at 10 Hz. The dye laser pulses, typically >1 mJ in energy and 7 ns in duration, are focused down to ∼1 mm in diameter, somewhat larger than the MOT clouds. The molecules are produced continuously, but because they are untrapped, they expand ballistically and fall. A larger ionization beam means that a larger volume of this expanding cloud is illuminated, thus increasing the signal. Ions are accelerated into a nearby Channeltron detector. The intense pulsed laser can produce atomic ions by multiphoton ionization of the much more numerous atoms, and the MOTs themselves can produce homonuclear molecules that can be ionized. Because the ionization process is pulsed, the TOF to the detector, typically a few microseconds, is + + well defined for each ion mass (K+ , Rb+ , K+ 2 , Rb2 , KRb ). We use a box-car averager + to select the ion of interest, in this case KRb , and discriminate against the background of other ions. The ion signal can be used to determine the temperature of the ultracold © 2009 by Taylor and Francis Group, LLC
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molecules. Turning off the PA laser stops the formation of molecules. By varying the delay of the detection laser pulse, the ballistic escape of these molecules from the detection volume, due to their thermal motion (and gravity), can be mapped out. The high-v levels of KRb(X) can be efficiently ionized through the 4 1 Σ+ state, which correlates to the K(4s) + Rb(4d) asymptote, using light at ∼16,500 cm−1 . This is shown in Figure 5.20. A second photon of the same wavelength produces KRb+ ions. The first resonant excitation step is usually saturated, while the ionization step is not. We perform various types of spectroscopy using this pulsed ionization. For mapping out the PA spectrum, we set the detection laser frequency on a transition between ground-state and excited-state vibrational levels, and scan the PA laser. On the other hand, if we fix the PA laser on resonance, and scan the detection laser, we can determine the vibrational populations of the ultracold molecules [51] and also perform spectroscopy of the excited states used in the detection [108]. Because
32
28
KRb ++e–
5 1S+
20 3 3P
4 1S+ u¢
Energy (103 cm–1)
16 4
3S+
K(4S1/2) + Rb(5PJ)
uPA
12 1 1P 2 3S+
8
2 1S+ R2PI
1 3P
4
PA
SE
u¢¢
0
a 3S+
K(4S1/2) +
X 1S+
–4 2
4
6
Rb(5S1/2) 8
10 R (Å)
12
14
16
18
FIGURE 5.20 Scheme for producing and detecting ultracold KRb molecules. The molecules are formed by PA followed by spontaneous emission (SE), and subsequently detected by resonant two-photon ionization (R2PI). For high-v levels of the X 1 Σ+ state, the 4 1 Σ+ state is used as the intermediate state. (From Wang, D. et al., Phys. Rev. A, 72, 032502, 2005. With permission.)
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the high-v levels of the ground state are closely spaced compared to the relevant vibrational levels of the excited state, the two structures are easily distinguished. The ground-state pattern, with its typical spacing of 5 cm−1 , repeats with the excited-state spacing, which is typically 20 cm−1 . Examples of these widely spaced and closely spaced progressions are shown in Figures 5.21 and 5.22, respectively. By recording detection spectra at different PA detunings, we can observe how the ground-state vibrational populations vary with the PA process. As the PA laser is detuned farther below the dissociation limit, the PA takes place at smaller R, and as expected, the decay is to more deeply bound levels of the ground state. For example, we find that for PA to the 3(0+ ) state, which correlates to the K(4s1/2 ) + Rb(5p3/2 ) asymptote, a PA detuning of ΔPA = −246.66 cm−1 gives an X-state vibrational distribution that peaks at v = 89, while for ΔPA = −307.69 cm−1 , the population maximizes at v = 88. This comparison is shown in Figure 5.22. This trend, and even the details of the vibrational distribution for a given ΔPA , agree rather well with the calculated Franck–Condon factors for decay from the PA level. A very similar detection scheme is used for the a state of KRb. In this case, the pulsed detection laser excites the 4 3 Σ+ state, which is then ionized. Once again the vibrational structure due to the a state is readily distinguishable from that of the excited state because of their disparate spacings. Higher intensities are generally required for
KRb+ signal (arb. units)
v¢ 42
45
50
55
60
DPA = –246.66 cm–1
16,500
16,600
16,700
16,800
Detection frequency (cm–1)
FIGURE 5.21 Typical detection spectrum for KRb X 1 Σ+ molecules. Groups of lines are labeled according to the vibrational levels of the 4 1 Σ+ state used in the ionization process. (From Wang, D. et al., Phys. Rev. A, 72, 032502, 2005. With permission.)
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KRb+ signal (arb. units)
X1S+, u¢¢: 91
90
89
87
(a) DPA = –246.66 cm–1
16,520 X1S+, u¢¢: 91 KRb+ signal (arb. units)
88
90
16,540
16,530 89
88
87
(b)
86
DPA = –307.69 cm–1
16,520
16,530
16,540
Detection frequency (cm–1)
FIGURE 5.22 An expanded view of the detection spectrum shown in Figure 5.21, for v = 45. Here the lines are labeled according to the vibrational levels of the X 1 Σ+ state that are populated by PA. Two different PA detunings are shown in (a) and (b). The solid circles indicate calculated Franck–Condon factors for the spontaneous emission (SE) step in Figure 5.20. (From Wang, D. et al., Phys. Rev. A, 72, 032502, 2005. With permission.)
detection of triplet vs. singlet molecules. This causes the appearance in the spectra of some atomic lines due to two-photon excitation to Rydberg states (e.g., 5s → 13d). This signal leaks into the KRb+ TOF window because the atomic Rb+ signals are large and thus broadened in time by space charge. These features are easily identified as atomic by repeating the scan with the PA laser off, and can be useful as wavelength calibrations. The pulsed ionization detection method described above reveals which vibrational levels are populated, but the pulsed laser bandwidth is too broad to resolve rotation. We use a different technique, ion depletion [109], to observe individual rovibrational transitions, as shown in Figure 5.23. Pulsed ionization is used to monitor the population in a given vibrational level (e.g., X 1 Σ+ (v = 89)) and a single-frequency cw laser is used to deplete this population. When this depletion laser is resonant with a particular rovibrational transition (e.g., X 1 Σ+ (v = 89, J = 2) → 3 1 Σ+ (v , J )), a fraction of the v = 89 population will be depleted by optical pumping. As the depletion laser is scanned, rotationally resolved resonances are manifest as dips in the ion signal, examples of which are shown in Figure 5.24. Analysis of these spectra provides not only the rotational populations in the ground state, but also the rotational spacings in both the ground and excited states. The initial task of locating these depletion © 2009 by Taylor and Francis Group, LLC
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Ultracold Molecule Formation by Photoassociation
32
KRb ++e–
28
4 1S+
Energy (103 cm–1)
20
K(4S1/2) +
16
u¢
12
3
1 1P
1 S+
Rb(5PJ)
2 3S+
8
R2PI
2 1S+ 1
4
PA
DEP
3P
SE
DUMP u¢¢
0 a
3S+
K(4S1/2) +
X
1S+
Rb(5S1/2)
–4 2
4
6
8
10
12
14
16
18
R (Å)
FIGURE 5.23 Scheme for rotationally resolved depletion spectroscopy. The molecule formation (PA and SE) and ionization detection (R2PI) processes are as in Figure 5.20. A cw depletion laser (DEP) optically pumps molecules out of a particular X 1 Σ+ (v , J ) level via a bound–bound transition to 3 1 Σ+ (v , J ), resulting in a resonant dip in the ion signal. (From Wang, D. et al., Phys. Rev. A, 75, 032511, 2007. With permission.)
transitions was greatly facilitated by first performing low-resolution spectroscopy via pulsed ionization detection. In this spectral region (∼11,500 cm−1 ), a slight variation of the method described above is required. Two photons from the pulsed dye laser do not have sufficient energy to ionize, so 532 nm light from the frequency-doubledYAG laser is used for the ionization step. This two-color detection scheme is very versatile because it decouples the resonant excitation step, which can be easily saturated, from the ionization step, which requires high intensity. We have also applied vibrationally resolved ionization detection to the homo3 + nuclear case of Rb2 , investigating both the X 1 Σ+ g [33] and a Σu [110] states. The 3 + relevant excited states are 2 1 Σ+ u and 2 Σg , respectively. The technique is identical to that described above for the high-lying states of KRb, but the results are rather different. As discussed earlier, for a given PA detuning, the long-range C3 /R3 excitedstate potential for Rb2 should result in less deeply bound vibrational levels than for © 2009 by Taylor and Francis Group, LLC
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KRb+ ions (u ≤ = 89) (arb. units)
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1200 1000 1¨0
800 1¨2
600
3¨2
400 200 0 11,500.7
11,500.8
11,500.9
Pump laser frequency (cm–1)
FIGURE 5.24 Depletion spectrum showing dips in the X 1 Σ+ (v = 89) ion signal as the depletion laser is scanned through rotationally resolved resonances to 3 1 Σ+ (v = 40, J ). (From Wang, D. et al., Phys. Rev. A, 75, 032511, 2007. With permission.)
KRb. This is indeed observed: for KRb X 1 Σ+ , levels bound by up to ∼30 cm−1 are typically seen, while for Rb2 X 1 Σ+ g , population is restricted to states with binding −1 energies less than ∼12 cm . However, the difference is predicted to be much more pronounced than this. There are two main reasons for this discrepancy. First, for the specific case of PA to Rb2 0u+ that we have investigated, resonant coupling between the two 0u+ states gives excited-state wavefunction amplitude at both long range and short range. The long-range portion enhances the PA process, while the short-range portion favors decay to more deeply bound levels. This is discussed in more detail in Section 5.2.2.3 below. The second reason for the discrepancy is that the distribution we measure is not necessarily the nascent one produced by the PA process. Molecules in very high vibrational levels of X 1 Σ+ g can be off-resonantly excited back to the PA level by the PA laser, from which they can decay into the continuum. Although this destructive reexcitation is off-resonant by approximately the binding energy, the molecules are exposed to the intense PA light for several milliseconds. We estimate that under our conditions, levels bound by <0.8 cm−1 will be efficiently destroyed before being detected. Indeed, in the experiment, these levels are missing in the observed distribution. For trapped molecules this effect would be even more dramatic, because of their longer exposure to the PA light. Although we have focused here on our own work in KRb and Rb2 , pulsed laser ionization has been applied by a number of groups to a variety of systems for detecting near-dissociation molecules produced by PA, or in some cases by the MOT light itself. Without attempting a comprehensive review, we briefly mention a few examples. In some cases, the detection has been state-selective and the spectra have been assigned, allowing specific vibrational populations to be determined. Pulsed ionization of ultracold Rb2 in high-lying levels has been realized by several groups [111–113]. © 2009 by Taylor and Francis Group, LLC
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This detection scheme has also been fruitfully applied to a number of heteronuclear systems: KRb [114], NaCs [115], RbCs [43,116], and LiCs [117]. Investigations of inelastic collisional processes in Cs2 [118,119] and RbCs [105] have taken advantage of pulsed ionization to make state-selective measurements. In contrast to the work with pulsed lasers, a cw version of photoionization has allowed observation of the highest-lying levels in Na2 with both vibrational and rotational resolution [120].
5.2.2 5.2.2.1
STRONGLY BOUND X STATE LEVELS Production by Photoassociation
Although PA followed by spontaneous emission tends to favor the formation of molecules in levels near dissociation, deeply bound molecules have been produced directly by PA or by transfer from higher-lying levels. In K2 , PA to the A 1 Σ+ u state was used to form ultracold molecules in v ∼ 36 of the X 1 Σ+ state [31]. These g −1 levels are bound by 1900 cm , almost half the well depth. Vibrationally stateselective detection was accomplished by pulsed laser REMPI through the B 1 Πu state at λ ∼ 710 nm, using a second photon at 532 nm to ionize. Production rates were rather low, ∼1000 molecules/s, due to the rather small Franck–Condon factors for the decay. The inner turning points of the ground-state and excited-state wavefunctions are responsible for this overlap. In order to improve the ultracold molecule formation rate, a two-step “R-transfer” technique, first suggested by Band and Julienne [24], was implemented in K2 [25]. The idea is to improve the overlap with deeply bound levels of the ground state by photoassociating at long range, followed by a second excitation to a higher-lying electronic state at shorter range, as shown in Figure 5.25. Specifically, the 1 1 Πg state was used for PA and the “R-transfer” step involved excitation to either the 5 1 Πu or 6 1 Πu states with a cw laser at 853 nm. The detection step again utilized pulsed ionization through the B 1 Πu state, although at shorter wavelengths (630–640 nm) to probe more deeply bound X 1 Σ+ g molecules. Peaks in the detection spectrum were clearly observed, but in many cases overlapping transitions from different v levels prevented a definitive assignment. By introducing an additional cw laser, tuned to photodissociate X-state molecules with v > 17, we were able to clarify the assignment and clearly observe v = 0 molecules. As expected, the “R-transfer” technique provided a much greater (by a factor of ∼100) production rate as well as lower vibrational levels. The price to pay for this improved efficiency is the incorporation of another cw laser, maintained on resonance, and the complexity of the associated spectroscopy. In a remarkable recent result, ultracold LiCs has been produced in v = 0 of the X 1 Σ+ state by single-step PA followed by spontaneous emission [39]. The PA is to low-lying levels of the B 1 Π state, which then decays to low-v levels of the X 1 Σ+ ground state. The molecules are detected by pulsed-laser REMPI through the B 1 Π state. For PA to v = 4, the Franck-Condon factor for decay to X 1 Σ+ (v = 0) is very favorable and a production rate of 5 × 103 molecules/s in J = 2 is estimated. A different PA rotational level (J = 1 instead of J = 2) has been used to produce the absolute ground state (v = 0, J = 0), but at a 50× slower rate. What is surprising is © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications 35 K(4s) + K(4d or 4f ) 30
5 1Pu or 6 1Pu
Energy (¥103 cm–1)
25
w2 = 853 nm
20 K(4s) + K(4p) 1 1Pg
15 Radiative decay
10
w1 = 769 nm (PA)
K(4s) + K(4s)
5 X 1Sg+ 0
0
10
20 R (Å)
30
40
FIGURE 5.25 Scheme of two-step “R-transfer” for the production of deeply bound levels of K2 . (From Nikolov, A.N. et al., Phys. Rev. Lett., 84, 246, 2000. With permission.)
how well the PA works for the v = 4 level, because it is situated 1167 cm−1 below the Li (2s1/2 ) + Cs(6p3/2 ) asymptote, and therefore has its outer turning point at very short range (∼5 Å). Another system in which deeply bound levels have been produced by single-step PA is NaCs. Vibrational levels v = 19–25 of the X 1 Σ+ state were formed by photoassociating at moderate detunings (∼30 cm−1 ) below the Na(3s1/2 ) + Cs(6p3/2 ) asymptote. The resulting ground-state molecules have been trapped in an electrostatic TWIST trap [121]. Detection in this case was also by pulsed-laser ionization. 5.2.2.2
Enhancement in Double-Minimum Potentials
Historically, the first method for enhancing one-photon PA so that lower-v, more deeply bound vibrational levels could be directly produced was the use of upperstate potential energy curves having two minima with an intervening barrier [107, 122]. Such double-minimum potentials are well known in the alkali metal dimers (e.g., [123–125]). The 0g− state of Cs2 has been studied in some detail because the production of a 3 Σ+ u molecules in more deeply bound levels is favored [126,127]. Other promising double-minimum states have been proposed for study [40,128]. Of particular significance in each case is the energy and internuclear distance at which © 2009 by Taylor and Francis Group, LLC
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the barrier in the double-minimum potential occurs. For vibrational levels close in energy to the barrier maximum, there is a significant amplitude of the vibrational wavefunction near the barrier; an example is shown in Figure 5.26. This amplitude is significantly greater than the typical amplitude at an inner turning point because the slope of the potential goes to zero at the barrier maximum; moreover, there is significant tunneling near the top of the barrier. The level v = 94 shown in Figure 5.26 can emit spontaneously to deeply bound levels in the potential (e.g., 27% to levels with v = 70–80, corresponding to (2) in the figure, and 8% to the v = 0 level, corresponding to (3) in the figure) [40].
19,000 18,000
31Su+ u ≤ = 94
17,000 16,000
1000
(1)
(2)
(3)
E (cm–1)
0 u ≤ = 75
–1000 X1Sg+
–2000 –3000
u≤ = 0
D (au)
–4000 0.3 0.2 0.1 0.0 –0.1 –0.2 –0.3 10
15
20
25
R (bohr)
FIGURE 5.26 Potential energy curves of the ground and double-minimum states of Cs2 , with corresponding wavefunctions and radiative transitions showing: (1) PA to the bound 1 + 1 + 3 1 Σ+ u state, (2) the 3 Σu → X Σg transition at the barrier dividing the two wells, and (3) + + 1 1 the 3 Σu → X Σg transition at the inner wall of the 3 1 Σ+ u potential. (From Pichler, M. et al., Phys. Rev. A, 69, 013403, 2004. With permission.)
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We have spoken so far only in terms of an adiabatic double-minimum potential. In fact, most double-minimum potentials arise from the avoided crossing of two diabatic potentials with different electronic characters. The vibrational levels near the barrier maximum can be considered as a linear combination of a short-distance (inner well) wavefunction and a longer range (outer well) wavefunction. It is the mixing of shortand long-distance components that enhances the emission to short-distance levels of the X state, including v = 0. Double-minimum potentials are not the only case in which this can occur. In the next section we discuss another case, “resonant enhancement.” Here electronic mixing of selected upper-state vibrational wavefunctions can simultaneously facilitate strong PA (a significant long-range component) and strong emission to deeply bound levels of the X state (a significant short-range component). 5.2.2.3
Enhancement by Resonant Coupling of Excited States
Excited molecular states of the same symmetry, but coming from different asymptotes, can be strongly coupled by the spin–orbit interaction. A prime example is the pair of 0u+ states in heavy alkali dimers that converge to the s + p1/2 and s + p3/2 limits. Just below the s + p1/2 limit, the barely bound vibrational levels in the 0u+ ( p1/2 ) potential are closely spaced in energy and have outer turning points at long range. In contrast, levels in the 0u+ ( p3/2 ) potential, because they are deeply bound, are more widely spaced and shorter range. Each of these 0u+ ( p3/2 ) levels can be perturbatively mixed with nearby 0u+ ( p1/2 ) levels. When this mixing is strong (“resonant coupling”), it gives rise to vibrational wavefunctions with both a long-range component, corresponding to the 0u+ ( p1/2 ) state, and a short-range component due to 0u+ ( p1/2 ). Resonant coupling thus provides the best of both worlds: amplitude at long range for photoassociation, combined with amplitude at short range for decay to deeply bound levels. This phenomenon was first exploited in Cs2 , where the population in deeply-bound (>10 cm−1 ) X 1 Σ+ g levels was strongly enhanced when using + PA to a resonantly coupled 0u ( p1/2 ) level [18,130]. More recently, a similar effect −1 was seen in 85 Rb2 : the population in X 1 Σ+ g (v = 112–116), bound by 2–9 cm , was enhanced by a factor of 5 for the case of resonant coupling [129], as shown in Figure 5.27. In closely related work, marked isotopic differences in Rb2 molecule formation rates were attributed to the effects of resonant coupling [131]. It is worth examining resonant coupling from the perspective of enhancing the formation of the v = 0 level of the X state. To do so by spontaneous or stimulated emission, one wants strong overlap of the upper-state wavefunction with the v = 0 wavefunction centered on Re , the equilibrium internuclear distance of the groundstate potential. A vertical line from Re will identify levels with strong overlap, as in the example for KRb shown in Figure 5.28. From this figure, it is clear that the inner turning points of selected vibrational levels of the 2 and 3 1 Σ+ and 1 and 2 1 Π states have good overlap with the v = 0 level. However, it is also necessary to have efficient PA, so we also need to consider the energy of the upper state. For example, in Figure 5.28 the levels accessed by PA are indicated by the horizontal line just below the K(4s) + Rb(5p1/2 ) asymptote at 12,579.0003 cm−1 . The intersection of the vertical and horizontal lines is then the optimal region for resonant coupling, if an electronic state exists here that has sufficient coupling with a longer-range state © 2009 by Taylor and Francis Group, LLC
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Ion signal (arb. units)
150
100
50
0 8
6
10
12
14
16
18
20
PA laser detuning (cm–1)
FIGURE 5.27 Enhancement of molecule formation by resonant coupling in 85 Rb2 . The relative population in X 1 Σ+ g (v = 112–116) is shown as a function of the PA detuning below the 5s + 5p1/2 limit. The solid curve is the theoretical prediction. The two peaks correspond to PA levels where strong coupling between the two 0u+ states results in both efficient PA and favorable decay to bound levels. (From Pechkis, H.K. et al., Phys. Rev. A, 76, 022504, 2007. With permission.)
33S+
15
k
23P
4p + 5s 4s + 5p
21P
Energy (×103 cm–1)
Rb
11P 31S+
10
23S+ 21S+ PA
13P
5
Radiative decay a3S+ 4s + 5s
0
X1S+ –5
5
10
15 R (a0)
20
25
FIGURE 5.28 Potential energy curves of KRb from Figure 5.14, showing a possible resonant coupling scheme. The horizontal dotted line indicates the K(4s) + Rb(5p1/2 ) asymptote. The vertical arrow at 7.7a0 indicates excellent overlap of the X, v = 0 wavefunction with the inner turning point of near-dissociation levels in the 2 1 Π state.
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accessible by PA. In the case of KRb the 2 1 Π state should be suitable. This state cannot be accessed by PA in the absence of resonant coupling, because there is a potential barrier in the potential energy curve as it approaches the K(4p3/2 ) + Rb(5s) asymptote. Because of this barrier, the long-range behavior of the highest bound-state vibrational wavefunctions of the adiabatic 2 1 Π state is a rapid exponential decrease outside the outer turning point at ∼11a0 . Fortunately, the 2 1 Π state is in fact resonantly coupled with the 1 1 Π state. The mutual perturbations of these two states was first established by conventional doubleresonance laser spectroscopy [49,132]. The detunings of these levels from the K(4s) + Rb(5p1/2 ) asymptote are given in Refs. [44,45] for the v = 61–63 levels of the 1 1 Π state, although no assignments are made to the 2 1 Π state. In a recent reexamination of the raw data from these measurements, we closely studied the region where the v = 17 level of the 2 1 Π state and the v = 60 level of the 1 1 Π state are predicted to be resonantly coupled, and were able to identify the rotational levels J = 1 to 4 of both vibrational levels. We predict that these levels (and other coupled levels at larger detunings) will emit strongly to the v = 0 level of the X state. Experiments to test these predictions are under way. It is expected that analogous coupled states can be identified in other systems, including the heteronuclear alkali metal dimers, using the same reasoning. 5.2.2.4
Stimulated Raman Transfer to Deeply Bound Levels
In situations where the initial molecule formation process, either PA or MA, does not form deeply bound levels, it is possible to transfer the population from an initial high-v level to a final low-v level by a two-photon Raman process. This requires an intermediate excited state with reasonable Franck–Condon factors to both initial and final states. Such a transfer has been performed in RbCs [133], using pulsed lasers (0.2 cm−1 linewidth) to transfer from a high-lying level (v = 37) of the a 3 Σ+ state, produced by PA, to v = 0 of the X 1 Σ+ state. A mixed 3 Σ+ − B 1 Π − b 3 Π intermediate state possessed the required singlet and triplet character to enable the transfer. REMPI with the same pulsed lasers (plus pulsed 532 nm light for the ionization step) was used to detect the populations in both the initial and final states. Vibrational, but not rotational resolution was achieved. A transfer efficiency of 6% was estimated. Starting with very weakly bound molecules produced by MA, coherent transfer to more deeply bound levels has been achieved in both Rb2 [134] and KRb [135]. The stimulated Raman adiabatic passage (STIRAP) process, incorporating pump and dump pulses in the counterintuitive order (see Figure 5.29), enabled high transfer efficiency. Pulses of cw laser light were used in order for the process to be phase coherent. In this initial work, the final states were not very deeply bound, ∼600 MHz for Rb2 and ∼10 GHz for KRb. In more recent work, very deeply bound states have −1 have been populated been reached. In Cs2 , X 1 Σ+ g levels bound by >1000 cm efficiently [136], while with KRb and Rb2 , molecules have been transferred to the a 3 Σ+ v = 0 level [75,137] and for KRb, to the X 1 Σ+ v = 0 level [137], the latter being the absolute ground state. Because of the rather disparate wavelengths required for the pump and dump transitions to these deeply bound states, the lasers were locked to separate teeth of a frequency comb in order to maintain coherence. In all of these © 2009 by Taylor and Francis Group, LLC
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D
ÁbÒ
d gb
Laser 1 Laser 2
W1
W2
ÁaÒ
ÁgÒ ga gg
FIGURE 5.29 Scheme for STIRAP transfer to more deeply bound levels in 87 Rb2 . The pump (1) and dump (2) lasers couple the initial state |a to the final state |g through the intermediate state |b. The sloped “laser 1” line should actually be vertical, because the electronic transition takes place at a nearly fixed internuclear distance (Franck–Condon approximation). (From Winkler, K. et al., Phys. Rev. Lett., 98, 043201, 2007. With permission.)
experiments, populations and transfer efficiencies were monitored by transferring back to the initial molecular state, reversing the MA, and measuring the resulting atoms by absorption imaging. A rather different technique was recently demonstrated to transfer PA-produced Cs2 molecules to the lowest vibrational level of the X 1 Σ+ g state [37]. Ultrafast (100 fs) pulses were used for broadband pumping of the initial v = 1–7 molecules back to the excited state. These excited molecules would decay back into X 1 Σ+ g . The spectrum of this pumping light was shaped to eliminate any components that could excite from v = 0. As a result, if a molecule decayed into v = 0, it would stay there. After a sequence of many of these shaped broadband pulses, a large fraction, 70%, of the initial v = 1–7 population would thus accumulate in v = 0. The various X 1 Σ+ g vibrational level populations were monitored by vibrationally state-selective pulsed ionization. © 2009 by Taylor and Francis Group, LLC
208
Cold Molecules: Theory, Experiment, Applications 3s + Na+ 45 Na+2 X2Sg+ 40
nl 1Lu(±)
35 L3
L'3 3 1S+g
3s + 4s
E (x103 cm–1)
30
1 1P g
25
4s + 4p
L2 20 A 1Su+ 15 LPA Na2
10
L1
3s + 3s
5 X1Sg+
2
4
6
8
10
12
14
16
18
20
R (Å)
FIGURE 5.30 Three-color (L1 –L2 –L3 ) doubly-resonant photoionization of Na2 and proposed photoassociative ionization (LPA –L3 ) of ultracold Na atoms via the 1g (4p3/2 ) state, correlated with the 1 1 Πg state at short range. Both processes should access the same autoionization resonance, which energetically can produce only v = 0, J = 0 Na+ 2 . (From Stwalley, W.C. and Wang, H., J. Mol. Spectrosc., 195, 194, 1999. With permission.)
5.3 5.3.1
FURTHER CONNECTIONS AND DIRECTIONS ULTRACOLD MOLECULAR IONS
Chapter 18 of this book, by Roth and Schiller, is an extensive discussion of cold molecular ions. One topic that is not discussed, however, is the possibility of forming untrapped molecular ions at ultracold temperatures and low densities by photoionization of ultracold neutral molecules formed by PA or MA. Indeed, as discussed above, the ionization of ultracold neutral molecules has already been extensively used for © 2009 by Taylor and Francis Group, LLC
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detection of the molecules. An interesting approach is to state-selectively ionize to form ultracold molecular ions. A possible route is shown in Figure 5.30, where it is shown how photoassociative ionization [7] could reach the same autoionizing level of Na2 that was reached by all-optical triple resonance spectroscopy of Na2 at high temperature [138]. The level reached in Ref. [138] could only autoionize to the lowest rovibrational level of the ground X state of the Na+ 2 molecular ion. It is expected that this lowest level could also be formed by autoionization in many of the other homonuclear and heteronuclear alkali dimers. Studies of the collisions of this lowest rovibrational level would also be very interesting because there would be no inelastic collisions that would lower the internal energy of the molecular ion (excluding possibly hyperfine-changing collisions).
5.3.2 TESTS OF FUNDAMENTAL PHYSICAL CONSTANTS AND SYMMETRIES The ultracold molecules formed by PA are well suited to precision measurements and tests of fundamental physics because they are nearly at rest, and can readily be retained for long interrogation times in magnetic or optical traps. In tests of fundamental physics, molecules are preferred over atoms when either (1) an intrinsic dipole moment or very large polarizability is needed, or (2) a high density of states is beneficial. There is an excellent summary of molecular tests of fundamental principles in Section 15.2 of this book. Clearly there is no inherent link between PA and precision spectroscopy, and in some cases other cold molecule formation techniques are clearly preferable. As a general rule, PA or MA are the most logical approaches when MOTs are possible for the required atoms. Some experiments work best in molecules whose constituent atoms lack accessible cycling transitions. For example, searches for an intrinsic electron dipole moment (EDM) require a molecule with relativistic electrons and a large polarizability. Methods for dealing with some of the most-preferred molecules, especially YbF, are detailed in Chapter 15. Similar tests using molecular ions such as HfF+ are also being considered [139]. Techniques for spectroscopy of molecular ions are discussed further in Chapter 18. In many cases the best choice of the molecule and its preparation method is a complex tradeoff between molecular properties, signal-to-noise issues, and experimental accessibility. A good example is the search for time dependence of the fundamental constants, the subject of Chapter 16. The possibility of very small variations in the electron-to-proton mass ratio μ = me /mp is of great current interest, and is most easily investigated in molecules because it is relatively easy to find pairs of states with differing dependencies on dμ/dt. A variety of possible laboratory tests in various molecules are discussed in detail in Chapter 16, among them two recent proposals [140,141] involving ultracold Cs2 and Sr2 that would utilize precision measurements in vibrationally excited photoassociated molecules. Much of the sensitivity in these schemes would come from the possibility of making optical measurements with extremely high resolution. It will be interesting to see how the balance shifts now that optically trapped groundstate molecules such as RbCs and KRb are becoming available, including the very recent announcement of optically trapped KRb produced by MA and Raman transfer © 2009 by Taylor and Francis Group, LLC
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in the absolute ground state (X 1 Σ+ (v = 0, J = 0)) at 350 nK [137]. It may be that the long interaction times and narrow linewidths possible for these heteronuclear alkali dimers will now partially outweigh their generally lesser sensitivity to fundamental symmetries and dμ/dt. It is also interesting to consider the prospect of using optically trapped ultracold molecules as secondary frequency standards in the microwave or IR region. Apparently no detailed analysis of this possibility has so far been published. Pure rotational-vibrational transitions are weakly dipole-allowed in heteronuclear species due to their dipole moments. In most of the alkali dimers the lifetimes are extremely long [50], leading to sub-milliHertz linewidths potentially suitable for metrology, although potential systematic issues due to cold collisions or ac Stark shifts have yet to be investigated. It would probably be necessary to keep the molecules in a quasi-electrostatic optical trap (QUEST) to attain sufficient interrogation times, so trap shifts are also of concern. Fortunately a recent analysis of rotational mixing of heteronuclear alkali dimers in electric fields [142] indicates that for many of the molecules, the vibrational dependence of the dc Stark shift passes through zero at a level between v = 30 and 50, suggesting that a field-independent transition should be possible between suitably chosen vibrational levels.
5.3.3
QUANTUM COMPUTING WITH ULTRACOLD POLAR MOLECULES
Ultracold polar molecules offer a combination of controllable dipolar interactions and long decoherence times that is well suited to quantum information applications. Several years ago a pivotal paper by David DeMille [143] stimulated intense interest that has continued unabated, so much so that it has become an oft-cited motivation for pursuing research on production of ultracold molecules by PA. This scheme, as well as other ideas put forth more recently, is described in detail in Chapter 17 of this book.
5.3.4
ULTRACOLD COLLISIONS AND CHEMISTRY
It is clear that there are many interesting issues having to do with the collisions of ultracold molecules [144–147]. Chapters 1 to 4 of this book cover a majority of these issues. It is worth noting, however, that photoassociative samples will usually contain both atoms and molecules formed by PA. Thus a photoassociated sample (starting with atoms M) of M2 molecules will ordinarily contain atoms M as well, making M + M2 collisions particularly easy to study. Moreover, the atoms can readily be removed by pushing them with a resonant laser, allowing the study of M2 + M2 collisions. For heteronuclear molecules MM , similar considerations apply, and either M or M or both can readily be removed. Usually only one atom will be reactive with the molecular ground state; in the case of X-state KRb (v = 0, J = 0), for example, K will react to form K2 , but Rb will not react to form Rb2 [45]. Similar considerations will apply to collisions of M2 + and MM + molecular ions. Another interesting consideration is the effect of the total nuclear spin of a homonuclear molecule [1]. In the absence of collisions with paramagnetic species such as M atoms, the homonuclear molecule M2 will exist in two forms, para (with even J if the © 2009 by Taylor and Francis Group, LLC
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atomic nuclear spin is half-integral) and ortho (with odd J if the atomic nuclear spin is half-integral). It is possible that X-state v = 0 molecules in both the J = 0 paraM2 and J = 1 ortho-M2 levels can ultimately be Bose–Einstein condensed, because M2 + M2 collisions will not change the nuclear spin and relax the J = 1 level.
5.3.5
FINAL SPECULATIONS
The applications of ultracold molecules formed by PA, by MA, or otherwise, will undoubtedly expand far beyond the topics just discussed. Promising ideas include molecule optics and the “molecule laser” (the molecular analogs of atom optics and the “atom laser”); studies of nucleation of atoms and molecules to form clusters (nanoparticles), including highly metastable species; artificial “solids” comprised of ultracold molecules in optical lattices; and integration of ultracold molecules into integrated circuitry, hollow fiber-optic networks, and other devices for practical applications. Clearly there is a great deal of opportunity for creativity in “quantum nanoscience” based on these new “quantum materials,” dilute gases of ultracold molecules.
ACKNOWLEDGMENTS We gratefully acknowledge funding support from NSF awards PHY-0354869 and PHY-0555481.
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46. 47. 48. 49.
50.
51.
52. 53. 54. 55. 56.
57.
58. 59. 60. 61. 62.
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States Near 6 aMolecular Collision Threshold Paul S. Julienne CONTENTS 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Properties for a Single Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Interactions for Multiple Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Magnetically Tunable Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Photoassociation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1
221 223 230 233 238 241
INTRODUCTION
Real atoms are typically complex, having ground and excited states with spin structure. The molecules formed from the atoms typically have a rich spectrum of nearthreshold bound and quasi-bound molecular states when the molecular spin, rotational, and vibrational structure is taken into account. When an ultracold gas of atoms is produced, the atoms are prepared in specific quantum states, and collisions between the atoms occur with an extremely precisely defined energy close to the E = 0 collision threshold of the interacting atoms, where E denotes energy. The collision then makes the near-threshold spectrum of the molecular complex of the two atoms accessible to electromagnetic probing. An external magnetic or electromagnetic field can be precisely tuned to couple the colliding atoms to a specific molecular state, which can be viewed as a scattering resonance. This permits both extraordinary spectral accuracy in probing near-threshold level positions (order of E/h = 10 kHz accuracy for 1 μK atoms) and precise resonant control of the collisions that determine both static and dynamical macroscopic properties of quantum gases. Consequently, understanding the near-threshold bound and scattering states is essential for understanding the collisions and interactions of ultracold atoms. This is also true for interactions of ultracold molecules. This chapter concentrates on understanding molecules that can be made by combining two cold atoms using either magnetically tunable Feshbach resonance states [1] or optically tunable photoassociation (PA) resonance states [2]. Such resonances provide a mechanism for the formation of ultracold molecules from already cold atoms. In addition, magnetically tunable resonances have been used very successfully
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to control the properties of ultracold quantum gases. This chapter treats both magnetically and optically tunable molecular resonances with the same scattering theory framework. The viewpoint from quantum defect theory is emphasized of conceptually separating the interaction of the atoms into short-range and long-range regions. These regions are characterized by very different energy and length scales. Much insight about near-threshold collisions and bound states, as well as practical tools for their study, can be gained by taking advantage of this separation [3–11]. While molecular physics is typically concerned with strong short-range interactions associated with “ordinary molecules,” ultracold physics is concerned with scattering states and very weakly bound molecular states in the threshold domain near E = 0. The long range potential, which has a lead term that varies as 1/Rn , plays an important role in connecting these two regimes. We briefly summarize here the theory of cold collisions, which is described in detail in Chapter 1. The scattering wavefunction is expanded in states of relative angular momentum of the two atoms characterized by partial wave quantum number = 0, 1, 2, . . . . Generally, the atoms can be initially prepared in one of several quantum states, and the scattering “channels” can be specified by a collective set of quantum numbers α representing the state of each atom and the partial wave. Upon solving the Schrödinger equation for the system, the effect of all short-range interactions during a collision with E > 0 is summarized in the scattering wavefunction for R → ∞ by a unitary S-matrix. Only the lowest few partial waves can contribute to cold collisions, and in the limit E → 0, only s-wave channels with = 0 have non-negligible collision cross-sections. Using the complex scattering length a − ib to represent the s-wave S-matrix element Sαα = exp [−2ik(a − ib)] in the limit E → 0, the contribution to the elastic scattering cross-section from s-wave collisions in channel α is σcl = lim g E→0
π 2 2 2 |1 | = 4gπ a + b , − S αα k2
(6.1)
√ where k = 2μE is the relative collision momentum in the center-of-mass frame for the atom pair with reduced mass μ. The rate coefficient Kloss = σloss v for E → 0 s-wave inelastic collisions that remove atoms from channel α is Kloss = lim g E→0
π h 1 − |Sαα |2 = 2g b, μk μ
(6.2)
where v = k/μ is the relative collision velocity. The symmetry factor g is equal to 1 when the atoms are bosons or fermions that are not in identical states, g = 2 or g = 1 respectively for two bosons in identical states in a normal thermal gas or a Bose– Einstein condensate, and g = 0 for two fermions in identical states. If there are no exoergic inelastic channels present, then b = 0 and only elastic collisions are possible. The Schrödinger equation also determines the bound states with discrete energies Ei < 0. While the conventional picture of molecules counts the bound states by vibrational quantum number v = 0, 1, . . . from the lowest energy ground-state up, it is more helpful for the present discussion to count the near-threshold levels from the E = 0 dissociation limit down by quantum numbers i = −1, −2, . . . . In the special © 2009 by Taylor and Francis Group, LLC
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case where a → +∞, the energy of the last bound s-wave state of the system with i = −1 depends only on a and μ and takes on the following “universal” form: E−1 = −
2 as a → +∞. 2μa2
(6.3)
Section 6.2 describes the bound and scattering properties of a single potential with a van der Waals long range form. Section 6.3 extends the treatment to multiple states and scattering resonances. Sections 6.4 and 6.5 respectively discuss the properties of magnetically and optically tunable molecular resonance states.
6.2
PROPERTIES FOR A SINGLE POTENTIAL
In this section let us ignore any complex internal atomic structure and first consider two atoms A and B that interact by a single adiabatic Born–Oppenheimer interaction potential V (R), illustrated schematically in Figure 6.1. The wavefunction for the system is |α|ψ /R, where |α represents the electronic and rotational degrees of freedom, and the wavefunction for relative motion is found from the radial Schrödinger equation 2 ( + 1) 2 d2 ψ ψ = Eψ . + V (R) + (6.4) − 2μ dR2 2μR2 Solving Equation 6.4 gives the spectrum of bound molecular states ψi with energy 2 /(2μ) < 0 and the scattering states ψ (E) with collision kinetic energy Ei = −2 ki Short-range
V(R) E=0
Long-range
AB
Separated
A+B V ~ –Cn / Rn
1 – 10 eV
kB(1 mK) = h(21 kHz) = 0.86 neV
a
Rbond R
FIGURE 6.1 Schematic figure of the potential energy curve V (R) as a function of the separation R between two atoms A and B. The horizontal lines labeled AB indicate a spectrum of molecular bound states leading up to the molecular dissociation limit at E = 0, indicated by the dashed line. The long-range potential varies as −Cn /Rn . See text for a definition of Rbond and a¯ . © 2009 by Taylor and Francis Group, LLC
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E = 2 k 2 /(2μ) > 0, where ki and k have units of (length)−1 . As R → ∞, the bound states decay as e−ki R and the scattering states approach ψ (E) → c sin(kR − π/2 + η )/k 1/2 .
(6.5)
Bound states are normalized to unity, |ψi |ψj |2 = δij δ . We choose the normal ization constant c = 2μ/2 π so that scattering states are normalized per unit energy, ψ (E)|ψ (E ) = δ(E − E )δ . Thus, the energy density of states is included in the wavefunction when taking matrix elements involving scattering states. The long-range potential between the two atoms varies as −Cn /Rn . We are especially interested in the case of n = 6 for the van der Waals interaction between two neutral atoms. This is the lead term in the long-range expansion of the potential in inverse powers of R that applies to many atoms that are used in ultracold exper iments. This potential has a characteristic length scale of Rvdw = 4 2μC6 /2 /2, which depends only on the values of μ and C6 [2]. Values of C6 are tabulated by Derevianko [12] for alkali–metal species and by Porsev and Derevianko [13] for alkaline–earth species. We prefer to use a closely related van der Waals length introduced by Gribakin and Flambaum [14]. a¯ = 4π/Γ(1/4)2 Rvdw = 0.955978 . . . Rvdw ,
(6.6)
where Γ(x) is the Gamma function. This length defines a corresponding energy scale E¯ = 2 /(2μ¯a2 ). The parameters a¯ and E¯ occur frequently in formulas based on the van der Waals potential. The wavefunction approaches its asymptotic form when R a¯ and is strongly influenced by the potential when R a¯ . Table 6.1 gives the values of a¯ and E¯ for several species used in ultracold experiments. Samples of cold atoms can be prepared with kinetic temperatures on the order of nK to mK. The energy associated with temperature T is kB T where kB is the Boltzmann constant. For example, at T = 1 μK, kB T = 0.86 neV and kB T /h = 21 kHz. This ultracold energy scale is 9 to 10 orders of magnitude smaller than the energy scale of 1 to 10 eV associated with ground- or excited-state interaction energies when a
TABLE 6.1 Characteristic van der Waals Scales a¯ and E¯ for Several Atomic Species Species
Mass (amu)
C6 (au)
a¯ (a0 )
E¯ /h (MHz)
E¯ /kB (mK)
6 Li
6.015122 22.989768 39.963999 86.909187 87.905616 132.905429 173.938862
1393 1556 3897 4691 3170 6860 1932
29.88 42.95 62.04 78.92 71.76 96.51 75.20
671.9 85.10 23.46 6.668 7.974 2.916 3.670
32.25 4.084 1.126 0.3200 0.3827 0.1399 0.1761
23 Na 40 K 87 Rb 88 Sr 133 Cs 174Yb
Note: 1 amu = 1/12 mass of a 12 C atom, 1 au = 1 Eh a06 where Eh is a hartree and 1 a0 = 0.0529177 nm.
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molecule is formed at small interatomic separation Rbond on the order of a chemical bond length. In a cold collision, the initially separated atoms have very low collision energy E = 2 k 2 /(2μ) ≈ 0 and very long de Broglie wavelength 2π/k. The atoms come together from large distance R and are accelerated by the interatomic potential V (R), so that when they reach distances on the order of Rbond they have very high kinetic energy on the order of |V (Rbond )|. The local de Broglie wavelength 2π/k(R, E) √ in the short-range classical part of the potential, where k(R, E) = 2μ(E − V (R))/, is orders of magnitude smaller than the separated atom de Broglie wavelength and is nearly independent of the value of E, which is close to 0. This separation of scales is illustrated in Figure 6.2, which shows examples of s-wavefunctions at a collision energy E/kB = 1 μK, where dividing E by kB allows us to express energy in temperature units. This example uses three isotopic combinations of pairs of Yb atoms, which have a spinless 1 S0 electronic configuration and a single ground-state electronic Born–Oppenheimer potential V (R). The species Yb makes a good example case to illustrate the principles in this section, because it has seven stable isotopes and 28 different atom pairs of different isotopic composition for which the threshold properties have been worked out [15]. All combinations have the same V (R) but different reduced masses. This mass-scaling approximation, which ignores very small mass-dependent corrections to the potential, is normally quite good except for very light species such as Li. Figure 6.2 shows that the three examples have similar phase-shifted sine waves with a common long de Broglie wavelength of 2π/k √ = 6300a . For small R where kR 1 the sine function vanishes as c sin k(R − a)/ k→ 0 √ c k(R − a). The actual wavefunction oscillates rapidly at small R due to the influence of the potential. Because the asymptotic form for kR 1 varies as k 1/2 as k → 0,
y (R)
30 20
174–174 171–171 170–173
10
E/kB = 1 mK
0 –10 4
a
–20 –30 0 –40
0 0
100 2000
200 4000
6000
8000
R [a0]
FIGURE 6.2 Radial wavefunction ψ0 (R) for = 0 at E/kb = 1 μK for the pairs 174Yb– 174Yb (solid), 171Yb–171Yb (dashed), and 170Yb–173Yb (dot-dashed), which have respective scattering lengths of 105a0 , −3a0 , and −81a0 [15]. The inset shows an expanded view of the wavefunction on a smaller length scale on the order of a¯ , the characteristic length of the van der Waals potential. The 174Yb–174Yb case shows the oscillations that develop when R < a¯ .
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the short-range oscillating part also has an amplitude proportional to k 1/2 in order to connect smoothly to the asymptotic form as k → 0. This property ensures that the threshold matrix elements that characterize Feshbach resonances and s-wave inelastic scattering are proportional to k 1/2 . Figure 6.3 illustrates more clearly the nature of threshold short-range scattering and bound-state wavefunctions. When given an appropriate short-range normalization, near-threshold scattering and bound-state wavefunctions have a common amplitude and phase in the region of R small compared to the range a¯ of the long-range potential. Although this can be put on a rigorous quantitative ground within the framework of quantum defect theory [8], it is easy to show using the familiar JWKB approximation [2,3,7]. We can always write the wavefunction in phase-amplitude form ψ (R, E) = α (R, E) sin β (R, E) and transform the Schrödinger equation 6.4 into a set of equations for α and β . The asymptotic ψ (R, E) in Equation 6.5 clearly corresponds to this form with α → c/k 1/2 as R → ∞. Another familiar form is the JWKB semiclassical wavefunction ψJWKB (R, E), for which (R, E) = c/k (R, E)1/2 αJWKB R π (R, E) = k (R , E)dR + , βJWKB 4 Rt
(6.7) (6.8)
where Rt is the inner classical turning point of the potential.
2
1
y (R)
0
–1 i=–1 bound state E/kB=1 mK scattering
–2
–3
a
10
100 R (a0)
FIGURE 6.3 wavefunctions for the last s-wave i = −1 bound state (solid line) with E−1,0 /h = −10.6 MHz and for the s-wave scattering state (dashed line) for E/h = 0.02 MHz (E/kB = 1 μK) for two 174Yb atoms. Both wavefunctions are given a common JWKB normalization at small R a¯ and are nearly indistinguishable for R < a¯ . The potential supports N = 72 bound states, and the wavefunction for this i = −1 and v = 71 level has N − 1 = 71 nodes.
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When the collision energy E is sufficiently large, so there are no threshold effects, the JWKB approximation is an excellent approximation at all R, and the form of αJWKB (R, E) in Equation 6.7 applies at all R, transforming into the correct quantum limit as R → ∞. On the other hand, the JWKB approximation fails for s-waves with very low collision energy. This failure occurs in a region of R near a¯ and for collision energies E on the order of E¯ or less. The consequence is that the JWKB wavefunction, with the normalization in Equation 6.7, is related to the actual wavefunction, with the asymptotic form in Equation 6.5, by a multiplicative factor C (E), so that as E → 0 (R, 0). ψ (R, E) = C (E)−1 ψJWKB
(6.9)
As k → 0 for a van der Waals potential varying as 1/R6 , the s-wave threshold form is C0 (E)−2 = k a¯ [1 + (r − 1)2 ], where r = a/¯a is the dimensionless scattering length in units of a¯ [8]. Equation 6.9 gives an excellent approximation for the threshold ¯ C0 (E)−1 approaches ψ0 (R, E) for R < a¯ and k < 1/a. At high energy, when E E, unity and the JWKB approximation for ψ0 (R, E) applies at all R. The unit normalized bound-state wavefunction ψi (R) can be converted to an “energy normalized” form by multiplying by |∂i/∂Ei |1/2 , where −∂i/∂Ei > 0 is the energy density of states. Away from threshold, this is just the inverse of the mean spacing between levels, whereas for s-wave levels near threshold for a van der Waals ¯ −1 as k−1,0 = 1/a → 0 [8]. The relation of ψi to the potential, ∂i/∂Ei0 → r/(2πE) energy-normalized JWKB form in the classically allowed region of the potential is ∂i ψi (R, Ei ) = ∂E
−1/2 ψJWKB (R, Ei ).
(6.10)
i Ei
Figure 6.3 plots C0 (E)ψ0 (R, E) ≈ ψJWKB (R, 0) for the scattering state and 0 JWKB 1/2 (R, 0) for the i = −1 bound-state. Thus the near|∂i/∂Ei0 | ψi0 (R, Ei0 ) ≈ ψ0 threshold bound and scattering wavefunctions, when given a common short-range normalization, are nearly identical and are well approximated by ψJWKB (R, 0) in the 0 region R < a¯ . For R > a¯ the wavefunctions begin to take on their asymptotic form as R → ∞. The shape of the wavefunction at very small R on the order of Rbond is usually independent of E for ranges of E/kB on the order of many K. The short-range shape is even independent of for small , because the rotational energy is very small compared to typical values of V (Rbond ). However, the amplitudes of the wavefunctions depend strongly on the whole potential, which determines a, and are analytically related to the form of the long range potential. The separation of scales for R > a¯ and R < a¯ is a key feature of ultracold physics that enables much physical insight as well as practical approximations to be developed about molecular bound and quasibound states and collisions. Given that C6 , μ, and the s-wave scattering length a are known, the Schrödinger equation 6.4 can be integrated inward using the form of Equation 6.5 as k → 0 as a boundary condition, thus giving the wavefunction and nodal pattern for R < a¯ as E → 0. Assume that it is possible to pick some R = Rm such that Rbond Rm a¯ and V (Rm ) is well-represented by its van der Waals form. Then the log of the derivative of the wavefunction at Rm , which also can be calculated, provides an inner boundary condition, independent of E over © 2009 by Taylor and Francis Group, LLC
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a wide range of E, for matching the wavefunction at E propagated from large R. All that is needed to do this is to know a, μ, and the long-range potential. Thus, it is readily seen that all of the near-threshold bound and scattering states, even those for > 0, can be calculated to a very good approximation for R > Rm once C6 , μ, and a are known. ¯ for up to 5 Figure 6.4 shows the spectrum of bound states Ei , in units of E, for two cases of scattering length, based on the van der Waals quantum defect theory of Gao [9,10]. Panel (a) shows the case of a = ±∞, where there is a bound state at E = 0. The locations of the bound states for a = ±∞ define the boundaries of the “bins” in which, for any a, there will be one and only one s-wave bound state, ¯ The panel also for example, −36.1E¯ < E−1,0 < 0 and −249E¯ < E−2,0 < −36.1E. shows the rotational progressions for each level as increases. The a = ±∞ van der Waals case also follows a “rule of 4,” where partial waves = 4, 8, . . . also have a bound state at E = 0. Panel (b) shows how the spectrum changes when a = a¯ , for which the there is a d-wave level at E = 0. Similar spectra can be calculated for any a. Gribakin and Flambaum [14] showed that the near-threshold s-wave bound state for a van der Waals potential in the limit a a¯ is modified from the universal form in Equation 6.3 as E−1 = −
2 . 2μ(a − a¯ )2
(6.11)
This approaches the universality limit when a √ a¯ , in which case the s-wave wavefunction takes on the universal form ψ0 (R, E) = 2/ae−R/a . Such an exotic bound state, known as a “halo molecule,” exists primarily in the nonclassical domain beyond
a = +•
a=a
(a)
(b) 0
0
–50
–50
–100
s
p
d
f
g
h
–100
E/E –150
–150
–200
–200
–250
–250
–300
–300
s
p
d
f
g
h
FIGURE 6.4 Dimensionless bound-state energies Ei /E¯ for partial waves = 0 . . . 5 (s, p, d, f , g, h). (a) The case where a = ±∞; (b) The case for a = a¯ .
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229
the outer classical turning point of the long-range potential with an expectation value of R of a/2, which grows without bound as a → +∞ [1]. Bound-state and scattering properties are closely related. It is instructive to imagine that there is some control parameter λ that can be varied to make the scattering length vary over its whole range from +∞ and −∞, changing the corresponding boundstate spectrum. One way to do this would be to vary the reduced mass. Of course, this is not physically possible. However, there are elements with many isotopes, so that a wide range of discrete reduced masses are possible. An excellent physical system to illustrate this is the Ytterbium atom, used in the examples of Figures 6.2 and 6.3. The stable isotopes with masses 168, 170, 172, 174, and 176 are all spinless bosons and the 171 and 173 isotopes are fermions with spin 1/2 and 5/2, respectively. Yb atoms can be cooled into the μK domain and all isotopes, including the fermionic ones in different spin states, have s-wave interactions. The locations of several = 0 and 2 threshold bound states of different isotopic combinations of Yb atoms in Yb2 dimer molecules have been measured, and the long-range potential parameters and scattering lengths determined [15]. Figure 6.5 shows the s-wave scattering length and bound state binding energies versus the continuous control parameter λ = 2μ. Physically, there are 28 discrete values between λ =168 and 176. The scattering length has a singularity, and a new bound state occurs with increasing λ, at λ = 167.3, 172.0, and 177.0. The range between 167.3 and 172 corresponds to exactly N = 71 bound states in the model potential used. Near λ = 167.3 the last s-wave bound state energy E−1,0 → 0 as −2 /(2μa2 ) as a → +∞. The binding energy |E−1,0 | gets larger as λ increases and a decreases, so that for a van der Waals potential E−1,0 approaches the lower edge of its “bin” at −36.1E¯ as a → −∞. As λ increases beyond 172.0, the i = −1 level becomes the i = −2 level as a new i = −1 “last” bound state appears in the spectrum. The variation of scattering length with 2μ is given by a remarkably simple formula. While semiclassical theory breaks down at threshold, Gribakin and Flambaum [14] showed that the correct quantum-mechanical relation between a and the potential is π a = a¯ 1 − tan Φ − , 8 where
Φ=
∞
Rt
−2μV (R)/2 = βJWKB (∞, 0) − π/4. 0
(6.12)
(6.13)
The number of bound states in the potential is N = [Φ/π − 5/8] + 1, where [. . .] means the integer part of the expression. These expressions work remarkably well in practice. Although the results in Figure 6.5 are obtained by solving the Schrödinger equation for a realistic potential, virtually identical results are obtained for a from Equation 6.12. In fact, a and Ei0 are nearly the same on the scale of Figure 6.5 if the simple hard-core van der Waals model of Ref. 14 is used for the potential, namely V (R) = −C6 /R6 if R ≥ R0 and V (R) = +∞ if R < R0 , where the cutoff R0 is chosen to fit a or E−1,0 data from two different isotopes. With the mass scaling √ ∝ μ in Equation 6.13, knowing C6 and E−1,0 for two isotopic pairs determines a © 2009 by Taylor and Francis Group, LLC
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a(l) (a0)
200 0 –200
(b)
1
–Ei0(l)/h (MHz)
–400
10
100
1000 166
i = –1
i = –1
i = –2
i = –2
168
170
172 174 l = 2μ (amu)
176
178
FIGURE 6.5 (a) s-Wave scattering length and (b) bound-state binding energies −Ei0 (λ) for Yb2 molecular dimers vs. the control parameter λ = 2μ. The vertical dashed lines show the points of singularity of a(λ). The horizontal dashed lines show the boundaries of the bins in which the i = −1 and i = −2 levels must lie.
and E−1,0 for all isotopic pairs. The approximation is fairly good even for levels with larger |i| or > 0, although it will become worse as |i| or increase. In summary, it is very useful to take advantage of the enormous difference in energy and length scales associated with the cold separated atoms and deeply bound molecular potentials. This allows us to introduce a generalized “quantum defect” approach for understanding threshold physics [3,6–8,10]. Threshold bound state and scattering properties are determined mainly by the long-range potential, once the overall effect of the whole potential is known through the s-wave scattering length. A similar analysis can be developed for other long-range potential forms, for example, 1/R4 ion-induced dipole or 1/R3 dipole–dipole interactions.
6.3
INTERACTIONS FOR MULTIPLE POTENTIALS
Generally, the cold atoms used in experiments have additional angular momenta (electron orbital and/or electron spin and/or nuclear spin), so that more than one scattering channel α can be involved in a collision. Each channel has a separated atom © 2009 by Taylor and Francis Group, LLC
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channel energy Eα . Figure 6.1 could be modified to illustrate such channels by adding additional potentials and their corresponding spectra dissociating to the Eα limits. If Etot is the total energy of the colliding system, the designation open or closed is used for channels with Etot > Eα or Etot < Eα , respectively. Inelastic collisions from entrance channel α are possible to open exit channels β when Eα > Eβ , whereas closed channels β can support quasibound states as scattering resonances when Eα < Etot < Eβ . The ability to tune resonance states to control scattering properties or to convert them into true molecular bound states is an important aspect of ultracold physics that has been exploited in a wide variety of experiments with bosonic or fermionic atoms [1]. Let us first examine the basic magnitude of the s-wave inelastic collision rates that are possible when open channels are present. The rate constant is determined by the magnitude of b in Equation 6.2, for which a typical order of magnitude is b ≈ a¯ for an allowed transition, that is, one with relatively large short-range interactions in the system Hamiltonian. The rate constant can be written Kloss = 0.84 × 10−10 g
b[au] cm3 /sec, μ[amu]
(6.14)
where b is expressed in atomic units (1 au = 0.0529177 nm) and μ in atomic mass units (μ = 12 for 12 C). Allowed processes will typically have the order of magnitude of 10−10 cm3 /sec for Kloss . The s-wave Kloss can be even larger, with an upper bound of bu = 1/(4k) being imposed by the unitarity property of the S-matrix, that is, 0 ≤ 1 − |Sαα |2 ≤ 1. Because the lifetime relative to collision loss is τ = 1/(Kloss n), where n is the density of the collision partner, allowed processes result in fast loss with τ 1 msec at typical quantum degenerate gas densities. This applies to atom– molecule and molecule–molecule collisions as well as atom–atom collisions. Such losses need to be avoided by working with atomic or molecular states that do not experience fast loss collisions, such as the lowest energy ground-state level, which does not have exoergic 2-body exit channels. Alternatively, placing the species in a lattice cell that confines a single atom or molecule can offer protection against collisional loss. An alternative formulation of the collision loss rate is possible by rewriting Equation 6.2, not taking the E → 0 limit but introducing a thermal average over a Maxwellian distribution of collision energies E, Kloss = g
+ 1 kB T * 1 − |Sαα |2 , T QT h α
(6.15)
where QT is the translational partition function, 1/QT = (2πμkB T /h2 )3/2 = Λ3T where ΛT is the molecular thermal de Broglie wavelength. The . . .T expression implies a thermal average over the velocity distribution. The sum represents a dynamical factor fD that varies as T 1/2 as T → 0 and has an upper bound of unity for s-waves and ≈ 2max if max partial waves contribute at the unitarity limit. Although Equation 6.15 reduces to Equation 6.14 in the T → 0 s-wave limit, it lets us see that the collision rate is given by an expression having the form τ−1 = Kloss n = g(nΛ3T )
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kB T fD . h
(6.16)
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This form embodies some general principles for any collisions of atoms and molecules. The dimensionless nΛ3T factor shows that the collision rate is proportional to phase space density of the collision partner (scale by mass ratios to convert to an atomic phase space density). The kB T /h factor sets an intrinsic rate scale (dimension of inverse time) associated with T . The dimensionless factor fD embodies all of the detailed collision dynamics. Even using fast time-dependent manipulations to control fD does not change the fundamental thermodynamic limits imposed by the phase space density and kB T /h factors. Given Equations 6.14 and 6.15 and plausible assumptions about b or fD , it is possible to estimate the time scales for a wide variety of atomic and molecular collision processes under various kinds of conditions. Now we will examine the important case of tunable resonant scattering when a closed channel is present. Assume that open entrance channel α, with Eα chosen as Eα = 0, is coupled through terms in the system Hamiltonian to a closed channel β with 0 < E < Eβ . Then a molecular bound state in channel β becomes a quasibound state that acts as a scattering resonance in channel α. Using Fano’s form of resonant scattering theory [16], let us assume a “bare” or uncoupled approximate bound state |C = ψc (R)|c with energy Ec in the closed channel β = c and a bare or background scattering state |E = ψbg (R, E)|bg at energy E in the entrance channel α = bg. The scattering phase shift η(E) = ηbg (E) + ηres (E) of the coupled system picks up a resonant part due to the Hamiltonian coupling W (R) between the bare channels. Here ηbg is the phase shift due to the uncoupled single background channel, as described in the last section, and 1 −1 2 Γ(E) , (6.17) ηres (E) = − tan E − Ec − δE(E) has the standard Breit–Wigner resonance scattering form. The two key features of the resonance are its width Γ(E) = 2π|C|W (R)|E|2 , and its shift
δE(E) = P
∞ −∞
|C|W (R)|E |2 dE . E − E
(6.18)
(6.19)
The primary difference between an “ordinary” resonance and a threshold one as E → 0 is that for the former we normally make the assumption that Γ(E) and δE(E) are evaluated at E = Ec and are independent of E across the resonance. By contrast, the explicit energy dependence of Γ(E) and δE(E) are key features of threshold resonances [11,17,18]. In the special case of the E → 0 limit for s-waves, 1 Γ(E) → (kabg )Γ0 2 Ec + δE(E) → E0 ,
(6.20) (6.21)
where Γ0 and E0 are E-independent constants. Note that Γ(E) is positive definite, so that Γ0 has the same sign as abg . Assuming an entrance channel without inelastic loss, © 2009 by Taylor and Francis Group, LLC
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so that ηbg (E) → −kabg , and for the sake of generality, adding a decay rate γc / for the decay of the bound state |C by irreversible loss processes, gives in the limit of E → 0, abg Γ0 a˜ = a − ib = abg − . (6.22) E0 − i(γc /2) This formalism accounts for both kinds of tunable resonances that are used for making cold molecules from cold atoms, namely, magnetically or optically tuned resonances. We now give our attention to each of these in turn.
6.4
MAGNETICALLY TUNABLE RESONANCES
Cold alkali metal atoms have a variety of magnetically tunable resonances that have been exploited in a number of experiments to control the properties of ultracold quantum gases or to make cold molecules. For the most part, experiments have succeeded with species that either do not have inelastic loss channels, or, if they do, the loss rates are very small. Thus, for practical purposes, we can set the resonance decay rate γc = 0 in examining a wide class of magnetically tunable resonances. While general coupled channel methods can be set up to solve the multichannel Schrödinger equation [1], we will use simpler models to explain the basic features of tunable Feshbach resonance states. Many resonances occur for alkali metal species in their 2 S electronic ground state because of their complex hyperfine and Zeeman substructure with energy splittings very large compared to kB T . Thus, closed spin channels that have bound states near Eα of an entrance channel α can serve as tunable scattering resonances for threshold collisions in that channel. The key to magnetic tuning of a resonance is that the resonance state |C has a different magnetic moment μc than the moment μatoms of the pair of separated atoms in the entrance channel. The bare bound-state energy can be tuned by varying the magnetic field B Ec (B) = δμ(B − Bc ),
(6.23)
where δμ = μatoms − μc is the magnetic moment difference and Bc is the field where Ec (Bc ) = 0 at threshold. The scattering length is real with b = 0 and takes on the following resonant form Δ , B − B0
(6.24)
B0 = Bc + δB.
(6.25)
a(B) = abg − abg where Δ=
Γ0 δμ
and
Note that the interaction between the entrance and closed channels shifts the point of singularity of a(B) from Bc to B0 . Such magnetically tunable Feshbach resonances are characterized by four parameters, namely, the background scattering length abg , the magnetic moment difference δμ, the resonance width Δ, and position B0 . © 2009 by Taylor and Francis Group, LLC
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Figure 6.6 shows an example of the scattering length and bound-state energies for the 40 K87 Rb molecule near the lowest energy spin channel of the separated atoms. The spin quantum numbers and hyperfine splitting in their respective electronic ground states are 1, 2, and 6.835 GHz for 87 Rb and 9/2, 7/2, and −1.286 GHz (inverted) for 40 K. There are 11 other closed spin channels in this system with E > E that have α 1 the same total projection quantum number as the lowest energy α = 1 spin channel. Because of their different magnetic moments the energy of a bound state of one of these closed channels can be tuned relative to the energy of the two separated atoms in the α = 1 s-wave channel, as shown in the figure. Due to coupling terms in the Hamiltonian among the various channels, bound states that cross threshold couple to the entrance channel and give rise to resonance structure in its a(B). The resonance with B0 near 54.6 mT (546 G) has been used to associate a cold 40 K atom and a cold 87 Rb atom to make a 40 K87 Rb molecule in a near-threshold state with a small binding energy on the order of 1 MHz or less [19]. It is extremely useful to introduce the properties of the long-range van der Waals potential and take advantage of the separation of short- and long-range physics discussed in the previous section. Assuming that the interaction W (R) is confined
–100
a (a0)
–150 –200 0 E/h (GHz)
2(–1) –0.2 –0.4
5(–2)
4(–2)
12(–3)
1(–1) 11(–3)
40
50
60
70
B (mT)
FIGURE 6.6 Molecular bound state energies (lower panel) and scattering length (upper panel) vs. magnetic field B in mT (1 mT = 10 G) for the lowest energy α = 1 s-wave spin channel of the 40 K87 Rb fermionic molecule. The bound-state energies are shown relative to the channel energy E1 of the two separated atoms taken to be zero. This α = 1 spin channel has respective 40 K and 87 Rb spin projection quantum numbers of −9/2 and +1, giving a total projection of −7/2. In this species there are 11 additional closed s-wave channels with Eα > E1 and with the same projection of −7/2. The bound-state quantum numbers are α(i), where i is the vibrational quantum number relative to the dissociation limit of closed channel α = 2, . . . , 12. Four bound states cross threshold in this range of B, giving rise to singularities in the scattering length.
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to distances R a¯ , the matrix element in Equation 6.18 defining Γ(E) can be factored as ¯ Γ(E) = Cbg, (E)−2 Γ, (6.26) where Γ¯ is a measure of resonance strength that depends only on the energyindependent short-range physics near E = 0, and is completely independent of the asymptotic boundary conditions. It thus can be used in characterizing the properties of both scattering and bound states when E = 0. The extrapolation of resonance properties away from E = 0 depends on two additional parameters associated with the long-range potential, μ and C6 , which determine ¯ Let us define a dimensionless resonance strength parameter a¯ and E. sres =
abg δμΔ Γ0 = rbg a¯ E¯ E¯
(6.27)
where rbg = abg /¯a. Using the threshold van der Waals form of Cbg,0 (E)−1 given in the previous section, we can write Γ¯ 1 ¯ . = (sres E) 2 1 + (1 − rbg )2
(6.28)
The above-threshold scattering properties are found from the scattering phase shift η(E) = ηbg (E) + ηres (E), where ηres (E) is found from Equation 6.17 once Ec , Γ(E), and δE(E) are known. The first two are given by Equations 6.23 and 6.26, and δE(E) =
Γ¯ tan λbg (E), 2
(6.29)
where tan λbg (E) is a function determined by the van der Waals potential, given abg . It has the limiting form tan λbg (E) = 1 − rbg as E → 0, and tan λbg (E) = 0 for E E¯ [3,8]. Thus the position of the scattering length singularity is shifted by δB = B0 − Bc = Δ
rbg (1 − rbg ) 1 + (1 − rbg )2
(6.30)
from the crossing point Bc of the bare bound state. Scattering phase shifts calculated from the van der Waals potential with the “quantum defect” forms in Equations 6.26 and 6.29 are generally in excellent agreement with complete coupled channels methods for energy ranges on the order of E¯ and even larger [11]. The properties of bound molecular states near threshold can also be calculated from the general coupled-channels quantum defect method using the properties of the longrange potential. When the energy Eb (B) = −2 kb (B)2 /(2μ) of the threshold s-wave bound state is small, that is, |Eb (B)| E¯ or kb (B)¯a 1, then the expression for Eb (B) from the quantum defect method is
1 Γ¯ − kb (B)¯a = . (Ec (B) − Eb (B)) rbg − 1 2 © 2009 by Taylor and Francis Group, LLC
(6.31)
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Cold Molecules: Theory, Experiment, Applications
If Γ¯ = 0, we recover the uncoupled, or bare, bound states of the system, whereas when Γ¯ > 0, this equation gives the coupled, or “dressed,” bound states. The threshold bound state “disappears” into the continuum at B = B0 , where a(B) has a singularity. The shift in Equation 6.30 follows immediately upon solving for Ec (B0 ) where Eb (B0 ) = 0. Threshold bound-state properties are strongly affected by the magnitudes of sres and rbg . When the coupled bound-state wavefunction is expanded as a mixture of closed and background channel components, |c and |bg respectively, an important property is the norm Z(B) of the closed channel component; the norm of the entrance channel component is 1 − Z(B). The value of Z can be calculated from a knowledge of Eb (B), because Z = |δμ−1 ∂Eb /∂B| [1]. There are two basic classes of resonances. One, for which sres 1, are called entrance channel dominated resonances. These have Z(B) 1 as B − B0 varies over a range that is a significant fraction of |Δ|. In addition, the bound-state energy is given by Equation 6.11 over a large part of this range. On the other hand, closedchannel-dominated resonances are those with sres 1. They have Z(B) large, on the order of unity, as |B − B0 | varies over a large fraction of |Δ|, and only have a “universal” bound state (Equation 6.3) over a quite small range |Δ| near B0 . ¯ so that Entrance-channel-dominated resonances have Γ(E, B) > E when 0 < E < E, no sharp resonance feature persists above threshold, where a(B) < 0 and the last bound state has disappeared. By contrast, closed-channel-dominated resonances with ¯ so that a sharp resonant |rbg | not too large will have Γ(E, B) < E when 0 < E < E, feature emerges just above threshold, continuing as a quasibound state with E > 0 into the region where a(B) < 0. Figure 6.7 shows an expanded view of the 4(−2) resonance of 40 K87 Rb near 54.6 mT. The figure shows the character of the bound state as it merges into threshold at B0 . It tends to be a universal “halo” bound state over a range of |B − B0 | that is less than about 1/3 of Δ. As |B − B0 | increases, the bound state increasingly takes on the character of the closed channel 4(−2) level as Z increases toward unity. Figure 6.8 shows an example of the very broad 6 Li resonance in the lowest energy α = 1 s-wave channel, which requires two 6 Li fermions in different spin states. This is a strongly entrance-channel-dominated resonance, where Z 1 over a range of |B − B0 | nearly as large as Δ. The last bound state is a universal halo molecule over a range larger than 100 G. The corrected Equation 6.11 is a good approximation over an even larger range. The scattering length graph shows that the size ≈ a(B)/2 of the halo state is very large compared to a¯ = 30a0 (see Table 6.1) over this range. Magnetically tunable scattering resonances have proven very useful in associating two cold atoms to make a molecule in the weakly bound states near threshold. This work is reviewed in detail in Ref. 1. The magnetoassociation (MA) process works by first preparing a gas with a mixture of both atomic species at B > B0 (assuming δμ > 0), where there is no threshold bound state. By ramping the B field down in time so that B < B0 , colliding pairs of atoms with E > 0 can be converted to diatomic molecules in a bound state with energy E < 0. The conversion efficiency will depend on both the ramp rate and the phase space density of the initial gas. If the initial atom pair is held in a single cell of an optical lattice instead of a gas, the conversion efficiency can approach 100%. A simple Landau–Zener picture has been found to be © 2009 by Taylor and Francis Group, LLC
237
Molecular States Near a Collision Threshold 1.0
Z
0.5
Eb/h (MHz)
0.0
B0 –5
–10 54.2
54.4
54.6
54.8
B (mT)
FIGURE 6.7 The lower panel shows an expanded view of Eb (B) near B0 for the 40 K87 Rb resonance with B0 = 54.693 mT (546.93 G) in Figure 6.6. The solid line comes from a coupled-channels calculation that includes all 12 channels with the same −7/2 projection quantum number. The dashed and dotted lines respectively show the universal energy of Equation 6.3 and the van der Waals corrected energy of Equation 6.11. The upper panel shows the closed channel norm Z(B). The width Δ = 0.310 mT (3.10 G), abg = −191a0 , and δμ/h = ¯ = 13.9 MHz, this is a marginal 33.6 MHz/mT (3.36 MHz/G). With a¯ = 68.8a0 and E/h entrance-channel-dominated resonance with sres = 2.08.
quite accurate for such lattice cells, where the conversion probability of the atom pair in the trap ground state i = 0 is 1 − e−A , where A=
2π Wci2 . E˙c
(6.32)
Here i ≥ 0 represents the above-threshold levels of the atom pair confined by the trap, continuing the below threshold series of dimer levels with i ≤ −1. For a three-dimensional harmonic trap with frequency ωx = ω√ y = ωz = ω, √the matrix element Wci = C|W (R)|i is well-approximated as Wci = Γ(Ei )/2π ∂Ei /∂i, where ∂Ei /∂i = 2ω and√Γ(Ei ) = 2ki abg δμΔ for the i = 0 trap ground state of relative motion with ki = 3μω/ (see Equations 6.18, 6.20, and 6.25). The trick used here in getting a matrix element Wci between two bound states from the matrix element C|W (R)|E involving an energy-normalized scattering state is to introduce the density of states as in Equation 6.10. In a similar manner, the matrix element can be obtained between the bare closed channel state and the bound states i < 0 of the entrance channel. Such matrix elements characterize avoided crossings like the one in Figure 6.6 for E/h near −0.4 MHz and B near 43 mT. Finally, it should be noted that a Landau–Zener model can also be used for molecular dissociation by a fast magnetic field ramp. An alternative phenomenological model has been developed to describe © 2009 by Taylor and Francis Group, LLC
238
Cold Molecules: Theory, Experiment, Applications (a) a(B) (a0)
8000 4000 0
Eb/h (MHz)
(b)
0 –1
B0
–2 –3 60
70
80
90
B (mT)
FIGURE 6.8 (a) Scattering length and (b) molecular bound state energy vs. magnetic field B for the lowest energy α = 1 s-wave spin channel of the 6 Li2 molecule. This channel has one 6 Li atom in the lowest +1/2 projection state and the other in the lowest −1/2 projection state for a total projection of 0. There are four additional closed channels with projection 0. In this range of B there is only one bound state that crosses threshold at B0 = 83.4 mT (834 G). The lower panel shows Eb (B) from a coupled channels calculation (solid circles), the universal limit of Equation 6.3 (dashed line) and the corrected limit of Equation 6.11 (solid line). The width Δ = 30.0 mT (300 G), abg = −1405a0 , and δμ/h = 28 MHz/mT (2.8 MHz/G). This is a strongly entrance-channel-dominated resonance with sres = 59, and Z < 0.06 over the range of B shown.
molecular association in cold gases, which are more complex than two atoms in a lattice cell [20].
6.5
PHOTOASSOCIATION
Cold atoms can also be coupled to molecular bound states through photoassociation (PA), as discussed in Chapters 5, 7, 8, and 9. Figure 6.9 gives a schematic description of PA, a process by which the colliding atoms can be coupled to such bound-state resonances through one or two photons. Reference 2 reviews theoretical and experimental work on PA spectroscopy and molecule formation. Molecules made using the magnetically tunable resonances described in the last section are necessarily very weakly bound, with binding energies limited by the small range of magnetic tuning. Photoassociation has the advantage that laser frequencies are widely tunable, so that a range of many bound states becomes accessible to optical methods, even the lowest v = 0 vibrational level of the ground state. In addition, the light can be turned off and on or varied in intensity for time-dependent manipulations. Photoassociation naturally lends itself to the resonant scattering treatment of a decaying resonance in Equation 6.22, which applies to the one-color case with position Ec = Ev∗ − hν1 , strength abg Γ0 (I) = Γ(E, I)/(2k), and shift δE(I). The latter two are linear in laser intensity I when I is low enough. Photoassociation is usually detected © 2009 by Taylor and Francis Group, LLC
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Molecular States Near a Collision Threshold
Ev*
A* + B
n2
n1
Decay gc
V(R)
E RC
A+B
R
FIGURE 6.9 Schematic representation of one- and two-color photoassociation (PA). The two colliding ground-state atoms at energy E can absorb a laser photon of frequency ν1 and be excited to an excited molecular bound state at energy Ev∗ . The bound state decays via spontaneous emission at rate γc /. If a second laser is present with frequency ν2 , the excited level can also be coupled to a ground-state vibrational level v at energy Ev , if h(ν2 − ν1 ) = E − Ev . The PA process depends on the ground-state wavefunction at the Condon point RC of the transition, where hν1 equals the difference between the excited- and ground-state potentials.
by the inelastic collisional loss of cold atoms it causes, due to the spontaneous decay of the excited state to make hot atoms or deeply bound molecules. In the limit E → 0 the complex scattering length is a(ν1 , I) = abg − Lopt b(ν1 , I) =
γc E 0 E02 + (γc /2)2
γc2 1 Lopt 2 , 2 E0 + (γc /2)2
(6.33)
(6.34)
where E0 = Ev∗ − hν1 + δE(I) is the detuning from resonance, including the intensity-dependent shift, and the optical length is defined by Lopt = abg Γ0 (I)/γc . Photoassociation spectra, line shapes, and shifts have been widely studied for a variety of like and mixed alkali–metal species. At the higher temperatures often encountered in magneto-optical traps, contributions to PA spectra from higher partial waves, such as, p- or d-waves, have been observed in a number of cases. The theory can be readily extended to higher partial waves. By introducing an energy-dependent complex scattering length the theory for s-waves can be extended to finite E away from threshold and to account for effects due to reduced dimensional confinement in optical lattices [21]. The optical length formulation of resonance strength is very useful for a decaying resonance. It also applies to decaying magnetically tunable resonances, if Γ0 from © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
Equation 6.25 is used to define a resonance length abg δμΔ/γc equivalent to Lopt [22]. The scattering length has its maximum variation of abg ± Lopt when the laser is tuned to E0 = ±γc /2, and losses are maximum at E0 = 0 where b = Lopt . When detuning is small, on the order of γc , significant changes to the scattering length on the order of a¯ are thus normally accompanied by large loss rates (see Equation 6.14). Losses can be avoided by going to large detuning, because when (γc /E0 ) 1, b = (Lopt /2)(γc /E0 )2 , whereas the change in a only varies as a − abg = −Lopt (γc /E0 ). To make the change a − abg large enough while requiring (γc /E0 ) 1 means that Lopt has to be very large compared to a¯ . The magnitude of Lopt depends on the matrix element C|Ω1 (R)|E where Ω1 (R) represents the optical coupling between the ground and excited state. Using Equations 6.18 and 6.20 and the above definition of Lopt , and factoring out the relatively constant Ω1 value, Lopt = π
|Ω1 |2 F(E) . γc k
(6.35)
The Franck–Condon overlap factor is F(E) =
0
≈
∞
2 ∗ ψv (R)ψ0 (R, E) dR
∂Ev∗ 1 |ψ0 (RC , E)|2 , ∂v DC
(6.36) (6.37)
where DC is the derivative of the difference between the excited- and ground-state potentials evaluated at the Condon point RC , and ∂Ev∗ /∂v is the excited-state vibrational spacing. Equation 6.37 is known as the reflection approximation, generally an excellent approximation where F(E) is proportional to the square of the groundstate wavefunction at RC , the Condon point where the molecular potential difference matches hν1 (see Figure 6.9). Thus, F(E) can be evaluated using expressions like Equations 6.5 or 6.9 for RC a¯ or RC a¯ , respectively. The reflection approximation is quite good over a wide range of E and for higher partial waves than the s-wave. By selecting a range of excited levels v by changing laser frequency ν1 , thus changing RC , the shape and nodal structure of the ground-state wavefunction can be mapped out over a range of R. The optical length has several important properties evident from Equation 6.35. First, because both Ω1 and γc are proportional to the same squared transition dipole moment, Lopt does not depend on whether the transition is strong or weak, but can be large for both kinds of transitions. Second, Lopt ∝ |Ω1 |2 so it can be increased by increasing laser intensity. Third, because F(E) ∝ k as E → 0 for entrance channel s-waves, Lopt ∝ F(E)/k is independent of E or k at low energy. However, it does depend strongly on the molecular structure through the Franck–Condon factor. In practice, using strong transitions with large decay rates such as those in alkali–metal species leads to the requirement to use excited molecular levels far from threshold with large binding energies. This is necessary to achieve large detuning from atomic and molecular resonance. This requirement means such levels have small F(E) factors, © 2009 by Taylor and Francis Group, LLC
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due to the very large value of DC in Equation 6.37. On the other hand, weak transitions with small decay rates, such as those associated with the 1 S0 → 3 P1 intercombination line transition for alkaline earth species such as Sr, can lead to quite large values of Lopt . This is because large detuning in γc units can be achieved for levels that are still quite close to the excited-state threshold. Such levels typically have large Franck– Condon factors. In fact, PA transitions near the weak intercombination line of Sr have been observed to have Lopt several orders of magnitude larger than was observed for strongly allowed molecular transitions involving Rb [23]. Thus, there are good prospects for some degree of optical resonant control of collisions in ultracold gases of species like Ca, Sr, or Yb. Two-color PA is also possible when a second laser with frequency ν2 is added, as shown in Figure 6.9. When the the frequency difference is chosen so that h(ν2 − ν1 ) = E − Ei , the ground i molecular level is in resonance with collisions at energy E. By keeping ν1 fixed and tuning ν2 , two-color PA spectroscopy can be used to probe the level energies. This is how the data on binding energies of the Yb2 molecule were obtained [15] so as to be able to construct Figure 6.5. In this case, 12 different levels from different isotopic species were measured, among which were levels with i = −1 and −2 and = 0 and 2. Two-color spectroscopy has also been carried out for several alkali–metal homonuclear species. Two-color processes are also an excellent way to assemble two cold atoms into a translationally cold molecule. Early work along these lines was done using the spontaneous decay of the excited level to populate a wide range of levels in the ground state. The disadvantage of spontaneous decay is that it is not selective. However, by using a laser with a precise frequency, a specific level can be chosen as the target level. One early experiment did this to associate two 87 Rb atoms in a Bose–Einstein condensate to make a molecular level at a specific energy of h(−636) MHz [24]. It is highly desirable to be able to make translationally cold molecules in their vibrational ground state v = 0. This is especially true of polar molecules, which have large dipole moments in v = 0. On the other hand, threshold levels have negligible dipole moments, because there is no charge transfer because of the large average atomic separation ≈ a¯ . A promising technique is to use MA using a tunable Feshbach resonance to associate the atoms into a threshold molecular level, then use a twocolor Raman process to move the population in that state to a much more deeply bound level. Although molecules in a gas are subject to fast destructive collisions with cold atoms or other molecules in the gas (see Equation 6.14), the molecules can be protected against such collisions by forming them in individual optical lattice trapping cells. Then the two-color Raman process could be used to produce much more deeply bound molecules that are stable against destructive collisions. This has been done successfully with 87 Rb2 [25] molecules. In the future, such methods are likely to produce v = 0 polar molecules, with which a range of interesting physics can be explored [26,27].
REFERENCES 1. Köhler, T., Góral, K., and Julienne, P.S., Production of cold molecules via magnetically tunable Feshbach resonances, Rev. Mod. Phys., 78, 1311, 2006.
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2. Jones, K.M., Tiesinga, E., Lett, P.D., and Julienne, P.S., Photoassociation spectroscopy of ultracold atoms: long-range molecules and atomic scattering, Rev. Mod. Phys., 78, 483, 2006. 3. Julienne, P.S. and Mies, F.H., Collisions of ultracold trapped atoms, J. Opt. Soc. Am. B, 6, 2257, 1989. 4. Moerdijk, A.J., Verhaar, B.J., and Axelsson, A., Resonances in ultracold collisions of 6 Li, 7 Li, and 23 Na, Phys. Rev. A, 51, 4852, 1995. 5. Vogels, J.M., Verhaar, B.J., and Blok, R.H., Diabatic models for weakly bound states and cold collisions of ground-state alkali-metal atoms, Phys. Rev. A, 57, 4049, 1998. 6. Burke, J.J.P., Greene, C.H., and Bohn, J.L., Multichannel cold collisions: simple dependences on energy and magnetic field, Phys. Rev. Lett., 81, 3355, 1998. 7. Vogels, J.M., Freeland, R.S., Tsai, C.C., Verhaar, B.J., and Heinzen, D.J., Coupled singlet-triplet analysis of two-color cold-atom photoassociation spectra, Phys. Rev. A, 61, 043407, 2000. 8. Mies, F.H. and Raoult, M., Analysis of threshold effects in ultracold atomic collisions, Phys. Rev. A, 62, 012708, 2000. 9. Gao, B., Zero-energy bound or quasibound states and their implications for diatomic systems with an asymptotic van der Waals interaction, Phys. Rev. A, 62, 050702, 2000. 10. Gao, B., Angular-momentum-insensitive quantum-defect theory for diatomic systems, Phys. Rev. A, 64, 010701, 2001. 11. Julienne, P.S. and Gao, B., Simple theoretical models for resonant cold atom interactions, in Atomic Physics 20, Roos, C., Häffner, H., and Blatt, R., Eds., AIP, Melville, New York, 2006, p. 261–268, physics/0609013. 12. Derevianko, A., Johnson, W.R., Safronova, M.S., and Babb, J.F., High-precision calculations of dispersion coefficients, static dipole polarizabilities, and atom–wall interaction constants for alkali–metal atoms, Phys. Rev. Lett., 82, 3589, 1999. 13. Porsev, S.G. and Derevianko, A., High-accuracy calculations of dipole, quadrupole, and octupole electric dynamic polarizabilities and van der Waals coefficients C6 , C8 , and C10 for alkaline-earth dimers, JETP 102, 195, 2006, [Pis’ma Zh. Eksp. Teor. Fiz., 129, 227–238 (2006)]. 14. Gribakin, G.F. and Flambaum, V.V., Calculation of the scattering length in atomic collisions using the semiclassical approximation, Phys. Rev. A, 48, 546, 1993. 15. Kitagawa, M., Enomoto, K., Kasa, K., Takahashi, Y., Ciurylo, R., Naidon, P., and Julienne, P.S., Two-color photoassociation spectroscopy of ytterbium atoms and the precise determinations of s-wave scattering lengths, Phys. Rev. A, 77, 012719, 2008. 16. Fano, U., Effects of configuration interaction on intensities and phase shifts, Phys. Rev. A, 124, 1866, 1961. 17. Bohn, J.L. and Julienne, P.S., Semianalytic theory of laser-assisted resonant cold collisions, Phys. Rev. A, 60, 414, 1999. 18. Marcelis, B., van Kempen, E.G.M., Verhaar, B.J., and Kokkelmans, S.J.J.M.F., Feshbach resonances with large background scattering length: Interplay with open-channel resonances, Phys. Rev. A, 70, 012701, 2004. 19. Ospelkaus, C., Ospelkaus, S., Humbert, L., Ernst, P., Sengstock, K., and Bongs, K., Ultracold heteronuclear molecules in a 3D optical lattice, Phys. Rev. Lett., 97, 120402, 2006. 20. Hodby, E., Thompson, S.T., Regal, C.A., Greiner, M., Wilson, A.C., Jin, D.S., Cornell, E.A., and Wieman, C.E., Production efficiency of ultracold Feshbach molecules in Bosonic and Fermionic systems, Phys. Rev. Lett., 94, 120402, 2005. 21. Naidon, P. and Julienne, P.S., Optical Feshbach resonances of alkaline–earth atoms in a 1D or 2D optical lattice, Phys. Rev. A, 74, 022710, 2006.
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22. Hutson, J.M., Feshbach resonances in ultracold atomic and molecular collisions: threshold behaviour and suppression of poles in scattering lengths, New J. Phys., 9, 152, 2007. 23. Zelevinsky, T., Boyd, M.M., Ludlow, A.D., Ido, T., Ye, J., Ciurylo, R., Naidon, P., and Julienne, P.S., Narrow line photoassociation in an optical lattice, Phys. Rev. Lett., 96, 203201, 2006. 24. Wynar, R., Freeland, R.S., Han, D.J., Ryu, C., and Heinzen, D.J., Molecules in a BoseEinstein condensate, Science, 287, 1016, 2000. 25. Winkler, K., Lang, F., Thalhammer, G., van der Straten, P., Grimm, R., and Hecker Denschlag, J., Coherent optical transfer of Feshbach molecules to a lower vibrational state, Phys. Rev. Lett., 98, 043201, 2007. 26. Lewenstein, M., Polar molecules in topological order, Nature Phys., 2, 309, 2006. 27. Büchler, H.P., Micheli, A., and Zoller, P., Three-body interactions with cold polar molecules, Nature Phys., 3, 726, 2007.
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Prospects for Control of 7 Ultracold Molecule Formation via Photoassociation with Chirped Laser Pulses Eliane Luc-Koenig and Françoise Masnou-Seeuws CONTENTS 7.1
7.2
Introduction: Can Ultrafast Meet Ultracold?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Making Ultracold Molecules by Photoassociation of Ultracold Atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Link with the Coherent Control Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Outline of the Present Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling of Photoassociation with Chirped Laser Pulses . . . . . . . . . . . . . . . . 7.2.1 The Physical Problem: Example of Photoassociation into a Long-Range Well of Cs2 and Choice of a Pulse in the 100 picosecond Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1.1 Choice of Cs2 as a Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1.2 A Qualitative Interpretation of Photoassociation . . . . . . . . 7.2.1.3 Timescales and Characteristic Distances for the Vibrational Motion in the Excited State. . . . . . . . . . . . . . . . . . 7.2.1.4 Description of the Initial Collision State . . . . . . . . . . . . . . . . . 7.2.1.5 Position of the “Last” Node RN . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Choice of a Linearly Chirped Pulse in the Picosecond Domain . . 7.2.2.1 The Chirped Pulse, Central Frequency, Energy, Spectral and Temporal Widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2.2 The Photoassociation Process: Time Window, Resonance Window, Concept of a Photoassociation Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 The Two-Channel Coupled Equations and the Choice for a Rotating Wave Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
247 247 248 248 249
249 249 251 251 252 254 254 254
256 256
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7.3
7.4
7.5
Cold Molecules: Theory, Experiment, Applications
7.2.3.1 Rotating Wave Approximation with the Instantaneous Frequency: Definition of the Photoassociation Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3.2 Rotating Wave Approximation with the Central Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of Numerical Simulations: Interpretation as Adiabatic Transfer Within a Photoassociation Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Results of Calculations for One Collision Energy. . . . . . . . . . . . . . . . . 7.3.2.1 Photoassociation within the Resonance Window . . . . . . . . 7.3.2.2 Formation of Halo Molecules via Optically Induced Feshbach Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2.3 Selectivity of the Resonance Window: Dependence of the Final Distribution of Population on the Pulse Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Analysis within a Two-State Model: The Concept of Adiabatic Transfer within a Photoassociation Window . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Averaging Over Initial Velocity Distribution: Use of Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4.1 Boltzmann Average on a Finite-Size Grid . . . . . . . . . . . . . . . 7.3.4.2 Average Introducing Box-Independent Energy-Normalized States: Use of a Scaling Law Near Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 What is the Absolute Number of Photoassociated Molecules? . . . 7.3.6 Transient Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shaping Vibrational Wavepackets in the Excited State to Optimize Stabilization into Deeply Bound Levels of the Lower State . . . . . . . . . . . . . . 7.4.1 Shaping the Vibrational Wavepacket in the Excited State . . . . . . . . 7.4.2 Proposal for a Two-Color Pump–Dump Experiment . . . . . . . . . . . . . 7.4.2.1 The Time-Dependent Franck–Condon Overlap . . . . . . . . . 7.4.2.2 A Two-Color Experiment for Creating Stable Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Dynamical Hole in the Initial State Wavefunction: Compression Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Phenomenological Observation of a Depletion Hole, a Momentum Kick, and a Compression Effect . . . . . . . . . . . . . . . . . . . . 7.5.2 Analysis of the Momentum Transfer with Partially Integrated Mass Current and Population. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Advantage of the Compression Effect for Photoassociation with a Second Pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Redistribution of Population in the Lower State after a Photoassociation Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4.1 Redistribution in the a3 Σ+ u State in the Case of Cesium Photoassociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4.2 Correlated Pairs of Hot Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . .
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258 259 260 260 261 261 261
262 264 266 267
267 268 270 271 271 272 273 274 276 276 278 279 281 281 281
Control of Ultracold Molecule Formation via Photoassociation
Beyond the Impulsive or Adiabatic Approximations: New Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Controlling the Compression Effect with a Nonimpulsive Pulse Inducing Many Rabi Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Nonadiabatic Broadband Pulse: Excitation at Large Distances. . . 7.6.2.1 Large Transfer of Population Outside the PA Window . . 7.6.2.2 Thermal Average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Conclusion and Prospects for the Near Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.1 7.1.1
283 283 284 284 284 286 287 287
INTRODUCTION: CAN ULTRAFAST MEET ULTRACOLD? MAKING ULTRACOLD MOLECULES BY PHOTOASSOCIATION OF ULTRACOLD ATOMS
The formation of ultracold molecules in the lowest rovibrational level of the ground electronic state is presently a subject of intense investigation. By ultracold, we mean molecules at a temperature well below 1 mK. As discussed in Chapters 5 and 6 of the present book by W. Stwalley and colleagues, and by Paul Julienne, the photoassociation (PA) route [1,2] is a very efficient one. Whereas laser cooling techniques cannot directly be applied to molecules, it is possible to start from an assembly of laser-cooled atoms, and then make cold molecules in an excited electronic state by photoassociating atom pairs. For that purpose, most experiments use a cw laser red-detuned, relative to the atomic resonance line, to a frequency coinciding with the transition to a loosely bound vibrational level of the dimer. Being in an excited electronic state, the photoassociated molecule is short-lived, and a second step is necessary to form a stable molecule, hereafter named the stabilization step. When it decays by spontaneous emission, the molecule may either break apart again into a pair of cold atoms, or reach various vibrational levels of the ground electronic state (or of the lowest triplet state in the case of alkali dimers) [3–5]. In the latter case stable molecules are indeed formed, and schemes to increase their formation rate depend very much upon details in their spectroscopy (such as the existence of long-range wells in the potential curves of the excited state, or of resonant coupling between two excited channels [6,7]). Therefore, the molecular spectroscopy foundation of the field reveals itself as essential. Compared to nonoptical routes toward cold molecules discussed in this book, the main advantage of the PA route is the ability to produce a large number of stable ultracold molecules, at the same translational temperature (a few μK, or even less) than the precursor atoms. In contrast with the “halo” molecules produced by sweeping magnetic Feshbach resonances [8], the stable molecules are formed in vibrational levels, which can be relatively deeply bound. The drawback is that stabilization through the spontaneous emission process spreads the population into a variety of excited vibrational levels, so that the product molecules are not in a pure vibrational state. Schemes to cool down the vibrational and rotational degrees of freedom have to be implemented. The present chapter explores one scheme, making use of chirped laser pulses, or more generally shaped pulses, rather than cw lasers, for PA, and implementing © 2009 by Taylor and Francis Group, LLC
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stabilization through induced emission. The idea is to fully exploit the possibilities of the optical techniques in order to gain control of the PA and stabilization processes. Besides improving molecule formation rates, the use of ultrafast lasers (picosecond and femtosecond range) and of laser shaping techniques is expected to make dynamical phenomena observable and controllable in real time.
7.1.2
LINK WITH THE COHERENT CONTROL FIELD
The formation of a bound molecule out of two colliding ultracold atoms is a simple example of a laser-induced chemical reaction at very low temperature. At room or even higher temperatures, as discussed in Chapter 8 by Evgueny Shapiro and Moshe Shapiro, the well-established field of coherent control relies upon the possibility of shaping laser pulses to control the output of chemical reactions. Whether similar schemes can be applied to the PA and stabilization reactions at low temperatures has been an open question. To give an answer, one should carefully explore the feasibility and the efficiency of pump–dump experiments to selectively create ultracold molecules in the v = 0 level of their ground electronic state. On the experimental side, the implementation of short pulse techniques looks controversial: indeed, the ultracold field being linked to precision spectroscopy, the cw-laser technology seems at first sight to be the best adapted. Changing to short and shaped pulses, one might lose the sensitivity to spectroscopic details that has been a key to success. Moreover, the spectroscopic analysis of the photoassociated or stabilized molecules, identifying which rovibrational levels of which electronic states are populated, is not well established in time-dependent experiments. Nevertheless, several groups [9–11] have started experiments to create molecules in a rubidium magneto-optical trap by photoassociating with femtosecond lasers. Although such experiments are not yet fully convincing, because the first pulse mainly destroys the molecules already present in the trap, rapid progress is occurring as the field develops. On the theoretical side, the novelty in the ultracold field lies in the description of the initial state of two colliding cold atoms. At room temperature, the Hamiltonian is dominated by kinetic energy, and time-dependent treatments using Gaussian wavepackets are well established. In the microKelvin range, a Gaussian wavepacket spreads more rapidly than it moves, so that the initial state must be a distribution of stationary collision states. The adequacy of such states to describe the collision dynamics has been revealed in early PA spectroscopy experiments [12,13]: when sweeping the frequency of the cw laser, the minima in the experimental signal are a signature of the nodes in the radial wavefunction associated with s-wave scattering. In this chapter, we shall therefore present numerical treatments calculating the time evolution, once a laser pulse is turned on, of a stationary collision wavefunction, delocalized over a wide range of internuclear distances.
7.1.3
OUTLINE OF THE PRESENT CHAPTER
An important theoretical effort has been devoted to the subject by our theoretical group in Orsay, in collaboration with Ronnie Kosloff and the Jerusalem group. It started as early as 1996 with the thesis work of Mihaela Vatasescu [14–16] and later © 2009 by Taylor and Francis Group, LLC
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on with the contribution of Jiri Vala [17]. Owing to the creation at the end of 2002 of a European Research Training Network, funded by the European Commission under contract HPRNCT 2002 00290, and named “Cold Molecules: Formation, Trapping and Dynamics” this activity was able to develop further. The first calculations using a stationary collision wavefunction for the initial state of time-dependent calculations were published in 2004 [18,19]. The active participation of network post-docs Christiane Koch [20–24], Jordi Mur-Petit [25] and Shimshon Kallush [26–28], as well as the thesis works of Pascal Naidon [29,30] and Kai Willner [31,32], led to many developments. This chapter presents a synthesis of most of the latter work, and of the ongoing debate about the possibility of overlap between the two fields of coherent control and ultracold molecules. It is organized as follows: • Section 7.2 presents modeling of PA with a chirped laser pulse, choosing cesium as a case study. • Section 7.3 presents results of the numerical calculations. It is shown that, for well-chosen pulse parameters, total adiabatic population transfer can be achieved, so that all the pairs of atoms with interatomic distances within a range defining a “photoassociation window” are transformed into bound molecules. An estimation of the total number of such photoassociated molecules is given. • Section 7.4 presents the possibility of shaping vibrational wavepackets in the excited state, in view of a focusing effect that optimizes the formation of stable ground-state molecules in a pump–dump experiment. • Section 7.5 analyzes the dynamical hole formed in the initial-state wavepacket after PA, and the compression effect due to a momentum kick. It is suggested that one should design a second pulse red-detuned relative to the first one in order to populate efficiently more deeply bound levels of the excited state. • Section 7.6 is devoted to another possible PA mechanism involving nonadiabatic transfer at large distances. • Section 7.7 gives a conclusion and prospects for future theoretical as well as experimental work.
7.2
MODELING OF PHOTOASSOCIATION WITH CHIRPED LASER PULSES
7.2.1 THE PHYSICAL PROBLEM: EXAMPLE OF PHOTOASSOCIATION INTO A LONG-RANGE WELL OF Cs2 AND CHOICE OF A PULSE IN THE 100 PICOSECOND RANGE 7.2.1.1
Choice of Cs2 as a Case Study
The Cs2 molecule is a good candidate because the formation of a large number of stable ultracold molecules has been observed in experiments [3,13]. The PA reaction 2Cs(6S, F = 4) + (ω(t)) → Cs2 (0g− (6S + 6P3/2 ); v, J) © 2009 by Taylor and Francis Group, LLC
(7.1)
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into the external well of the 0− g (6S + 6P3/2 ) electronic state is considered here as a case study. In this reaction two ultracold cesium atoms, in their ground 6S state, colliding at a temperature in the 50 μK range, absorb a photon red-detuned from the atomic resonance line to form a bound level (v, J) in the outer potential well of the excited state. The frequency of the laser is time-dependent. As for many PA experiments, the spectroscopy is very well known [33]. The potential curves have been described elsewhere [18] and are displayed in Figure 7.1. We use vibrational numbering vtot for the double-well potential and v for the external well; the latter supports at least 226 levels, from v = 0 (vtot = 25) to v = 225 (vtot = 256). The initial electronic state is the a3 Σ+ u (6S + 6S); only s-wave scattering is considered, and rotational excitation of the molecule is not allowed. Therefore, the calculations are performed with realistic potentials, fitted to the experimental spectra, but it is still a simplified model, because we neglect the hyperfine structure and also the possibility of two- (or more) photon absorption with population of Rydberg excited electronic states. tP+tc
tP–tc
tP
c<0
t
dmin
Excited-state potential
umax
dL dmax
umin
ћwat
ћwL
Lower-state potential Rmin RL
Rmax
Rc(t)
FIGURE 7.1 Scheme of the PA process with a chirped pulse. The potential curves for the − lower triplet a3 Σ+ u state and for the double well 0g (6S + 6P3/2 ) excited state of the Cs2 dimer are represented. Population of vibrational levels of the external well is the most efficient PA scheme studied to date. Also indicated are the time window t (upper horizontal scale) of a typical pulse with negative chirp parameter χ, the resonance window (down vertical arrow) in the energy domain [δmin , δmax ] and the PA window [Rmin , Rmax ] (lower horizontal scale), defined in Section 7.2.3.1. (From Luc-Koenig, E. et al., Phys. Rev. A, 70, 033414, 2004. With permission.)
© 2009 by Taylor and Francis Group, LLC
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7.2.1.2 A Qualitative Interpretation of Photoassociation During PA with a cw laser, a level v, J is populated when the laser is at resonance, and this can be achieved by varying the detuning. The PA probability is governed by the matrix element of the dipole transition moment between the initial continuum level and the final bound vibrational level, and therefore by the Franck–Condon overlap between the corresponding wavefunctions. Both wavefunctions are delocalized over a wide range of internuclear distances, because, for a long-range molecule, the vibrational motion extends to very large distances. In this asymptotic region of the excited-state potential, the vibrational motion becomes very slow, so that during a significant part of the vibrational period the interatomic distance stays close to Rout , the outer classical turning point; assuming the reflection principle to be valid [2], it is convenient to consider a vertical transition at this distance. One may think of PA as a pair of atoms at distance Rout absorbing a photon. Such a model explains why the PA probability is small [12,13] when Rout is located at a node of the stationary collision radial wavefunction. Turning to a laser pulse that is no longer narrowband, several levels can be at resonance, and we shall define a resonance window in the energy domain. Similarly, Rout now spans a range of distances, leading to the definition of a PA window (see Section 7.2.2.2). 7.2.1.3 Timescales and Characteristic Distances for the Vibrational Motion in the Excited State We give in Table 7.1 the values of some parameters associated with typical vibrational levels populated in PA experiments. According to the Le Roy–Bernstein law [34], for a potential with R−3 asymptotic behavior the binding energy |Ev | scales as (vD − v)6 , where vD is a constant, most often noninteger. The classical vibrational period is estimated from Tvib (v) ≈ 2π
∂v 4π . ≈ ∂E Ev+1 − Ev−1
(7.2)
and lies in the range of a few hundred picoseconds for |Ev | < 25 cm−1 or v > 42 in case of the cesium dimer. An estimation for other alkali dimers [15], using the scaling laws for a potential with −C3 /R3 asymptotic behavior, has shown that for a binding energy of 1 cm−1 Tvib varies typically from ∼120 psec for Li2 to ∼550 psec for Cs2 . Since the dependence of the period on the binding energy is |Ev |−5/6 , the ∼100 psec order of magnitude is typical of many PA experiments. The potentials being highly anharmonic, we note the significant variation of the level spacing and vibrational period with the vibrational quantum number. For the discussion on optimization of the chirped pulse, it is relevant to consider the revival period, defined [35] as 4π Trev (v) ≈ , (7.3) |Ev+1 − 2Ev + Ev−1 | and characterizing the coincidences in the vibrational motion of neighboring levels with different periods. Indeed, because the vibrational periods Tvib (v) and Tvib (v − 1) © 2009 by Taylor and Francis Group, LLC
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TABLE 7.1 Characteristic Constants for Several Levels in the Long-Range Potential Well of the Cs2 0− g (6S + 6P3/2 ) Electronic State, Located within Two Resonance Windows vtot
v
Rout (a0 )
Ev (cm−1 )
159 153 149 137 129 122 29
128 122 118 106 98 92 4
135 RL122 = 148.5 176 107.5 RL98 = 93.7 85.5 30
−0.456 −0.675 −0.869 −1.74 −2.65 −3.57 −70.3
Tvib (psec) 1095 784 635 350 250 196 20.2
Trev (nsec) 37.0 28.5 24.3 15.7 15.3 10 24.5
vtot is the vibrational number, while the numbering v is restricted to the levels in the outer well. Rout is the outer classical turning point, showing the extension of the vibrational motion up to distances of ∼100 a0 . For the two levels v = 98 and v = 122, the outer turning points are labeled RL98 and RL122 , respectively, and the radial wavefunction is drawn in Figure 7.2. Ev is minus the binding energy, Tvib is the classical vibrational period, defined in Equation 7.2, and Trev is the revival period, defined in Equation 7.3. Note the marked variation of the vibrational spacing and of Tvib as a function of the vibrational index, due to the strong anharmonicity of the potential. The set of levels [92–106] defines the resonance window for the 98 defined below, while the set [118–128] defines the resonance window for P 122 . The last line pulses P± ± shows the characteristic constants for a much deeper level.
differ, the motion of the wavepackets for two neighboring levels are out of phase until revival occurs. Also indicated in the table and the figure are the distances Rout , outer classical turning point of the levels within the resonance window. Note the large value of Rout , typical of a long-range molecule. For two levels considered in the discussion below, v = 98 (Rout = RL98 = 93.7a0 ), and v = 122 (Rout = RL122 = 150.4a0 ), the radial vibrational wavefunctions ϕe,v (R) are drawn in Figure 7.2a and d. The strong probability maximum in the vicinity of Rout justifies the picture of the “photoassociation distance.” 7.2.1.4
Description of the Initial Collision State
We have chosen for the initial state at t = tinit a stationary collision wavefunction Ψg (R, tinit ) = ϕg,E (R). Because the kinetic energy E is very small, there is a wide range of distances where the relative motion is completely governed by the potential energy. This is illustrated in Figures 7.2b and c, where the radial wavefunctions ϕg,E for various collision energies are drawn together with the radial wavefunction of the last bound level, v = 53, in the a3 Σ+ u potential. (Note that, as in Refs. 18 and 19, the data for this potential are such that the scattering length is 539 a0 , to be compared with the experimental value of 2440 a0 [36].) © 2009 by Taylor and Francis Group, LLC
253
Control of Ultracold Molecule Formation via Photoassociation R (a0) 0
200
400
(a)
600
800
1000 0.2
ϕe,u
u = 122
0
RL122
–0.2 (b)
u″ = 53 0.32 mK
ϕg,E, l=0(R)
0.33 μK
0.005 0 –0.005 –0.01
37 nK RN
Ra(54 μK)
0.11mK
54 μK
a
0.01
(c) ϕg,E, l=0(R)
0.005 0 RN
–0.005
RL98
u = 98
ϕe,u
(d)
0
20
40
60
80 R (a0)
100
120
140
–0.01 0.2 0 –0.2 –0.4
FIGURE 7.2 Wavefunctions for the initial collisional state and the resonantly photoassociated level and Franck–Condon overlap. (a) Radial wavefunction ϕe,v (R) for the v = 122 vibrational level in the external potential well of the Cs2 0− g (6S + 6P3/2 ) electronic 122 , is indicated by a dashed box, and state. The PA window, centered at RL122 , for the pulses P± coincides with a maximum of ϕg,E (R), drawn in the figure below. (b) Radial wavefunctions ϕg,E (R) for s-wave scattering at various energies E, and for the last vibrational level (v = 53) in the a3 Σ+ u potential. (c) Short-range behavior of the initial states: for R ≤ RN = 82.3a0 , the nodal structure is independent of the energy. For R > RN , the position of the nodes depends upon the energy. Because a > RN , the first node after R > RN is located at R = a for zero energy, and at Ra (54 μK) for E = 54 μK. (d) Same as (a) for the v = 98 level. The 98 , for the pulses P 98 , and indicated by a dashed box, now is close PA window, centered at RL ± to a node of ϕg,E (R). (b and c from Koch, C.P. et al., J. Phys. B: Atom. Mol. Opt. Phys., 39, S1017–S1041, 2006.)
It is clear that for R ≤ RN = 82.3a0 , the nodal structure of ϕg,E is independent of the energy. The energy dependence is concentrated in the normalization factor: this property will be used below in Section 7.3.4 to perform the Boltzmann averaging from calculations at a single energy. It is also clear that because a node of ϕg,E (R) is located at RN , not far from RL98 = 93.7a0 , the Franck–Condon overlap between the level v = 98 and the continuum levels of the initial state will be small (see Figure 7.2d). In contrast, for the level v = 122, the outer turning point coincides with a maximum of the ϕg,E functions, and a larger overlap is expected (see Figure 7.2a). © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
Position of the “Last” Node RN
The region where the nodal structure of ϕg,E is energy independent is bounded by RN , which is the “last” node of the threshold wavefunction E = 0, except for systems with a scattering length a > RN . For a low-energy s-wave radial scattering wavefunction, in a potential with asymptotic −C6 /R6 behavior, RN can be estimated [37] as a function of the scattering length a through 1 2
b RN
2 = arctan
ηb − a 3π + + pπ. ηb 8
(7.4)
2π 6 1/4 In Equation 7.4, b = ( 2mC ) is a characteristic length, η = Γ(1/4) 2 = 0.477989 is 2 a constant, and m the reduced mass. The integer number p is the smallest integer ensuring a positive value for the right-hand side of Equation 7.4.
7.2.2
CHOICE OF A LINEARLY CHIRPED PULSE IN THE PICOSECOND DOMAIN
In order to deal with a reasonable number of photoassociated levels, to be able to follow the behavior of the various components of the vibrational wavepacket during and after the pulse, and in some cases to shape this wavepacket (see Section 7.4), we have chosen to perform calculations with pulses in the 10 to 100 psec range. The dynamics is studied on a timescale that is still much smaller than the lifetime of the excited state, and therefore spontaneous emission will be neglected. It is also shorter than Tvib , so that the relative motion of the atoms is negligible during the pulse. 7.2.2.1 The Chirped Pulse, Central Frequency, Energy, Spectral, and Temporal Widths Let us consider a Gaussian chirped pulse of energy Epulse , centered at time tP , with a frequency ωL (Figure 7.3b) ω(t) = ωL + χ · (t − tP ) =
d (ωL t + φ(t)), dt
(7.5)
that varies linearly around the carrier frequency ωL . χ is the linear chirp rate in the time domain, so that the phase of the electric field is φ(t) =
1 χ(t − tP )2 + φ(tP ). 2
(7.6)
The laser is red-detuned from the atomic D2 line at ωat by δL = (ωat − ωL ),
(7.7)
and at resonance with bound levels v in the excited state of energy |Ev | ∼ δL . The spectral bandwidth δω, defined as the full-width at half-maximum (FWHM) of the intensity profile, is related to the duration τL of the transform limited pulse with the same bandwidth by δω = 4 ln 2/τL ≈ 14.7 cm−1 /τL (in psec). The instantaneous © 2009 by Taylor and Francis Group, LLC
255
Control of Ultracold Molecule Formation via Photoassociation Amplitude of the electric field
(a) 1
χ=0
0.8
f (t)
0.6 |χ| ≠ 0
0.4 2τL 2τC tP
0.2 0 50
100
(b)
150 t (psec)
0.25
200
250
Instantaneous frequency sweeping 4 χ>0
ћ χ (t–tP) cm–1
2
0
2ћ |χ|τC 2τC
–2
2τL
χ<0
tP –4 50
100
150 t (psec)
200
250
98 as an example. (a) Stretching FIGURE 7.3 Properties of a chirped pulse, choosing P± of the temporal width from 2τL to 2τC . Starting from a transform-limited pulse of width 2τL = 2 × 15 psec, the envelope f (t) of the electric field (see Equation 7.11) (solid line) is stretched (broken line) to 2τC = 2 × 34.8 psec when the linear chirp parameter √ is set equal to |χ| = 4.79 × 10−3 psec−2 , while its maximum at t = tP decreases from 1 to τL /τC . The time window [tP − τC , tP + τC ] is indicated by a horizontal broken line. (b) Central frequency sweeping: the central frequency ω(t), defined in Equation 7.5 varies linearly as χ(t − tP ), where the chirp parameter χ is positive (thin solid line) or negative (thick solid line). The resonance window 2|χ|τC is spanned from higher to lower energies in the case of a negative chirp and from lower to higher energies in the case of a positive chirp. Here we assume that the carrier frequency ωL is at resonance with the v = 98 level in the external well of Cs2 0− g (6S + 6P3/2 ), and we have indicated with a vertical solid line the variation of ω(t) around ωL , which spans 1.83 cm−1 during the time window. (From Luc-Koenig, E. et al., Phys. Rev. A, 70, 033414, 2004. With permission.)
intensity of the pulse illuminating an area σ involves a Gaussian envelope f (t) (see Figure 7.3a), Epulse 4 ln 2 t − tP 2 = IL [ f (t)]2 , (7.8) exp −4 ln 2 I(t) = στC π τC © 2009 by Taylor and Francis Group, LLC
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with a FWHM equal to τC (≥ τL ) [19], the parameters τC and τL being related by 2 τ C χ2 τ4C = (4 ln 2)2 −1 (7.9) τL [ f (tP )]2 =
τL . τC
(7.10)
This corresponds to an electric field E(t) = E0 f (t) cos(ωL t + φ(t)), 2 Epulse 1 4 ln 2 2 E0 = , c0 σ τL π
(7.11) (7.12)
where c is √ the speed of light and 0 the vacuum permittivity. The energy of the pulse Epulse = σ π/(4 ln 2)IL τL does not depend upon χ. In fact, 98% of this energy is concentrated in the window [tP − τC , tP + τC ]. 7.2.2.2 The Photoassociation Process: Time Window, Resonance Window, Concept of a Photoassociation Window For physical interpretation, it is convenient to consider that the energy of the pulse is transferred to the atom pair during the time window [tP − τC , tP + τC ]. Within this 2τC period, the instantaneous laser frequency is resonant with all the excited levels with a binding energy in the range [δmin = δL − |χ|τC , δmax = δL + |χ|τC ]. This defines a window 2|χ|τC in the energy domain, hereafter called the resonance window and a set of resonant levels [vmin , vmax ]. Assuming that the PA is a vertical transition at the outer turning point of the resonant level, the resonance window can be translated into a window [Rmin , Rmax ] in the R−domain, hereafter labeled PA window, which will be defined more precisely in Section 7.2.3.1. The time window, resonance window, and PA window have been indicated in Figure 7.1. The sensitivity of the results to the choice of the pulse parameters (detuning, intensity, spectral width, linear chirp rate) is discussed in detail in Refs. [18] and [19], where many different pulses have been considered. In the present work, we 98 and P 122 , with a central concentrate on two typical pulses, hereafter referred to as P± ± frequency chosen to be resonant with the vibrational levels v = 98 and 122, described above in Table 7.1 and in Figure 7.2. Two other pulses will also be considered in the discussion of Section 7.6. We give in Table 7.2 the parameters for the various pulses, as well as the associated PA windows. Another characteristic timescale associated with the radiation is the Rabi period, discussed in detail in Ref. [18], which is larger than 20 psec for the chirped pulses considered here.
7.2.3 THE TWO-CHANNEL COUPLED EQUATIONS AND THE CHOICE FOR A R OTATING WAVE APPROXIMATION The vibrational dynamics in the ground state and in the excited state coupled by the laser field is described by a nonperturbative solution of the time-dependent © 2009 by Taylor and Francis Group, LLC
v Considered in Refs. [18,19,22,24] and [38] Parameters for Several Pump Pulses P±,0
Label
δL (cm−1 )
IL (kW cm−2 )
δω (cm−1 )
τL (psec)
τC (psec)
100χ (cm−1 psec−1 )
Photoassociation Window (a0 )
98 P− 98 P+ P098 122 P− 122 P+ opt P− nad P−
2.656 2.656 2.656 0.675 0.675 0.675 2.656
120 120 120 120 120 750 120
0.98 0.98 0.98 0.26 0.26 0.26 2.45
15 15 15 57.5 57.5 57.5 6
34.8 34.8 15 110 110 376 96.3
−2.5 +2.5 0 −0.2 +0.2 −6.7 −2.5
107.5 → 93.7 → 85.5 85.5 → 93.7 → 107.5 Transform limited 176 → 148.5 → 135 135 → 148.5 → 176 Nonimpulsive Nonadiabatic
104 Pebox 3.2 3.5 0.4 20 30 3 600
A 95% 95% 140% 140% 150% 50%
Central detuning δL , assumed to be resonant with the level v; maximum intensity IL τL /τC (where IL is the maximum intensity of the transform limited pulse P0v ); energy range associated with the spectral width δω; temporal widths (τL and τC ); linear chirp parameter χ; distances within the PA window (in the case of a positive chirp, the crossing distance RC (t) of the dressed potential curves increases; in the case of a negative chirp it decreases, see Section 7.2.3.1); Pebox is the probability of population transfer from the ground to the excited state, by the PA pulse, equal to the number of photoassociated molecules for a unity-normalized state in a box of length L = 19, 250 a0 ; A is the ratio between the adiabaticity window (discussed below in Section 7.3.3) and the time window: for total adiabatic population transfer during the time window, the condition A > 100% should be fulfilled. The two last lines correspond to pulses for which the impulsive approximation or the opt nad provides an example of nonadiabatic adiabatic approximation are no longer valid, as discussed in Section 7.6. P− optimizes the compression effect, while P− population transfer at large distances.
Control of Ultracold Molecule Formation via Photoassociation
TABLE 7.2
257
© 2009 by Taylor and Francis Group, LLC
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Schrödinger equation. In such a unitary treatment, the population is conserved, so that the sum of populations in the lower and in the excited state remains constant. Complete sets of vibrational (bound + continuum) levels in the two electronic states are introduced. Therefore, quantum threshold effects (see Section 7.3.4.2) are automatically accounted for. In this chapter, only s-wave scattering is considered, and one introduces a two-component radial wavefunction Ψ(R, t) describing the relative motion of the nuclei both in the lower electronic state (Ψground (R, t)) and in the excited state (Ψexc (R, t)). ∂ ˆ ˆ mol + W(t))Ψ(t) ˆ HΨ(t) = (H = i Ψ(t). ∂t
(7.13)
ˆ mol = Tˆ + Vˆ el is the sum of the kinetic energy operThe molecular Hamiltonian H ator Tˆ and electronic potential energy operator Vˆ el , with components Vground and Vexc on the lower and excited surfaces, respectively. The coupling term is written in the dipole approximation: ˆ = −D ge (R ) · eL E(t) W (7.14) ge (R ) between the lower and excited elecinvolving the transition dipole moment D tronic state of the molecule and the electric field. The latter is defined by a polarization vector eL assumed to be constant and by an amplitude E(t) (see Equation 7.11). The ˆ due to the oscillations in E(t) is rapid temporal dependence of the Hamiltonian H eliminated in the framework of the rotating wave approximation. Two choices for the reference frequency have been considered. 7.2.3.1
Rotating Wave Approximation with the Instantaneous Frequency: Definition of the Photoassociation Window
One possibility is to use the instantaneous frequency defined in Equation 7.5 to introduce new radial wavefunctions in the two channels Ψg (R, t) = Ψground (R, t) exp(−i[ωL t + φ(t)]/2)
(7.15)
Ψe (R, t) = Ψexc (R, t) exp(+i[ωL t + φ(t)]/2),
(7.16)
where the phase φ(t) was defined in Equation 7.6. When the high-frequency component in the coupling term can be neglected (rotating wave approximation), the coupled radial equations now read ⎛ ⎞ dφ ˆ (R) + W T + V g ∂ Ψg (R, t) ⎜ ⎟ Ψg (R, t) 2 dt i =⎝ dφ ⎠ Ψe (R, t) , ∂t Ψe (R, t) W Tˆ + Ve (R) − 2 dt (7.17) where the potentials Vg (R) and Ve (R), dressed with the carrier frequency, Vg (R) = Vground + ωL /2, Ve (R) = Vexc − ωL /2,
© 2009 by Taylor and Francis Group, LLC
(7.18)
Control of Ultracold Molecule Formation via Photoassociation
259
cross at the distance RL . The coupling term is real and its time-dependence is restricted to the Gaussian envelope f (t) 1 R ) · eL E0 f (t). W (R, t) = − D( 2
(7.19)
Considering the diagonal terms in Equation 7.17, such that dφ = χ(t − tP ), it is also dt convenient to define time-dependent dressed potentials, χ(t − tP ) = Vground + ω(t)/2, 2 V¯ e (R, t) = Ve − χ(t − tP ) = Vexc − ω(t)/2. 2
V¯ g (R, t) = Vg +
(7.20)
During the time window, their crossing point at a distance RC (t) spans the range of distances χ > 0 → Rmin = RC (tP − τC ),
Rmax = RC (tP + τC ),
(7.21)
χ < 0 → Rmax = RC (tP − τC ),
Rmin = RC (tP + τC ),
(7.22)
defining the PA window [Rmin , Rmax ]. 7.2.3.2
Rotating Wave Approximation with the Central Frequency
An alternative possibility is to consider the central laser frequency ωL , so that the new radial wavefunctions in the two channels are ˜ g (R, t) = Ψground (R, t) exp(−iωL t/2) Ψ
(7.23)
˜ e (R, t) = Ψexc (R, t) exp(+iωL t/2). Ψ
(7.24)
When the high-frequency component in the coupling term can be neglected (rotating wave approximation), the radial coupled equations become ∂ i ∂t
˜ g (R, t) Ψ Tˆ + Vg (R) ˜ e (R, t) = W exp(−iφ) Ψ
W exp(iφ) Tˆ + Ve (R)
˜ g (R, t) Ψ ˜ e (R, t) , Ψ
(7.25)
where the potentials dressed with the carrier frequency have been defined in Equation 7.18, and where the coupling term is no longer real, 1 i R ) · eL E0 f (t) exp − χ(t − tP )2 exp(−iφ(tP )), W (R, t) exp(−iφ(t)) = − D( 2 2 (7.26) but includes a time-dependent phase varying as the phase of the electric field (see Equation 7.6). This formulation of the dynamics has been used in the calculations, while Equation 7.17 was used for the physical interpretation. In fact, for a linearly chirped pulse, © 2009 by Taylor and Francis Group, LLC
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when the rotating wave approximation is valid, the results of the calculations do not depend upon the choice of the reference frequency, because the probability density ˜ e |2 ) is the same, and because the slowly in each channel (e.g., |Ψexc |2 = |Ψe |2 = |Ψ ∗ ˆ Ψexc exp (−iωL t) is oscillating contribution to the nondiagonal quantity Ψground W ∗ equal to Ψg W Ψe exp(iφ) independent of the model. It is the phase of this nondiagonal quantity that is responsible for the momentum control, described in Section 7.5. The advantage of the rotating wave approximation at the instantaneous frequency is that it leads to the more familiar picture of a two-level system, as described in Section 7.3.3. The coupling term then has a slow time dependence determined by the envelope of the electric field.
7.3
7.3.1
RESULTS OF NUMERICAL SIMULATIONS: INTERPRETATION AS ADIABATIC TRANSFER WITHIN A PHOTOASSOCIATION WINDOW NUMERICAL METHOD
The main ingredients in the numerical solution of the time-dependent Schrödinger equation are given as follows: 1. The numerical representation of the wavefunctions is based on a collocation scheme [39], involving the value of the function at grid points and a set of interpolation functions. For applications to long-range molecules, Kokoouline and colleagues [40] have implemented a Mapped Fourier Grid method, where the grid step is rescaled to the value of the local de Broglie wavelength. In the present work, the interpolation functions are a set of sine functions [31,32]. Using a grid of finite extension L discretizes the continuum, with levels such that ϕg,E (R) has a node at both ends of the grid, and a level density dn L mL 2 = , where m is the reduced mass. A very large extension L is dE En n2 π2 thus necessary to represent an initial continuum state with an energy in the μK range, as well as the neighboring continuum levels and the last bound level of the a3 Σ+ u potential populated during the PA process. For the Cs2 problem considered, a typical grid has an extension L ≈= 20,000a0 and 1024 grid points: energies as low as 36 nK can be represented. 2. The propagation is performed in discrete steps, choosing Δt much shorter than the characteristic times of the problem (pulse durations, vibrational periods, Rabi periods) all larger than 10 psec: in typical calculations, we have chosen Δt ∼0.05 psec. 3. The time-dependent Schrödinger equation is then solved between t and ˆ t + Δt by expanding the evolution operator exp[−iHΔt/] in Chebychev p ˆ polynomials [39,41]. This requires the evaluation of H Ψ for all values of the index p between 1 and the number N of Chebychev polynomials, typically of the order of 100. © 2009 by Taylor and Francis Group, LLC
Control of Ultracold Molecule Formation via Photoassociation
7.3.2
261
RESULTS OF CALCULATIONS FOR ONE COLLISION ENERGY
After the pulse, bound molecules are formed both in the excited state and in the ground state. As an example, we now present results of numerical calculations for excitation of an ensemble of ground-state cesium atoms, at a temperature T = 54 μK, by the 98 pulse, as described in Table 7.2. P− 7.3.2.1
Photoassociation within the Resonance Window
In Figure 7.4a, we have displayed the probability of population transfer to vibrational levels in the Cs2 0− g (6S + 6P3/2 ) excited state. During the pulse, many levels are populated, including highly nonresonant levels, due to time-energy uncertainty at short times. Rabi oscillations are visible during the pulse. Pulse “transients” are also observed, due to interferences between amplitudes of population coherently transferred to one level at different times: they are discussed in Section 7.3.6. After the pulse, only ≈15 vibrational levels in the excited state remain populated. Such levels lie in the energy range swept by the instantaneous laser frequency, defined previously in Section 7.2.2 as the resonance window. In contrast, the population does not remain in the levels outside the resonance window. The very small value (in the range of 10−4 ) of the population transferred is linked to our choice of a unity normalized initial continuum state in a very large box; most of the pairs of atoms are at very large distances, and are not affected by excitation in the range of the PA window. To be more realistic, on the right vertical scale of Figure 7.4, the number of molecules for a typical trap of volume V = 10−3 cm3 , containing 108 atoms, is estimated including the thermal average procedure described below in Section 7.3.4. The order of magnitude of one molecule per pulse will be discussed below. 7.3.2.2
Formation of Halo Molecules via Optically Induced Feshbach Resonance
After the pulse, there is also population of the two upper vibrational levels of the lower state, v = 52 and v = 53 (see Figure 7.4b); stable molecules are thus formed in a one-color scheme, because the time-dependent frequency of the pulse is sweeping an optical Feshbach resonance. In the dressed potential picture, the initial continuum level of the ground potential V¯ g (R, t) is at resonance with a bound level of the excited potential V¯ e (R, t). Note the efficiency of the process: the number of molecules formed in these two levels of the lower state is equivalent to the number of photoassociated molecules in 15 levels in the excited state. The efficiency of various PA pulses for this population transfer has been discussed in Ref. [19]. The levels v = 53 and v = 52 are respectively bound by 5 × 10−6 and 0.042 cm−1 , to be compared with the resonance window of ∼1.74 cm−1 in the excited state. Due to the very small value of the binding energy, these molecules are “halo molecules,” as defined by Koehler and colleagues [8]: their creation as a byproduct of the PA process should be further investigated. Recently, Kallush and Kosloff [28] have discussed the nonperturbative character of the PA process, where the conservation of the total population requires a © 2009 by Taylor and Francis Group, LLC
262
Cold Molecules: Theory, Experiment, Applications Excited-state population
(a)
Peb,oux(t)
(b) Nue(t) 0.8
Pulse
Lower-state population
Pbg,oux″(t)
u≤ = 52
u = 95 – 107 3 × 10–4
0.6 1.5 × 10–4
2 × 10–4
0.4
1 × 10–4
u≤ = 53
1 × 10–4
0.2 0.5 × 10–4
100
150 200 t (psec)
250
0.0 50
0.3
0.2
0.1 tP
tP u = 122 – 135
0.0 50
Nug″ (t) 0.4
Pulse
100
150 200 t (psec)
250
FIGURE 7.4 Molecules formed by PA within the resonance window and by an optical Feshbach resonance. (a) Probability of population transfer Pbox e,v (t), during and after the pump 98 pulse P− , into several vibrational levels v of the excited state. The levels v = 95–107 (solid line) lie in the resonance window and remain populated after the pulse. In contrast, for the levels outside the resonance window (dash–dotted line), no population remains after the pulse. (b) Population transferred Pbox g,v (t) in the two last vibrational levels v = 52 (solid line) and
v = 53 (dash–dotted line) of the a3 Σ+ u state by sweeping an optical Feshbach resonance. The initial state is a stationary collision state, at energy E = kB T , where T = 54 μK, unity normalized in a box with L = 19,250a0 . The scale on the left vertical axis of each figure is the population transfer probability, for an initial state unity normalized in the box, while the scale on the right vertical axis is the number of photoassociated molecules for a trap of volume V = 10−3 cm3 , containing 108 atoms, after a thermal average for T = 50 μK either in g the excited state Nve (t) or in the ground state Nv (t)(see Section 7.3.4). (From Luc-Koenig, E. et al., Phys. Rev. A, 70, 033414, 2004. With permission.)
significant modification of the initial state, with population of bound levels (discussed here) and of continuum levels (as discussed further in Section 7.5.4). 7.3.2.3
Selectivity of the Resonance Window: Dependence of the Final Distribution of Population on the Pulse Parameters
We have shown an example of results where the photoassociated molecules are selectively formed in vibrational levels within the resonance window. It is interesting to discuss the robustness of this concept by varying the pulse parameters, as illustrated in Figure 7.5. 122 show that the levels populated For negative chirp, calculations with the pulse P− − in 0g (6S + 6P3/2 ) (see Figure 7.5a) stay within the resonance window when the intensity IL is increased to 9 IL . Moreover, the population depends weakly upon IL . © 2009 by Taylor and Francis Group, LLC
263
Control of Ultracold Molecule Formation via Photoassociation (a)
(b) 5 × 10–3 PA window
1 × 10–4
4 × 10–3
Ng(u") = |<Ψg |ϕg(u") >|2
Ne(u) = |<Ψe |ϕe(u) >|2
2 × 10–4
3 × 10–3
2 × 10–3
1 × 10–3
–3
–2 –1 Eu (cm–1)
0
0
49
50
51
52
53
0
u"
FIGURE 7.5 Control of the vibrational distribution of the photoassociated molecules with the pulse parameters (sign of the chirp, intensity). (a) Distribution, as a function of their binding energy, of vibrational levels in the excited state, when illuminating with the pulse 122 (circles) and P 122 (squares) with intensity I = 120 kW cm−2 . In the case of a negaP− L + tive chirp, the population (circles) is restricted to the PA window, indicated by a horizontal 122 is changed to 9 I (up triangles) the population distriarrow. When the intensity of P− L bution remains unchanged. In the case of a positive chirp (squares) the PA efficiency is slighly better and levels below the resonance window are also populated. Moreover, this effect increases when the intensity is changed to 9 I (inverted triangles). (b) Population of the last bound levels of the a3 Σ+ u state (halo molecules) as a function of the vibrational 122 (circles), P 122 (squares); it is not modified index v . Excitation is the same with P− + when the intensity is increased by a factor 9 for negative (up triangles) or positive (inverted triangles) chirp. 122 , the distribution extends to levels outside the PA window, and this In contrast, for P+ effect increases with intensity. The population is slightly larger for a positive chirp, and this effect also increases with IL . As discussed in Refs [9,42], for χ < 0 the instantaneous frequency “follows” the motion of the wavepacket in the attractive potential Ve , so that for sufficient intensity the population is brought back to the initial state and up again. This phenomenon is labelled “multiple interaction” in Ref. [42]. For χ > 0, this recycling effect is much weaker: once a level is populated, the instantaneous frequency is no longer resonant with its energy, and the population remains in this photoassociated level. In contrast, the population of the last bound level in the lower state is insensitive both to the sign of the chirp and to the intensity (see Figure 7.5b). Finally, we must stress that the concept of the resonance window is linked to an adiabatic transfer, as will be explained in the following section (Section 7.3.3), and that a minimum intensity is required. © 2009 by Taylor and Francis Group, LLC
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7.3.3 ANALYSIS WITHIN A TWO-STATE MODEL: THE CONCEPT OF ADIABATIC TRANSFER WITHIN A PHOTOASSOCIATION WINDOW When the duration of the pulse is short compared to the vibrational period, that is, when τC Tvib , which is fulfilled in the present calculations, the impulsive approximation [43] is valid. One can then ignore the motion of the nuclei during the pulse, so that the kinetic energy term may safely be neglected in Equation 7.17, allowing interpretation of the numerical results in the framework of an R- and t-dependent two-level system. Following the notation of Ref. [44], we define for each internuclear distance R and each time t a splitting between the lower and the excited state 2Δ(R, t) = Ve (R) − Vg (R) − χ(t − tP ) = Vexc (R) − Vground (R) − ωL − χ(t − tP ),
(7.27)
so that 2Δ(∞, tP ) = δL , and a coupling term, due to the laser, written W (t) = WL × f (t),
(7.28)
provided we neglect the R-dependence of the transition dipole moment. It is then straightforward to diagonalize the matrix
Δ(R, t) W (t)
W (t) , −Δ(R, t)
(7.29)
by introducing a mixing angle for the adiabatic basis through tan θ(R, t) =
|W (t)| . Δ(R, t)
(7.30)
If the adiabatic picture is valid, at distances where Δ(R, t) changes sign, that is within the PA window, the two dressed states should change their character between the beginning and the end of the pulse, allowing population inversion. The dressed potential curves are drawn in Figure 7.6a, and their crossing point RC (t) spans the PA window [Rmin , Rmax ], as discussed above. The mixing angle θ(R, t), computed from Equation 7.30, is displayed in Figure 7.6b, for three values of the time (t = tP − τC , tP , tP + τC ); within the PA window, θ(R, t) varies by π, thus verifying the condition for population inversion. Outside this range, the population transferred to the excited state during the pulse goes back to the lower electronic state. This simple model, combined with the reflection approximation, where a vibrational level in the excited state is assumed to be populated through a vertical transition at © 2009 by Taylor and Francis Group, LLC
265
Control of Ultracold Molecule Formation via Photoassociation Δ(R,t) < 0
(a)
Resonance window 2ћ|χ|τC
Δ(R,t) > 0 − V g(t)
χ<0
t=tP–τC
t=tP+τC
− V e(t)
Photoassociation window Rmax
Rmin
(b)
Mixing angle for the adiabatic basis tan[θ(R,t)] = |W(t)|/Δ(R,t) 1.2 1
θ(R, t)/π(rad)
0.8
t=tP–τC
0.6 t=tP+τC
0.4 0.2
t=tP
0 −0.2
Rmin 0
50
Rmax 100 R (a0)
150
200
FIGURE 7.6 (a) The resonance window, obtained by fixing V¯e (R, t) (solid curve) and drawing V¯g (t) = V¯e (R, t) − 2Δ(R, t) at the beginning (t = tP − τC , short dashed and dash–dotted line) and at the end (t = tP + τC , long dashed and dash–dotted line) of the time window for a negative chirp parameter, χ < 0. The long and short dashed lines correspond to negative values of the splitting Δ(R, t), while the dash–dotted lines correspond to positive values. The crossing point spans the PA window from Rmax to Rmin . (b) Mixing angle θ(R, t) (see Equation 7.30 in text) in units of π, characteristic of the adiabatic basis for the two-level system at fixed internuclear distance R. θ(R, t) is drawn as a function of R, for three values of the time t: t = tP − τC , at the beginning of the time window (short dashed line), t = tP , at the maximum of the pulse (solid line), and t = tP + τC , at the end of the time window (long dashed line). (From Luc-Koenig, E. et al., Phys. Rev. A, 70, 033414, 2004. With permission.)
the outer turning point, is consistent with the results of our numerical calculations. The latter can be interpreted as a total adiabatic population inversion within the PA window [Rmin , Rmax ]. This means that all the pairs of atoms initially lying within this range of interatomic distances are transformed into bound molecules. However, this interpretative model is not fully quantitative. © 2009 by Taylor and Francis Group, LLC
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Indeed, as discussed in detail in Refs. [18,19], the condition for adiabatic evolution is ∂θ (R, t) 4 Δ2 (R, t) + W 2 (t), (7.31) ∂t and requires a strong laser coupling in the vicinity of the instantaneous crossing point RC (t). It cannot be fulfilled in the wings of the pulse or when the chirp rate becomes too large. During the time window, at the instantaneous crossing point RC (t), a necessary condition for the pulse parameters and transition dipole moment is τL (7.32) |χ|τC WL2 , 2 requiring that the coupling (laser intensity and dipole transition moment) is large enough, and restricting the chirp parameter. The adiabaticity condition (7.31) can be verified only during an adiabaticity window, when tP − ατC ≤ t ≤ tP + ατC , where the parameter α has an upper limit αmax defined in Equation 28 of Ref. [19]. Therefore, the model discussed above requires αmax 1 to ensure that the full time window is included within the adiabaticity window. We have indicated in the last column of Table 7.2 the maximum percentage A of the PA window that can be adiabatic, where A = 100αmax . 98 , A = 95% only, so that some nonadiabatic effects are manifested For the pulse P± in the wings. As illustrated in Figure 7.6, the variation of the mixing angle by π from tP − τC until tP + τC is strictly observed only at the center of the PA window. The adiabaticity criterion is not fulfilled at the instantaneous crossing point in the wings of the pulse; there is not a fully adiabatic population transfer at both ends of the time window. In addition, some population transfer still occurs outside the PA window, at distances slightly smaller than Rmin or larger than Rmax . Nonadiabatic effects and population transfer outside the PA window will be further discussed in Sections 7.3.6 and 7.6.
7.3.4 AVERAGING OVER INITIAL VELOCITY DISTRIBUTION: USE OF SCALING LAWS In order to compare with experiments, the total PA probability per pulse P(T ) has to be calculated for an ensemble of ultracold atoms in thermal equilibrium at temperature T . As shown in Ref. [24], this probability is given by P(T ) =
1 ˆ f , t0 )e−βHˆ g ], ˆ + (tf , t0 )P ˆ e U(t Tr[U Zeq
(7.33)
ˆ f , t0 ) denotes the where Pˆ e is the projector on the electronically excited state, U(t time-evolution operator (including the interaction with the PA laser) and ρˆ T (t0 ) = ˆ e−βHg /Zeq is the initial density operator. Here β = 1/(kB T ) (kB is the Boltzmann © 2009 by Taylor and Francis Group, LLC
Control of Ultracold Molecule Formation via Photoassociation
267
ˆ ˆ g the lower-state Hamiltoconstant), Zeq = Tr[e−βHg ] the partition function, and H nian. Because in the calculations reported above the initial stationary state is a single ˆ g in a box of extension L, a procedure to perform unity-normalized eigenstate of H Boltzmann averaging has been implemented [24]. Two different methods are checked and compared.
7.3.4.1
Boltzmann Average on a Finite-Size Grid
A purely numerical procedure considers the complete set of initial three-dimensional box-states corresponding to different partial-waves up to lmax and checks the converL in Equation 7.33 as a function of the gence of the numerator and denominator of Zeq box size L and the number of partial waves. At low temperature T ∼ 100 μK, a box length L ∼ 5000a0 corresponds to a density of states sufficiently high to correctly L /L 3 is very slow describe the thermal distribution. The convergence with lmax of Zeq (typically 100 partial waves have to be included), but the converged value becomes identical to the analytically known partition function of the ideal gas. In contrast, for the numerator, the convergence with lmax is rapid. Typically, due to the height of the centrifugal barrier, at most 10 partial waves should be considered, the correct treatment of higher l waves becoming more important when shape resonances occur as in the example of 87 Rb2 . Therefore, it is sufficient to perform analytical calculations for the numerator of Equation 7.33, using for the denominator the analytic partition function of the ideal gas. 7.3.4.2 Average Introducing Box-Independent Energy-Normalized States: Use of a Scaling Law Near Threshold Another semi-analytic method was implemented in Ref. [19] for s-wave scattering. It relies on a scaling law for the probability of finding a pair of atoms separated by a distance within the PA window. It is valid for pulses with detuning large enough that the PA window occurs at distances shorter than the distance RN defined in Section 7.2.1.5. There, as displayed in Figure 7.2 and discussed in Section 7.2.1.4, the nodal structure of ϕg,E is independent of E in the energy range considered. The energy dependence is concentrated in the scaling factor C −2 (E) introduced by F. Mies in the generalized form of multichannel quantum defect theory [45,46]. There are various ways to define the norm of the continuum functions ϕg,E . In numerical calculations the radial wavefunctions of the discrete levels in a box can be box (R), where normalized to 1. Those unity box-normalized functions are written ϕg,E the dependence on the size L of the box is explicitly written. At large distances they behave as sine functions. Alternatively, the energy-normalized radial wavefunctions ϕg, E (R) are related to the previous ones by the density of states in the box at the energy Edn/dE|box E , so that ϕg,E (R) =
box dn box ϕg,E (R). dE E
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Cold Molecules: Theory, Experiment, Applications
For sufficiently low collision energy E, the WKB approximation remains valid at short distance R ≤ RN and ϕg,E (R) reads 1 ϕg,E (R) = C(E)
2m cos πp(R)
R
p(R )dR −
Rin
π , 2
R < RN ,
(7.34)
where Rin is the inner turning point for the classical motion and p(R) the local classical momentum. In the region R ≤ RN , an important property of Equation 7.34 is that the momentum p(R) = 2m(E − Vground (R)) is determined almost exclusively by the potential energy, and is nearly independent of E. This is why, as illustrated in Figure 7.2, for R ≤ RN , the energy dependence of ϕg,E (R) is concentrated in the scaling factor C(E)−2 . The latter was estimated by Crubellier and Luc-Koenig [37] in the general case of a potential characterized by an asymptotic R−6 behavior and a scattering length a. A strong departure from the √ Wigner threshold law, where C −2 (E) ∝ E for E → 0, was obtained for large values of |a|. In such a situation, due to a resonance effect, a marked enhancement of the short-range density is observed, which is interpreted as a threshold quantum effect. In Ref. [37] the semi-analytical calculations give a general estimation in reduced units for a variety of cold collision systems. To compare with real situations, Figure 7.7 displays the energy dependence of C(E)−2 for collisions in the ground X1 Σ+ g poten133 Cs . The high sensitivity to potential of tials of 85 Rb2 and 87 Rb2 and in the a3 Σ+ 2 u the value of the scattering length a is manifested by the quasi-disappearance of the quantum enhancement effect when the triplet Cs scattering length is reduced from a = 2440a0 [36] to a = 538a0 . Making use of Figure 7.7, it is sufficient to compute the probability of PA Pebox (E0 ) in a box of length L for a single energy E0 and to use the scaling law, ⎡ P¯ e (E) = Pebox (E0 )
C(E) C(E0 )
−2 ⎢ ⎢ ⎢ ⎣
⎤ dn box dE E0 ⎥ ⎥ box ⎥ . dn ⎦ dE E
(7.35)
to obtain the probability P¯ e (E) per unit energy range that a pair of atoms with energy E is photoassociated after the pulse. This markedly reduces the computational effort. The total PA probability per pulse for a pair of atoms is obtained by integrating P¯ e (E) over E, with a weighting factor e−βE /Z using the ideal gas partition function Z. The method, initially restricted to s-waves, could be generalized to higher l-waves. The contribution of the various partial waves is discussed in Ref. [24].
7.3.5 WHAT IS THE ABSOLUTE NUMBER OF PHOTOASSOCIATED MOLECULES? Considering a typical cesium MOT with a density of 1011 cm−3 , containing N = 108 atoms at a temperature of 50 μK, and restricting the calculations to s-wave, about 98 , and six molecules per pulse with one molecule per pulse is formed by PA with P− 122 P− [19,24], with a more convenient detuning (see Table 7.2). © 2009 by Taylor and Francis Group, LLC
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Control of Ultracold Molecule Formation via Photoassociation
–2
4
1
C(E)
85
20
9
E/kB (mK) 25 40
15 C(E)–2 3
Rb
133
2
15
70
100
130
__ Singlet ..... Triplet
Cs
133Cs
1 87Rb
10
0 133
Cs
0
10
20 30 T1/2 (mK)1/2
40
5 133
Cs
87
0 0
2
4
6 T1/2 (μK)1/2
Rb 8
10
12
√ FIGURE 7.7 Quantum enhancement effect: variation with E of the density factor C −2 (E), which determines the short-range probability for energy-normalized s-wave radial scattering wavefunctions describing cold collisions at energy E in the ground singlet potentials of 85 Rb2 (thick full lines), 87 Rb2 (thin full lines), and in the lowest triplet potential of 133 Cs2 , for two values of the scattering length (long broken and dot-dashed medium lines). In all cases, the WKB value C(E)√= 1 is reached for large energy, E/kB > 50 μK, and the Wigner threshold law (C −2 (E) ∼ E) is observed at small energy. For sufficiently large scattering length |a|, a strong enhancement of the scaling factor is observed in the intermediate energy region. The values of the scattering length are equal to, respectively, 3650a0 for 85 Rb2 [47], 90.6a0 for 87 Rb [48] and 2440a for 133 Cs (long broken medium line) [36]. The dot-dashed medium 2 0 2 line corresponds to the value a = 538a0 for the 133 Cs2 scattering length, chosen in the present chapter and markedly reducing the quantum enhancement effect. A systematic description of collisions in ground singlet and lowest triplet potentials of Rb2 and Cs2 dimers is presented in Ref. [37]. (From Koch, C.P. et al., J. Phys. B: Atom. Mol. Opt. Phys., 39, S1017–S1041, 2006. With permission.) 98 to P 122 is 1. Effect of the detuning. The increase of the PA efficiency from P− − mainly due to a larger density of atom pairs in the region of the PA window; the ratio 0.0026/0.0003 in the probability density is comparable to the ratio 20/3.2 obtained for the PA probability Pebox in Table 7.2. As illustrated in Figure 7.2, in the first case the PA window is located in the vicinity of a node of the initial stationary wavefunction ϕg,E , in the second case in the vicinity of a maximum, leading to a better Franck–Condon overlap. 2. Effect of the chirp sign. Changing the sign of the chirp does not significantly 98 , since total adiabatic population transfer change the results obtained for P− within the PA window is effective and since the number of Rabi oscillations during the pulse (1.5) is small; the final population transferred by the pulse
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Cold Molecules: Theory, Experiment, Applications 98 is slightly larger (7% increase). In contrast, for P 122 , the final population P+ + 122 in agreement with the transferred is much larger (45% increase) than for P− increase in Pebox (see Table 7.2). Indeed, although the adiabaticity condition is better satisfied for this pulse (parameter A in the table), the large number of Rabi oscillations (5.3) results in an important recycling of the population for the negative chirp. The resonant condition of the laser excitation RC (t) is “following” the wavepacket motion, toward short distances, in the excited potential. In contrast, for a positive chirp, the population resonantly excited to Vexc at time t is out of resonance with the optical frequencies for t > t, preventing recycling.
When all partial waves are considered the number of molecules per pulse increases to 1.4 and 10, respectively. In Ref. [24] results for rubidium PA are also presented, revealing a slightly less favorable situation for the typical number of molecules formed by one pulse. In the latter case the PA occurs at shorter distances, with a different mechanism (resonant coupling), so that direct comparison is not possible. The previous discussion is restricted to one pulse. In an experiment, the total number of molecules will depend upon the repetition rate of the laser and the efficiency with which the dynamical hole in the initial scattering wavefunction, carved out due to full population transfer within the PA window, and discussed below in Section 7.5, will be refilled. Nevertheless, for the conditions of a typical MOT, the number of molecules formed by PA with a cw laser is about 106 per second. Using the chirped pulses we have considered, with a repetition rate of 108 Hz (10 nsec between two pulses), and assuming that one pulse is not destroying the molecules formed by the previous one, the number of molecules should be one to two orders of magnitude larger than for cw PA.
7.3.6 TRANSIENT EFFECTS During the pulse there is a coherent transfer of population from the initial state to the electronically excited state. The partial amplitude of population transferred during the time-interval [t, t + dt] coherently interferes with the amplitude of population already transferred at t < t. For a level v, the partial transfers are maxima at tP , where v , when the level is resonantly excited, the instanthe intensity is maximum and at tres taneous crossing point RC (t) being located at its outer turning point. As illustrated in Figure 7.8, oscillations are observed for the levels that come to resonance with the instantaneous frequency of the laser ω(t) before the maximum of the pulse. The amplitudes of the oscillations are strong for χ < 0, where the motion of the wavepacket in the excited state follows the laser excitation. For χ > 0 the vibrational motion in the excited state prevents deexcitation at t > t of the population transferred at time t. Such oscillations are a signature of significant nonadiabatic effects in the population transfer, which explains that they are observed during a time larger than the time window. The adiabaticity criterion is not fulfilled at the instantaneous crossing point in © 2009 by Taylor and Francis Group, LLC
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Control of Ultracold Molecule Formation via Photoassociation box(t) Pe,u
box(t) Pe,u
tP
c<0
tP
c<0 4 × 10–5
4 × 10–5 u=102
u=102 u=107 2 × 10–5
0 50
100
u=107
u=98
u=98
u=95
u=95
150 200 t (psec)
250 50
100
150 200 t (psec)
2 × 10–5
0 250
FIGURE 7.8 Coherent transients in the population of vibrational levels in the excited state. Time variation during the pulse of the relative population for vibrational levels v of the excited 98 pulses. v = 107, dashed line; electronic state populated during the time window for the P± v = 102, thin solid line, v = 98, thick solid line; v = 95, dash–dotted line. The left (right) panel corresponds to χ < 0 (χ > 0). The level v = 98 is resonant at t = tP , corresponding to the maximum intensity of the pulses. For a negative (positive) chirp, levels v > 98 (v < 98) are resonant before tP , and levels v < 98 (v > 98) are resonant after tP . Strong oscillations in the population are observed for levels resonantly excited before tP , especially for a negative chirp rate χ, when the crossing point RC (t) of the laser excitation follows the vibrational motion of the excited wavepacket toward shorter distances R. (From Luc-Koenig, E. et al., Phys. Rev. A, 70, 033414, 2004. With permission.)
the wings of the pulse. “Coherent transients” have been previously studied in the perturbative limit in the coherent excitation of a two-level system with a linearly chirped pulse [49], and have been recently observed in time-dependent PA of rubidium [50].
7.4
SHAPING VIBRATIONAL WAVEPACKETS IN THE EXCITED STATE TO OPTIMIZE STABILIZATION INTO DEEPLY BOUND LEVELS OF THE LOWER STATE
When a chirped laser pulse is used for the PA process, a coherent wavepacket is formed in the excited state, and has components in all the vibrational levels within the resonance window. After the pulse, this wavepacket propagates toward short distances. Because population transfer back to the initial state with a second (dump) pulse is a coherent process, it is convenient to use this property to optimize the formation of ultracold stable molecules. For instance, in the chosen example of Cs2 0− g (6S + 6P3/2 ), the time-dependent Franck–Condon overlap with the bound levels in the lower electronic state can be optimized by achieving a focused wavepacket. © 2009 by Taylor and Francis Group, LLC
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7.4.1
Cold Molecules: Theory, Experiment, Applications
SHAPING THE VIBRATIONAL WAVEPACKET IN THE EXCITED STATE
Indeed, for a given vibrational level v of a photoassociated molecule, the transfer of population to bound levels of the lower electronic state is optimal at the inner turning point, half a vibrational period after the excitation. Therefore the dump pulse should be delayed by Tvib /2. However, because the 0− g (6S + 6P3/2 ) potential is strongly anharmonic, the vibrational period Tvib varies markedly from one level to the next one (see Table 7.1). This can be compensated by choosing a negative chirp parameter for the PA pulse, in order to populate first the levels with a larger vibrational period (see Figure 7.9). Due to a scaling law on Tvib , and on the revival time Trev , a linear chirp χfoc can be chosen [18,23] so that after the pulse all the components reach the inner barrier at the same time td = tP + 21 Tvib (vL ), where vL is the level resonant at the maximum of the pulse. One thus obtains a focused wavepacket, as displayed in Figure 7.10a. Previous work has suggested ways of designing laser fields for generation of spatially squeezed molecular wavepackets [51,52]. Because they considered low vibrational levels, a numerical optimization procedure appeared to be necessary. Here, we are dealing with long-range molecules, where the motion is controlled by the asymptotic potential, and the availability of scaling laws makes the optimization easier, as for atomic Rydberg wavepackets.
7.4.2
PROPOSAL FOR A TWO-COLOR PUMP–DUMP EXPERIMENT
In Ref. [22], the possibility of creating stable molecules via a two-color pump–dump experiment has been discussed in detail. The vibrational wavepacket Ψe (R, t = tP +
Vibration of the wavepacket
Rd(u1) Rd(u2)
u1 u2
RC (u1)
E1 E2
RC (u2)
t1
t2 Photoassociation td Stabilization
Chirped pulse
FIGURE 7.9 How to create a focused vibrational wavepacket: the PA pulse coherently excites vibrational levels with slightly different periods Tvib . With a negative chirp, higher levels (v1 ) with a longer period Tvib (v1 ) are populated first (t1 ), lower levels with a shorter period Tvib (v2 ) being populated later (t2 ). After half a vibrational period, provided the time delay t2 − t1 compensates for the difference (Tvib (v1 ) − Tvib (v2 ))/2, the partial wavepackets corresponding to various vibrational levels simultaneously reach the inner turning point at the same time td . (From Koch, C.P. et al., Phys. Rev. A, 73, 033408, 2006. With permission.)
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Control of Ultracold Molecule Formation via Photoassociation
273
(a) 0.005 | Ψe (R, td = tP + Tvib/2) | 0.004 0.003
χ = χfoc
0.002 χ=0
0.001 0.000
0
50
100
150
200
R (a0) (b)
Maximum probability to form bound stable molecules
χ = χfoc
0.2
0.1 tP
td
χ=0
0 –100
0
100
200 t (psec)
300
FIGURE 7.10 (a) Example of a focused wavepacket |Ψe (R, td )| (thick solid line) obtained in the excited state half a vibrational period, td = tP + 21 Tvib (vL ), after PA with the negatively 98 , which corresponds to the chirp χ chirped pulse P− foc and has a maximum intensity at t = tP . For comparison, the wavepacket obtained by photoassociating with a transform-limited pulse P098 with χ = 0 is drawn in a thin solid line. Note the strong increase of population (typically ∼8) due to the chirp. (From Luc-Koenig, E. et al., Phys. Rev. A, 70, 033414, 2004. With permission.) (b) Variation of the time-dependent Franck–Condon factor F (tdyn ), defined in Equation 7.36. (From Koch, C.P. et al., Phys. Rev. A, 73, 033408, 2006. With permission.) 98 , hereafter referred to as the pump pulse, moves freely τC ), created by the pulse P− in the excited state potential for a while; then a conveniently delayed second (dump) pulse transfers the population to the lowest triplet state. The choice for the optimal delay is discussed below.
7.4.2.1 The Time-Dependent Franck–Condon Overlap According to the value chosen for the time tdyn = t − tP , one finds very different values for the Franck–Condon overlap matrix elements between the wavepacket Ψe (R, t = tP + tdyn ) and the stationary vibrational wavefunctions ϕg,v (R) of the bound levels in the initial state. We define time-dependent Franck–Condon overlap © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
factors Fv (tdyn ) = | < Ψe (R, t = tP + tdyn )|ϕg,v (R) > |2 F(tdyn ) =
=53 v
Fv (tdyn ),
(7.36)
v =0
where F(tdyn ) is the sum of the overlap factors with all the bound levels in the lower state. Its time dependence is displayed in Figure 7.10b. Just after the PA pulse, at time tP + 40 psec ∼ tP + τC , the factors Fv (tdyn ) are negligible for most bound levels of the lower state, except for the last level v = 53, which is very loosely bound, so that F(tdyn ) is negligible. During the motion of the wavepacket Ψe (R, t = tP + tdyn ) toward short distances, the time-dependent Franck–Condon factors increase, especially when the wavepacket is localized in the vicinity of the inner turning point. Due to the focusing effect, F(tdyn = td ) has indeed a strong maximum in a short time period around td = Tvib /2. However, this maximum is far from 1; at most 20% of the population of the excited state can be transferred to bound levels of the a3 Σ+ u state. The rest is transferred to continuum levels, creating pairs of ground-state atoms. This conclusion is due to the marked difference between the excited and the a 3 Σ+ u state potentials. A situation where the two potentials are more alike (for instance in the case of heteronuclear alkali dimers) would allow a larger percentage of transfer to bound levels. The importance of the spectroscopy is again manifest. 7.4.2.2 A Two-Color Experiment for Creating Stable Molecules The proposed experiment, as described in Ref. [22], is given schematically in Figure 7.11. A first (pump) pulse creates a shaped wavepacket in the excited state. After half a vibrational period, when the focusing effect is maximum, a second (shorter) pulse transfers the population into bound levels v of the a3 Σ+ u state. Two strategies, described in Ref. [22] are possible: 1. Choosing short dump pulses, it is possible to transfer a maximum of population toward several bound levels in the a3 Σ+ u electronic state. For instance, 98 , a short π pulse in the 15 fsec range can dump all when the pump pulse is P− the population to the lower state. Then, at most 20% of this population stays in bound levels, while the remaining 80% goes into the continuum, mainly as pairs of hot ground-state atoms. Chirping the dump pulse is efficient in suppressing Rabi oscillations in the intensity dependence. 2. By narrowing the spectral width of the dump pulse, it is possible to populate only bound levels, and even to select a single level. In the chosen example, a dump pulse in the 5 psec range selectively transfers population to the rather deeply bound level v = 14 (binding energy = 113 cm−1 ). It is not possible to go directly to the v = 0 level, because the outer well in the
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Control of Ultracold Molecule Formation via Photoassociation
tP + 40 psec
11,900
tP + 250 psec
tP + 120 psec = td 11,800
Energy (cm–1)
11,700
t=0
100
tP + 40 psec
0 –100 –200 20
40
60
80
100
120
Internuclear distance (a0)
FIGURE 7.11 A two-color pump–dump experiment. The two potentials Vground (R) and Vexc (R) are drawn as thin lines. The square of the stationary wavefunction in the a3 Σ+ u triplet state before the pulse is drawn as a full line. The PA pulse (on the right side with an up arrow) transfers population to the 0− g (6S + 6P3/2 ) electronic state. It has a maximum intensity at t = tP . After this pump pulse, at t = tP + 40 psec, a vibrational wavepacket has been created in the excited state, the modulus |Ψe (R, td )|2 is drawn as a long dashed line, while a hole has been carved in |Ψg (R, t)|2 (long dashed line). Half a vibrational period later, at t = td , a focused wavepacket (thin line) has reached the inner turning point and is optimal for population transfer to the a3 Σ+ u state by the dump pulse (left side down arrow). After one vibrational period at t = tP + 250 psec, the wavepacket in the excited state is back to the outer turning point (dash–dotted line). (From Koch, C.P. et al., Phys. Rev. A, 73, 033408, 2006. With permission.)
excited curve is located at large distances from the minimum in Vground (R). In such a situation, the dump pulse at td = tP + Tvib /2 transfers only 12% of the population to the a3 Σ+ u state; the remaining part stays in the excited state, can decay by spontaneous emission, or could be transferred, one vibrational period after td , at t = tP + 3Tvib /2, when the vibrational wavepacket is focused for the second time, by a another dump pulse toward bound levels in the lower state. Designing a series of pump and dump pulses then seems an interesting perspective. The previous discussion shows again the crucial importance of the spectroscopy of the systems considered.
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7.5 THE DYNAMICAL HOLE IN THE INITIAL STATE WAVEFUNCTION: COMPRESSION EFFECT If a second pump pulse identical to the first one is illuminating the system after a short delay, its efficiency should be markedly reduced, because all the pairs of atoms at distances R within the PA window [Rmin , Rmax ] have been transformed into excited molecules. This void in the pair wavefunction of the initial state, hereafter identified as the “dynamical hole,” is well known from previous work on coherent control [43,53] as one of the main outcomes of excitation of a molecular system with a short laser pulse. It is associated to a “momentum kick.” The effect is also present in the ultracold regime [27] and has been analyzed in detail in Ref. [38]. A look at Figure 7.1 shows that in the region of the PA window, the potential in the initial state is negligible, so that the classical acceleration is zero, whereas this is not the case for the excited state where the Vexc (R) potential has an R−3 asymptotic behavior.
7.5.1
PHENOMENOLOGICAL OBSERVATION OF A DEPLETION HOLE, A MOMENTUM K ICK, AND A COMPRESSION EFFECT
The generation of the depletion hole on the initial a3 Σ+ u electronic state due to PA with a short pulse can be understood by employing a phase-amplitude representation for both the lower- and the excited-state wavepackets,
iSg/e (R, t) Ψg/e (R, t) = Ag/e (R, t) exp R pg/e (R, t)dR i = Ag/e (R, t) exp ,
(7.37)
where, within a semiclassical interpretation, pg (R, t) (pe (R, t)) is the local momentum in the lower (excited) state [54,55]. Let us emphasize that a nonvanishing local momentum exists in the spatial range where there is a R-variation in the phase Sg/e (R, t) of the wavepacket. From the numerical calculations reported above, the evolution of the amplitude 122 is displayed in Figure 7.12. It is clear Ag (R, t) after illuminating with the pulse P− that after the pulse a hole has been carved out in the region of the PA window, defined in Table 7.2 as 148a0 < R < 176a0 . In this region, the potential and the classical acceleration are negligible in the lower state. After the pulse, the hole starts moving to smaller internuclear distances at a velocity ∼4.2 m/sec, two orders of magnitude larger than the classical velocity of the atoms colliding at T = 54 μK in the asymptotically R−6 a3 Σ+ u potential. The motion is restricted to a localized part of the wavepacket, the wavefunction at distances larger than Ra = 314a0 not being modified, and the node at Ra remaining fixed. (As indicated in Figure 7.2, Ra (54 μK), denoted as Ra below, is the first node in the “outer” region for the scattering state at 54 μK, corresponding to the node at R ∼ a in the threshold wavefunctions at T ∼ 0 K. In the outer region the position of the nodes is energy-dependent). © 2009 by Taylor and Francis Group, LLC
Control of Ultracold Molecule Formation via Photoassociation |Ψg(R,t)|
RN
PA window
tinit
1 × 10–2
277
t = tP
5 × 10–3 0 1×
10–2
t = tP + τC = tP + 110 psec
5 × 10–3 0 1×
10–2
t = tP + 200 psec
5 × 10–3 0 1×
10–2 t = tP + 350 psec
5 × 10–3 0 1×
10–2
5 × 10–3
t = topt = tP + 850 psec
0 1×
10–2 t = tP + 1300 psec
5 × 10–3 0
0
100 Rmin
200
300
Rmax
400
Ra R (a0)
FIGURE 7.12 Dynamical hole and compression effect. Top panel: amplitude |Ψg (R, t = tinit )| (dotted grey line) of the initial stationary scattering wavefunction for a pair of ground-state cesium atoms colliding with energy E = kB T , with T = 54 μK, and zero angular momentum, in the absence of an external field. Note the position of the “first outer node” at Ra and the last inner node at RN . Top panel: amplitude |Ψg (R, t = tP )|(thick black line), when illuminating 122 , at the maximum of the pulse t = t ; the pulse has carved out a hole in the with the pulse P− P initial wavefunction. This hole is indeed located in the region [Rmin , Rmax ] of the PA window, delimited by the two vertical lines. Next panels: after the pulse, the Ψg (R, t) “wavepacket” moves inwards: note the increase of the maximum value of its amplitude in the inner region. At t − tP ∼ 850 psec, the wavepacket has been partly reflected by the a3 Σ+ u inner wall located at R ∼ 10a0 . Note that this reflection takes place on a timescale close to the classical vibrational half-period (∼[300,600] psec) for the photoassociated levels in the excited state. (From Mur-Petit, J. et al., Phys. Rev. A, 75, 061404(R), 2007. With permission.)
In contrast, we observe an increase of the probability density at distances shorter than the PA window, manifesting a “compression” of the wavepacket. A secondary maximum is created, which moves toward the repulsive wall of the lower state potential on a timescale of a few hundred picoseconds, typical of half the vibrational period © 2009 by Taylor and Francis Group, LLC
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in the excited state. Due to the laser coupling between the two states, a “momentum kick” has indeed been given to the initial state wavepacket. The “compression” effect is maximum at t = topt , and later the wavepacket is reflected on the inner wall of the potential.
7.5.2 ANALYSIS OF THE MOMENTUM TRANSFER WITH PARTIALLY INTEGRATED MASS CURRENT AND POPULATION An analysis related to the classical limit of the Bohmian dynamics [56] has been proposed in Ref. [38]. It considers the time variation of the local probability current in each channel defined as ∗ ∂Ψg/e (R, t) ∂Ψg/e (R, t) ∗ mjg/e (R, t) = Ψg/e (R, t) − Ψg/e (R, t) 2i ∂R ∂R = pg,e (R, t)|Ψg,e (R, t)|2 ,
(7.38)
and the local probability density |Ψg/e (R, t)|2 = [Ag/e (R, t)]2 . Both quantities are partially integrated in the region R < Ra (54 μK) which includes the PA window. The position Ra (54 μK) of the first node in the outer region, hereafter referred to Ra , has been discussed in the previous section. We define a partially integrated mass current Ra part mIg/e (t) 0
=
Ra
pg/e (R, t)|Ψg/e (R, t)|2 dR,
(7.39)
0
and a partial population, between 0 and Ra , in each channel: Ra part Ng/e (t) = 0
Ra
|Ψg/e (R, t)|2 dR.
(7.40)
0
The average value of the momentum gained during the pulse by the inner part of the lower- or excited-state wavepacket, that is, the momentum kick, can be evaluated from the mass current and the partial population by part
part
< pg/e (t) >=
mIg/e (t) part
.
(7.41)
Ng/e (t)
As illustrated in Figure 7.13, such analysis has been performed in the case of PA 122 with negative chirp. If one compares the partially integrated mass by the pulse P− current and population to the same quantities integrated over the complete extension L of the box, mIg/e (t) and Ng/e (t), one observes that during the time window there is a large exchange of population between the two channels mainly located at © 2009 by Taylor and Francis Group, LLC
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Control of Ultracold Molecule Formation via Photoassociation
large distances R > Ra . In contrast, the momentum exchange is localized to a short part distance because mIg/e (t) ∼ mIg/e (t). Indeed for R > Ra , the amplitude Ag/e (R, t) and the phase Sg/e (R, t) in the wavepackets become R-independent. This formalism is consistent with the concept of the PA window, most convenient once the delocalized character of the initial state is introduced in the model. It is more efficient than the procedure described in Refs [9,27], which consider physical quantities integrated over the whole R-domain and use the Heisenberg equation to implement local control theory. part
part
A comparison of the time variation of mIg/e (t), Ng/e (t), in Figure 7.13.
part
dNe dNe dt , and dt
is reported
1. At the beginning of the time window, for tP − τC < t < tP , the mass currents mIg and mIe in the two channels are oscillating below and above their mean value with a π phase difference with opposite phase, while their sum is not oscillating. Due to the large negative acceleration in the excited potential this negative current is smoothly increasing in absolute value. On average, the two channels are equally sharing the strong classical acceleration −3 asymptotic behavior. due to the 0− g (6S + 6P3/2 ) potential with R 2. Then, we observe that for tP < t < tP + τC the mass current gain in the lower state becomes larger than in the excited state: the acceleration is no longer equally shared between the two channels. part 3. After the pulse, for t > tP + τC , Ig remains constant during a long time interval of ∼600 psec. This can be explained by the negligible value of the part classical force in the lower state. Because the population Ng no longer varies, this mass current corresponds to a constant mean velocity of the order of −2.7 m/sec. By comparing the results obtained with various pulses, Ref. [38] concludes that the creation of an important flux of population toward short distances is favored both by an important population cycling during the pulse and by a significant transfer of this population back to the initial state. In other words, the optimal pulse should be as close as possible to a (2n)π-like pulse, with a negative chirp, and a large number n of cycles. This can be achieved with a pulse that is “following” the excited wavepacket over a long time and causes many complete cycles of Rabi oscillations. The choice of an “optimal” pulse is described in Section 7.6.1.
7.5.3 ADVANTAGE OF THE COMPRESSION EFFECT FOR PHOTOASSOCIATION WITH A SECOND PULSE To explore the possibilities offered by the compression of the wavepacket, we have represented in Figure 7.14 the variation of the Franck–Condon overlap between Ψg (R, t = topt ), when the compression effect is maximum, and various stationary wavefunctions ϕe,v of the bound vibrational levels in the external well of the excited state. Important overlap is obtained with almost all levels in the outer well
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Cold Molecules: Theory, Experiment, Applications Time window 2 × 10–3 1 × 10–3 mI part(t)
0 –1 × 10–3 –2 × 10–3 –3 × 10–3
N part(t)
8 × 10–3 6 × 10–3 4 × 10–3 2 × 10–3
dNe/dt(t)
0 5 × 10–3
0 –5 × 10–3 0
200
400
600
800
1000
t (psec) 122 maximum at t = t = 350 psec. The time FIGURE 7.13 Analysis of PA with the pulse P− P window [tP − τC , tP + τC ] is indicated by the horizontal arrow: outside this time window, part there is no radiative coupling. Upper panel: the variation of the mass currents mIg (R, t) in part
the lower state (long dashed line), mIe (R, t) in the excited state (solid line), and of their sum (dashed–dotted line) is displayed as a function of time. Note the Rabi oscillations during the pulse. In the region of the PA window, the classical acceleration is zero in the ground state. part After the pulse, Ig remains constant but negative over a long time delay. The inner part of the wavepacket is moving to short distances with a constant average velocity of ∼2.7 m/sec. Later (not visible in this figure) it is accelerated by the short-range potential and reflected. In the excited state, the wavepacket is moving to short distances with a large velocity, and reflected after 600 psec. Middle panel: time variation of the partial populations in the lower part part (Ng (t), long dashed line) and excited (Ne (t), full line) state during the pulse. Note the part
part
Rabi oscillations with opposite phase, while the sum Ng (t) + Ne (t) (dash–dotted line) remains constant. Lower panel: comparison of the time variation of the total population in part
dNe e the excited state dN dt (solid line) and of the partial population dt (broken line). During the pulse there is a large temporary transfer of population, mainly at large distances, outside the PA window. (From Mur-Petit, J. et al., Phys. Rev. A, 75, 061404(R), 2007. With permission.)
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of Vexc . The deepest level populated is v = 29, vext = 4, with a binding energy of 70 cm−1 and an inner turning point at R = 30a0 . Such a level has a good Franck– Condon overlap with the v = 42 and 33 levels of the a3 Σ+ u potential, respectively bound by 3.7 and 20.5 cm−1 , which could improve the efficiency of the stabilization step. In contrast, for the initial wavepacket Ψg (R, t = tinit ), non-negligible overlap is obtained only with vibrational levels close to the dissociation limit in the Vexc potential. As discussed in Ref. [38], the compression effect suggests PA with a second PA pulse red-detuned relative to the first one, and populating deeply bound levels in the excited state. In favorable cases, the latter levels could contribute efficiently to the stabilization step toward deeply bound levels of the lower electronic state, and hence to the formation of vibrationally cold stable molecules. Indeed, the bottleneck in the PA reaction being the small probability density for pairs of cold atoms at short relative distance, the existence of a compression effect seems an interesting procedure to improve PA.
7.5.4
REDISTRIBUTION OF POPULATION IN THE LOWER STATE PHOTOASSOCIATION PULSE
AFTER A
After the pulse, the presence of a dynamical hole means that the wavepacket in the a 3 Σ+ u state differs from the initial stationary collision state, with well defined collision energy Einit ; it has now significant projection on bound levels ϕg,v and on continuum levels with different energy.
7.5.4.1
Redistribution in the a3 Σ+ u State in the Case of Cesium Photoassociation
Population of the two last bound levels has already been mentioned in Section 7.3.2.2 and illustrated in Figure 7.4. Redistribution of the population in various continuum levels of the Vground potential is analyzed in Figure 7.15 for different intensities and sign of the chirp. The density of population dNg (E )/dE for energy-normalized continuum levels is independent of the sign of the chirp; in contrast, when the maximum intensity of the pulse is increased by augmenting IL (see Equation 7.8), the width of the redistribution of collision energies is clearly reduced, the area below the curve being roughly conserved. As for the creation of halo molecules (see Section 7.3.2.2), it is independent of the pulse parameters.
7.5.4.2
Correlated Pairs of Hot Atoms
Due to Rabi cycling during the pulse, the redistribution of population in the lower state creates correlated pairs of hot atoms, as illustrated in Figure 7.15. Possible applications to condensates are being discussed. © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications 1 × 10–2 8 × 10–3
Ngpart|0R(t)
PA window
6 × 10–3 4 × 10–3 2 × 10–3
|Ψg(R,t)|
0 2 × 10–2
1 × 10–2
0
25
50
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0
100 125 150 175 200 225 250 275
|<ϕe,u|Ψg(R,t) >|2
R (a0)
2 × 10–4 1 × 10–4 0
–80
–70
–60
–50
–40
–30
–20
–10
Eu (cm–1)
FIGURE 7.14 The compression effect and its optimization; various quantities are compared related to the initial wavepacket for two colliding atoms (dashed lines), and to the compressed wavepackets when the compression effect is maximum, at topt = tP + 950 psec, after illu122 (solid lines). Also displayed are the compressed wavepackets, minating with the pulse P−
= t + 350 psec, after illuminating with the optimized pulse P now at topt P − (shaded curves). opt
part
Upper panel: integrated population Ng |R 0 (t) = Int for the initial collision state (dashed line); due to a quantum reflection effect ([57] and Chapter 6 by Paul Julienne), the probability of finding two cesium atoms at distances shorter than ∼ 100a0 is negligible; after a PA pulse, at (shaded curve)), this the maximum of the compression effect (t = topt (solid line) and t = topt probability is no longer negligible. Middle panel: wavepacket amplitude |Ψg (R, t = tinit )| of the initial collision state (dashed line); lower state wavepacket |Ψg (R, topt )|, after illuminat122 (solid line), and |Ψ (R, t )| after illuminating with the pulse P opt ing with the pulse P− g − opt (shaded curve). Lower panel: variation of the overlap | < ϕe,v |Ψg (R, t) > |2 with the eigenfunctions for all the bound vibrational levels v in excited potential curve, as a function of their binding energy Ev , for the initial state at t = tinit (dashed line) and for the wavepackets created (shaded curve). In contrast with the initial wavepacket, the at t = topt (solid line), and t = topt compressed wavepackets display large overlaps with the lower vibrational levels of the excited potential. The shaded areas in the three panels demonstrate how the compression effect can be opt markedly improved with use of the nonimpulsive pulse P− , by a duration τC comparable to half the vibrational period in the excited state, described in Section 7.6.1. (From Mur-Petit, J. et al., Phys. Rev. A, 75, 061404(R), 2007. With permission.) © 2009 by Taylor and Francis Group, LLC
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7.6
BEYOND THE IMPULSIVE OR ADIABATIC APPROXIMATIONS: NEW MECHANISMS
The present chapter has focused on an excitation mechanism where population transfer from the lower to the excited state takes place in the impulsive approximation within a PA window. As discussed in Refs [18,19,23,38], other mechanisms do exist, and might have interesting applications.
7.6.1
CONTROLLING THE COMPRESSION EFFECT WITH A NONIMPULSIVE PULSE INDUCING MANY RABI CYCLES
From the conclusion of Section 7.5, a pulse has been designed to optimize the comopt pression effect in the ground-state wavepacket. The new pulse, described as P− 122 by increasing the chirp rate χ and therefore the in Table 7.2, is obtained from P− duration. For this pulse, the duration τC = 376.13 psec is comparable to half the vibration period Tvib in the excited state. The intensity was increased to obtain 24 Rabi oscillations, close to a (2n)π-pulse. The probability of excitation Pebox is markedly decreased (3 × 10−4 instead of 2 × 10−3 ). As displayed in Figure 7.14, a spectacular = t + 350 psec, is observed. The compressed compression effect, maximum at topt P wavepacket is associated with large values for the time-dependent Franck–Condon factor with low-lying levels in the external well of the excited state, suited for PA with opt a second pulse red-detuned from P− and delayed by 350 psec.
dNg/dE' (E' )= |<Ψg |ϕg(E' ) >|2
3 × 107
2 × 107
1 × 107
Einit=54 μK 0
100
200
300 E' (μK )
400
500
600
FIGURE 7.15 Population redistribution in the lower state: analysis of the wavepacket after 122 (solid line) and with a similar pulse with higher intensity, illuminating with the pulse P− changing the factor IL in Equation 7.8 to 9IL (broken line). The results do not depend upon the sign of the chirp.
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7.6.2 7.6.2.1
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NONADIABATIC BROADBAND PULSE: EXCITATION AT LARGE DISTANCES Large Transfer of Population Outside the PA Window
Photoassociation restricted to a PA window requires a sufficiently large value of the pulse duration τL , when the spectral width δω and the energy range swept by the resonant instantaneous frequency |χ|τC are sufficiently small to satisfy the condition (δω + |χ|τC ) = δω(1 +
1 − (τC /τL )−2 ) < δL ,
(7.42)
ensuring that there is no population of continuum levels of the excited state. This condition is often not valid. As an example, we give in Table 7.2 the paramnad , identical to P 98 for the detuning δ , the intensity I and the eters of the pulse P− L L − chirp rate χ ensuring the focusing condition, but with a bandwith δω increased up to 2.45 cm−1 , nearly equal to the detuning δL . 98 The dynamics of the PA reaction corresponding to the narrowband pulse P− 98 nad and broadband pulse P− are compared in Figure 7.16. Whereas for P− , adiabatic nad , the time evolution of the population transfer occurs within the PA window, for P− box in the excited state Pe (t) presents Rabi oscillations, which are the signature of significant nonadiabatic effects. After the end of the pulse, about 6% of the population remains in the excited electronic state, and the radial distribution of the photoassociated wavepacket Ψe (R, td = tP + Tvib /2) at the focusing time occurs mainly at large distances. In this region, the wavepacket in the excited state has an amplitude proportional to that of the initial scattering wavefunction Ψg (R, tinit ) = ϕg,Einit (R), with a proportionality constant √β, independent of R and of Einit in the threshold range. The modulus |β| = 0.060 is time-independent after the end of the pulse. Therefore we can easily separate the short-range contribution to the excited wavepacket by substracting the long-range contribution. The amplitude of |Ψe (R, t = tP + Tvib /2) − βΨg (R, tinit )| represents the wavepacket transferred at short-range Ψewindow (R, t = tP + Tvib /2) and is drawn in Figure 7.16. This short-range contribution extends in the spatial range [Rmin , Rmax ] of the PA window. However, because the levels populated in the excited state are mainly in the continuum, corresponding to a pair of atoms Cs(6S) + Cs(6P) rather than to a bound molecule, this contribution should be avoided in a PA experiment. 7.6.2.2 Thermal Average Even when population is transferred at long range, it is possible to estimate the total probability per pump pulse for a pair of atoms to be photoassociated (see Equation 7.33) provided the interference term between the short- and long-range contributions can be neglected. Indeed, the first contribution Pwindow (T ), associated with Ψewindow (R, t), can be evaluated by averaging over the initial velocity distribution as reported in Section 7.3.4.2. The Boltzmann average for the long-range contribution, associated with β(t)Ψg (R, tinit ), is trivial, because it reflects the thermal average of the initial distribution of atom pairs. However, the molecules formed in such a way may easily decay into pairs of hot ground-state atoms. The evaluation of the number of these photoassociated molecules is therefore of interest © 2009 by Taylor and Francis Group, LLC
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(a) 0.15
Focalized wavepacket |Ψe (R, td=tP+Tvib/2)|
(c)
Pe
0.006 Nonadiabatic transfer at large distance 0.004 0.002 0
0.1 5000 R(a0)
0 0.05
0.004
10000 0.002
tP 0
300
500
200
700
(b) 0.15
400
600
0 800 1000 1200 1400 1600 1800 2000
(d) Adiabatic transfer within a window 0.004
0.1 0.002 0.05 tP 0 300
500 700 t (psec)
0 200
400
600
800 1000 1200 1400 1600 1800 2000 R (a0)
FIGURE 7.16 Two mechanisms of photoassociation: (a, c) Nonadiabatic transfer at large nad with broad spectral bandwidth (δω = 2.453 cm−1 , τ = 6 psec distances, for the pulse P− L 98 τC = 96 psec); (b, d) Adiabatic transfer within a PA window (for the narrowband pulse P− −1 with narrow bandwith (δω = 0.981 cm , τL = 15 psec, τC = 36 psec). Both pulses have the same detuning, the same chirp rate χ and correspond to the same intensity IL for the associated transform limited pulse (see Table 7.2). (a, b) Time evolution of the population in the excited state Pebox (t) (for a total population normalized to 1 on the grid) during the PA process, tP corresponding to the maximum of the pulse. Due to the vertical scale chosen here (compare to Figure 7.4) the population transferred to bound levels of the excited state is not visible in (b). (c, d) Excited-state wavepacket |Ψe (R, t)| at the time td = tP + Tvib /2, half a vibrational period after the pulse maximum t = tP , when the wavepacket focuses at the inner turning point (see also Figure 7.10) (bold black full line). The thin line in (c) shows the contribution of the short-range excitation Ψewindow (R, t) defined in text. Also displayed for comparison in the inset is the initial scattering wavefunction ϕg,Einit in the lower potential at large distance and with energy E/kB = 54 μK (dash–dotted line in the inset).
only if efficient stabilization mechanisms can be found. In situations where the long-range contribution dominates, the description of the initial state with thermal Gaussian wavepackets [23] is more appropriate than the description with stationary scattering states. © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
CONCLUSION AND PROSPECTS FOR THE NEAR FUTURE
In this chapter, we have discussed the possibility of controlling the formation of ultracold molecules by PA of ultracold atoms, using chirped laser pulses. The choice of the optimal pulse is governed by the spectroscopy of the molecule considered. Here we have presented and analyzed results of numerical calculations in the particular case of PA of ultracold Cs atoms. A linear chirp parameter was used, with pulses of typical duration a few tens of picoseconds, well suited to PA at small detuning. In the calculations, the initial state is not a Gaussian wavepacket but a stationary scattering wavefunction describing the collision of a pair of ultracold atoms. Ways of performing Boltzmann averages have been described, including use of scaling laws relying on threshold behavior of the wavefunction. The conclusion of the calculations is that it is apparently possible to substantially enhance the PA rate. An efficient mechanism is adiabatic population transfer within a PA window. The latter is defined by the range of distances swept during the pulse by the crossing point of the two potential curves dressed with the instantaneous frequency. Full adiabatic population transfer means that all the pairs of atoms with relative distance within the PA window are transformed into bound excited molecules. In the excited state, a coherent vibrational wavepacket is created. It is easy to choose the chirp parameter in order to shape this wavepacket. In particular, the difference between the vibrational periods due to the strong anharmonicity of the potential can be corrected by use of a convenient negative chirp. After half a vibrational period, a focused wavepacket is then obtained in the vicinity of the inner turning point. In a two-color pump–dump experiment, a maximum of population can then be transferred by a dump pulse to bound levels of the lower state, forming ultracold stable molecules with a low vibrational temperature. During the PA step, there is also an important population transfer to the highest bound vibrational levels of the initial electronic state. The formation of halo molecules by an optically induced Feshbach resonance should be further exploited. The redistribution of population in the continuum is creating pairs of hot atoms. Since all the pairs of atoms within the PA window have been transformed into bound molecules in an excited state, the chirped laser pulse creates a depletion hole in the initial state wavefunction. After the pulse, due to a negative momentum kick, the probability density is pushed inwards, markedly increasing the number of atom pairs at short distances. The physical interpretation makes use of the time dependence of the mass current and population in each channel to understand the role of the parameters of the PA pulse. By varying such parameters, optimization of the compression effect in the lower-state wavepacket is demonstrated. Due to an increase of the short-range probability density by more than two orders of magnitude, we predict a significant gain in the PA rates into deeply bound levels of the excited state by a second pulse, red-detuned relative to the first one, and conveniently delayed. The latter levels could contribute efficiently to the stabilization step toward deeply bound levels of the ground state, and hence to the formation of vibrationally cold molecules. The conclusion is that a sequence of different pulses for pump, dump then pump again should be optimized in view of efficient experimental achievements. Future
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work should determine whether a phase relation between the different pulses would further improve the process. The present work has been focused on the design of one or two chirped pulses, the important problem of avoiding the destruction of the stable molecules by the following pulses not yet being addressed. The next step is therefore a careful design of a sequence of chirped pulses. After this chapter was written, new experimental results have been published [58] concerning the formation of ultracold Cs2 molecules in the v = 0 level of the ground state by PA of cesium with a sequence of identical noncoherent femtosecond pulses. In the latter work, following the ideas developed in Refs. [59,60], the pulses are amplitude-shaped to remove the excitation frequency band of the v = 0 level, preventing reexcitation from that state. It will be interesting to combine the ideas presented in this chapter concerning the advantages of chirped pulses with this method.
ACKNOWLEDGMENTS The present work has been supported by the European Commission in the frame of the Cold Molecule network under contract HPRN-CT-2002-00290. We are grateful to Ronnie Kosloff, Christiane Koch, Jordi Mur-Petit, Pascal Naidon, Mihaela Vatasescu, and Kai Willner for efficient and pleasant collaboration. Laboratoire Aimé Cotton is part of Fédération Lumière Matière (LUMAT, FR 2764).
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46. Julienne, P.S. and Mies, F.H., Collisions of ultracold trapped atoms, J. Opt. Soc. Am. B, 6, 2257, 1989. 47. Roberts, J.L., Burke, Jr., J.P., Claussen, N.R., Cornish, S.L., Donley, E.A., and Wieman, C., Improved characterization of elastic scattering near a Feshbach resonance in Rb-85, Phys. Rev. A, 64, 024702, 2001. 48. Marte, A., Volz, T., Schuster, J., Dürr, S., Rempe, G., van Kempen, E.G.M., and Verhaar, B.J., Feshbach resonances in rubidium 87: Precision measurement and analysis, Phys. Rev. Lett., 89, 283202, 2002. 49. Zamith, S., Degert, J., Stock, S., de Beauvoir, B., Blanchet, V., Bouchene, M.A., and Girard, B., Observation of coherent transients in ultrashort chirped excitation of an undamped two-level system, Phys. Rev. Lett., 87, 033001, 2001. 50. Salzmann, W., Mullins, T., Eng, J., Albert, M., Wester, R., Weidemüller, M., Merli, A., Weber, S.M., Sauer, F., Plewicki, M., Weise, F., Woeste, L., and Lindinger, A., Coherent transients in the femtosecond photoassociation of ultracold molecules, Phys. Rev. Lett., 100, 233003, 2008. 51. Averbukh, I.S. and Shapiro, M., Optimal squeezing of molecular wave-packets, J. Chem. Phys., 47, 5086, 1993. 52. Abrashkevich, D.G., Averbukh, I.S., and Shapiro, M., Optimal squeezing of vibrational wave packets in sodium dimers, J. Chem. Phys., 101, 9295, 1994. 53. Ashkenazi, G., Banin, U., Bartana, A., Kosloff, R., and Ruhman, S., Quantum description of the impulsive photodissociation dynamics of I− 3 in solution, Adv. Chem. Phys., 100, 229–315, 1997. 54. Messiah, A. Mécanique Quantique, Dunod, Paris, 1960. 55. Schiff, L.I., Quantum Mechanics, McGraw-Hill, New York, Toronto, London, 1955. 56. Bohm, D., A suggested interpretation of the quantum theory in terms of “hidden” variables. ii, Phys. Rev., 85, 180, 1952. 57. Côté, R., Heller, E.J., and Dalgarno, A., Quantum suppression of cold atom collisions, Phys. Rev. A., 53, 234, 1996. 58. Viteau, M., Chotia, A., Allegrini, M., Bouloufa, N., Dulieu, O., Comparat, D., and Pillet, P., Optical pumping and vibrational cooling of molecules, Science, 321, 232–234, 2008. 59. Tannor, D.J., Bartana, A., and Kosloff, R., Laser cooling of internal degrees of freedom of molecules by dynamically trapped states, Faraday Discussions, 113, 365, 1999. 60. Bartana, A., Kosloff, R., and Tannor, D.J., Laser cooling of molecules by dynamically trapped states, Chem. Phy., 267, 195, 2001.
© 2009 by Taylor and Francis Group, LLC
Adiabatic Raman 8 Photoassociation with Shaped Laser Pulses Evgeny A. Shapiro and Moshe Shapiro CONTENTS 8.1 8.2 8.3
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adiabatic Raman Photoassociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory of Multichannel Photoassociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Two-State Description of Multichannel Photoassociation . . . . . . . . . 8.3.2 ARPA as a Projective Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Single-Channel ARPA of a Chosen Wave Form. . . . . . . . . . . . . . . . . . . . 8.4.2 Thermal Averaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 PA of a Superposition State: Determining the Multichannel Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Molecular Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1
291 292 294 294 300 302 302 304 306 308 311 312 312
INTRODUCTION
Adiabatic Raman photoassociation (ARPA) is a method of recombining pairs of ultracold atoms to form ultracold molecules using coherent Raman scattering [1]. ARPA is essentially an extension of three-level stimulated Raman adiabatic passage (STIRAP) [2–6] to a situation in which the initial state corresponds to two colliding atoms. The basic idea of using coherent Raman with the promise of making substantial population transfers from the initial manifold to the lowest-lying diatomic bound state has caught the attention of many researchers over the last decade. A substantial number of theoretical [1,7–18] and experimental [19–23] papers have been devoted to this subject. In this chapter we concentrate on the case where the initial state of the colliding atoms is a wavepacket composed of molecular continuum states, or a dense set of closely lying bound levels such as those encountered in a trap. We also address 291 © 2009 by Taylor and Francis Group, LLC
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the “mixed-state” situation in which the atomic sample is initially in a thermal ensemble with a non-negligible energy spread. We then show that in the “pure-state” case, the ARPA process may be viewed as a way of executing a projective measurement of the incoming wavepacket of the colliding atoms, with the measurement basis being controlled by the laser pulses inducing the photoassociation (PA). This process is related to the application of quantum control [24–26] and wavepacket reconstruction techniques [27–32] to gain information about properties of wavefunctions that cannot be revealed by executing intensity-type measurements. The interest in such wavepacket reconstruction techniques has been intensified by recent experiments on multiparticle dynamics of atoms and molecules in small-size traps, such as atomic chips [33,34], dipole traps [35], and optical lattices [36,37]. We illustrate the ideas discussed above by presenting numerical simulations of the PA of Rb + Rb to form Rb2 and K + Rb to form KRb. These simulations demonstrate the ability to measure a qubit of information encoded in the temporal shape or in the two-channel structure of an incoming wavepacket.
8.2 ADIABATIC RAMAN PHOTOASSOCIATION Many processes in molecular physics relying on quantum-mechanical effects necessitate cooling the translational degrees of freedom of molecules. In most of these applications the molecules are prepared in a single (electronic, vibrational, and rotational) energy eigenstate. Unlike the atomic case, due to the involvement of many (vibrational and rotational) levels, direct laser cooling of the translational motion of molecules to ultracold temperatures has so far proved to be an impossible task. It was realized early on [38] that it should however be possible to create translationally cold diatomic molecules by photoassociating pairs of ultracold atoms. According to the two-step PA scheme, two colliding atoms, energetically residing in the near threshold region of a molecular continuum (Figure 8.1a), absorb a “pump” laser photon while undergoing a transition to an excited bound molecular state |1. In the second step,
(a)
(b)
>
(c)
>
|1
>
>
|0
>
|E2
>
|0
>
|1
|1 |E
>
|2
|2
> > |3> |0>
|E1
FIGURE 8.1 The PA scenarios discussed in the text. (a) Single-channel PA. (b) ARPA with two incoming channels and two intermediate bound states. (c) Realistic arrangement in the two-channel PA of KRb. The dotted lines show undesired transitions driven by the strong pump pulses.
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293
the population of state |1 is (partially) transferred to the ground molecular electronic state by either a spontaneous-emission [38] process, or a stimulated-emission [39] process, induced by an additional (“dump”) laser pulse. Although quite successful as a spectroscopic tool [40–48] (see Chapter 5 by W. Stwalley, P. Gould, and E. Eyler), as a preparatory tool, two-step PA suffers from losses due to spontaneous emission. Even in the presence of the stimulated emission process induced by a “dump” pulse [39], sponaneous emission populates in an incoherent fashion a large number of vibrational and rotational levels of the ground electronic state (or the low metastable electronic state), resulting in a translationally cold but internally hot ensemble of molecules. In contrast, the single step adiabatic Raman photoassociation technique [1] is capable of overcoming these difficulties. In this scheme PA is executed coherently using the “dump-before-pump” pulse ordering of STIRAP [2–6]. In its simplest form, STIRAP involves the “Λ-type” configuration of states, consisting of an initial state |i, a final state |f, and an intermediate, energetically higher, state |1. The task of transferring, as completely as possible, the population from state |i to state |f is achieved by coupling the three states with two laser fields: a “pump” field, tuned to be in near resonance with the |i ↔ |1 transition, and a “dump” field, preceding the pump field, tuned to be in near resonance with the |1 ↔ |f transition. When a two-photon resonance with the |i ↔ |f transition is maintained, and the pulses are long enough, this so-called “counter-intuitive” pulse sequencing [3] brings about a complete population transfer from state |i to state |f: as the dump pulse is turned off and the pump pulse is turned on, the field-dressed state correlating initially with state |i (the so-called “dark state”) is observed to execute a smooth passage to state |f while leaving the intermediate state |1 unpopulated at all times, thus eliminating the unwanted spontaneous emissions from state |1. The ARPA process in a thermal ensemble of atoms as well as in a Bose–Einstein condensate of atoms, has been studied theoretically in great detail [1,7,8,11–18]. Its first experimental demonstrations [21–23] have verified the existence of the fielddressed “dark state,” predicted in Ref. [1] to arise from a strongly coupled Λ-type system made up of an initial-continuum and two (intermediate and final) bound states. In addition, the closely related process of adiabatic passage from a loosely bound excited molecular state, produced by the “Feshbach resonance” switching process, to a deeply bound low-lying state, was observed [49–51]. Initially, the lack of a clear understanding of the nature of the adiabatic transfer from the continuum during ARPA led to conflicting opinions [7,11,12] about its viability. The answer lies, as pointed out in Ref. [11], in going beyond the three-level STIRAP-like picture. In ARPA distribution of population between the (continuum) energy eigenstates is of great importance. The conflicting opinions [1,7,11,12] have resulted from the different procedures employed for taking into account this effect. It turns out, as shown below, that the nature of the initial population spread affects the nature of the (pump and dump) pulses that maximize the efficiency of the ARPA process. The theory presented below describes ARPA in two situations: (1) an initial continuum of states and (2) an initial (dense) set of bound states. In either case the PA yield is determined by the projection of the incoming multichannel wavefunction onto a © 2009 by Taylor and Francis Group, LLC
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wavepacket correlating adiabatically with the (deeply bound) target state. The shape of such wavepackets is defined by the shapes of the pump and dump laser pulses. Thus, ARPA may be viewed as a process in which one measures a continuum wavefunction by projecting it onto a set of (experimentally controllable) wavepackets of known form. Our approach builds on the work presented in Refs. [1] and [17]. It also extends the recent work [52], which explored coherently controlled adiabatic passage in molecular dissociation. In the present study we describe the influence of temperature on the PA process and demonstrate how ARPA can be used to read the information encoded in the incoming wavefunction. Based on the calculations of Ref. [17], we discuss the role of incoherent initial conditions. This chapter is organized as follows. Section 8.3 presents the general theory of the adiabatic PA of a multichannel wavepacket. Section 8.4 illustrates our approach by simulating the measurement of the shape of the incoming single-channel wavepacket in the PA of a Rb2 molecule. We also study the possibility of reading a two-channel superposition in the PA of KRb, and discuss the role of temperature. The modeling of the input molecular data for laser-driven formation of 85 Rb2 and 40 K-87 Rb molecules is given in Section 8.5. Section 8.6 presents the conclusions and discussion.
8.3 THEORY OF MULTICHANNEL PHOTOASSOCIATION 8.3.1 TWO-STATE DESCRIPTION OF MULTICHANNEL PHOTOASSOCIATION Figure 8.2 shows the molecular potentials describing a PA for two alkali atoms, using the Rb2 potentials as an example. We consider the ARPA scenarios illustrated in Figure 8.1a and 8.1b, starting with two atoms colliding on either the X 1 Σ+ g potential 1 + alone (for PA of a homonuclear molecule), or on both the X Σ and the a3 Σ+ diatomic potentials (for PA of a heteronuclear molecule). To realize the process shown in Figure 8.1a, one could use a pair of laser pulses: a pump pulse, coupling the 3 initial wavepacket |Ψi to state |1, chosen to be one of the LS-coupled A1 Σ+ u -b Πu bound states, and an anti-Stokes dump pulse, coupling state |1 to the target state |0, chosen to be one of the deeply bound vibrational states of the ground electronic state X 1 Σ+ g. In the two-channel case shown in Figure 8.1b, there are two incoming continuum channels, two intermediate states (|1 and |2) and two pairs of pump and dump pulses, each pair acting simultaneously, with the pump pulses delayed relative to the dump pulses, in accordance with the counterintuitive scheme. In general, a collision between two atoms takes place on a superposition of several potential surfaces, distinguished by the total angular momentum F and/or its projection mF . In the derivation below we assume that there are N incoming open channels (for which Eth , the threshold energy, is less than E, the total energy). Alternatively, for PA of atoms in a trap, we assume that there are N manifolds of nondegenerate bound trap states. As the intermediate state we choose N nondegenerate bound levels of the excited electronic potentials. We use N pump and N (anti-Stokes) dump laser pulses. © 2009 by Taylor and Francis Group, LLC
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Adiabatic Raman Photoassociation with Shaped Laser Pulses
Potential energy (au)
0.06
A1Σu+
0.04
b3Πu
0.02
+
a3Σu
0
+
X1Σg –0.02 7.5
10
12.5 15 17.5 20 Internuclear distance (a0)
22.5
25
FIGURE 8.2 The Rb2 Born–Oppenheimer potentials of Ref. [53] involved in our calculations. The dotted line shows the energy region of the intermediate ARPA states used in our calculations.
The total Hamiltonian of the system can be written as ˆ Hˆ = Hˆ M − 2μ
N
{P,n (t) cos(ωP,n t + φP,n ) + D,n (t) cos(ωD,n t + φD,n )},
(8.1)
n=1
where Hˆ M is the field-free Hamiltonian of the atoms; P,n (t), ωP,n , φP,n , D,n (t), ωD,n , and φP,n are slowly varying dump-and-pump amplitudes, frequencies, and phases, and −2μ ˆ is the electronic dipole moment for the transition between the ground and excited electronic states. As in the theory of single-channel photodissociation and PA [1,24,54], the wavefunction of the atoms can be represented (in atomic units, au) by |Ψ(t) = b0 (t)e−iE0 t |0 +
N
n=1
bn (t)e−iEn t |n +
N
∞
dE bE (t)e−iEt |E, k +
k=1 Eth,k
(8.2) for case 1 (an initial continuum). Here |0 denotes the target state, |n the intermediate bound state(s), and |E, k + a scattering state correlating as t → −∞ with the product of a free-translational state of energy E, and the kth internal (e.g., electronic) state [24,55]. En and E are the field-free energies of the states: (E0 − Hˆ M )|0 = (En − Hˆ M )|n = (E − Hˆ M )|E, k + = 0. © 2009 by Taylor and Francis Group, LLC
(8.3)
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For case 2 (atoms in a trap), |Ψ(t) = b0 (t)e−iE0 t |0 +
N
bn (t)e−iEn t |n +
N
bmk (t)e−iEmk t |mk ,
(8.4)
k=1 mk
n=1
where Emk and |mk are, respectively, the eigenvalues and eigenstates of an atom in a trap, satisfying (Emk − Hˆ M )|mk = 0. We assume that all the dump laser fields are in exact resonance with the corresponding transitions, ω0,n ≡ En − E0 . We also assume that all the pump laser fields are in near resonance with the transitions from a given energy E (or Emk ) of the incoming states. The detunings are defined as Δn,E ≡ ωP,n − (En − E),
Δn,mk ≡ ωP,n − (En − Emk ),
for case 1,
for case 2. (8.5)
Given the above choice of pump pulse, it follows from the rotating wave approximation (RWA) [56] that the pump laser couples each kth component of the incoming wavepacket to a single intermediate bound state. The expansion coefficients of Equation 8.2 are obtained by solving a matrix Schrödinger equation of the form b˙ 0 = i
N
Ω∗n,0 bn
(8.6)
n=1 f b˙ n = i Ωn,0 b0 − Γn bn + i
N
∞
bE,k Ωn,E,k ei Δn,E t dE
(8.7)
k=1 Eth,k
b˙ E,k = i
N
bn Ω∗n,E,k e−i Δn,E t ,
(8.8)
n=1
For case 2 (Equation 8.4), the last two equations are replaced by f b˙ n = i Ωn,0 b0 − Γn bn + i
N
bmk Ωn,mk ei Δn,mk t
(8.9)
k=1 mk
b˙ mk = i
N
bn Ω∗n,mk e−i Δn,mk t .
(8.10)
n=1
The complex Rabi frequencies of Equations 8.6 to 8.10 are defined as Ωn,0 = D,n μn,0 e−iφD,n ,
Ωn,mk = P,n μn,mk e−iφP,n ,
Ωn,E,k = P,n μn,E,k e−iφP,n . (8.11)
The lower limit Eth,k of the integration in Equation 8.7 is the threshold continuum f energy of the kth incoming channel. The empirical term Γn bn in Equations 8.7 and 8.9 © 2009 by Taylor and Francis Group, LLC
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Adiabatic Raman Photoassociation with Shaped Laser Pulses
describes nonradiative decay of the excited bound state |n. The initial |E, k + or |mk states are assumed to be unaffected by the analogous nonradiative decay process. The field-induced decay of the input and target states observed in the experiments [49] is neglected for notational simplicity. Solving formally for either bmk or bE,k , bE,k (t) = bE,k (0) + i
N
t
n=1 0
bmk (t) = bmk (0) + i
N
t
n=1 0
dt Ω∗n,E,k bn (t )e−i Δn,E t
dt Ω∗n,mk bn (t )e−i Δn,mk t
(8.12)
we obtain upon substitution in Equations 8.7 and 8.9 the equations of motion, which can be written in a matrix form as b˙0 = i Ω†D · bexc t b˙ exc = i b0 ΩD − dt Γ(full) (t − t ) · bexc (t ) + if (full) (t).
(8.13)
0
Here bexc ≡ (b1 . . . bN )T is the column vector (hence the T transpose sign) of ampliT tudes of the excited bound states, ΩD ≡ Ω1,0 . . . Ωn,0 is the vector of Rabi frequencies between the |0 state and the excited bound states, and Ω†D is its conjugate transpose. The matrix Γ(full) is given by f
(full)
Γn n (τ) = Γn δn,n +
N
k=1 Eth,k
Ωn,E,k Ω∗n ,E,k exp(iΔn,E τ)dE,
(8.14)
for case 1, and by f
(full)
Γn n (τ) = Γn δn,n +
N
Ωn,mk Ω∗n ,mk exp(iΔn,mk τ)
(8.15)
k=1 mk
for case 2. The N-component vector f (full) (t) describing the pumping from the initial multichannel wavepacket into the manifold of excited bound levels is given as fn(full) (t) =
fn(full) (t) =
k
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Ωn,E,k bE,k (0) eiΔn,E t dE,
for case1,
Eth,k
k
or
∞
mk
Ωn,mk bmk (0) eiΔn,mk t ,
for case 2.
(8.16)
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Cold Molecules: Theory, Experiment, Applications
Equations 8.13 can be further simplified using the “flat continuum” or “slowly varying continuum” approximation (SVCA). This approximation assumes that the variation of the continuum-bound dipole matrix elements with energy over the pump laser bandwidth is negligible. With this assumption, and after changing the lower limit of integration over each continuum channel to −∞, we can rewrite the second term in Equation 8.14 as N
∞
k=1 Eth,k
Ωn,E,k Ω∗n ,E,k exp(iΔn,E τ)dE ≈
N
Ωn,E0 ,k Ω∗n ,E0 ,k
k=1
≈ 2πδ(τ)
N
∞ −∞
exp(iΔn,E τ)dE
Ωn,E0 ,k Ω∗n ,E0 ,k
(8.17)
k=1
where E0 is some energy within the bandwidth of the pump pulse. Equation 8.13 now becomes b˙ exc = i b0 ΩD − Γ(SVCA) · bexc (t) + if (full) (t)
(8.18)
where (SVCA) Γn,n
=
f Γn δn,n
+π
N
Ωn,E0 ,k Ω∗n ,E0 ,k .
(8.19)
k=1
(Note ∞ that when substituting Equation 8.18 into Equation 8.13, we use the relation 0 δ(τ)dτ = 1/2.) The applicability of the SVCA in the energetic region of interest here (which for cold collisions is near the threshold) depends on variation of the continuum spectrum in this region over the pump pulse bandwidth. In most cases studied, the SVCA provides a reasonably good description of the PA process. In case 2 an approximation analogous to SVCA implies that the summation over each mk goes from −∞ to ∞, and that, for each k, the Rabi frequencies depend only weakly on mk : Ωn,mk ≡ Ωn,k . Equation 8.15 now becomes N
Ωn,mk Ω∗n ,mk exp(iΔn,mk τ) ≈
k=1 mk
N
Ωn,k Ω∗n ,k
k=1
≈ 2π
N
k=1
∞
exp(iΔn,mk τ)
mk =−∞
Ωn,k Ω∗n ,k
∞
(vib) . δ t − jTk
j=−∞
(8.20) −1 (vib) Here Tk = 2π × dEmk /dmk is the vibrational period of the wavepacket in the (vib) exceeds the duration of our kth manifold of states in the trap. Assuming that Tk © 2009 by Taylor and Francis Group, LLC
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laser pulse sequence (i.e., the trap energy levels are densely spaced compared to the bandwidths of the laser pulses), we arrive at Equation 8.18 with f
(SVCA)
Γn,n
= Γn δn,n + π
N
Ωn,k Ω∗n ,k .
(8.21)
k=1 (full)
The source vector fk
(t) factorizes within SVCA as
fn(full) (t) =
(0)
Ωn,E0 ,k fk (t) = ΩP f (0) (t),
n
(0)
fk (t) =
∞
−∞
bE,k (0) eiΔn,E t dE,
for case 1
(8.22)
and fn(full) (t) =
(0)
Ωn,k fk (t) = ΩP f (0) (t),
n (0)
fk (t) =
∞
bmk (0) eiΔn,k t ,
for case 2.
(8.23)
mk =−∞
T (0) (0) Unlike f (full) (t), the N-component vector f (0) (t) ≡ f1 (t), . . . , fk (t) . . . describes the initial state of the system without any reference to a particular pump transition. The matrix ΩP is composed of the Rabi frequencies for the transitions between each input channel and each intermediate state. Thus one can write the inhomogeneous source term as a product of the pump line shape and the Fourier transform of the initial wavepacket components. In the semiclassical regions, f (0) (t) describes the envelope of the incoming wavepacket as it moves along the classical phase space trajectory [58–61]. We now introduce the effective average Rabi frequency of the bound–bound transitions, Ωbb ≡ Ω†D · ΩD , (8.24) and the effective average amplitude of the excited bound states: beff ≡
Ω†D · bexc . Ωbb
(8.25)
With these notations, the Schrödinger equation 8.18 assumes the form b˙0 = i Ωbb beff b˙ eff = i Ωbb b1 −
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(8.26) Ω†D · Γ(SVCA) · bexc (t) Ω† · ΩP · f (0) (t) +i D . Ωbb Ωbb
(8.27)
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In the beginning of the next section we consider the problem with only one incoming channel and only one intermediate state |1. In that case, beff = b1 Ω∗1,0 /|Ω1,0 |, and the decay term in Equation 8.27 has a simple form: −Ω†D · Γ · bexc /Ωbb = −Γ1,1 b1 . In the general case of multichannel PA, this decay term may have a rather complicated time dependence as it is determined simultaneously by all the amplitudes of the excited bound states. Nevertheless, because in the adiabatic process studied here the excited bound states remain almost unpopulated, we omit the rapid dynamics of the decay term in our qualitative analysis. The validity of this approximation is verified by numerical simulations in the next section. For now, we replace the term −Ω†D · Γ · bexc /Ωbb in Equation 8.27 by the effective decay of the form −Γeff (t) beff . With the above approximation the Schrödinger equation assumes a form resembling the single-channel wavepacket photodissociation and PA case [1,24,54] d b = i{H · b(t) + f} dt where
b(t) =
and
b0 (t) , beff (t)
H=
0 Ωbb
(8.28) Ω∗bb , iΓeff
0 . f(t) = Ω†D · ΩP · f (0) /Ωbb
(8.29)
(8.30)
8.3.2 ARPA AS A PROJECTIVE MEASUREMENT We now proceed to solve Equation 8.28 in the adiabatic approximation, following the procedure described in Refs. [1] and [54]. In order to do that, we first obtain E, the matrix of eigenvalues of H, satisfying, U · H = E · U.
(8.31)
E has two nonzero elements, E1,1 = E+ , E2,2 = E− where E± =
3 2 1 iΓeff ± 4|Ωbb |2 − Γ2eff , 2
(8.32)
The diagonalizing matrix U is a complex orthogonal transformation which can be parameterized as cos θ sin θ U= . (8.33) −sin θ cos θ where θ is a complex angle. Using Equation 8.31 we see that tan θ = E+ /Ωbb . © 2009 by Taylor and Francis Group, LLC
(8.34)
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Adiabatic Raman Photoassociation with Shaped Laser Pulses
The adiabatic approximation (which was thoroughly investigated in Ref. [57]) is obtained by neglecting the nonadiabatic coupling matrix U−1 · dU/dt. This allows us to write Equation 8.28 in the form d a = iE(t) · a + ig, dt
(8.35)
sin θ Ω†D · ΩP · f (0) /Ωbb . g(t) = cos θ Ω†D · ΩP · f (0) /Ωbb
(8.36)
where a(t) = U(t) · b(t), and
We can solve Equation 8.35 by making the following substitution,
t a(t) = exp i dt E(t ) · α(t),
(8.37)
0
to obtain that b(t) = U−1 · a(t) is given as
t dt exp i E+ (t ) dt sin θ(t ) Ω†D · ΩP · f (0) /Ωbb
t
b0 (t) = icos θ(t)
t
0
t
− isin θ(t)
t
0
beff (t) = isin θ(t) 0
t
t dt exp i E− (t ) dt cos θ(t ) Ω†D · ΩP · f (0) /Ωbb
t dt exp i E+ (t ) dt sin θ(t ) Ω†D · ΩP · f (0) /Ωbb
(8.38)
t
t
+ icos θ(t) 0
t dt exp i E− (t ) dt cos θ(t ) Ω†D · ΩP · f (0) /Ωbb . t
(8.39) The final yield of the ARPA process is defined as the probability P0 = |b0 (t → ∞)|2 . Using Equations 8.32 and 8.34, we see that cos θ(t → ∞) = 0, and all the excited bound-state amplitudes, as well as beff , vanish. Substituting cos θ(t → ∞) = 0 and sin θ(t → ∞) = 1 in Equation 8.38 we obtain b0 (t → ∞) = 0
where
∞
∗ fARPA (t)f (0) (t) dt ≡ fARPA |f (0) t ,
t E− (t ) dt cos θ(t) Ω†D · ΩP /Ωbb . fARPA (t) = −i exp i
(8.40)
(8.41)
0
Thus, the PA amplitude b0 (t → ∞) is given by a projection of f (0) onto the specific wavepacket that correlates adiabatically with the target molecular state |0. © 2009 by Taylor and Francis Group, LLC
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∗ The shape of this wavepacket, given by fARPA , is controlled by the amplitudes and the ∗ phases of ΩD (t) and ΩP (t). Wavepackets orthogonal to fARPA do not photoassociate ∗ in the ARPA process, while the ones that project well onto fARPA do. We conclude that one must be able to measure the shape of the initial wavepacket by choosing the proper fARPA via pulse shaping, and using the ARPA to find the projection of f (0) onto it. To illustrate this, we explore in the next section the measurement of both the temporal and multichannel structure of the incoming continuum wavepacket in PA with one and two incoming continua.
8.4 8.4.1
NUMERICAL EXAMPLES SINGLE-CHANNEL ARPA OF A CHOSEN WAVE FORM
We first consider the ARPA scenario illustrated in Figure 8.1a in which we address PA of an initial wavepacket composed of near-threshold s-wave scattering states of two 85 Rb atoms colliding on the X 1 Σ+ potential. (The numerical parameters are given in g Section 8.5 and Ref. [17].) The pump pulse couples this initial state to an excited inter3 mediate bound state belonging to the A1 Σ+ u and b Πu spin–orbit-coupled states, |1 = 1 + 3 A Σu -b Πu (v = 133, J = 1) (E1 = 0.042848 au). The anti-Stokes dump pulse couples this intermediate state to the target state |0 = X 1 Σ+ g (v = 4, J = 0) (E = −0.01823 au). Figure 8.3 shows the envelopes of two choices for the initial continuum wavepacket. Panel (a) shows the envelope f (0+) (t) of a Ψ+ state whose expansion coefficients are chosen as explained below, to be (+) bE (t = 0) = (δ2E π)−1/4 exp −(E − E0 )2 /2δ2E + i(E − E0 )t0 ,
(8.42)
with E0 = 100 μK, δE = 70 μK, and t0 = 1220 nsec. Panel (b) of the same figure shows the envelope f (0−) (t) of the state Ψ− . It is obtained by multiplying f (0+) (t)
Envelope (au)
(a) 2×10–5 1×10–5 0 Envelope (au)
(b) 2×10–5
–2×10–5 0
0.5
1 Time (μsec)
1.5
2
FIGURE 8.3 The envelopes of the two continuum wavepackets, f (0+) (a) and f (0−) (b), taken as the initial conditions in the simulations.
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by sin[2δE (t − t0 )] and scaling the wavefunction up to satisfy the normalization condition f (0) | f (0) t = 2π. The symmetries of wavepackets viewed on a progressively finer scale offer a temperature-robust way of encoding several qubits of information [62–64]. The encoding and often the full control over the quantum evolution of the wavepacket [65] can be implemented by alternating periods of free motion with phase kicks imposed by coordinate-dependent Stark shifts. Distinguishing odd from even wave forms is the essence of the decoding of the qubits of information encoded in the wavefunction by the dynamics of atoms in a trap. The calculations below demonstrate the possibility to distinguish between the even wave form f (0+) and the odd one f (0−) . In the first calculation we examine PA with a pair of simple sin2 αt-shaped laser pulses. The dump pulse has σ− type polarization, an intensity of 1.12 × 105 W/cm2 , and a central wavelength 733 nm, in resonance with the transition between the target state |0 and the intermediate state |1. The pump pulse has σ+ type polarization, an intensity of 1.6 × 105 W/cm2 , and a central wavelength of 1063.4 nm, in resonance with the transition between a scattering state on the X 1 Σ+ g potential with a mean energy 100 μK and state |1. Both pulses are of 750 nsec duration (full-width at halfmaximum in the amplitude profile), with the peak of the pump pulse delayed by 600 nsec relative to the peak of the dump pulse. Setting 1/Γf = 30 nsec, we numerically solve the Schrödinger equation in the SVCA. Figure 8.4a displays the envelopes of the two laser pulses. Figure 8.4b shows Re( fARPA (t)) for this sequence, with Im( fARPA (t)) remaining negligible at all times except for a short period at t = 0.963 μsec, where 2ΩD = Γ, and cos θ diverges. It follows from Equation 8.40 that the ARPA process must be able to distinguish between f (0+) , the even-shaped initial continuum state, and f (0−) , the odd-shaped ∗ initial continuum state. This is due to the different projections onto fARPA : ∗ ∗ (0+) 2 (0−) | fARPA | f t | = 0.9 whereas fARPA | f t = 0.05. Figure 8.4c,d shows the calculated evolution of the target population P0 (t) for the initial state given by, respectively, Ψ+ and Ψ− . The final PA probability is equal to 0.89 for the Ψ+ initial state, and 0.001 for the Ψ− initial state. The PA probability of the even incoming state is thus 900 times larger than that of the odd state. The shapes of the pump and dump laser pulses define the shape of fARPA . Therefore by shaping the pulses it may be possible to choose the waveform transferred from the continuum during PA. According to Equation 8.41 fARPA is a function of E− (t), cos θ(t), ΩP (t), and the ratio Ω∗D /Ωbb . Both E− (t) and cos θ(t) depend only on the absolute values of the laser intensities, and their phases are difficult to control. On the other hand, both the amplitude of the pump Rabi frequency ΩP and the phase between the pump and the dump pulses (i.e., the phase of the product Ω∗D ΩP ) can be easily controlled by standard experimental methods. Figure 8.5a shows the envelopes of the pump and dump laser pulses tailored to photoassociate an incoming odd wavepacket. The preferential PA of the odd state is achieved by flipping the phase of P,1 (t) at t = t0 . Panel (b) of Figure 8.5 shows Re( fARPA (t)) for this sequence. Panels (c) and (d) respectively show the dynamics of the target state population for the Ψ+ and Ψ− initial states. ∗ ∗ | f (0+) t |2 = 0.04, and | fARPA | f (0−) t |2 = 0.7. The squared projections are | fARPA © 2009 by Taylor and Francis Group, LLC
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Field envelope (au)
(a) 2×10–6 1×10–6 0
Re ( f ARPA) (au)
(b)
0 –2×10–6 –4×10–6
(c) P0
0.8 0.4
0.4
P0
(d)
0.2
0
0
0.5
1 Time (μsec)
1.5
2
FIGURE 8.4 ARPA with Gaussian laser pulses. (a) Amplitudes of the pump (solid line) and the dump (dashed line) laser pulses. (b) Re(fARPA (t)). (c) Population of the target state |0 during PA of the continuum wavepacket Ψ+ . (d) Population of the target state during the PA of Ψ− .
The PA probability P0 (τARPA ) in the simulations is 0.002 for the even initial continuum envelope, and 0.68 for the odd one. The PA probability of the even incoming wave form is thus 340 times lower than that of the odd one. We see that by simply flipping the phase of the pump laser pulse we have chosen to photoassociate an odd incoming waveform instead of an even one. It is interesting to note that the numerical ratios of the PA probabilities in the above examples are higher than the analytical ones. As seen in Figure 8.3, the numerical optimization of the PA probabilities requires a slight temporal shift of the input wavepackets relative to the maximum of the theoretical fARPA (Figures 8.4b and 8.5b).
8.4.2 THERMAL AVERAGING We now discuss the effect of thermal averaging on the efficiency of ARPA when the initial state is a thermal ensemble. We show below that the energetic spread due to © 2009 by Taylor and Francis Group, LLC
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Field envelope (au)
(a)
4×10–6
Re ( f ARPA) (au)
(b)
10–6 0 –6 –10
0
–4×10–6 0.4
P0
(c)
0.2
0 (d) P0
0.6 0.3 0 0
0.5
1 Time (μsec)
1.5
2
FIGURE 8.5 ARPA with a shaped pump pulse. (a) Amplitudes of the two laser pulses. The sudden change of the sign of the pump amplitude at t0 = 1150 nsec is due to a flip in the phase of the pulse. (b) Re( fARPA (t)). (c) Population of the target state during the PA of Ψ+ . (d) Population of the target state during the PA of Ψ− .
the existence of a nonzero temperature defines the optimal parameters of the pump and dump laser pulse shapes. These in turn define fARPA of Equation 8.41 and set the upper limit for the PA yield. The laser pulse parameters used in our ARPA simulations are optimized [17] for a 100 μK thermal ensemble of Rb atoms. The pump and dump overlap time is chosen to match the average coherence time of a colliding atom pair at 100 μK, and the spectral widths of the pulses are chosen as the energy spread of the thermal ensemble. The laser pulse durations define their intensities because the area under the pulse envelope must be sufficiently large to ensure adiabaticity and efficient population transfer. (Note, the pulse intensities in this work are higher than those in Ref. [17], our main interest being the study of interference effects rather than searching for the lowest acceptable values of laser power.) The initial ensemble can be represented as a mixed state composed of many energy eigenstates, or as a mixed-state Gaussian wavepacket in phase space. While the first representation allows for an accurate numerical averaging over the ensemble [17], the second one is more convenient for simple estimates of the ARPA yield. © 2009 by Taylor and Francis Group, LLC
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In order to estimate the fraction of atoms photoassociated per pulse pair [1,17], we need to multiply P(E), the PA probability per collision at energy E, by the number of collisions experienced by a given atom during the pulses (this is equivalent to averaging over all possible values of t0 ). The number of collisions during the pulses is calculated as follows: at a given energy E, the velocity of an atom is 1 v = (2E/m) 2 and the distance traversed by the atom during a pulse of τlaser duration is vτlaser . The collision cross-section is πb2 where b is the impact parameter, related 1 to the partial wave angular momentum J as b = (J + 1/2)/p = (J + 1/2)/(2mE) 2 . Hence, the number of collisions experienced by the atom during the two pulses is N = ρπb2 vτlaser where ρ is the density of atoms. We conclude that for J = 0 the fraction of atoms photoassociated per pulse pair is 1
f (E) =
P(E)πρ(2E/m) 2 τlaser P(E)πρτlaser . = 1 8mE 4m3/2 (2E) 2
(8.43)
Using the estimate P(E) = 1, τlaser = 750 nsec, ρ = 1011 cm−3 , E = 100 μK, and the reduced atomic mass m = 1823 × 85/2 au, we obtain f ≈ 6 × 10−7 per pulse-pair. This estimate needs to be adjusted to take into account that approximately threequarters of near-threshold collisions between Rb atoms occur on the a3 Σ+ u potential surface, and these are not affected in our arrangement due to the selection rules for the transitions at the chosen frequencies. It is possible to increase the PA yield by increasing the phase space density of the initial sample by forming a Bose–Einstein condensate, or pairing two atoms in a single potential well of an optical lattice [66,67]. Some ways of using laser pulses to drive the phase space density of colliding atoms toward shorter internuclear distance are discussed in Refs. [68] to [70], and in Chapter 7 (by E. Luc-Koenig and F. MasnouSeeuws) of this book. Alternatively, one could repeat the PA pulse sequence many times, employing an irreversible process to prevent the molecules created by one pulse pair from being destroyed by another. This might possibly be implemented either by physically removing the molecules from the laser focus, or by exciting the molecules created via ARPA by an additional laser pulse, followed by a decay into low vibrational states of the X 1 Σ+ g potential [1,17].
8.4.3
PA OF A SUPERPOSITION STATE: DETERMINING THE MULTICHANNEL STRUCTURE
Equations 8.40 and 8.41 show that the time-dependent amplitudes and phases of the laser pulses define the temporal structure of the photoassociated waveform. This is true in the multichannel as well as in the single-channel ARPA. We now assume that all the laser pulses have similar simple time profiles, and concentrate on using PA for determining the multichannel structure of the input wavepacket. The measurement is based on controlling the interference of quantum pathways during ARPA, each pathway corresponding to the adiabatic passage via one of the intermediate |1 . . . |n states. © 2009 by Taylor and Francis Group, LLC
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As shown in Equations 8.25 and 8.27, the effect of the pump is equivalent to the production of a single coefficient beff that determines the multichannel dynamics. The source term for the production of beff is the (scaled) projection of the incoming vector f (0) on the multichannel (Ω†D · ΩP )† vector. Alternatively, the source term can be viewed as the projection of the vector ΩP · f (0) on the vector of the dump Rabi frequencies Ω∗D . Both the matrix ΩP and the vector Ω∗D are controlled by the pump and dump laser amplitudes. Therefore there is a large choice of possibilities to control the projective measurement of the incoming vector f (0) performed in the PA process. As an example, we assume that at a certain moment in time the incoming wave(0) function is composed of only the jth channel: fj = (0, . . . , 1j , . . . , 0)T . In that case (0) the source vector, ΩP · f , coincides with the jth column of the ΩP matrix ΩP, j , and the maximal PA probability for a fixed dump pulse area is obtained when the Ω∗D vector is parallel to ΩP · f (0) , which is realized when Ω∗D ∝ ΩP, j .
(8.44)
If, on the other hand, the vector Ω∗D is orthogonal to ΩP,j , then the PA rate would be zero. These conclusions parallel those made for adiabatic Raman dissociation in Ref. [52]. There, Equation 8.44 was a result of the definition of the prevailing dark state, which correlates at t → −∞ with the initial bound state and at t → ∞ with the final jth continuum channel. The above assessment (8.44) does not include the effect of the uncontrolled decay of the photoassociated waveform. In its absence the above analysis was verified numerically for PA of a superposition of two KRb continuum channels belonging to the singlet X 1 Σ+ and the triplet a3 Σ+ electronic states (see Figure 8.1b and 8.1c). The PA was conducted in the near-threshold region via a pair of intermediate levels belonging to the LS coupled A1 Σ+ and b3 Πu potentials. A detailed description of the KRb molecular data used in our simulation is given in Section 8.5. According to our observations, if the rates of decay from the two intermediate levels are artificially set to Γf , then Equation 8.44 correctly predicts the ratio of the two dump fields needed to provide the maximal and the minimal PA probabilities. If, however, the field-induced decay is taken into account, then the maximal and minimal PA rates are achieved at the intensity ratios, which are different from those predicted by Equation 8.44. Moreover, there are always undesired satellite transitions. For example, as shown in Figure 8.1c (the black dotted arrow), the pump frequency component that couples the lower-energy intermediate level |1 to the incoming continuum also couples the higher-energy |2 state to a more energetic continuum level, which does not exist in the initial wavepacket. In a similar fashion, as depicted in Figure 8.1c, the pump frequency component that couples the incoming continuum with the higher level |2 can also couple the lower level |1 with one of the highly excited bound states of the X 1 Σ+ g potential (denoted as state |3 in Figure 8.1c). (0)
In order to decompose the incoming wavepacket coefficient vector into its f1 (t) (0) 3 + (belonging to X 1 Σ+ g ) and f2 (t) (belonging to a Σu ) components, we need to find ∗ the complex dump-pulse component ΩD(1) which projects well and exclusively onto
ΩP f1 , and Ω∗D(2) , which projects well and exclusively onto ΩP f2 . Comparing the (0)
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(0)
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results of PA with the dump Ω∗D = Ω∗D(1) with those with Ω∗D = Ω∗D(2) one can then determine the multichannel composition of the incoming wavepacket. The above numerical analysis can be significantly simplified if we choose as the intermediate state |1 a state that couples mainly to the singlet continuum and as intermediate state |2 a state that couples mainly to the triplet continuum. The problem is further simplified if the states |1 and |2 have the same order of magnitude Franck– Condon overlaps with the target state |0. In KRb, we choose the intermediate state |1 3 as the A1 Σ+ u − b Πu (v = 108, J = 1) state (E1 = 0.04292 au), and the intermediate 3 state |2 as the A1 Σ+ u − b Πu (v = 141, J = 1) (E2 = 0.04659 au) state. State |1 couples almost exclusively to the a3 Σ+ u low-energy continuum, and state |2 couples continuum. Transitions from the incoming threshold-energy mostly to the X 1 Σ+ g continua into these levels are driven by, respectively, the 1062 nm and 977.7 nm pump laser components. The target state |0 is chosen to be the X 1 Σ+ g (v = 13, J = 0) (E = −0.01434 au) state. The dump transitions from the two intermediate states |1 and |2 to |0 occur at wavelengths of 796 nm and 748 nm, respectively. As in the above, the presence of satellite transitions leads to complications. Thus the strong 1062-nm pump component also couples state |2 to E ≈ 0.00367 au (1160 K) scattering states, and the strong 976-nm pump component couples state |1 to the level v = 55 of the X 1 Σ+ potential surface with energy E3 = −3.61 × 10−3 au. These additional transitions and states (shown in Figure 8.1c) were included in our simulation. In the first set of calculations, both pump pulses had the peak intensity IP = 5 × 107 W/cm2 . In the first simulation, I1D = 3 × 105 W/cm2 , I2D = 0. The final PA proba(0) bilities are P0(1) = 7 × 10−6 when the initial wavepacket is given by f1 , and P0(2) = (0) 0.14 when the initial state is f2 . This huge difference (a factor of 2 × 104 ) is (0) due to the fact that the X 1 Σ+ g continuum, described by f1 is practically decoupled from the intermediate state |1. In the second simulation, IP = 2 × 108 W/cm2 , I2D = 3 × 105 W/cm2 , I1D = 0. Now P0(1) = 0.04, and P0(2) = 7 × 10−5 , with P0(1) /P0(2) = 570. In the above calculations, we have neglected the off-diagonal elements of the decay matrix (Equation 8.19) which couple the intermediate states to one another via the continua. Inclusion of these terms may lead to superpositions of intermediate states that appear to be either “dark” or “bright” with respect to the decay [24,71–73], reminiscent of the interference stabilization of atoms and molecules in strong laser fields [74–76]. The possibility of utilizing such states for optimization and control of PA is an interesting open question. Adding temporal shaping to the laser pulses in order to process information encoded both in the multichannel structure and in the temporal structure of the incoming wavepacket is another unexplored possibility.
8.5
MOLECULAR DATA
The set of Rb2 Born–Oppenheimer (BO) potentials used in the above calculations are shown in Figure 8.2. Our choice of the intermediate bound states in Rb2 PA was motivated by the possibility of generating large-area microsecond-long pulses at the wavelengths needed for the continuum-bound transitions, and by the requirement that © 2009 by Taylor and Francis Group, LLC
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the chosen intermediate states be well separated from the neighboring bound states. Thus we choose the intermediate states to belong to the spin–orbit-coupled A1 Σ+ u and b3 Πu electronic potential curves of the diatomic molecules under consideration. 3 We choose those A1 Σ+ u -b Πu vibrational levels that have reasonably large Franck– Condon (FC) overlap with the incoming continuum states and are separated in energy from the incoming continua by approximately the energy of 1064-nm photons. The target vibrational levels of the X 1 Σ+ g potential are chosen such that the dump transitions can also be driven by readily available laser sources. We used the analogous set of BO potentials for the proof-of-principle simulations of multichannel PA of KRb molecules. An alternative choice of the intermediate levels [40,68–70,77,78] would be the 3 set of the highly extended “subcontinuum” bound states just below the A1 Σ+ u –b Πu continuum threshold. The continuum-bound FC factors associated with these states are much larger, allowing for the use of weaker pump lasers. This scenario necessitates either using the incoherent decay populating a wide range of excited levels and 3 + continua of X 1 Σ+ g and a Σu potentials, or controlling the long-range behavior of the subcontinuum states, which in turn depends very strongly on the (poorly known) couplings between the many potentials that determine the dynamics. Another potential difficulty in coherent control of the excited subcontinuum evolution is the necessity to use strong broadband laser pulses with frequencies close to those used to cool and trap the atoms. Recent attempts to optimize PA by controlling behavior of these states have yet not brought a clear answer as to whether a significant optimization is possible [77–80]. Photoassociation scenarios of the nonpolar Rb2 and polar KRb molecules differ due to the presence of the gerade–ungerade symmetry: the dipole transition from the 1 + 3 a 3 Σ+ u electronic state onto either A Σu or b Πu states in Rb2 is forbidden. Therefore, 1 + no ARPA to the X Σg electronic state is possible for collisions occurring on the a3 Σ+ u potential surface of Rb2 . The analysis of ARPA in Rb2 takes into account only the part of the continuum 1 + 3 + wavefunction that belongs to the X 1 Σ+ g channel. In KRb, both the X Σ and a Σ 1 + 3 continua are dipole coupled to the vibrational states of the A Σ -b Πu potentials. In general, the incoming wavefunction is given by a superposition of the X 1 Σ+ and a3 Σ+ components. The single-channel coefficients in such a superposition depend on the internuclear separation due to the hyperfine coupling and the effects of external fields. The modeling of the KRb molecular data follows the same route described in detail in Ref. [17] for Rb2 . The short-range potentials for Rb2 and KRb molecules were taken from Refs. [81] and [82], respectively. The long-range X 1 Σ+ and a3 Σ+ potentials were modeled as a −C6 /r 6 dispersion term, with the value C6 = 4426 au for Rb2 [83], and C6 = 4106.5 au for KRb [84]. The long-range and short-range potentials were smoothly joined in the region 38–52 au to ensure the correct scattering lengths for 85 Rb [85,86], and for 40 K-87 Rb [87]. The joining procedure is described in more 2 detail in Ref. [17] The coordinate-dependent spin–orbit coupling between the A1 Σ+ u and b3 Πu electronic states of Rb2 was taken from Ref. [81]. The average value of this coupling is of the order of 50 cm−1 . In KRb calculations, we set this value to 335 cm−1 in order to correctly reproduce the avoided crossing between these potential surfaces [82]. © 2009 by Taylor and Francis Group, LLC
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(a)
(b) 0
–2 –2 –4 Rb2, continuum–bound
0
KRb, continuum–bound
0
(d)
(c)
–4
–2
–2
–4
–4 Rb2, bound–bound
–6
0.045 0.04 0.03 0.035 Energy of the excited bound state (au)
Log10 of the Franck–Condon factor
Log10 of the Franck–Condon factor
0
KRb, bound–bound
0.035 0.03 0.045 0.04 Energy of the excited bound state (au)
FIGURE 8.6 Franck–Condon (FC) factors for transitions in Rb2 and KRb. (a) FC factors 1 + 3 for the X 1 Σ+ g − A Σu − b Πu continuum-bound transitions in Rb2 vs. the energy of the + 1 3 A Σu − b Πu bound state. (b) FC factors for the transitions between the continuum states of the X 1 Σ+ (open squares) and a3 Σ+ potentials (filled squares), vs. the energy of the A1 Σ+ − b3 Πu bound state of KRb. (c) FC factors for transitions between the X 1 Σ+ g (v = 4) bound 3 Π bound states vs. the energy of the A1 Σ+ − b3 Π bound state − b state and various A1 Σ+ u u u u in Rb2 . (d) FC factors for transitions between the X 1 Σ+ (v = 13) bound state and various A1 Σ+ − b3 Πu bound states vs. the energy of the A1 Σ+ − b3 Πu bound state in KRb.
The bound-state energies, wavefunctions, and the bound–bound Franck–Condon factors were found using the finite-difference algorithm FDEXTR [88]. The obtained energies of the A1 Σ+ -b3 Πu vibrational states were used as an input in the artificial channel calculation [89] of the complex FC factors for transitions from the X 1 Σ+ and a3 Σ+ continua to the bound states of the coupled A1 Σ+ and b3 Πu potentials. The results of these calculations are shown in Figure 8.6. The continuum-bound FC factors are taken for the scattering at 100 μK energy. Note that the continuumbound FC factor T (E) can indeed exceed one, as long as the normalization condition, |T (E)|2 dE = 1, is fulfilled. The s-wave scattering of two 85 Rb atoms on the X 1 Σ+ g potential at the energies of interest is influenced by a resonance enhancing the scattering length to above 2400 au [85,86]. The resonance arises from the last (quasi-) bound state lying very close to the continuum threshold [90]. Hence, the amplitude of the continuum wavefunction in the inner region is enhanced relative to the nonresonant case. This enhances the Franck– 1 + 3 Condon factors for the X 1 Σ+ g − A Σu − b Πu continuum-bound transitions in Rb2 . The required laser intensities are therefore lower than those for the nonresonant case. While the continuum-bound FC factors in KRb depend on the collision energy as © 2009 by Taylor and Francis Group, LLC
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T (E) ∼ E 1/4 [91], the behavior of the continuum-bound FC factor in Rb2 is more complicated due to the presence of the resonance [17]. 1 + In the calculations of the Rb2 PA, we estimate the X 1 Σ+ g − A Σu electronicdipole moment to be μ = 3 au. This value corresponds to the 5S1/2 (m = 1/2) − 5P3/2 (m = 3/2) transition in Rb in a circularly polarized field, and is consistent with all the other 5S − 5P matrix elements for the Rb atom in a polarized laser field. The exact values of the atomic reduced dipole matrix elements can be found in Ref. [92]. The electronic-dipole transition matrix elements in KRb are taken as 3.5 ea0 for the X 1 Σ+ ↔ A1 Σ+ transition, and 0.5 ea0 for the a3 Σ+ ↔ 13 Π transition, the order-of-magnitude estimate based on the data from Refs. [93,94].
8.6
CONCLUSIONS
In this chapter, we have addressed a question of some controversy in the past [1,7, 11,12]. Our work elucidates what, exactly, is transferred from the incoming channels during an ARPA process. We have shown that ARPA projects the incoming wavefunction onto a certain multichannel waveform, whose shape is determined by the amplitudes and the phases of the pump and dump laser pulses (Equations 8.40 and 8.41). In this way one can choose the basis of the projective measurement performed by the PA process. We have also discussed the role of temperature of the initial sample in the ARPA process and the extent to which it can limit the process. The above idea has been applied to measuring the temporal structure of a singlechannel incoming wavepacket in the PA of Rb atoms to form Rb2 , and the multichannel structure in the two-channel PA of K + Rb to form KRb. We have demonstrated the possibility of measuring a qubit of information encoded in the wavepacket’s shape, and a qubit of information encoded in the two-channel superposition. In our numerical simulations, the temporal shapes of the driving fields needed to perform PA agreed well with the predictions of our analytical theory. On the other hand, the simple rule (Equation 8.44) for the ratio of the intensities and the phases needed to control two-channel PA often disagrees with the simulations. We attribute this discrepancy to the effects of complex decay, not taken in account in the simple analytical theory leading to Equation 8.44. We have shown that the the multichannel structure of the wavefunction can be measured with the help of a numerical parameter search based on the knowledge of the level couplings and decay rates. Alternatively, one can use intermediate levels coupled to only one (say, a3 Σ+ u ) incoming channel and not to the other(s). The presence of additional bound states not included in the simplified theory does not destroy the effectiveness of the detection scheme. Such robustness was quite unexpected in the KRb simulations because the central wavelength of one of the pump pulses is just 1.2 nm away from an exact resonance with an undesired transition between the states |1 and |3 (see Figure 8.1c). We attribute the high robustness of the scheme to the counterintuitive structure of the pulse sequence: by the time the pump pulse couples the states |3 and |1, state |1 is already mixed with the target state |0 in such a way that the population of |1 remains negligible at all times. © 2009 by Taylor and Francis Group, LLC
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The most significant approximation used in our theory is the the factorization fARPA = ΩP f (0) (t) of Equation 8.41 which is justified by, but is not a necessary part of, the slowly varying continuum approximation (SVCA). If this factorization cannot be assumed, ARPA performs a measurement of the initial state filtered through the coordinate-dependent Franck–Condon window.
ACKNOWLEDGMENTS The authors thank S. Lunell and D. Edvardsson for sharing the output data of their calculation [81], and S. Kotochigova for consultations regarding molecular data. We are also pleased to thank I. Thannopulos, J. Ye and A. Pe’er for many discussions.
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Part III Few- and Many-Body Physics
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Ultracold Feshbach 9 Molecules Francesca Ferlaino, Steven Knoop, and Rudolf Grimm CONTENTS 9.1
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Ultracold Atoms and Quantum Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Basic Physics of a Feshbach Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Binding Energy Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Making and Detecting Feshbach Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Bosons and Fermions: Role of Quantum Statistics . . . . . . . . . . . . . . . . . 9.2.2 Overview of Association Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Purification Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Detection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Internal-State Manipulation Near Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Avoided Level Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Cruising Through the Molecular Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Halo Dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Halo Dimers and Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Collisional Properties and Few-Body Physics . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Efimov Three-Body States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Molecular BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Toward Ground-State Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Stimulated Raman Adiabatic Passage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 STIRAP Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Further Developments and Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1
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INTRODUCTION
The coldest molecules available for experiments are diatomic molecules produced by association techniques in ultracold atomic gases. The basic idea is to bind atoms together when they collide at extremely low kinetic energies. If any release of internal energy is avoided in this process, the molecular gas just inherits the ultralow 319 © 2009 by Taylor and Francis Group, LLC
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temperature of the atomic gas, which can in practice be as low as a few nanokelvin. If moreover a single molecular quantum state is selectively populated, there will be no increase of the system’s entropy. A very efficient method to implement such a scenario in quantum-degenerate gases, Bose–Einstein condensates, and Fermi gases relies on the magnetically controlled association via Feshbach resonances. Such resonances serve as the entrance gate into the molecular world and allow for the conversion of atomic into molecular quantum gases. In this chapter, we give an introduction into experiments with Feshbach molecules and their various applications. Our illustrative examples are mainly based on work performed at Innsbruck University, but the reader will also find references to related work of other laboratories.
9.1.1
ULTRACOLD ATOMS AND QUANTUM GASES
The development of techniques for cooling and trapping of atoms has led to great advances in physics, which have already been recognized by two Nobel prizes. In 1997 the prize was jointly awarded to Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips “for their developments of methods to cool and trap atoms with laser light” [1–3]. In 2001, Eric A. Cornell, Wolfgang Ketterle, and Carl E. Wieman jointly received the Nobel prize “for the achievement of Bose–Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates” [4,5]. Laser cooling techniques allow researchers to prepare atoms in the microkelvin range. The key techniques have been developed since the early 1980s and a detailed account can be found in Refs. [6,7]. In brief, resonance radiation pressure is used to decelerate atoms from a thermal beam to velocities low enough to be captured into a socalled magneto-optical trap or, alternatively, low-velocity atoms can be captured into such a trap directly from a vapor. In the trap the atoms are further cooled by Doppler cooling to temperatures of the order of one millikelvin. Then, for some species, subDoppler cooling methods can be applied, reducing the temperatures deep into the microkelvin range. Typical laser-cooled clouds contain up to about 1010 atoms, and the atomic number densities are of the order of 1011 cm−3 . Such laser-cooled atoms have found numerous applications; a particularly important one is the realization of ultraprecise atomic clocks [8,9]. The quantum-degenerate regime requires atomic de Broglie wavelengths to be similar to or larger than the interparticle spacing. As in laser-cooled clouds the phasespace densities are several orders of magnitude too low to reach this condition, and further cooling needs to be applied. The method of choice is evaporative cooling [10], which can be applied in conservative traps like magnetic traps. Evaporative cooling relies on the selective removal of the most energetic atoms in combination with thermal equilibration of the sample by elastic collisions. The process can be forced by continuously lowering the trap depth. By trading one order of magnitude in the particle number, one can gain typically up to three orders of magnitude in phasespace densities, and eventually reach the quantum-degenerate regime. The typical conditions are then a trapped gas of roughly a million atoms with densities of the order of 1014 cm−3 and temperatures in the nanokelvin range. © 2009 by Taylor and Francis Group, LLC
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Quantum degeneracy has been achieved with bosonic and fermionic atoms. The attainment of Bose–Einstein condensation (BEC) in dilute ultracold gases marked the starting point of a new era in physics [11–13], and degenerate Fermi gases entered the stage a few years later [14–16]. The reader may be referred to the proceedings of the Varenna summer schools in 1998 and 2006 [17,18], which describe these exciting developments. For reviews on the theory of degenerate quantum gases of bosons and fermions see Refs. [19,20], respectively, and the textbooks of Refs. [21,22]. The trapping environment plays an important role for the applications of ultracold matter, including the present one to create molecules. Most of the experiments on molecule formation are performed with optical dipole traps [23] for two main reasons. Such optical traps can confine atoms in any Zeeman or hyperfine substate of the electronic ground state; this includes the particularly interesting lowest internal state, which cannot be trapped magnetically. Moreover, optical dipole traps leave full freedom to apply any external magnetic fields, whereas in magnetic traps the application of additional magnetic fields would substantially affect the trapping geometry. A special optical trapping environment can be generated in optical lattices [24,25]. Such lattices are created in optical standing-wave patterns, and in the three-dimensional case they provide an array of wavelength-sized microtraps, each of which may contain a single molecule. The new field of ultracold molecular quantum gases rapidly emerged in 2002/2003, when several groups reported on the formation of Feshbach molecules in BECs of 85 Rb [26], 133 Cs [27], 87 Rb [28], 23 Na [29], as well as in degenerate or near-degenerate Fermi gases of 40 K [30] and 6 Li [31–33]. Signatures of molecules in a BEC had been seen before using a photoassociative technique [34]. In the fall of 2003, the fast progress culminated in the creation of molecular Bose–Einstein condensates (mBEC) in atomic Fermi gases [35–37]. Heteronuclear Feshbach molecules entered the stage a few years later. So far, such molecules have been produced in Bose–Fermi mixtures of 87 Rb–40 K [38], and in Bose–Bose mixtures of 85 Rb–87 Rb [39] and 41 K–87 Rb [40]. The field of ultracold Feshbach molecules is developing rapidly, and there is great interest in homo- and heteronuclear molecules both in weakly bound and deeply bound regimes.
9.1.2
BASIC PHYSICS OF A FESHBACH RESONANCE
Feshbach resonances represent the essential tool to control the interaction between the atoms in ultracold gases, which has been key to many breakthroughs. Here we briefly outline the basic physics of a Feshbach resonance and its connection to the underlying near-threshold molecular structure. For more detailed discussions of the theoretical background see Chapters 6 and 11. The reader is also referred to two recent review articles [41,42]. In a simple picture, we consider two molecular potential curves Vbg (R) and Vc (R), as illustrated in Figure 9.1. For large internuclear distances R, the background potential Vbg (R) asymptotically correlates with two free atoms in an ultracold gas. For a collision process with a very small energy E, this potential represents the energetically open channel, usually referred to as the entrance channel. The other potential, Vc (R), representing the closed channel, is important as it can support bound molecular states near the threshold of the open channel. © 2009 by Taylor and Francis Group, LLC
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Closed channel EC
E
Energy
0 Entrance channel or open channel
Vc(R)
Vbg(R)
0
Atomic separation R
FIGURE 9.1 Basic two-channel model for a Feshbach resonance. The phenomenon occurs when two atoms colliding at energy E in the entrance channel resonantly couple to a molecular bound state with energy Ec supported by the closed channel potential. In the ultracold domain, collisions take place near zero energy, E → 0. Resonant coupling is then conveniently realized by magnetically tuning Ec near 0 if the magnetic moments of the closed and open channel differ.
A Feshbach resonance occurs when the bound molecular state in the closed channel is energetically close to the scattering state in the open channel, and some coupling leads to a mixing between the two channels. The energy difference can be controlled via a magnetic field when the corresponding magnetic moments are different. This leads to a magnetically tunable Feshbach resonance, which is the common way to achieve resonant coupling in the experiments with ultracold gases where the collision energy is practically zero. A magnetically tuned Feshbach resonance can be described by a simple expression for the s-wave scattering length a as a function of the magnetic field B, a(B) = abg 1 −
Δ B − B0
.
(9.1)
Figure 9.2a illustrates this resonance expression. The background scattering length abg , which is the scattering length of Vbg (R), represents the off-resonant value. It is directly related to the energy of the last bound vibrational level of Vbg (R). The parameter B0 denotes the resonance position, where the scattering length diverges (a → ±∞), and the parameter Δ is the resonance width. The energy of the weakly bound molecular state near the resonance position B0 is shown in Figure 9.2b, relative to the threshold of two free atoms with zero kinetic energy. The energy approaches the threshold on the side of the resonance where a is large and positive. Away from the resonance, the energy varies linearly with B with a slope given by δμ, the difference in magnetic moments of the open and © 2009 by Taylor and Francis Group, LLC
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4
(a)
a/abg
2 D
0 –2 –4 0.0 (b)
E/(dm D)
Eb
0.00
–0.01
–0.5 –2
–1
0 (B–B0)/D
–0.1
1
0.0
2
FIGURE 9.2 Scattering length a (a) and molecular state energy E (b) near a magnetically tuned Feshbach resonance. The inset shows the universal regime near the point of resonance where a is very large and positive.
closed channels. Near the resonance the coupling between the two channels mixes in entrance-channel contributions and strongly bends the molecular state. In the vicinity of the resonance position at B0 , where the two channels are strongly coupled, the scattering length is very large. For large positive a, a “dressed” molecular state exists with a binding energy given by Eb =
2 , 2mr a2
(9.2)
where mr is the reduced mass of the atom pair. In this limit, Eb depends quadratically on the magnetic detuning B − B0 and results in the bend depicted in the inset of Figure 9.2. This region is of particular interest because of its universal properties; see discussions in Chapters 6 and 11. Molecules formed in this regime are extremely weakly bound, and they are described by a wavefunction that extends far out of the classically allowed range. Such exotic molecules are therefore commonly referred to as halo dimers; we shall discuss their intriguing physics in Section 9.4. These considerations show how a Feshbach resonance is inherently connected with a weakly bound molecular state. The key question for experimental applications is how to prepare the molecular state in a controlled way; we shall address this in Section 9.2.
9.1.3
BINDING ENERGY REGIMES
The name Feshbach molecule emphasizes the production method, as it commonly refers to diatomic molecules made in ultracold atomic gases via Feshbach resonances. © 2009 by Taylor and Francis Group, LLC
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But what are the physical properties associated with a Feshbach molecule, and what is the exact definition of such a molecule? It is quite obvious that a Feshbach molecule is a highly excited molecule, existing near the dissociation threshold and having an extremely small binding energy as compared to the one of the vibrational ground state. To give an exact definition, however, is hardly possible as the properties of molecules gradually change with increasing binding energy, and there is no distinct physical property associated with a Feshbach molecule. Molecules created via Feshbach resonances can be transferred to many other states near threshold (Section 9.3) or to much more deeply bound states (Section 9.5), thus being converted to more conventional molecules. There is no really meaningful definition of when a molecule can still be called a Feshbach molecule, or when it loses this character. We therefore use a loose definition and call a Feshbach molecule any molecule that exists near the threshold in the energy range set by roughly the quantum of vibrational energy. Any other molecule we consider a deeply bound molecule, although views may differ on this. Figure 9.3 illustrates the vast range of molecular binding energies for the example of Cs2 dimers. According to our definition Feshbach molecules may have binding energies of up to roughly h × 100 MHz for the heavy Cs2 dimer and up to h × 2.5 GHz for the light Li2 dimer. These values are very small compared to the binding energies of ground-state molecules, which are of the order of h × 10 THz or h × 100 THz for the triplet and singlet potential, respectively. Nevertheless, ultracold Feshbach molecules can serve as the starting point for state transfer to produce very deeply bound, and even ground-state molecules, as we shall discuss in Section 9.5. 1 kHz Halo dimers
Binding energy/h
1 MHz
Feshbach molecules Evdw Vibrational quantum
1 GHz
Deeply bound molecules 1 THz Triplet Singlet
Ground-state molecules
1 PHz
FIGURE 9.3 Binding energy regimes for the example of Cs2 dimers, illustrated on a logarithmic scale. For the long-range van der Waals attraction between the atoms, a characteristic energy Evdw is introduced; see Chapter 6 and Section 9.4. The vibrational quantum, here defined as the energy range below threshold in which one finds at least one bound state, corresponds to about 40 Evdw ; see Figure 6.4 in Chapter 6. The binding energies of the vibrational ground state of the triplet and the single potential are typically five or six orders of magnitudes larger than this vibrational quantum.
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Two particular regimes of Feshbach molecules can be realized close to the dissociation threshold. The halo regime requires binding energies well below the so-called van der Waals energy Evdw (Section 6.2 in Chapter 6), which corresponds to about 3 MHz for Cs2 and about 600 MHz for Li2 . Molecules in high partial-wave states (Section 9.3) can exist above the dissociation threshold, as the large centrifugal barrier prevents them from decaying; such metastable Feshbach molecules have a negative binding energy.
9.2
MAKING AND DETECTING FESHBACH MOLECULES
This section discusses how to create ultracold molecules, starting from a degenerate or near-degenerate atomic gas. The three main experimental steps are illustrated in Figure 9.4. In a first step (a) atoms are converted into molecules, which can be done by sweeping the magnetic field across a Feshbach resonance. In a second step (b) a purification scheme can be applied that removes all the remaining atoms. In a third step (c) the molecules are forced to dissociate in a reverse sweep across the resonance in order to detect the resulting atoms. As an example, Figure 9.5 shows an image taken after the Stern–Gerlach separation of an atom–dimer mixture released from a trap; the smaller cloud represents a pure sample of Feshbach molecules. Similar techniques can also be applied to trapped atoms. In the following we describe the three steps of production, purification, and dissociation after a discussion of the important role of quantum statistics.
9.2.1
BOSONS AND FERMIONS: ROLE OF QUANTUM STATISTICS
Ultracold diatomic molecules can be composed either of two bosons, two fermions, or a boson and a fermion. In the first two cases, the molecules have bosonic character, while in the third case the molecules are fermions. Bosons are described by (b) Purification
(c) Dissociation
Energy
(a) Production
Magnetic field
FIGURE 9.4 Illustration of a typical experimental sequence to create, purify, and detect a sample of Feshbach molecules: (a) association via a magnetic-field sweep across the resonance, (b) selective removal of the remaining atoms, and (c) forced dissociation through a reverse magnetic field sweep. The dissociation is usually followed by imaging of the resulting cloud of atoms. The solid line corresponds to the bound molecular state, which intersects the threshold (gray horizontal line) and causes the Feshbach resonance; see also Figure 9.2. The dashed line indicates the molecular state above threshold, where it has the character of a quasi-bound state coupling to the scattering continuum.
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FIGURE 9.5 Example for the preparation of a pure molecular cloud. Here an optically trapped BEC of 87 Rb atoms was subjected to a magnetic-field sweep across a Feshbach resonance. The Stern–Gerlach technique was then applied after release from the trap to spatially separate the atoms (left) from the molecules (right). Finally the molecules were dissociated by a reverse Feshbach ramp, and an absorption image was taken. The field of view is 1.7 × 0.7 mm. (Courtesy of S. Dürr and G. Rempe.)
wavefunctions, which are symmetric with respect to exchange of identical particles, while, in the fermionic case, the wavefunctions are antisymmetric. This fundamental difference has important consequences. Identical bosons (fermions) can collide in even (odd) partial waves. The role of partial waves is discussed in Chapter 1. Here we use as the corresponding angular momentum quantum number. The first partial-wave contribution for bosons (fermions) is then the s-wave with = 0 (p-wave with = 1) [43]. At ultralow temperatures, only s-wave collisions are dominant with the consequence that collisions between identical fermions are suppressed. The absence of s-wave collisions is also the reason why, for instance, direct evaporative cooling cannot be applied to a sample of identical fermions. The situation changes when fermionic atoms are in different internal states, like hyperfine or Zeeman sublevels. The distinguishable particles can then interact in any partial wave. Fermion-composed molecules in s-wave states are therefore generally associated from ultracold two-component spin mixtures. Interactions in all partial waves are obviously allowed if two different atomic species are involved, regardless of their fermionic or bosonic character. The quantum-statistical character of a molecule can crucially influence its collisional stability. Feshbach molecules in highly excited vibrational states are in principle very sensitive to vibrational relaxation induced by atom–molecule and molecule– molecule collisions. If such an inelastic relaxation process occurs, the energy release is very large as compared to the trap depth and will result in an immediate loss of all collision partners from the trap. A remarkable exception to this general behavior are Feshbach molecules composed of fermionic atoms in a halo state, as they exhibit an extremely high stability against inelastic decay. The reason of this different behavior is a Pauli suppression effect. In a two-component Fermi mixture, relaxation processes are unlikely because they necessarily involve at least two identical fermions; for more details see Chapter 10 and Ref. [44]. An intermediate case can be found for molecules composed in a Bose–Fermi mixture of two atomic species. Here, the collisional stability depends on the atomic partner involved in a collision, either a boson © 2009 by Taylor and Francis Group, LLC
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or a fermion. Only in collisions with fermionic atoms can a similar Pauli suppression effect be expected. The different degree of stability of bosons and fermions leads to a surprising result. One might think that the best way to achieve mBEC is the direct conversion of an atomic BEC. However, the stability of fermions turned out to be a key ingredient to experimentally achieve the condensation of molecules; see Section 9.4.4. For attaining a collisionally stable mBEC of boson-composed molecules, their transfer into the rovibrational ground state (Section 9.5) may be the only feasible way.
9.2.2
OVERVIEW OF ASSOCIATION METHODS
The optimum association strategy depends on several factors connected to the specific system under investigation. The main factors to consider are the quantum-statistical nature of the atoms that form the molecule and the particular Feshbach resonance employed. Ultralow temperatures and high phase-space densities are essential requirements for an efficient molecule creation. Therefore all association techniques start with an ultracold trapped atomic sample. The most widespread technique for molecule association uses time-varying magnetic fields. The magnetic field modifies the energy difference between the scattering state and the molecular state. When the two states have the same energy, the scattering length diverges and a molecular state can be accessed in the region of positive scattering length (a > 0); see Section 9.1.2. Magnetic field time sequences include linear ramps, fast jumps, or oscillating magnetic fields. Figure 9.4a illustrates a magnetic-field ramp, commonly referred to as “Feshbach ramp” [45–47]. A homogeneous magnetic field is swept across a Feshbach resonance from the side where a < 0 to the side where a > 0. The coupling between atom pairs and molecules removes the degeneracy at the crossing of the two corresponding energy levels. The resulting avoided level crossing can be used to adiabatically convert atom pairs into molecules by sweeping the magnetic field across the resonance, as indicated by the arrow in Figure 9.4a. The atomic sample is partially converted into a molecular gas. In the fermionic case, the conversion efficiency does solely depend on the ramp speed and on the atomic phase-space density [48], and can reach values close to one. In the case of bosons, the conversion efficiency is affected by inelastic collision processes. Near the center of a Feshbach resonance, the atoms experience a huge increase of the three-body recombination rate, while, after the association, the atom–dimer mixture can undergo fast collisional relaxation. The best parameters for molecule association require a compromise for the optimum ramp speed, which should be slow enough for an efficient conversion, but fast enough to avoid detrimental losses. Inelastic collisional loss can be reduced by using more elaborate time sequences for the magnetic field or, alternatively, resonant radio-frequency techniques. The key idea is to introduce an efficient atom–molecule coupling while minimizing the time spent near the resonance, where strong loss and heating results from inelastic interactions. An efficient technique is to apply a small sinusoidal modulation to the homogeneous field. The oscillating field then induces a corresponding modulation of the energy difference between the scattering state and the molecular state. The molecules © 2009 by Taylor and Francis Group, LLC
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are associated when the modulation frequencies match the molecular binding energies [49]. Typical modulation frequencies range from a few kHz to a few 100 kHz. The signature of molecule production is a resonant decrease of the atom number, as observed in gases of 85 Rb [50] and 40 K [51], and in a mixture of the two isotopes 85 Rb and 87 Rb [39]. A resonant coupling between atom pairs and molecules can also be induced by a radio-frequency excitation, which stimulates a transition between the atom pair state and the molecular state without modulating the energy difference. Heteronuclear 40 K87 Rb molecules were produced with this technique, both in an optical lattice [38] and in an optical dipole trap [52]. Collisional losses can also be suppressed by using a suitable trapping environment. This is the case when atoms are prepared in an optical lattice with double site occupancy. A stable system with a single molecule in each lattice site can then be created using, for instance, a Feshbach ramp [53,54]. A completely different association strategy can be applied to create molecules in an atomic Fermi gas of 6 Li atoms. This special situation is particularly important as it has opened up a simple and efficient route toward mBEC [36,37]; see also Section 9.4.4. Near a Feshbach resonance, in the regime of very large positive scattering lengths, atomic three-body recombination efficiently produces halo dimers. In a three-body recombination event in a two-component atomic Fermi gas, two nonidentical fermions bind into a molecule while the third atom carries away the leftover energy and momentum. Usually, the energy released in three-body collisions is very large so that all collision partners immediately leave the trap. However, for the special case of halo dimers, the very small binding energies can be on the order of the
250
Atom number (104)
200
150 Nat 100 50 2Nmol 0
0
1
2 Time (sec)
3
FIGURE 9.6 Formation of 6 Li2 halo dimers via three-body recombination. The molecules are created in an optically trapped spin mixture of atomic 6 Li at a temperature of 3 μK. The experiment is performed near a broad Feshbach resonance, where a = +1420a0 and Eb = kB × 15 μK = h × 310 kHz. Nat and Nmol denote the number of unbound atoms and the number of molecules, respectively. (Adapted from Jochim, S. et al., Phys. Rev. Lett., 91, 240402, 2003.)
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typical thermal energies of the trapped particles and thus less than the trap depth [18]. The formation of halo dimers via three-body recombination then proceeds with small heating and negligible trap losses. This way of forming molecules can also be interpreted in terms of a thermal atom–molecule equilibrium [55,56]. Typical timescales for the atom–molecule thermalization are on the order of 100 msec to 1 sec, requiring the high collisional stability of fermion-composed molecules in the halo regime. As an example, Figure 9.6 shows how an initially pure sample of 6 Li atoms changes into an atom–molecule mixture [32].
9.2.3
PURIFICATION SCHEMES
Purification is an important step in many experiments with Feshbach molecules. The removal of atoms can be required so as to avoid fast collisional losses of molecules. In other cases, pure molecular samples are needed for particular applications. Most of the purification schemes act selectively on atoms and they are applied on timescales short compared to the typical molecular lifetimes. A possible purification strategy exploits the difference in magnetic moments of molecules and atoms. The two components of the gas can be spatially separated using the Stern–Gerlach technique, which uses magnetic field gradients [27,28]; for an example see Figure 9.5. This method is usually applied to gases in free space after release from the trap. A fast and efficient purification method, well suited for application to a trapped cloud, uses resonant light to push the atoms out of the sample by resonant radiation pressure [29,53,57]. The “blast” light needs to be resonant with a cycling atomic transition, which allows for the repeated scattering of photons. In many cases, the atoms are not in a ground-state sublevel connected to a cycling optical transition, so an intermediate step is necessary. In the most simple implementation, this is an optical pumping step that selectively acts on the atoms without affecting the molecules. Alternatively a microwave pulse can be used to transfer the atoms into the appropriate atomic level. The high selectivity of this double-resonance method minimizes residual heating and losses of molecules from the trap.
9.2.4
DETECTION METHODS
A very efficient way to detect Feshbach molecules is to convert them back into atom pairs and to take an image of the reconverted atoms with standard absorption imaging techniques [27,28]. In principle, each of the above-described association methods can be reverted and turned into a corresponding dissociation scheme. A robust and commonly used dissociation scheme is to simply reverse the Feshbach ramp, as illustrated in Figure 9.4c. The reverted ramp is usually chosen to be much faster than the one employed for association. The molecules are then brought into a quasi-bound state above the atomic threshold. Here they quickly dissociate into atom pairs, converting the dissociation energy into kinetic energy. The back-ramp is usually done in free space and can be performed either immediately after release from the trap or after some expansion time (≈10–30 msec). The choice of the delay time between the dissociation and the detection sets the type of information that can be © 2009 by Taylor and Francis Group, LLC
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extracted from the absorption image. For a long delay time, the sample expands thus mapping the information on the momentum distribution onto the spatial distribution. The image then contains quantitative information on the dissociation energy [59]. For a short delay time, the absorption image simply reflects the spatial distribution of the molecules before the dissociation. Fast and efficient dissociation requires a sufficiently strong coupling of the molecular states to the scattering continuum. Molecular states with high rotational quantum number typically have weak couplings and high centrifugal barriers. A striking example is provided by Cs2 Feshbach molecules with = 8. (Partial waves associated with even angular momentum quantum numbers = 0, 2, 4, 6, 8 are by convention denoted as s, d, g, i, and l-waves, respectively. In the case of odd angular momentum quantum numbers = 1, 3, 5, 7 the partial waves are called p, f , h, and k-waves, respectively [43].) These molecules do not sufficiently couple to the atomic continuum and dissociation is prevented. There are two ways to detect them. The first is based on a time reversal of the association path (Section 9.3.2) [60]. The second exploits the crossing and the mixing with a quasi-bound state above threshold with ≤ 4 to force the dissociation [61]. Dissociation patterns can also provide additional spectroscopic information. An example is shown in Figure 9.7. Here the dissociation patterns of 87 Rb2 show strong modifications by the presence of a d-wave shape resonance [58]. In the halo regime, molecules can be detected by direct imaging [37,62]. The imaging frequency for halo molecules is very close to the atomic frequency. Presumably, the first photon dissociates the molecule into two atoms, which then absorb the subsequent photons. In the case of heteronuclear molecules, directed imaging is possible in a wider range of binding energies. This happens because the excited and ground molecular potentials both vary as 1/R6 ; see Chapter 5. In the homonuclear case, the excited potential varies as 1/R3 [63].
(a) 0.1 G
(b) 0.7 G
(c) 1.4 G
FIGURE 9.7 Dissociation patterns of 87 Rb2 molecules near a d-wave shape resonance. The molecules can dissociate either directly into the continuum or indirectly by passing through a d-wave shape-resonance state located behind the centrifugal barrier. Initially, the direct dissociation process dominates and preferentially populates isotropic s-wave states (a). Approaching the shape resonance, the pattern reveals first an interference between s- and d-partial waves (b), and then shows a pure outgoing d-wave (c). The given magnetic field values are relative to the field where dissociation is observed to set in. (Adapted from Volz, T. et al., Phys. Rev. A, 72, 010704(R), 2005.)
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9.3
INTERNAL-STATE MANIPULATION NEAR THRESHOLD
Below the atomic threshold a manifold of molecular states exists. They correspond to different vibrational, rotational and magnetic quantum numbers, and belong to potentials correlated with different hyperfine states. Having different magnetic moments, many levels cross when the magnetic field is varied. When some coupling between two molecular states that cross is present an avoided crossing occurs. The concept of avoided level crossings is very general and has found important applications in many fields of physics. We have already discussed the application to associate molecules in Section 9.2. In the field of ultracold molecules, the information on the avoided crossings is fundamentally important to fully describe the molecular spectrum and to understand the coupling mechanisms between different states. At the same time, controlled state transfer near avoided crossings opens up the intriguing possibility of populating many different molecular states that are not directly accessible by Feshbach association.
9.3.1 AVOIDED LEVEL CROSSINGS If an avoided crossing is well separated from any other avoided crossing, including the one resulting near threshold from the coupling to the scattering continuum, a simple two-state model can be applied. Consider two molecular states (i = 1, 2), for which the magnetic field dependencies of the binding energies Ei of the diabatic states ϕi are given by Ei (B) = μi (B − Bc ) + Ec ,
(9.3)
where μi are the magnetic moments, and Bc and Ec , the magnetic field and energy at which the two states cross, respectively (Figure 9.8). The crossing is conveniently
E2
Energy
E+
Ec V E1
E–
Bc Magnetic field
FIGURE 9.8 Schematic representation of an avoided crossing between two molecular states. The dashed lines represent the diabatic states with energies E1 and E2 , with a crossing at Bc . The solid lines represent the adiabatic states with energies E+ and E− , caused by a coupling V between the two diabatic states.
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described in terms of adiabatic states if the two states are coupled by the interaction Hamiltonian H, ϕ1 |H|ϕ2 = 0. To obtain the corresponding adiabatic energies one has to solve the following eigenvalue problem, E1 V ϕ1 ϕ1 =E , ϕ2 ϕ2 V E2 where V = ϕ1 |H|ϕ2 is the coupling strength between the diabatic states. The adiabatic energies are given by 2E± = (E1 + E2 ) ±
(E1 − E2 )2 + 4V 2 ,
(9.4)
where the energies E+ and E− correspond to the upper and lower adiabatic levels of the avoided crossing. The energy difference between the adiabatic levels is ΔE = |E+ − E− | =
(μ1 − μ2 )2 (B − Bc )2 + 4V 2 ,
(9.5)
which at the crossing is twice the coupling strength, that is, ΔE(Bc ) = 2V . Let us illustrate the occurrence of avoided level crossings by the example of Cs2 , for which the near-threshold spectrum is shown in Figure 9.9. The molecular spectrum of Cs2 provides many avoided crossings, basically for two reasons. First, there exists a high density of molecular levels because its large mass gives rise to a small vibrational and rotation spacing, and its large nuclear spin gives rise to many hyperfine substates. Second, relatively strong relativistic spin–spin dipole and second-order spin–orbit interactions couple many of these states. Most crossings in Figure 9.9 are indeed avoided crossings and the inset shows one in an expanded view as an example. As they are well separated from each other, the simple two-state model applies very well. The crossing position Bc and the coupling strength V can be measured by different methods. Magnetic moment spectroscopy relies on measuring the effect of a magnetic field gradient after the molecular cloud is released from the trap. The difference in the position of a molecular cloud after a fixed time of flight with and without a magnetic field gradient is directly proportional to the magnetic moment. In the avoided crossing region the magnetic moment of the adiabatic molecular states shows a rapid change with magnetic field. This technique has been applied to map out several avoided crossings of the Cs2 spectrum, from which one is shown in Figure 9.10a. Another method relies on direct probing of ΔE by radio-frequency excitation between the adiabatic states, as demonstrated for 87 Rb2 [64]. More sophisticated methods rely on Ramsey-type or Stückelberg interferometry, which have been applied to avoided crossings in 87 Rb2 [64] and Cs2 [60], respectively. In the Ramsey scheme a radio-frequency pulse creates a coherent superposition. After a hold time and a second pulse, the population in one of the molecular branches is measured. This population shows an oscillation as a function of the hold time with a frequency of ΔE/h. In the Stückelberg scheme a coherent superposition is obtained by a magnetic field sweep over the avoided crossing; see Section 9.3.2. After a hold time, a reverse magnetic field sweep is applied and the population in both molecular branches is measured. Here the oscillation frequency in the population is also equal © 2009 by Taylor and Francis Group, LLC
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) 4g(4
4 d(4)
6s(6)
2g (2)
6 g (6) 5
6l( 5)
6l(4 )
4
6l(3)
Binding energy Eb/h (MHz)
4g(3)
0
5
6g(6) 4g(4)
6 –12
10 10
20
30 Magnetic field (G)
40
–14
–16 50
FIGURE 9.9 Molecular energy structure of Cs2 below the threshold of two free Cs atoms in the absolute hyperfine ground-state sublevel. The molecular states near threshold are well characterized by quantum numbers | f , mf ; , m [41], where f represents the sum of the total atomic spins F1,2 of the individual atoms, and is the rotational quantum number. The respective projection quantum numbers are given by mf and m . The interaction Hamiltonian conserves f + at zero magnetic field, and always conserves mf + m . In this example, Feshbach association is performed with pair of atoms that has mf + m = 6, and only molecular states with mf + m = 6 have to be considered. Therefore the labeling f (mf ) is sufficient to characterize the molecular states. Due to relatively strong relativistic spin-spin dipole and second-order spinorbit interactions, molecular states with rotational quantum number up to l-wave ( = 8) have to be taken into account in the case of Cs. The inset shows one of the narrow avoided crossing between two molecular states. (Adapted from Mark, M. et al., Phys. Rev. A, 76, 042514, 2007.)
to ΔE/h. In Figure 9.10b the mapping of a very narrow avoided crossing with the Stückelberg scheme is shown.
9.3.2
CRUISING THROUGH THE MOLECULAR SPECTRUM
A magnetic field ramp over the avoided crossing can be used for state transfer. When the magnetic field is ramped slowly, the population follows the adiabatic states. This is called an adiabatic ramp. If the change in magnetic field is very fast, the molecules do not experience the coupling between the diabatic states, and therefore remain in their initial state. This is a nonadiabatic ramp. The well-known Landau–Zener model describes the final population in both molecular states after the ramp. The probability of adiabatic transfer is given by ˙ , p = 1 − exp −B˙c /|B| © 2009 by Taylor and Francis Group, LLC
(9.6)
334 (a)
Mol. fraction of 6g(6) state
Cold Molecules: Theory, Experiment, Applications (b) 300
6g(6)
1.5
m(mB)
DE/h (kHz)
250
1 0.5 0
50 75 100 125 150 175 Hold time (msec)
200 150 100
1.0 50
4g(4) 12
13 14 Magnetic field (G)
15
0
11.1
11.2 11.3 11.4 11.5 Magnetic field (G)
11.6
FIGURE 9.10 Examples of two different techniques to map avoided crossings, both applied to Cs2 . (a) Magnetic moment spectroscopy on the two molecular states around the 6g(6)/4g(4) avoided crossing, measuring the magnetic moment as function of the magnetic field. A fit to the data gives Bc = 13.29(4) G and V = h × 164(30) kHz. (From Mark, M. et al., Phys. Rev. A, 76, 042514, 2007. With permission.) (b) Stückelberg interferometry on the two molecular states around the 6g(6)/6l(3) avoided crossing. The solid curve is a fit according to Equation 9.5, resulting in Bc = 11.339(1) G and V = h × 14(1) kHz. Inset: raw data showing an oscillation of the population in state 6g(6) right in the center of the crossing. (From Mark, M. et al., Phys. Rev. Lett., 99, 113201, 2007. With permission.)
where B˙ is the linear ramp speed and B˙c = 2πV 2 /(|μ1 − μ2 |). An adiabatic ramp ˙ B˙c , while a nonadiabatic ramp requires corresponds to a ramp speed of |B| ˙ B˙c . |B| Manipulation of the population over avoided crossings is fully coherent as there is no dissipation. Intermediate ramp speeds result in a coherent splitting of the population over the two molecular states. This property was demonstrated by a Stückelberg interferometer, in which two subsequent passages through an avoided crossing in combination with a variable hold time led to high-contrast population oscillations, as shown in the inset of Figure 9.10b. The required ramp speed for equal splitting is on the order of 1 G/μsec for an avoided crossing with V ∼ h × 100 kHz. The magnetic field ramps over avoided crossings allow to populate states that are not directly accessible by Feshbach association. This includes states that do not cross the atomic threshold, for example the 6g(6) state in Figure 9.9. There are also states that do cross the atomic threshold, but do not induce Feshbach resonances as the coupling with the atomic continuum is too weak. This occurs for molecular states with high rotational quantum number. For Cs2 molecular states with > 4, for example, the l-wave states shown in Figure 9.9, no Feshbach resonances exist. In Refs. [57] and [61] the population of the l-wave states using avoided crossings was demonstrated. As an example, the population scheme for one of the l-wave states is shown in Figure 9.11. The ability to populate molecular states that do not induce Feshbach resonances raises the question on the lifetime of these states above the atomic threshold, as the vanishing coupling with the atomic continuum implies that dissociation is suppressed. By studying the lifetime of the molecular sample above threshold it was found that l-wave molecules are indeed stable against dissociation on a timescale of 1 sec [61]. The © 2009 by Taylor and Francis Group, LLC
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4) 6l(
2
4g (4)
Eb/h (MHz)
0
4 6g(6) 6
15
20 25 Magnetic field (G)
30
FIGURE 9.11 Population scheme for the Cs2 6l(4) state. Starting with Feshbach association on the 19.8 G g-wave Feshbach resonance, the resulting 4g(4) Feshbach molecules are brought to larger binding energies by ramping down the magnetic field. At a binding energy of about h × 5 MHz, the avoided crossing with the 6g(6) state is followed adiabatically. Then a fast ramp in the opposite direction is made, jumping the avoided crossing, after which a slow magnetic field ramp toward higher magnetic fields leads to adiabatic transfer from the 6g(6) to the 6l(4) state. The molecular sample in the 6l(4) state can be brought above the atomic threshold, as the large centrifugal barrier prohibits dissociation. (From Knoop, S. et al., Phys. Rev. Lett., 100, 083002, 2008. With permission.)
preparation of long-lived Feshbach molecules above the atomic threshold opens up the possibility to create novel metastable quantum states with strong pair correlations. The method of jumping avoided crossings is in practice limited to crossings with energy splittings up to typically h × 200 kHz. For stronger crossings, fast ramps are technically impracticable and the crossings are followed adiabatically when the magnetic field is ramped, leaving no choice for the state transfer. This problem can be overcome with the help of radio-frequency excitation [64]. For 87 Rb2 the transfer of molecules over nine level crossings was demonstrated when the magnetic field was ramped down from a Feshbach resonance near 1 kG to zero. This efficiently produced molecules with a binding energy Eb = h × 3.6 GHz at zero magnetic field. The combination of these elaborate ramp and radio-frequency techniques allows for cruising through the complex molecular energy structure. The level spectrum serves as a molecular “street map” and by passing straight through crossings or performing left or right turns one can reach any desired destination.
9.4
HALO DIMERS
Very close to resonance the Feshbach molecules are extremely weakly bound and become halo dimers. This halo regime is interesting because of its universal properties, © 2009 by Taylor and Francis Group, LLC
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regarding its intrinsic behavior (Section 9.4.1) and its few-body interaction properties (Section 9.4.2), in particular in the context of Efimov physics (Section 9.4.3). The properties of fermion-composed halo dimers, extensively discussed in Chapter 10, allow the realization of molecular Bose–Einstein condensation (mBEC), which is the topic of Section 9.4.4.
9.4.1
HALO DIMERS AND UNIVERSALITY
For large scattering lengths a, the Feshbach molecule enters the halo regime, where its binding energy is given by Equation 9.2. Its magnetic field dependence is illustrated in the inset of Figure 9.2. The halo state can be described in terms of a single effective molecular potential having scattering length a. In Figure 9.12 a typical halo wavefunction is shown as function of the interatomic distance R. This highlights its remarkable property to extend far into the classically forbidden range, which is at the heart of its universal properties. The asymptotic form of the halo wavefunction is proportional to e−R/a and the average distance between the two atoms is a/2; see also Section 11.5 of Chapter 11. The halo regime requires a to be much larger than the range of the two-body potential. For ground-state alkali atoms a characteristic range can be described by the van der Waals length Rvdw = 21 (2mr C6 /2 )1/4 , where C6 is the van der Waals dispersion coefficient; see Chapters 6 and 10. The corresponding van der Waals 2 ). Thus a halo dimer can be defined through the criteria energy is Evdw = 2 /(2mr Rvdw a Rvdw and Eb Evdw . For some examples the van der Waals scales are given in Table 9.1, where Rvdw is expressed in units of the Bohr radius a0 = 0.529 × 10−10 m. The halo regime corresponds to the universal regime for a > 0. The concept of universality is that the properties of any physical system in which |a| is much larger than the range of the interaction is determined by a, and therefore all these systems exhibit the same universal behavior [66]. In the universal regime the details of the shortrange interaction become irrelevant because of the dominant long-range nature of the wavefunction. The existence of the halo dimer at large positive a is a manifestation
E/h (MHz)
200 Rf(R) ~ e–R/a
100 0 –100 0
50
100
R (a0)
150
200
250
FIGURE 9.12 Radial wavefunction of a halo dimer, where R is the interatomic distance, extending far into the classically forbidden range where the wavefunction is proportional to e−R/a . The wavefunction is calculated for a Lennard–Jones potential and mr = 3 amu [65]. The potential depth is tuned such that the highest vibrational level, in this case v = 5, is in the halo regime.
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TABLE 9.1 Characteristic Van Der Waals Scales Rvdw and Evdw for Some Selected Examples 6 Li
Rvdw (a0 ) Evdw /h (MHz)
2
31 614
40 K 2
40 K87 Rb
65 21
72 13
85 Rb
2
82 6.2
133 Cs 2
101 2.7
of universality in two-body physics [67]. Halo states are known in nuclear physics, with the deuteron being a prominent example. In molecular physics, the He dimer has for many years served as the prime example of a halo state. Feshbach molecules with halo character are special to other halo states in the sense that their properties are magnetically tunable. In particular, one can scan from the universal regime of halo dimers to the nonuniversal regime of nonhalo Feshbach molecules. In principle, every Feshbach resonance features a halo region. In practice, however, only a few resonances are suited to experimentally realize this interesting regime. Those are broad resonances with large widths |Δ| (typically 1 G) and large background scattering lengths |abg | that exceed Rvdw [41,42]. Prominent examples are found in the two fermionic systems 6 Li and 40 K, and the bosonic systems 39 K, 85 Rb and 133 Cs, some of which are discussed in Section 6.4 of Chapter 6 and Section 11.5 of Chapter 11. Another universal feature of a halo dimer regards spontaneous dissociation, which is possible when its atomic constituents are not in the lowest internal states and open decay channels are present. In the halo regime the dissociation rate scales as a−3 , as has been observed for 85 Rb2 [68], and can be understood as a direct consequence of the large halo wavefunction [69].
9.4.2
COLLISIONAL PROPERTIES AND FEW-BODY PHYSICS
The concept of universality extends beyond two-body physics to few-body phenomena, including the low-energy scattering properties of halo dimers. The inelastic collisional properties are important as inelastic collisions usually lead to trap loss, determining the lifetime of a sample of trapped halo dimers. At the same they provide a convenient experimental observable to study few-body physics. In general, the loss of dimers can be described by the following rate equation: 2 n˙ D = −αnD − βnD nA ,
(9.7)
where α and β are the loss rate coefficients for dimer–dimer and atom–dimer collisions, respectively, and nA (nD ) is the atomic (molecular) density. Fast trap loss has been observed for Feshbach molecules in the nonuniversal regime, with α and β on the order of 10−11 to 10−10 cm3 /sec; see Section 3.3 of Chapter 3. The loss coefficient α is most conveniently measured using a pure molecular sample. To determine β an atom–dimer mixture is required, the atom number greatly exceeding the molecule number being beneficial. © 2009 by Taylor and Francis Group, LLC
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In the universal regime, simple scaling laws for α and β can be derived. For halo dimers composed of two identical bosons, β scales linearly with a [70,71], resulting in an enhancement of dimer loss near a Feshbach resonance. In contrast, for halo dimers made of two fermions in different spin states an a−2.55 and an a−3.33 scaling are derived for α and β [44], respectively, connected to the collisional stability of the molecular sample near a Feshbach resonance, as discussed in Section 9.2.1. Universal scaling laws are also derived for other atom–dimer systems [52,72], as well as for the related problem of three-body recombination in atomic samples [71,72]. Inelastic dimer–dimer scattering represents an elementary four-body process. For four identical bosons a universal prediction has so far not been derived, as the four-body problem is substantially more challenging than the three-body problem. Experimentally one can investigate this process by measuring α in a pure molecular sample of halo dimers made of identical bosons, which has been done in the case of Cs2 [73]. The results are given in Figure 9.13, showing a strong scattering length dependence of α. A pronounced loss minimum around 500a0 is observed, followed by a linear increase toward larger a. The interpretation of this striking behavior is still an open issue and might stimulate progress in the theoretical description of the four-body problem. For the elastic scattering properties, universal scaling laws have been derived for fermion-composed halo dimers. Both the atom–dimer and dimer–dimer scattering lengths simply scale with the atom–atom scattering length a, namely as 1.2 a [74] and 0.6 a [44], respectively. Therefore the elastic cross-sections, being proportional to the square of these scattering lengths, are very large in the halo region. This leads to
10
α (10–11cm3/sec)
8 6 4 2 Halo
Nonhalo 0
–1.0
0.0
a (1000a0)
0.5
1.0
FIGURE 9.13 Experimental study of the interaction between halo dimers. The relaxation rate coefficient α for inelastic dimer–dimer scattering is measured as function of the scattering length a for a pure sample of optically trapped Cs Feshbach molecules. The experiment makes use of the wide tunability of the scattering length for Cs, in particular into the halo regime. The solid line is a linear fit to the data in the region a ≥ 500 a0 . The shaded region indicates the a < Rvdw regime, where the Feshbach molecules are not in a halo state. (From Ferlaino, F. et al., Phys. Rev. Lett., 101, 023201, 2008. With permission.)
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fast thermalization in an atom–dimer mixture and a pure dimer sample, opening the possibility of evaporative cooling. For boson-composed halo dimers the few-body problem is more complicated and universal predictions for their elastic scattering properties are so far lacking.
9.4.3
EFIMOV THREE-BODY STATES
Efimov quantum states in a system of three identical bosons [75,76] are a paradigm for universal few-body physics. These states have attracted considerable interest, fueled by their bizarre and counter-intuitive properties, and by the fact that they had been elusive to experimentalists for more than 35 years. In 2006, experimental evidence for Efimov states in an ultracold gas of Cs atoms was reported [77]. In the context of ultracold quantum gases, Efimov physics manifests itself in three-body decay properties, such as resonances in the three-body recombination and atom–dimer relaxation loss rates. These resonances appear on top of the nonresonant “background” scattering behavior, given by the universal scaling laws discussed in Section 9.4.2. Efimov’s scenario is illustrated in Figure 9.14, showing the energy spectrum of the three-body system as a function of the inverse scattering length 1/a. For a < 0, the natural three-body dissociation threshold is at zero energy. States below are trimer states and states above are continuum states of three free atoms. For a > 0, the dissociation threshold is given by the binding energy of the halo dimer; at this threshold a trimer dissociates into a halo dimer and an atom. Efimov predicted an infinite series of weakly bound trimer states with universal scaling behavior. When the scattering 0
1/a < 0
1/a > 0
A+A+A
A+A+A
×eπ/s0 T
Energy
×e2π/s0
A+D
T
Inverse scattering length 1/a
FIGURE 9.14 Efimov’s scenario. Appearance of an infinite series of weakly bound Efimov trimer states (T) for resonant two-body interaction, showing a logarithmic-periodic pattern with universal scaling factors eπ/s0 and e2π/s0 for the scattering length and the binding energy, respectively. The binding energy is plotted as a function of the inverse two-body scattering length 1/a. The gray regions indicate the scattering continuum for three atoms (A + A + A) and for an atom and a halo dimer (A + D). To illustrate the series of Efimov states, the universal scaling factor is artificially set to 2.
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length is increased by a universal scaling factor eπ/s0 , a new Efimov state appears, which is just larger by this factor and has a weaker binding energy by a factor e2π/s0 . For three identical bosons s0 ≈ 1.00624, so that the scaling factors for the scattering length and the binding energy are 22.7 and 22.72 ≈ 515, respectively. For a < 0, Efimov states are “Borromean” states [67], which means that a weakly bound three-body state exists in the absence of a weakly bound two-body state. This property that three quantum objects stay together without pairwise binding is part of the bizarre nature of Efimov states. The Efimov trimers influence the three-body scattering properties. When an Efimov state intersects the continuum threshold for a < 0 three-body recombination loss is enhanced [79,80], as the resonant coupling of three atoms to an Efimov state opens up fast decay channels into deeply bound dimer states plus a free atom. Such an Efimov resonance has been observed in an ultracold, thermal gas of Cs atoms [77]. For a > 0 a similar phenomenon is predicted, namely an atom–dimer scattering resonance at the location at which an Efimov state intersects the atom–dimer threshold [81,82]. Resonance enhancement of β has been observed in a mixture of Cs atoms and Cs2 halo dimers [78]; see Figure 9.15. The asymmetric shape of the resonance can be explained by the background scattering behavior, which here is a linear increase as a function of a.
40 nK 170 nK
b (cm3/sec)
10–10
10–11 –1.5
–1.0
–0.5 0.0 a (1000a0)
0.5
1.0
FIGURE 9.15 Observation of a scattering resonance in an ultracold, optically trapped mixture of Cs atoms and Cs2 halo dimers. As for Figure 9.13, one benefits from the wide tunability of the scattering length provided by Cs. The data show the loss rate coefficient for atom– dimer relaxation β for temperatures of 40(10) nK and 170(20) nK. A pronounced resonance is observed at about +400 a0 , corresponding to a magnetic field of 25 G. The solid curve is a fit of an analytic model from effective field theory [66] to the data for a > Rvdw . (Adapted from Knoop, S. et al., Nature Phys. (in press, 2009), http://arxiv.org/abs/0807.3306. With permission.) © 2009 by Taylor and Francis Group, LLC
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Efimov physics impacts not only the scattering properties of three identical bosons, but any three-body system in which at least two of the three pairwise two-body interactions are large. Interestingly, depending on the mass ratio of the different components and the particle statistics, the scaling factor eπ/s0 can be much smaller than 22.7, facilitating the observation of successive Efimov resonances in future experiments [83].
9.4.4
MOLECULAR BEC
The achievement of Bose–Einstein condensation of molecules out of degenerate atomic Fermi gases is arguably one of the most remarkable results in the field of ultracold quantum gases. The experimental realization was demonstrated in ultracold Fermi gases of 6 Li and 40 K and turned out to be an excellent starting point to investigate the so-called BEC–BCS crossover and the properties of strongly interacting Fermi gases [18,20]. The realization of an mBEC of Feshbach molecules is only possible in the halo regime, where the collisional properties are favorable; see Section 9.4.2. In a spin mixture of 6 Li, containing the lowest two hyperfine states, the route to mBEC is particularly simple [36]. Evaporative cooling toward BEC can be performed in an optical dipole trap at a constant magnetic field in the halo region near the Feshbach resonance. In the initial stage of evaporative cooling the gas is purely atomic and molecules are produced via three-body recombination; see Section 9.2.2. With decreasing temperature the atom–molecule equilibrium favors the formation of molecules and a purely molecular sample is cooled down to BEC. The large atom– dimer and dimer–dimer scattering lengths along with strongly suppressed relaxation loss facilitate an efficient evaporation process. In this way, mBECs are achieved with a condensate fraction exceeding 90%. The experiments in 40 K followed a different approach to achieve mBEC [35]. For 40 K the halo dimers are less stable because of less favorable short-range three-body interaction properties. Therefore the sample is first cooled above the Feshbach resonance, where a is large and negative, to achieve a deeply degenerate atomic Fermi gas. A sweep across the Feshbach resonance then converts the sample into a partially condensed cloud of molecules. The emergence of the mBEC in 40 K is shown in Figure 9.16. For very large a the size of the halo dimers becomes comparable to the interparticle spacing, and the properties of the fermion pairs start to be determined by manybody physics. For a < 0 two-body physics no longer supports the weakly bound molecular state and pairing is entirely a many-body effect. In particular, in the limit of weak interactions on the a < 0 side of the resonance, pairing can be understood in the framework of the well-established BCS theory, developed in the 1950s to describe superconductivity. Here the fermionic pairs are called Cooper pairs. The BEC and BCS limits are smoothly connected by a crossover regime where the gas is strongly interacting. This BEC–BCS crossover has attracted considerable attention in many-body quantum physics [18,20,84]. A theoretical description of this challenging problem is very difficult and various approaches have been developed. With tunable Fermi gases, a unique testing ground has become available to quantitatively investigate the crossover problem. © 2009 by Taylor and Francis Group, LLC
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(a)
Optical density
(b)
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5 0.0
0.0 –200 –100 0 100 200 Position (μm)
–200 –100 0 100 200 Position (μm)
FIGURE 9.16 Emergence of a molecular BEC in an ultracold Fermi gas of 40 K atoms, observed in time-of-flight absorption images. The density distribution on the left-hand side ((a) 2D surface plot; (b) 1D cross-section) was taken for a Fermi gas that was cooled down to 19% of the Fermi temperature. After ramping across the Feshbach resonance no mBEC was observed as the sample was too hot. The density distribution on the right-hand side was observed for a colder sample at 6% of the Fermi temperature. Here the ramp across the Feshbach resonance resulted in a bimodal distribution, revealing the presence of an mBEC with a condensate fraction of 12%. (From Greiner, M. et al., Nature, 426, 537, 2003. With permission.)
9.5 TOWARD GROUND-STATE MOLECULES Ultracold gases of ground-state molecules hold great prospects for novel quantum systems with more complex interactions than existing in atomic quantum gases. For the experimental preparation of such molecular systems, the collisional stability is an important prerequisite. This makes rovibrational ground states the prime candidates, as vibrational relaxation is energetically suppressed. Stable BECs of dimers may for instance serve as a starting point to synthesize mBECs of more complex constituents. A particular motivation arises from the large electric dipole moments of heteronuclear dimers; see Chapter 2. This leads to the long-range dipole–dipole interaction, which is anisotropic and can be modified by an electric field. As a result, a rich variety of phenomena can be expected, with conceptual challenges and intriguing experimental opportunities. Proposals range from the study of new quantum phases and quantum simulations of condensed-matter system (Chapter 12), to fundamental physics tests (Chapters 15 and 16) and schemes for quantum-information processing (Chapter 17). The STIRAP method has attracted considerable interest as a highly efficient way to coherently transfer atom pairs or Feshbach molecules into a more deeply bound state [85]. The method offers a high transfer efficiency without heating of the molecular © 2009 by Taylor and Francis Group, LLC
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sample, thus allowing the preservation of the high phase-space density of an ultracold gas. Several groups are actively pursuing such experiments with the motivation to create quantum gases of collisionally stable ground-state molecules.
9.5.1
STIMULATED RAMAN ADIABATIC PASSAGE
The basic idea of STIRAP to transfer population between two quantum states relies on a particularly clever implementation of a coherent two-photon Raman transition, involving a dark state during the transfer. For a detailed description of its principles and an overview of early applications, the reader may refer to Ref. [86]. Let us consider a three-level system with |a and |b representing two different ground-state levels, and |e being an electronically excited state, as shown in Figure 9.17. Laser 1 (2) couples the state |a (|b) to the state |e and the Rabi frequency Ω1 (Ω2 ) describes the corresponding coupling strength [87]. The Rabi frequency is defined as d · E/, with E being the electric-field amplitude of the laser field and d the dipole matrix element. The states |a and |b are long lived, whereas the excited state |e can undergo spontaneous decay into lower states. An essential property of such a coherently coupled three-level system is the existence of a dark state |D as an eigenstate of the system. This state generally occurs if both laser fields have the same resonance detuning with respect to the corresponding transition, that is, if the two-photon detuning is zero. The state is dark in the sense that it is decoupled from the excited state |e and thus not influenced by its radiative decay. The dark state can be understood as a coherent superposition of state |a and state |b, |D =
1 Ω1 + Ω 2 2 2
(Ω2 |a − Ω1 |b) ,
(9.8)
with state |e not being involved. The key idea of STIRAP is to slowly change Ω1 and Ω2 in a way that the system is always kept in a dark state. This avoids losses caused by spontaneous light scattering in the excited state |e [88]. One may intuitively expect that the transfer occurs by first applying Laser 1 and then Laser 2, as in a conventional two-photon Raman transition. STIRAP instead uses a counterintuitive laser pulse scheme, in which Laser 2 is applied first. Initially, only state |a is populated Figure 9.17a; this corresponds to the dark |e>
|e>
|e>
|e>
|e>
Laser 1
Laser 2
Laser 1
Laser 2 |a>
|a> |b>
|b>
(a)
|a> |b>
(b)
|a> |b>
(c)
|a> |b>
(d)
(e)
FIGURE 9.17 Illustration of the basic idea of STIRAP. Transfer from state |a to state |b, keeping the population always in a dark state (see Equation 9.8). Initially Ω2 = 1 and Ω1 = 0, and finally Ω2 = 0 and Ω1 = 1.
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state |D if only Laser 2 is active. If one now slowly introduces Laser 1, the dark state |D begins to evolve into a superposition of |a and |b (Figure 9.17b). By adiabatically ramping down the power of Laser 2 and ramping up the intensity of Laser 1, the character of the superposition state changes and it acquires an increasing |b admixture (Figure 9.17c, d). When finally Laser 1 is on and Laser 2 is turned off, the population is completely transfered to the state |b (Figure 9.17e). This adiabatic passage through the dark state |D does not involve any population in the radiatively decaying state |e. Note that the relative phase coherence of the two laser fields is an essential requirement for STIRAP. Moreover, it is important to note that the coherent sequence can be time-reversed to achieve transfer from |b to |a.
9.5.2
STIRAP EXPERIMENTS
In the experiments to create quantum gases of collisionally stable ground-state molecules, the initially populated state |a corresponds to a Feshbach molecule and |b is a deeply bound molecular state. The state |e is a carefully chosen electronically excited level, which has to fulfil two important conditions. To obtain large enough optical Rabi frequencies Ω1 and Ω2 the state must provide sufficiently strong coupling to both states |a and |b, demanding sufficiently strong dipole matrix elements besides the technical requirement of sufficiently intense laser fields at the particular wavelengths. Moreover, state |e needs to be well separated from other excited states, as a significant coupling to these would deteriorate the dark character of the superposition state. Proof-of-principle experiments have been performed on homonuclear 87 Rb2 molecules [89] and heteronuclear 40 K87 Rb molecules [90], demonstrating the transfer by one or a few vibrational quanta. The STIRAP transfer of 87 Rb2 started with trapped Feshbach molecules. In order to characterize and optimize the experimental parameters, the study of a “dark resonance” was an essential step (Figure 9.18). STIRAP could then be efficiently implemented to transfer the molecules from the last bound vibrational level (Eb = h × 24 MHz) to the second-to-last state (h × 637 MHz) with an efficiency close to 90%. In the experiment with the heteronuclear 40 K87 Rb molecules, STIRAP was demonstrated with final molecular binding energies of up to h × 10 GHz. Subsequent experiments then moved on to more deeply bound molecules and in particular to the rovibrational ground states of the triplet and the singlet potential. To bridge a large energy range with STIRAP the experiments become more challenging both for technical and physical reasons. In particular, the optical transition matrix elements become an important issue as they enter the optical Rabi frequencies. In molecular physics, the Franck–Condon principle states that an optical transition does not change the internuclear separation. As a consequence, the overlap of the wavefunctions between ground state and excited state, quantified by the so-called Franck–Condon factors, crucially enters into the matrix elements. A general rule of thumb can be given for finding a large Franck–Condon overlap based on a simple classical argument. The dominant part of the wavefunction occurs near the turning points of the classical oscillatory motion in a specific molecular potential. A large © 2009 by Taylor and Francis Group, LLC
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Normalized molecule number
1 0.8 0.6 0.4 0.2 0
−40
−20
0 δ/2π (MHz)
20
40
FIGURE 9.18 Dark resonance observed as a signature of a dark state between two different vibrational levels in a trapped sample of 87 Rb2 Feshbach molecules. Here both laser fields are turned on simultaneously, and the frequency of Laser 1 is varied while Laser 2 is kept at a fixed frequency, thus varying the two-photon detuning δ. The population of the optically excited state |e leads to a strong background loss caused by spontaneous decay into other levels in a region corresponding to the single-photon transition linewidth. If the two-photon resonance condition is met (δ = 0), a suppression of loss occurs in a very narrow frequency range, which indicates the decoupling from the excited state. The behavior can be modeled on the basis of a three-state system, which provides full information on the relevant experimental parameters. In practice, the study of such a dark resonance is an important step to implement STIRAP. (Adapted from Winkler, K. et al., Phys. Rev. Lett., 98, 043201, 2007.)
Franck–Condon factor for an optical transition can be expected if the classical turning points approximately coincide for the ground and excited state. Molecules with large binding energies of about h × 32 THz were demonstrated in an experiment on 133 Cs2 [91]. The experiment starts with the creation of a pure sample of Feshbach molecules in a weakly bound state |a with a binding energy of h × 5 MHz. Here the standard methods for association and purification are applied as described in Section 9.2.A suitable Franck–Condon overlap was experimentally found for the particular choice of |e that is depicted in Figure 9.19a. The vertical arrows indicate the optical transitions at internuclear separations where maximum wavefunction overlap occurs. The applied timing sequence is shown in Figure 9.19b. In a first STIRAP pulse sequence, the molecules are transferred from |a to |b within 15 μsec. After a variable holding time in state |b, the deeply bound molecules are reconverted into Feshbach molecules by a time-reversed STIRAP pulse sequence. The Feshbach molecules are then detected by the usual dissociative imaging. The experimental results in Figure 9.19c show that 65% of the initial number of molecules reappear after the full double-STIRAP sequence, pointing to a single STIRAP efficiency of about 80%. In the final stage of preparation of the present manuscript, breakthroughs have been achieved on the creation of rovibrational ground-state molecules in three different experiments. While Ref. [92] reports on polar 40 K87 Rb molecules in both the © 2009 by Taylor and Francis Group, LLC
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2300 cm–1 u¢ = 225
|e>
8000
0+ u
Transfer
Rabi frequency
(a)
Holding time
Reverse transfer
Laser 1
4000
STIRAP time
Laser 1 ~1126 nm
(c)
a3∑+ u
0
u = 155
6S + 6S
|a>
–1061
u = 73
|b>
X1∑+ g u=0
–3650 6
8 10 12 14 16 18 20 22 24 Internuclear distance (a0)
Molecule number (103)
Energy (cm–1)
Laser 2 Laser 2 ~1006 nm
8 7 6 5 4 3 2 1 0 0
5
10 15 20 25 30 STIRAP time (μsec)
35
40
FIGURE 9.19 STIRAP experiment starting with ultracold 133 Cs2 Feshbach molecules. (a) The relevant molecular potential curves and energy levels are shown schematically. The vertical arrows show the optical transitions and their positions indicate the internuclear distances where maximum Franck–Condon overlap is expected near the classical turning points. (b) Variation of the Rabi frequencies during the pulse sequence. (c) Experimental results showing the measured molecule population in |a as a function of time through a double-STIRAP sequence. (Adapted from Danzl, J.G. et al., Science, 321, 1062, 2008. With permission.)
ground state of the triplet potential (Eb = h × 7.2 THz) and that of the singlet potential (h × 125 THz), Ref. [93] demonstrates homonuclear 87 Rb2 molecules in the rovibrational ground state of the triplet potential (h × 7.0 THz). A further experiment [94] has succeeded in the production of 133 Cs2 molecules in the rovibrational ground state of the singlet potential (h × 109 THz). The successful 40 K87 Rb experiment also provides a further remarkable demonstration of the importance of the Franck–Condon overlap arguments. At first glance, it may be surprising that the huge binding energy difference to the singlet ground state could be bridged in a single STIRAP step. However, an excited molecular state |e could be found for which the outer classical turning point provides good overlap with the Feshbach molecular state, while the internuclear distance at the inner turning point matches the rovibrational ground state. In the absence of such a fortunate coincidence, transfer may not be feasible in a single STIRAP sequence. Two consecutive STIRAP steps involving different excited states [85,94] promise a general way to achieve efficient ground-state transfer for any homo- or heteronuclear Feshbach molecule. © 2009 by Taylor and Francis Group, LLC
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9.6
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FURTHER DEVELOPMENTS AND CONCLUDING REMARKS
In the rapidly developing field of cold molecules, Feshbach molecules play a particular role as the link to research on ultracold quantum matter. In this chapter, we have presented the basic experimental techniques and discussed some major developments in the field. In this final Section, we briefly point to some other recent developments not discussed so far, adding further interesting aspects to our overview of the research field. Intriguing connections to condensed-matter physics can be found when Feshbach molecules are trapped in optical lattices. In this spatially periodic environment, a Feshbach molecule can be used both as a well-controllable source of correlated atom pairs, and as an efficient tool to detect such pairs. A recent experiment [95] shows how Feshbach molecules, prepared in a three-dimensional lattice, are converted into “repulsively bound pairs” of atoms when forcing their dissociation through a Feshbach ramp. Such atom pairs stay together and jointly hop between different sites of the lattice, because the atoms repel each other. This counterintuitive behavior is due to the fact that the bandgap of the lattice does not provide any available states for taking up the interaction energy. Another fascinating example along the lines of condensed-matter physics is a quantum state with exactly one Feshbach molecule per lattice site, as illustrated in Figure 9.20. Such a state has vanishing entropy and closely resembles a Mott-insulator state [96]. It offers an excellent starting point for experiments on strongly correlated many-body states (Chapter 12) and on quantum information processing (Chapter 17), and for the production of a BEC of molecules in the internal ground state [85]. In the
FIGURE 9.20 Illustration of a quantum state with one molecule at each site of an optical lattice. In the central region of the three-dimensional lattice one finds exactly one molecule per site, trapped in the vibrational ground state. Such a state was prepared with 87 Rb2 Feshbach molecules. (From Volz, T. et al., Nature Phys., 2, 692, 2006. With permission.)
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latter case, the basic idea is to convert the collisionally unstable Feshbach molecules into collisionally stable molecules in the rovibrational ground state. Then the lattice is no longer needed for isolating the molecules from each other and dynamical melting of the ordered state by ramping down the lattice may eventually lead to mBEC. Novel phenomena can also be observed with Feshbach molecules in lowdimensional trapping schemes, such as one-dimensional tubes realized in twodimensional optical lattices. Here a two-body bound state can exist also for negative scattering lengths, contrary to the situation in free space [97]. Low-dimensional traps also offer very interesting environments to create strongly correlated many-body regimes. Counterintuitively, the sensitivity of Feshbach molecules to inelastic decay can inhibit loss and lead to a stable correlated state, as recently demonstrated with 87 Rb in a one-dimensional trap [98]. 2 Various mixtures of ultracold atomic species offer a wide playground for ultracold heteronuclear molecules, and the basic interaction properties of many different combinations involving bosonic and fermionic atoms are currently being explored. A new twist to the field is given by ultracold Fermi–Fermi systems, which have recently been realized in mixtures of 6 Li and 40 K [99,100]. Molecule formation in such systems will lead to bosonic molecules, which in the many-body context may lead to new types of pair-correlated states and novel types of strongly interacting fermionic superfluids. Another interesting question is whether more complex ultracold molecules can be created than simple dimers. A first step into this direction is the observation of scattering resonances between ultracold dimers [101] which have been found in the collisional decay of an optically trapped sample of 133 Cs2 molecules and interpreted as a result of a resonant coupling to tetramer states. All the examples presented in this chapter highlight how great progress is currently being made to fully control the internal and external degrees of freedom of various types of ultracold molecules under conditions near quantum degeneracy. This will soon enable many new applications, ranging from high-precision measurements and quantum computation to the exploration of few-body physics and novel correlated many-body quantum states. Several major research themes are obvious and will undoubtedly lead to great success, but we are also confident that the field also holds the potential for many surprises and developments that we can hardly imagine now.
ACKNOWLEDGMENTS We would like to thank all members of the Innsbruck group “Ultracold Atoms and Quantum Gases” (www.ultracold.at) for contributing to our research program on Feshbach molecules. In particular, we would like to thank Johannes Hecker Denschlag and Hanns-Christoph Nägerl for long-standing collaboration on this topic. We are also indebted to Cheng Chin, Paul Julienne, and Eite Tiesinga for the many insights gained when jointly preparing a review article on Feshbach resonances together with one of us (R.G.). We are grateful to the Austrian Science Fund (FWF) for supporting our various projects related to ultracold molecules. F.F. is a Lise-Meitner fellow of the FWF, and S.K. is supported by the European Commission with a Marie Curie Intra-European Fellowship. © 2009 by Taylor and Francis Group, LLC
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Molecular Regimes in 10 Ultracold Fermi Gases Dmitry S. Petrov, Christophe Salomon, and Georgy V. Shlyapnikov CONTENTS 10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Feshbach Resonances and Diatomic Molecules . . . . . . . . . . . . . . . . . . . 10.2 Homonuclear Diatomic Molecules in Fermi Gases. . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Weakly Interacting Gas of Bosonic Molecules: Molecule–Molecule Elastic Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Suppression of Collisional Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Collisional Stability and Molecular BEC . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Heteronuclear Molecules in Fermi–Fermi Mixtures. . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Effect of Mass Ratio on Elastic Intermolecular Interaction . . . . . . . 10.3.2 Collisional Relaxation for Moderate Mass Ratios. . . . . . . . . . . . . . . . . 10.3.3 Born–Oppenheimer Picture of Collisional Relaxation . . . . . . . . . . . . 10.3.4 Molecules of Heavy and Light Fermionic Atoms . . . . . . . . . . . . . . . . . 10.3.5 Trimer States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.6 Collisional Relaxation of Molecules of Heavy and Light Fermions and Formation of Trimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Crystalline Molecular Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Born–Oppenheimer Potential in a Many-Body System of Molecules of Heavy and Light Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Gas–Crystal Quantum Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Molecular Superlattice in an Optical Lattice . . . . . . . . . . . . . . . . . . . . . . 10.5 Concluding Remarks and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1 10.1.1
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and achieving various regimes of superfluidity. The initial idea was to achieve the Bardeen–Cooper–Schrieffer (BCS) superfluid phase transition in a two-component Fermi gas, which requires attractive interactions between the atoms of different components. Then, in the simplest version of this transition, at sufficiently low temperatures, fermions belonging to different components and with opposite momenta on the Fermi surface form correlated (Cooper) pairs in the momentum space. This leads to the appearance of a gap in the single-particle excitation spectrum and to the phenomenon of superfluidity (e.g., Ref. [1]). In a dilute ultracold two-component Fermi gas, the most efficient formation is that of Cooper pairs due to the attractive intercomponent interaction in the s-wave channel (negative s-wave scattering length a). However, for typical values of a, the superfluid transition temperature is extremely low. For this reason, the efforts of many experimental groups have been focused on modifying the intercomponent interaction using Feshbach resonances. The scattering length a near a Feshbach resonance can be tuned from −∞ to +∞. This has led to exciting developments (see Ref. [2] for review), such as the direct observation of superfluid behavior in the strongly interacting regime (n|a|3 1, where n is the gas density) through vortex formation [3], and the study of the influence of imbalance between the two components of the Fermi gas on superfluidity [4–8]. We focus here on the remarkable physics of weakly bound diatomic molecules of fermionic atoms. This initially unexpected physics connects molecular and condensed matter physics. The weakly bound molecules are formed on the positive side of the resonance (a > 0) [9–12] and they are the largest diatomic molecules obtained so far. Their size is of the order of a and it has reached thousands of angstroms in current experiments. Accordingly, their binding energy is exceedingly small (10 μK or less). Being composite bosons, these molecules obey Bose statistics, and they have been Bose-condensed in experiments with 40 K2 [13,14] at JILA and with 6 Li2 at Innsbruck [15,16], MIT [17,18], ENS [19], Rice [20], and Duke [21]. Nevertheless, some of the interaction properties of these molecules reflect Fermi statistics of the individual atoms forming the molecule. In particular, these molecules are found to be remarkably stable with respect to collisional decay. Being in the highest rovibrational state, they do not undergo collisional relaxation to deeply bound states on a timescale exceeding seconds at densities of about 1013 cm−3 . This is more than four orders of magnitude longer than the lifetime of similar molecules consisting of bosonic atoms. The key idea of our discussion of homonuclear diatomic molecules formed in a two-component Fermi gas by atoms in different internal (hyperfine) states is to show how one obtains an exact universal result for the elastic interaction between such weakly bound molecules and how the Fermi statistics for the atoms provides a strong suppression of their collisional relaxation into deep bound states. It is emphasized that the repulsive character of the elastic intermolecular interaction and remarkable collisional stability of the molecules are the main factors allowing for their Bose–Einstein condensation and for prospects related to interesting manipulations with these molecular condensates. Currently, a new generation of experiments is being developed for studying degenerate mixtures of different fermionic atoms [22,23], with the idea of revealing the influence of the mass difference on superfluid properties and finding novel types of superfluid pairings. On the positive side of the resonance one expects the formation © 2009 by Taylor and Francis Group, LLC
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of heteronuclear weakly bound molecules; this attracts a great deal of interest, in particular for creating dipolar gases. The 6 Li –40 K weakly bound molecules have already been created in experiments [24]. We present an analysis of how the mass ratio for constituent atoms influences the elastic interaction between the molecules and their collisional stability. The discussion is focused on molecules of heavy and light fermions, where one expects the formation of trimer bound states and the manifestation of the Efimov effect. We then show that a many-body system of such molecules can exhibit a gas–crystal quantum transition. Remarkably, the atomic system itself remains dilute, and the crystalline ordering is due to a relatively long-range interaction between the molecules originating from the exchange of light fermions. Realization of the crystalline phase requires a very large mass ratio for the atoms forming a molecule in order to suppress the molecular kinetic energy. This can be achieved in an optical lattice for heavy atoms, where the crystalline phase of a dilute molecular system emerges as a superlattice, and we discuss the related physics. The chapter is concluded by an overview of prospects for manipulations with the weakly bound molecules of fermionic atoms. The leading ideas include the achievement of ultralow temperatures and BCS transition for atomic fermions, the creation of dipolar quantum gases, as well as observations of peculiar trimer bound states in an optical lattice.
10.1.2
FESHBACH RESONANCES AND DIATOMIC MOLECULES
At ultralow temperatures, when the de Broglie wavelength of atoms greatly exceeds the characteristic radius of interatomic interaction forces, atomic collisions and interactions are generally determined by the s-wave scattering. Therefore, in twocomponent Fermi gases one may consider only the interaction between atoms of different components, which can be tuned by using Feshbach resonances. The description of a many-body system near a Feshbach resonance requires a detailed knowledge of the two-body problem. In the vicinity of the resonance, the energy of a colliding pair of atoms in the open channel is close to the energy of a molecular state in another hyperfine domain (closed channel). The coupling between these channels leads to a resonant dependence of the scattering amplitude on the detuning δ of the closed channel state from the threshold of the open channel, which can be controlled by an external magnetic (or laser) field. Thus, the scattering length becomes field dependent (see Figure 10.1). The Feshbach effect is a two-channel problem that can be described in terms of the Breit–Wigner scattering [25,26], and various aspects of such problems have been discussed by Feshbach [27] and Fano [28]. In cold atom physics the idea of Feshbach resonances was introduced in Ref. [29], and optically induced resonances have been analyzed in Refs. [30–33]. At resonance, the scattering length changes from +∞ to −∞, and in the vicinity of the resonance one has the inequality n|a|3 1, where n is the gas density. The gas is then said to be in a strongly interacting regime. It is still dilute in the sense that the mean interparticle separation greatly exceeds the characteristic radius of the interparticle interaction Re . However, the amplitude of binary interactions (scattering © 2009 by Taylor and Francis Group, LLC
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a>0
BCS
Weakly bound molecules BEC
II
B
I
III
FIGURE 10.1 The dependence of the scattering length on the magnetic field near a Feshbach resonance. The symbols I, II, and III label the regime of a weakly interacting degenerate atomic Fermi gas, the strongly interacting regime of BCS–BEC crossover, and the regime of weakly bound molecules. At sufficiently low temperatures region I corresponds to the BCS superfluid pairing, and region III to the Bose–Einstein condensation of molecules.
length) is larger than the mean separation between particles, and in the quantum degenerate regime the conventional mean field approach is no longer valid. For large detuning from resonance the gas is in the weakly interacting regime, that is, the inequality n|a|3 1 is satisfied. On the negative side of the resonance (a < 0), at sufficiently low temperatures of the two-species Fermi gas one expects the BCS pairing between distinguishable fermions, which is well described in the literature [1]. On the positive side (a > 0), two fermions belonging to different components form diatomic molecules. For a Re , these molecules are weakly bound and their size is of the order of a. The crossover from BCS to BEC behavior has recently attracted a great deal of interest, in particular with respect to the nature of superfluid pairing, transition temperature, and elementary excitations. This type of crossover has been earlier discussed in the literature in the context of superconductivity [34–37] and in relation to superfluidity in two-dimensional films of 3 He [38,39]. The idea of resonant coupling through a Feshbach resonance for achieving a superfluid phase transition in ultracold two-component Fermi gases has been proposed in Refs. [40] and [41], and for the two-dimensional case it has been discussed in Ref. [42]. The two-body physics of the Feshbach resonance is the most transparent if the (small) background scattering length is neglected. Then, for low collision energies ε, the scattering amplitude is given by [26]: √ γ/ 2μ F(ε) = − √ , ε + δ + iγ ε
(10.1)
√ where the quantity γ/ 2μ ≡ W characterizes the coupling between the open and closed channels and μ is the reduced mass of the two atoms. The scattering length is © 2009 by Taylor and Francis Group, LLC
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a = −F(0). In Equation 10.1 the detuning δ is positive if the bound molecular state is below the continuum of the colliding atoms. Then, for δ > 0, the scattering length is positive, and for δ < 0 it is negative. Introducing a characteristic length R∗ = 2 /2μW
(10.2)
and expressing the scattering amplitude through the relative momentum of particles √ k = 2με/, we can rewrite Equation 10.1 in the form F(k) = −
a−1
1 . + R∗ k 2 + ik
(10.3)
The validity of Equation 10.3 does not require the condition kR∗ 1. At the same time, this equation formally coincides with the amplitude of scattering of slow particles by a potential with the same scattering length a and an effective range R = −2R∗ , obtained under the condition k|R| 1. The length R∗ is an intrinsic parameter of a Feshbach resonance. It characterizes the width of the resonance. From Equations 10.1 and 10.2 we see that small W and, consequently, large R∗ correspond to narrow resonances, whereas large W and small R∗ lead to wide resonances. The term “wide” is generally used when the length R∗ drops out of the problem, which, according to Equation 10.3, requires the condition kR∗ 1. In a quantum degenerate atomic Fermi gas the characteristic momentum of particles is the Fermi momentum, kF = (3π2 n)1/3 . Thus, in the strongly interacting regime and on the negative side of the resonance (a < 0), for a given R∗ the condition of wide resonance depends on the gas density n and takes the form kF R∗ 1 [43–47]. For a > 0 one has weakly bound molecular states (it is certainly assumed that the characteristic radius of interaction Re a), and for such molecular systems the criterion of the wide resonance is different [48,50]. The binding energy of the weakly bound molecule state is determined by the pole of the scattering amplitude (Equation 10.3). One then finds [48,50] that this state exists only for a > 0, and under the condition R∗ a
(10.4)
ε0 = 2 /2μa2 .
(10.5)
the binding energy is given by
The wavefunction of such a weakly bound molecular state has only a small admixture of the closed channel, and the size of the molecule is ∼a. The characteristic momenta of the atoms in the molecule are of the order of a−1 , and in this respect the inequality 10.4 represents the criterion of a wide resonance for the molecular system. Under these conditions atom–molecule and molecule–molecule interactions are determined by a single parameter—the atom–atom scattering length a. In this sense, the problem becomes universal. It is equivalent to the interaction problem for the twobody potential, which is characterized by a large positive scattering length a and has © 2009 by Taylor and Francis Group, LLC
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a potential well with a weakly bound molecular state. The picture remains the same when the background scattering length cannot be neglected, although the condition of a wide resonance can be somewhat modified [51]. Most ongoing experiments with Fermi gases of atoms in two different internal (hyperfine) states use wide Feshbach resonances [52]. For example, weakly bound molecules 6 Li2 and 40 K2 have been produced in experiments [10–20] by using Feshbach resonances with a length R∗ 20 Å, and for the achieved values of the scattering length a (from 500 to 2000 Å) the ratio R∗ /a was smaller than 0.1. In this review we will consider the case of a wide Feshbach resonance.
10.2
HOMONUCLEAR DIATOMIC MOLECULES IN FERMI GASES
10.2.1 WEAKLY INTERACTING GAS OF BOSONIC MOLECULES: MOLECULE–MOLECULE ELASTIC INTERACTION As we have shown in the previous section, the size of weakly bound bosonic molecules formed at a positive atom–atom scattering length a in a two-species Fermi gas (region III in Figure 10.1) is of the order of a. Therefore, at densities such that na3 1, the atoms form a weakly interacting gas of these molecules. Moreover, under this condition, at temperatures sufficiently lower than the molecular binding energy ε0 and for equal concentrations of the two atomic components, practically all atoms are converted into molecules [53]. This is definitely the case at temperatures below the temperature of quantum degeneracy, Td = 2π2 n2/3 /M (the lowest one in the case of fermionic atoms with different masses, with M being the mass of the heaviest atom). One can clearly see this by comparing Td with ε0 , given by Equation 10.5. Thus, one has a weakly interacting molecular Bose gas and the first question is related to the elastic interaction between the molecules. For a weakly interacting gas the interaction energy in the system is equal to the sum of pair interactions and the energy per particle is ng (2ng for a noncondensed Bose gas), with g being the coupling constant. In our case this coupling constant is given by g = 4π2 add /(M + m), where add is the scattering length for the molecule– molecule (dimer–dimer) elastic s-wave scattering, and M, m are the masses of heavy and light atoms, respectively. The value of add is important for evaporative cooling of the molecular gas to the regime of Bose–Einstein condensation and for the stability of the condensate. The Bose–Einstein condensate is stable for repulsive intermolecular interactions (add > 0), and for add < 0 it collapses. We thus see that for analyzing macroscopic properties of the molecular Bose gas, one should first solve the problem of elastic interaction (scattering) between two molecules. In this section we present the exact solution of this problem for homonuclear molecules formed by fermionic atoms of different components (different internal states) in a two-component Fermi gas. The case of M = m will be discussed in Section 10.3. The solution for M = m was obtained in Refs. [49] and [50] assuming that the atom–atom scattering length a greatly exceeds the characteristic radius of interatomic potential: a Re . © 2009 by Taylor and Francis Group, LLC
(10.6)
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Then, as in the case of the three-body problem with fermions [54–57], the amplitude of elastic interaction is determined only by a and can be found in the zero-range approximation for the interatomic potential. This approach was introduced in the two-body physics by Bethe and Peierls [58]. The leading idea is to solve the equation for the free relative motion of two particles by placing a boundary condition on the wavefunction ψ at a vanishing interparticle distance r: (rψ) 1 =− , rψ a
r → 0,
(10.7)
which can also be rewritten as ψ ∝ (1/r − 1/a),
r → 0.
(10.8)
One then gets the correct expression for the wavefunction at distances r Re . When a Re , Equation 10.8 correctly describes the wavefunction of weakly bound and continuum states even at distances much smaller than a. We now use the Bethe–Peierls approach for the problem of elastic molecule– molecule (dimer–dimer) scattering, which is a four-body problem described by the Schrödinger equation: 2
2 2 (∇ − ∇r22 − ∇R2 ) + U(r1 ) + U(r2 ) m r1 3
√ + U[(r1 + r2 ± 2R)/2] − E Ψ = 0,
−
(10.9)
±
where m is the atom mass. Labeling fermionic atoms in different internal states by the symbols ↑ and ↓, the distance between two given ↑ and ↓ fermions is r1 , and r2 is the distance √between the other two. √ The distance between the centers of mass of these pairs is R/ 2, and (r1 + r2 ± 2R)/2 are the separations between ↑ and ↓ fermions in the other two possible ↑↓ pairs (see Figure 10.2). The total energy is E = −2ε0 + ε, with ε being the collision energy, and ε0 = 2 /ma2 the binding energy of a dimer. The wavefunction Ψ is symmetric with respect to the permutation of bosonic ↑↓ pairs and antisymmetric with respect to permutations of identical fermions: Ψ(r1 , r2 , R) = Ψ(r2 , r1 , −R) √ √ r1 + r2 ± 2R r1 + r2 ∓ 2R r1 − r2 = −Ψ , ,± √ . 2 2 2
(10.10)
For the weak binding of atoms in a molecule, assuming that the two-body scattering length satisfies inequality 10.6, at all interatomic distances (even much smaller than a) except for very short separations of the order of or smaller than Re , the © 2009 by Taylor and Francis Group, LLC
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r1 1 R
1
2
r2
2
FIGURE 10.2 Set of coordinates for the four-body problem.
motion of atoms in the four-body system is described by the free-particle Schrödinger equation,
mE 2 2 2 − ∇r1 + ∇r2 + ∇R + 2 Ψ = 0. (10.11) The correct description of this motion requires the four-body wavefunction Ψ to satisfy the Bethe–Peierls boundary condition for the vanishing distance in any pair √ of ↑ and ↓ fermions, that is, for r1 → 0, r2 → 0, and r1 + r2 ± 2R → 0. Due to the symmetry condition 10.10 it is necessary to require a proper behavior of Ψ only at one of these boundaries. For r1 → 0 the boundary condition reads Ψ(r1 , r2 , R) → f (r2 , R)(1/4πr1 − 1/4πa).
(10.12)
The function f (r2 , R) contains the information about the second pair of particles when the first two are on top of each other. In the ultracold limit, where ka 1, (10.13) the molecule–molecule scattering is dominated by the contribution of the s-wave channel. Inequality 10.13 is equivalent to ε ε0 and, hence, the s-wave scattering can be analyzed from the solution of Equation 10.11 with E = −2ε0 < 0. For large R the corresponding wavefunction takes the form √ Ψ ≈ φ0 (r1 )φ0 (r2 )(1 − 2add /R), R a, (10.14) where the wavefunction of a weakly bound molecule is given by φ0 (r) = √
© 2009 by Taylor and Francis Group, LLC
1 2πa r
exp(−r/a).
(10.15)
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Combining Equations 10.12 and 10.14, we obtain the asymptotic expression for f at large distances R: √ R a. (10.16) f (r2 , R) ≈ (2/r2 a) exp (−r2 /a)(1 − 2add /R), In the case of s-wave scattering, the function f depends only on three variables: the absolute values of r2 and R, and the angle between them. We now derive and solve the equation for f . The value of the molecule–molecule scattering length add is then deduced from the behavior of f at large R determined by Equation 10.16. We first establish a general form of the wavefunction Ψ satisfying Equation 10.11, with the boundary condition 10.12 and symmetry relations 10.10. In our case the total energy E = −22 /ma2 < 0, and the Green function of Equation 10.11 reads √ √ (10.17) G(X) = (2π)−9/2 (Xa/ 2)−7/2 K7/2 ( 2 X/a), √ where X = |S − S |, K7/2 ( 2X/a) is the decaying Bessel function, and S= {r1 , r2 , R} is a nine-component vector. Accordingly, |S − S | = (r1 − r 1 )2 + (r2 − r 2 )2 + (R − R )2 . The four-body wavefunction Ψ is regular everywhere except for vanishing distances between ↑ and ↓ fermions. Therefore, it can be expressed through G(|S − S |) with coordinates S corresponding to a vanishing distance the ↑ and ↓ fermions, that is, for r 1 → 0, r 2 → 0, and √ between (r 1 + r 2 ± 2R )/2 → 0. Thus, for the wavefunction Ψ satisfying the symmetry relations 10.10 we have Ψ(S) = Ψ0 + d3 r d3 R G(|S − S1 |) + G(|S − S2 |) − G(|S − S+ |) − G(|S − S− |) h(r , R ),
(10.18)
√ √ where S1 = {0, r , R }, S2 = {r , 0, −R }, and S± = {r /2 ± R / 2, r /2 ∓ R / 2 ∓ √ r 2}. The function Ψ0 is a properly symmetrized finite solution of Equation 10.11, regular at any distances between the atoms. For E < 0, nontrivial solutions of this type do not exist and we have to put Ψ0 = 0. The function h(r2 , R) has to be determined by comparing Ψ in Equation 10.18 at r1 → 0, with the boundary condition 10.12. Considering the limit r1 → 0, we extract the leading terms on the right-hand side of Equation 10.18. These are the terms that behave as 1/r1 or remain finite in this limit. The last three terms in the square brackets in Equation 10.18 provide a finite contribution: (10.19) d3 r d3 R h(r , R ) G(|S¯ 2 − S2 |) − G(|S¯ 2 − S+ |) − G(|S¯ 2 − S− |) , where S¯ 2 = {0, r2 , R}. To find the contribution of the first term in the square brackets, we subtract and add an auxiliary quantity: √ h(r2 , R) exp (− 2r1 /a). (10.20) h(r2 , R) G(|S − S1 |)d3 r d3 R = 4πr1 © 2009 by Taylor and Francis Group, LLC
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The result of the subtraction yields a finite contribution, which for r1 → 0 can be written as d3 r d3 R [h(r , R ) − h(r2 , R)]G(|S − S1 |) =P
d3 r d3 R [h(r , R ) − h(r2 , R)]G(|S¯ 2 − S1 |),
r1 → 0,
(10.21)
with the symbol P denoting the principal value of the integral over dr (or dR ). A detailed derivation of Equation 10.21 and the proof that the integral in the second line of this equation is convergent are given in Ref. [50]. In the limit r1 → 0, the right-hand side of Equation 10.20 is equal to √ h(r2 , R)(1/4πr1 − 2/4πa). (10.22) We thus find that for r1 → 0 the wavefunction Ψ of Equation 10.18 takes the form Ψ(r1 , r2 , R) =
h(r2 , R) + R, 4πr1
r1 → 0,
(10.23)
where R is the sum of regular r1 -independent terms given by Equations 10.19 and 10.21, and by the second term on the right-hand side of Equation 10.22. Equation 10.23 must coincide with Equation 10.12, and comparing the singular terms of these equations we find h(r2 , R) = f (r2 , R). As the quantity R must coincide with the regular term of Equation 10.12, equal to −f (r2 R)/4πa, we obtain the following equation for the function f : & d3 r d3 R G(|S¯ − S1 |)[f (r , R ) − f (r, R)] + G(|S¯ − S2 |) −
' √ G(|S¯ − S± |) f (r , R ) = ( 2 − 1)f (r, R)/4πa.
(10.24)
±
Here S¯ = {0, r, R}, and we omitted the symbol of the principal value for the integral in the first line of Equation 10.24. As we have already mentioned above, for s-wave scattering the function f (r, R) depends only on the absolute values of r and R and on the angle between them. Thus, Equation 10.24 is an integral equation for the function of three variables. In order to find the molecule–molecule scattering length, it is more convenient to transform √ the momentum–space function, Equation 10.24 into an equation for f (k, p) = d3 rd3 Rf (r, R) exp(ik · r/a + ip · R/ 2a), which yields the following expression:
f (k ± (p − p)/2, p ) d3 p 2 + p2 /2 + (k ± (p − p)/2)2 + (k ± (p + p)/2)2 ±
=
2π2 (1 + k 2 + p2 /2)f (k, p) f (k , −p) d3 k − . 2 2 2 2 + k + k + p /2 2 + k 2 + p2 /2 + 1
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(10.25)
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By making the substitution f (k, p) = (δ(p) + g(k, p)/p2 )/(1 + k 2 ) we reduce Equation 10.25 to an inhomogeneous equation for the function g(k, p): 1 2π2 (1 + k 2 + p2 /2)g(k, p) + (1 + k 2 + p2 /4)2 − (kp)2 p2 (1 + k 2 )( 2 + k 2 + p2 /2 + 1)
g(k ± (p − p)/2, p )d3 p =− 2 2 p (2 + p /2 + (k ± (p − p)/2)2 + (k ± (p + p)/2)2 ) ± ×(1 + (k ± (p − p)/2)2 ) g(k , −p)d3 k + . (10.26) p2 (2 + k 2 + k 2 + p2 /2)(1 + k 2 ) In the case of s-wave scattering, the function g(k, p) depends on the absolute values of k and p and on the angle between these vectors. For p → 0 this function tends to a finite value independent of k. As one can easily establish on the basis of Equation 10.16 and the definition of g(k, p), the molecule–molecule scattering length is given by add = −2π2 a limp→0 g(k, p). Numerical calculations from Equation 10.26 give, with 2% accuracy [48], add = 0.6a > 0.
(10.27)
This result was first obtained in Refs. [49] and [50] from the direct numerical solution of Equation 10.24 by fitting the obtained f (r, R) with the asymptotic form 10.16 at large R. The calculations show the absence of four-body weakly bound states, and the behavior of f at small R suggests a soft-core repulsion between dimers, with a range ∼a. The result of Equation 10.27 is exact, and it indicates the stability of molecular BEC with respect to collapse. Compared to earlier studies, which assumed add = 2a [37,59], Equation 10.27 gives almost twice as small a sound velocity of the molecular condensate and a rate of elastic collisions smaller by an order of magnitude. The result of Equation 10.27 has been confirmed by Monte Carlo calculations [60] and by calculations within the diagrammatic approach [61,62]. An approximate diagrammatic approach leading to add = 0.75a has been developed in Ref. [59].
10.2.2
SUPPRESSION OF COLLISIONAL RELAXATION
The weakly bound dimers that we are considering are diatomic molecules in the highest rovibrational state (see Figure 10.3). They can undergo relaxation into deeply bound states in their collisions with each other: for example, one of the colliding molecules may relax to a deeply bound state while the other one dissociates (Including p-wave interactions, one can think of the formation of deeply bound states as two identical (↑ or ↓) fermions. So, the collision of two weakly bound molecules can lead to the creation of a deep bound state by two ↑ (or ↓) fermionic atoms, and two ↓ (or ↑) atoms become unbound.). The released energy is the binding energy of the final deep state, which is of the order of 2 /mRe2 . It is transformed into the kinetic energy of the particles in the outgoing collision channel and they escape from the trapped © 2009 by Taylor and Francis Group, LLC
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2 e0= ћ 2 ma
R
FIGURE 10.3 Interaction potential U as a function of the distance R between two distinguishable fermionic atoms. The dashed line shows the energy level of the weakly bound molecule, and the solid line the energy level of a deeply bound state.
sample. Therefore, the process of collisional relaxation of weakly bound molecules determines the lifetime of a gas of these molecules and possibilities to Bose-condense this gas. We now show that collisional relaxation is suppressed due to Fermi statistics for atoms in combination with a large size of weakly bound molecules [49,50]. The binding energy of the molecules is ε0 = 2 /ma2 and their size is ∼a Re . The size of deeply bound states is of the order of Re . Therefore, the relaxation process may occur when at least three fermionic atoms are at distances ∼Re with respect to each other. As two of them are necessarily identical, due to the Pauli exclusion principle the relaxation probability acquires a small factor proportional to a power of (qRe ), where q ∼ 1/a is a characteristic momentum of the atoms in the weakly bound molecular state. Relying on the inequality a Re we outline a method that allows us to establish the dependence of the relaxation rate on the scattering length a, without going into a detailed analysis of the short-range behavior of the system. It is assumed that the amplitude of the inelastic relaxation process is much smaller than the amplitude of elastic scattering. Then the dependence of the relaxation rate on a is related only to the a-dependence of the initial-state four-body wavefunction Ψ. We again consider the ultracold limit described by the condition 10.13, where the relaxation process is dominated by the contribution of the s-wave molecule–molecule scattering. The key point is that the relaxation process requires only three atoms to approach each other in short distances of the order of Re . The fourth particle can be far away from these three and, in this respect, does not participate in the relaxation process. This distance is of the order of the size of a molecule, which is ∼ a Re . We thus see that the configuration space contributing to the relaxation probability can be viewed as a system of three atoms at short distances ∼ Re from each other and a fourth atom separated from this system by a large distance ∼a (see Figure 10.4). In this case the four-body wavefunction decomposes into a product: Ψ = η(z)Ψ(3) (ρ, Ω), © 2009 by Taylor and Francis Group, LLC
(10.28)
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Re<
a
FIGURE 10.4 Configuration space contributing to the relaxation probability.
where Ψ(3) is the wavefunction of the three-fermion system, ρ and Ω are the hyperradius and the set of hyperangles for these fermions, and z is the distance between their center of mass and the fourth atom. The wavefunction η(z) describes the motion of this atom and is normalized to unity. Note that Equation 10.28 remains valid for any hyperradius ρ |z| ∼ a. The transition to a deeply bound two-body state occurs in the system of three atoms and does not change the wavefunction of the fourth atom, η(z). Therefore, averaging the transition probability over the motion of the fourth particle, the rate constant for inelastic relaxation in dimer–dimer collisions can be written as αrel = α(3) |η(z)|2 d3 z = α(3) , (10.29) where α(3) is the rate constant for relaxation in the three-atom system. At interatomic distances ∼Re , where the relaxation occurs, as well as at all distances where the hyperradius ρ a, the wavefunction Ψ(3) is determined by the Schrödinger equation with zero energy and, hence, depends on the scattering length a only through a normalization coefficient: Ψ(3) = A(a)ψ,
ρ a,
(10.30)
where the function ψ is independent of a. The probability of relaxation and, hence, the relaxation rate constant are proportional to |Ψ|2 at distances ∼Re . We thus have αrel = α(3) ∝ |A(a)|2 .
(10.31)
The goal then is to find the coefficient A(a), which determines the dependence of the relaxation rate on a. For this purpose it is sufficient to consider distances where a ρ Re and Equation 10.30 is still valid. Then, using the zero-range approximation we find the coordinate dependence of the three-body wavefunction Ψ(3) . The derivation is presented in Ref. [50] and the result is Ψ(3) = A(a)Φν (Ω)ρν−1 ,
ρ a,
(10.32)
where Φν (Ω) is a normalized function of hyperangles, and the coefficient ν depends on the symmetry of Ψ(3) . The a-dependence of the prefactor A(a) can be determined from the following scaling arguments for the four-body problem. The scattering © 2009 by Taylor and Francis Group, LLC
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length a is the only length scale in our problem and we can measure all distances in units of a. Using two rescaled coordinates, ρ = aρ and z = az , we see that Ψ(3) in Equations 10.32 and 10.28 becomes a function of ρ/a, multiplied by A(a)aν−1 . The wavefunction η(z) is normalized to unity and hence it is a function of z/a, multiplied by a−3/2 . Accordingly, the four-body wavefunction Ψ of Equation 10.28 is a function of rescaled coordinates, multiplied by the coefficient A(a)aν−5/2 . By applying the same rescaling to Equation 10.14 and using Equation 10.15, we see that the same coefficient should be proportional to a−3 . Therefore, A(a) ∝ a−ν−1/2 and αrel ∝ a−s , where s = 2ν + 1. The strongest relaxation channel corresponds to the lowest value of ν. It is achieved in the case of p-wave symmetry in the three-body system described by the wavefunction Ψ(3) and is equal to ν = 0.773, which leads to s = 2.55. Assuming that the short-range physics is characterized by the length scale Re and the energy scale 2 /mRe2 we can restore the dimensions and write αrel = C(Re /m)(Re /a)s ,
s = 2.55,
(10.33)
where the coefficient C depends on a particular system and cannot be obtained using the zero-range approximation. Note that the p-wave symmetry in the three-body system corresponds to the p-wave scattering of a fermionic atom of one of the molecules (referred to as the third fermion) on the other molecule. Then the fourth particle also undergoes p-wave scattering on this molecule in such a way that the total orbital angular momentum of the molecule– molecule collision is equal to zero. Because the third and fourth fermions are bound to each other in the molecular state with a size ∼a, the relative momentum of their collisions with the other molecule is ∼1/a and such p-wave collisions are not at all suppressed. The relaxation channel corresponding to the s-wave scattering of the third fermion on the molecule leads to ν = 1.1662 and hence to the relaxation rate proportional to a−3.33 , as in the case of ultracold atom–molecule collisions [49,50]. Thus, for large a this mechanism can be omitted. The channels where the third fermion (and the fourth one) scatters on the molecule with orbital angular momentum l > 1, lead to an even stronger decrease of the relaxation rate with increasing a and hence can be neglected.
10.2.3
COLLISIONAL STABILITY AND MOLECULAR BEC
Equation 10.33 implies a remarkable collisional stability of weakly bound molecules consisting of fermionic atoms in two different internal states and a counterintuitive decrease of the relaxation rate with increasing two-body scattering length a. For currently achieved values of the scattering length a ∼ 1000 Å, the suppression factor (Re /a)s for the relaxation process is about four orders of magnitude. This effect is due to Fermi statistics for the atoms. It is not present for weakly bound molecules of bosonic atoms, even if they have the same large size. Indeed, as the size of weakly bound molecules is ∼a, identical fermionic atoms participating in the relaxation process have very small relative momenta, k ∼ 1/a. Hence, the probability that they approach each other in short distances ∼Re , where the relaxation transitions occur, © 2009 by Taylor and Francis Group, LLC
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should be suppressed as (kRe )2 ∼ (Re /a)2 compared to the case of molecules of bosonic atoms. The exponent s in Equation 10.33 is different from 2 due to the Frank–Condon factor for the relaxation transition and three-body dynamics. The remarkable collisional stability of weakly bound molecules K2 and Li2 consisting of two fermionic atoms has been observed in experiments at JILA [12–14], Innsbruck [11,15,16], MIT [17,18], ENS [10,19], Rice [20], and Duke [21]. At molecular densities n ∼ 1013 cm−3 the lifetime of the gas ranges from tens of milliseconds to tens of seconds, depending on the value of the scattering length a. A strong decrease of the relaxation rate with increasing a, following from Equation 10.33, is consistent with the experimental data. The potassium experiment at JILA [12] and the lithium experiment at ENS [19] give the relaxation rate constant αrel ∝ a−s , with s ≈ 2.3 for K2 , and s ≈ 1.9 for Li2 , in agreement with theory (s ≈ 2.55) within experimental uncertainty. The experimental and theoretical results for potassium and lithium are shown in Figures 10.5 and 10.6. The absolute value of the rate constant for a 6 Li2 condensate is αrel ≈ 1 × 10−13 cm3 /sec for the scattering length a ≈ 110 nm. For K2 it is an order of magnitude higher at the same value of a [12], which can be a consequence of a larger value of the characteristic radius of interaction Re . The suppression of the relaxation decay rate of weakly bound molecules of fermionic atoms has a crucial consequence for the physics of these molecules. At realistic temperatures the relaxation rate constant αrel is much smaller than the rate 2 v , where v is the thermal velocity. For example, constant of elastic collisions, 8πadd T T for the Li2 weakly bound molecules at a temperature T ∼ 3 μK and a ∼ 800 Å, the corresponding ratio is of the order of 10−4 or 10−5 . This opens wide possibilities for reaching BEC of the molecules and cooling the Bose-condensed gas to temperatures of the order of its chemical potential. Long-lived BECs of weakly bound molecules have been observed for 40 K2 at JILA [13,14] and for 6 Li2 at Innsbruck [15,16], MIT [17,18], ENS [19], Rice [20], Duke [21], and recently at Melbourne [63] and Tokyo [64]. Measurements of the molecule–molecule scattering length confirm the result add = 0.6a with the accuracy up to 30% [16,19].
N/N (msec−1)
1 0.1 0.01
0.001 −3
−2
−1 ΔB (G)
0
FIGURE 10.5 Two-body decay rate of a 40 K2 ultracold molecular gas as a function of the magnetic field detuning from the 40 K Feshbach resonance at 202 G. The dots indicate experimental values, and the solid line shows the theoretical results normalized to the experimental value at ΔB = −1.6 G.
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arel (10–13 cm3/s)
10
1
0.1
40
100 a (nm)
200
FIGURE 10.6 Two-body decay rate αrel of a 6 Li2 molecular condensate as a function of interatomic scattering length near the 6 Li Feshbach resonance at 834 G. Solid line: least square fit, αrel ∝ a−1.9±0.8 . Dashed line, theory: αrel ∝ a−2.55 . The theoretical relaxation rate has been normalized to the experimental value at a = 78 nm.
10.3 10.3.1
HETERONUCLEAR MOLECULES IN FERMI–FERMI MIXTURES EFFECT OF MASS RATIO ON ELASTIC INTERMOLECULAR INTERACTION
We now focus on the novel physics of heteronuclear molecules, which are expected to be formed in a mixture of two different fermionic atoms (Fermi–Fermi mixture) at a large positive two-body scattering length a. In several aspects, the physics is similar to that discussed above for homonuclear molecules of fermionic atoms in different internal states. However, for a large mass ratio of the atoms, the situation changes drastically. This is related to the existence of three-body bound Efimov states, which in general makes it impossible to describe molecule–molecule scattering using only the value of the two-body scattering length a. We start with calculating the amplitude of the elastic interaction (scattering) between weakly bound heteronuclear molecules consisting of a heavy (mass M) and light (mass m) fermionic atoms, assuming that the atom–atom scattering length satisfies the inequality a Re and again considering the ultracold limit determined by the condition 10.13. In this case the scattering is dominated by the contribution of the s-wave channel, and we present here the exact results obtained in Ref. [65] using the zero-range approximation. Under the condition ka 1 the collision energy is much smaller than the molecular binding energy ε0 . Hence, the s-wave molecule– molecule elastic scattering can be analyzed after setting the total energy equal to −2ε0 = −2 /μa2 . In the zero-range approximation one should solve the four-body free-particle Schrödinger equation, which again can be written in the form (10.11):
−∇r21 − ∇r22 − ∇R2 + 2/a2 Ψ = 0,
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where r1 is the distance between two given heavy and light fermions, and r2 , the distance between the other two (Figure 10.7). However, it is now more convenient to define the distance between the centers of mass of these pairs as βR, and the separations between the heavy and light fermions√in the other two possible heavy– light pairs as r± = α± r1 + α∓ r2 ± βR, with β = 2α+ α− , α+ = μ/M, α− = μ/m, and μ = mM/(m + M) being the reduced mass. The symmetry condition (10.10) then takes the form Ψ(r1 , r2 , R) = Ψ(r2 , r1 , −R) = −Ψ(r± , r∓ , ±β(r1 − r2 ) ∓ (α+ − α− )R), (10.34) and the Bethe–Peierls boundary condition should be applied for a vanishing distance in any pair of heavy and light fermions, that is, for r1 → 0, r2 → 0, and r± → 0. For r1 → 0 it is again given by Equation 10.12. Due to the change in the definition of the coordinates, the asymptotic expression for the wavefunction Ψ at large distances R now reads Ψ ≈ φ0 (r1 )φ0 (r2 )(1 − add /βR),
R a,
(10.35)
where the notation add is again used for the molecule–molecule scattering length and the wavefunction of a weakly bound molecule is given by Equation 10.15. Then the asymptotic expression for the function f (r2 , R) at large R is given by f (r2 , R) ≈ (2/r2 a) exp (−r2 /a)(1 − add /βR),
R a.
(10.36)
For s-wave scattering the function f depends only on three variables: the absolute values of r2 and R, and the angle between them. Using the procedure described in Section 10.2, we obtain for f the same integral Equation 10.24. The effect of m
r1
m
r2
βR
M M
FIGURE 10.7 Set of coordinates for the four-body problem with two heteronuclear molecules.
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different masses is contained in the expressions for the vectors S± , which now read S± = {α∓ r ± βR , α± r ∓ βR , ∓βr ∓ (α+ − α− )R }. In order to find the molecule–molecule scattering length as a function of the mass ratio M/m it is again more convenient to transform the integral equation for the function f (r, R) into an equation in the momentum space. Introducing the Fourier transform of the function f (r, R) as f (k, p) = d3 rd3 Rf (r, R) exp(ik · r/a + iβp · R/a), we obtain the following momentum–space equation:
±
f (k ± α∓ (p − p), p ) d3 p 2 + β2 p2 + (k ± α∓ (p − p))2 + (k ± α± (p + p))2
=
f (k , −p) d3 k 2π2 (1 + k 2 + β2 p2 )f (k, p) . − 2 + k 2 + k 2 + β2 p2 2 + k 2 + β2 p2 + 1
(10.37)
By making the substitution f (k, p) = (δ(p) + g(k, p)/p2 )/(1 + k 2 ) we reduce Equation 10.37 to an inhomogeneous equation for the function g(k, p). This equation is similar to Equation 10.26 and we do not present it here because of its complexity. As well as in the case of M = m, for p → 0 the function g(k, p) tends to a finite value independent of k. The molecule–molecule scattering length is again given by add = −2π2 a limp→0 g(k, p). In Figure 10.8 we display the ratio add /a versus the mass ratio M/m, found in Ref. [65]. For the case of homonuclear molecules (m = M) we recover the molecule–molecule scattering length add = 0.6a. The universal dependence of add /a on the mass ratio, presented in Figure 10.8, can be established in the zero-range approximation only if M/m is smaller than 13.6. Calculations then show the absence of four-body weakly bound states, and for M/m ∼ 1 the behavior of f suggests a soft-core repulsion between molecules, with a range ∼a. For a mass ratio larger than the limiting value 13.6, the description of the molecule–molecule scattering requires a three-body parameter coming from the short-range behavior of the three-body subsystem consisting of one light and two 1.2 1.1
add/a
1.0 0.9 0.8 0.7 0.6 1
2
4 M/m
FIGURE 10.8 The ratio add /a vs. M/m.
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heavy fermions [54,55]. A qualitative explanation of this behavior will be given in Section 10.3.3.
10.3.2
COLLISIONAL RELAXATION FOR MODERATE MASS RATIOS
The most exciting physics with weakly bound (heteronuclear) bosonic molecules consisting of different fermionic atoms is related to their collisional stability. In addition to the homonuclear molecules discussed above, they are in the highest rovibrational state and hence undergo relaxation into deeply bound states in molecule–molecule collisions, which leads to decay of the sample. The collisional relaxation determines the lifetime of the Bose gas of weakly bound molecules and there is a subtle question of whether and how the mass ratio M/m can influence the suppression of this process [65], originating from the Fermi statistics for the atoms and playing a crucial role in the case of homonuclear molecules. In a similar way, when they behave as pointlike bosons at large intermolecular distances, heteronuclear molecules ”remember” that they consist of fermions when the intermolecular separation becomes smaller than the molecule size (∼a). The relaxation requires the presence of at least three fermions at separations ∼Re from each other. Two of them are necessarily identical, so that due to the Pauli exclusion principle the relaxation probability acquires a small factor proportional to a power of (qRe ), where q ∼ 1/a is a characteristic momentum of the atoms in the weakly bound molecular state. What changes in this picture when the fermionic atoms forming the molecule have different masses? We first consider molecule–molecule relaxation collisions for the case where the mass ratio is smaller than the limiting value 13.6 and short-range physics is not supposed to influence the dependence of the relaxation rate on the two-body scattering length a. In addition to the case of homonuclear molecules in Section 10.2, we assume the inequality a Re and consider the ultracold limit described by Equation 10.13. The configuration space contributing to the relaxation probability can again be viewed as a system of only three atoms at short distances ∼Re from each other and a fourth atom separated from this system by a large distance ∼a. Hence, the four-body wavefunction decomposes into a product according to Equation 10.28: Ψ = η(z)Ψ(3) (ρ, Ω), with Ψ(3) being the wavefunction of the three-fermion system, and ρ a and Ω being the hyperradius and the set of hyperangles for these fermions. The distance between their center of mass and the fourth atom is z, and the function η(z) describes the motion of this atom. In the case of fermionic atoms with different masses there are two possible choices of a three-body subsystem out of four fermions. The most important is the relaxation in the system of one atom with mass m and two heavier atoms with masses M. We then use the same arguments as in Section 10.2 and obtain Equation 10.32 for the function Ψ(3) at distances where Re ρ a: Ψ(3) = A(a)Φν (Ω)ρν−1 , with the coefficient A(a) determining the a-dependence of the relaxation rate according to Equation 10.31: αrel = α(3) ∝ |A(a)|2 . A similar scaling procedure as in Section 10.2 again leads to αrel ∝ a−s , where s = 2ν + 1, and by restoring the dimensions we can write the relaxation rate in the form of Equation 10.33: αrel = C(Re /m)(Re /a)s , with a coefficient C depending on the mass ratio and on short-range physics. © 2009 by Taylor and Francis Group, LLC
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2
s
1
0
–1 2
4
6
8 M/m
10
12
14
FIGURE 10.9 The dependence of the exponent s = 2ν + 1 in Equation 10.33 on the mass ratio M/m.
However, the exponent s now depends not only on the symmetry of the three-body wavefunction Ψ(3) , but also on the mass ratio M/m. The smallest value of ν, that is, the one corresponding to the leading relaxation channel at large a, is achieved for the p-wave symmetry in the system of one light and two heavy fermions [55]. In the interval −1 ≤ ν < 2 it is given by the root of the function [55]: λ(ν) =
ν(ν + 2) πν ν sin γ cos(νγ + γ) − sin(νγ) cot + , ν+1 2 (ν + 1) sin2 γ cos γ sin(πν/2)
(10.38)
where γ = arcsin[M/(M + m)]. A detailed derivation of Equation 10.38 is given in Ref. [55]. In the case of equal masses we recover s = 2ν + 1 ≈ 2.55 obtained in Section 10.2, and it slowly decreases with increasing mass ratio (Figure 10.9). For M/m ∼ 1 nothing dramatic happens: the suppression of the relaxation rate with increasing twobody scattering length a becomes slightly weaker than for homonuclear molecules. However, for the mass ratio approaching the limiting value 13.6, the exponent s first reaches zero and then becomes negative, showing even an increase of the relaxation rate with a. We will give a qualitative explanation of this phenomenon using the Born–Oppenheimer approximation for the system of two heavy and one light atoms.
10.3.3
BORN–OPPENHEIMER PICTURE OF COLLISIONAL RELAXATION
In the Born–Oppenheimer approximation one assumes that the state of a fast light atom adiabatically adjusts itself to the positions R1 and R2 of the slow heavy atoms. One then finds the wavefunction and energy of a bound state of the light atom with two heavy atoms at a given separation between them, R = R1 − R2 . For convenience, from this point on we change the notations and use R to specify the distance between © 2009 by Taylor and Francis Group, LLC
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the heavy atoms, and r the coordinate of a light atom relative to their center of mass. In general, there are two states of a light atom in the field of two heavy ones: the gerade state (+) with the wavefunction remaining unchanged under permutation of the heavy atoms (R → −R), and the ungerade state (−) with the wavefunction changing its sign under this operation. The corresponding wavefunctions are given by ψ± R (r)
= N±
e−κ± (R)|r−R/2| e−κ± (R)|r+R/2| ± |r − R/2| |r + R/2|
,
(10.39)
where N± are normalization coefficients that depend on R. The corresponding binding energies are ± (R) = −2 κ2± (R)/2m,
(10.40)
and the parameters κ± (R) follow from the equation κ± (R) ∓ exp −κ± (R)R /R = 1/a.
(10.41)
Equation 10.41 is obtained by using the Bethe–Peierls boundary condition 10.8 for the wavefunctions ψ± R of Equation 10.39 at vanishing light–heavy atom separations |r ± R/2|. The ungerade (−) state energy is always higher than the energy of the gerade (+) state. Moreover, for R < a, the ungerade state is no longer bound and we are dealing only with the gerade bound state. In the limit of R a Equation 10.41 gives κ+ = 0.56. Then the energy of the gerade bound state, representing an effective potential for the relative motion of the heavy atoms, is given by + (R) = −0.162 /mR2 .
(10.42)
We thus see that when the heavy atoms are separated from each other by a distance R a, the light atom mediates an effective 1/R2 attraction between them. Actually, the same result follows from the Efimov picture of effective interaction in a three-body system [54] and the Born–Oppenheimer approximation only gives a physically transparent illustration of this picture [65,66]. For a large mass ratio the mediated attractive potential + (R) = −0.162 /mR2 strongly modifies the physics of the relaxation process. It competes with the Pauli principle, which in terms of effective interaction manifests itself in the centrifugal 1/R2 repulsion between the heavy atoms. The presence of this repulsion is clearly seen from the fact that the light-atom wavefunction ψ+ R (r) does not change sign under permutation of heavy fermions. As the total wavefunction of the three-body system ψ+ R (r)χ(R) is antisymmetric with respect to this permutation, the wavefunction of the relative motion of heavy atoms χ(R) should change its sign. Therefore, χ(R) contains only partial waves with odd angular momenta, and for the lowest angular momentum ( p-wave) the centrifugal barrier is Uc (R) = 22 /MR2 . For comparable masses it is significantly stronger than + (R). Thus, we have the physical picture discussed in the case of homonuclear molecules: the Pauli principle (centrifugal barrier) reduces the probability for the © 2009 by Taylor and Francis Group, LLC
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atoms to be at short distances and, as a consequence, the relaxation rate decreases with increasing the atom–atom scattering length a. The role of the effective attraction increases with M/m. As a result, the decrease of the relaxation rate with increasing a becomes weaker. The exponent s in Equation 10.33 continuously decreases with increasing M/m and becomes zero for M/m = 12.33 (see Figure 10.9). In the Born–Oppenheimer picture this means that at this point one has a balance between the mediated attraction and the centrifugal repulsion. A further increase in M/m makes s negative and it reaches the value s = −1 for the critical mass ratio M/m = 13.6. Thus, in the range 12.33 < M/m < 13.6 the relaxation rate increases with a. For an overcritical mass ratio M/m > 13.6 we have a well-known phenomenon of the fall of a particle to the center in an attractive 1/R2 potential [26]. In this case the shape of the wavefunction at distances of the order of Re can significantly influence the large-scale behavior, and a short-range three-body parameter is required to describe the system. The wavefunction of heavy atoms χ(R) acquires many nodes at short distances R, which indicates the appearance of three-body bound Efimov states.
10.3.4
MOLECULES OF HEAVY AND LIGHT FERMIONIC ATOMS
The discussion of the previous subsection shows that weakly bound molecules of heavy and light fermions become collisionally unstable for the mass ratio M/m close to the limiting value 13.6. The effect of the Pauli principle becomes weaker than the attraction between heavy atoms at distances R a, mediated by light fermions. However, this picture explains only the dependence of the relaxation rate on the twobody scattering length a. At the same time, for heteronuclear molecules the relaxation rate and the amplitude of the elastic molecule–molecule interaction can also depend on the mass ratio irrespective of the value of a and short-range physics. To elucidate this dependence, we will look at the interaction between the molecules of heavy and light fermions at large intermolecular separations. We consider the interaction between two such molecules in the Born–Oppenheimer approximation and calculate the wavefunctions and binding energies of two light fermions in the field of two heavy atoms fixed at their positions R1 and R2 . The sum of the corresponding binding energies gives an effective interaction potential Ueff for the heavy fermions as a function of the separation R = |R1 − R2 | between them. For R > a, there are two bound states, gerade (+) and ungerade (−), for a light atom interacting with a pair of fixed heavy atoms. Their wavefunctions are given by Equation 10.39, and the corresponding binding energies follow from Equations 10.40 and 10.41. For large R satisfying the condition exp(−R/a) 1, Equation 10.40 yields ± (R) ≈ −|ε0 | ∓ 2|ε0 |
Uex (R) a exp(−R/a) + , R 2
(10.43)
where the binding energy of a single molecule, ε0 , is given by Equation 10.5 with the reduced mass μ very close to the light atom mass m: Uex (R) = 4|ε0 | © 2009 by Taylor and Francis Group, LLC
a a 1− exp(−2R/a). R 2R
(10.44)
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Because the light fermions are identical, their two-body wavefunction is an antisymmetrized product of gerade and ungerade wavefunctions: √ − + − ψR (r1 , r2 ) = [ψ+ (10.45) R (r1 )ψR (r2 ) − ψR (r2 )ψR (r1 )]/ 2. The Born–Oppenheimer adiabatic approach is valid at distances R > a, where the effective interaction potential between the molecules, Ueff , is the sum of + (R) and − (R) (one should add 2|ε0 | so that Ueff (R) → 0 for R → ∞). This potential is displayed in Figure 10.10. At sufficiently large interheavy separations where Equation 10.43 is valid, the effective potential can be written as Ueff (R) = + (R) + − (R) + 2|ε0 | ≈ Uex (R).
(10.46)
The potential Uex originates from the exchange of light fermions and thus can be treated as an exchange interaction. It is purely repulsive and, according to Equation 10.44, has the asymptotic shape of a Yukawa potential at large R. Direct calculations show that Uex is a very good approximation of Ueff for R 1.5a. We now demonstrate the calculation of the dimer–dimer scattering length add in the limit of M/m 1 [67]. In the Born–Oppenheimer approach the Schrödinger equation for the relative motion of two molecules reads (−(2 /m)∇R2 + Ueff (R) − ε)Ψ(R) = 0,
(10.47)
where ε is the collision energy. Note, that the repulsive effective potential is inversely proportional to the light mass m, whereas the kinetic energy operator in Equation 10.47 has a prefactor 1/M. Therefore, for a large mass ratio M/m, the heavy atoms approach each other at distances smaller than a with an exponentially small √ tunneling probability P ∝ exp(−B M/m), where B ∼ 1. This leads to the relaxation rate constant αrel ∝ exp(−B M/m), (10.48) which strongly decreases with increasing mass ratio M/m. Ueff
a
R
FIGURE 10.10 Interaction potential for two molecules of heavy and light fermions as a function of the separation R between the heavy atoms. © 2009 by Taylor and Francis Group, LLC
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The analysis shows that the elastic part of the scattering amplitude can be calculated with a very high accuracy using Equation 10.47 for M/m 20 and is practically insensitive to the way we choose the boundary condition for the wavefunction at R = a. The dominant contribution to the scattering comes from distances in the vicinity of R = add a, where the effective potential can be approximated by Equation 10.44 with a constant preexponential factor: Ueff (R) ≈ 22 (maadd )−1 exp(−2R/a).
(10.49)
Then, the zero-energy solution of Equation 10.47, which decays at smaller R, reads a Ψ(R) = K0 R
2M a −R/a , e m add
(10.50)
where K0 is the decaying Bessel function. Comparing the result of Equation 10.50 at large R with the asymptotic behavior Ψ(R) ∝ (1 − add /R), we obtain an equation for add : a e2γ M a add = ln . (10.51) 2 2 m add This gives add ≈ a ln
M/m,
(10.52)
and the scattering cross-section is 2 σdd = 8πadd .
(10.53)
From Equation 10.50 we see that the interval of distances near R = add , where the wavefunction changes, is of the order of a. This justifies the use of Equation 10.49. In fact, the corrections to Equation 10.51 can be obtained by treating the difference between Equations 10.44 and 10.49 perturbatively. In this way the first-order correction to the dimer–dimer scattering length is −(3/4)a2 /add , where add is determined from Equation 10.51. Qualitatively, Ueff (R) can be viewed as a hard-core potential with radius add , where the edge is smeared out on a length scale ∼a add . Therefore, the ultracold limit for dimer–dimer collisions, required for the validity of Equation 10.53, is realized for relative momenta of the dimers, k, satisfying the inequality kadd 1.
(10.54)
It can be useful (see Ref. [68]) to approximate the potential Ueff by a pure hard core with radius add . This approximation works under the condition ka 1, which is less strict than Equation 10.54. © 2009 by Taylor and Francis Group, LLC
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Let us now mention that numerical calculations [67] show that there are no resonances in the dimer–dimer scattering amplitude, which could appear in the presence of a weakly bound state of two dimers. Here we give a qualitative explanation of the absence of these bound states. Suppose there is such a state with energy ε → 0. Then, at distances R > a the wavefunction of the heavy atoms should exponentially decay √ on the distance scale ∼a m/M a, because Ueff represents a barrier with height ∼1/ma2 . This means that the heavy atoms in such a bound state should be localized mostly at distances smaller than a. The gerade light atom is also localized at these distances, as seen from the shape of the function ψ+ . The motion of the ungerade light atom relative to the localized trimer can be viewed as scattering with odd values of the angular momentum, and due to the centrifugal barrier, the bound states of this atom with the trimer should be localized at distances ∼a from the heavy atoms. In this case one would expect the Born–Oppenheimer approximation to work, because the ungerade light atom is moving much faster than the heavy atoms. However, this leads to a contradiction, because in the Born–Oppenheimer approach discussed above the ungerade state at interheavy separations R < a is unbounded. We thus conclude that weakly bound states of two dimers are absent. Although there are no resonances in the dimer–dimer collisions, there are branchcut singularities in the scattering amplitude. They are related to the presence of inelastic processes in molecule–molecule collisions. These represent the relaxation of one of the colliding dimers into a deeply bound state, with the other dimer being dissociated, and the formation of bound trimers consisting of two heavy and one light atom, the other light atom carrying away the released binding energy.
10.3.5 TRIMER STATES The trimer states, which in most cases can be called Efimov trimers, are interesting objects. Their existence can be seen from the Born–Oppenheimer picture for two heavy atoms and one light atom in the gerade state. Within the Born–Oppenheimer approach the three-body problem reduces to the calculation of the relative motion of the heavy atoms in the effective potential created by the light atom. For the light atom in the gerade state, this potential is + (R), found in the previous subsection. The Schrödinger equation for the wavefunction of the relative motion of the heavy atoms, χν (R), reads ˆ (10.55) Hχ(R) = −(2 /M)∇R2 + + (R) χν (R) = ν χν (R). The trimer states are nothing more than the bound states of heavy atoms in the effective potential + (R). Accordingly, they correspond to the discrete part of the spectrum ν , where the symbol ν denotes a set containing angular (l) and radial (n) quantum numbers. For R a the potential + (R) is proportional to −1/R2 (see Equation 10.42) and, if this effective attraction overcomes the centrifugal barrier, we arrive at the wellknown phenomenon of the fall of a particle to the center in an attractive 1/R2 potential. Then, for a given orbital angular momentum l, the radial part of χν can be written as χν (R) ∝ R−1/2 sin(sl ln R/r0 ), © 2009 by Taylor and Francis Group, LLC
R a,
(10.56)
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where sl =
0.16M/m − (l + 1/2)2 .
(10.57)
The three-body parameter r0 determines the phase of the wavefunction at small distances and, in principle, depends on l. The wavefunction 10.56 has infinitely many nodes, which means that in the zero-range approximation there are infinitely many trimer states. This is one of the properties of three-body systems with resonant interactions discovered by Efimov [54]. We see that the fall to the center is possible in many angular momentum channels, provided the mass ratio is sufficiently large. However, for practical purposes and for simplicity, it is sufficient to consider the case where the Efimov effect occurs only for the angular momentum channel with the lowest possible l for a given symmetry. This implies that when the heavy atoms are fermions and one has odd l, in order to confine ourselves to l = 1 we should have the mass ratio in the range 14 M/m 76. For bosonic heavy atoms where l is even, we set l = 0 and consider M/m 39 to avoid the Efimov effect for l ≥ 2. In both cases we need a single three-body parameter r0 . The formation of Efimov trimers in ultracold dimer–dimer collisions is energetically allowed only if ν < −2|0 |. This means that the trimers that we are interested in are relatively well bound and their size is smaller than a. Therefore, the process of trimer formation is exponentially reduced for large mass ratios as the heavy atoms have to tunnel under the repulsive barrier Ueff (R). Moreover, this process requires all of the four atoms to approach each other at distances smaller than a, and its rate decreases with trimer size because it is more difficult for two identical light fermions to be in a smaller volume. From Equation 10.56 one sees that the behavior of the three-body system does not change if r0 is multiplied by λl = exp(π/sl ).
(10.58)
On the other hand, the dimensional analysis shows that the quantity ν /ε0 depends only on the ratio a/r0 . This means that, except for a straightforward scaling with a, properties of the three-body system do not change when a is multiplied or divided by λl . This discrete scaling symmetry of a three-body system, which shows itself in the log-periodic dependence of three-body observables, has yet to be observed experimentally. In the case of three identical bosons, where the Born–Oppenheimer approach does not work and it is necessary to solve the three-body problem exactly [54], the observation of the consequences of the discrete scaling requires a to be changed by a factor of λ ≈ 22.7, which is technically very difficult in ongoing experiments with cold atoms. In this respect three-body systems with a very large mass difference can be more favorable because of smaller values of λ. For example, in order to see one period of the log-periodic dependence in a Cs–Cs–Li three-body system, a has to be changed only by a factor of λ ≈ 5. At this point it is worth emphasizing that three-body effects can be observed in a gas of light–heavy dimers, where the interdimer repulsion originating from the exchange of the light fermions strongly reduces the decay rate associated with the relaxation of the dimers into deep bound states. The trimer formation in dimer–dimer collisions is © 2009 by Taylor and Francis Group, LLC
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very sensitive to the positions and sizes of the Efimov states, and the measurement of the formation rate can be used to demonstrate the discrete scaling symmetry of a three-body system. Indeed, this rate should have the log-periodic dependence on a and is detectable by measuring the lifetime of the gas of dimers. Besides the Efimov trimers, one light and two heavy atoms may form “universal” trimer states well described in the zero-range approximation without introducing the three-body parameter [69]. In particular, they exist for the orbital angular momentum l = 1 and mass ratios below the critical value, where the Efimov effect is absent and short-range physics drops out of consideration. One of such states emerges at M/m ≈ 8 and crosses the trimer formation threshold (tr = −2|ε0 |) at M/m ≈ 12.7. The existence of this state is already seen in the Born–Oppenheimer picture. It appears as a bound state of fermionic heavy atoms in the potential + (R) for l = 1. The other state exists at M/m even closer to the critical mass ratio and never becomes sufficiently deeply bound to be formed in cold dimer–dimer collisions. The universal trimer states also exist for l > 1 and M/m > 13.6 [69]. However, trimer formation in dimer–dimer collisions at such mass ratios is dominated by the contribution of Efimov trimers with smaller l. Therefore, below, we focus on the formation of Efimov trimers. The calculation of the intrinsic lifetime of a trimer requires a detailed knowledge of short-range physics and is a tedious task. Estimates of the imaginary part of the trimer energy, τ−1 , from the experimental data on Cs3 trimers [70] show that it is smaller approximately by a factor of 4 than the real part ν (in this case η∗ ≈ 0.06). From a general point of view, we do not expect that the trimers with a binding energy ν < −2|ε0 | are very long-lived. However, one can have relatively narrow resonances, and we demonstrate calculations for various values of the elasticity parameter η∗ [67].
10.3.6
COLLISIONAL RELAXATION OF MOLECULES OF HEAVY AND LIGHT FERMIONS AND FORMATION OF TRIMERS
Let us now discuss inelastic processes in dimer–dimer collisions and start with relaxation of the dimers into deeply bound states. The typical size of a deeply bound state is of the order of the characteristic radius of the corresponding interatomic potential. We first consider the relaxation channel that requires one light and two heavy atoms to approach each other at distances ∼Re a. Unlike trimer formation, this decay mechanism is a purely three-body process. The other light atom is just a spectator. A qualitative scenario of this process is the following. With the tunneling probability, which is exponentially suppressed for large M/m, two dimers approach each other at distances R ∼ a. Then the heavy atoms are accelerated toward each other in the potential + (R), and the light atom in the gerade state is always closely bound to the heavy ones, as is seen from the shape of the function ψ+ . The relaxation transition occurs when the heavy atoms (and the gerade light fermion) are at interatomic separations ∼Re . The relaxation rate constant certainly satisfies Equation 10.48, but we should also find out how to take into account the relaxation process in the description of the trimer states and their formation. The most convenient way to do so is to consider the three-body parameter r0 as a complex quantity and introduce the so-called elasticity © 2009 by Taylor and Francis Group, LLC
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parameter η∗ = −sl Arg(r0 ) [71]. As follows from the asymptotic expression for the wavefunction 10.56, a negative argument of r0 ensures that the incoming flux of heavy atoms is not smaller than the outgoing one: Φout /Φin = exp[4sl Arg(r0 )] = exp[−4η∗ ] ≤ 1.
(10.59)
This mimics the loss of atoms at small distances due to the relaxation into deeply bound states. In the analysis of Efimov states, the imaginary part of r0 leads to the appearance of an imaginary part of ν . This means that any Efimov state has a finite lifetime τ due to the relaxation. For small |Arg(r0 )| and for trimer states that are localized at distances smaller than a, we get τ−1 /|ν | = 4|Arg(r0 )| = 4η∗ /sl . Strictly speaking, this fact indicates that it is not possible to separate the relaxation process from trimer formation because the trimers that are formed in dimer–dimer collisions will eventually decay due to relaxation. Nevertheless, both the modulus and the argument of the three-body parameter can be determined by measuring the lifetime of a gas of dimers, leading to a number of quantitative predictions concerning the structure of Efimov states in the three-body subsystem of one light and two heavy atoms. Another relaxation channel is the one in which two light atoms approach a heavy atom at distances ∼Re a. This channel is, however, suppressed due to the Fermi statistics for the light atoms, which strongly reduces the probability of having them in a small volume. As a result, for realistic parameters this relaxation mechanism is much weaker than the one in the system of one light and two heavy atoms [67]. The study of the formation of trimer states in molecule–molecule collisions requires us to go beyond the conventional Born–Oppenheimer approximation, because this approximation breaks down for the ungerade light atom at separations between the heavy atoms R < a. Within the recently developed “hybrid Born– Oppenheimer” approach [67] the Born–Oppenheimer method is applied to the gerade light fermion, which is characterized by the wavefunction ψ+ R (r) and energy + (R) adiabatically ajusting themselves to the motion of the heavy atoms. The gerade light atom is then integrated out by introducing the potential + (R) for the heavy atoms. Once this is done the original four-body problem is reduced to a three-body problem described by the Schrödinger equation [Hˆ − (2 /2μ3 )∇r2 − E]Ψ(R, r) = 0,
(10.60)
where Hˆ is given by Equation 10.55, μ3 = 2mM/(2M + m), E = −2|ε0 | + ε is the total energy of the four-body system in the center-of-mass reference frame, and ε is the dimer–dimer collision energy. This problem is then treated exactly the same as in the Bethe–Peierls approach, and the interaction of the light atom with the heavy atoms is included in the form of the Bethe–Peierls boundary condition 10.8 for Ψ at vanishing light–heavy separations |r ± R/2|. The ungerade symmetry for this atom is taken into account by the condition Ψ(R, r) = −Ψ(R, −r). © 2009 by Taylor and Francis Group, LLC
(10.61)
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As the heavy atoms are identical fermions, we have Ψ(R, r) = −Ψ(−R, r). Combined with Equation 10.61, this leads to the condition Ψ(R, r) = Ψ(−R, −r). Therefore, Ψ(R, r) describes atom–dimer scattering with even angular momenta, and for ultracold collisions we have to solve an s-wave atom–dimer scattering problem. In order to solve Equation 10.60 we follow the method of Ref. [55] and introduce an auxiliary function f (R) and write down the wavefunction Ψ(R, r) in the form
Ψ(R, r) = χν (R)χ∗ν (R )Kκν (2r, R )f (R ), (10.62) R
ν
where Kκν (2r, R ) = and
e−κν |r−R /2| e−κν |r+R /2| − 4π|r − R /2| 4π|r + R /2|
(√ 2μ3 (ν − E)/, ν > E, κν = √ −i 2μ3 (E − ν )/, ν < E.
(10.63)
(10.64)
For ν < E, the trimer can be formed in the state ν. In this case κν is imaginary and the function 10.63 describes an outgoing wave of the light atom moving away from the trimer. The choice of the sign in Equation 10.64 ensures that there is no incoming flux in the atom–trimer channel. Using the Bethe–Peierls boundary condition 10.8 for the wavefunction 10.62 at |r ± R/2| → 0 we obtain an integral equation for the function f (R): 2
Lˆ − Lˆ + sin2 θ
√
3 2μ(−ε0 − E)/ − 1/a f (R) = 0, 4π
(10.65)
√ where μ = mM/(m + M), θ = arctan 1 + 2M/m, and
Lˆ f (R) = [χν (R)χ∗ν (R )Kκν (R, R ) − χ0ν (R)χ0∗ ν (R )Kκ0ν (R, R )] f (R ), R
ν
(10.66) & ' ˆ (R) = P Lf G(|R − R |) f (R)−f (R ) ± G( R2 +R2 −2R · R cos 2θ)f (R ) , R
√
G(X) =
sin 2θM(−ε0 − E)K2 ( M(−ε0 − E)X/ sin θ) , 82 π3 X 2
(10.67) (10.68)
with K2 (z) being the exponentially decaying Bessel (Macdonald) function. A detailed derivation is given in Ref. [67] and is omitted here. The operators Lˆ and Lˆ conserve angular momentum and, expanding the function f (R) in spherical harmonics, we arrive at a set of uncoupled one-dimensional integral equations for each of the radial functions fl (R). Below we present the results for s-wave dimer–dimer scattering. © 2009 by Taylor and Francis Group, LLC
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At large distances (R a) the reduced wavefunction Ψ(R, r) takes the form: Ψ(R, r) ≈ Ψ(R)ψ− R (r),
(10.69)
Ψ(R) ∝ f (R),
(10.70)
and it can be shown that R a.
Therefore, f (R) can serve as the wavefunction for the dimer–dimer motion at large distances. In particular, it contains the dimer–dimer scattering phase shift. The dimer–dimer s-wave scattering amplitude add is determined from the asymptotic behavior of the solution of Equation 10.65 at large distances for E = 2ε0 , which should be matched with f0 (R) ∝ (1/R − 1/add ) (10.71) √ at R a ln M/m. In Figure 10.11 we compare the resulting add /a with that following from Equation 10.51. The results agree quite well even for moderate values of M/m. These results also agree with the calculations based on the exact four-body equation for M/m < 13.6 [65] and displayed in Figure 10.8, and with the Monte Carlo results for M/m < 20 [72]. It is straightforward to extend this theory to account for inelastic processes of the trimer formation and the relaxation of dimers into deeply bound states. Let us first assume that the rate of the relaxation into deep molecular states is negligible and neglect this process. Then the three-body parameter is real, and the trimer formation rate is determined by the imaginary part of the s-wave scattering length. The rate constant is given by [26] 16π α=− (10.72) Im(add ). M
2
1.5
add/a (Equation 10.51) add/a (HBO)
1
20
40
60 M/m
80
100
FIGURE 10.11 The dimer–dimer s-wave scattering length add /a. The solid curve shows the results obtained in the hybrid Born–Oppenheimer (HBO) approximation, and the dotted line the results of Equation 10.51.
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Alternatively, if it is necessary to know the rate of the trimer formation in the state ν, one can substitute the solution of Equation 10.65 into Equation 10.62 and calculate the flux of light atoms at r → ∞. The summation over ν gives the same result as Equation 10.72. We find that the contribution of the highest “dangerous” trimer state is by far dominant and α is very sensitive to its position. We now include the relaxation of the dimers into deeply bound states. As we have mentioned above, the light–heavy–heavy relaxation process can be taken into account by adding an imaginary part to the three-body parameter. The total inelastic decay rate is then still given by Equation 10.72. However, strictly speaking, we can no longer distinguish between the formation of the trimer in a particular state and the collisional relaxation since the trimers ultimately decay due to the relaxation process. In this sense the only decay channel is the relaxation. However, for a sufficiently long lifetime of a trimer, that is, if the trimer states are narrow resonances, we can still observe a pronounced dependence of the total inelastic decay rate on the position of the highest “dangerous” trimer state (see below). Figure 10.12 shows the results for the inelastic collisional rate in the case of bosonic molecules with the mass ratio M/m = 28.5 characteristic of 171Yb–6 Li dimers. The solid line corresponds to the case of a real three-body parameter. It is convenient to introduce a related quantity, a0 , defined as the value of a at which the energy,
0.008 0.06
(M/ha)α
0.004 0.04 0
1
1.02
1.04
0.02
0 1
10 a/a0
FIGURE 10.12 The inelastic rate constant for bosonic dimers with M/m = 28.5 as a function of the atom–atom scattering length a. The solid line corresponds to the case of a real threebody parameter. The results plotted in dashed, dotted, dash–dotted, and dash–dot–dot lines are obtained by taking into account the light–heavy–heavy relaxation processes for values of the elasticity parameter of η∗ = 0.1, 0.5, 1, and ∞, respectively (see text). The quantity a0 is the value of a at which the energy of a trimer state equals E = −2ε0 and a new inelastic channel opens. The inset shows the region a ≈ a0 in greater detail in order to see the threshold behavior (10.73).
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ν , of a trimer state exactly equals E = −2ε0 . This new “dangerous” trimer state becomes more deeply bound for a > a0 and the rate constant rapidly increases. It is proportional to the density of states in the outgoing atom–trimer channel. The corresponding orbital angular momentum is equal to 1 and the threshold law reads (see also inset in Figure 10.12): α ∝ const + (E − ν )3/2 ∝ const + (a − a0 )3/2 .
(10.73)
The constant term in Equation 10.73 describes the contribution of more deeply bound states, which is typically very small. In fact, as trimer states become more compact, both light atoms should approach the heavy atoms and each other at small distances where the trimer formation takes place. Because they are identical fermions, there is a strong suppression of the trimer formation to these deeply bound states. The dependence of α on a/a0 is periodic on the logarithmic scale, with the multiplicative factor being equal to λ1 ≈ 7.3. The dashed, dotted, and dash–dotted curves are obtained for η∗ = 0.1, 0.5, and 1, respectively. The corresponding values of the ratio Φout /Φin are 0.67, 0.14, and 0.02. The horizontal line represents the limiting case of η∗ = ∞ or Φout = 0. This case is universal in the sense that physical observables depend only on the masses and the atomic scattering length. For a very weak light–heavy–heavy relaxation, the dimer–dimer inelastic collision can be viewed as the formation of a trimer (with the rate constant α) followed by its slow decay due to the relaxation. In this case one can think of detecting the trimers spectroscopically. We note, however, that even for the conditions corresponding to the dashed curve in Figure 10.12, that is, for η∗ as small as 0.1, the decay rate of the trimer, τ−1 ≈ 0.25|ν |/ is rather fast, which will likely make its direct detection difficult. For larger η∗ it is impossible to separate the formation of trimers from their intrinsic relaxational decays, and α is practically the relaxation rate constant. Remarkably, it remains quite sensitive to the positions of the trimer states (in this case resonances) for values of η∗ up to 0.5 and even larger. This suggests that measuring the lifetime of a gas of dimers as a function of a may provide important information on three-body observables. Moreover, for small η∗ it may be possible to create a stable molecular gas in sufficiently broad regions of a, where “dangerous” trimer states are far from the trimer formation threshold. The inelastic rate α for other mass ratios in the range 20 < M/m < 76 has also been found [67]. Its dependence on the scattering length has the same form as depicted in Figure 10.12. The maximum of the rate constant is well fitted by the formula αmax = √ 5.8(a/M) exp(−0.87 M/m) √ and the position of the horizontal line (η∗ = ∞) by α∞ = 1.6(a/M) exp(−0.82 M/m). The multiplicative factor in the log-periodic dependence is given by Equations 10.58 and 10.57 with l = 1. The same method was employed to estimate the formation of the universal trimer state with the orbital angular momentum l = 1 at mass ratios M/m > 12.7 but below the critical value for the onset of the Efimov effect [67]. The rate constant increases with M/m and reaches α = 0.2(a/M) close to the critical mass ratio. This corresponds to the imaginary part of the scattering length Imadd ≈ −4 × 10−3 a, which is smaller by a factor of 300 than the real part of add obtained from four-body calculations © 2009 by Taylor and Francis Group, LLC
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[66]. Thus, the formation of this state does not change the elastic scattering amplitude add shown in Figure 10.8. We can now estimate the collisional rates for 171Yb–6 Li dimers. On the basis of the results in Figure 10.12 we find that for a = 20 nm the upper bound of the inelastic rate constant is αmax ≈ 4 × 10−13 cm3 /sec. The elastic rate constant for a thermal gas with a ∼ 20 nm and T ∼ 100 nK equals αel ≈ 8π|add |2 2T /M ∼ 4 × 10−11 cm3 /sec.
(10.74)
Here we used the calculated s-wave dimer–dimer scattering length add ≈ 1.4a. We see that αel , is much larger than α and this inequality becomes even more pronounced for larger a due to the scaling relations αel ∝ a2 and αmax ∝ a. Thus, the gas of molecules of heavy and light fermions is well suited for evaporative cooling toward their Bose–Einstein condensation.
10.4 10.4.1
CRYSTALLINE MOLECULAR PHASE BORN–OPPENHEIMER POTENTIAL IN A MANY-BODY SYSTEM OF MOLECULES OF HEAVY AND LIGHT FERMIONS
A strong long-distance repulsive interaction between weakly bound molecules of light and heavy fermionic atoms has an important consequence not only for the relaxation process, but also for the macroscopic properties of the molecular system. In contrast to two-component Fermi gases of atoms in different internal states, heteronuclear Fermi–Fermi mixtures can form a molecular crystalline phase even when the mean interparticle separation greatly exceeds the size of the molecule: R¯ a.
(10.75)
Let us consider a mixture of heavy and light fermionic atoms with equal concentrations and a large positive scattering length for the interaction between them, satisfying the inequality a Re . At zero temperature all atoms will be converted into weakly bound molecules and under the condition 10.75 the molecular size (∼a) will be much smaller than the mean intermolecular separation. Then, using the Born–Oppenheimer approximation and integrating out the motion of light atoms we are left with a system of identical (composite) bosons that is described by the Hamiltonian: Hˆ = −
2
1
ΔRi + Ueff (Rij ), 2M 2 i
(10.76)
i=j
where the indices i and j label the bosons, their coordinates are denoted by Ri and Rj , and Rij = |Ri − Rj | is the separation between the ith and jth bosons. Assuming that the motion of light fermions is three-dimensional, the effective repulsive potential Ueff is given by Equation 10.44 and is independent of the mass of the heavy atom M. Therefore, at a large mass ratio M/m it dominates over the kinetic © 2009 by Taylor and Francis Group, LLC
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energy, which is inversly proportional to M, and which may lead to the formation of a crystalline phase. We will discuss the case where the motion of heavy atoms is confined to two dimensions, while the motion of light atoms can be either two- or three-dimensional. It will be shown that the Hamiltonian 10.76 with Ueff in 10.44 supports the first-order quantum gas-crystal transition at T = 0 [68]. This phase transition resembles the one for the flux lattice melting in superconductors, where the flux lines are mapped onto a system of bosons interacting via a two-dimensional Yukawa potential [73]. In this case Monte Carlo studies [74,75] identified the first-order liquid-crystal transition at zero and finite temperatures. Aside from the difference in the interaction potentials, a distinguished feature of our system is related to its stability. The molecules can undergo collisional relaxation into deeply bound states, or form weakly bound trimers. Another subtle question is how dilute the system should be to enable the use of the binary approximation for the molecule–molecule interaction, leading to Equations 10.76 and 10.44. Let us first consider the system of N molecules and derive the Born–Oppenheimer interaction potential for this system. Omitting the interaction between light (identical) fermions, it is sufficient to find N lowest single-particle eigenstates, and the sum of their energies will give the interaction potential for the molecules. For the interaction between light and heavy atoms we use the Bethe–Peierls approach, and the wavefunction of a single light atom then reads
Ψ({R}, r) =
N
Ci Gκ (r − Ri ),
(10.77)
i=1
where r is its coordinate and R denotes the set of coordinates of the heavy atoms. The Green function Gκ satisfies the equation (−∇r2 + κ2 )Gκ (r) = δ(r). The energy of the state 10.77 equals = −2 κ2 /2m, and here we only search for negative singleparticle energies. The dependence of the coefficients Ci and κ on {R} is obtained using the Bethe–Peierls boundary condition: Ψ({R}, r) ∝ Gκ0 (r − Ri ),
r → Ri .
(10.78)
Up to a normalization constant, Gκ0 is the wavefunction of a bound state of a single molecule with energy −ε0 = −2 κ20 /2m and molecular size κ−1 0 . The Bethe–Peierls boundary condition 10.78 written through the Green function Gκ0 can be used for both two- and three-dimensional motion of light atoms. In the latter case one has κ0 = a−1 , the Green function is Gκ0 (r − Ri ) ∝ (|r − Ri |−1 − a−1 ) for |r − Ri | → 0, and Equation 10.78 takes the form of Equation 10.8. 4 From Equations 10.77 and 10.78 we get a set of N equations: j Aij Cj = 0, where Aij = λ(κ)δij + Gκ (Rij )(1 − δij ), Rij = |Ri − Rj |, and λ(κ) =limr→0 [Gκ (r) − Gκ0 (r)]. The single-particle energy levels are determined by the equation det Aij (κ, {R}) = 0. © 2009 by Taylor and Francis Group, LLC
(10.79)
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For Rij → ∞, Equation 10.79 gives an N-fold degenerate ground state with κ = κ0 . At finite large Rij , the levels split into a narrow band. Given a small parameter, ˜ 0 |λκ (κ0 )| 1, ξ = Gκ0 (R)/κ
(10.80)
where R˜ is a characteristic distance at which heavy atoms can approach each other, and the bandwidth is Δ ≈ 4|ε0 |ξ |ε0 |. It is important for the adiabatic approximation that all lowest N eigenstates have negative energies and are separated from the continuum by a gap ∼|ε0 |. We now calculate the single-particle energies up to the second order in ξ. To this order we write κ(λ) ≈ κ0 + κλ λ + κλλ λ2 /2 and turn from Aij (κ) to Aij (λ): Aij = λδij + [Gκ0 (Rij ) + κλ λ∂Gκ0 (Rij )/∂κ](1 − δij ),
(10.81)
where all derivatives are taken at λ = 0. Using Aij (10.81) in Equation 10.79 gives a polynomial of degree N in λ. Its roots4λi give the light-atom energy spectrum i = −2 κ2 (λi )/2m. The total energy, E = N i=1 i , is then given by N N
2 λi + (κκλ )λ λ2i . E = −( /2m) Nκ0 + 2κ0 κλ 2
i=1
(10.82)
i=1
Keeping only the terms up to the second order in ξ and using the basic properties of determinants and polynomial roots 4we find that the first-order terms vanish and the energy reads E = −Nε0 + (1/2) i=j U(Rij ), where U(R) = −
∂G2κ0 (R) 2 κ0 (κλ )2 + (κκλ )λ G2κ0 (R) . m ∂κ
(10.83)
Thus, up to the second order in ξ, the interaction in the system of N molecules is the sum of the binary potentials (10.83). If the motion of light atoms is three-dimensional, the Green function is Gκ (R) = (1/4πR) exp(−κR), and λ(κ) = (κ0 − κ)/4π, with the molecular size κ−1 0 equal to the three-dimensional scattering length a. Equation 10.83 then gives a repulsive potential 3D : Uex (Equation 10.44) that we now denote as Uex 3D Uex (R) = 4|ε0 |(1 − (2κ0 R)−1 ) exp(−2κ0 R)/κ0 R,
(10.84)
and the criterion (Equation 10.80) reads (1/κ0 R) exp(−κ0 R) 1. For the twodimensional motion of light atoms we have Gκ (R) = (1/2π)K0 (κR) and λ(κ) = −(1/2π) ln(κ/κ0 ), where K0 is the decaying Bessel function and κ−1 0 follows from Ref. [76]. This leads to a repulsive intermolecular potential: 2D Uex (R) = 4|ε0 |[κ0 RK0 (κ0 R)K1 (κ0 R) − K02 (κ0 R)],
(10.85)
with the validity criterion K0 (κ0 R) 1. In both cases, which we denote 2 × 3 and 2 × 2 for brevity, the validity criteria are well satisfied already for κ0 R ≈ 2. © 2009 by Taylor and Francis Group, LLC
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GAS–CRYSTAL QUANTUM TRANSITION
The inequality κ0 R¯ 2 may be considered as the condition under which the system is described by the Hamiltonian 10.76, with Ueff given by Equation 10.84 for the three-dimensional motion of light atoms, or by Equation 10.85 in the case where this motion is two-dimensional. The state of the system is then determined by two parameters: the mass ratio M/m and the rescaled two-dimensional density nκ−2 0 . At a large M/m, the potential repulsion dominates over the kinetic energy, which should lead to the formation of a crystalline ground state. For separations Rij < κ−1 0 the adiabatic approximation breaks down. However, the interaction potential U(R) is strongly repulsive at larger distances. Hence, even for an average separation between −1 heavy atoms R¯ close to 2/κ0 , they approach each √ other at distances smaller than κ0 with a small tunneling probability P ∝ exp(−β M/m) 1, where β ∼ 1. It is then 3D (R) (or U 2D (R)) to R κ−1 in a way that provides a proper possible to extend Uex ex 0 molecule–molecule scattering phase shift in vacuum and verify that the phase diagram for the many-body system is not sensitive to the choice of this extension [69]. In Figure 10.13 we display the zero-temperature phase diagram obtained by the Diffusion Monte Carlo method [68]. Simulations were performed with 30 particles and showed that the solid phase is a two-dimensional triangular lattice. For the largest density it has been verified that using more particles has little effect on the results. For both 2 × 3 and 2 × 2 cases the (Lindemann) ratio γ of the rms displacement of molecules to R¯ on the transition lines ranges from 0.23 to 0.27.At low densities n the de 3D 2D Broglie wavelength of molecules is Λ ∼ γR¯ κ−1 0 , and Uex (R) (or Uex (R)) can be approximated by a hard-disk potential with the diameter equal to the two-dimensional
M m Crystal
103 2 × 3, γ = 0.24 2 × 2, γ = 0.23
Gas 102 0.00
0.05
0.10
0.15
0.20
0.25
nk0–2
FIGURE 10.13 Diffusion Monte Carlo gas–crystal transition lines for three-dimensional (triangles) and two-dimensional (circles) motion of light atoms. Solid curves show the lowdensity hard-disk limit, and dashed curves the results of the harmonic approach (see text).
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391
scattering length. Then, using the Diffusion Monte Carlo (DMC) results for hard-disk bosons [77], we obtain the transition lines shown by solid curves in Figure 10.13. At larger n, we have Λ < κ−1 0 and, using the harmonic expansion of U(R) around equlibrium positions in the crystal, we calculate the Lindemann ratio and select γ for the best fit to the Monte Carlo data points (dashed curves in Figure 10.13).
10.4.3
MOLECULAR SUPERLATTICE IN AN OPTICAL LATTICE
The mass ratio above 100, required for the observation of the crystalline order, can be achieved in an optical lattice with a small filling factor for heavy atoms. Their effective mass in the lattice, M∗ , can be made very large, and the discussed solid phase should appear as a superlattice. There is no interplay between the superlattice order and the shape of the underlying optical lattice, in contrast to the recently studied solid and supersolid phases in a triangular lattice with the filling factor of order one [78–80]. The superlattice discussed in our review remains compressible and supports two branches of phonons. The gaseous and solid phases of weakly bound molecules in an optical lattice are metastable. As well as in the gas of such molecules in free space, the main decay channels are the relaxation of molecules into deep bound states and the formation of trimer states by one light and two heavy atoms. The relaxation into deeply bound states turns out to be rather slow, with a relaxation time exceeding 10 sec even at two-dimensional densities of ∼109 cm−2 [68]. The most interesting is the formation of the trimer states. 4 In an optical lattice the trimers are eigenstates of the Hamiltonian H0 = −(2 /2M∗ ) i=1,2 ΔRi + + (R12 ). In a deep lattice it is possible to neglect all higher bands and regard Ri as discrete lattice coordinates and Δ as the lattice Laplacian. Then, the fermionic nature of the heavy atoms prohibits them to be on the same lattice site. For a very large mass ratio M∗ /m the kinetic energy term in H0 can be neglected, and the lowest trimer state has energy tr ≈ + (L), where L is the lattice period. It consists of a pair of heavy atoms localized at neighboring sites and a light atom in the gerade state. Higher trimer states are formed by heavy atoms localized in sites separated by distances R > L. This picture breaks down at large R, where the spacing between trimer levels is comparable with the tunneling energy 2 /M∗ L 2 and the heavy atoms are delocalized. In the many-body molecular system the scale of energies in Equation 10.76 is much smaller than |ε0 |. Thus, the formation of trimers in molecule–molecule “collisions” is energetically allowed only if the trimer binding energy is tr < −2ε0 . Because the lowest trimer energy in the optical lattice is + (L), the trimer formation requires the condition + (L) −2ε0 , which is equivalent to κ−1 0 1.6L in the 2 × 3 case and −1 κ0 1.25L in the 2 × 2 case. This means that for a sufficiently small molecular size or large lattice period L the formation of trimers is forbidden. At a larger molecular size or smaller L the trimer formation is possible. The formation rate has been calculated in Ref. [68] by using the hybrid Born–Oppenheimer approach, and here we only present the results and give their qualitative explanation. In order to form a bound trimer state two molecules have to tunnel toward each other at distances R κ−1 0 . This can be viewed as tunneling of particles with mass M∗ in the repulsive potential Ueff (R). Therefore, the probability of approaching at © 2009 by Taylor and Francis Group, LLC
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interheavy √ separations where the trimer formation occurs, acquires a small factor exp(−J M∗ /m) with J ∼ 1, and so does the formation rate. Thus, one can suppress the trimer formation by increasing the ratio M∗ /m. On the other hand, for M∗ /m 100 these peculiar bound states can be formed on the time scale τ 1 sec. Note that the trimer states in an optical lattice, at least the lowest ones, are much more long-lived than in the gas phase. An intrinsic relaxational decay is strongly suppressed as it requires the two heavy atoms of the trimer to approach each other and occupy the same lattice sites. The trimer state can also decay when one of the heavy atoms of the trimer is approached by its own light atom and another light atom at light–heavy separations ∼Re . However, this decay channel turns out to be rather slow, with a decay time exceeding tens of seconds even at two-dimensional densities of n ∼ 109 cm−2 [68].
10.5
CONCLUDING REMARKS AND PROSPECTS
The most distinguishing feature of weakly bound bosonic molecules formed of fermionic atoms is their remarkable collisional stability, despite the fact that they are in the highest rovibrational state. As we mentioned in the Introduction, the lifetime of such molecules can be of the order of seconds or even tens of seconds at densities ∼1013 cm−3 , depending on the value of the two-body scattering length. This allows for interesting manipulations with these molecules. One of the ideas relates to reaching extremely low temperatures in a gas of fermionic atoms at a < 0 and achieving the superfluid BCS regime. This regime has not been obtained so far because of difficulties with evaporative cooling of fermionic atoms due to Pauli blocking of their elastic collisions. The route to BCS may be the following. In the first stage, one arranges a deep evaporative cooling of the molecular Bose-condensed gas to temperatures of the order of the chemical potential. Then one converts the molecular BEC into fermionic atoms by adiabatically changing the scattering length to negative values. This provides an additional cooling, and the obtained atomic Fermi gas will have extremely low temperatures, 10−2 TF , where TF is the Fermi temperature. The gas can then enter the superfluid BCS regime [81]. Moreover, at such temperatures elastic collisions are suppressed by a very strong Pauli blocking and the thermal cloud is in the collisionless regime. This is promising for identifying the BCS-paired state through the observation of collective oscillations or free expansion [16,19,82–84]. It will also be interesting to transfer weakly bound molecules of fermionic atoms to their ground (or less excited) rovibrational state. For molecules of bosonic atoms this has been done using two-photon spectroscopy [85–87] and by magnetically tuned mixing of neighbouring molecular levels, which enables otherwise forbidden radiofrequency transitions [88]. Long lifetimes of weakly bound molecules of fermionic atoms at densities ∼1013 cm−3 may ensure an efficient production of ground-state molecules compared to the case of more short-lived molecules of bosonic atoms. One could then extensively study the physics of molecular Bose–Einstein condensation. Moreover, the ground-state heteronuclear molecules have a relatively large permanent dipole moment and can be polarized by an electric field. This may be used to create a gas of dipoles interacting via anisotropic long-range forces, which drastically changes the physics of Bose–Einstein condensation (e.g., Ref. [89] and references therein). In © 2009 by Taylor and Francis Group, LLC
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experiments at JILA weakly bound fermionic 40 K –87 Rb molecules have been created and transferred to low lying rovibrational states [90], and recently to the ground state [91,92]. They have been cooled to temperatures very close to the regime of quantum degeneracy, which opens possibilities for future studies of unconventional superfluid pairing in dipolar fermionic gases. In the last few years, the observation of the Efimov effect was one of the important goals in cold atom studies. As we discussed in Section 10.3, the Efimov trimers in the gas phase are short-lived and rather represent narrow resonances. The Efimov effect then manifests itself in the log-periodic dependence of collision rates on the two-body scattering length. In particular, this is the case for the rate of three-body recombination of atoms [70] and for the rate of trimer formation in molecule–molecule collisions [67]. In this sense, the trimer formation in gases of bosonic molecules consisting of heavy and light fermions (such as LiYb) attracts great interest as the observation of the Efimov oscillations requires a much smaller change of the two-body scattering length (by a factor of 7 or 5) than in the case of identical bosons. Of particular interest are the trimer states of two heavy and one light fermion in an optical lattice. For two-dimensional densities ∼108 cm−2 the rate of trimer formation can be of the order of seconds, and these states can be detected optically. As we mentioned in Section 10.4, the lattice trimers are long-lived, with a lifetime that can be of the order of tens of seconds. Thus, it is interesting to study to what extent these nonconventional states, in which the heavy atoms are localized in different sites and the light atom is delocalized between them, can exhibit the Efimov effect. The creation of a superlattice of molecules in an optical lattice also looks feasible. A promising candidate is the 6 Li–40 K mixture, as the Li atom may tunnel freely in a lattice while localizing the heavy K atoms to reach high mass ratios. A lattice with period 250 nm and K effective mass M ∗ = 20 M provide a tunneling rate ∼103 sec−1 sufficiently fast to let the crystal form. Near a Feshbach resonance, a value a = 500 nm gives a binding energy 300 nK, and lower temperatures should be reached in the gas. The parameters nκ−2 0 of Figure 10.13 are then obtained at two-dimensional densities in the range 107 –108 cm−2 easily reachable in experiments.
ACKNOWLEDGMENTS The work on this review was financially supported by the IFRAF Institute, by ANR (grants 05-BLAN-0205 and 06-Nano-014), by the EuroQUAM program of ESF (project Fermix), by Nederlandse Stichtung voor Fundamenteel Onderzoek der Materie (FOM), and by the Russian Foundation for Fundamental Research. LKB is a research unit no. 8552 of CNRS, ENS, and of the University of Pierre et Marie Curie. LPTMS is a mixed research unit no. 8626 of CNRS and the University Paris-Sud.
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82. Menotti, C., Pedri, P., and Stringari, S., Expansion of an interacting Fermi gas, Phys. Rev. Lett., 89, 250402, 2002. 83. O’Hara, K.M., Hemmer, S.L., Gehm, M.E., Granade, S.R., and Thomas, J.E., Observation of a strongly interacting degenerate Fermi gas of atoms, Science, 298, 2179, 2002. 84. Kinast, J., Hemmer, S.L., Gehm, M.E., Turlapov, A., and Thomas, J.E., Evidence for superfluidity in a resonantly interacting Fermi gas, Phys. Rev. Lett., 92, 150402, 2004. 85. Kerman, A.J., Sage, J.M., Sainis, S., Bergeman, T., and DeMille, D., Production and state-selective detection of ultracold RbCs molecules, Phys. Rev. Lett., 92, 153001, 2004. 86. Sage, J.M., Sainis, S., Bergeman, T., and DeMille, D., Optical production of ultracold polar molecules, Phys. Rev. Lett., 94, 203001, 2005. 87. Winkler, K., Lang, F., Thalhammer, G., Straten, P.v.d., Grimm, R., and Hecker Denschlag, J., Coherent optical transfer of Feshbach molecules to a lower vibrational state, Phys. Rev. Lett., 98, 043201, 2007. In this experiment the presence of an optical lattice suppressed inelastic collisions between molecules of bosonic 87 Rb atoms, which provided a highly efficient transfer of these molecules to a less excited ro-vibrational state and a long molecular lifetime of about 1 second. 88. Lang, F., Straten, P.v.d., Brandstatter, B., Thalhammer, G., Winkler, K., Julienne, P.S., Grimm, R., and Hecker Denshlag, J., Cruising through molecular bound-state manifolds with radiofrequency, Nature Physics, 4, 223, 2008. 89. Santos, L. and Pfau, T., Spin-3 chromium Bose–Einstein condensates, Phys. Rev. Lett., 96, 190404, 2006. 90. Ospelkaus, S., Pe’er,A., Ni, K.-K., Zirbel, J.J., Neyenhuis, B., Kotochigova, S., Julienne, P.S., Ye, J., and Jin, D.S., Ultracold dense gas of deeply bound heteronuclear molecules, Nature Physics, 4, 622, 2008. 91. Ni, K.-K., Ospelkaus, S., de Miranda, M.H.G., Pe’er, A., Neyenhuis, B., Zirbel, J.J., Kotochigova, S., Julienne, P.S., Jin, D.S., and Ye, J., A high phase-space-density gas of polar molecules, Science, 322, 231, 2008. 92. Ospelkaus, S., Ni, K.-K., de Miranda, M.H.G., Neyenhuis, B., Wang, D., Kotochigova, S., Julienne, P.S., Jin, D.S., and Ye, J., Ultracold polar molecules near quantum degeneracy, arXiv:0811.4618.
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Theory of Ultracold 11 Feshbach Molecules Thomas M. Hanna, Hugo Martay, and Thorsten Köhler CONTENTS 11.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Microscopic Theory of Feshbach Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Zeeman Effect in the Hyperfine Structure of Alkali–Metal Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Interatomic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Singlet and Triplet Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Bound States and Scattering Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Two-Channel Two-Potential Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Two-Channel Single-Resonance Approach . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Magnetic Tuning of the Scattering Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Resonance Width and Background-Scattering Length . . . . . . . . . . . . 11.4.2 Relation between Bound-State Energy and Resonance Position . 11.5 Classification of Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Closed-Channel Dominated Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Entrance-Channel Dominated Resonances . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1
399 400 400 402 402 404 406 406 407 409 409 410 412 412 414 416 416 416
INTRODUCTION
Resonances in the collision cross-sections of particles have a long history of theoretical studies in atomic and molecular physics [1–3], as well as in nuclear physics [4,5]. Their recent applications to Feshbach-resonance phenomena in cold gases often refer to a situation in which a dilute assembly of atoms is exposed to a spatially homogeneous magnetic field. Such an experimental setup allows magnetic tuning of the interatomic interactions. Early suggestions referred to the possibility of manipulating collision cross-sections in dilute vapors of spin-polarized hydrogen 399 © 2009 by Taylor and Francis Group, LLC
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and deuterium [6], as well as lithium [7]. At temperatures in the sub-microkelvin regime, the proposed applications of Feshbach resonances involved magnetic tuning of the s-wave scattering length [8], as well as formation of molecules [9], in trapped gases of alkali–metal atoms at the threshold between scattering and molecular binding. While the possibility of manipulating scattering lengths was demonstrated in a sodium Bose–Einstein condensate [10], the production of ultracold molecules using magnetically tunable Feshbach resonances has been the subject of experimental studies in Bose [11–14] and Fermi [15–19] gases. A review on cold molecules in general can be found, for instance, in Ref. [20]. Their applications in the context of the cross-over from Bardeen–Cooper–Schrieffer (BCS) pairing to molecular Bose–Einstein condensation are detailed in Refs. [21–26]. This chapter gives an overview of theoretical methods for describing the properties of diatomic molecules produced from ultracold atomic gases using the technique of magnetically tunable Feshbach resonances. Section 11.2 summarizes the microscopic description of diatomic energy spectra in terms of the coupled-channels theory. This summary involves brief descriptions of the Zeeman effect in the hyperfine structure of alkali–metal atoms and of the basic interatomic interactions relevant to ultracold gases. Section 11.3 gives an overview of two-channel approximations modeling Feshbach resonances at the threshold between scattering and molecular binding. These methods are applied in Section 11.4 to describe the relation between singularities of the scattering length and the energy of the highest excited vibrational molecular bound state. Section 11.5 extends these applications to the Feshbach-resonance-enhanced collision physics, as well as the associated boundstate properties, at finite energies about the scattering threshold in ultracold gases. Section 11.6 presents the conclusions of this chapter.
11.2 11.2.1
MICROSCOPIC THEORY OF FESHBACH MOLECULES ZEEMAN EFFECT IN THE HYPERFINE STRUCTURE OF ALKALI–METAL ATOMS
Magnetic tuning of diatomic bound-state and scattering properties relies on the Zeeman effect in the hyperfine structure of alkali–metal atoms. The splitting into sublevels of the 2 S1/2 electronic ground state of such an atom which is exposed to a magnetic field B can be described by the following Hamiltonian comprised of hyperfine and Zeeman interactions [27]: Hint =
Chf s · i + μB (ge s + gn i) · B. 2
(11.1)
Here s denotes the spin of the single unpaired valence electron, i is the nuclear spin, Chf is the hyperfine-structure constant, ge and gn are the electronic and nuclear gyromagnetic factors, respectively, and μB is the Bohr magneton. The sign convention for gn used in Equation 11.1 follows Ref. [27]. At zero magnetic field, the electronic ground state is split into two hyperfine levels characterized by the quantum number f associated with the total atomic angular © 2009 by Taylor and Francis Group, LLC
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momentum, f = s + i. Using the general relation, s·i=
1 2 f − s2 − i 2 , 2
(11.2)
the energy shifts of the individual levels as obtained from the hyperfine interaction of Equation 11.1 are determined by [28] ΔE =
Chf f ( f + 1) − s(s + 1) − i(i + 1) . 2
(11.3)
Here i is the angular-momentum quantum number associated with the nuclear spin, while the electron spin is characterized simply by s = 1/2. In the presence of an external magnetic field of strength B = |B| the hyperfine levels of total angular-momentum quantum number f are split into Zeeman sublevels. Because the rotational symmetry is still preserved about the field direction, these sublevels can be characterized by the magnetic quantum numbers mf associated with the component of f along this direction. This is illustrated in Figure 11.1 for the example of an i = 3/2 alkali–metal atom. Although the Hamiltonian of Equation 11.1 does not commute with f 2 , the Zeeman sublevels are often labeled by the pair of quantum numbers ( fmf ) associated with the degenerate hyperfine state they correlate with adiabatically in the limit of zero magnetic field. Nuclear-spin quantum numbers, i, and numerical values of ge and gn for the stable isotopes of all alkali– metal atomic species, as well as the hyperfine transition frequencies for their 2 S1/2 electronic ground states are tabulated in Ref. [27]. 6000 4000
mf = +2 mf = +1 mf = 0
f=2
mf = –1
E/h (MHz)
2000 0
mf = –2
Ehf/h
–2000 –4000
mf = –1
f=1
mf = 0
–6000 0
500
1000 B (G)
1500
mf = +1 2000
FIGURE 11.1 Zeeman splitting of the f = 1 (solid curves) and f = 2 (dashed curves) hyperfine-energy levels vs. magnetic field strength, B, referring to the 2 S1/2 electronic ground state of 87 Rb. Given the nuclear-spin quantum number, i = 3/2 [27], as well as s = 1/2, the hyperfine constant Chf and the hyperfine transition frequency between the f = 1 and f = 2 levels at zero magnetic field, Ehf /h = 6835 MHz [27], are related by Chf = Ehf /2 in accordance with Equation 11.3.
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INTERATOMIC INTERACTIONS
Using the Born–Oppenheimer approximation, the Hamiltonian of the relative motion describing bound molecular states as well as collisions of two ground-state alkali– metal atoms in the presence of a magnetic field is given by [8,29] 2 2 int ∇ + Hn + Vint . 2μ 2
H=−
(11.4)
n=1
Here μ is the reduced mass, H1int and H2int are the Hamiltonians defined in Equation 11.1 for each atom, and Vint is the effective interaction potential depending on the relative position of the atoms, r. For many applications, such as the description of broad scattering resonances and their associated Feshbach molecules, it is sufficient to include in Vint only the rotationally symmetric singlet and triplet Born–Oppenheimer potentials, VS=0 and VS=1 , respectively. Their labels S = 0 and S = 1 refer to the possible values of the angular-momentum quantum number associated with the total spin of the two atomic valence electrons, S = s1 + s2 . In this approximation, the interaction part of Equation 11.4 can be represented by [8,29] Vint = V0 P0 + V1 P1 ,
(11.5)
where P0 and P1 project onto the subspaces of definite total electron-spin quantum number, S = 0, 1. Several applications, such as predictions of narrow scattering resonances for heavy alkali–metal atomic species or spin relaxation in collisions of atoms in excited Zeeman states, may require the inclusion of higher-order corrections (e.g., Refs. [29–32]) in Equation 11.5. Whereas the singlet and triplet potentials depend only on the inter-atomic distance, r = |r|, these additional interactions are generally directional, that is, they couple different partial waves in the relative motion of the atoms. All examples of bound-state and scattering phenomena throughout this chapter might be described, however, at least qualitatively, on the basis of the isotropic Born–Oppenheimer potentials only.
11.2.3
SINGLET AND TRIPLET POTENTIALS
The inset of Figure 11.2 shows the typical shapes of singlet and triplet potentials associated with interactions of alkali–metal atoms of the same chemical species. At short distances, their functional forms are governed by Pauli repulsion of the two electron clouds. Their behavior at asymptotically large interatomic distances is determined by an attractive van der Waals potential, that is, VS (r) ∼ −C6 /r 6 , r→∞
(11.6)
where the van der Waals coefficient C6 is the same for the singlet and triplet states. To provide accurate predictions on the magnetic-field locations of low-energy scattering resonances using Equation 11.4, the Born–Oppenheimer potentials are © 2009 by Taylor and Francis Group, LLC
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Energy (au)
rf0(r)
0.02
3Su+
0.01 0 1S + g
–0.01 0
10
20
Triplet
Singlet 0 0
1000
at = –388 a0
2000 r (a0)
3000
4000
as = +2795 a0
FIGURE 11.2 Sketch of the relation between singlet (dashed curve) and triplet (solid curve) zero-energy scattering wavefunctions plotted vs. the interatomic distance r in units of the Bohr radius a0 = 0.0529 nm, and the associated scattering lengths, as and at , respectively. The values of as and at , referring here to the 85 Rb isotope [33], are determined by the zeros of the linear asymptotes of the scattering wavefunctions. Inset: Typical shapes of singlet (dashed curve) and triplet (solid curve) Born–Oppenheimer potentials for rubidium atoms in their 2 S1/2 electronic ground states [34,35].
usually calibrated in such a way that they recover the correct C6 coefficient (e.g., Refs. [36–39]) as well as the associated scattering lengths (e.g., Refs. [33,40–52]). The singlet and triplet scattering lengths, as and at , respectively, are determined by the solutions of the stationary Schrödinger equation
2 d 2 − + VS (r) rφ0 (r) = 0, 2μ dr 2
(11.7)
which exhibit the following long-range asymptotic behavior at zero energy [53]: rφ0 (r) ∝ r − a.
(11.8)
Here a refers to either as or at . Such wavefunctions and their linear asymptotes determining the singlet and triplet scattering lengths are sketched in Figure 11.2. Table 11.1 summarizes the parameters as , at , and C6 of the Born–Oppenheimer potentials for pairs of identical alkali–metal atoms. Ideally, the singlet and triplet potentials depend only on the chemical element, while the differences in as and at for different isotopes are only due to their different reduced masses in Equation 11.7. © 2009 by Taylor and Francis Group, LLC
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TABLE 11.1 Singlet and Triplet Scattering Lengths as and at , Respectively, as well as Long-Range van der Waals Dispersion Coefficients C6 Associated with the Interactions of Several Homonuclear Alkali–Metal Atom Pairs Species 6 Li 7 Li 23 Na 39 K 40 K 41 K 85 Rb 87 Rb 133 Cs
as (a0 )
at (a0 )
45.167(8) [50] 33(2) [41] 19.20(30) [46] 138.49(12) [52] 104.41(9) [52] 85.53(6) [52] 2795 +420 −290 [33] 90.4(2) [33] 280.37(6) [49]
−2140(18) [50] −27.6(5) [41] 62.51(50) [46] −33.48(18) [52] 169.67(24) [52] 60.54(6) [52] −388(3) [33] 98.98(4) [33] 2440(25) [49]
C6 (au) 1393.39(16) [37] 1393.39(16) [37] 1561 [38] 3927.171 [52] 3927.171 [52] 3927.171 [52] 4703(9) [33] 4703(9) [33] 6860(25) [49]
All quantities are given in atomic units.
11.2.4
BOUND STATES AND SCATTERING RESONANCES
The solutions of the stationary Schrödinger equation associated with the Hamiltonian of Equation 11.4 determine the molecular energy levels as well as the scattering resonances for a pair of alkali–metal atoms in their electronic ground states. In order to calculate molecular bound and continuum levels, it is convenient to perform a partial-wave analysis of the stationary wavefunction with respect to the orbital angular √ momentum of the relative motion of the atoms, . Given a definite modulus ( + 1) of and the projection m onto the magnetic field direction, an asymptotic scattering channel is defined through a channel state, 5 6 |α = f1 mf1 , f2 mf2 , m .
(11.9)
Here the pairs of indices ( f1 mf1 ) and ( f2 mf2 ) determine the Zeeman levels of the atoms at asymptotically large separations. The curly brackets on the right-hand side of Equation 11.9 indicate that the channel state is symmetric if the atoms are identical bosons and antisymmetric if they are identical fermions [29]. The minimum energy of an atom pair at asymptotically large separations determines an associated channel energy, Eα = Ef1 mf1 + Ef2 mf2 ,
(11.10)
where Ef1 mf1 and Ef2 mf2 are the Zeeman energies of the individual atoms. Given the energy E of a stationary wavefunction associated with the Hamiltonian of Equation 11.4, a scattering channel with index α is said to be open when the channel energy, Eα , is below E. Otherwise the channel is referred to as closed. Stable molecular bound states and continuum levels are separated by the scattering (or dissociation) © 2009 by Taylor and Francis Group, LLC
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threshold determined by the lowest possible channel energy. Long-lived metastable molecular states might exist, however, also in the presence of open decay channels (e.g., Ref. [54]). The stationary wavefunction of an atom pair, Ψ(r, E), can be determined using the coupled-channels method [29,31,55]. To this end, Ψ(r, E) is expanded in terms of basis-set components ψα (r, E) associated with the channel states defined in Equation 11.9. Using the radial wavefunctions, Fα (r, E) = r ψα (r, E),
(11.11)
the molecular bound states and continuum levels are determined by the following set of coupled equations: ∂ 2 Fα (r, E) 2μ + 2 E δαβ − Vαβ (r) Fβ (r, E) = 0. ∂r 2
(11.12)
β
Here α and β are the channel indices and 2 ( + 1) int δαβ + Vαβ (r) Vαβ (r) = Eα + 2μr 2
(11.13)
includes the interatomic interaction potentials. Using the approximation of Equation 11.5, the scattering channels are coupled in Equation 11.12 simply because the product of Zeeman states in Equation 11.9 is not necessarily associated with a definite total electronic-spin quantum number S. For this reason, the off-diagonal part of Equation 11.13 involves linear combinations of the singlet and triplet potentials. Additional directional forces lead to coupling also between different partial waves in the relative motion of the atoms [29]. Due to rotational symmetry about the direction of the magnetic field, however, the quantum number, M = mf1 + mf2 + m , representing the projection of the total diatomic angular momentum Ftotal = f1 + f2 + onto this direction is strictly conserved [29]. Figure 11.3 shows energies of 87 Rb2 molecular bound states for = 0 vs. B below the ( f1 = 1, mf1 = +1; f2 = 1, mf2 = +1) scattering threshold calculated using only the singlet and triplet potentials in Equation 11.13. Due to the conservation law M = mf1 + mf2 = 2, there are five coupled s-wave scattering channels in this example, namely ( f1 , mf1 ; f2 , mf2 ) = (1, 1; 1, 1), (1, 1; 2, 1), (1, 0; 2, 2), (2, 1; 2, 1), and (2, 0; 2, 2). The bound-state labels ( f1 , f2 )v at zero magnetic field strength indicate the vibrational quantum number, v = −1, −2, −3, . . . , counted downward from the ( f1 , f2 ) threshold which gives rise to the series of molecular energy levels [56]. Intersections of such bound-state energies with the scattering threshold, as indicated by the filled circles in Figure 11.3, are expected to lead to resonant enhancement of the zero-energy collision cross-section for a pair of 87 Rb atoms prepared in their ( f = 1, mf = +1) Zeeman ground state. © 2009 by Taylor and Francis Group, LLC
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(1,1) –2 (1,2) –4 (1,2) –4
–10,000 E/h (MHz)
–11,000 (1,1) –3 (2,2) –5 –12,000
(2,2) –5
–13,000 0
200
400
600 B (G)
800
1000
FIGURE 11.3 Sketch of several coupled-channel bound-state energies (solid curves) associated with 87 Rb2 molecules below the ( f1 = 1, mf1 = +1; f2 = 1, mf2 = +1) scattering threshold (dotted curve) versus the magnetic field strength, B. The calculations shown were performed using a five s-wave channel spherical-box model including only the central interaction of Equation 11.5. Bound-state labels to the left refer to the quantum numbers ( f1 , f2 )v. Intersections of bound-state energies with the scattering threshold lead to resonant enhancement in collisions at zero kinetic energy between 87 Rb atoms in their lowest energetic Zeeman levels. Their experimentally observed positions are indicated by filled circles. Precise predictions on these and several other scattering resonances as well as their associated measurements can be found in Ref. [56].
11.3
FESHBACH RESONANCES
11.3.1 TWO-CHANNEL TWO-POTENTIAL APPROACH Resonant enhancements of scattering cross-sections in multichannel collision physics are often described in terms of the Feshbach theory of closed-channel resonance states [57]. Feshbach’s general formalism involves projecting the stationary Schrödinger equation onto complementary subspaces associated with the open and closed scattering channels. This theory has been applied in the context of the nearthreshold collision physics of ultracold gases consisting of alkali–metal atoms in a variety of different approaches (e.g., Refs. [9,30,58]). Within a limited range of energies and magnetic field strengths, the Feshbachresonance enhanced diatomic interactions in ultracold gases can often be described parametrically by simpler two-channel two-potential or two-channel single-resonance methods. Such approaches might be based on the following general form of a twochannel Hamiltonian [59]: Hbg W (r) H= . (11.14) W (r) Hcl (B) Here r is the interatomic distance and B denotes the strength of the homogeneous magnetic field. The diagonal matrix elements of Equation 11.14 involve kinetic- and © 2009 by Taylor and Francis Group, LLC
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potential-energy contributions of the following form: Hbg = − Hcl (B) = −
2 2 ∇ + Vbg (r), 2μ
(11.15)
2 2 ∇ + Vcl (B, r). 2μ
(11.16)
Here Hbg is referred to as the bare entrance-channel Hamiltonian describing, in the hypothetical absence of interchannel coupling, the configuration of Zeeman states in which the individual atoms of s-wave interacting pairs in an ultracold gas are prepared. The potential Vbg (r) thus mimics a bare entrance-channel interaction. Two-channel approaches can be used when the entrance channel is coupled significantly only to a single Zeeman-state configuration of atom pairs that belongs to an energetically closed channel. Accordingly, the bare closed-channel Hamiltonian, Hcl (B), describes such atom pairs in the hypothetical absence of interchannel coupling. This implies that the dissociation thresholds of Vcl (B, r) and Vbg (r) are separated by the positive difference in energies associated with the closed-channel and entrance-channel Zeeman-state configurations. Throughout the remainder of this chapter, the zero of energy is the dissociation threshold of the entrance-channel potential. Due to this convention, the Hamiltonian of Equation 11.14 depends on the magnetic field strength B only through the bare closed-channel Hamiltonian. The off-diagonal matrix elements W (r) on the right-hand side of Equation 11.14 provide the interchannel coupling. Although Vbg (r) and Vcl (B, r) are often chosen to have the typical form of diatomic potentials, governed by Pauli repulsion at short distances and by the van der Waals attraction, −C6 /r 6 , in the limit r → ∞, the coupling-matrix element W (r) mimics the microscopic spin-exchange or dipolar interactions. Figure 11.4 illustrates such a two-channel two-potential model [60] determined by Equation 11.14, in combination with Equations 11.15 and 11.16. This example has been set up to describe weakly bound 23 Na2 molecular states leading to resonant enhancement in s-wave collision cross-sections of 23 Na atom pairs prepared in the ( f = 1, mf = +1) Zeeman ground states at magnetic field strengths in the vicinity of 907 G [60]. In this model, Vbg (r) and Vcl (B, r) have been chosen to be of exactly the same shape, with an energy offset determined by the Zeeman effect in the 23 Na hyperfine structure. Their potential well and long-distance behavior have been calibrated in such a way that they recover the exact scattering length at and position of the highest excited vibrational energy level, as well as the C6 coefficient of the sodium triplet potential. For simplicity, the number of vibrational levels supported by Vbg (r) and Vcl (B, r) is reduced to only five as compared to the 16 levels supported by the complete triplet potential. The off-diagonal matrix elements W (r) used in this model to describe the interchannel coupling have the arbitrary form of a decaying exponential function.
11.3.2 TWO-CHANNEL SINGLE-RESONANCE APPROACH Strong interchannel interactions can occur in ultracold diatomic collisions when the magnetically tunable energy of a closed-channel vibrational state, Eres (B), is © 2009 by Taylor and Francis Group, LLC
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nearly degenerate with the entrance-channel dissociation threshold. Such a bare closed-channel Feshbach-resonance state, |φres , is given by the solution of the following Schrödinger equation: Hcl (B)|φres = Eres (B)|φres .
(11.17)
In the example of Figure 11.4, the resonance state is determined by the second vibrational level labeled v = −2 from the top of the sodium triplet√potential, which has been modeled by Vcl (B, r). Its associated radial wavefunction, 4π rφres (r), is indicated by the dotted curve. Given the assumption that the interaction between the channels is caused mainly by |φres , the closed-channel Hamiltonian might be replaced simply by the following one-dimensional term: Hcl (B) → |φres Eres (B)φres |.
(11.18)
100
0
u = –4
–100
–200 u = –5
(4π)1/2rφ b(r)
E/h (GHz)
u = –3
u = –2 0.2 0.1 0 –0.1 –0.2
–300
15
100 r (a0) 20
25
30
35 r (a0)
40
1000
45
50
FIGURE 11.4 Sketch of effective entrance-channel (solid curve) and closed-channel (dashed curve) potentials plotted vs. the interatomic separation r. Horizontal lines refer to the locations of several bare entrance-channel and closed-channel vibrational energy levels counted downward from the top of the potential wells (excluding v = −1 due to its proximity to the dissociation thresholds). In the situation shown, the v = −2 closed-channel vibrational energy level is degenerate with the entrance-channel zero-energy scattering threshold. The associated √ radial resonance-state wavefunction, 4π rφres (r), is indicated by the dotted curve. Inset: Entrance-channel (solid curve) and closed-channel (dashed curve) radial wavefunctions of the highest excited vibrational bound state as obtained from this two-channel two-potential model at a resonance detuning of Eres (B)/h = −10 MHz. The model potentials shown, as well as the off-diagonal couplings and resonance slope ∂Eres /∂B used in this figure, refer to the description of 23 Na2 molecules at magnetic field strengths in the vicinity of 907 G given in Ref. [60].
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Here the resonance state |φres is assumed to be unit normalized, that is φres |φres = 1. Such a replacement in a two-channel Hamiltonian shall be referred to in the following as a single-resonance or configuration-interaction [3] model. Contrary to the two-channel two-potential method, Equation 11.18 always allows the stationary Schrödinger equation associated with the Hamiltonian of Equation 11.14 to be solved analytically in terms of the bare bound and continuum states of Hbg [3,59].
11.4 11.4.1
MAGNETIC TUNING OF THE SCATTERING LENGTH RESONANCE WIDTH AND BACKGROUND-SCATTERING LENGTH
Ultracold diatomic collisions can often be described by a single interaction parameter, the s-wave scattering length, a, which can depend sensitively on B through Eres (B) in Equation 11.18. Within the limited range of magnetic field strengths in which a two-channel single-resonance approach is applicable, the resonance-state energy can usually be approximated by a linear function, Eres (B) =
∂Eres (B − Bres ). ∂B
(11.19)
Here ∂Eres /∂B is the, by assumption, constant difference in magnetic moments between atom pairs in the closed-channel and entrance-channel Zeeman-state configurations, and Bres is the magnetic field strength at which Eres (B) crosses the entrance-channel dissociation threshold. Given that Equation 11.19 is valid, the analytic solution of the stationary Schrödinger equation at zero energy recovers the following parametric expression for the scattering length of Ref. [30]: a(B) = abg 1 −
ΔB . B − B0
(11.20)
Here abg is referred to as the background scattering length, which can be derived, in the hypothetical absence of interchannel coupling, from Equations 11.7 and 11.8 using the entrance-channel potential Vbg (r) instead of VS (r). According to Equation 11.20, the scattering length has a singularity at the magnetic field strength B0 , which is referred to as the resonance position, as well as a zero, whose distance from B0 is given by the resonance width ΔB. Figure 11.5 illustrates these parameters, as well as abg , for pairwise collisions of 87 Rb atoms prepared in the ( f = 1, mf = +1) Zeeman ground state at magnetic field strengths in the vicinity of 1007 G. According to Figure 11.3, it is the vibrational molecular energy level labelled ( f1 = 2, f2 = 2)v = −5 at B = 0 which approaches the scattering threshold at the measured resonance position of B0 = 1007.4 G [61]. The energy and magnetic moment of the near-resonant vibrational level below this threshold [13], as well as the low-energy diatomic collision physics directly above it, can often be described approximately by a properly adjusted two-channel model. © 2009 by Taylor and Francis Group, LLC
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800 600 ΔB
400
abg
a(B) (a0)
200 0
a=0
–200 –400 –600
B0
–800 1006
1006.5
1007
1007.5 B (G)
1008
1008.5
FIGURE 11.5 Illustration of Equation 11.20 describing resonant enhancement of the s-wave scattering length a(B) (solid curve) as a function of the magnetic field strength B. Associated parameters include the resonance position B0 (vertical dashed line), resonance width ΔB, and background scattering length abg (horizontal dotted line). The resonance width determines the distance between the resonance position and the zero crossing of the scattering length as indicated by the vertical dotted line. The specific resonance displayed refers to collisions between 87 Rb atoms in the ( f = 1, m = +1) Zeeman ground state using parameters determined in the f experiments of Refs. [61,62].
11.4.2
RELATION BETWEEN BOUND-STATE ENERGY AND RESONANCE POSITION
Figure 11.6 shows a measured order of magnitude variation of the scattering length for pairs of 23 Na atoms prepared in the ( f = 1, mf = +1) Zeeman ground state in the vicinity of a singularity detected at B0 = 907 G [10]. According to the lower panel, the resonance position exactly coincides with the magnetic field strength at which the energy of the highest excited diatomic vibrational bound state Eb (B) becomes degenerate with the threshold for dissociation. This coupled-channel bound state, |φb , often termed the Feshbach molecule, is determined by the stationary Schrödinger equation H|φb = Eb |φb .
(11.21)
Here H might be either the microscopic Hamiltonian of Equation 11.4 or the two-channel Hamiltonian of Equation 11.14, depending on the level of approximation. As Eb (B) approaches the dissociation threshold, the scattering length is always positive and changes sign at the singularity where the Feshbach molecule vanishes into the continuum. Such a scenario in general is often referred to as a zero-energy resonance [53] or, in the context of ultracold gases, as a Feshbach or Fano–Feshbach resonance. Equation 11.20 gives a parametric description of such singularities of the scattering length within the range of magnetic field strengths in which a single-resonance © 2009 by Taylor and Francis Group, LLC
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a/abg
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–40 Eb(B)
–10–1 –1
–60
890
–10 895
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906.6 905
906.8 910
907 915
B (G)
FIGURE 11.6 (a) Dependence of the scattering length a(B) on the magnetic field strength B for a pair of 23 Na atoms in the ( f = 1, mf = +1) Zeeman ground state in the vicinity of B0 = 907 G. The circles indicate measured data [10] normalized to the background scattering length, while the curve refers to predicted resonance parameters [63]. (b) Energies of the Feshbach molecular state (solid curve labeled Eb ) vs. the magnetic field strength. The inset shows an enlargement of the bound-state energies near resonance as obtained from the twochannel two-potential approach of Ref. [60] (filled circles), a two-channel single-resonance approach (solid curve), in addition to the universal estimate, Eb = −2 /(2μa2 ) (dotted curve). The position of the singularity of a(B) in (a) coincides with the magnetic field strength at which Eb (B) becomes degenerate with the zero-energy entrance-channel dissociation threshold. (From Inouye, S. et al., Nature, 392, 151, 1998. With permission.)
picture is applicable. Using a two-channel single-resonance approach, Figure 11.7 sketches the behavior of the Feshbach molecular bound-state energy and of the s-wave collision cross-section away from the scattering threshold for the narrow (ΔB = 1 G; Refs. [60,63]) B0 = 907 G zero-energy resonance of 23 Na and for the broad (ΔB = 10.71 G; Ref. [64]) B0 = 155 G zero-energy resonance of 85 Rb. Due to © 2009 by Taylor and Francis Group, LLC
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(a) 20
(b)
0.8
15
0.8 4
0.6 10
0.6
0.4
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0.2 0
0 Eb(B) Eres(B) –5
0.4 E/h (MHz)
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2 0.2 0
0 Eb(B)
Eres(B)
–2
–10 –4 −15 23Na −20
900
905 B (G)
910
915
85Rb –6 140
150
160 B (G)
170
180
FIGURE 11.7 Sketch of the relation between the Feshbach molecular bound-state energy below the dissociation threshold and the s-wave collision cross-section above it as a function of the magnetic field strength, B. (a) The B0 = 907 G zero-energy resonance of 23 Na of Figure 11.6. (b) The B0 = 155 G zero-energy resonance associated with a pair of 85 Rb atoms prepared in the ( f = 2, mf = −2) Zeeman state. The s-wave collision cross-section, σ(k), is presented in terms of the dimensionless quantity, sin2 ϑ(k) = k 2 σ(k)/(8π), where k is the angular wavenumber associated with the relative motion of an atom pair and ϑ(k) is the s-wave scattering phase shift [53]. Positive energies E = 2 k 2 /(2μ) refer to diatomic collisions, while negative energies, E < 0, are associated with bound states.
the interchannel coupling, the energy of an unperturbed resonance state Eres (B) above the scattering threshold is generally shifted and broadened [1–5]. For the narrow resonance in Figure 11.7a, the maximum collision cross-section, as well as the bound-state energy Eb (B) follow the linear trend of Eres (B), apart from an unresolved energy range about the dissociation threshold. For the broad resonance in Figure 11.7b, the slope and width of the collision cross-section, as well as the dependence of the bound-state energy on the magnetic field strength, B, differ qualitatively from Eres (B) over the entire range of energies shown. A detailed analysis [58,65] indicates that the associated strong interchannel interactions result from the comparatively large width of the B0 = 155 G zero-energy resonance of 85 Rb, in combination with the large negative background scattering length of abg = −443 a0 [64].
11.5 11.5.1
CLASSIFICATION OF RESONANCES CLOSED-CHANNEL DOMINATED RESONANCES
Signatures of resonant enhancement in such s-wave collision cross-sections at finite kinetic energy have been observed, for instance, in experiments on Feshbach molecule © 2009 by Taylor and Francis Group, LLC
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Ediss/kB (μK)
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n(E/kB) (1/mK)
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Sudden 1 G/msec 0.2 G/msec 0.1 G/msec Asymptotic
3 2 1
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100 150 . B (G/msec)
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FIGURE 11.8 Calculated probability densities for the dissociation of 87 Rb2 Feshbach molecules into free atom pairs with kinetic energy of the relative motion E for different ramp speeds B˙ at the end of a linear magnetic field sweep. In accordance with associated experiments [62], the sweep starts at a magnetic field strength Bi = B0 − 50 mG and terminates at Bf = B0 + 40 mG in the vicinity of the B0 = 685 G zero-energy resonance of 87 Rb atoms prepared in the ( f = 1, mf = +1) Zeeman ground state. Solid and dotted curves refer to the idealized scenarios of an infinitely fast switch of the magnetic field strength from Bi to Bf , and an asymptotically wide sweep across the zero-energy resonance, starting and ending infinitely far away from B0 , respectively. Inset: Comparison between calculated average kinetic energies of the atomic constituents and measurements on Feshbach molecular dissociation [66] using asymptotically wide linear magnetic field sweeps across the 23 Na zero-energy resonance of Figure 11.6. (Reprinted data from Figure 11.2 from Mukaiyama, T. et al., Phys. Rev. Lett. 92, 180402, 2004. With permission. Copyright (2004) by the American Physical Society.)
dissociation using sufficiently fast magnetic field sweeps across B0 , from positive to negative scattering lengths [62]. As an example, Figure 11.8 shows calculated energy spectra of Feshbach molecule fragments dissociated by linear sweeps of variable speed B˙ across a narrow (ΔB = 6.2(6) mG; Ref. [62]) zero-energy resonance for pairs of 87 Rb atoms prepared in their Zeeman ground state. As the ramp speed increases, a pronounced peak appears at an energy determined by the final magnetic field strength, given that the resonance is sufficiently narrow, such as the rubidium zero-energy resonance of this example, which is visible in Figure 11.3 at B0 = 685 G, or the sodium resonance of Figure 11.6. The position of this peak in the dissociation-energy spectra of Figure 11.8 coincides, to a good approximation, with the kinetic energy of maximum s-wave collision cross-section at the final magnetic field strength, Bf = B0 + 40 mG. Apart from a narrow energy range about the scattering threshold, the associated resonance state is closed-channel dominated, similar to the resonance in the s-wave collision cross-section shown in the left image of Figure 11.7a. © 2009 by Taylor and Francis Group, LLC
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As the ramp speed B˙ decreases for given initial and final magnetic field strengths, the dissociation-energy spectrum approaches its universal limit [66], as indicated by the dotted curve in Figure 11.8, which depends only on the product abg ΔB and on ˙ The inset of Figure 11.8 shows measured kinetic energies of the relative motion B. of Feshbach molecule fragments originating from dissociation by linear magnetic field sweeps across the sodium B0 = 907 G zero-energy resonance in the universal limit [66]. Whereas this asymptotic shape of a dissociation-energy spectrum is common to all zero-energy resonances, the appearance of a sharp peak as in Figure 11.8 depends on the width of the resonance above the scattering threshold being much smaller than its energy, as in Figure 11.7a.
11.5.2
ENTRANCE-CHANNEL DOMINATED RESONANCES
Broadness of zero-energy resonances tends to indicate strong interchannel interactions, as illustrated in Figure 11.7b. The strength of interactions between the entrance and closed scattering channels might be estimated on the basis of the resonance width ΔB, in addition to abg , ∂Eres /∂B, and a single parameter characterizing the long-distance behavior of the Born–Oppenheimer potentials, the van der Waals length [67], 1 lvdW = (2μC6 /2 )1/4 . (11.22) 2 Given the parameters ΔB, abg , and ∂Eres /∂B, Wigner’s threshold law [68] yields the following estimate for the energy width of a resonance [30,60] above the scattering threshold [66]: Γres (E) = k abg (∂Eres /∂B) ΔB. (11.23) Here k is the angular wavenumber related to the kinetic energy of the relative motion of the colliding atoms by E = 2 k 2 /(2μ). The interchannel interactions might be considered to be strong when the width of a resonance exceeds its energy within a 2 ), as displayed in range limited by the van der Waals energy, EvdW = 2 /(2μlvdW Figure 11.7. Using Equation 11.23 and k = 1/lvdW , this requirement leads to the following criterion characterizing a broad resonance [65,69]: 2 ) 2 /(2μlvdW 1. abg (∂Eres /∂B) ΔB/lvdW
(11.24)
The Feshbach molecule bound states associated with broad resonances, in the sense of Equation 11.24, tend to be entrance-channel dominated over an experimentally significant range of magnetic field strengths and energies below the dissociation threshold. Figure 11.9 shows the bound-state energy Eb as a function of the magnetic field strength B for a typical example, the B0 = 155 G zero-energy resonance of 85 Rb, which is shown in Figure 11.7b. As B decreases toward the resonance position B0 , that is, when a(B) diverges to infinity, the energy Eb (B) smoothly approaches the dissociation threshold in accordance with the universal formula [71,72], Eb = −2 /(2μa2 ). © 2009 by Taylor and Francis Group, LLC
(11.25)
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–0.2
E/h (MHz)
0 –0.4
–0.6
B0
–2 –4 –6
–0.8 –8 –1 156
–10
156
160
157
158
164 159 B (G)
160
161
162
FIGURE 11.9 Bound-state energies of 85 Rb2 Feshbach molecules vs. magnetic field strength B in the vicinity of B0 = 155 G displayed similarly to Figure 11.6b. The squares with error bars refer to measurements near resonance [64] using the atom–molecule Ramsey interferometry technique of Ref. [11], while filled circles indicate a coupled-channels calculation [70]. For comparison, the dotted curve represents the universal estimate for Eb (B) of Equation 11.25. Inset: Comparison between the different theoretical approaches showing their increasing deviations with increasing magnetic field strength away from the resonance position. (Data reprinted from Figure 11.2 from Claussen, N.R. et al., Phys. Rev. A, 67, 060701(R), 2003. With permission. Copyright (2003) by the American Physical Society.)
This dependence of Eb (B) only on the scattering length indicates that the associated bound-state wavefunction consists almost entirely of its entrance-channel component of asymptotic form [72], 1 e−r/a , (11.26) φb (r) = √ 2πa r in the limit a → +∞. As illustrated in the inset of Figure 11.6, the asymptotic boundstate energy of Equation 11.25 also applies to 23 Na2 Feshbach molecules associated with the narrow, in the sense of Equation 11.24, closed-channel dominated zeroenergy resonance, at B0 = 907 G. At magnetic field strengths just below B0 , the 23 Na Feshbach molecular wavefunction, as sketched in the inset of Figure 11.4, also 2 eventually acquires a universal entrance-channel component given by Equation 11.26. In the limit a → +∞, the bound-state wavefunction of Equation 11.26 can have an arbitrarily long range determined only by the scattering length, probing a region of distances where the classical attractive force between the atoms may be neglected. The bond length of such a universal Feshbach molecule, that is, its average interatomic distance, is found to be ∞ r = 4π r 2 dr r |φb (r)|2 = a/2. (11.27) 0
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This quantum physical spatial extent might be compared to the outer turning point of a hypothetically classical motion determined by the bound-state energy of Equation 11.25 and by the long-range van der Waals potential through the relation 6 Eb = −C6 /rclassical , which yields 1/3 rclassical = a(2lvdW )2 a/2 = r.
(11.28)
In this sense, the Feshbach molecules associated with the measured bound-state energies of Figure 11.9 are in classically forbidden states. For this reason, their universal long-range entrance-channel wavefunction of Equation 11.26, in the limit a → +∞, formally refers to the interaction-free stationary Schrödinger equation with energy Eb of Equation 11.25.
11.6
CONCLUSIONS
In summary, this chapter has given an overview of several theoretical methods to describe magnetic tuning of the s-wave scattering length in ultracold gases of alkali– metal atoms and its relation to the physics of diatomic Feshbach molecules. These methods involve the microscopic coupled-channels theory, its reduction to effective two-channel approaches, as well as the universal description of Feshbach molecules. The presentation of these models accounts for their decreasing ranges of validity with respect to the energy about the scattering threshold and to the magnetic field strength in the vicinity of zero-energy resonances.
ACKNOWLEDGMENTS We are grateful to Chris Greene, Eleanor Hodby, Wolfgang Ketterle, Servaas Kokkelmans, Takashi Mukaiyama, Sarah Thompson, and Carl Wieman for allowing us to present their experimental and theoretical data. This work has been supported by the United Kingdom Engineering and Physical Sciences Research Council (grant number EP/E025935/2) and by a University Research Fellowship of the Royal Society.
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43. Bohn, J.L., Burke, J.P., Greene, C.H., Wang, H., Gould, P.L., and Stwalley, W.C., Collisional properties of ultracold potassium: Consequences for degenerate Bose and Fermi gases, Phys. Rev. A, 59, 3660, 1999. 44. Burke, J.P., Greene, C.H., Bohn, J.L., Wang, H., Gould, P.L., and Stwalley, W.C., Determination of 39 K scattering lengths using photoassociation spectroscopy of the 0− g state, Phys. Rev. A, 60, 4417, 1999. 45. Wang, H., Nikolov, A.N., Ensher, J.R., et al., Ground-state scattering lengths for potassium isotopes determined by double-resonance photoassociative spectroscopy of ultracold 39 K, Phys. Rev. A, 62, 052704, 2000. 46. Samuelis, C., Tiesinga, E., Laue, T., Elbs, M., Knöckel, H., and Tiemann, E., Cold atomic collisions studied by molecular spectroscopy, Phys. Rev. A, 63, 012710, 2000. 47. Modugno, G., Ferrari, G., Roati, G., Brecha, R.J., Simoni, A., and Inguscio, M., Bose– Einstein condensation of potassium atoms by sympathetic cooling, Science, 294, 1320, 2001. 48. Loftus, T., Regal, C., Ticknor, C., Bohn, J., and Jin, D., Resonant control of elastic collisions in an optically trapped Fermi gas of atoms, Phys. Rev. Lett., 88, 173201, 2002. 49. Chin, C., Vuletic, V., Kerman, A.J., Chu, S., Tiesinga, E., Leo, P.J., and Julienne, P.S., Precision Feshbach spectroscopy of ultracold Cs2 , Phys. Rev. A, 70, 032701, 2004. 50. Bartenstein, M., Altmeyer, A., Riedl, S., et al., Precise determination of 6 Li cold collision parameters by radio-frequency spectroscopy on weakly bound molecules, Phys. Rev. Lett., 94, 103201, 2005. 51. D’Errico, C., Zaccanti, M., Fattori, M., Roati, G., Inguscio, M., Modugno, G., and Simoni, A., Feshbach resonances in ultracold 39 K, New J. Phys., 9, 223, 2007. 52. Falke, S., Knöckel, H., Friebe, J., Riedmann, M., Tiemann, E., and Lisdat, C., Ground-state scattering lengths for potassium isotopes determined by double-resonance photoassociative spectroscopy of ultracold 39 K, Phys. Rev. A, 78, 012503, 2008. 53. Taylor, J.R., Scattering Theory, Wiley, New York, 1972. 54. Thompson, S.T., Hodby, E., and Wieman, C.E., Spontaneous dissociation of 85 Rb Feshbach molecules, Phys. Rev. Lett., 94, 020401, 2005. 55. Gao, B., Theory of slow-atom collisions, Phys. Rev. A, 54, 2022, 1996. 56. Marte, A., Volz, T., Schuster, J., Dürr, S., Rempe, G., van Kempen, E.G.M., and Verhaar, B.J., Feshbach resonances in rubidium 87: Precision measurement and analysis, Phys. Rev. Lett., 89, 283202, 2002. 57. Feshbach, H., Theoretical Nuclear Physics, Wiley, New York, 1992. 58. Marcelis, B., van Kempen, E.G.M., Verhaar, B.J., and Kokkelmans, S.J.J.M.F., Feshbach resonances with large background scattering length: Interplay with open-channel resonances, Phys. Rev. A, 70, 012701, 2004. 59. Child, M.S., Molecular Collision Theory, Academic, London, 1974. 60. Mies, F.H., Tiesinga, E., and Julienne, P.S., Manipulation of Feshbach resonances in ultracold atomic collisions using time-dependent magnetic fields, Phys. Rev. A, 61, 022721, 2000. 61. Volz, T., Dürr, S., Ernst, S., Marte, A., and Rempe, G., Characterization of elastic scattering near a Feshbach resonance in 87 Rb, Phys. Rev. A, 68, 010702(R), 2003. 62. Dürr, S., Volz, T., and Rempe, G., Dissociation of ultracold molecules with Feshbach resonances, Phys. Rev. A, 70, 031601(R), 2004. 63. van Abeelen, F.A. and Verhaar, B.J., Unpublished, quoted by Inouye, S. et al. in Ref. [10], 1998. 64. Claussen, N.R., Kokkelmans, S.J.J.M.F., Thompson, S.T., Donley, E.A., Hodby, E., and Wieman, C.E., Very-high-precision bound-state spectroscopy near a 85 Rb Feshbach resonance, Phys. Rev. A, 67, 060701(R), 2003.
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65. Julienne, P.S. and Gao, B., Simple theoretical models for resonant cold atom interactions. In Roos, C., Häffner, H., and Blatt, R. Eds., Atomic Physics, Vol. 20, American Institute of Physics, Conference Proceedings 869, 2006, p. 261–268; e-print physics/0609013. 66. Mukaiyama, T., J.R. Abo-Shaeer, Xu, K., Chin, J.K., and Ketterle, W., Dissociation and decay of ultracold sodium molecules, Phys. Rev. Lett., 92, 180402, 2004. 67. Jones, K.M., Tiesinga, E., Lett, P.D., and Julienne, P.S., Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering, Rev. Mod. Phys., 78, 483, 2006. 68. Wigner, E.P., On the behavior of cross-sections near thresholds, Phys. Rev., 73, 1002, 1948. 69. Petrov, D.S., Three-boson problem near a narrow Feshbach resonance, Phys. Rev. Lett., 93, 143201, 2004. 70. Servaas Kokkelmans, private communication, quoted by Donley, E.A. et al. in Ref. [11], 2002. 71. Braaten, E. and Hammer, H.W., Universality in few-body systems with large scattering length, Phys. Rep., 428, 259, 2006. 72. Bethe, H.A., Theory of the effective range in nuclear scattering, Phys. Rev., 76, 38, 1949.
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Condensed Matter 12 Physics with Cold Polar Molecules Guido Pupillo, Andrea Micheli, Hans-Peter Büchler, and Peter Zoller CONTENTS 12.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Overview: Strongly Interacting Systems of Cold Polar Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Effective Many-Body Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Self-Assembled Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Blue-Shielding and Three-Body Interactions . . . . . . . . . . . . . . . . . . . . . 12.2.4 Hubbard Lattice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.5 Lattice Spin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.6 Hubbard Models in Self-Assembled Dipolar Lattices . . . . . . . . . . . . 12.3 Engineering of Interaction Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Molecular Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1.1 Rotational Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1.2 Coupling to External Electric Fields . . . . . . . . . . . . . . . . . . . . 12.3.2 Two Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2.1 Designing the Repulsive 1/r 3 Potential in 2D . . . . . . . . . 12.3.2.2 Designing ad hoc Potentials with ac Fields . . . . . . . . . . . . 12.4 Many-Body Physics with Cold Polar Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Two-Dimensional Self-Assembled Crystals . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Floating Lattices of Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 Three-Body Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.4 Lattice Spin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1
421 423 423 425 427 430 431 433 436 436 436 437 438 439 444 447 447 450 455 459 463
INTRODUCTION
The realization of Bose–Einstein condensates (BEC) and quantum degenerate Fermi gases with cold atoms has been one of the highlights of experimental atomic physics during the last decade [1]. In view of recent progress in the experimental work on the production of cold molecules we expect a similarly spectacular 421 © 2009 by Taylor and Francis Group, LLC
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development for molecular physics [2–22]. The outstanding features of the physics of cold atomic and molecular gases are the microscopic knowledge of the manybody Hamiltonians, as realized in the experiments, combined with the possibility to control and tune system parameters via external fields. External field control can be achieved by confining ultracold gases with magnetic, electric and optical traps, allowing for the formation of quantum gases in one-, two-, and threedimensional geometries, and tuning contact interparticle interactions by varying the scattering length via Feshbach resonances [23,24]. This control is key to the experimental realization of fundamental quantum phases, as illustrated by the BEC– BCS (Bardeen–Cooper–Schrieffer) crossover in atomic Fermi gases [25–29], the Kosterlitz–Thouless transition [30–32] and the superfluid Mott insulator quantum phase transition with cold bosonic atoms in an optical lattice [33,34]. A recent highlight has been the realization of a degenerate magnetic dipolar gas of 52 Cr atoms [35–37]. In this chapter we will mainly discuss heteronuclear molecules prepared in the electronic and vibrational ground state. Polar molecules are characterized by large electric dipole moments associated with rotational excitations. This gives rise to large dipole– dipole interactions between molecules, which can be manipulated with external dc and ac microwave fields. The possibility to tune these strong, long-range and anisotropic interactions raises interesting prospects for cold ensembles of polar molecules as strongly correlated systems [38–54]. Much work has recently been devoted to the study of cold collisions in dipolar gases [55–61], which in this book is reviewed in Chapters 1, 2, 3, and 4 contributed by Hutson, Bohn, Dalgarno, and Krems. In the context of degenerate molecular gases many recent studies have focused on the regime of weak interactions, where the isotropic contact interaction potential competes with the anisotropic long-range dipole–dipole interaction. For example, the existence of rotons in weakly interacting dipolar gases has been predicted [62–70], while exciting prospects have been envisioned for rotating systems [71–76] and polar molecules in optical lattices [77–85]. In this chapter we will be interested in the many-body dynamics of polar molecules in the strongly interacting limit. In particular, we will develop a toolbox for engineering interesting many-body Hamiltonians based on the manipulation of the electric dipole moments with external dc and ac fields, and thus of the molecular interactions. This forms the basis for the realization of novel quantum phases in these systems. Our emphasis will be on condensed-matter physics, while we refer to the contribution by Yelin, DeMille, and Côté in the present book for applications in the context of quantum information processing [86–91]. This chapter is organized as follows. In Section 12.2 we give a qualitative tour through some of the key ideas of engineering Hamiltonians and of the associated quantum phases. This is followed by two slightly more technical sections, Section 12.3.1 and Section 12.3.2, where we provide details of the realization of a two-dimensional setup where particles interact via purely repulsive 1/r 3 potentials, and where we sketch how to design more complicated interactions by using a combination of ac and dc fields. Section 12.4 illustrates the possibility of inducing strongly correlated phases and realizing quantum simulations by tuning intermolecular interactions. © 2009 by Taylor and Francis Group, LLC
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423
OVERVIEW: STRONGLY INTERACTING SYSTEMS OF COLD POLAR MOLECULES
In this section we give a qualitative overview of many-body physics of cold polar molecules with emphasis on strongly interacting systems. In the following sections we will revisit some of the topics for a more indepth discussion.
12.2.1
EFFECTIVE MANY-BODY HAMILTONIANS
Hamiltonians underlying condensed-matter physics of N structureless bosonic or fermionic particles have the generic form Heff =
N
i=1
p2i 3D + Vtrap (ri ) + Veff ({ri }), 2m
(12.1)
where p2i /2m is the kinetic energy term, and Vtrap (ri ) is a confining potential for the 3D ({r }) represents an effective N-body interaction, which can particles. The term Veff i be expanded as a sum of two-body and many-body interactions 3D Veff ({ri }) =
N
N
V 3D ri −rj + W 3D ri , rj , rk + · · · ,
i<j
(12.2)
i<j
3D ({r }) where in most cases only two-body interactions are considered. Typically, Veff i must be interpreted as effective interactions valid in a low-energy theory, obtained by integrating out the high-energy degrees of freedom of the system. In the following we will discuss the derivation of such many-body Hamiltonians for a cold ensemble of polar molecules in the electronic and vibrational ground state, and show how the effective N-body interactions in these systems can be “designed” by controlling rotational excitations with external fields, in a form unique to polar molecules. Our starting point is the Hamiltonian for a gas of cold heteronuclear molecules prepared in their electronic and vibrational ground state,
H(t) =
N
i
+
N
p2i (i) + Vtrap (ri ) + Hin − di E(t) 2m Vdd (ri − rj ).
(12.3)
i<j
Here the first term in the single-particle Hamiltonian corresponds to the kinetic energy of the molecules, while Vtrap (ri ) represents a trapping potential, as provided, for (i) example, by an optical lattice, or an electric or magnetic trap. The term H in describes the internal low-energy excitations of the molecule, which for a molecule with a closed electronic shell 1 Σ(ν = 0) (e.g., SrO, RbCs, or LiCs) correspond to the rotational degree of freedom of the molecular axis. This term is well described by a rigid rotor © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications (i)
Hin ≡ Hrot = BJi2 where B is the rotational constant (in the few to tens of gigahertz regime) and Ji is the dimensionless angular momentum. The rotational eigenstates |J, M for a quantization axis z, and with eigenenergies BJ(J + 1) can be coupled by a static (dc) or microwave (ac) field E via the electric dipole moment di , which is typically of the order of a few Debye. For distances outside the molecular core, the two-body interaction as given by the second part Equation 12.3 is the dipole–dipole interaction di · dj − 3(di · er ) er · dj Vdd (r) = . (12.4) r3 Here r ≡ |r| = |ri − rj | denotes the distance between two polar molecules, and er is the unit vector along the collision axis. Due to the large electric dipole moments this term provides a comparatively strong, long-range anisotropic interaction between the molecules. The many-body dynamics of cold polar molecules is thus governed by an interplay between dressing and manipulating the rotational states with dc and ac fields, and strong dipole–dipole interactions. In the absence of electric fields, the molecules prepared in a ground rotational state J = 0 have no net dipole moment, and interact via a van der Waals attraction VvdW ∼ −C6 /r 6 , reminiscent of the interactions of cold alkali metal atoms in the electronic ground states. Electric fields admix excited rotational states and induce static or oscillating dipoles, which interact via strong dipole–dipole interactions Vdd with the characteristic 1/r 3 dependence given in Equation 12.4. Note that two parallel dipoles repel each other, while dipoles aligned along the collision axis will attract each other, which may possibly lead to instabilities in a many-body system. Thus, stable many-particle phases may often be possible only in reduced geometries, that is, in combination with an external trapping potential Vtrap (ri ). Finally, we emphasize that microwave excited rotational states of polar molecules are long-lived, which prevents significant decoherence. This is in contrast to atomic systems, where spontaneous emission from laser excited electronic states is one of the main contributions to decoherence. The connection between the full molecular N-particle Hamiltonian of Equation 12.3 including rotational excitations and dressing fields, and the effective Hamiltonian of (Equation 12.1) can be made using a Born–Oppenheimer approxi 4N (i) mation. The diagonalization of the Hamiltonian [92] HBO = i Hin − di E + 4N i<j Vdd (ri − rj ) for frozen spatial positions {ri } of the N molecules yields a set of 3D ({r }), which can be interpreted as the effective N-particle energy eigenvalues Veff i potential in the single-channel many-body Hamiltonian (Equation 12.1). The depen3D ({r }) on the electric fields E provides the basis for the engineering of dence of Veff i the many body interactions in (Equation 12.2). The validity of this adiabatic approximation and of the associated decoupling of the Born–Oppenheimer channels will be discussed below. The above considerations set the stage for a discussion of engineering many-body Hamiltonians for polar molecules, and associated quantum phases. In Sections 12.2.2 and 12.2.3 we will discuss several examples of dc and ac field configurations for designing specific two-body and three-body interactions. Our discussion can also © 2009 by Taylor and Francis Group, LLC
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be adapted to optical lattices, and thus to a derivation of Hubbard models for polar molecules (Section 12.2.4). Section 12.2.5 extends these derivations to molecules with nonzero electron spin. Extended Hubbard models with couplings to phonons are presented in Section 12.2.6 for molecules trapped in self-assembled dipolar cystals.
12.2.2
SELF-ASSEMBLED CRYSTALS
The conceptually simplest example, although remarkably rich from a physics point of view, is a system of cold polar molecules in a dc electric field under strong transverse confinement. The setup is illustrated in Figure 12.1a. A weak dc field along the z-direction induces a dipole moment d in the ground state of each molecule. 3D (r) = D(r 2 − These molecules interact via the effective dipole–dipole interaction Veff 2 5 2 3z )/r according to their induced dipoles, with D = d . For molecules confined to the (x, y)-plane perpendicular to the√ electric field this interaction is purely repulsive. For molecules displaced by z > r/ 3 the interaction becomes attractive, resulting in an instability in the many-body system. This instability can be suppressed by a sufficiently strong two-dimensional confinement with the potential Vtrap (zi ) along z, due to, for example, an optical force induced by an off-resonant light field [46]. The two-dimensional dynamics in this pancake configuration is described by the Hamiltonian
p2ρi
2D 2D = Heff + Veff (ρij ), (12.5) 2m i
i<j
which is obtained by integrating out the fast z-motion. Equation 12.5 is the sum of the two-dimensional kinetic energy in the (x, y)-plane and a repulsive two-dimensional dipolar interaction 2D (ρ) = D/ρ3 , Veff
(12.6)
with ρij ≡ (xj − xi , yj − yi ) a vector in the (x, y)-plane (solid line in Figure 12.1b). The distinguishing feature of the system described by the Hamiltonian (Equation 12.5) is that tuning the induced dipole moment d drives the system from a weakly interacting gas (a two-dimensional superfluid in the case of bosons), to a crystalline phase in the limit of strong repulsive dipole–dipole interactions. This transition and the crystalline phase have no analog in the atomic bose gases with short-range interactions modeled by a pseudopotential of a given scattering length. A crystalline phase corresponds to the limit of strong repulsion where particles undergo small oscillations around their equilibrium positions, which is a result of the balance between the repulsive long-range dipole–dipole forces and an additional (weak) confining potential in the (x, y)-plane. The relevant parameter is rd ≡
Epot D/a3 Dm = 2 = 2 , 2 Ekin /ma a
(12.7)
which is the ratio of the the interaction energy and the kinetic energy at the mean interparticle distance a. This parameter is tunable as a function of d from small rd to © 2009 by Taylor and Francis Group, LLC
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(a)
+kL
Eac(t)
Edc
ϑ
d2 r
d1
j ez
ez
ϑ
–kL
d1
j
ey
ex
ex
(b) 1
d2 r
ey
V2D eff (r) (arb. units)
C3 / r3
0 0.5
1
1.5
2
2.5 r (arb. units)
FIGURE 12.1 (a) System setup. Polar molecules are trapped in the (x, y)-plane by an optical lattice made of two counter propagating laser beams with wavevectors ±kL = ±kL ez . The dipoles are aligned in the z-direction by a dc electric field Edc ≡ Edc ez . An ac microwave field is indicated. Inset: Definition of polar (ϑ) and azimuthal (ϕ) angles for the relative orientation of the intermolecular collision axis r12 with respect to a space-fixed frame, with axis along 2D (ρ) for polar molecules z. (b) Qualitative sketch of effective two-dimensional potentials Veff confined in a two-dimensional (pancake) geometry. Here, ρ = r12 sin ϑ(cos ϕ, sin ϕ, 0) is the two-dimensional coordinate in the plane z = 0 and ρ = r12 sin ϑ (see inset of (a)). Solid line: 2D (ρ) = D/ρ3 induced by a dc electric field. Dash–dotted line: Repulsive dipolar potential Veff “Step-like” potential induced by a single ac microwave field and a weak dc field. Dashed line: Attractive potential induced by the combination of several ac fields and a weak dc field. 2D (ρ) and the separation ρ are given in arbitrary units. (From Micheli, A. The potentials Veff et al., Phys. Rev. A, 76, 043604, 2007. With permission.) © 2009 by Taylor and Francis Group, LLC
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large. A crystal forms for rd 1, when interactions dominate. For a dipolar crystal, this is the limit of large densities, where collisions become harmful. However, the crystalline phase will protect a cold ensemble of polar molecules from (harmful) close-encounter collisions. This density dependence is different from that in Wigner crystals with 1/r-Coulomb interactions, as realized for example with laser cooled trapped ions [93]. In the latter case rc = (e2 /a)/2 /ma2 ∼ a and the crystal forms at low densities. In addition, the charge e is a fixed quantitiy, while d can be varied as a function of the dc field. Figure 12.2a shows a schematic phase diagram for a dipolar gas of bosonic molecules in two-dimensional as a function of rd and temperature T . In the limit of weak interactions rd < 1, the ground state is a superfluid with a finite (quasi-) condensate. In the opposite limit of strong interactions rd 1 the polar molecules are in a crystalline phase for temperatures T < Tm with Tm ≈ 0.09D/a3 [94]. The configuration with minimal energy is a triangular lattice with excitations given by acoustic phonons. In Ref. [46] we investigated the intermediate strongly interacting regime with rd 1, using a recently developed path integral Monte Carlo technique (PIMC) [95]. We determined the critical interaction strength rQM = 18 ± 4 for the quantum melting transition from the crystal into the superfluid, a result that has been confirmed with a number of quantum Monte Carlo techniques [47,48]. In Section 12.4 below we return to the discussion of these quantum phases, and in particular the crystalline phase, to show that the relevant regime of parameters for these phases to occur is accessible with polar molecules. Besides the fundamental interest in dipolar quantum gases, the crystalline phase has interesting applications, such as in the context of quantum information [91]. We will return to self-assembled dipolar lattices below in a discussion of Hubbard models.
12.2.3
BLUE-SHIELDING AND THREE-BODY INTERACTIONS
By combining dc and ac fields to dress the manifold ofrotational energy levels it is possible to design effective interaction potentials V 3D ri − rj with (essentially) any shape as a function of distance. For example, the addition of a single linearlypolarized ac field to the configuration of Figure 12.1a leads to the realization of the two-dimensional “step-like” potential of Figure 12.1b, where the character of the repulsive potential varies considerably in a small region of space. The derivation of this effective two-dimensional interaction is outlined in Figure 12.3 and discussed in more detail in Section 12.3.2.2 below [46,61]. The (weak) dc field splits the first excited rotational (J = 1) manifold of each molecule by an amount δ, while a linearly polarized ac field with Rabi frequency Ω is blue-detuned from the (|g − |e) transition by Δ (Figure 12.3a). Because of δ and the choice of polarization, for distances ρ (d 2 /δ)1/3 the relevant single-particle states for the two-body interaction reduce to the states |g and |e of each molecule. Figure 12.3b shows that the dipole–dipole interaction splits the excited state manifold of the two-body rotational spectrum, making the detuning Δ position-dependent. As a consequence, the combined energies of the bare ground state of the two-particle spectrum and of a microwave photon become degenerate with the energy of a (symmetric) excited state at a characteristic resonant (Condon) point ρC = (d 2 /Δ)1/3 , which is represented © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications (a)
Ta3/D
0.15
Normal
Tm
0.1
0.05 Superfluid
Crystal
0 0
rQM
10
30
rd
40
(b1) 1.25W0
0.17W0
0.02W0
(b2)
1.71W0
1.25W0
0.47W0
0.23W0
0.21W0
1.38W0
0.58W0
0.58W0
0.34W0
0.28W0
(b3)
FIGURE 12.2 (a) Tentative phase diagram in the T − rd plane: crystalline phase for interactions rd > rQM and temperatures below the classical melting temperature Tm (dashed line) [94]. The superfluid phase appears below the upper bound T < π2 n/2 m (dotted line) [31,32]. The quantum melting transition is studied at fixed temperature T = 0.014 D/a3 with interactions rd = 5 − 30 (dash–dotted line). The crossover to the unstable regime for small replusion and finite confinement ω⊥ is indicated (hatched region). (b) Strengths of the dominant three-body interactions Wijk appearing in the Hubbard model of Equation 12.9 for different lattice geometries: (b1) one-dimensional setup; (b2) two-dimensional square lattice; (b3) two-dimensional honeycomb lattice. The characteristic energy scale W0 = γ2 DR06 /a6 is discussed later in Equation 12.38. (From Micheli, A. et al., Phys. Rev. A, 76, 043604, 2007; Büchler, H.P., Nature Phys., 3, 726, 2007. With permission.)
by an arrow in Figure 12.3b. At this Condon point, an avoided crossing occurs in a field-dressed picture, and the new (dressed) ground-state potential inherits the character of the bare ground and excited potentials for distances ρ ρC and ρ ρC , respectively. Figure 12.3c shows that the dressed ground-state potential (which has © 2009 by Taylor and Francis Group, LLC
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Condensed Matter Physics with Cold Polar Molecules (a)
(b)
ħD ˙ eÒ
ħwe
~ E(r, z = 0)/ħD
(c)
E(r, z = 0)/ħwe
2
˙ e; eÒ
+2
ħd +1 ˙ e–1Ò
˙ e+1Ò
W
D/we
˙ g; eÒ+
1 ˙ g; eÒ–
W
0
˙ g; gÒ
2W/D
˙ g; eÒ+ –1 ˙ g; eÒ–
˙ gÒ
0
˙ g; gÒ
1 0.5
1
r/rc
2
–2 0.5
˙ e; eÒ 1
r/rc
2
FIGURE 12.3 Design of the step-like potential of Figure 12.1b: (a) Rotational spectrum of a molecule in a weak dc field. The dc field splits the (J = 1) manifold by an amount δ. The linearly polarized microwave transition with detuning Δ and Rabi frequency Ω is shown as an arrow. (b) Born–Oppenheimer√ potentials for the internal states for Ω = 0 (bare potentials), where |g; e± ≡ (|g; e ± |e; g)/ 2. The resonant Condon point ρC is indicated by an arrow. (c) ac-field-dressed Born–Oppenheimer potentials. The dressed ground-state potential has the largest energy. (From Büchler, H.P., et al., Phys. Rev. Lett., 98, 060404, 2007. With permission.)
the largest energy) is almost flat for ρ ρC and it is strongly repulsive as 1/ρ3 for ρ ρC , which corresponds to the realization of the step-like potential of Figure 12.1b. We remark that, due to the choice of polarization, this strong repulsion is present only in the plane z = 0, while for z = 0 the ground-state potential can become attractive. The optical confinement along z of Figure 12.1a is therefore necessary to ensure the stability of the system. The interactions in the presence of a single ac field are described in detail in Ref. [61], where it is shown that in the absence of external confinement, this case is analogous to the (three-dimensional) optical blue-shielding developed in the context of ultracold collisions of neutral atoms [96–98], however with the advantage of the long lifetime of the excited rotational states of the molecules, as opposed to the electronic states of cold atoms. The strong inelastic losses observed in three-dimensional collisions with cold atoms [96–98] can be avoided via a judicious choice of the field’s polarization, eventually combined with a tight confinement to ensure a twodimensional geometry (as, e.g., in Figure 12.3). For example, in Ref. [99] it is shown that in the presence of a dc field and of a circularly polarized ac field the attractive timeaveraged interaction due to the rotating (ac-induced) dipole moments of the molecules allows for the cancelation of the total dipole–dipole interaction. The residual interactions remaining after this cancelation are purely repulsive three-dimensional 3D (r) ∼ (d 4 /Δ)/r 6 . This interactions with a characteristic van der Waals behavior Veff three-dimensional repulsion provides for a shielding of the inner part of the interaction potential and thus it will strongly suppress inelastic collisions in experiments. This will possibly lead to the realization of quantum degenerate systems of molecules and perhaps to tightly packed crystalline structures in three dimensions [99]. © 2009 by Taylor and Francis Group, LLC
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A cancelation of the leading effective two-body interaction similar to the one described above in a dense ensemble of molecules can lead to the realization of systems where the effective three-body interaction W 3D (ri , rj , rk ) of Equation 12.2 dominates over the two-body term V 3D (ri − rj ) and determines the properties of the system in the ground state. This is interesting, because model Hamiltonians with strong three-body and many-body interactions have attracted a lot of interest in the search for microscopic Hamiltonians exhibiting exotic ground-state properties. Well-known examples are the fractional quantum Hall states described by the Pfaffian wavefunctions which appear as ground states of a Hamiltonian with three-body interactions [100–102]. These topological phases support anionic excitations with nonabelian braiding statistic. Three-body interactions are also an essential ingredient for systems with a low energy degeneracy characterized by string nets [103,104], which play an important role in models for nonabelian topological phases. The possibility of realizing a Hamiltonian where the two-body interaction can be manipulated independently of the three-body term has been studied in Ref. [85]. There, it is shown that a stable system of particles interacting via purely repulsive three-body potentials can be realized by combining the setup above with a tight optical confinement provided by an optical lattice. In fact, the latter serves the two-fold purpose of ensuring the collisional stability of the setup and of defining a characteristic length scale (the lattice spacing) where the exact cancelation of the two-body term occurs. The details of this derivation are given in Section 12.4.3, in connection with the derivation of an extended Hubbard model with three-body interactions introduced in Section 12.2.4.
12.2.4
HUBBARD LATTICE MODELS
Hubbard Hamiltonians are model Hamiltonians describing the low-energy physics of interacting fermionic and bosonic particles in a lattice [105]. They have the general tight-binding form H=−
i,j,σ
† Jijσ bi,σ bj,σ
+
Uijσσ i,j,σ,σ
2
ni,σ nj,σ .
(12.8)
† ) are the destruction (creation) operators for a particle at site i in the Here bi,σ (bi,σ internal state σ, Jijσ describes coherent hopping of a particle from site i to site j (typically
the nearest neighbor), and Uijσσ describes the on-site (i = j) or off-site (i = j) two-
† bi,σ . Hubbard models have a long body interactions between particles, with ni,σ = bi,σ history in condensed-matter physics, where they have been used as tight-binding approximations of strongly correlated systems. For example, for a system of electrons in a crystal hopping from the orbital of a given atom to that of its nearest neighbor, σ represents the electron spin. A (fermionic) Hubbard model comprising electrons in a two-dimensional lattice with interspecies on-site interactions is thought to be responsible for the high-temperature superconductivity observed in cuprates [106]. In recent years, Hubbard models have been shown to properly describe the lowenergy physics of interacting bosonic and fermionic atoms trapped at the bottom
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of an optical lattice [107,108]. The resulting low-energy Hamiltonians are of the form of Equation 12.8. Spectacular experiments with ultracold atoms have led to the realization of the superfluid/Mott insulator quantum phase transition for bosonic atoms [33,34], and further experimental work with fermions may resolve the phase diagram of the fermionic Hubbard model in two dimensions by performing an analog quantum simulation of Equation 12.8 with two-species cold fermions [109,110]. Since the interactions between cold atoms are short-ranged, in these systems σ,σ Hubbard Hamiltonian typically have on-site interactions only (Ui,i in Equation 12.8). However, it has been shown that the presence of moderately long-range interactions in Equation 12.8, such as nearest-neighbor interactions, can lead to interesting phases such as checkerboard solids and two-dimensional supersolids [83,84]. Polar molecules in optical lattices can provide for off-site interactions [78–82] that are strong, of the order of hundreds of kilohertz, and long-range; that is they decay with distance as 1/r 3 . Due to these strong interactions, two molecules cannot hop onto the same site, and thus the particles may be treated as effectively “hard-core.” An intriguing possibility is offered by the interaction engineering discussed above in the context of the realization of effective lattice models where particles interact via exotic (extended) Hubbard Hamiltonians. An example of this is given in Ref. [85], where it is shown how to engineer the following Hubbard-like Hamiltonian H = −J
bi† bj +
ij
Uij i=j
2
ni nj +
Wijk ni nj nk , 6
(12.9)
i=j=k
where Wijk ni nj nk is an off-site three-body term. The latter is tunable independently of the two-body term Uij ni nj , to the extent that it can be made to dominate the dynamics and determine the properties of the system in the ground state. In contrast to the common approach to derive effective many-body terms from Hubbard models involving two-body interactions, which are obtained in the J U perturbation limit, and are thus necessarily small [111], the derivation of the Hubbard model (Equation 12.9) is based directly on the effective many-particle potential given by (Equation 12.2). Thus, all the energy scales in Equation 12.9 can be tuned independently, which allows one to obtain comparatively large hopping rates determining the time and temperature scales to observe exotic quantum phases. This is important, because analytical calculations in one dimension suggest that the Hamiltonian (Equation 12.9) has a rich ground-state phase diagram, supporting valence-bond, charge-density-wave, and superfluid phases [85]. In Section 12.4 below we provide the microscopic derivation of the effective interaction potentials of Equation 12.9.
12.2.5
LATTICE SPIN MODELS
The Hamiltonian in Equation 12.3 can be generalized to include other internal degrees of freedom for each molecule in addition to rotation. This offers new possibilities to engineer effective interactions and novel many-body phases. For example, the addition of a spin-1/2 (qubit) degree of freedom to polar molecules trapped in an optical lattice allows us to construct a complete toolbox for the simulation of any permutation symmetric lattice spin models [51]. Lattice spin models are ubiquitous in © 2009 by Taylor and Francis Group, LLC
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condensed-matter physics where they are used to describe the characteristic behavior of complicated interacting physical systems. The basic building block is a system of two polar molecules strongly trapped at given sites of an optical lattice, where the spin-1/2 (or qubit) is represented by a single unpaired electron outside a closed shell of a 2 Σ1/2 heteronuclear molecule in its rotational ground state, as provided, for example, by alkaline-earth monohalogenides. As dicussed above, heteronuclear molecules possess large permanent electric dipole moments, which are responsible for strong, long-range and anisotropic dipole–dipole interactions, whose spatial dependence can be manipulated using microwave fields. The spin-rotation coupling in the molecules makes these dipole–dipole interactions spin-dependent. General lattice spin models can then be readily built from these binary interactions. Although in this review we will present results for spin-1/2 models only, we notice that the inclusion of hyperfine effects offers extensions to spin systems with larger spin. For example, the design of a large class of spin-1 interactions for polar molecules has been shown in Ref. [52], which allows, for example, for the realization of a generalized Haldane model in one dimension [112]. Two highly anisotropic models with spin-1/2 particles that can be simulated are illustrated in Figures 12.4a and 12.4b, respectively. The first takes place on a square two-dimensional lattice with nearest-neighbor interactions (I) Hspin
−1
−1
z z x x = J σi,j σi,j+1 + cos ζσi,j σi+1,j .
(12.10)
i=1 j=1
Introduced by Douçot and colleagues [113] in the context of Josephson junction arrays, this model (for ζ = ±π/2) admits a two-fold degenerate ground subspace that (a)
(b)
S2 Ÿ
D2
S1
y
z
r
E(t) E(t)
D1 Ÿ
z
Ÿ
x
Si
Sj
FIGURE 12.4 Example anisotropic spin models that can be simulated with polar molecules trapped in optical lattices. (a) Square lattice in two-dimensional with nearest-neighbor orientation-dependent Ising interactions along xˆ and zˆ . Effective interactions between the spins S1 and S2 of the molecules in their rovibrational ground states are generated with a microwave field E(t) inducing dipole–dipole interactions between the molecules with dipole moments D1 and D2 , respectively. (b) Two staggered triangular lattices with nearest neighbors oriented along orthogonal triads. The interactions depend on the orientation of the links with respect to the electric field. (Dashed lines are included for perspective.) (From Micheli, A. et al., Nature Phys., 2, 341, 2006. With permission.)
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is immune to local noise up to th order and hence is a good candidate for storing a protected qubit. The second occurs on a bipartite lattice constructed with two two-dimensional triangular lattices, one shifted and stacked on top of the other. The interactions are indicated by nearest-neighbor links along the xˆ , yˆ , and zˆ directions in real space:
y y
(II) Hspin = J⊥ σjx σkx + J⊥ σj σk + Jz σjz σkz . (12.11) x -links y-links z-links This model has the same spin dependence and nearest-neighbor graph as the model on a honeycomb lattice introduced by Kitaev [114]. He has shown that by tuning the ratio of interaction strengths |J⊥ |/|Jz | one can tune the system from a gapped phase carrying abelian anyonic excitations to a gapless phase, which, in the presence of a magnetic field, becomes gapped with nonabelian excitations. In the regime |J⊥ |/|Jz | 1 the Hamiltonian can be mapped to a model with four-body operators on a square lattice with ground states that encode topologically protected quantum memory [115]. One proposal [116] describes how to use trapped atoms in spin-dependent optical lattices (II) to simulate the spin model Hspin . There the induced spin couplings are obtained via spin-dependent collisions in second-order tunneling processes. Larger coupling strengths as provided by polar molecules are desirable. In both spin models (I and II) above, the signs of the interactions are irrelevant, although it is possible to tune the sign if needed.
12.2.6
HUBBARD MODELS IN SELF-ASSEMBLED DIPOLAR LATTICES
In Hubbard models with cold atoms or molecules in optical lattices there are no phonon degrees of freedom corresponding to an intrinsic dynamics of the lattice, as the back action on the optical potentials is typically negligible. Thus, atomic and molecular Hubbard models allow for the study of strong correlations in the absence of phonon effects. However, the simulation of models where the presence of (crystal) phonons strongly affects the (Hubbard) dynamics of the particles remains a challenge. These models are of fundamental interest in condensed-matter physics, where they describe polaronic and/or superconducting materials [117]. In the context of atoms, one example is to immerse atoms moving on a lattice into a BEC of a second atomic species, representing a bath of Bogoliubov excitations [118–120]. A second example is a self-assembled floating lattice of molecules as discussed in Section 12.2.2, which provides a periodic potential for extra atoms or molecules, whose dynamics can again be described in terms of a Hubbard model [53]. Phonon degrees of freedom enter as vibrations of the dipolar lattice. The Hamiltonian for extra atoms or molecules in a self-assembled dipolar lattice is H = −J
ci† cj +
0 1
† Vij ci† cj† cj ci + Hc + Mq eiq·Rj cj† cj (aq + a−q ). 2 i,j
q,j
(12.12) Here, the first and second terms define a Hubbard-like Hamiltonian for the extraparticles of the form of Equation 12.8, where the operators ci (ci† ) are destruction © 2009 by Taylor and Francis Group, LLC
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~ J
dp dc
dp
~ Vi, j
dp Edc
a
dc
dc
ex
ez ey
FIGURE 12.5 Floating lattices of dipoles. A self-assembled crystal of polar molecules with dipole moment dc provides a two-dimensional periodic honeycomb lattice Vcp (darker shading corresponds to deeper potentials) for extra molecules with dipole dp dc giving rise to a lattice model with hopping J˜ and long-range interactions V˜ i,j . (From Pupillo, G. et al., Phys. Rev. Lett., 100, 050402, 2008. With permission.)
(creation) operators of the extra-particles. However, the third and fourth terms describe the acoustic phonons of the crystal and the coupling of the extra-particles to the crystal phonons, respectively. In particular, Hc is the phonon Hamiltonian
Hc = ωq aq† aq , (12.13) q
where aq destroys a phonon of quasimomentum q in the mode λ. Tracing over these phonon degrees of freedom in a strong coupling limit provides effective Hubbard models for the extra-particles dressed by the crystal phonons H˜ = −J˜
ci† cj +
1
V˜ ij ci† cj† cj ci . 2
(12.14)
i,j
The hopping of a dressed extra-particle between the minima of the periodic potential occurs at a rate J˜ , which is exponentially suppressed due to the copropagation of the lattice distortion, while off-site particle–particle interactions V˜ i,j are now a combination of direct particle–particle interactions and interactions mediated by the coupling to phonons. The setup we have in mind is depicted in Figure 12.5, © 2009 by Taylor and Francis Group, LLC
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Condensed Matter Physics with Cold Polar Molecules EJ,M – E0,0 ħw ħD
ħw + 2ħd/3
˙ f1,0Ò
ħw + ħd/3 ħd ħw ħw – ħd/3
q=0 ˙ f1,–1Ò
q = –1
q = +1
˙ f1,1Ò
0 ˙ f0,0Ò
FIGURE 12.6 Solid lines: energies EJ,M (left) and states |φJ,M (right) with J = 0, 1, for a molecule in a weak dc electric field Edc = Edc e0 with β ≡ dEdc /B 1. The dc-field-induced splitting δ and the average energy separation ω ¯ are δ = 3Bβ2 /20 and ω ¯ = 2B + Bβ2 /6, respectively. Dashed and dotted lines: energy levels for a molecule in combined dc and ac fields. (The ac-Stark shifts of the dressed states are not shown.) Dashed line: the ac field is monochromatic, with frequency ω, linear polarization q = 0, and detuning Δ = ω − (ω ¯ + 2δ/3) > 0. Dotted lines: schematics of energy levels for an ac field with polarization q = ±1 and frequency ω = ω. (From Micheli, A. et al., Phys. Rev. A, 76, 043604, 2007. With permission.)
where extra-particles, which are molecules with a dipole moment dp dc interact repulsively with the molecules in the crystal, and thus see a periodic (honeycomb) lattice potential. The distinguishing features of this realization of lattice models are as follows: (i) Dipolar molecular crystals constitute an array of microtraps with its own quantum dynamics represented by phonons (lattice vibrations), while the lattice spacings are tunable with external control fields, ranging from a micrometer down to the hundred nanometer regime, that is, potentially smaller than for optical lattices. (ii) The motion of the extra particles is governed by an interplay of Hubbard (correlation) dynamics in the lattice and coupling to phonons. The tunability of the lattice allows access to a wide range of Hubbard parameters and phonon couplings. Compared with optical lattices, for example, a small-scale lattice yields significantly enhanced hopping amplitudes, which set the relevant energy scale © 2009 by Taylor and Francis Group, LLC
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for the Hubbard model, and thus also the temperature requirements for realizing strongly correlated quantum phases.
12.3
ENGINEERING OF INTERACTION POTENTIALS
In this section we show in some detail how to realize a collisionally stable twodimensional setup where particles interact via a purely repulsive 1/r 3 potential, by using a combination of a dc field and of tight optical confinement in the field’s direction. We then sketch how to design more complicated interactions using a combination of ac, dc, and optical fields, by focusing on the step-like potential of Figure 12.1b. This engineering of interaction potentials is at the core of the realization of the strongly correlated phases and quantum simulations discussed in Section 12.4.
12.3.1
MOLECULAR HAMILTONIAN
We consider spinless polar molecules in the electronic and vibrational ground state X 1 Σ(0). In the following, we are interested in manipulating the rotational states of the molecules using dc and ac electric fields and in confining their motion using a (optical) far off-resonance trap (FORT). The application of these external fields will serve as a key element in engineering effective interaction potentials between the molecules. The low-energy effective Hamiltonian for the external motion and internal rotational excitations of a single molecule is H(t) =
p2 + Hrot + Hdc + Hac (t) + Hopt (r), 2m
(12.15)
where p2 /2m is the kinetic energy for the center-of-mass motion of a molecule of mass m, Hrot accounts for the rotational degrees of freedom, while the terms Hdc , Hac (t), and Hopt (r) refer to the interaction with electric dc and ac (microwave) fields and to the optical trapping potential for the molecule in the ground electronic-vibrational manifold, respectively. In the following we consider tight harmonic optical traps with the frequency ω⊥ = 2π × 150 kHz, the same for all the relevant rotational states of the molecule. That is, we neglect possible tensor-shifts induced by the optical trapping field in the energies of the excited rotational states of the molecules, which in general can be compensated for by an appropriate choice of additional laser fields [61]. Thus, for a confinement along ez , Hopt (r) reads Hopt (r) = mω2⊥ z2 /2, independent of the internal (rotational) quantum state [44,121]. 12.3.1.1
Rotational Spectrum
The term Hrot in Equation 12.15 is the Hamiltonian describing a rigid spherical rotor [122] Hrot = B J2 ,
(12.16)
which accounts for the rotation of the internuclear axis of a molecule with total angular momentum J [122–124]. Rotations are the lowest-energy internal excitations of the © 2009 by Taylor and Francis Group, LLC
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molecule. Here B is the rotational constant for the electronic-vibrational ground state, which is of the order of B ∼ h 10 GHz [125]. We denote the energy eigenstates of the Hamiltonian 12.16 by |J, M, where J is the quantum number associated with the total internal angular momentum and M is the quantum number associated with its projection onto a space-fixed quantization axis. The excitation spectrum is EJ = BJ(J + 1), which is anharmonic. Each J-level is (2J + 1)-fold degenerate. A polar molecule has an electric dipole moment d, which for Σ-state molecules is directed along the internuclear axis eab , that is, d = deab . Here, d is the “permanent” dipole moment of a molecule in its electronic-vibrational ground state. This dipole moment gives rise to the dipole–dipole interaction between two molecules. The spherical component dq = eq · d of the dipole operator in the space-fixed √ spherical basis {e−1 , e0 , e1 }, with eq=0 ≡ ez and e±1 = ∓(ex ± iey )/ 2, couples the rotational states |J, M and |J ± 1, M + q according to J ± 1, M + q|dq |J, M = d(J, M; 1, q|J ± 1, M + q) 2J + 1 × (J, 0; 1, 0|J ± 1, 0) , 2(J ± 1) + 1 where (J1 , M1 ; J2 , M2 |J, M) are the Clebsch–Gordan coefficients. This means that for a spherically symmetric system the eigenstates of the rotor have no net dipole moment, J, M|d|J, M = 0. However, the dipole coupling to an external electric field breaks this spherical symmetry by aligning each molecule along the field’s direction. This induces an interaction between the rotational energy levels of the molecule, and a corresponding finite dipole moment in each rotational state, as explained below. 12.3.1.2
Coupling to External Electric Fields
The terms Hdc and Hac (t) in Equation12.15 are the electric dipole interactions of a molecule with an external dc electric field Edc = Edc ez directed along e0 ≡ ez , and with ac microwave fields Eac (t) = Eac e−iωt eq + c.c., which are linearly (q = 0) or circularly polarized (q = ±1) relative to ez , respectively. Here we have neglected the spatial dependence of Eac because in the following we are interested in dressing the rotational states of the molecules with microwave fields, whose wavelengths are of the order of centimeters, and thus much larger than the size of our system. Then, the terms Hdc and Hac (t) in Equation 12.15 read Hdc = −d · Edc = −d0 Edc , Hac (t) = −d · Eac (t) = −dq Eac e−iωt + h.c.
(12.17a) (12.17b)
In the presence of a single dc electric field Edc (Eac , ω⊥ = 0), the internal Hamiltonian is that of a rigid spherical pendulum [122] H = Hrot + Hdc = BJ2 − d0 Edc which conserves the projection of the angular momentum J on the quantization axis, that is, M is a good quantum number. Thus, the energy eigenvalues and eigenstates are labeled EJ,M and |φJ,M , respectively, where each eigenstate |φJ,M is a superposition © 2009 by Taylor and Francis Group, LLC
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of various states |J, M mixed by the electric dipole interaction (here J, not conserved, is used as a simple label). The effects of a dc electric field on a single polar molecule are shown in Figure 12.6, and they amount to (1) a split in the (2J + 1)-fold degeneracy in the rotor spectrum, and (b) aligning the molecule along the direction of the field. The latter corresponds to inducing a finite dipole moment in each rotational state. For weak fields β ≡ dEdc /B 1, the state |φJ,M and its associated induced dipole moment approximately read √ β β J2 − M2 (J + 1)2 − M 2 |φJ,M = |J, M − |J − 1, M + |J + 1, M, 2 J 3 (2J + 1) 2 (J + 1)3 (2J + 1) (12.18) and φJ,M |d|φJ,M = dβ
3M 2 /J(J + 1) − 1 e0 , (2J − 1)(2J + 3)
respectively. Thus, the ground state acquires a finite dipole moment φ0,0 |d0 |φ0,0 = dβ/3 along the field axis, which is at the origin of ground-state dipole–dipole interactions between polar molecules. We notice that for a typical rotational constant B ∼ h 10 GHz and dipole-moment d ≈ 9 D, the condition β 1 corresponds to dc fields (much) weaker than B/d ≈ 2 kV/cm. Individual transitions of the internal Hamiltonian can be addressed by applying one (or several non-interfering) microwave field Eac (t), which can be linearly or circularly polarized. This is shown in Figure 12.6 for transitions coupling the J = 0 and J = 1 manifolds, where the Rabi frequency Ω and the detuning Δ are Ω ≡ Eac φ1,q |dq |φ0,0 / and Δ ≡ ω − (E1,q − E0,0 )/, respectively. Dressed energy levels of a molecule are obtained by diagonalizing the Hamiltonian H = Hrot + Hdc + Hac (t) in a Floquet picture. That is, first, the Hamiltonian matrix is evaluated in the 4basis |φJ,M , which diagonalizes the time-independent part of H as Hrot + Hdc = J,M |φJ,M EJ,M φJ,M |, and then the time-dependent wavefunction is expanded in a Fourier series in the ac frequency ω. After applying a rotating wave approximation, that is, keeping only the energy conserving terms, one obtains a time˜ whose eigenvalues correspond to the dressed energy independent Hamiltonian H, levels [61].
12.3.2 TWO MOLECULES We now consider the interactions of two polar molecules j = 1, 2 confined to the (x, y) plane by a tight harmonic trapping potential of frequency ω⊥ , directed along z. The interaction of the two molecules at a distance r ≡ r2 − r1 = rer is described by the Hamiltonian H(t) =
2
j=1
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Hj (t) + Vdd (r),
(12.19)
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where Hj (t) is the single-molecule Hamiltonian Equation 12.15, and Vdd (r) is the dipole–dipole interaction of Equation 12.4. In the absence of external fields Edc = Eac = 0, the interaction of the two molecules in their rotational ground state is determined by the van der Waals attraction Vvdw ∼ C6 /r 6 with C6 ≈ −d 4 /6B. This expression for the interaction potential is valid outside of the molecular core region r > rB ≡ (d 2 /B)1/3 , where rB defines the characteristic length at which the dipole–dipole interaction becomes comparable to the splittings of the rotational levels. In the following we show that it is possible to induce and design long-range interaction potentials, by dressing the interactions with appropriately chosen static and/or microwave fields. In fact, the combination of the latter with low-dimensional trapping allows us to engineer effective potentials whose strength and shape can both be tuned. The derivation of the effective interactions proceeds in two steps: 1. We derive a set of Born–Oppenheimer potentials by first separating Equation 12.19 into center-of-mass and relative coordinates, and diagonalizing the Hamiltonian Hrel for the relative motion for fixed molecular positions. Within an adiabatic approximation, the corresponding eigenvalues play the role of an effective three-dimensional interaction potential in a given state manifold dressed by the external field. 2. We eliminate the motional degrees of freedom in the tightly confined z-direction to obtain an effective two-dimensional dynamics with interaction 2D (ρ). Veff In the following we show how to design interaction potentials, presenting in some details the simplest case of a purely repulsive 1/r 3 potential in two-dimensional, obtained using a static electric field (Section 12.3.2.1). We then sketch how to design more elaborate potentials using a combination of static and microwave fields coupling the lowest rotor states of each molecule (Section 12.3.2.2). 12.3.2.1
Designing the Repulsive 1/r 3 Potential in 2D
Collisions in a dc field. We consider a weak static electric field applied in the z-direction E = Edc e0 with β = dEdc /B 1, and in the absence of an optical trapping potential (ω⊥ = 0). The effective interaction potentials for the collision of two particles can be obtained in the adiabatic approximation by neglecting the kinetic energy and by diagonalizing the following Hamiltonian Hrel for fixed particle positions Hrel =
2
BJj2 − Edc d0;j + Vdd (r)
j=1
=
|Φn (r)En (r)Φn (r)|,
(12.20)
n
where En (r) and |Φn (r) are the nth adiabatic energy eigenvalues and two-particle eigenfunctions, respectively. In the limit r → ∞ the latter are symmetrized products © 2009 by Taylor and Francis Group, LLC
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of the single-particle states |φJj ,Mj j of Equation 12.18, while for finite r they are superpositions of several single-particle states, which are mixed by the dipole–dipole interaction Vdd (r). The quantity n ≡ (J; M; σ) is the collective quantum number labeling the eigenvalues En (r), with J = J1 + J2 being the total number of rotational excitations shared by two molecules, M ≡ |M1 | + |M2 | the total projection of angular momentum onto the electric field direction, and σ = ± the permutation symmetry associated with the exchange of the two particles. We note that, due to the presence of the dc field, J is not conserved here and used as a simple label for the various energy manifolds. Because we are mainly interested in collisions of molecules in the ground state, in the following we restrict our discussion to the Jj = 0 and 1 manifolds of each molecule, which include 16 rotational states of the combined two-particle system. Figure 12.7 shows the corresponding eigenvalues En (r) as a function of the interparticle distance r, for β = 1/5. The vector r is expressed in spherical coordinates r = (r, ϑ, ϕ), with ϑ and ϕ the polar and azimuthal angles, respectively, and z = r cos ϑ. Figure 12.7a shows that the energy spectrum has a markedly different behavior in the molecular core region r < rB and for r > rB , with rB ≡ (d 2 /B)1/3 . In fact, for r < rB the energy spectrum is characterized by a series of level crossings and anticrossings, so that the adiabatic approximation is generally not valid. For r > rB the energy levels group into well-defined manifolds approximately spaced by an energy 2B, corresponding to the energy of rotational excitation. In the following we focus on this region r > rB , where the adiabatic approximation can be used. Figures 12.7a through 12.7e present an expanded view of the two lowest-energy manifolds of Figure 12.7a in the region r > rB , for ϑ = π/2 and ϑ = 0, respectively. Figures 12.7b and 12.7d show that the excited-state manifold with one quantum of rotation (J1 + J2 = 1) is asymptotically split into two sub-manifolds. This separation corresponds to the electric-field-induced splitting of the Jj = 1 manifold of each molecule, and it is thus given by δ = 3Bβ2 /20 (see caption of Figure 12.6). More importantly, Figure 12.7c and 12.7e shows that the effective ground-state potential E0 (r) has a very different character for the cases ϑ = π/2 and ϑ = 0, respectively. In fact, in the case ϑ = π/2 (Figure 12.7c), corresponding to collisions in the (z = 0)plane, the potential is attractive for r < r , while for r > r it becomes repulsive and decays at large distances as 1/r 3 , where r is a characteristic length defined below. On the other hand, for ϑ = 0 (see Figure 12.7e) the potential is purely attractive, with dipolar character. This change in character of the ground-state potential as a function of ϑ is captured by the following analytic expression for E0;0;+ (r), derived using a perturbation expansion in Vdd (r)/B, 3D (r) ≡ E0;0;+ (r) ≈ Veff
C C3 6 1 − 3 cos2 ϑ + 6 . 3 r r
(12.21)
Here, the constants C3 ≈ d 2 β2 /9 and C6 ≈ −d 4 /6B are the dipolar and van der Waals coefficients for the ground-state potential, respectively, and the constant term 2E0,0 = −β2 B/3 due to single-particle dc Stark-shifts has been neglected. Equation 12.21 is 3D (r) has a local valid for r rB and Vdd (r)/B 1, and it shows that the potential Veff © 2009 by Taylor and Francis Group, LLC
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Condensed Matter Physics with Cold Polar Molecules (a)
10–2
E2; M;+
2
– [EJ; M;s (r, p/2)–ħw–2E 0,0]/2B
(b)
EJ; M;s (r, 0)/2B
E1; 0;– E 1; 0;+
E2; M;– E1;1 ±;+
ħd
0 E1;1 ±;–
E1; 0;–
E1;1 ±;+
1 E1;1 ±;– 0
E1;0 ;+
–10–2
E0;0 ;+
(c)
1
rd
10
[EJ; M;s (r, p/2)–2E0,0]/2B
10–5 0.5
1
1.5
–10–5
ħd E1;1 ±;+
10–2 1
10
1 (e)
r
10
r/rB
[EJ; M;s (r, 0)–2E0,0]/2B
10–5
E0;0 ;+
0
E1;1 ±;– rd
E0;0 ;+
0
r/rB
– (d) [E J; M;s (r, 0)–ħw–2E0,0]/2B 10–2 E1; 0;– E1; 0;+ 0
r/rB
–10–5 r/rB
1
r
10
r/rB
FIGURE 12.7 Born–Oppenheimer potentials EJ;M;σ (r, ϑ) for two molecules colliding in the presence of a dc field, with β ≡ dEdc /B = 1/5. The solid and dashed curves correspond to symmetric (σ = +) and antisymmetric (σ = −) eigenstates, respectively. (a): Born–Oppenheimer potentials for the 16 lowest-energy eigenstates EJ;M;σ (r, ϑ). The molecular-core region is identified as the region r < rB = (d 2 /B)1/3 , while for r rB the eigenstates group into manifolds separated by one quantum of rotational excitation 2B. (b) and (d) Enlargements of the first excited energy manifold of panel (a) in the region r rB for ϑ = π/2 and ϑ = 0, respectively. Note the electric-field-induced splitting δ ≡ 3Bβ2 /20. The distance rδ where the dipole– dipole interaction becomes comparable to δ is rδ = (d 2 /δ)1/3 . (c) and (e) Enlargements of the ground-state potential E0,0;+ (r, ϑ) of panel (a) in the region r rB for ϑ = π/2 and ϑ = 0, respectively. The distance r of Equation 12.22, where the dipole–dipole interaction becomes comparable to the van der Waals attraction is indicated. Note the repulsive (attractive) character of the potential for ϑ = π/2 (ϑ = 0) and r > r . (From Micheli, A., et al., Phys. Rev. A, 76, 043604, 2007. With permission.)
maximum in the plane z = r cos ϑ = 0 at the position r , defined as 1/3 2|C6 | 1/3 3d 2 r ≡ ≈ , C3 Bβ2
(12.22)
where the dipole–dipole and van der Waals interactions become comparable. The height of this maximum is V =
© 2009 by Taylor and Francis Group, LLC
C3 2 Bβ4 ≈ , 4|C6 | 54
(12.23)
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Cold Molecules: Theory, Experiment, Applications
and the curvature of the potential along z [∂z2 V (r = r , z = 0) = −6C3 /r5 ≡ −mω2c /2] defines a characteristic frequency ωc ≡
12C3 mr5
1/2 ,
(12.24)
to be used below. The latter has a strong dependence β8/3 = (dEdc /B)8/3 on the applied electric field. For distances r r the dipole–dipole interaction dominates 3D (r) ∼ C (1 − 3 cos2 ϑ)/r 3 (see over the van der Waals attractive potential, and Veff 3 Ref. [46]). Thus, if it were possible to confine the collisional dynamics to the (z = 0)plane with r r , rB , purely repulsive long-range interactions with a characteristic dipolar spatial dependence ∼1/r 3 could be attained. In the following we analyze the conditions for realizing this setup, using a strong confinement along z as provided, for example, by an optical trapping potential. Parabolic confinement. The presence of a finite trapping potential of frequency ω⊥ in the z-direction provides for a position-dependent energy shift of Equation 12.21. The new total potential reads V (r) =
C C3 1 6 2 ϑ + 6 + mω2⊥ z2 . 1 − 3 cos 3 4 r r
(12.25)
As noted before, for z = 0 the repulsive dipole–dipole interaction dominates over the attractive van der Waals potential at distances r r given by Equation 12.22. In addition, for ω⊥ > 0 the harmonic potential confines the particle’s motion in the z-direction. Thus, the combination of the dipole–dipole interaction and the harmonic confinement yields a repulsive potential that provides for a three-dimensional barrier separating the long-distance from the short-distance regime. If the collision energy is much smaller than this barrier, the particles’ motion is confined to the long-distance region, where the potential is purely repulsive. Figure 12.8 is a contour plot of V (r) in units of V , for β > 0 and ω⊥ = ωc /10, with r ≡ (ρ, z) = r(sin ϑ, cos ϑ) (the angle ϕ is neglected due to the cylindrical symmetry of the problem). Darker regions correspond to a stronger repulsive potential, while the white region at ρ ≈ 0 is the short-range, attractive part of the interaction. The repulsion due to the dipole–dipole and harmonic potentials is distinguishable at |z|/r ∼ 0 and 7, respectively. The less dark regions located at (ρ⊥ , ±z⊥ ) ≡ ⊥ (sin ϑ⊥ , ± cos ϑ⊥ ) correspond to the existence of two saddle points positioned in between √ the maxima of 2 1/5 3 V (r), with ⊥ = (12C3 /mω⊥ ) and cos ϑ⊥ = 1 − (r /⊥ ) / 5, (see circles in Figure 12.8a). These saddle points act as an effective potential barrier separating the attractive part of the potential present at r < l⊥ from the region r ⊥ ≥ r , rB where the effective interaction potential given by Equation 12.25 is purely repulsive. For collision energies smaller than the height of this barrier the dynamics of the particles can be reduced to quasi two-dimensional, by averaging over the fast particle motion in the z-direction. We notice that the existence of two saddle points at distances r ∼ ⊥ separating the long-distance from the short-distance regimes is a general feature of systems with a comparatively weak transverse trapping potential ω⊥ /ωc < 1, with ωc defined in Equation 12.24. In fact, for a strong transverse trapping potential ω⊥ ≥ ωc © 2009 by Taylor and Francis Group, LLC
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Condensed Matter Physics with Cold Polar Molecules (a)
(b) 7
+5
5.78
6
0.01
0
SE/S0
5 0.01
0 –z^
0.01
4 3
7.01(w^/wc)1/5
2 1
–5 0
r^
0 10 r/r
15
20
10–6 10–5 10–4 10–3 10–2 10–1 100 10+1 10+2 w^/wc
FIGURE 12.8 (a) Contour plot of the effective potential V (ρ, z) of Equation 12.25, for two polar molecules interacting in the presence of a dc field β > 0, and a confining harmonic potential in the z-direction, with trapping frequency ω⊥ = ωc /10, where ωc ≡ (12C3 /mr5 )1/2 of Equation 12.24 and r = (2|C6 |/C3 )1/3 of Equation 12.22. The contour lines are shown for V (ρ, z)/V ≥ 0, with V = Bβ4 /54. Darker regions represent stronger repulsive interactions. The combination of the dipole–dipole interactions induced by the dc field and of the harmonic confinement leads to realizing a three-dimensional repulsive potential. The repulsion due to the dipole–dipole interaction and the harmonic confinement is distinguishable at z ∼ 0 and z/r ∼ ±7, respectively. Two saddle points (circles) located at (ρ⊥ , ±z⊥ ) separate the longdistance region where the potential is repulsive ∼ 1/r 3 from the attractive short-distance region. The gradients of the potential are indicated by dash–dotted lines. The thick dashed line indicates the instanton solution for the tunneling through the potential barrier. (b) The euclidian action SE as a function of ω⊥ /ωc (solid line). For ω⊥ < ωc ≈ 0.88 ωc (ω⊥ > ωc ) the “bounce” occurs for z(0) = 0 (within the plane z(0) = 0), see text. ωc is signaled by a circle. √ The point 2 For ω⊥ > ωc the action is SE ≈ 5.78S0 , with S0 = m|C6 |/r , which is ω⊥ -independent, consistent with the “bounce” occurring in the (z = 0)-plane (see text). (From Micheli, A., et al., Phys. Rev. A, 76, 043604, 2007. With permission.)
the two saddle points collapse into a single one located at z = 0, and ρ = ⊥ ∼ r . In this limit the dynamics is purely two-dimensional, with the particles strictly confined to the (z = 0)-plane. Collisional stability. Inelastic collisions and three body recombination in an ensemble of polar molecules may lead the system to a potential instability, associated with the attractive character of the dipole–dipole interaction [62–70]. In our discussion, this instability is associated with the population of the short-distance region r < ⊥ , which can be efficiently suppressed by strong dipole–dipole interactions and transverse confinement. In fact, for collision energies smaller than the potential barrier V (ρ⊥ , ±z⊥ ) the particles are mostly confined to the long-distance regime, where they scatter elastically. That is, when a cold ensemble of molecules is considered the barrier ensures the stability of the system by “shielding” the short-distance attractive part of the two-body potential. In this limit, residual losses are due to the tunneling through the potential barrier at a rate Γ, which can be effectively suppressed for reasonable values of β and ω⊥ , as shown below. The tunneling rate Γ = Γ0 e−SE / through the barrier V (ρ⊥ , ±z⊥ ) can be calculated using a semi-classical/instanton approach [126]. The euclidian action SE , © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
which is responsible for the exponential suppression √ of the tunneling, is plotted in Figure 12.8 as a function of ω⊥ /ωc , in units of S0 = m|C6 |/r2 . The figure shows that the form of SE is different for ω⊥ ωc and ω⊥ ωc . In fact, for ω⊥ ωc the action increases with increasing ω⊥ as SE ≈ 7.01S0 (ω⊥ /ωc )1/5 = 1.43(⊥ /a⊥ )2 (dotted line), which depends on the confinement along z, via a⊥ = (/mω⊥ )1/2 . On the other hand, for ω⊥ ωc it reads SE ≈ 5.78 S0 , which is ω⊥ -independent. The transition between the two different regimes mirrors the change in the nature of the underlying potential V (r). In particular, for ω⊥ ωc the dynamics is strictly confined to the plane z = 0 and thus it becomes independent of ω⊥ . The constant Γ0 is related to the quantum fluctuations around the semiclassical trajectory, and its value is strongly system-dependent. For the crystalline phase of Ref. [46], it is the collisional “attempt frequency,” proportional to the characteristic phonon frequency Γ0 ≈ C3 /ma5 , where a is the mean interparticle distance. In the limit of strong interactions and tight transverse confinement Γ rapidly tends to zero. We illustrate this for the example of SrO molecules, which have a permanent dipole moment of d ≈ 8.9 D and mass m = 104 amu. Then, for a tight transverse optical lattice with harmonic oscillator frequency ω⊥ = 2π × 150 kHz and for a dc-field β = dEdc /B = 1/3 we have (C32 m3 ω⊥ /85 )1/5 ≈ 3.39 and obtain Γ/Γ0 ≈ e−5.86×3.39 ≈ 2 × 10−9 . Even for a dc field as weak as β = 1/6 we still obtain a suppression by five orders of magnitudes, as Γ/Γ0 ≈ e−5.86×1.94 ≈ 10−5 . This calculation confirms that a collisionally stable system of polar molecules in the strongly interacting regime can be realized by combining the strong dipole–dipole interactions with a tight transverse confinement. Effective two-dimensional interaction. The effective two-dimensional interaction potential is obtained by integrating out the fast particle motion in the transverse direction z. For r > ⊥ a⊥ , the two-particle eigenfunctions in the z-direction approximately factorize into products of single-particle harmonic oscillator wavefunctions 2D /ω , the effective two-dimensional ψk1 (z1 )ψk2 (z2 ), and, to first order in Veff ⊥ 2D interaction potential Veff reads 2D Veff (ρ) ≈ √
1 2πa⊥
dze−z
2 /2a2 ⊥
3D Veff (r).
(12.26)
For large separations ρ ⊥ the two-dimensional potential reduces to 2D Veff (ρ) =
C3 , ρ3
which is a purely repulsive two-dimensional interaction potential. The derivation of 2D (ρ) is one of the central results of this section. We show below (Section 12.4) that Veff this interaction potential can be used to realize interesting many-body phases, in the context of condensed-matter applications with cold molecular quantum gases. 12.3.2.2
Designing ad hoc Potentials with ac Fields
We have shown how to design purely repulsive two-dimensional effective groundstate interactions that decay as ∼1/r 3 . The use of one or several noninterfering ac © 2009 by Taylor and Francis Group, LLC
Condensed Matter Physics with Cold Polar Molecules
445
fields allows us to engineer more complicated interactions by combining the spatial texture of the adiabatic ground-state potential of the two-particle spectrum with that of selected excited potentials, in a field-dressed picture. This mixing of ground- and excited-state potentials is favored by the dipole–dipole interactions that split the degeneracy of the excited-state manifolds of the two-particle spectrum and render the state-selectivity of the ac fields space-dependent, as explained below. In combination with a strong optical confinement, and due to the long lifetimes of excited rotational states [127], this allows for the realization of collisionally stable ensembles of molecules in a strongly interacting regime. We exemplify the situation above by considering the case of a single ac field Eac (t) = Eac e−iωt eq + c.c. which is added to the configuration of Figure 12.7 (interactions in the presence of a static electric field Edc = βBez ). The field’s polarization is chosen to be linear (q = 0) and the frequency ω is blue-detuned from the (|φ0,0 → |φ1,0 )-transition of the single-particle spectrum by an amount Δ = ω − 2B/ > 0. The ac field induces: (1) oscillating dipole moments in each molecule, which give rise to long-range dipole–dipole interactions (in addition to those determined by the static field Edc ), whose sign and angular dependence are given by the polarization q; (2) a coupling between the ground and excited-state manifolds of the two-particle spectrum at a resonant (Condon) point rC = (d 2 /3hΔ)1/3 , where the dipole–dipole interaction becomes comparable to the detuning Δ. This coupling is responsible for an avoided crossing of the field-dressed energy levels at rC , whose properties depend crucially on the polarization q. This fact is at the core of the engineering of interaction potentials, in that the three-dimensional effective dressed adiabatic ground-state interaction potential inherits the character of the bare ground and excited potentials for r rC and r rC , respectively. The setup described above is illustrated in Figure 12.9a and 12.9b, where the continuous and dashed lines are the bare (Eac = 0) symmetric and antisymmetric potentials EJ;M;σ (r) of Figure 12.7, respectively, and the presence of the ac field is signaled by a black arrow at the resonant Condon point rC . The presence of the weak dc field splits the (J = 1)-manifold asymptotically by an amount δ as in Figure 12.7b, which makes it possible the apply to adiabatic approximation in the excited-state manifold for distances r rδ = (d 2 /δ)1/3 . In fact, the energy of the E1;0;+ (r) potential becomes degenerate with the energy of other bare symmetric potentials only at distances r rδ . In addition, we notice that the presence of the splitting δ also shifts the level crossing with antisymmetric states to small distances r rδ . For distances r rδ = (d 2 /δ)1/3 , we are allowed to consider only the four states of Figure 12.9b, because all the other potentials of the (J = 1)- and (J = 2)-manifolds are far detuned by an amount that is (at least) of order δ Δ and they are not coupled by the ac field to the bare ground state E0,0;+ (r), due to the choice of the field’s polarization. Figure 12.9b shows that the splitting induced by the dipole–dipole interaction in the (J = 1)-manifold renders the detuning Δ position-dependent, so that at rC the energy of the bare ground state and that of the symmetric bare excited state become degenerate. The resulting dressed ground-state potential is outlined in Figure 12.9b (bold black line) and it roughly corresponds to the bare E0,0;+ (r) and E1,0;+ (r) potentials for r > rC and r < rC , respectively. Accordingly, Figure 12.9c shows that the dressed ground-state potential E˜ 0;0;+ (r), which has the highest energy, © 2009 by Taylor and Francis Group, LLC
446
Cold Molecules: Theory, Experiment, Applications (a)
(b) [EJ; M;s (r,p/2)–2E0,0]/ħ E2;0;+
– + 4d/3 2w
[EJ; M;s (r,p/2)–2E0,0]/ħ
– 2w +4d/3
E2;0;+
E2;1 ±;±
– + d/3 2w
w +2D
E2;2m;+
– + 2d/3 2w
w +D
– + 5d/3 w
w – w +2d/3
E1;0;±
– + 2d/3 w –– d/3 w w – 4d/3
d
w –2D
E1;1 ±;±
w –2D
W
E1;0;– W
w –4D E0;0;±
0 0
E0;0;+
0
10 20 30 40 50 rc rd r/rB
(c)
E1;0;+
D
0 10 20 30 40 50 60 70 80 90 100 rd rc r/rB
~ [EJ; M;s (r,p/2)–2E0,0]/ħD
2
1 ~ E0;0;+
÷8W
0
~ E1;0;–
–1
~ E1;0;+
~ E2;0;+
–2 0
10 20 30 40 50 60 70 80 90 100 rc r/rB
FIGURE 12.9 (a) Schematic representation of the effects of dc and ac microwave fields on the interaction of two molecules. The solid and dashed lines are the bare potentials En (r) ≡ EJ;M;σ (r, ϑ) of Section 12.3.2.1 with ϑ = π/2 for interactions in the presence of the dc field only, for the symmetric (σ = +) and antisymmetric (σ = −) states, respectively. The dc field induces a splitting δ of the first-excited manifold of the two-particle spectrum. A microwave field of frequency ω = ω + 2δ/3 + Δ is blue-detuned by Δ > 0 from the single-particle rotational resonance. The dipole–dipole interaction further splits the excitedstate manifold, making the detuning space-dependent. Eventually, the combined energy of the bare ground-state potential E0;0;+ (r) and of an ac photon (black arrow) becomes degenerate with the energy of the bare symmetric E1;0;+ (r, π/2). The resonant point rC = (d 2 /3Δ)1/3 occurs at r ≈ 46 rB . (b) Enlarged view of the potentials of panel (a) with M = 0. The dressed ground-state potential is indicated by a thick solid line. (c) The four potentials of panel (b) in the field-dressed picture. The dressed ground-state potential E˜ 0;0;+ (r, π/2) has the largest energy and is indicated by a thick solid line. (From Micheli, A. et al., Phys. Rev. A, 76, 043604, 2007. With permission.)
turns from weakly to strongly repulsive for r rC and r rC , respectively. This change in the character of the ground-state interaction potential corresponds to the design of a “step-like” interaction. This example shows that the three-dimensional ground-state interaction for two molecules can be strongly modified by combined ac © 2009 by Taylor and Francis Group, LLC
Condensed Matter Physics with Cold Polar Molecules
447
and dc fields, which is the central result of this section. More complicated potentials can be engineered using multiple ac fields and different polarizations. Analogous to the case (Eac = 0) of Section 12.3.2.1, the interaction potential of Figure 12.9 is actually repulsive along certain directions (e.g., for θ = π/2, as shown in the figure), while it becomes attractive at other configurations (e.g., for θ = 0, not shown). As for the (Eac = 0) case of Section 12.3.2.1, when more than two particles are considered this attraction can lead to many-body instabilities. Moreover, here the dressed potential E˜ 0;0;+ (r) of Figure 12.9c is not the lowest-energy potential, which in general can introduce additional loss channels. The latter correspond to diabatic couplings to symmetric states for particles approaching distances r rC , and are therefore present even in the simple two-particle collision process, and to couplings to antisymmetric states, which can be induced, for example, by three-body collisions or by non-compensated tensor shifts for two optically trapped particles. The presence of all of these loss channels may render impractical the realization of collisionally stable setups for strongly interacting molecular gases (see, however, Ref. [99] for a solution involving the use of a circularly polarized ac field). At the same time, we have shown above that for the system depicted in Figure 12.9 the presence of the static field shifts the various resonance points with the potentials that are responsible for these loss channels in the (ϑ = π/2)-plane (z = 0) to distances r rC . This suggests that by confining the motion of the particles to the plane z = 0 using a strong optical transverse confinement analogous to that of Section 12.3.2.1 it may be possible to realize collisionally stable systems in the region r > rC . This scheme has been shown to work in Ref. [61] and thus the main message here is that a judicious combination of the dipole–dipole interactions and of the optical confinement can act as an effective “shield” of the region r < rC where losses occur and the collision system may be stable. The use of the step-like potential above and of other engineered potentials can lead to the realization of interesting phases for an ensemble of polar molecules in the strongly interacting regime [46,61].
12.4
MANY-BODY PHYSICS WITH COLD POLAR MOLECULES
12.4.1 TWO-DIMENSIONAL SELF-ASSEMBLED CRYSTALS The above discussion of the intermolecular potentials and of the stability of collision systems in reduced dimensionality provides the microscopic justification for studying an ensemble of polar molecules in two-dimensions interacting via (modified) dipole– dipole potentials. At low temperatures T < ω⊥ , the general many-body Hamiltonian has the form of Equation 12.5. As an example of the possibilities offered by potential engineering to realize novel many-body quantum phases, we here focus on bosonic 2D (ρ) = D/ρ3 of Equation 12.6, particles interacting via the effective potential Veff derived in Section 12.3.2.1. The Hamiltonian of Equation 12.5 gives rise to novel quantum phenomena, which have not been accessed so far in the context of cold neutral atoms and molecules. The phase diagram for the two-dimensional system of bosonic dipoles is outlined in Figure 12.2a. In the limit of weak interactions rd < 1, the ground state is a superfluid (SF) characterized by a finite superfluid fraction ρs (T ), which depends © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
on temperature T with ρs (T = 0) = 1. In the opposite limit of strong interactions rd 1 the polar molecules are in a crystalline phase for temperatures T < Tm with Tm ≈ 0.09D/a3 ! 0.018rd ER,c , with ER,c ≡ π2 2 /2ma2 the crystal recoil energy, typically a few to tens of Kilohertz. The configuration with minimal energy is thus a triangular lattice with spacing aL = (4/3)1/4 a. Excitations of the crystal are acoustic phonons with the Hamiltonian given by Equation 12.13, and characteristic Debye √ frequency ωD ∼ 1.6 rd ER,c . At T = 0 the static structure factor S diverges at a reciprocal lattice vector K, and thus S(K)/N acts as an order parameter for the crystalline phase. In Ref. [46] we investigated the intermediate strongly interacting regime with rd 1, and we determined the critical interaction strength rQM for the quantum phase transition between the superfluid and the crystal. In our analysis we used a recently developed PIMC-code based on the Worm algorithm [95], which is an exact Monte (b) 0 50
7
100
4 3 2
3D
1
6
50
2D crystal
5
0 0
4
r qm
3 2D superfluid
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a/1 mm
rs
(d) 1 0.8 0.6 0.4 0.2 0
4
5 x/a
0.5 1
0 0
1
N = 36
(e) 0.8
N = 36
N = 90
0.6
N = 90
S(K)/N
0
3
1
1 0.1
0
2
(c)
5
2
1
g2
(d/÷m) / (D/÷ 200 amu)
8
y/a
(a) 9
2 r/d
0.4 0.2 0
5
10 15 20 25 rd
5
10 15 20 25 rd
FIGURE 12.10 (a) Quantum phases of two-dimensional dipoles. Contour plot of the interaction strength rd = Dm/2 a as a function of the dipole moment d (in Debye) and of the interparticle distance a (in μm), with m the mass of a molecule (in atomic units 200 × amu). The regions of stability of the two-dimensional superfluid and crystalline phases where ω⊥ > D/a3 are indicated, with ω⊥ = 2π × 150 kHz the frequency of the transverse confinement. (b) PIMC-snapshot of the mean particle positions in the crystalline phase for N = 36 at rd ≈ 26.5. (c) Density–density (angle-averaged) correlation function g2 (r), for N = 36 at rd ∼ 11.8. (d) Superfluid density ρs and (e) static structure factor S(K)/N as a function of rd , for N = 36 (circles) and N = 90 (squares). (From Büchler, H.P. et al., Phys. Rev. Lett., 98, 060404, 2007. With permission.)
© 2009 by Taylor and Francis Group, LLC
Condensed Matter Physics with Cold Polar Molecules
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Carlo method for the determination of thermodynamic quantities in continuous space at small finite temperature. In Figure 12.10d and 12.10e, the order parameters ρs and S(K)/N are shown at a small temperature T = 0.014D/a3 for different interaction strengths rd and particle numbers N = 36, 90. We find that ρs exhibits a sudden drop to zero for rd ≈ 15, while at the same position S(K) strongly increases. In addition, during the Monte Carlo simulations we observed that in a few instances ρs suddenly increased from 0 to 1, and then returned to 0, in the interval rd ≈ 15–20, which suggests a competition between the superfluid and crystalline phases. These results indicate a superfluid to crystal phase transition at rqm = 18 ± 4.
(12.27)
The step-like behavior of ρs and S(K)/N is consistent with a first-order phase transition, a result that has been confirmed in Refs. [47] and [48]. Notice that the superfluid with rd ∼ 1 is strongly interacting, and in particular the density-density correlation function is quenched at lengths R < a (see Figure 12.10c). This observation is consistent with the validity condition of the effective two-dimensional interaction potential given in Equation 12.6, indicating that two particles never approach each other to distances smaller than l⊥ . Having determined the low-temperature phase diagram, the remaining question is whether these phases, and in particular the crystalline phase emerging at strong dipole–dipole interactions, are in fact accessible with polar molecules. This question is addressed in Figure 12.10a, which is a contour plot √ of the interaction strength rd as a function of the induced dipole moment d = D (in units of Debye) and of √ the mean interparticle distance a (in μm). The dimensionless quantity m/200 amu depends on the mass m of the molecules (in atomic units, au), and it is of order one for characteristic molecules like SrO or RbCs. In the figure, stable two-dimensional configurations for the molecules exist in the parameter region where the transverse (optical) trapping frequency ω⊥ = 2π × 150 Hz exceeds the dipole–dipole interaction (⊥ ω > D/a3 ), such that l⊥ = (12D/mω2⊥ )1/5 < a, consistent with the stability discussion of Section 12.3.2.1 (notice that l⊥ ∼ (D/ω⊥ )1/3 for realistic parameters). The figure shows that for a given induced dipole d the ground state of an ensemble of polar molecules is a crystal for mean interparticle distances l⊥ a amax , where amax ≡ d 2 m/2 rqm corresponds to the distance at which the crystal melts into a superfluid. For SrO (RbCs) molecules with the permanent dipole moment d = 8.9 D (d = 1.25 D), amin ≈ 200 nm (100 nm), while amax can be several micrometers. Since for large enough interactions the melting temperature Tm can be of the order of several microkelvin, the self-assembled crystalline phase should be accessible for reasonable experimental parameters using cold polar molecules. The zero-temperature phases can be observed using Bragg scattering with optical light, which allows for probing the crystalline phase. The detection of vortices can be used as a definitive signature of superfluidity. We notice that the two-dimensional (quasi) condensate involves a fraction of the total density only, and therefore we expect only small coherence peaks in a time-of-flight experiment. Finally, we notice that by adding an additional in-plane optical confinement, it may be possible to realize strongly interacting one-dimensional phases © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
analogous to the two-dimensional crystals discussed above [49,50,91]. For large enough interactions r 1, the phonon frequencies have the simple form ωq = 1/2 4 ER,c , with fq = j>0 4 sin(qaj/2)2 /j5 . The Debye frequency (2/π2 ) 12rd fq √ is ωD ≡ ωπ/a ∼ 1.4 rd ER,c , while the classical melting temperature can be estimated to be of the order of Tm ! 0.2rd ER,c /kB [91].
12.4.2
FLOATING LATTICES OF DIPOLES
An interesting possibility offered by the realization of the self-assembled crystals discussed above is to utilize them as floating mesoscopic lattice potentials to trap other particles, which can be atoms or polar molecules of a different species. We show below that within an experimentally accessible regime of parameters extended Hubbard models with tunable long-range phonon-mediated interactions describe the effective dynamics of the extra-particles dressed by the lattice phonons. The systems that we have in mind are shown in Figures 12.5, 12.11a, and 12.11b, where extra particles confined to a two-dimensional 4 crystal plane or a one-dimensional tube scatter from the periodic lattice potential j Vcp (Rj − r). Here, r and Rj = Rj0 + uj are the coordinates of the particle and crystal molecule j, respectively, with Rj0 the (a)
dc
a
b
Vcp ~ J dp
~ Vi, j
(b) Vcp
~ J
dc
~ Vi, j
a
~ Vi, i
FIGURE 12.11 A dipolar crystal of polar molecules provides a periodic lattice Vcp for extra atoms or molecules giving rise to a lattice model with hopping J˜ and long-range interactions V˜ i,j (see text and Figure 12.5). (a) A one-dimensional dipolar crystal with lattice spacing a provides a periodic potential for a second molecular species moving in a parallel tube at distance b (Configuration 1). (b) One-dimensional setup with atoms scattering from the dipolar lattice (configuration 2). (From Pupillo, G. et al., Phys. Rev. Lett., 100, 050402, 2008. With permission.)
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equilibrium positions and uj small displacements. If the extra-particles are molecules, this potential is given by the repulsive dipole–dipole interaction Vcp (Rj − r) = dp dc /| Rj − r |3 with dp dc the induced dipole moment, and for atoms we assume that the interaction is modeled by a short range pseudopotential proportional to an elastic scattering length acp . In addition, extra molecules and atoms will interact via dipolar, or short-range interactions, respectively. We are interested in a situation where the extra particles in the lattice are described by a single band Hubbard Hamiltonian coupled to the acoustic phonons of the lattice as given in Equation 12.12 [128]. In the latter equation, the first and second terms describe the nearest-neighbor hopping of the extra-particles with hopping amplitudes J, and interactions V , computed for each microscopic model by band-structure calculations for uj = 0, respectively. The third term is the phonon Hamiltonian. The fourth term is the phonon coupling obtained in lowest order in the displacement uj = i
0 † (/2mc Nωq )1/2 ξq (aq + a−q )eiq·Rj , q
with Mq = V¯ q q · ξq (/2Nmc ωq )1/2 βq . Here, ξq and N are the phonon polarization and the number of lattice molecules, ¯ the Fourier transform of the particle–crystal interaction Vcp , respectively, while Vq 2is iqr and βq = dr|w0 (r)| e , with w0 (r) the Wannier function of the lowest Bloch band [128]. The validity of the single-band Hubbard model requires that J, V < Δ, and the temperature kB T < Δ with Δ the energy gap to the first excited Bloch band. The Hubbard parameters of Equation 12.12 are of the order of magnitude of the recoil energy, J, V ≈ ER,c , which is (much) smaller than the Debye frequency √ ωD ∼ ER,c rd , for rd 1 [129]. This separation of timescales J, V ωD , combined with the fact that the coupling to phonons is dominated by high frequencies ω > J, V (see the discussion of Mq below) is reminiscent of polarons as particles dressed by (optical) phonons, where the dynamics is given by coherent and incoherent hopping on a lattice [117,128]. This physical picture is brought out in a master equation treatment within a strong coupling perturbation theory. The starting point is a Lang–Firsov transformation of the Hamiltonian H → SHS † with a density-dependent displacement ⎡
⎤
Mq iqR0 † † ⎦. S = exp⎣− e j cj cj aq − a−q ωq q,j
This eliminates the phonon coupling in the second line of Equation 12.12 in favor of a transformed kinetic energy term −J
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ci† cj Xi† Xj ,
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Cold Molecules: Theory, Experiment, Applications
where the displacement operators ⎤
Mq iqR0 † ⎦ Xj = exp⎣ e j aq − a−q ω q q ⎡
can be interpreted as a lattice recoil of the dressed particles in a hopping process. In addition, the bare interactions are renormalized according to (1) V˜ ij = Vij + Vij ,
4 (1) with Vij = −2 q cos(q(Ri0 − Rj0 ))Mq2 /ωq , that is, the phonon couplings induce and modify off-site interactions. The onsite interaction is given by V˜ j,j = Vj,j − 2Ep with Ep =
Mq2 q
ωq
,
the polaron self-energy or polaron shift. For J = 0 the new Hamiltonian is diagonal and describes interacting polarons and independent phonons. The latter are vibrations of the lattice molecules around new equilibrium positions with unchanged frequencies. A stable crystal requires the variance of the displacements Δu around these new equilibrium positions to be small compared to a. A Born–Markov approximation with the phonons the finite-temperature heatbath with J, V ωD (see above), and the transformed kinetic energy −J
ci† cj (Xi† Xj − Xi† Xj )
as the system-bath interaction with Xi† Xj the equilibrium bath average, provides the master equation for the reduced density operator of the dressed particles ρt in the Lindblad form [130]
Γδ,δ i j,l ˜ + ρ˙ t = [ρt , H] [bjδ , ρt blδ ] + [blδ , ρt bjδ ] , 2
(12.28)
j,l,δ,δ
† with bjδ = cj+δ cj . The effective system Hamiltonian H˜ takes the form of Equation 12.14, which is of the extended Hubbard type, valid for J˜ , V˜ ij , Ep < Δ. Coherent hopping of the dressed particles is then described by
J˜ = JXi† Xj ≡ J exp(−ST ),
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where ST =
Mq 2 [1 − cos(qa)](2nq (T ) + 1) ωq q
characterizes the strength of the particle-phonon interactions, and nq (T ) is the thermal occupation at temperature T [128]. The dissipative term in Lindblad form in Equation 12.28 corresponds to ther mally activated incoherent hopping with rates Γδ,δ j,l , which can be made negligible for the energies of interest kB T min(Δ, Ep , kB TC ) [117,131]. Corrections to Equation 12.14 proportional to J 2 are small relative to H˜ provided J Ep [117] (also J ωD in one-dimensional [131,132]). Thus, in the parameter regime of interest the dynamics of the dressed particles is described by the extended Hubbard Hamil˜ In the following, we verify the existence of this parameter regime and tonian H. calculate the effective Hubbard parameters from the microscopic model for the onedimensional configuration of Figure 12.11a, where extra-particles are polar molecules of a different species. An analogous calculation for the configuration of Figure 12.11b is reported in Ref. [53]. In the configuration of Figure 12.11a molecules of a second species are trapped in a tube at a distance b from the crystal tube under one-dimensional trapping conditions. For crystal molecules fixed at the equilibrium positions with lattice spacing a, the extra particles feel a periodic potential Vcp (x) = dc dp
b2 + (x − ja)2
−3/2
,
j
which determines the band structure. The lattice depth V0 ≡ Vcp (a/2) − Vcp (0) ∼ rd
dp mp e−3b/a ER,p dc m (b/a)3
is shown in Figure 12.12a as a function of b/a, where the thick solid lines indicate the parameter regime 4J < Δ, and ER,p = 2 π2 /2mp a2 . The potential is comb-like for b/a < 1/4, because the particles resolve the individual molecules forming the crystal, while it is sinusoidal for b/a 1/4. The strong dipole–dipole repulsion between the extra-particles acts as an effective hardcore constraint [85]. We find that for 4J < Δ and dp dc the bare off-site interactions satisfy Vij ∼ dp2 /(a|i − j|)3 < Δ, which justifies a single-band approximation for the dynamics of the extra-particles in the static potential. The particle–phonon coupling is dc dp Mq = ab
© 2009 by Taylor and Francis Group, LLC
2 Nmc ωq
1/2 q2 K1 (b|q|)βq
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Cold Molecules: Theory, Experiment, Applications (a)
V0 /ER,p
S0, E0/J
(b)
rd = 500
100 100
rd = 500
10 1
10
0.1 1
rd = 50
0.01
rd = 50
0.1 b/a
0 (1)
(c)
0
1
Vj, j+1/ER,p
b/a
1
(d)
0
rd 1
10
200
rd = 500
1
100 rd = 50
50
0.1
–0.1
0
20
–0.5 1
b/a
0.2
0.4
0
0.5
0
10
b/a
0.8
FIGURE 12.12 Configuration 1 (Figure 12.11a): Hubbard parameters for dp /dc = 0.1 and m = mp . (a) Lattice depth V0 in units of ER,p vs. b/a for rd = 50 and 500. Thick continuous lines: tight-binding region 4J < Δ. (b) Reduction factor S0 (dashed–dotted lines) and polaron shift Ep /J (solid lines), for 4J < Δ. (c) Continuous lines: phonon-mediated (1) interactions V . Horizontal (dashed) lines: Vj,j+1 . (d) Contour plot of V˜ j,j+1 /2J˜ (solid j,j+1
lines) as a function of b/a and rd . A single-band Hubbard model is valid left of the dashed region (4J˜ , V˜ ij < Δ), and right of the black region (Ep < Δ).
√ with K1 being the modified Bessel function of the second kind, and Mq ∼ q for q → 0. In the regime of interest b/a < 1 where the single-band approximation is valid (4J, Vij < Δ), we find that Mq is peaked at large q ∼ π/a, so that the main contribution to the integrals in the definition of ST and Ep is indeed dominated by large frequencies ωq > J. Together with the separation of timescales J, Vij ωD , this is consistent with the picture of the system’s dynamics as given by particles dressed by fast (optical) phonons, as discussed above. We note that this so-called antiadiabatic regime is generally hard to achieve in cold atomic setups [129]. A plot of S0 as a function of b/a is shown in Figure 12.12b. We find the scaling S0 ∝
√ rd (dp /dc )2 ,
and within the regime of validity of the single-band approximation, S0 can be tuned from S0 1 (J˜ ∼ J) to S0 1 (J˜ J) corresponding to the large and small polaron limit, respectively. The polaron shift Ep generally exceeds the bare hopping rate J, © 2009 by Taylor and Francis Group, LLC
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and in particular, Ep J for S0 1; see Figure 12.12b. Together with the condition ωD J this ensures that the corrections to Equation12.14 which are proportional to J 2 , are indeed small, and thus Equation12.14 fully accounts for the coherent dynamics of the dressed particles. The extended Hubbard model corresponding to the configuration of Figure 12.11a is characterized by tunable off-site interactions, which are a combination of the direct dipole–dipole interactions between the extra-particles and of the phonon-mediated (1) (1) interactions Vi,j . For b/a 1/4 we find that the interactions Vi,j decay slowly with (1)
the interparticle distance as ∼1/|i − j|2 , and are thus long-ranged. The sign of Vi,j is a function of the ratio b/a. Thus, depending on b/a the phonon-mediated interactions can enhance or reduce the direct dipole–dipole repulsion of the extra particles. As an (1) example, Figure 12.12c shows that the sign of the term Vj,j+1 alternates between attractive and repulsive as a function of the ratio b/a, and that for small enough values of b/a the phonon-mediated interactions can become larger than the direct dipole–dipole interactions. The effective Hubbard parameters V˜ j,j+1 and J˜ are summarized in Figure 12.12d, which is a contour plot of V˜ j,j+1 /2J˜ as a function of rd and b/a. The ratio V˜ j,j+1 /2J˜ increases as b/a decreases or rd increases, and can be much larger than one. This appearance of strong off-site interactions in the effective dynamics is a necessary ingredient for the realization of a variety of new quantum phases [78–82]. As an example of the possible quantum phases that can be realized in this setup, at half filling, and considering nearest-neighbor interactions only, the particles in the configuration described above undergo a transition from a (Luttinger) liquid (V˜ i,i+1 < 2J˜ ) to a charge-density wave (V˜ i,i+1 > 2J˜ ) as a function of b/a and rd . Figure 12.12d shows that the parameter regime V˜ i,i+1 ≈ 2J˜ (see Ref. [133]), where this transition occurs can be satisfied for various choices of rd and b/a, for example, for rd = 100 and b/a ≈ 0.5.
12.4.3 THREE-BODY INTERACTIONS As discussed in Section 12.2, it is of interest to design systems where effective manybody interactions dominate over two-body interactions, and determine the properties of the system in the groundstate. Here we describe how an effective low-energy 3D of the form given in Equation 12.2 can be derived in the interaction potential Veff Born–Oppenheimer approximation for 1 Σ polar molecules interacting via dipole– dipole interactions by dressing low-energy rotational states of each molecule with external static and microwave fields, in analogy to the discussion of Section 12.3.2. We here focus on a system of molecules in a static electric field E = Eez directed along the z-axis (Figure 12.13), where the two states |gi ≡ |φ0,0 i and |e+ i ≡ |φ1,+1 i with energies Eg and Ee,± are coupled by a circularly polarized microwave field propagating along the z-axis. The microwave transition is characterized by the (blue) detuning Δ > 0 and the Rabi frequency Ω/. While the following discussion can be readily generalized to include the degenerate case [85], here we assume that the degeneracy of the states |e− ≡ |φ1,−1 i and |e+ is lifted, for example, by an additional microwave field coupling the state |e− near-resonantly to © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
(a)
(b)
˙ e Òi Ee
6B 2B
D
Ee , ± Wd
˙ e+ Òi ˙ e Òi
2B
D ˙ e– Ò i
W
˙ e+ Òi
0
W
0
1
2
3
dE/B
Eg
Eg ˙ g Òi
˙ g Òi
FIGURE 12.13 Spectrum of a polar molecule. (a) Level structure for Ed/B = 3: the circular polarized microwave field couples the ground state |g with the excited state |e+ with Rabi frequency Ω/ and detuning Δ. The excited state |e+ is characterized by a finite angular momentum Jz |e+ = |e+ . Applying a second microwave field with opposite polarization and Rabi frequency Ωd / allows us to lift the degeneracy in the first excited manifold by resonantly couple the state |e− to the next manifold. (b) Internal excitation energies for a single polar molecule in a static electric field E = Eez . (From Büchler, H.P. et al., Nature Phys., 3, 726, 2007. With permission.)
the next state manifold (see Figure 12.13). Then, the internal structure of a single polar molecule reduces to a two-level system and is described as a spin-1/2 particle via the identification of the state |gi (|e+ i ) as eigenstate of the spin operator Siz with positive (negative) eigenvalue. Using the rotating frame and the rotating wave approximation, the Hamiltonian describing the internal dynamics of the polar molecule can be written as (i)
H0 =
1 Δ 2 Ω
Ω = hSi −Δ
(12.29) y
with the effective magnetic field h = (Ω, 0, Δ) and the spin operator Si = (Six , Si , Siz ). The eigenstates of this Hamiltonian √ are denoted as |+i = α|gi + β|e+ i and |−i = −β|gi + α|e+ i with energies ± Δ2 + Ω2 /2. For distances |rij | (D/B)1/3 with D = |g|di |e+ |2 and di the dipole operator, the dipole–dipole interaction in Equation 12.4 between two polar molecules can be mapped onto the effective spin interaction Hamiltonian Hd = Hdint + Hdshift . The first term describes an effective spin–spin interaction Hdint = −
1
y y Dν(rij ) Six Sjx + Si Sj − η2− Siz Sjz , 2 i=j
© 2009 by Taylor and Francis Group, LLC
(12.30)
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Condensed Matter Physics with Cold Polar Molecules
√ where η± = ηg ± ηe is determined by the induced dipole moments ηg = ∂E Eg / D √ and ηe = ∂E Ee,+ / D. The anisotropic behavior of the dipole–dipole interaction is accounted for by ν(r) = (1 − 3 cos2 ϑ)/r 3 with ϑ the angle between r and the z-axis. In addition, the asymmetry of the induced dipole moments gives rise to a positiondependent renormalization of the effective magnetic field and an energy shift Hdshift
1
η− η+ z η2+ = Dν(rij ) Si + . 2 2 4
(12.31)
i=j
Within the Born–Oppenheimer approximation, an analytic expression for the effective interaction Veff ({ri }) between two polar molecules each prepared in the state |+i can be derived in second-order perturbation theory in the dipole–dipole interaction Vdd (r)/h as 3D Veff ({ri }) = E (1) ({ri }) + E (2) ({ri })
(12.32)
where D/(a3 |h|) = (R0 /a)3 1 is the (small) parameter controlling the perturbative expansion, a is the characteristic length scale of the interparticle separation, and R0 = √ (D/ Δ2 + Ω2 )1/3 is a Condon point, analogous to that discussed in Section 12.3.2.2. The energy shift E (1) ({ri }) =
2 1 2 α ηg +β2 ηe −α2 β2 Dν(rij ), 2
(12.33)
i=j
gives rise to a dipole–dipole interaction between the particles, while the term 2
|N|2 Dν rij D2 ν (rik ) ν rjk + √ 2 2 Δ2 + Ω 2 i<j 2 Δ + Ω k=i,k=j (12.34) corresponds to a correction to the two-particle interaction potential and an additional three-body interaction. The matrix elements M and N take the form E (2) ({ri }) =
√
|M|2
α2 ηg + β2 ηe ηe − ηg − (α2 − β2 )/2 , 2 N = α2 β2 ηe − ηg + 1 .
M = αβ
Therefore, the effective interaction potential Veff up to second order in (R0 /a)3 reduces to the form in Equation 12.2 with the two-particle interaction potential V (r) = λ1 D ν (r) + λ2 DR03 [ν (r)]2 ,
(12.35)
and the three-body interaction W (r1 , r2 , r3 ) = γ2 R03 D [ν(r12 )ν(r13 ) + ν(r12 )ν(r23 ) + ν(r13 )ν(r23 )]. © 2009 by Taylor and Francis Group, LLC
(12.36)
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2 The dimensionless coupling parameters are λ1 = α2 ηg + β2 ηe − α2 β2 , λ2 = 2|M|2 + |N|2 /2, and γ2 = 2|M|2 . These parameters can be tuned by varying the strength of the electric field Ed/B and the ratio of the Rabi frequency and the detuning, Ω/Δ (Figure 12.14). Of particular interest are the values of the external fields where the leading two-particle interaction vanishes, that is, λ1 = 0. Then, the interaction is dominated by the second-order contribution with λ2 and γ2 , which includes the three-body interaction (Figure 12.14d), while a small deviation away from the line λ1 = 0 allows us to change the character of the two-particle interaction. Note, that an n-body interaction term (n ≥ 4) appears as (n − 1)-th order perturbation in the small parameter (R0 /a)3 . Therefore, the contribution of these terms is suppressed and can be safely ignored.
l1 (a) 4
10l2 (b) 4
1 0.8 0.4
2
3.0
3
0.6 dE/B
dE/B
3
0.2
1.8
2
1.2
0
1
2.4
1
0.6
–0.2 0
1
2
3
0
4
1
W/D (c)
2
3
4
3
4
W/D
10g2
(d)
4
3.0
0.1 l2
2.4
3 dE/B
1.8 2
0.05
1.2
g2
0.6 1
0
1
2 W/D
3
4
0
1
2 W/D
FIGURE 12.14 Parameters of the effective interaction potential. (a)–(c): Strength of the interaction parameters λ1 , λ2 , and γ2 as a function of the external fields Ed/B and Ω/Δ. The leading dipole–dipole interaction vanishes for λ1 = 0 [dashed line in (b) and (c)], and the second order contributions dominate the interaction. (d) Strength of λ2 (dashed line) and γ2 (solid line) along the line in parameter space with λ1 = 0. (From Büchler, H.P. et al., Nature Phys., 3, 726, 2007. With permission.)
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459
The perturbative expansion requires that a R0 . We notice that particles can be confined to interparticle distances larger than R0 by combining repulsive dipole–dipole interactions with a strong (optical) transverse confinement ω⊥ , in analogy to Section 12.3.2.1. Then, for a repulsive two-particle potential with λ1 −λ2 (R0 /a)3 and for ω⊥ > D/R03 , the particles approach distances |ri − rj | < R0 at an exponentially small rate Γ ∼ (/ma2 ) exp(−2SE /), with SE / ∼ Dm/R0 2 . This exponential suppression ensures the stability of the collision system for the duration of an experiment. Applying an optical lattice provides a periodic structure for the polar molecules 3D given by Equation 12.32. described by the Hamiltonian of Equation 12.1, with Veff In the limit of a deep lattice, a standard expansion of the field operators ψ† (r) = 4 † i w(r − Ri )bi in the second-quantized expression of Equation 12.1 in terms of lowest-band Wannier functions w(r) and particle creation operators bi† [107] leads to the realization of the Hubbard model of Equation 12.9, characterized by strong nearest-neighbor interactions [85]. We notice that the particles are treated as hardcore because of the constraint a R0 . The interaction parameters Uij and Vijk in Equation 12.9 derive from the effective interaction V ({ri }), and in the limit of well-localized Wannier functions reduce to a3 a6 + U1 , 3 |Ri − Rj | |Ri − Rj |6
Uij = U0 and
Wijk = W0
a6 + perm , |Ri − Rj |3 |Ri − Rk |3
(12.37)
(12.38)
respectively, with U0 = λ1 D/a3 , U1 = λ2 DR03 /a6 , and W0 = γ2 DR03 /a6 . The dominant contributions and strengths of the three-body terms in different lattice geometries are shown in Figure 12.2b. For LiCs with a permanent dipole moment d = 6.3 D trapped in an optical lattice with spacing a ≈ 500 nm, the leading dipole–dipole interaction can give rise to very strong nearest-neighbor interactions with U0 ∼ 55Ekin , and Ekin = 2 /ma2 . On the other hand, tuning the parameters via the external fields to λ1 = 0 results in the characteristic energy scale for the three-body interaction W0 ≈ (R0 /a)3 Ekin . Then, controlling the hopping energy J by the strength of the optical lattice allows for the regime with dominant three-body interactions. For bosonic particles, an analytic calculation has suggested that the ground-state phase diagram of Equation 12.9 with Uij = 0 in one-dimensional is characterized by the presence of valence bond states at specific rational fillings of the lattice, charge-density waves and superfluid phases [85].
12.4.4
LATTICE SPIN MODELS
Cold gases of polar molecules can be used to construct in a natural way a complete toolbox for any permutation symmetric two spin-1/2 (qubit) interaction, based on techniques of interaction engineering discussed in the previous sections. © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
The system we consider in this section is comprised of heteronuclear molecules in the 2 Σ1/2 electronic state, such as, for example, alkaline-earth monohalogenides with a single electron outside a closed shell. We adopt a model molecule where the rotational excitations are described by the Hamiltonian Hm = BN2 + gN · S,
(12.39)
with N the rotational angular momentum of the nuclei, and S the dimensionless electronic spin (assumed to be S = 1/2 in the following). Here B denotes the rotational constant and γ is the spin-rotation coupling constant. The typical values of B are a few tens of gigahertz, and γ is usually in the hundred megahertz regime. The coupled basis of a single molecule i corresponding to the eigenbasis of i is {|N , S , J ; M } where J = N + S with eigenvalues E(N = 0, 1/2, 1/2) = Hm i i i Ji i i i 0, E(1, 1/2, 1/2) = 2B − γ, and E(1, 1/2, 3/2) = 2B + γ/2. The Hamiltonian describing the internal and external dynamics of a pair of molecules trapped in wells of an optical lattice is denoted by H = Hin +4 Hex . The i , interaction describing the internal degrees of freedom is Hin = Hdd + 2i=1 Hm where Hdd is the dipole–dipole interaction. The Hamiltonian describing the external, 4 or motional, degrees of freedom is Hex = 2i=1 Pi2 /(2m) + Vi (xi − x¯ i ), where Pi is the momentum of molecule i with mass m, and the potential generated by the optical lattice Vi (x − x¯ i ) describes an external confinement of molecule i about a local minimum x¯ i with one-dimensional rms width z0 . We assume that the traps are approximately harmonic near the trap minimum with a vibrational spacing ωosc . Furthermore, we assume that the molecules can be prepared in the motional ground state of each local potential using dissipative electromagnetic pumping [3]. It is convenient to direct the quantization axis zˆ along the line connecting the two molecules, x¯ 2 − x¯ 1 = Δzˆz with Δz corresponding to a multiple of the lattice spacing. The ground subspace of each molecule is isomorphic to a spin-1/2 particle. Our goal is to obtain an effective spin–spin interaction between two neighboring molecules. Static spin–spin interactions due to spin-rotation and dipole–dipole couplings do exist but are very small in our model: HvdW (r) = −(d 4 /2Br 6 ) 1 + (γ/4B)2 1 + 4S1 · S2 /3 − 2S1z S2z . The first term is the familiar van der Waals 1/r 6 interaction, while the spin-dependent part is strongly suppressed as γ/4B ≈ 10−3 1. However, dipole–dipole coupled excited states can be dynamically mixed using a microwave field. The molecules are assumed to be trapped with a separation Δz ∼ rγ ≡ (2d 2 /γ)1/3 , where the dipole–dipole interaction is d 2 /rγ3 = γ/2. In this regime the rotation of the molecules is strongly coupled to the spin and the excited states are described by Hunds case (c) states in analogy to the dipole–dipole coupled excited electronic states of two atoms with fine structure. The ground states are essentially spin-independent. In the subspace of one rotational quantum (N1 + N2 = 1), there are 24 eigenstates of Hin which are linear superpositions of two electron spin states and properly symmetrized rotational states of the two molecules. There are several symmetries that reduce Hin to block diagonal form. First, Hdd , conserves the quantum number Y = MN + MS where MN = MN1 + MN2 and MS = MS1 + MS2 are the total rotational and spin projections along the intermolecular axis. Second, parity, defined as the interchange of the two © 2009 by Taylor and Francis Group, LLC
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Condensed Matter Physics with Cold Polar Molecules
molecules followed by parity though the center of each molecule, is conserved. The σ = ±1 eigenvalues of parity are conventionally denoted g(u) for gerade(ungerade). Finally, there is a symmetry associated with reflection R of all electronic and rotational coordinates through a plane containing the intermolecular axis. For |Y | > 0 all eigenstates are even under R but for states with zero angular momentum projection there are ±1 eigenstates of R. The 16 distinct eigenvalues correspond to degenerate subspaces labeled |Y |± σ (J), with J indicating the quantum number in the r → ∞ asymptotic manifold (N = 0, J = 1/2; N = 1, J). Remarkably, the eigenvalues and eigenstates can be computed analytically yielding the Movre–Pichler potentials [134] plotted in Figure 12.15a. In order to induce strong dipole–dipole coupling we introduce a microwave field E(x, t)eF with a frequency ωF and Rabi-frequency Ω tuned near resonance with the N = 0 → N = 1 transition. The effective Hamiltonian acting on the lowest-energy states is obtained in second-order perturbation theory as Heff (r) =
gf |Hmf |λ(r)λ(r)|Hmf |gi i,f λ(r)
ωF − E(λ(r))
|gf gi |,
(12.40)
where {|gi , |gf } are ground states with N1 = N2 = 0 and {|λ(r)} are excited eigenstates of Hin with N1 + N2 = 1 and with excitation energies {E(λ(r))}. The reduced interaction in the subspace of the spin degrees of freedom is then obtained by tracing over the motional degrees of freedom. For molecules trapped in the ground motional states of isotropic harmonic wells with rms width z0 , the wave function is separable in center of mass and relative coordinates, and the effective spin–spin Hamiltonian is Hspin = Heff (r)rel . The Hamiltonian in Equation 12.40 is guaranteed to yield some entangling interaction for appropriate choice of field parameters, but it is desirable to have a systematic way to design a spin–spin interaction. The model presented here possesses sufficient structure to achieve this essentially analytically. The effective Hamiltonian of molecules 1 and 2 in a microwave field is Heff (r) =
3 |Ω| α β σ1 Aα,β (r)σ2 , 8
(12.41)
α,β=0
where {σα }3α=0 ≡ {1, σx , σy , σz } and A is a real symmetric tensor. Equation 12.41 describes a generic permutation symmetric two-qubit Hamiltonian. The components A0,s describe a pseudo magnetic field which acts locally on each spin and the components As,t describe two-qubit coupling. The pseudo magnetic field is zero if the microwave field is linearly polarized but a real magnetic field could be used to tune local interactions and, given a large enough gradient, could break the permutation invariance of Hspin . For a given field polarization, tuning the frequency near an excited state induces a particular spin pattern on the ground states. These patterns change as the frequency is tuned though multiple resonances at a fixed intermolecular separation. Table 12.1 presents the parameters needed to simulate the Ising and Heisenberg interactions in © 2009 by Taylor and Francis Group, LLC
462 (a)
Cold Molecules: Theory, Experiment, Applications
g
0+g
2g
1+ B
1g 1u
E/2B
1
g
1– B
(b) z/D z 1u
4
0u–
1 (0, 2 ;
3 1, 2 )
3
0–g
3g 4B
1g 2u
0+g
0u+
1
E(t)
1
(0, 2 ; 1, 2 )
2
Ÿ
z 1u
0+ 0– 1g u 0u– g
1 1
Ÿ
0
0 –1
–1 0 0.5 D z/rg (g /4B)1/3
1
1.5
Ÿ
x
1
(0, 2 ; 0, 2 ) 2
x/D z
1
0
1
y –2 y/D z
r/rg
FIGURE 12.15 (a) Movre–Pichler potentials for a pair of molecules as a function of their separation r. The potentials E(gi (r)) for the four ground states (dashed lines) and the potentials E(λ(r)) for the first 24 excited states (solid lines). The symmetries |Y |± σ of the corresponding excited manifolds are indicated, as are the asymptotic manifolds (Ni , Ji ; Nj , Jj ). (b) Implemen(II)
tation of spin model Hspin . Shown is the spatial configuration of 12 polar molecules trapped by two parallel planes) with separation normal to the plane √ triangular lattices (indicated by shaded √ of Δz/ 3 and in plane relative lattice shift of Δz√ 2/3. Nearest neighbors are separated by b = Δz and next nearest neighbor couplings are at 2b. The graph vertices represent spins and the edges correspond to pairwise spin couplings. The edge width and shading indicate the nature of the dominant pairwise coupling for that edge (thick dark edge = σz σz , thin dark edge = σy σy , thin light edge = σx σx , black = “other”). For nearest neighbor couplings, the edge width indicates the relative strength of the absolute value of the coupling. For this implementation, the nearest-neighbor separation is b = rγ . Three fields all polarized along zˆ were used to generate the effective spin–spin interaction with frequencies and intensities optimized to approximate the (II) ideal model Hspin . The field detunings at the nearest-neighbor spacing are ω1 − E(1g (1/2)) =
−0.05γ/2, ω2 − E(0g− (1/2)) = 0.05γ/2, ω3 − E(2g (3/2)) = 0.10γ/2 and the amplitudes are |Ω1 | = 4|Ω2 | = |Ω3 | = 0.01γ/. For γ = 40 MHz this generates effective coupling strengths Jz = −100 kHz and J⊥ = −0.4Jz . The magnitude of residual nearest-neighbor couplings are less than 0.04|Jz | along x- and y-links and less than 0.003|Jz | along z-links. The size of longer-range couplings Jlr are indicated by edge line style (dashed: |Jlr | < 0.01|Jz |, dotted: |Jlr | < 10−3 |Jz |). Treating pairs of spins on z-links as a single effective spin in the low-energy sector, the model approximates Kitaev’s 4-local Hamiltonian [115] on a square grid (shown here are one plaquette on the square lattice and a neighbor plaquette on the dual lattice) with an effective coupling strength Jeff = −(J⊥ /Jz )4 |Jz |/16 ≈ 167 Hz. (From Micheli, A. et al., Nature Phys., 2, 341, 2006. With permission.)
this way. Using several fields that are sufficiently separated in frequency, the resulting effective interactions are additive creating a spin texture on the ground states. The anisotropic spin model HXYZ = λx σx σx + λy σy σy + λz σz σz can be simulated using three fields: one polarized along zˆ tuned to 0u+ (3/2), one polarized along yˆ tuned to 0g− (3/2), and one polarized along yˆ tuned to 0g+ (1/2). The strengths λj can be tuned by adjusting the Rabi frequencies and detunings of the three fields. Using © 2009 by Taylor and Francis Group, LLC
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TABLE 12.1 Some Spin Patterns that Result from Equation 12.41 Polarization
Resonance
Spin Pattern
xˆ zˆ zˆ yˆ yˆ √ ( yˆ − xˆ )/ 2 cos ξˆx + sin ξˆz
2g 0u+ 0g− 0g− 0g+ 0g+ 1g
cos ξˆy + sin ξˆz
1g
σz σz σ · σ σx σx + σy σy − σz σz σx σx − σy σy + σz σz −σx σx + σy σy + σz σz −σx σy − σy σx + σz σz λ1 (σx σz + σz σx ) + λ2 σz σz +λ3 (σx σx + σy σy ) λ1 (σy σz + σz σy ) + λ2 σz σz +λ3 (σx σx + σy σy )
The field polarization is given with respect to the intermolecular axis zˆ , and the frequency ωF is chosen to be near resonant with the indicated excited state potential at the internuclear separation Δz. The sign of the interaction will depend on whether the frequency is tuned above or below resonance. Source: Micheli, A. et al., Nature Physics, 2, 341, 2006.
an external magnetic field and six microwave fields with, for example, frequencies and polarizations corresponding to the last six spin patterns in Table 12.1, arbitrary permutation symmetric two-qubit interactions are possible. The Kitaev model of Equation 12.11 (Spin model II) can be obtained in the following way. Consider a system of four molecules connected by edges of three length b forming an orthogonal triad in space. There are several different microwave field (II) configurations that can be used to realize the interaction Hspin along the links. One choice is to use two microwave fields polarized along zˆ , one tuned near resonance with a 1g potential and one near a 1u potential. A realization of model II using a different set of three microwave fields is shown in Figure 12.15b. The obtained interaction is close to ideal with small residual coupling to next nearest neighbors.
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110. Jördens, R., Strohmaier, N., Günter, K., Moritz, H., and Esslinger, T., A Mott insulator of Fermionic atoms in an optical lattice, arXiv:0804.4009. 111. Tewari, S., Scarola, V.W., Senthil, T.S.D., and Sarma, S. Emergence of artificial photons in an optical lattice, Phys. Rev. Lett., 97, 200401, 2006. 112. Haldane, F.D.M., Two-Dimensional Strongly Correlated Electron Systems, Gan, Z.Z. and Su, Z.B., Eds., Gordon and Breach, 1988. 113. Douçot, B., Feigel’man, M.V., Ioffe, L.B., and Ioselevich, A.S., Protected qubits and Chern-Simons theories in Josephson junction arrays, Phys. Rev. B, 71, 024505, 2005. 114. Kitaev, A.Yu., Anyons in an exactly solved model and beyond, Annals of Physics, 321, 2, 2006. 115. Dennis, E., Kitaev, A.Yu., Landahl, A., and Preskill, J., Topological quantum memory, J. Math. Phys., 43, 4452, 2002. 116. Duan, L.M., Demler, E., and Lukin, M.D., Controlling spin exchange interactions of ultracold atoms in optical lattices, Phys. Rev. Lett., 91, 090402, 2003. 117. Alexandrov, A.S., Theory of Superconductivity, IoP Publishing, Philadelphia, 2003. 118. Illuminati, F. and Albus, A., High-temperature atomic superfluidity in lattice BoseFermi mixtures, Phys. Rev. Lett., 93, 090406, 2004. 119. Wang, D.-W., Lukin, M.D., and Demler, E., Engineering superfluidity in Bose-Fermi mixtures of ultracold atoms, Phys. Rev. A, 72, R051604, 2005. 120. Bruderer, M., Klein, A., Clark, S.R., and Jaksch, D., Polaron physics in optical lattices, Phys. Rev. A, 76, 011605(R), 2007. 121. Friedrich, B. and Herschbach, D., Alignment and trapping of molecules in intense laser fields, Phys. Rev. Lett., 74, 4623, 1995. 122. Herzberg, G., Molecular Spectra and Molecular Structure I, Spectra of Diatomic Molecules, Van Nostrand Reinhold, New York, 1950. 123. Brown, J.M. and Carrington, A., Rotational Spectroscopy of Diatomic Molecules, Cambridge University Press, New York, 2003. 124. Judd, B.R., Angular Momentum Theory for Diatomic Molecules, Academic Press, New York, 1975. 125. See e.g. http://physics.nist.gov/PhysRefData/MolSpec/ 126. Coleman, S., Fate of the false vacuum: Semiclassical theory, Phys. Rev. D, 15, 2929, 1977. 127. Kotochigova, S., Tiesinga, E., and Julienne, P.S., Photoassociative formation of ultracold polar KRB molecules, Eur. Phys. J. D, 31, 189, 2004. 128. Mahan, G.D., Many Particle Physics, Kluwer Academic/Plenum Publishers, NewYork, 2000. 129. This antiadiabatic regime is hard to achieve with cold atoms, see, e.g., Refs. [118] and [119]. 130. Carmichael, H.J., Statistical Methods in Quantum Optics 1, Springer-Verlag, Berlin, 1999. 131. Ortner, M., Micheli, A., Pupillo, G., and Zoller, P., Quantum simulations of extended Hubbard models with dipolar crystals, 2009 (submitted for publication). 132. For single-frequency phonons in 1D, see: Datta, S., Das, A., and Yarlagadda, S., Manypolaron effects in the Holstein model, Phys. Rev. B, 71, 235118, 2005. 133. Hirsch, J.E. and Fradkin, E., Phase diagram of one-dimensional electron-phonon systems. II. The molecular-crystal model, Phys. Rev. B, 27, 4302, 1983; Niyaz, P., Scalettar, R.T., Fong, C.Y., and Batrouni, G.G., Phase transitions in an interacting boson model with near-neighbor repulsion, Phys. Rev. B, 50, 362, 1994. 134. Movre, M. and Pichler, G., Resonant interaction and self-broadening of alkali resonance lines I. Adiabatic potential curves, J. Phys. B: Atom. Molec. Phys., 10, 2631, 1977.
© 2009 by Taylor and Francis Group, LLC
Part IV Cooling and Trapping
© 2009 by Taylor and Francis Group, LLC
Cooling, Trap Loading, 13 and Beam Production Using a Cryogenic Helium Buffer Gas Wesley C. Campbell and John M. Doyle CONTENTS 13.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Buffer-Gas Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Loading of Species into the Buffer Gas. . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1.1 Laser Ablation and LIAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1.2 Beam Injection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1.3 Capillary Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1.4 Discharge Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Rotational and Vibrational Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Buffer-Gas Loading of Magnetic Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Lifetime of Trapped Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1.1 Evaporative Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1.2 Buffer Gas Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1.3 Spin Relaxation Loss (Atoms) . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Zeeman Relaxation Collisions between Molecules and Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2.1 Inelastic Collisions of 2Σ Molecules with He . . . . . . . . . 13.3.2.2 Inelastic Collisions of 3Σ Molecules with He . . . . . . . . . 13.4 Buffer-Gas Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Thermalization and Extraction Conditions . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Boosting Condition and Slow Beam Constraints . . . . . . . . . . . . . . . . . 13.4.3 Studies with Diffusively Extracted Beams . . . . . . . . . . . . . . . . . . . . . . . 13.4.4 Studies with Hydrodynamically Extracted Beams . . . . . . . . . . . . . . . 13.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
474 474 476 476 478 480 480 483 484 485 485 487 488 491 493 494 496 497 498 499 501 504 504 504
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Cold Molecules: Theory, Experiment, Applications
INTRODUCTION
Trapping and cooling of neutral particles, starting 23 years ago with the first magnetic trapping of an atom [1], has completely transformed the toolbox of atomic, molecular, and optical physics. The breadth of scientific impact is astounding, ranging from the creation of new quantum systems, observation of new collisional processes, enhancements in precision measurement techniques, and new approaches to quantum information and simulation. However, we feel that the current situation is only a beginning. Compared to the situation with atomic and molecular beams, the number of trapped or cooled species is paltry—no more than 30 different species in comparison to hundreds for beams. With new species come new interactions (and new complications). There is much to look forward to. Examples of “unfinished business” include the creation of polar molecules in optical lattices (predicted to be useful as a tunable Hubbard model system), strongly interacting dipolar gases, dramatic improvements in the search for permanent dipole moments, in-laboratory study of the myriad of astrophysical cold collisional processes, quantum computers based on single atoms or molecule qubits, and arbitrary species cooling for precision studies such as variation of fundamental constants (see Chapter 16 by Flambaum and Kozlov). All of these are on the near horizon, thanks to the continued large efforts toward expanding trapping and cooling techniques. The development of these new techniques and their rapid application to new systems has brought a continuing harvest of scientific findings. This chapter describes the experimental techniques of buffer-gas cooling, loading, and slow beam formation, the latter two building upon the first. Buffer-gas cooling uses a helium refrigerator to cool a gas of helium, which, in turn, cools atoms or molecules. The extraordinary generality and simplicity of buffer-gas cooling results from a combination of (1) helium being chemically inert and effectively structureless, (2) the typical cold elastic cross-section of helium with any atom or molecule being around 10−14 cm2 (allowing for centimeter size cooling cells), (3) the appreciable saturated vapor pressure of helium at temperatures down to 200 mK and (4) existing techniques to create gas-phase samples of nearly any species, including radicals. There are three main sections to this chapter. First, we review buffer-gas cooling, in which hot atoms or molecules are introduced into a cold helium gas, where they thermalize. Second, we discuss buffer-gas loading of traps. Third, we describe the production of cold molecular and atomic beams made by adding a hole in the side of the cryogenic buffer-gas cell. These three processes are depicted in Figure 13.1. Although buffer-gas cooling could be used with other gases to cool to temperatures above 4 K (e.g., H2 or Ne to temperatures around 15 K), this review covers only buffergas cooling to temperatures around and below the boiling point of liquid helium, 4.2 K. Much of the buffer gas physics described in this review would apply to higher temperature systems, especially those using neon.
13.2
BUFFER-GAS COOLING
The technique of buffer-gas cooling [2] relies on collisions with cold buffer gas atoms to thermalize atoms or molecules to low temperature. The buffer gas serves to dissipate translational energy of the target species and, in the case of molecules, © 2009 by Taylor and Francis Group, LLC
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Cooling, Trap Loading, and Beam Production (a)
(b)
Cold buffer-gas cell
Molecule
(c)
Helium atom
FIGURE 13.1 Buffer-gas (a) cooling, (b) loading, and (c) beam creation. (a) Buffer-gas cooling occurs when the hot target species A (molecules or atoms) enters cold buffer gas inside a cold cell, which are both at a low temperature T . The A particles collide with the buffer gas, cooling to a temperature close to T in approximately 100 collisions. The particles of A then diffuse to the wall, where they stick and are lost. The sizes of the A–He and He–He cross-sections determine the efficiency of indirect cooling of A by the cell wall after the initial thermalization. (b) In buffer-gas loading, thermalized A particles feel a potential due to some type of trapping field (e.g., magnetic or electric). The A particles in trapping states feel a force pulling them to the center of the trap, whereas those in untrapped (or antitrapped) states move to the walls where they stick. The time for this “fall in” is about the diffusion time of A particles across the cell with the trapping field turned off. (c) Thermalized A particles can exit a hole in the side of a cryogenic cell and form a cold beam of A. The extraction efficiency and velocities of A in the beam depend on the density of helium, size of the cell, and size of the exit hole.
their rotational energy as well. Because this dissipation scheme does not depend on any particular energy level pattern, any target species is amenable to it. As with the case of evaporative cooling of a trapped ensemble, buffer-gas loading relies on elastic collisions. At temperatures of ∼1 K, all substances except for He (and certain spin-polarized species, such as atomic hydrogen) have negligible vapor pressure, so the question arises as to how to bring the species to be cooled into the gas phase and then into the buffer gas. Five methods have been used to accomplish this: laser ablation, beam injection, capillary filling, discharge etching, and laser-induced atom desorption (LIAD). The translational thermalization process of the target species with cold helium can be modeled by assuming elastic collisions between two mass points, m (buffer gas atom) and M (target species). From energy and momentum conservation in a hardsphere model, we find, after thermal averaging, that the difference ΔT in temperature of the atom or molecule before and after a collision with the buffer gas atom is given by ΔT = (T − T )/κ, with T denoting the temperature of the buffer gas, T the initial temperature of the atom or molecule, and κ ≡ (M + m)2 /(2Mm). The equation for the temperature change can be generalized and recast in differential form: dT = −(T − T )/κ d © 2009 by Taylor and Francis Group, LLC
(13.1)
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where T is the temperature of the atom or molecule after collisions with the buffer gas atom. Equation 13.1 has a solution T /T = (T /T − 1)e−/κ + 1.
(13.2)
Under the conditions of T ≈ 1000 K and M/m ≈ 50, of order 100 collisions are required for the atoms or molecules to fall to within 30% of the He buffer-gas temperature T = 0.25 K. This 100 collisions typically corresponds to a time of order 0.1–10 msec, depending on the buffer-gas density; this is consistent with our observations of buffer-gas cooling. Figure 13.2 shows the thermalization of laser-ablated VO in cold helium buffer gas for different delays after the ablation pulse. The spectra show complete translational thermalization in less than 10 msec, in accordance with the above simple model. In order to ensure that the target species thermalizes before impinging on the wall of the cell, it is necessary that the density of the buffer gas be large enough to allow for thermalization on a path smaller than the size of the cell. Cells are typically of order 1 cm in diameter. Assuming an elastic collision cross-section of about 10−14 cm2 between the target species and helium (an assumption accurately borne out by numerous experiments [3–6]), the minimum density required is typically 3 × 1014 cm−3 . This requirement puts a lower limit on the temperature of the buffer gas. Figure 13.3 shows the dependence of number density on temperature for 3 He [7] and 4 He [8] at about 1 K. One can see that 3 He can be used at temperatures as low as 180 mK and 4 He as low as 500 mK. In the above discussion of thermalization, it was assumed that the temperature of the buffer gas was fixed to its final temperature (typically around 1 K). For short times (less than the diffusion time of helium to the cell walls) one must consider the heat capacity of the buffer gas, that is, when there is no other mechanism present to take energy away from the hot molecules. To cool to a temperature within 30% of T , the ratio Υ of the minimum number of (precooled) helium atoms to the number of initially hot (T ) target species is given by Υ = (T /T − 1)/0.3. For T ≈ 4 K and T ≈ 1000 K this amounts to Υ ≈ 1000. This is a kind of “worst case.” Depending on the experimental setup, goals, and elastic cross-sections, it may be possible to use the cold walls of the chamber to cool the buffer gas with little loss in the number of molecules. This is especially true while, for example, molecules are being drawn through the buffer gas into the center of a magnetic trap.
13.2.1
LOADING OF SPECIES INTO THE BUFFER GAS
We know of five demonstrated species introduction methods (the first four of which have been used in our laboratory): laser ablation, beam injection, capillary filling, discharge etching and LIAD. A schematic of the first four methods is depicted in Figure 13.4. 13.2.1.1
Laser Ablation and LIAD
Numerous species have been laser-ablated and buffer-gas cooled. Laser ablation of solid materials is well established as an important tool in many scientific and technological endeavors, including surface processing, surgery, mass spectrometry and © 2009 by Taylor and Francis Group, LLC
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Absorption
0.1 0 0.4 1–2 msec 0.3 0.2 0.1 0 0.4 0.3
4–10 msec 3/2
1/2
P1 (0)
R1 (–1)
0.2
3/2
P1 (0)
3/2
P1 (0)
0.1 0
–3
–2
–1
0 Frequency (GHz)
1
2
3
FIGURE 13.2 Laser absorption spectra showing the thermalization of laser-ablated VO in helium buffer gas. The molecules thermalize to the temperature of the cell walls in less than 4 msec. (From Weinstein, J.D. et al., J. Chem. Phys., 109(7), 2656–2661, 1998. With permission.)
growth of materials. Despite prolific applications, sorting out the fundamental mechanisms for laser ablation and identifying all the physical processes involved in laser ablation has proven quite difficult. For example, the coupling mechanism between the laser light and the sample can be very complex because the optical and thermal properties may change upon laser exposure due to the formation of excited states and plasma. These processes become even more complicated and essentially intractable © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications 1022 3He
1020
4He
Density (cm–3)
1018
1016
1014
1012
1010
0.2
0.5 1 Temperature (K)
2
5
FIGURE 13.3 The saturated vapor density of helium at buffer-gas cooling temperatures. The 4 He curve is extrapolated from ITS-90 [8] and the 3 He curve is from Ref. [7] with the old ITS90 curve (thin line) shown for reference. The shaded region represents the range of saturated vapor densities used for buffer-gas loading, which sets a minimum temperature of 180 and 500 mK for 3 He and 4 He buffer gas, respectively.
when chemical reactions are required to produce the species of interest. For example, CaF, CaH, and VO radicals were produced by the same 532 nm (4 nsec pulse length, approximately 10 mJ energy) laser ablation of stable solids, CaF2 , CaH2 , and V2 O5 , respectively. The yields, however, varied by five orders of magnitude. In the case of CaH, only 10−8 of the pulse energy went into breaking the single chemical bond to produce CaH from CaH2 . Because of the heating limitations of buffer-gas cooling, the pulse energies used are typically below 20 mJ. LIAD is a laser-induced desorption process that has been demonstrated exclusively with atomic Rb and K. We refer the reader to Ref. [9] for more details. In short, laser light excites a plasmon resonance in the solid metal, driving atoms off the surface and into the buffer gas. 13.2.1.2
Beam Injection
Beam injection is perhaps the most general way of introducing target species into a cold buffer gas. However, it is also the most difficult. The idea is to have an orifice in the side of the buffer-gas cell, typically millimeter to centimeter size, to allow a beam of molecules or atoms to pass into the cell. Making beams of essentially any atom or molecule is possible. The density of helium in the cell is not high enough to form clusters (at least for atoms and simple molecules). However, with the addition of the molecular beam input orifice, a new host of problems arises. First, the escaping © 2009 by Taylor and Francis Group, LLC
479
Cooling, Trap Loading, and Beam Production (a)
(b) Cryostat
Cryostat Room temp. enclosure
Solid precursor
Helium buffer gas
High temp. beam source
Molecules of interest
Ablation laser beam (d)
(c)
Cryostat
Cryostat
Room temp. gas reservoir
Capillary tube
RF discharge coil
FIGURE 13.4 Introduction of molecules into the buffer gas through (a) laser ablation of a solid precursor, (b) molecular beam injection, (c) capillary injection, and (d) discharge etching.
helium can build up density just outside the cell, knocking incoming molecules out of the molecular beam, preventing them from entering the cell. Second, even if the molecules enter the cell, they may be hydrodynamically pulled back out of the cell as helium is continually moving through the orifice into the vacuum region. Third, helium must continually be replenished, adding a potential heat load to the cryogenic system. Despite these difficulties, we have been able to show efficient beam injection loading and buffer-gas cooling of Rb, N, NH, and ND3 , including trapping of N and © 2009 by Taylor and Francis Group, LLC
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NH and electric guiding of ND3 (Tables 13.1 and 13.2). The key to success was rapid pumping of the buffer gas in the beam region (outside the orifice) via a cryogenic sorption pump for helium made of charcoal. 13.2.1.3
Capillary Filling
Capillary filling is, perhaps, the first idea that comes to mind when thinking about how to introduce atoms or molecules into a buffer gas. Indeed, it was the first method used to buffer-gas cool a molecule [10]. In essence, it is guiding using solid walls. Gas-phase species are transported from high temperature (typically at or above 300 K) to 4 K via a fill capillary. This arrangement has its own peculiar technical problems. First, because one end of the fill capillary is exposed to 4 K buffer gas, not only is there a heat load from the tube into the buffer gas, but the tube near the buffer-gas cell could become cold enough to freeze the incoming species. Second, because species require the fill tube to be maintained above their freezing point, most species require temperatures at or above room temperature, which can become a problem due to the cell capillary depositing heat into the cold buffer-gas cell. Third, depending on the flow conditions the species coming down the tube can suffer intraspecies collisions and collisions with the walls of the tube. As such, this method may be of limited utility for chemically reactive species. However, it is possible to partially overcome the wall issues by using high enough flow of gas through the capillary so that the target species does not hit the tube wall before entering the cold cell. One variation on this approach is coflow of an inert gas in the capillary. If the flow is rapid enough, the diffusion time of a molecule through the inert gas to the wall of the capillary can be less than the time the molecule spends in the capillary (the “residence time”). One can show that the condition for the diffusion time to be equal to the residence time in a capillary of length L (cm) and with a flow rate N˙ (sccm) is approximately given by L ≈ 0.1N˙ at T ≈ 4 K. Without resorting to coflow the capillary method is generally limited to a small subset of atomic and molecular species. However, that small subset contains some very important molecules, including O2 , NH3 , HCN, and N2 . Although in principle atomic samples could be introduced into a buffer-gas cell through a heated capillary (particularly for low boiling point metals like Hg and Cs), to our knowledge no atom has ever been buffer-gas cooled using capillary filling. 13.2.1.4
Discharge Etching
A final method for introducing species into the cold buffer gas is discharge etching, in which an electrical discharge plasma in the buffer gas “etches” off species that are frozen to the cell wall, bringing them into the gas phase. Once liberated, the discharge plasma can even excite/dissociate a fraction of them (e.g., into a metastable state). To date we have only used this for one species, He*, but based on the success with He* and previous work with low-temperature production of atomic hydrogen, it is likely that this method will find further application. The type of discharge used for this purpose in our lab is a λ/4 radio-frequency helical resonator coil discharge. The coil is wound around or inside the buffer-gas cell [11]. Our He∗ experiments begin with an entirely empty cell with an inner surface precoated with a few monolayers of atomic helium and with one end open to high © 2009 by Taylor and Francis Group, LLC
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TABLE 13.1 A List of Molecules That have been Helium Buffer-Gas Cooled to Less than 10 K Species
Na
CaF CaHf CH3 Fg CrH CO DClg H2 Sg H2 COg HCN MnH NDh NH NHh,i (v = 1)j ND3 ND3 ND3 NOg O2 PbO PbO SrO SrO SrO ThOk ThOl VO
5 × 1013
a b
c d e f g h i j k l
1 × 108
1 × 105
1 × 105 1 × 108 5 × 1011 2 × 108 2 × 107
na (cm−3 ) 8 × 107 ≤2 × 1013 1 × 106 ≤5 × 1012 ≤5 × 1012 ≤5 × 1012 ≤1 × 1013 ≤7 × 1013 1 × 106
1 × 109 ≤2 × 1012 1 × 1012 2 × 1012 1 × 1013
Fa (sec−1 )
1 × 1015 1 × 1012 7 × 1010 3 × 1012 1 × 109
1 × 1013 5 × 1010
1 × 1012
Tb (K) 2 0.40 1.2 0.65 1.3 1.85 1 1.7 1.3 0.65 0.55 4 0.55 0.615 7 7 5 1.8 1.6–25 4 4 4 Beam Guide 4
1 × 1015 1.5
Result
τc (sec)
γd [×104 ]
Loadinge
Cold 0.08 1.3 Ablation Trap 0.50 103 Ablation Cold – – Capillary Trap 0.12 0.9 Ablation Cold – – Capillary Cold – – Capillary Cold – – Capillary Cold – – Capillary Cold – – Capillary Trap 0.18 0.05 Ablation Trap 0.50 7 From beam Cold – – From beam Trap 0.95 20 From beam Trap 0.033 >5 From beam Beam – – Capillary Guide – – Capillary Guide Capillary Cold – – Capillary Guide – – Capillary Cold 0.020 – Ablation Cold – – Ablation Cold >0.02 – Ablation – – Ablation [57] – – Ablation [57] Cold 0.03 – Ablation Beam – – Ablation Cold 0.06 – Ablation
Ref. [33,42] [4,18] [43,44] [45] [46,47] [48] [13,49] [15] [50] [45] [30,51] [6] [52] [16,51] [53] [53] [54] [14,55] [39] [42] [37] [57]
[58] [58] [18,59]
N, n, F, maximum number, density, and flux. T , lowest temperature reached with this species in a helium buffer-gas-based experiment. This temperature does not necessarily correspond to the maximum number or density given in the N, n, F columns. In the case of trapping this temperature indicates the lowest trapped temperature achieved. In the case of a beam, it generally indicates the transverse temperature of the beam. Results indicate whether the molecules were just cooled or cooled and trapped, made into a beam, or made into a guided beam. τ is the maximum lifetime observed. In the case of trapping, this is the approximate maximum lifetime observed in the trap. γ is the ratio of diffusion to spin relaxation cross-section. Loading refers to the method used to introduce species into the buffer gas—either laser ablation, capillary injection, LIAD, or loading from a beam. Data from Refs. [4,18] were reanalyzed for γ. Steady-state flow of molecules into the cell, we estimate helium densities used equivalent to about 100 msec diffusion lifetime. 15 NH and 15 ND also trapped. γ = 7 × 104 for 3 He. Lifetime limited by spontaneous emission. Simultaneous metastable and ground-state production and observation. Instantaneous flux over a 3 msec wide pulse.
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TABLE 13.2 Atoms That have been Helium Buffer-Gas Cooled Species Ag Au Bi Ceh Cr Cr Cs Cu Dy Er Eu Fe Gd He∗i Hf Ho Ho K Lij Mn Mo N Nak Na Nd Nil Pr Rbm Rbn Rb Re Sc Tb Ti Tm Y Yb Yb Zr a b
Na
na (cm−3 )
Fa (sec−1 )
4 × 1013 1 × 1013 5 × 1011 1 × 1012 1 × 1012 1 × 1011 2 × 109 3 × 1012 2 × 1012 2 × 1011 1 × 1012 5 × 1011 1 × 1010 5 × 1011 1 × 1012 1 × 1010 9 × 1011 4 × 109 2 × 1013 2 × 1012 2 × 1010 1 × 1011
1 × 1013 1 × 1012
5 × 1012
5 × 1011
2 × 1012
5 × 1012 1 × 1015
5 × 1012 1 × 1012 1 × 1013 3 × 1011 1.2 × 1012
1.5 × 1011 8 × 109 1 × 109
1 × 1012 1 × 1011 2 × 1011 4 × 1010 2 × 1011 1 × 1011 2 × 1013
Tb (K) 0.42 0.40 0.50 0.60 <0.01 0.35 4 0.32 0.60 0.80 0.25 0.60 0.80 <0.001 0.35 0.60 0.80 4 0.09 0.85 0.20 0.55 4 0.48 0.80 0.60 0.80 4.5 4 1.85 0.50 0.80 0.80 0.80 0.80 0.80 4
5 × 1014 1 × 1010
0.80
τc γd Result (sec) [×104 ] Loadingg Trap Cold Cold Cold Trap Trap Cold Trap Trap Trap Trap Cold Cold Trap Cold Trap Trap Cold Cold Trap Trap Trap Cold Trap Trap Cold Trap Cold Cold Cold Cold Cold Trap Cold Trap Cold Cold Beam Cold
2.3 0.12 0.09 0.08 >100 60 0.002 8 >20 0.05 >100 <0.01 0.07 100 0.00? 4 >20 0.008 100 >100 >100 12 – 0.3 1 0.02 0.12 0.018 – 10 0.45 0.14 0.12 0.15 0.03 0.18 0.10 – 0.10
300 >10 <4 4 e e – 800 50 4 e <0.5 nmf e <0.3 nm 30 – e e e e – e 9 1 10 – – – <30 <1.6 10 ∼4 3 <3 – – nmf
Ref.
Ablation Ablation Ablation Ablation Ablation Ablation Ablation Ablation Ablation Ablation Ablation Ablation Ablation Discharge Ablation Ablation Ablation Ablation Ablation Ablation Ablation From beam Ablation Ablation Ablation Ablation Ablation Ablation From beam LIAD Ablation Ablation Ablation Ablation Ablation Ablation Ablation Ablation Ablation
[60] [60] [26] [61] [5,20] [62] [63] [60] [28] [28] [3] [22] [64] [11,65] [66] [61] [28] [63] [67] [68,69] [70] [71] [37] [72] [28] [22] [28] [73] [42] [9] [26] [29] [28] [29] [28] [74] [39,75] [39] [74]
N, n, F, maximum number, density, and flux. T , lowest temperature reached with this species in a helium buffer-gas-based experiment. This temperature does not necessarily correspond to the maximum number or density given in the N, n, F columns. continued
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TABLE 13.2 (continued)
c d e f g h i j k l m n
In the case of trapping this temperature indicates the lowest trapped temperature achieved. In the case of a beam, it generally indicates the transverse temperature of the beam. Results indicate whether the molecules were just cooled or cooled and trapped, made into a beam, or made into a guided beam. τ is the maximum lifetime observed. In the case of trapping, this is the approximate maximum lifetime observed in the trap. γ is the ratio of diffusion to spin relaxation cross-section. γ was very high and no spin relaxation was observed. nm, γ was observed to be low but was not measured. Loading refers to the method used to introduce species into the buffer gas—either laser ablation, capillary injection, LIAD, or loading from a beam. OD > 10. Surface evaporation down to 2 mK; radio-frequency evaporation to <100 μK. Evaporation from 160 mK. Flux in a 100 msec pulse. OD = 6. OD = 75. OD = 1.2.
vacuum. The helium on the walls of the cell is naturally tightly bound to the surface as all the weakly bound helium has already been pumped away. A short radio-frequency pulse (typically 300 μsec) is sent to the discharge coil. The discharge is ignited, pulling enough helium atoms off the surface to act as a buffer gas. A small fraction of the helium (about 10−5 ) is turned into metastable (magnetic 3 S-state) helium and is buffer-gas cooled. About 5 × 1011 metastable helium atoms are created, buffer-gas cooled, and magnetically trapped in this way [11].
13.2.2
ROTATIONAL AND VIBRATIONAL RELAXATION
A central issue with the technique of buffer-gas cooling of molecules is the efficiency of cooling rotational, vibrational, spin, or other internal degrees of freedom. For experiments limited by counting statistics, cooling of these degrees of freedom may be critical due to the drastic effect such cooling can have on the population of molecules in the desired state. It may also be desirable to use the buffer gas to cool only specific degrees of freedom while leaving others out of thermal equilibrium. Applications of this kind of selective thermalization include buffer-gas loading of magnetic traps (described in Section 13.3) where the electron spin temperature must remain thermally disconnected from the buffer-gas temperature. In order to assess the feasibility of such experiments, the timescales for relaxation must be understood. Generally, collision-induced quenching is far more efficient for rotation than vibration. Rotational quenching is driven by the angular anisotropy of the helium interaction with the molecule, and the timescale for a small impact parameter cold collision is similar to a rotational period. Vibrational relaxation, on the other hand, is driven by the dependence of the interaction potential on the internuclear separation in the © 2009 by Taylor and Francis Group, LLC
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molecule. The vibrational motion of the nuclei, being characterized by typical energy separations of 1000 K, is much faster than a cold helium atom collision. Typical quenching cross-sections by cold (∼1 K) helium yield a rotational quench on the order of every 10 to 100 elastic collisions whereas it can take more than 108 elastic collisions before a vibrational quench [12]. DeLucia and coworkers measured rotationally inelastic cross-sections with He of H2 S, NO, and H2 CO. Typical values were on the order of 1 to 10 × 10−16 cm2 at 1 K [13,15,56], which is about 10 to 100 times smaller than a typical diffusive cross-section at 1 K. We therefore expect buffer-gas cooling to effectively thermalize the rotational temperature of the target molecules while leaving the vibrational temperature out of thermal equilibrium. Because the rotational and translational energy transfer cross-sections are similar, thermalization of both happens rapidly and in tandem in buffer-gas cooling. We do not detect nonequilibrium rotational populations. Vibrationally hot molecules, on the other hand, have been created using laser ablation and molecular beam loading and have been observed in our group. We were able to translationally cool CaH(v = 1) to <500 mK and saw no evidence of vibrational quenching, giving a limit of σCaH,v=1 < 10−18 cm2 on the helium induced vibrational quenching cross-section [4]. We also magnetically trapped NH (v = 1) and found that the lifetime of the trapped vibrationally excited molecules was limited by spontaneous emission, not collisional thermalization. The fact that the vibrational temperature remained extremely hot allowed us to perform a precise measurement of the NH (X 3 Σ− , v = 1 → 0) spontaneous emission lifetime. We measured a limit of kNH,v=1 < 3.9 × 10−15 cm3 sec−1 for the helium induced vibrational quenching coefficient [16].
13.3
BUFFER-GAS LOADING OF MAGNETIC TRAPS
Buffer-gas loading can take place when buffer-gas cooling occurs within a trapping potential. Using the language of magnetic traps, the low-field seekers (LFS, meaning those states whose Zeeman energy increases with increasing magnetic field) cool to the temperature of the buffer gas and feel a force drawing them to the center of the trap (where the field magnitude is a minimum). This “fall-in” process takes place on a timescale comparable to the field-free diffusion time of molecules in the buffer gas. Depending on the helium density and initial conditions, this can take place over Nfall-in = 10 to 104 collisions. The high-field seekers (HFS) are, in contrast, pushed out to the edge of the trap on about the same timescale, where they stick to the cold walls and are lost from the trapping region. In order for trapping to work, the temperature of the buffer gas must be lower than the trap depth and the LFSs must not change to other (energetically favorable) states as they diffuse to the center of the trap and form a trapped distribution. As such, if there is an inelastic process such as reorientation of the magnetic moment of the molecule due to a collision with helium, the molecule may be lost before trapping occurs. The ratio of diffusion (elastic) to inelastic cross-sections, γ, for helium colliding with the target species must be above a certain value for trapping to take place, γ > Nfall-in . © 2009 by Taylor and Francis Group, LLC
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13.3.1
LIFETIME OF TRAPPED MOLECULES
The lifetime of a trapped sample may be limited by collisions with the helium buffer gas. Figure 13.5 schematically shows the lifetime of two hypothetical trapped molecular species. Four distinct regimes are represented, each with a different physical mechanism that limits the trap lifetime. Elastic collisions with helium can promote molecules to energies above the depth of the trap (evaporative loss, occurring near the “valley of death” and in the “diffusion enhancement” regions indicated in Figure 13.5). Helium collisions can also inelastically change the internal state of the molecule to an untrapped state, which limits the trap lifetime in the section of Figure 13.5 marked “spin relaxation.” The removal of the buffer gas is necessary in order to shut these processes down to obtain a thermally isolated sample that could be cooled to a temperature below that of the cell walls. This is shown in the area labeled “thermal isolation,” where the rate of He collisions is low, leading to long trap lifetimes and low heating rates for the trapped sample. A comparison of the lifetime of the trapped molecules to the pump-out time of the buffer gas out of the cell serves as a guide for the feasibility of achieving thermal isolation. 13.3.1.1
Evaporative Loss
isolation
D enh iffusi anc on em ent
Thermal
Trapped molecule lifetime (arb. units)
A thermal Boltzmann trap distribution of molecules always contains molecules whose energies allow them to access the spatial edge of the trap. As molecules reach this edge and stick to the cold walls of the buffer-gas cell, a truncation of the distribution takes place. Molecules that pass the edge of the trap are lost from the distribution and
Spin relaxation Valley of death Buffer-gas density (log scale)
FIGURE 13.5 Trapped molecule lifetime vs. buffer-gas density (note semilog scale) for two hypothetical molecular species. For the dashed curve, the helium-induced Zeeman relaxation cross-section has been increased by a factor of 10.
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this evaporative loss can be the limiting factor in determining the lifetime of trapped molecules. The evaporation-limited lifetime of trapped molecules depends sensitively on η, the trap depth measured in terms of the molecules’ translational energy η≡
μBmax . kB T
(13.3)
In the long mean free path regime, the average total distance λ traveled before the molecule’s trajectory is randomized by a collision (the diffusion mean free path) is large compared to the size of the trap. A lower limit for the evaporation-limited long mean free path trap lifetime can be estimated by assuming ballistic travel through the trap and a distribution of molecules that is continuously thermalized to the temperature of the cell walls. (This situation occurs in the “valley of death” region of Figure 13.5.) The evaporation rate is then given by flux of molecules to the trap edge, φ = (1/4)nv. ¯ This yields the expression 4Veff η τ≥ e (13.4) v¯molec A where Veff is the “effective volume” of the trap (the ratio of the total molecule number to peak molecule density), v¯molec is the average velocity of the molecules and A is the surface area of the trap edge [2]. For a spherical quadrupole trap (such as the anti-Helmholtz traps used in our group) whose depth is set by the radius of the cell, r0 , we have Veff ≈ 2.7 × V0 /η3 (where V0 = 4/3πr03 ). The long mean free path evaporation described by Equation 13.4 is the limiting loss mechanism at low helium density, where the assumption of ballistic travel through the trap is valid. At higher buffer-gas densities, where λ < r0 , the helium enforces diffusive motion of the molecules in the trap, which slows the evaporation rate and therefore increases the trap lifetime, as shown in the “diffusion enhancement” section of Figure 13.5. It is in this short mean free path regime that buffer-gas loading typically begins, and the modeling of the lifetime is somewhat more complicated than in the long mean free path regime. Starting with the field-free diffusion lifetime, the trap lifetime can be obtained by solving the diffusion equation with the boundary condition that the molecule density be zero at the walls of the cylindrical cell. The lifetime of the lowest-order field-free diffusion mode is given by 16nHe σd τ0 = √ 3 2π
mred kB T
α1 r0
2
+
π h0
2 −1 (13.5)
where σd is the thermal average of the diffusion cross-section, r0 and h0 are the internal radius and length of the buffer-gas cell, mred is the reduced mass of the collision complex, and α1 ≈ 2.40 is a numerical factor [17]. The effect of the trapping field can be accounted for by numerically solving the diffusion equation with a drift term from the trap [18]. Figure 13.6 shows the calculated lifetime using this approach (in units of the field-free diffusion lifetime, τ0 ) as © 2009 by Taylor and Francis Group, LLC
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Trap lifetime (t0 )
NH
100
10
1
0
2
4
6
8
10
12
14
h
FIGURE 13.6 Trapped molecule lifetime vs. trap depth in units of the field-free diffusion lifetime, τ0 (Equation 13.5). The points are data taken for NH [6].
a function of η. As can be seen from Figure 13.6, for η > 6, the trapping lifetime is 10 times greater than the field-free diffusion time. 13.3.1.2
Buffer Gas Removal
In the regime where the trap lifetime is limited by evaporation (the long mean free path regime, Section 13.3.1.1), the presence of the buffer gas shortens the trapped molecule lifetime. This is because the constant rethermalization of the trapped sample to the buffer-gas temperature continuously promotes molecules to energies above the trap edge. If all collisions could be turned off, the trapped molecules with total energy lower than the trap depth would stay in the trap for an extended period of time, indicated as “thermal isolation” in Figure 13.5. In this regime, the trapped molecule lifetime would likely become limited by longer timescale effects, such as Majorana transitions or molecule–molecule collisions. One way to achieve this thermal disconnect is to remove the buffer gas. This can be done either by cooling the cell walls [5] to lower the saturated vapor density of helium (see Figure 13.3) or by pumping the gas out through an orifice in the side of the cell [19]. In order for the molecule number after buffer gas removal to be large, the timescale for pump-out must be small compared to the “valley of death” lifetime in Figure 13.5. Such a pump-out can introduce two new complications. First, a helium film on the walls of the cell can desorb slowly, leading to longer times spent in the “valley of death.” The binding energy of the first (bottom) monolayer is typically of order 100 K and the timescale for desorption at 500 mK is extremely long. The binding energy of the least bound (top) monolayer is on the order of the cell temperature © 2009 by Taylor and Francis Group, LLC
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He density in buffer-gas cell (cm–3)
1015 1014 1013 1012 Film present 1011 1010 No film 109 108
0
1
2
3
4
5 6 Time (sec)
7
8
9
10
FIGURE 13.7 Calculated helium buffer-gas density vs. time after opening a high-conductance valve to pump out the cell [20]. The rapid pump-out in the absence of a film is compromised by the slow helium film desorption.
and therefore rapidly desorbs during the cell pump-out. The difficulty occurs with the layers in between, which desorb on the timescales of experiments, effectively softening the vacuum and maintaining the buffer-gas density near the valley of death, as shown in Figure 13.7. To combat this problem, the temperature of the cell can be lowered after some desorption, meaning that buffer-gas loading should occur above the base temperature of the refrigerator. This sort of heating while pumping is well known to the room-temperature UHV atom trapping community as a bakeout. A typical cryo-bakeout involves a temperature rise of less than 1 K above base temperature. The second possible complication is the helium wind pulling the trapped molecules out of the trapping region. We have studied this effect as well and found that, fortunately, it is possible to avoid it in most circumstances [20]. 13.3.1.3
Spin Relaxation Loss (Atoms)
A significant drawback of trapping via a LFS Zeeman sublevel is that this opens an exothermic inelastic collision channel. During a collision, a stretched-state [76] LFS molecule can undergo a Zeeman transition to a less-trapped or even HFS state. This process is referred to as either “spin relaxation” [77] (as in Figure 13.5) or collision-induced Zeeman relaxation. To utilize buffer-gas loading to trap a species, this process must be sufficiently unlikely that the external motion of the molecule can be thermalized and the experiment carried out before the Zeeman state changes. All of the inelastic collision processes seen in atoms are also possible with molecules, and they are categorized here to distinguish them from processes unique © 2009 by Taylor and Francis Group, LLC
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to molecules. Collision-induced Zeeman transitions in atoms can be induced by spinexchange, dipolar relaxation, interaction anisotropy, Feshbach resonances, metastable decay, shape resonances, and three-body recombination. The last of those will not be treated here, because achievable cold polar molecule densities are currently far too low for three-body collisions. In spin-exchange, an atom with a net spin can trade angular momentum with the spin of its collision partner. This is not a factor for collisions with ground-state 4 He because it has zero spin, but the 3 He nuclear spin and any other atom with net angular momentum can potentially participate in spin-exchange collisions. The dominant mechanism for alkali atom spin-exchange with 3 He is the overlap of the alkali valence electron with the 3 He nucleus, which results in a (molecular-type) hyperfine interaction. Spin-exchange cross-sections for alkali atoms colliding with 3 He at room temperature are typically 109 times smaller than the elastic cross-section [21], and should be no larger for molecule-3 He collisions. Dipolar relaxation also requires both collision partners to have magnetic moments. Although it is not a major concern for collisions with helium, it is ubiquitous in magnetically trapped samples. Briefly, the spin–spin interaction (which will be described in detail in the treatment of 3 Σ molecules) between magnetic moments of two colliding particles can exchange angular momentum between the electrons and the relative motion of the particles. This can arise as a result of the magnetic dipole–magnetic dipole interaction (which dominates for light particles) or as a result of a second-order interaction with a nonzero electron orbital angular momentum state (which tends to dominate for heavy particles). Dipolar relaxation rate coefficients are typically smaller than 10−13 cm3 sec−1 . The interaction between the atom of interest and a helium atom (VHe ) does not couple directly to the electron spin projection. This is in contrast to magnetic particles, which can cause Zeeman transitions through the spin–spin interaction. The long-range part of the helium interaction is typically dominated by the van der Waals potential and can be thought of as being an electrostatic interaction (although the shortrange portion involves exchange-type interactions). There is essentially no first-order probability for this to cause a spin reprojection: 7
S, MS |VHe | S, MS ∝ δMS M . S
(13.6)
The helium atom can, however, distort the spatial charge cloud. In this respect, both the radial and angular wavefunctions of the electrons must be considered. We will concentrate first on the angular charge distribution, some examples of which are depicted in Figure 13.8. For atoms, the angular distribution of electron charge is governed by the electron orbital angular momentum state |L, ML . For atoms with nonzero electron orbital angular momentum, the helium interaction can introduce an admixture of different ML states, which can be thought of as the exertion of a torque on the electron cloud. The spin–orbit interaction in the atom ensures that this admixing of ML states is accompanied by an admixing of MS states, potentially driving a Zeeman transition. S-state atoms, on the other hand, have only ML = 0 and are spherically symmetric. In order to drive Zeeman transitions, the helium atom must first mix in a state with nonzero electron orbital angular momentum, such as a p-orbital. The energy splitting © 2009 by Taylor and Francis Group, LLC
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FIGURE 13.8 Schematic depiction of the angular charge distribution in (a) an S-state atom, (b) a P-state atom, and (c) the rest frame of a diatomic molecule.
between these states tends to be much larger than the helium interaction energy. Numerous S-state atoms have been buffer-gas cooled and loaded into magnetic traps: He*, Cr, Mo, Mn, Na, Li, Cu, N, Ag, Eu. Only with Ag and Cu was γ small enough to be measurable in the experiments [22]. It is expected that atoms in non-S-states suffer inelastic collision loss rates much higher than S-state atoms. Indeed, calculations indicate that the main-group non-Sstate atoms Sr(P), Ca(P), and O(P) have γ ≈ 1 [23–25]. Furthermore, it has been observed that nominal S-state atoms that mix in large amounts of non-S shells have relatively low values of γ. Bi is the most prominent of these, with a measured γ < 5 × 104 [26]. The overall picture of non-S-state atoms having low γ therefore rests on strong experimental and theoretical footing. There are important exceptions to this well-founded characterization of non-Sstate spin relaxation. Due to the nontrivial behavior of the radial part of the electron wavefunction in many atoms, there is a collisional shielding that occurs. Specifically, if the partially filled orbitals of a non-S-state atom lie closer to the nucleus than an outer filled s-orbital, there is an effective shielding of the anisotropy. Many rare earth (and some transition-metal) atoms exhibit such a submerged shell structure [27]. The recognition of such structure as a possible suppression agent for spin relaxation in cold collisions led our group to pursue and demonstrate trapping of non-S-state atoms [28,29]. We found that in the case of the rare earth atoms, in which unpaired electrons are shielded by two filled outer s-shells, the shielding was dramatic. This resulted in γ > 104 (and as high as ≈106 ) for Tm, Er, Nd, Tb, Pr, Ho, and Dy, all in states with nonzero electron orbital angular momentum [28]. For the transition metal atom Ti, we found that even a single filled s-orbital effectively shields the unfilled d-orbital, resulting in γ = (4.0 ± 1.8) × 104 [29]. There can be an enhancement of the Zeeman transition probability if the interaction time between the trapped species and helium is extended by the formation of a quasibound complex, as is the case for a shape resonance (Figure 13.9). If the temperature of the colliding particles is high enough to permit non-s-wave collisions between particles, there may be quasibound states with nonzero angular momentum © 2009 by Taylor and Francis Group, LLC
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Energy
L=1
L=0
Collision complex separation
FIGURE 13.9 An L = 1 shape resonance can occur if the energy of the incoming particle matches the quasibound state of the L = 1 potential.
with lifetimes longer than a typical collision duration. The two particles stuck in such a quasi-bound state can orbit each other, forming a long-lived complex that increases the likelihood of Zeeman transitions in the species of interest due to the perturbations induced by the other particle. The probability of this occurring is maximized when the kinetic energy of the particles is equal to the energy of the quasibound state above the dissociation threshold. Shape resonances are very common in the few-partial-wave regime typical of buffer-gas cooling, and their effect on Zeeman relaxation is an active field of study [6,30]. Another type of resonant interaction that can lead to trap loss is a Feshbach resonance, which can occur through coupling to a true bound level of a higherenergy state of the collision complex. If the total energy of this bound state matches the energy of the colliding atoms, an enhancement of the inelastic rate can occur. Feshbach resonances have come into prominence in the field of cold molecules due to their usefulness for making ultracold molecules from laser-cooled atoms (see Chapter 9).
13.3.2
ZEEMAN RELAXATION COLLISIONS BETWEEN MOLECULES AND HELIUM
All of the mechanisms described above for atoms can and do lead to helium induced Zeeman relaxation of diatomic molecules. There are, however, additional pathways leading to trap loss that are unique to molecules. Figure 13.8c shows a hypothetical electron charge density for a diatomic molecule in the rest frame of the molecule. Comparison to Figure 13.8a,b suggests that a similar argument of © 2009 by Taylor and Francis Group, LLC
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helium induced Zeeman relaxation would be catastrophic to the potential of buffer-gas loading molecules. The tremendous anisotropy between the molecule and the helium atom leads to a potential minimum at some angle with respect to the internuclear axis, which would allow an incoming helium atom to exert a torque on the molecule. There is, however, an important difference between Figure 13.8c and Figure 13.8a,b. Panels (a) and (b) show eigenstates of JZ , the lab-frame projection of the total angular momentum. The molecule-fixed frame electron charge distribution, however, is not and the rotational motion of the nuclei in the lab frame must be taken into account. This degree of freedom is described by the rotational wavefunction, which gives the probability amplitude for finding the internuclear axis pointing in a particular direction (θ, φ) on the lab frame. We can write the rotational wavefunction in terms of R, the rotational quantum number of the nuclei, |R, MR = fR,MR (θ, φ),
(13.7)
where f is a function of the angles between the internuclear axis and the lab-fixed coordinate system (a rigid rotor wavefunction). Thus we find that even though there is a strongly anisotropic potential between the helium atom and the molecule in the molecule-fixed frame, this potential is averaged over the rotational wavefunction in the lab frame, where the interaction with the helium atom takes place. The strong molecule-fixed frame anisotropy depicted in Figure 13.8c is therefore effectively shielded from the helium atom by the rotation of the molecule in the lab frame. A sampling of rotational probability distributions are shown in Figure 13.10. The distance from the origin is the relative probability for the internuclear axis to be oriented at an angle θ with respect to the lab-frame Z-axis. It can immediately be seen that the pure |R = 0, MR = 0 ground state shown in Figure 13.10a is spherically symmetric, much like the S-state atom depicted in Figure 13.8a. Likewise, the first excited rotational state |R = 1, MR = 1 is highly nonspherical, similar to the Pstate atom shown in Figure 13.8b. It is therefore reasonable to expect a pure R = 0 state to be less likely to undergo helium induced Zeeman relaxation than an R = 1
(a)
(b)
Z
(c)
q X, Y
FIGURE 13.10 Schematic of the probability distribution that the internuclear axis makes an angle θ with the lab-frame Z-axis. Distance from the origin (times sin θ dθ if we wish to average over the polar angle φ) is the probability for the rotor to be pointing in the range θ, θ + dθ for (a) an |R = 0, MR = 0 state molecule, (b) an |R = 1, MR = 1 state, and (c) a rotational ground state 1 Π molecule. © 2009 by Taylor and Francis Group, LLC
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state, which exhibits significant anisotropy in the lab frame. It is the manipulation of the rotational wavefunction by the helium interaction that governs collision-induced Zeeman relaxation of molecules. The analogy between L in atoms and R in molecules can be understood by thinking of an atom as an ionically bonded molecule where the valence electron is one “ion” and the rest of the atom is the other. The angular wavefunction of the electron |L, ML , then, is nothing more than the rotational wavefunction |R, MR of this pseudomolecule, and indeed they are described by the same spherical harmonics. The conclusions that can be drawn from this analysis are very similar to those that were drawn for atoms. For buffer-gas loading, it is desirable to reduce the anisotropy between the molecule and the helium atom. The most “spherical” molecules are those with pure |R = 0 rotational ground states. Rotationally excited states and molecules with significant admixtures of R = 0 should be more likely to spin-relax. This immediately eliminates molecules with Λ = 0, because the rotational ground states for such molecules contain significant R = 0 contributions, as shown in Figure 13.10c. Furthermore, the presence of the nearby Λ- or Ω-doublet state of opposite parity is likely to be close enough in energy to be easily mixed in by the Stark effect caused by the helium atom, further increasing the anisotropy. For these reasons it is expected that Σ states will be far more robust against helium induced Zeeman relaxation than molecules in states with Λ = 0. It is still true that choosing molecules with reduced anisotropy in the moleculefixed frame is beneficial. After all, if there is no anisotropy in the molecule-fixed frame, the helium interaction in the lab frame is spherical for all rotational states. It is therefore desirable to seek molecules with short bond lengths, which should reduce the anisotropy of the electron charge cloud in the molecule frame for a fixed helium approach distance. There is potential for the Zeeman relaxation of superficially anisotropic molecules to be suppressed by spatially large, roughly spherical electron wavefunctions, much like was found with the submerged-shell rare earth elements. With this understanding in place, is is necessary to examine the properties of magnetic Σ state molecules that lead to Zeeman relaxation in order to assess the feasibility of buffer-gas loading of molecules. 13.3.2.1
Inelastic Collisions of 2 Σ Molecules with He
The simplest paramagnetic molecular state with Λ = 0 is 2 Σ. The rotational quantum number N is equal to R from the above discussion, and from here forward we will use N for the discussion of nuclear rotation, in accordance with standard notation for Σ states. The lone valence electron contributes 1 μB of magnetic moment, making 2 Σ molecules suitable for trapping. The rotational ground state of 2 Σ is a pure |N = 0 state, which eliminates helium-induced Zeeman relaxation from first-order effects, in accordance with Equation 13.6. In order to change the spin projection MS , there must be a mechanism by which the spin can be coupled to the rotational wavefunction, which can be altered by the electrostatic interaction. As developed formally by Krems and Dalgarno [31], the spin-rotation interaction γSR N · S is just such a mechanism. It can couple states with the same value of the sum MS + MN , which are the projections of the molecular spin and rotation on © 2009 by Taylor and Francis Group, LLC
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the lab-fixed Z-axis. This can be seen immediately with the use of the following identity: 1 HSR = γSR N · S = γSR NZ SZ + (N+ S− + N− S+ ) . (13.8) 2 The inner product N · S depends on the relative projection of N and S, so all states with the same value of MJ ≡ MS + MN also have the same energy under the γSR N · S interaction and can be coupled by it. It can be seen that for the rotational ground state, N = 0, MN = 0 and thus the spin–rotation interaction cannot change MS without changing MJ . Because MJ is a good quantum number, collisions between helium atoms and 2 Σ molecules in their rotational ground state cannot directly cause spin-depolarization (first-order perturbation theory predicts a Zeeman transition probability of zero). Despite these considerations, Krems and Dalgarno have pointed out [31] that there is a three-step process by which the electrostatic interaction of the colliding helium atom and the spin–rotation interaction in the molecule can cause spin depolarization. Even though the electrostatic interaction cannot couple to the electron spin, electric fields can mix rotational states. This means that the helium atom can perturb the rotational state distribution from pure N = 0 into a mixture of N states during the collision. In the second step, the spin–rotation interaction can mix in different MS states from the N > 0 portion of the rotational eigenstate that is perturbed by the proximity of the helium atom. Finally, the electrostatic interaction between the helium atom and these mixed states has off-diagonal elements in MJ , which lead to spindepolarization. The first cold molecule trapping was performed with a 2 Σ molecule, CaH [4]. About 108 molecules were loaded into a magnetic trap using buffer-gas loading. The ratio γ of the He–CaH elastic collision cross-section to the Zeeman relaxation crosssection was measured to be γ ≥ 107 [4,18,32,78]. Thermal isolation of the trapped species by cooling the cell walls to cryopump the buffer gas was unsuccessful due to the long pump-out time required (∼10 sec) compared to the evaporation limited lifetime, which illustrates the importance of a fast traversal of the “valley of death” in Figure 13.5. Species with a magnetic moment much higher than one Bohr magneton (the case of CaH), such as Cr and Eu, were thermally isolated by cooling the trap walls to lower the vapor pressure of helium. This worked because they experienced a deeper trap during the pump-out and therefore suffered less loss due to helium induced evaporation while traversing the “valley of death.” Another 2 Σ molecule, CaF, was also buffer-gas cooled and we measured γ = (1.3 + 1.3/–0.5) × 104 [33], which was likely dominated by the nonzero thermal population of N = 1 rotationally excited molecules. 13.3.2.2
Inelastic Collisions of 3 Σ Molecules with He
The low trap depth that 2 Σ molecules experience leads to difficulty in the attempt to achieve thermal isolation due to the required speed for buffer-gas removal. Because technological constraints currently limit the maximum magnetic trap depths to around 4 T, it makes sense to pursue molecules with larger magnetic moments to utilize the © 2009 by Taylor and Francis Group, LLC
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effectively deeper traps. For this and other reasons, extension of this type of study to 3 Σ molecules is a natural next step. The mechanism described above developed by Krems and Dalgarno [31] will still be present for 3 Σ molecules. As they have pointed out, however, the addition of the spin–spin interaction in 3 Σ molecules leads to another relaxation channel that tends to dominate the spin depolarization. The rotational ground state of a 3 Σ molecule, unlike a 2 Σ, is not a pure |N = 0 state, even in zero field. This is due to the spin–spin interaction, which can be written as √
2 q 4π HSS = − λSS 6 (13.9) Y2,−q (θ, φ)Tq(2) [S, S], (−1) 3 5 q (2)
where λSS is the spin–spin coefficient and Tq [S, S] is the spherical tensor product of S with itself [34]. The = 2 spherical harmonics in Equation 13.9 connect states with different spin projections and rotational quantum numbers that differ by ΔN = 2. The spin–spin interaction therefore mixes in some N = 2 character into the rotational ground state: |ψ, MJ ∝ |N = 0, MS +
λSS c N = 2, MN , MS , 6Be
(13.10)
where Be is the rotational constant. Here it can be seen mathematically that the spatial orientation of the internuclear axis in the lab frame is coupled to the projection of the electron spin through the spin–spin interaction, even before the helium interacts with the molecule. The amount of N = 2 present depends on the ratio λSS /Be . The helium collision induced Zeeman relaxation of 3 Σ molecules in the zero-field limit is therefore expected to scale as the square of this ratio [31]: 7 2 λ2 σZR ∝ ψ, MJ |VHe | ψ, MJ ∝ SS . Be2
(13.11)
In light of this scaling law, it is desirable for buffer-gas loading that a 3 Σ molecule have a large rotational constant and small spin–spin coefficient to minimize the helium interaction anisotropy and therefore the helium-induced Zeeman relaxation. This realization was one of the factors that led our group to the imidogen (NH) radical. This spin–spin driven helium induced Zeeman relaxation is likely to be the dominant relaxation mechanism for molecules for which the spin–spin coefficient (λSS ) is larger than the spin–rotation coefficient (γSR ). It is not clear from this qualitative model whether the additional 1 μB of magnetic moment gained in moving from 2 Σ molecules to 3 Σ states is worth the trouble. If the spin–spin driven Zeeman relaxation of 3 Σ molecules is too strong, the trap lifetime will be substantially limited by inelastic collisions and not the trap depth. In 2003, a quantitative calculation was performed by Krems and colleagues that predicted a favorable Zeeman relaxation rate coefficient for imidogen (NH) with helium [35,36]. Furthermore, experiments © 2009 by Taylor and Francis Group, LLC
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performed in our group indicate that thermal isolation of 2 μB species is dramatically easier than 1 μB [19]. Experimentally, our group demonstrated the first trapping of 3 Σ polar molecules by buffer-gas loading more than 108 imidogen radicals into a magnetic trap directly from a molecular beam [6]. We found that imidogen does have favorable collisional properties for buffer-gas loading. By measuring the trap lifetime for a range of 3 He buffer-gas densities, we found a ratio of γ = 7 × 104 , which is sufficiently large to observe trap lifetimes approaching 1 sec. Nonetheless, there is reason to believe that this ratio is lowered by the presence of a shape resonance. Krems and coworkers [35,36] have predicted an l = 3 shape resonance between imidogen and 3 He in the collision energy range around 0.5 K. Furthermore, our single-point measurement only partially supports the spin–spin interaction driven relaxation mechanism described below. To investigate these issues further, we first changed the buffer-gas isotope to 4 He, for which the l = 3 level is a true bound state and therefore no longer corresponds to a shape resonance. We found that the imidogen–4 He system has a collision induced Zeeman relaxation crosssection about four times smaller than the more standard 3 He, supporting the shape resonance prediction. Furthermore, by changing isotopomers of NH (e.g., switching from NH to ND) we were able to verify the 1/B2 dependence of the Zeeman relaxation cross-section for the imidogen–4 He system [30].
13.4
BUFFER-GAS BEAM PRODUCTION
Buffer-gas cooling has been used to create cold molecular and atomic beams and demonstrated to produce higher fluxes of cold molecules than any other source. The general approach is to buffer-gas cool the target species in a centimeter-sized cryogenic cell that has an exit hole with a diameter on the order of millimeters to centimeters. Some of the cold target species will exit the hole, either effusively or “boosted,” depending on the experiment geometry and density of helium in the cell. Each regime has its peculiar features, as we discuss below. A simple outline of the operation of a buffer-gas beam source is as follows, and is depicted in Figure 13.11. Cold helium gas at temperature T is introduced into a buffer-gas-filled cryogenic cell that is also at temperature T . The cell is a tube with (a)
L
(b)
d
D l
FIGURE 13.11 Schematic of a simple buffer gas beam source being operated under (a) diffusive and (b) hydrodynamic cell conditions.
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diameter D and length L ≥ D. At one end (typically opposite the buffer gas entry location) there is an exit hole of diameter d. The helium density nHe is held constant by continuously flowing cold helium gas to replenish that lost through the exit hole. Hot particles (atoms or molecules) of the target species A are also introduced into the cell at a location we designate as the target species “entrance point.” Generally, the entrance point is at the end of the cell opposite the exit hole (i.e., closer to where the buffer gas enters), as shown in Figure 13.11.
13.4.1 THERMALIZATION AND EXTRACTION CONDITIONS The hot particles of A can come arbitrarily close to equilibrium with the buffer gas after a characteristic number of collisions N . Typically, N ≈ 100 collisions are sufficient for thermalization to√4 K, as discussed in Section 13.2. The particle’s mean free path is given by λt = 1/ 2nHe σt where σt is the thermal average of the diffusion crosssection. The thermalization time is given by τtherm = N λt /¯vA,cooling where v¯ A,cooling is the microscopic velocity of particles of A averaged over the temperature range they experience during the thermalization process. The particle’s thermalization length √ can be written as Rtherm = αλt where, according to simulations, α varies between N and N , depending upon the initial conditions of the particles of A. For full thermalization, the cell must be large enough to satisfy D ≥ Rtherm . For the purposes of this discussion we will assume that this thermalization condition is always fulfilled. Fully thermalized particles of A can leave the cell volume by two processes: by diffusing to the boundaries of the cell or by being entrained in the helium flow and hydrodynamically pulled out of the exit hole. In order to determine which regime is applicable, the timescales for each process need to be compared. After thermalization, the particle’s macroscopic longitudinal velocity coincides with the flow velocity of helium through the cell, given by Vflow ≈ (d/D)2 vHe,thermal . The time it takes for particles of A to traverse the cell at this macroscopic helium flow velocity is given by τpumpout = L/Vflow . The radial motion of fully thermalized A particles, on the other hand, is diffusive. The diffusion lifetime is given by τdiffusion =
D2 , 4λc vA,thermal
(13.12)
where D is the cell diameter √ (not the diffusion constant), λc is the cold mean free path, given by λc = 1/nHe σc 1 + mA /mHe and vA,thermal is the cold thermal velocity of the particles of A. The dimensionless parameter ξ≡
τdiffusion τpumpout
(13.13)
provides a quantitative characterization of which limit the system is in, with ξ 1 indicating diffusive and ξ 1 hydrodynamic cell conditions (see Figure 13.11). We note that the entrainment being discussed here is not the same as that seen in typical © 2009 by Taylor and Francis Group, LLC
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supersonic nozzle expansions where it is collisions within and just outside the nozzle that play the key role, and flow velocities are comparable to the sonic velocity. The mean flow velocity of helium within the buffer-gas cell is substantially less than the sonic velocity because, in general, d < D. For full thermalization of hot particles of A to take place, it must be that τtherm < τdiffusion for diffusive cell conditions and τtherm < τpumpout for the hydrodynamic regime. If ξ 1, then hydrodynamic effects can be ignored. In this diffusive limit, most of the A particles stick to the cell walls. In this case, the fraction of thermalized particles that exit the hole is set by the solid angle coverage fraction of the exit aperture and is generally much less than 1. If ξ 1 then a large fraction of A particles can be entrained inside the cell and be forced by the helium wind to move toward the exit hole and into the beam. Exact analysis depends on the geometrical details, but it can be shown that for D ≈ L ξ=
κd 2 κd 2 nHe σc 1 + mA /mHe = 4λD 4D
(13.14)
where κ is a dimensionless constant on the order of unity.
13.4.2
BOOSTING CONDITION AND SLOW BEAM CONSTRAINTS
The forward velocity vF of the beam of A particles depends on nHe and d, and can be between vA,thermal and vsupersonic . Specifically, vF is determined by the Reynolds number (or Knudsen number, Re = 1/Kn) for the orifice, Re = d/λ. In the effusive limit, λ d and vF ≈ vA,thermal , as shown in Figure 13.12. The A particles’ trajectories outside the exit hole experience no further collisions. By contrast, when λ d, the A particles are “boosted” because the outward flow of He outside the exit hole causes collisions with A particles, boosting them to higher forward velocities. This causes vF to increase from vA,thermal to vboosted as Re is increased from 1 to >100. Note that the regime of 1 < Re < 100 is often vA,thermal
vsupersonic
Boosted
Normalized number
Effusive
vHe,thermal
Fully boosted Supersonic
0 Molecule velocity in beam
FIGURE 13.12 Velocity distributions of a buffer-gas beam operated in four different forward velocity regimes. All four have roughly the same velocity spread, corresponding to vA,thermal .
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called the intermediate regime in room-temperature supersonic nozzle work, whereas Re > 1000 is called fully supersonic. A fully supersonic expansion is not necessary to produce high boosts. It turns out that the conditions are such that a buffer-gas beam extracted in the hydrodynamic regime will be boosted, meaning that the forward velocity of the molecules will be higher than their mean thermal velocity. This can be seen by noting that for ξ > 1, Equation 13.14 can be expressed as Re > D/d,
(13.15)
where we note that D/d is always greater than 1. Nonetheless, the huge flux enhancement attained by running hydrodynamically (as opposed to diffusively) makes this alternative attractive for certain experiments. The choice between running a cold beam source with diffusive vs. hydrodynamic cell conditions depends on the specific experimental goals. A hydrodynamically extracted beam yields about a factor of 1000 higher flux than a beam made from a diffusive-mode cell, but the forward velocity is higher, around 100 m/sec for a 4 K cell. On the other hand, an effusive cold beam (which must be extracted under diffusive cell conditions) can have forward velocities as low as around 10 m/sec, corresponding to forward kinetic energies 100 times lower than a hydrodynamically extracted beam.
13.4.3
STUDIES WITH DIFFUSIVELY EXTRACTED BEAMS
The diffusive regime was studied in detail using PbO molecules and Na atoms. A full description is given in Ref. [37]. The basic layout of the beam apparatus is shown in Figure 13.11a. The buffer-gas cell is a brass box of ∼10 cm on edge. The exit hole has d = 3 mm and is centered on one side face. Several ablation targets are mounted on the inside top face of the cell about 6 cm from the exit aperture. Na atoms are ablated from Na metal or NaCl targets, and PbO molecules are ablated from a vacuum hotpressed PbO target. Buffer gas continuously flows into the cell. A good vacuum is maintained in the beam region by means of a charcoal sorption pump with a pumping speed for helium of ∼1000 l/sec. For this particular cell, using an assumed cold diffusion cross-section σc ≈ 3 × 1015 cm−3 , the crossover between effusive and boosted flow (the condition d = λ) occurs for nHe ≈ 1015 cm−3 . We characterize the beam source for both species within a range of densities around the anticipated optimal condition for Na, namely nHe ≈ 0.2 − 5 × 1015 cm−3 . For Na, in-cell laser absorption spectroscopy is used to determine the number of thermalized Na atoms, NNa , and the cold collision diffusion cross-section, σc,Na . We find NNa ≈ 1014 /pulse for both the metallic Na and NaCl ablation targets. We measure diffusion lifetimes of τ (msec) ≈ 4 × 10−15 × nHe (cm−3 ). From this we infer σc,Na ≈ 3 × 10−15 cm2 . Previous work with PbO has measured an ablation yield of ≈ 1012 /pulse [38], and our in-cell LIF measurements indicate a comparable yield. In Figure 13.13 we plot the number NA,beam of thermalized particles of species A exiting the hole as a function of nHe . The ablation plume is not directed at the aperture in order to ensure that particles only exit the cell and form a beam by first © 2009 by Taylor and Francis Group, LLC
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Number in beam per pulse, NA,beam
1011 Na PbO Na simulation
Plateau
1010 3 nHe
109
108
107
0
1
2 3 Buffer-gas density, nHe (1015 cm–3)
4
5
FIGURE 13.13 The number of cold A particles, NA,beam , that emerge in the beam as a function of the buffer-gas density (nHe ) in the diffusive cell regime. Curves with specific functional forms have been inserted to show the different scaling regimes. (From Maxwell, S.E. et al., Phys. Rev. Lett., 95, 173201, 2005. Copyright 2005 by the American Physical Society. With permission.)
colliding with the cold buffer gas. We can therefore monitor the thermalization of the ablated particles by looking for their appearance in the beam. In the Na data 3 ) up and simulations, we find that NA,beam increases rapidly (approximately ∝ nHe to a critical value of nHe , above which NA,beam is roughly constant at its maximum 3 scaling is consistent with a simple picture in which A value. The low-density nHe 3 ≈ (N λ)3 after a near particles are uniformly distributed over a volume of Rtherm thermalization to T . The intersection of the solid curves corresponds to the condition where the thermalization length matches the distance between the ablation target and the aperture. This saturation corresponds to a maximum fraction in the beam ( fmax ) given by the fractional solid angle of the aperture compared to the rest of the cell, indicating that the motion of particles of A in the cell is fully diffusive and randomized. For this cell geometry we have fmax ≈ 3 × 10−4 . The condition for full thermalization (i.e., all of the hot Na produced is thermalized to a temperature close to T ) is apparent in both the experimental and simulated data for Na. We find that the highest effusive-regime helium density is almost exactly equal to the density necessary for thermalization of an ablated sample of Na. Thus it should be anticipated that this cell is near the optimal geometry for producing a maximal flux of slow (effusive), cold Na. By contrast, the different initial conditions produced by the ablation of PbO makes the helium density necessary for thermalization larger than for Na, implying that our cell geometry is not optimal for PbO. Nonetheless, we find the translational and rotational temperature of the PbO in the beam is close to T , because beam particles are produced only through collisions with buffer-gas atoms. © 2009 by Taylor and Francis Group, LLC
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Forward velocity, vF (m/sec)
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Mean velocity of an effusive He beam
120 100 80 Mean velocity of an effusive Na beam
60 40
Mean velocity of an effusive PbO beam
20 0
0
1
2
3
4
5
Buffer-gas density, nHe (1015 cm–3)
FIGURE 13.14 PbO and Na beam mean forward velocities vF as a function of buffer-gas density nHe . Extrapolation of the data to zero buffer-gas density is illustrated by best-fit lines (dashed). (From Maxwell, S.E. et al., Phys. Rev. Lett., 95, 173201, 2005. Copyright 2005 by the American Physical Society. With permission.)
In order to verify that the cell is being operated in the diffusive regime, we can calculate ξ. The emptying time of the cell can be calculated using the known cell and exit hole sizes. For this cell, this results in a pump-out time of τpumpout = 100 msec, which is longer than the 1 to 10 msec diffusion times for the buffer-gas densities accessed in these experiments. This verifies that this work is strongly in the diffusive cell regime, that is, ξ 1. Figure 13.14 shows the average forward velocity vF of the beams of A particles as the buffer-gas density nHe is varied. For both Na and PbO, the data show a linear increase of vF with nHe . Here we can see that for this diffusive-regime source, the output beam can be tuned all the way from effusive to strongly boosted (see Figure 13.12). The linear increase of vF with buffer-gas density is consistent with the following simple picture: a slowly moving particle of A takes a time te to exit the hole, where te ∼ d/vA ; during this time, it undergoes Ne collisions with fast, primarily forward-moving He atoms, where Ne ∼ nHe σc vHe te , resulting in a net velocity boost given by ΔvA ∼ vA d/λ ∝ nHe . This picture should be roughly valid for densities below the fully supersonic regime. The velocities we measure for Na are approximately reproduced by modeling the beam formation process with our measured value of σc .
13.4.4
STUDIES WITH HYDRODYNAMICALLY EXTRACTED BEAMS
The cell used for our studies of hydrodynamically extracted beams is smaller than that used in the diffusive regime beam studies, making it possible to achieve ξ = 1 © 2009 by Taylor and Francis Group, LLC
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while still operating in the convenient nHe ≈ 1015 cm−3 region. The heart of the experimental apparatus is a ≈2.5 cm size cell anchored to the cold plate of a cryostat. This cell is a copper box with two fill lines (one for helium and one for molecular oxygen) on one side, the exit aperture on the opposite side, a Yb ablation target inside, and windows for laser access. With this cell, we make beams of either atomic Yb or O2 . In order to produce beams of O2 , we flow He and O2 continuously into the cell where they mix and thermalize to the temperature of the cell. To produce beams of Yb, we flow He continuously into the cell and ablate the Yb target with a pulsed YAG laser. Helium typically flows into the cell at 1 × 1017 to 8 × 1018 atom/sec. Despite the large He gas flow, the vacuum in the beam region is maintained at a low 3 × 10−8 torr by two stages of differential pumping with high speed cryopumps made of activated charcoal. We run two different types of experiments with this cell, either magnetically guided (with O2 ) or unguided (with Yb). In the guided work, we couple O2 into a magnetic guide and measure the flux of O2 exiting the guide with a residual gas analyzer (RGA). This work is described in detail in Ref. [39] and we direct the reader there. In the unguided work, we use Yb to characterize the beam source. The flux and velocity profile of the Yb beam and the density and temperature of the Yb gas in the cell are all measured using laser absorption spectroscopy. Two exit hole configurations were demonstrated to produceYb beams. Figure 13.15 shows the output efficiency of a simple slit aperture (1 × 4 mm), where this efficiency is defined as the ratio of the number of Yb in the beam to the number of cold Yb produced by ablation in the cell. At high buffer-gas flows, ξ > 1 (the hydrodynamic cell regime) and up to 40% of the cold Yb atoms in the cell are detected in the resultant beam. The divergence of the beam is measured by comparing the Doppler shifts in the absorption spectra parallel and transverse to the atomic beam. The beam is more collimated than a pure effusive source, with a divergence of ≈0.1 steradian. This observed effect of angular peaking of the beam has been seen in room-temperature beam experiments using two species of different mass and is often called “Machnumber focusing” [40,41]. The higher mass species has its angular beam distribution narrowed due to the same boosting effect described earlier—the light species knocks the heavy species from behind, randomizing its transverse motion while always kicking it forward. This effect peaks in the middle of the intermediate regime, Re ≈ 30, becoming much less pronounced as one approaches either the effusive or fully supersonic regime. A peak on-axis flux per unit solid angle of 6 × 1015 atom/sec/sr (5 × 1012 atoms/pulse) has been measured. For trapping, where total kinetic energy of the trappable species must be less than the trap depth (typically a few Kelvin) the high forward velocity of a boosted beam (vF = 130 m/sec > 100 K effective temperature forYb) makes it unsuitable. Trapping work would ideally use a beam with both high flux (available for ξ > 1, implying Re > 1) and vF as close as possible to vA,thermal . In an attempt to achieve the best of the two worlds, we developed a two-stage aperture called the suppressor nozzle (by analogy to a suppressor or silencer for a gun, which works much the same way) [39]. With this two-stage aperture, the He-Yb mixture still passes through the 1 × 4 mm slit out of the buffer-gas cell, resulting in a large hydrodynamic flux enhancement but correspondingly boosted forward velocity (vF > vA,thermal ). To suppress this boost © 2009 by Taylor and Francis Group, LLC
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Fraction into beam
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
x = tdiffusion /tpumpout
FIGURE 13.15 The ratio of the number of ablated and cooled ytterbium atoms to the number emitted into the beam for a simple slit aperture (no suppressor nozzle). At high buffer-gas densities, which correspond to long diffusion times and therefore high ξ (see main text), up to 40% of the cold ablated atoms are detected in the peak instantaneous flux represented here of 5 × 1015 atom/sec sr−1 . (From Patterson, D., and Doyle, J.M., J. Chem. Phys., 126, 154307, 2007. With permission.)
of the forward beam velocity while maintaining the high brightness provided by the hydrodynamic regime, the cell is fitted with the suppressor nozzle just outside the slit aperture. The suppressor nozzle consists of a small chamber with a large exit hole opposite the cell slit. This second, larger exit hole is covered by a stainless steel mesh with pore size 140 μm (28% transparency) and creates a near-effusive beam (actually a large number of near-effusive beams in parallel) in the vacuum region, where there are no more collisions. The He density in the suppressor nozzle volume would be far too low to thermalize hot atoms entering the cell. However, this density is high enough to collisionally slow the internally cold (but forward boosted) Yb that comes through the first orifice and low enough so that ξ 1; that is, diffusion dominates the dynamics. In this sense, the suppressor nozzle itself is a small buffer-gas cooling cell operated in the diffusive cell regime. The Yb beam from the suppressor nozzle has a mean velocity of 35 m/sec, with a spread of 20 m/sec. The measured peak on-axis beam flux of 5 × 1012 atom/sec/sr represents about 1% of the cold atomic Yb produced in the ablation, or about 3% of the output of the one-stage aperture (compare to 0.03% for the diffusive cell regime). One of the future directions for producing cold beams is to inject hot molecules along the same direction and into the stream of cold helium gas flowing inside a long tube L > D. The tube length would be set so that particles of A would diffuse a distance D/2 in a time L/Vflow for a helium density set so that nHe = ΥnA , where © 2009 by Taylor and Francis Group, LLC
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nA is the density of particles of A and Υ is defined in Section 13.2. The cooling of particles of A in such a system could be enhanced with suppressed loss of A particles in a case where σHe−He < σHe−A .
13.5
SUMMARY
Buffer-gas cooling of molecules promises to continue to provide researchers with a general method for obtaining gas-phase samples of cold molecules. The applicability of the technique to a wide variety of atoms and molecules has already led to extensions of the fields of magnetic trapping, quantitative spectroscopy, and slow beam creation. Furthermore, due to the fact that buffer-gas cooling accepts the full Boltzmann distribution of the initial molecular sample, the high numbers of molecules that can be cooled with this method (often also translating to high densities or high beam fluxes) exceed other methods, often by orders of magnitude. Because diatomic molecules have rotational splittings that are often of the order of 1 K, the use of cryogenic helium buffer gas for collisional cooling is a conceptually straightforward process that seems to fit naturally into the world of diatomic molecules. For many types of experiments, the rotational state population enhancement alone is sufficient to make buffer-gas cooling an attractive option. Buffer-gas cooling is also useful for experiments requiring certain degrees of freedom to remain out of equilibrium with the buffer-gas temperature, such as magnetostatic trapping and metastable lifetime measurements. It is important to continue to study the collisional thermalization processes in detail to gain a thorough understanding of the relevant mechanisms. We expect the knowledge base that will be developed for cold collisional processes can be applied in the future to sympathetically or evaporatively cool trapped molecules down to the ultracold regime.
ACKNOWLEDGMENTS We would like to acknowledge the many members of the experimental and theoretical groups at Harvard who contributed to the development of buffer-gas cooling. Others who are deserving of acknowledgment for advancing the field include F. DeLucia for his pioneering work on collisional cooling, D. DeMille, G. Meijer, A. Peters, M. Stoll, D. Kleppner, T. Greytak, and W. Ketterle. This work was supported by the U.S. NSF, DOE, and ARO.
REFERENCES 1. Migdall, A.L., Prodan, J.V., Phillips, W.D., Bergeman, T.H., and Metcalf, H.J., First observation of magnetically trapped neutral atoms, Phys. Rev. Lett., 54, 2596–2599, 1985. 2. Doyle, J.M., Friedrich, B., Kim, J., and Patterson, D., Buffer-gas loading of atoms and molecules into a magnetic trap, Phys. Rev. A, 52(4), R2515–R2518, 1995. 3. Kim, J., Friedrich, B., Katz, D.P., Patterson, D., Weinstein, J.D., deCarvalho, R., and Doyle, J.M., Buffer-gas loading and magnetic trapping of atomic europium, Phys. Rev. Lett., 78, 3665, 1997.
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4. Weinstein, J.D., deCarvalho, R., Guillet, T., Friedrich, B., and Doyle, J.M., Magnetic trapping of calcium monohydride molecules at milikelvin temperatures, Nature, 395, 148, 1998. 5. Weinstein, J.D., deCarvalho, R., Hancox, C.I., and Doyle, J.M., Evaporative cooling of atomic chromium, Phys. Rev. A, 65, 021604(R), 2002. 6. Campbell, W.C., Tsikata, E., Lu, H.-I., van Buuren, L.D., and Doyle, J.M., Magnetic trapping and Zeeman relaxation of NH (X 3 Σ− ), Phys. Rev. Lett., 98, 213001, 2007. 7. Huang, Y.H. and Chen, G.B., A practical vapor pressure equation for helium-3 from 0.01 K to the critical point, Cryogenics, 46(12), 833–839, 2006. 8. Pobell, F., Matter and Methods at Low Temperatures, 2nd ed., Springer, 1996. 9. Hatakeyama, A., Enomoto, K., Sugimoto, N., andYabuzaki, T., Atomic alkali–metal gas cells at liquid–helium temperatures: Loading by light-induced atom desorption, Phys. Rev. A, 65, 022904, 2002. 10. Messer, J.K. and De Lucia, F.C., Measurement of pressure-broadening parameters for the CO–He system at 4 K, Phys. Rev. Lett., 53(27), 2555–2558, 1984. 11. Doret, S., Connolly, C., and Doyle, J.M., Ultracold metastable helium, 2008. Unpublished. 12. Forrey, R.C., Kharchenko, V., Balakrishnan, N., and Dalgarno,A., Vibrational relaxation of trapped molecules, Phys. Rev. A, 59(3), 2146–2152, 1999. 13. Ball, C.D. and De Lucia, F.C., Direct measurement of rotationally inelastic cross sections at astrophysical and quantum collisional temperatures, Phys. Rev. Lett., 81(2), 305–308, 1998. 14. Ball, C.D. and De Lucia, F.C., Direct observation of Λ-doublet and hyperfine branching ratios for rotationally inelastic collisions of NO–He at 4.2 K, Chem. Phys. Lett., 300, 227, 1999. 15. Mengel, M. and De Lucia, F.C., Helium and hydrogen induced rotational relaxation of H2 CO observed at temperatures of the interstellar medium, Astrophys. J., 543, 271–274, 2000. 16. Campbell, W.C., Groenenboom, G.C., Lu, H.-I., Tsikata, E., and Doyle, J.M., Timedomain measurement of spontaneous vibrational decay of magnetically trapped NH, Phys. Rev. Lett., 100, 083003, 2008. 17. Hasted, J.B., Physics of Atomic Collisions, 2nd ed., chap. 1.6, American Elsevier Publishing Company, 1972. 18. Weinstein, J.D., Magnetic trapping of atomic chromium and molecular calcium monohydride, Ph.D. thesis, Harvard University, 2001. 19. Harris, J.G.E., Michniak, R.A., Nguyen, S.V., Brahms, N., Ketterle, W., and Doyle, J.M., Buffer gas cooling and trapping of atoms with small effective magnetic moments, Europhys. Lett., 67(2), 198–204, 2004. 20. Michniak, R., Enhanced buffer gas loading: Cooling and trapping of atoms with low effective magnetic moments, Ph.D. thesis, Harvard University, 2004. 21. Walker, T.G. and Happer, W., Spin-exchange optical pumping of noble-gas nuclei, Rev. Mod. Phys., 69(2), 629–641, 1997. 22. Johnson, C., Brahms, N., Newman, B., Doyle, J., Kleppner, D., and Greytak, T., Zeeman relaxation of cold atomic Fe and Ni in collisions with 3 He, in preparation, 2008. 23. Krems, R.V. and Dalgarno, A., Disalignment transitions in cold collisions of 3 P atoms with structureless targets in a magnetic field, Phys. Rev. A, 68, 013406, 2003. 24. Kokoouline, V., Santra, R., and Greene, C.H., Multichannel cold collisions between metastable Sr atoms, Phys. Rev. Lett., 90(25), 253201, 2003. 25. Santra, R. and Greene, C.H., Tensorial analysis of the long-range interaction between metastable alkaline–earth–metal atoms, Phys. Rev. A, 67, 062713, 2003.
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26. Maxwell, S.E., Hummon, M.T., Wang,Y., Buchachenko, A.A., Krems, R.V., and Doyle, J.M., Spin-orbit interaction and large inelastic rates in bismuth-helium collisions. Phys. Rev. A, 78, 042706, 2008. 27. Mayer, M.G., Rare-earth and transuarnic elements, Phys. Rev., 60, 184–187, 1941. 28. Hancox, C.I., Doret, S.C., Hummon, M.T., Luo, L., and Doyle, J.M., Magnetic trapping of rare-earth atoms at millikelvin temperatures, Nature, 431, 281–284, 2004. 29. Hancox, C.I., Doret, S.C., Hummon, M.T., Krems, R.V., and Doyle, J.M., Suppression of angular momentum transfer in cold collisions of transition metal atoms in ground states with nonzero orbital angular momentum, Phys. Rev. Lett., 94, 013201, 2005. 30. Campbell, W.C., Tscherbul, T.V., Lu, H.-I., Tsikata, E., Krems, R.V., and Doyle, J.M., Mechanism of collisional spin relaxation in 3 Σ molecules. Phys. Rev. Lett., 102, 013003, 2009. 31. Krems, R.V. and Dalgarno, A., Quantum-mechanical theory of atom–molecule and molecular collisions in a magnetic field: Spin depolarization. J. Chem. Phys., 120(5), 2296–2307, 2004. 32. There is a typo in Ref. [4]; the quoted elastic-to-inelastic collision ratio should read σe /σs > 104 . 33. Maussang, K., Egorov, D., Helton, J.S., Nguyen, S.V., and Doyle, J.M., Zeeman relaxation of CaF in low-temperature collisions with helium, Phys. Rev. Lett., 94, 123002, 2005. 34. Mizushima, M., The Theory of Rotating Diatomic Molecules, Wiley, New York, 1975. 35. Cybulski, H., Krems, R.V., Sadeghpour, H.R., Dalgarno, A., Kłos, J., Groenenboom, G.C., van der Avoird, A., Zgid, D., and Chałasi´nski, G., Interaction of NH(X 3 Σ− ) with He: Potential energy surface, bound states, and collisional relaxation, J. Chem. Phys., 122, 094307, 2005. 36. Krems, R.V., Sadeghpour, H.R., Dalgarno, A., Zgid, D., Kłos, J., and Chałasi´nski, G., Low-temperature collisions of NH(X 3 Σ− ) molecules with He atoms in a magnetic field: An ab initio study, Phys. Rev. A, 68, 051401(R), 2003. 37. Maxwell, S.E., Brahms, N., deCarvalho, R., Glenn, D.R., Helton, J.S., Nguyen, S.V., Patterson, D., Petricka, J., DeMille, D., and Doyle, J.M., High-flux beam source for cold, slow atoms or molecules, Phys. Rev. Lett., 95(17), 173201, 2005. 38. Egorov, D., Weinstein, J.D., Patterson, D., Friedrich, B., and Doyle, J.M., Spectroscopy of laser-ablated buffer-gas-cooled PbO at 4 K and the prospects for measuring the electric dipole moment of the electron, Phys. Rev. A, 63, 030501(R), 2001. 39. Patterson, D. and Doyle, J.M., Bright, guided molecular beam with hydrodynamic enhancement, J. Chem. Phys., 126, 154307, 2007. 40. Anderson, J.B., Separation of gas mixtures in free jets, AIChE Journal, 13(6), 1188–1192, 1967. 41. Waterman, P.C. and Stern, S.A., Separation of gas mixtures in a supersonic jet, J. Chem. Phys., 31(2), 405–419, 1959. 42. Egorov, D.M., Buffer-gas cooling of diatomic molecules, Ph.D. thesis, Harvard University, 2004. 43. Crownover, R.L., Willey, D.R., Bittner, D.N., and De Lucia, F.C., Very low temperature spectroscopy: The pressure broadening coefficients for CH3 F between 4.2 and 1.9 K. J. Chem. Phys., 89, 6147–6156, 1988. 44. Beaky, M.M., Flatin, D.C., Holton, J.J., Goyette, T.M., and De Lucia, F.C., Hydrogen and helium pressure broadening of CH3 F between 1 K and 600 K, J. Mol. Structure, 352/353, 245, 1995.
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45. Stoll, M., Buffer-gas cooling and magnetic trapping of CrH and MnH molecules, Ph.D. thesis, Humboldt-Universität zu Berlin, 2008. 46. Willey, D.R., Crownover, R.L., Bittner, D.N., and De Lucia, F.C., Very low temperature spectroscopy: The pressure broadening coefficients for CO–He between 4.3 and 1.7 k. J. Chem. Phys., 89, 1923, 1988. 47. Beaky, M.M., Goyette, T.M., and De Lucia, F.C., Pressure broadening and line shift measurement of carbon monoxide in collision with helium from 1 to 600 K, J. Chem. Phys., 105, 3994, 1996. 48. Willey, D.R., Choong, V.E., and De Lucia, F.C., Very low temperature helium pressure broadening of DCl in a collisionally cooled cell, J. Chem. Phys., 96, 898–902, 1992. 49. Willey, D.R., Bittner, D.N., and De Lucia, F.C., Pressure broadening cross-sections for the H2 S–He system in the temperature region between 4.3 and 1.8 K, J. Molec. Spec., 134, 240, 1989. 50. Ronningen, T.J. and De Lucia, F.C., Helium induced pressure broadening and shifting of HCN hyperfine transitions between 1.3 and 20 K, J. Chem. Phys., 122, 184319, 2005. 51. Campbell, W.C., Magnetic trapping of imidogen molecules, Ph.D. thesis, Harvard University, 2008. 52. See, for example, Euro. Phys. J. D, 2004, Special Issue on Cold Molecules. 53. Patterson, D., Rasmussen, J., and Doyle, J.M., Intense atomic and molecular beams via neon buffer gas cooling, New Journal of Physics, 2009 (to be published). 54. van Buuren, L.D., Sommer, C., Motsch, M., Pohle, S., Schenk, M., Bayerl, J., Pinske, P.W.H., and Rempe, G., Electrostatic extraction of cold molecules from a cryogenic reservoir. Phys. Rev. Lett., 102, 033001, 2009. 55. Willey, D.R., Bittner, D.N., and De Lucia, F.C., Collisional cooling of the NO–He system: The pressure broadening cross-sections between 4.3 and 1.8 K. Mol. Phys., 66, 1, 1988. 56. Ball, C.D. and De Lucia, F.C., Direct observation of Λ-doublet and hyperfine branching ratios for rotationally inelastic collisions of NO–He at 4.2 K, Chem. Phys. Lett., 300, 227–235, 1999. 57. DeMille, D., et al. Cold beam of SrO for trapping studies. Unpublished. 58. Vutha, A.C., Baker, O.K., Campbell, W.C., DeMille, D., Doyle, J.M., Gabrielse, G., Gurevich, Y.V., and Jansen, M.A.H.M., Cold beam of ThO for EDM studies. Unpublished. 59. Weinstein, J.D., deCarvalho, R., Amar, K., Boca, A., Odom, B.C., Friedrich, B., and Doyle, J.M., Spectroscopy of buffer-gas cooled vanadium monoxide in a magnetic trapping field, J. Chem. Phys., 109(7), 2656–2661, 1998. 60. Brahms, N., Newman, B., Johnson, C., Kleppner, D., Greytak, T., and Doyle, J.M., Magnetic trapping of silver and copper, and anomolous spin–relaxation in the Ag–He system, submitted to PRL. 61. Newman, B., Brahms, N., Johnson, C., Kleppner, D., Greytak, T., and Doyle, J., Buffergas cooled cerium, 2008. Unpublished. 62. Bakker, J.M., Stoll, M., Weise, D.R., Vogelsang, O., Meijer, G., and Peters, A., Magnetic trapping of buffer-gas-cooled chromium atoms and prospects for the extension to paramagnetic molecules, J. Phys. B, 39, S1111, 2006. 63. Parsons, M., Chakraborty, R., Campbell, W., and Doyle, J.M., Ablation studies, Unpublished, 2008. 64. Hancox, C.I., Doret, S.C., Hummon, M.T., Luo, L., and Doyle, J.M., Buffer-gas cooling of gadolinium. Unpublished.
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65. Nguyen, S., Doret, S.C., Connolly, C., Michniak, R., Ketterle, W., and Doyle, J.M., Evaporation of metastable helium in the multi-partial-wave regime, Phys. Rev. A, 92, 060703(R), 2005. 66. Nguyen, S.V., Doret, S.C., Helton, J., Maussang, K., and Doyle, J.M., Buffer gas cooled hafnium, 2008. Unpublished. 67. deCarvalho, R., Brahms, N., Newman, B., Doyle, J.M., Kleppner, D., and Greytak, T., A new path to ultracold hydrogen, Can. J. Phys., 83, 293–300, 2005. 68. Nguyen, S.V., Helton, J.S., Maussang, K., Ketterle, W., and Doyle, J.M., Magnetic trapping of an atomic 55 Mn–52 Cr mixture, Phys. Rev. A, 71, 025602, 2005. 69. Nguyen, S.V., Harris, J.G.E., Doret, S.C., Helton, J., Michniak, R.A., Ketterle, W., and Doyle, J.M., Spin-exchange and dipolar relaxation of magnetically trapped Mn, Phys. Rev. Lett., 99, 2007. 70. Hancox, C.I., Hummon, M.T., Nguyen, S.V., and Doyle, J.M., Magnetic trapping of atomic molybdenum, Phys. Rev. A, 71, 031402, 2004. 71. Hummon, M.T., Campbell, W.C., Lu, H., Tsikata, E., Wang, Y., and Doyle, J.M., Magnetic trapping of atomic nitrogen (14 N) and cotrapping of NH (X 3 Σ− ), Phys. Rev. A, 78, 050702(R), 2008. 72. Nguyen, S.V., Michniak, R., and Doyle, J., Trapping of Na in the presence of buffer gas, 2004. 73. Hong, T., Gorshkov, A.V., Patterson, D., Zibrov, A.S., Doyle, J.M., Lukin, M.D., and Prentiss, M.G., Realization of coherent optically dense media via buffer-gas cooling. Phys. Rev. A, 79, 013806, 2009. 74. Hancox, C.I., Magnetic trapping of transition-metal and rare-earth atoms using buffer gas loading, Ph.D. thesis, Harvard University, 2005. 75. Patterson, D. and Doyle, J.M., Buffer gas cooled Yb in cell, 2008. Unpublished. 76. mJ = ±J. 77. We will focus here on the case where the magnetic moment comes entirely from electron spin, although it is possible for the magnetic moment to have contributions from electron orbital angular momentum. 78. A more recent analysis of the data gives this value, which is larger than that quoted in Refs. [4], [18], and [32].
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Slowing, Trapping, 14 and Storing of Polar Molecules by Means of Electric Fields Sebastiaan Y.T. van de Meerakker, Hendrick L. Bethlem, and Gerard Meijer CONTENTS 14.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Stark Deceleration of Neutral Polar Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 The Stark Decelerator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Phase Stability in the Stark Decelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.3 Transverse Focusing in a Stark Decelerator . . . . . . . . . . . . . . . . . . . . . . . 14.2.4 Stark Deceleration of OH Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.5 Longitudinal Focusing of a Stark Decelerated Molecular Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.6 Deceleration of Molecules in High-Field Seeking States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 The Zeeman, Rydberg, and Optical Decelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Trapping Neutral Polar Molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 DC Trapping of Molecules in Low-Field Seeking States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 Storage Ring and Molecular Synchrotron . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.3 AC Trapping of Molecules in High-Field Seeking States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.4 Trap Lifetime Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Applications of Decelerated Beams and Trapped Molecules . . . . . . . . . . . . . 14.5.1 High-Resolution Spectroscopy and Metrology . . . . . . . . . . . . . . . . . . . . 14.5.2 Collision Studies at a Tunable Collision Energy . . . . . . . . . . . . . . . . . . 14.5.3 Direct Lifetime Measurements of Metastable States . . . . . . . . . . . . . . 14.6 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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INTRODUCTION
The importance of the role that atomic and molecular beams have played in physics and chemistry cannot be overstated [1]. Nowadays, sophisticated laser-based techniques exist to sensitively and quantum-state-selectively detect atoms and molecules in a beam. In the early days, such detection techniques were lacking and the particles in a beam were detected, for instance, by a “hot wire” (Langmuir–Taylor) detector, by electron-impact ionization, or by deposition and ex situ inspection of the deposit on a substrate placed at the end of a beam-machine [2]. In order to achieve quantum-state selectivity of the detection process, these detection techniques were combined with inhomogeneous magnetic and/or electric fields to characteristically alter the trajectories of the particles on their way to the detector. This was the approach pioneered by Otto Stern and Walther Gerlach in 1922 [3]. The key concept of their experiment, that is, the sorting of quantum states via space quantization, has been extensively used ever since. The experimental geometries were devised to create strong magnetic or electric field gradients on the beam axis to efficiently deflect particles. In 1939, Isidor Rabi introduced the molecular beam magnetic resonance method by using two magnets in succession to produce inhomogeneous magnetic fields of oppositely directed gradients. In Rabi’s setup, the deflection of particles caused by the first magnet was compensated for by the second magnet such that the particles were directed on a sigmoidal path to the detector.A transition to “other states of space quantization,” induced between the two magnetic sections, could be detected via the resulting reduction of the detector signal [4]. Later, both magnetic [5,6] and electric [7] field geometries were designed to focus particles in selected quantum states onto a detector. An electrostatic quadrupole focuser, that is, an arrangement of four cylindrical electrodes alternately energized by positive and negative voltages, was used to couple a beam of ammonia molecules into a microwave cavity. Such an electrostatic quadrupole lens focuses ammonia molecules that are in the upper level of the inversion doublet while simultaneously defocusing those that are in the lower level. The inverted population distribution of the ammonia molecules in the microwave cavity that was thus produced led to the invention of the maser by James Gordon, Herbert Zeiger, and Charles Townes in 1954–1955 [8,9]. Apart from making it possible to observe a spectacular amplification of the microwaves by stimulated emission, these focusing elements made it possible to record, with high resolution and good sensitivity, microwave spectra in a molecular beam. By using several multipole focusers in succession, interlaced with interaction regions with electromagnetic radiation, versatile setups to unravel the quantum structure of atoms and molecules were developed. In scattering experiments, multipole focusers were exploited to study the steric effect, that is, how the orientation of an attacking molecule affects its reactivity [10]. Variants of the molecular beam resonance methods as well as scattering machines that employed state selectors were implemented in many laboratories, and have yielded a wealth of detailed information on stable molecules, radicals, and molecular complexes alike. Manipulation of beams of atoms and molecules by electric and magnetic fields is about as old as atomic and molecular beams themselves; atomic and molecular beam techniques could not have been developed without the ability to manipulate the atoms or molecules. In his autobiography, Norman Ramsey recalls that, when he arrived © 2009 by Taylor and Francis Group, LLC
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in Rabi’s lab in 1937, Rabi was rather discouraged about the future of molecular beam research; his discouragement only vanishing when he invented the molecular beam magnetic resonance method [11]. However, even though the manipulation of beams of molecules by external fields has been used extensively and with great success in the past, it concerned exclusively the transverse motion of the molecules. Only in 1999 was it experimentally demonstrated that appropriately designed arrays of timedependent electric fields can be used to also influence and control the longitudinal (forward) velocity of the molecules in a beam. This so-called “Stark decelerator” was used to slow down a beam of neutral polar molecules [12]. Since then, the ability to produce focused packets of state-selected accelerated or decelerated molecules has made possible a host of new experiments. How to achieve a full control over the three-dimensional motion of neutral polar molecules using an electric field, and how exerting this control can be made use of in a variety of novel experiments is the subject of this chapter. For the sake of simplicity, we will restrict our discussion to the interaction of polar molecules (i.e., of molecules that carry a permanent electric dipole moment) with electric fields. However, we note that the same arguments and principles hold for the interaction of particles with a magnetic dipole moment with magnetic fields. In a quadrupole or hexapole focuser, the magnitude of the electric field is zero on the symmetry axis, which is normally made to coincide with the molecular beam axis. Close to this axis, the electric field strength is—to a good approximation— cylindrically symmetrical, and it increases with distance r from the axis, proportionately to r or r 2 for the quadrupole or hexapole field, respectively (Figure 14.1). For polar molecules in a so-called low-field seeking (LFS) quantum state, that is, a state in which their space-fixed dipole moment is antiparallel to the external electric field, the Stark energy increases with increasing electric field strength. The force that a molecule is subjected to in an electric field is given by the negative gradient of the Stark energy, which implies that in a multipole field there is always a restoring force toward the molecular beam axis acting on a molecule in a LFS (b)
(a) +
–
–
–
– –
(c) +
+
+ +
+ –
|E|
FIGURE 14.1 The electrode geometries for generating (a) dipole, (b) quadrupole, and (c) hexapole fields. The lower part shows how the magnitude |E| of the electric field varies along a line passing perpendicularly through the center for each geometry. Whereas the dipole geometry generates a constant electric field, the quadrupole (hexapole) geometry generates an electric field that increases linearly (quadratically) with distance from the center.
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state. A multipole focuser operates like a perfect lens when this restoring force is linearly proportional to r. For a molecule whose Stark energy scales linearly with the strength of the electric field, it is the hexapole focuser that acts as a perfect lens; on the other hand, a quadrupole focuser is needed to perfectly focus a molecule whose Stark energy scales quadratically with the electric field strength. The depth of the confining potential depends on the properties of the molecule as well as on the magnitude of the electric field that can be experimentally realized. In Figure 14.2, the Stark shifts are shown for the relevant low rotational levels of OH, CO(a3 Π1 ), ND3 , and H2 CO molecules, which have been used in the studies highlighted in this chapter. It can be seen from these plots that at feasible electric field strengths of up to 100 kV/cm, potential energy wells with a depth of typically 1 cm−1 (≈1.44 K) can be created. This is enough to transversally confine such molecules about the beam axis because, in a typical molecular beam experiment, the transverse velocity distribution
3 2 1 0 –1 –2 –3
MJ W 2
–9/4 –3/4
J = 3/2
3/4 OH (X 2P3/2) 0
50
Energy (cm–1)
Energy (cm–1)
MJ W
9/4 100
150
J = 1, K = 1
0
0
–1 –2
ND3 0
1 50
100
150
Electric field strength (kV/cm)
200
0
–1 –2
CO (a3P1)
1
MJ K
MJ K –1 Energy (cm–1)
Energy (cm–1)
1
J=1 0
0 50 100 150 200 Electric field strength (kV/cm)
Electric field strength (kV/cm)
2
1
–3
200
–1
2 Jt 1 11 0 10 –1 –2 –3 –4 H2CO –5 0
–1
0
1 50
100
150
200
Electric field strength (kV/cm)
FIGURE 14.2 Stark energy curves of OH(X 2 Π3/2 ), CO(a 3 Π1 ), ND3 , and H2 CO in low rotational states. In zero field, there are two closely spaced levels with opposite parity. This zero-field energy splitting is caused by Λ-doubling (OH, CO), inversion doubling (ND3 ), or K-doubling (H2 CO). In an electric field, the opposite parity levels are coupled, leading to a large linear Stark splitting. Levels that undergo a positive (negative) Stark shift in an increasing electric field are referred to as “low-field seeking, LFS” (“high-field seeking, HFS”). If the Stark shift becomes comparable with the spacing of the rotational levels, the interaction with higher rotational states needs to be taken into account. Since this interaction pushes the rotational levels down, ultimately, all states become HFS at sufficiently high electric fields.
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is centered around zero, with a full-width at half-maximum (FWHM) spread of several tens of m/sec, corresponding to a sub-cm−1 transverse kinetic energy spread. Normally, the multipole focusers are made just long enough to focus a beam of molecules onto the interaction or detection point somewhere downstream. Inside the multipole, the molecules follow a sinusoidal path. This is schematically shown in Figure 14.3. The position of the focus can be controlled by varying the voltages on the multipole electrodes, or, for fixed voltages, by varying the time during which the voltages are switched on; with the second approach, one can eliminate the chromatic aberration. Figure 14.3a pertains to the case when the focusing lens is perfect, whereas Figure 14.3b shows what happens when the focusing lens is imperfect, due to a nonlinear behavior of the force. If the molecules spent a longer time inside the focusing lens or if the trapping potential were steeper, the molecules would undergo multiple transverse oscillations. The latter is the case, for instance, in a guide [13] or when a long multipole focuser is bent into a torus, confining the particles on a circular orbit, such as in an electrostatic storage ring [14]. In multipole focusers as well as in the deflection elements that had been used in atomic and molecular beam experiments in the past, the field gradients are perpendicular to the beam axis, and thus the velocity component of the particles along the beam axis is not affected. In a typical molecular beam experiment, the forward velocity distribution is centered at a high velocity (300 to 2000 m/sec) with a FWHM spread of about 10% of this central velocity. Even at the low end of this velocity range, the kinetic energy of the molecules, whose Stark energies are shown in Figure 14.2, is on the order of 100 cm−1 . This is much larger than the depth of any single potential well that can be realized for these molecules with an electrostatic field. Therefore, a direct (a)
(b)
4
r (mm)
3 2 1 0 A –1
0
2
B 4
A 6 z (cm)
8
10
0
2
B 4
6 z (cm)
8
10
FIGURE 14.3 Trajectories in the (r, z)-plane of molecules in a LFS state, traversing a hexapole at a fixed velocity. Here z is the hexapole’s axis of cylindrical symmetry. The hexapole is switched on (off) when the molecules are at position A (B). The molecules are assumed to originate from a single point. (a) The case when the force in the hexapole is perfectly linear. As a result, the molecular beam is focused onto a single point. (b) The case when a zero-field energy splitting due to Λ-doubling, for instance, is included. In this case, the force is nonlinear at low electric field strengths, resulting in a blurring of the focus.
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longitudinal confinement of the molecules in a static potential well is impossible. On the other hand, the FWHM longitudinal velocity spread of the beam molecules corresponds to a kinetic energy spread on the order of only 1 cm−1 , which is comparable to the FWHM of the transverse kinetic energy spread. As potentials with a depth of 1 cm−1 can be produced, such a potential—with a gradient along the molecular beam axis—can longitudinally confine the molecules, provided this potential moves along with the molecular beam. A multipole focuser would be able to produce a strong-enough gradient of the electric field along the molecular beam axis if it were mounted with its symmetry axis perpendicular to the molecular beam axis. If such a perpendicularly mounted multipole focuser could be moved along the molecular beam axis at the most probable velocity of the molecules, it would provide for the requisite longitudinal confinement; the molecules on the beam axis would oscillate, both in position and velocity, around the center of the multipole. Although the most probable velocity of the molecules in the beam would thereby not be changed, the pack of molecules would remain flocked together while moving in the forward direction. Such a device would thus have the same advantage for the longitudinal motion of the beam molecule as a multipole focuser normally has for their transverse motion. Moreover, if the velocity of this moving potential well could be gradually changed, (a fraction of ) the beam molecules could be brought to any desired final velocity. In order to, for instance, decelerate the beam molecules, the potential well would have to be gradually slowed down, such that the molecules in the beam would spend more time on the leading slope of the longitudinal potential well, thereby feeling a force opposing their motion. In order to accelerate the molecules, the potential well would have to be gradually sped up, thus pushing the molecules forward on the trailing slope of the potential well. The hypothetical situation just described is actually almost exactly what happens in a real Stark decelerator [12]. However, rather than mechanically moving an electrode geometry that generates the confinement potential along the molecular beam, a static array of electrode pairs that create field gradients along the molecular beam axis is used. By switching the fields on adjacent pairs of electrodes on and off at the appropriate times, a traveling potential well is created [15]. In a Stark decelerator, the molecules can either be transported along the beam axis at a constant velocity or gradually decelerated or accelerated to any desired final velocity. In the first experimental demonstration of Stark deceleration, a beam of metastable CO(a3 Π1 , J = 1) molecules was slowed down from 225 to 98 m/sec [12]. Experiments of this kind have been considered and tried before. Electric field deceleration of neutral molecules was first attempted by John King at MIT in 1958. King intended to produce a slow ammonia beam, to obtain a maser with an ultranarrow linewidth. Much better known, especially in the physical chemistry community, is the experimental effort of Lennard Wharton to demonstrate electric field acceleration of a molecular beam. In the 1960s, at the University of Chicago, he constructed an 11 m long molecular beam machine for the acceleration of LiF molecules in HFS states from 0.2 to 2.0 eV, with the goal of using these high-energy beams for reactive scattering studies [16]. Both of the above experiments were unsuccessful, and were not continued after the graduation of the PhD students involved [17,18]. Whereas the interest in slow molecules as a maser medium declined, owing to the invention of the © 2009 by Taylor and Francis Group, LLC
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laser, the molecular beam accelerator was superseded by gas-dynamic acceleration of heavy species in seeded supersonic beams of He or H2 [19]. The state-selected molecular beam, which exits a Stark decelerator, with a tunable velocity and a tunable velocity spread, is ideally suited for many applications. Decelerated beams can be used, for instance, for high-resolution spectroscopic studies [20,21], taking advantage of the increased interaction times. We also anticipate that these beams are advantageous for future molecular interferometry and molecule optics experiments. Decelerated beams also enable the study of (in)elastic collisions and reactive scattering as a function of collision energy, down to zero collision energy [22]. Last but not least, a Stark decelerator enables three-dimensional trapping of neutral polar molecules. “If one extends the rules of two-dimensional focusing to three dimensions, one possesses all ingredients for particle trapping.” This is how Wolfgang Paul put it in his Nobel lecture [23], and as far as the underlying physics principles of particle traps are concerned, it is indeed as simple as that. In order to experimentally realize the trapping of neutral particles, however, one has to face the challenge of producing sufficiently slow particles that can be trapped in a relatively shallow trap. When the particles are confined along a line, rather than around a point, the requirements on the kinetic energy of the particles are more relaxed, and storage of neutrons in a 1 m diameter magnetic hexapole torus could thus be demonstrated [24]. Trapping of atoms in a three dimensional trap only became feasible when Na atoms were laser cooled to sufficiently low temperatures to enable confinement in a quadrupole magnetic trap [25]. The Stark decelerator made possible the first demonstration of three-dimensional trapping of neutral ammonia molecules in a quadrupole electrostatic trap [26], even prior to the first demonstration of an electrostatic storage ring for neutral molecules [14]. The quadrupole trap permits the observation of packets of molecules, isolated from their environment, for times of up to several seconds. Among other things, this enables the direct measurement of lifetimes of metastable states [27]. The electrostatic trap, loaded from a Stark decelerator, also holds great promise for studies of cold collisions. More generally, electrostatic traps are key to a further development of the field of cold molecules, with the production and study of quantum degenerate gases of polar molecules as a prominent goal. In the remainder of this chapter, Stark deceleration of a molecular beam will be presented in more detail, followed by a description of the process of trapping of neutral polar molecules. An overview of the applications of slow beams and trapped samples of molecules will also be given. The experiments described in this chapter, which exemplify the operational characteristics of the various components, have been performed in different molecular beam machines and using different molecules. The deceleration and three-dimensional trapping of molecules in LFS states is explained using the OH radical as a model molecular system; a photograph of the Stark decelerator that was used in these experiments is shown in Figure 14.4. The buncher, used for longitudinal focusing of a molecular beam, and the storage ring and synchrotron were demonstrated using the ND3 molecule. The ND3 molecule, decelerated to a near standstill while in a LFS state, and then transferred, by microwave radiation, to a HFS state, was also used in the demonstration of an ac trap for neutral polar molecules. In order to explain the operation of a decelerator for molecules in HFS states, the © 2009 by Taylor and Francis Group, LLC
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FIGURE 14.4 Photograph of a Stark decelerator. The molecular beam passes through the 4 × 4 mm2 opening between the electrodes, shown enlarged in the inset from the perspective of the molecules. The forward velocity of the molecules is affected by switching, at appropriate times, a voltage difference of 40 kV between opposing electrodes. (From van de Meerakker, S.Y.T. et al., Annu. Rev. Phys. Chem., 57, 159–190, 2006. Copyright (2006) by Annual Reviews www.annualreviews.org. With permission.)
so-called alternating gradient (AG) decelerator, the experiments on metastable CO in the appropriate quantum state will be highlighted. Throughout this chapter we restrict ourselves to the discussion and demonstration of the basic concepts of the various elements; for more detail, we refer the reader to the original literature.
14.2
STARK DECELERATION OF NEUTRAL POLAR MOLECULES
14.2.1 THE STARK DECELERATOR The Stark decelerator (or accelerator) for neutral polar molecules is the equivalent of a linear decelerator (or accelerator) for charged particles. The Stark decelerator exploits the quantum-state specific force that a polar molecule is subjected to in an electric field. This force is rather weak, typically some eight to ten orders of magnitude weaker than the force that the molecule, when singly ionized, would experience in an equivalent electric field. Nevertheless, this force suffices to exert a complete control over the motion of polar molecules using principles akin to those developed to manipulate charged particles. In a Stark decelerator, the longitudinal velocity of a beam of polar molecules is manipulated using a longitudinally inhomogeneous electric field array. Let us consider a single electric field stage of such an array, itself composed of two opposing electrodes that are connected to power supplies of opposite polarity, as shown in Figure 14.5. A polar molecule in a LFS state, as it approaches the plane of the electrodes, experiences the increasing electric field as a potential hill, and thus loses kinetic energy as it climbs the upward slope of the hill. However, on leaving the high-field region along the longitudinal coordinate, the molecule regains the same amount of © 2009 by Taylor and Francis Group, LLC
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20 kV
W(z)
–20 kV
FIGURE 14.5 The potential energy W (z) of a polar molecule as a function of position z along the molecular beam axis. The molecule is in a LFS state and the electric field is created by two electrodes energized by high voltages of opposite polarity.
kinetic energy it lost during its ascent. The acceleration on the downward slope of the hill can be avoided if a time-varying electric field is used: if the electric field is switched off abruptly, before the molecule has left the region of high electric field, the kinetic energy or velocity of the molecule will not return to its original value. As noted in the previous section, the effect of a single electric field stage on the forward velocity of a beam molecule is rather small. In order to obtain a significant change of the velocity, the process of hill climbing needs to be repeated many times. Therefore, a Stark decelerator consists of an array of electric field stages, as shown in Figure 14.6. Each field stage consists of two parallel cylindrical metal rods of radius r, held apart at a distance 2r + d. One of the rods is connected to a positive, and the other to a negative switchable power supply. Alternating rods are connected to each other. The adjacent field stages are separated by a distance L from one another. At a given time, the even stages are switched to a high voltage while the odd stages are grounded. The potential energy W (z) of a molecule as a function of the position z along the beam axis is shown in Figure 14.6 as well. The deceleration procedure is now straightforward: when the molecule has reached a position that is close to the top of the first potential hill, the even stages are grounded and the odd stages are switched to a high voltage. As a result, the molecule will find itself in front of a potential hill again and will lose kinetic energy while climbing it. When the molecule has reached the high electric field region, corresponding to the hilltop, the voltages are switched back to the original configuration. By repeating this process many times, the velocity of the molecule can be arbitrarily reduced in a stepwise fashion. The amount of kinetic energy that is lost per stage, and thus the velocity with which the molecule exits the decelerator, depends on the exact position of the molecule © 2009 by Taylor and Francis Group, LLC
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20 kV
Time t1
–20 kV W(z)
–20 kV
20 kV
20 kV
Time t2
–20 kV
W(z)
–20 kV
FIGURE 14.6 The potential energy W (z) of a polar molecule as a function of position z along the axis of an array of field stages (electrode pairs) with voltages applied as shown. Repeated switching between the upper and the lower field configuration is performed after a time interval t2 − t1 .
at the time when the fields are switched. A key property of the Stark decelerator is that it not only decelerates a single molecule in the beam, but all molecules whose positions and velocities are from within a certain range, called the acceptance of the decelerator. As a result, one can decelerate (or accelerate) a part of the beam to any desired velocity while keeping the selected part of the beam flocked together as a compact packet. Note that it is essential that the molecules remain in the same quantum state throughout the deceleration process. In order to achieve this, the orientation of a molecule needs to adiabatically follow the field, which requires that the electric field varies slowly enough.
14.2.2
PHASE STABILITY IN THE STARK DECELERATOR
Central to the understanding of the operation principle of a Stark decelerator are the concepts of a synchronous molecule and of phase stability. Returning to Figure 14.6, we call the position z of a molecule at the time when the fields are switched the © 2009 by Taylor and Francis Group, LLC
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“phase angle,” φ = zπ/L (while the spatial periodicity of the field stages is 2L, the periodicity of the corresponding phase angle is 2π). We define the position corresponding to φ = 0◦ as a position between two adjacent field stages such that the electrodes at φ = 90◦ become grounded just after the fields were switched.A molecule with velocity v0 is called synchronous, if its phase φ0 is always the same at the time when the fields are switched; that is, φ0 remains constant. As a consequence, a synchronous molecule loses a constant amount of kinetic energy, ΔK(φ0 ), per stage. This happens because the synchronous molecule travels exactly a distance L during the time interval between two successive switchings. Hence the synchronous molecule is always “in phase” with the switching of the fields in the decelerator. A molecule that has a slightly different phase φ and/or velocity v than the synchronous molecule will experience an automatic correction toward the equilibrium values φ0 and v0 . For instance, a molecule whose phase is slightly higher than φ0 at a particular switching time will lose more kinetic energy than the synchronous molecule, and so will slow down with respect to the synchronous molecule. This, in turn, will decrease its phase, until it begins to lag behind the synchronous molecule, at which point the process reverses. Molecules from within a certain region in phase-space, bounded by the so-called separatrix, undergo stable phase-space oscillations around the synchronous molecule. This feature of the process is referred to as phase stability; it ensures that some nonsynchronous molecules will also be decelerated, and that these molecules will remain bunched together in a packet throughout the deceleration process. In order to better understand phase stability, it is helpful to consider the trajectories of the molecules along the molecular beam axis and to derive the corresponding longitudinal equation of motion. The most important elements of the derivation are reproduced here; more details can be found elsewhere [29,30]. A mathematically more rigorous derivation, based on a spatial and temporal Fourier representation of the potential, has also been given [31]. The Stark energy of a molecule W (zπ/L) is symmetric about the position of a field stage (a pair of electrodes) and can be conveniently written as a Fourier series: W
zπ L
∞ zπ π a0
an cos n + + 2 L 2 n=1 zπ zπ zπ a0 − a1 sin − a2 cos 2 + a3 sin 3 + ··· = 2 L L L
=
(14.1)
By definition, the synchronous molecule travels a distance L during the time interval between two successive switching times. The change in the kinetic energy per stage, ΔK(φ0 ) = −ΔW (φ0 ), for a synchronous molecule with phase φ0 and velocity v0 at a certain switching time is then given by the difference in the potential energy at the positions φ0 and φ0 + π: ΔW (φ0 ) = W (φ0 + π) − W (φ0 ) = 2a1 sin φ0 .
(14.2)
As a result, the average force, F, acting on the synchronous molecule is given by F(φ0 ) = −
© 2009 by Taylor and Francis Group, LLC
ΔW (φ0 ) 2a1 =− sin φ0 , L L
(14.3)
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provided terms with n > 2 in Equation 14.1 can be neglected. The average force acting on a nonsynchronous molecule with a phase φ = φ0 + Δφ, but with a velocity v0 , is given by − 2aL1 sin(φ0 + Δφ). Hence, to a good approximation, the equation of motion of a nonsynchronous molecule with respect to the synchronous one is mL d2 Δφ 2a1 + [sin(φ0 + Δφ) − sin(φ0 )] = 0 π dt 2 L
(14.4)
where m is the mass of the molecule. In the phase-stability diagrams of Figure 14.7, contours of constant energy are shown that result from a numerical integration of Equation 14.4 for OH radicals in the J = 3/2, MΩ = −9/4 state. The computations were carried out with the parameters of a Stark decelerator operated in our laboratory, at the values of the synchronous 60
f0 = 0°
vz (m/sec)
40 20 0
–20 –40 –60 –p
0
p
2p f (rad)
3p
4p
5p
0
p
2p f (rad)
3p
4p
5p
60 f = 70° 0
vz (m/sec)
40 20 0
–20 –40 –60 –p
FIGURE 14.7 Phase-stability diagram for OH(J = 3/2, MΩ = −9/4) radicals when the decelerator is operated at a synchronous phase angle φ0 = 0 ◦ (upper panel) or φ0 = 70 ◦ (lower panel), with vz the velocity of the nonsynchronous molecule along the longitudinal coordinate z and φ its phase. The value vz = 0 corresponds to the velocity of the synchronous molecule. The positions of the electrodes of the decelerator are indicated by dashed lines. (From van de Meerakker, S.Y.T. et al., Annu. Rev. Phys. Chem., 57, 159–190, 2006. Copyright (2006) by Annual Reviews www.annualreviews.org. With permission.)
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20 f0 = 0°
20°
40° 60° 0
80°
f (rad) p
–p –20
–40
FIGURE 14.8 Longitudinal acceptance of a Stark decelerator for OH radicals for different values of the synchronous phase angle φ0 . Here vz is the velocity of the nonsynchronous molecule along the longitudinal coordinate z and φ is its phase. The value vz = 0 corresponds to the velocity of the synchronous molecule.
phase angle φ0 = 0◦ and φ0 = 70◦ . The solid curves show the phase-space trajectories of the molecules. The dashed lines indicate the positions of the electrodes of the decelerator. The closed curves in the phase-space diagram correspond to bound orbits; molecules within such a “bucket,” bound by the thick contour, oscillate in the phase space about the phase and velocity of the synchronous molecule. Note that the operation of the decelerator at φ0 = 0◦ corresponds to a transport of (a part) of the molecular beam through the decelerator without a change of velocity. Acceleration or deceleration of the beam occurs for −90◦ < φ0 < 0 and 0 < φ0 < 90◦ , respectively. The separatrix defines the longitudinal acceptance of the Stark decelerator, and is shown for different phase angles φ0 in Figure 14.8. One can see that the acceptance is larger for smaller values of φ0 , while the deceleration per stage increases for higher values of φ0 . Because either is desirable, there is a tradeoff between the two. A more extensive description of phase stability in a Stark decelerator revealed that additional phase-stable regions exist; these have indeed been observed in an experiment [30]. The higher-order phase-stable regions can be understood as resulting from higher partial waves in the Fourier expansion of the time-dependent inhomogeneous electric field and from their interferences [31].
14.2.3 TRANSVERSE FOCUSING IN A STARK DECELERATOR Phase stability only ensures that the molecules remain in a “bucket” of the longitudinal phase space. However, what is indispensable is that the molecules also stay together, throughout the deceleration process, in the transverse direction. We have opted for a compact design of the decelerator, whose electric field stages serve simultaneously for deceleration and transverse focusing. Indeed, in the electrode geometry © 2009 by Taylor and Francis Group, LLC
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shown in Figure 14.6, molecules in LFS states remain transversely confined to the molecular beam axis. This is because the electric field is always weaker on the axis than at the electrodes. In order to focus the molecules in both transverse directions (x and y), the rod-electrode pairs that make up a deceleration stage are alternately positioned horizontally (along the x-axis) and vertically (along the y-axis); see inset of Figure 14.4. The three-dimensional trajectories of molecules passing through a Stark decelerator are rather complex. In the longitudinal direction, a molecule oscillates in position and velocity about the synchronous molecule, while in the transverse direction it oscillates about the longitudinal (molecular beam) axis z. The oscillation frequencies involved are generally similar in magnitude, but depend strongly on the phase angle φ. We studied the influence of the transverse motion of the molecules on the longitudinal phase stability, and found that, for high values of the synchronous phase φ0 , the transverse motion actually enhances the longitudinal phase-space region corresponding to phase-stable deceleration. For low values of φ0 , however, the transverse motion reduces the acceptance of the Stark decelerator and unstable regions in the longitudinal phase space appear. These effects can be quantitatively explained in terms of a coupling between the longitudinal and transverse motion, and coupled equations of motion have been derived that reproduce the observations. This coupling does not significantly impair the overall performance of the Stark decelerator, provided the number of the deceleration stages is limited [32]. At low longitudinal velocities, the assumptions used in deriving the analytical models of the longitudinal and transverse motion [29,30,32] are no longer valid. Therefore, in order to avoid losses, special care must be taken when designing the last few electrode pairs of the decelerator.
14.2.4
STARK DECELERATION OF OH RADICALS
A schematic of the Stark deceleration and trapping machine that has been used to decelerate and trap OH radicals in our laboratory is shown in Figure 14.9. A pulsed beam of OH radicals is produced by photodissociation of HNO3 molecules, which are coexpanded with a rare gas through a room-temperature pulsed solenoid valve. In most experiments, either Kr or Xe is used as the carrier gas, producing a beam with the most probable velocity of 450 m/sec or 360 m/sec, respectively. In the supersonic expansion, the beam is rotationally and vibrationally cooled, as a result of which, after the expansion, most of the OH radicals reside in the lowest rotational (J = 3/2) and vibrational level of the electronic ground state X 2 Π3/2 . This level has a Λ-doublet splitting of 0.055 cm−1 and each Λ-doublet component is split (neglecting hyperfine structure) into a MJ Ω = −3/4 and a MJ Ω = −9/4 component when an electric field is applied. This is shown by the Stark-energy diagram in Figure 14.2. The MJ Ω = −9/4 component offers a Stark shift three times larger than the MJ Ω = −3/4 component. Only molecules that are in the LFS MJ Ω = −3/4 or MJ Ω = −9/4 components of the upper Λ-doublet level participate in the electric field manipulation process. The molecular beam passes through a skimmer, and enters a second vacuum chamber, containing the decelerator. In the decelerator chamber, the beam of OH radicals enters a short hexapole that focuses the beam onto the Stark decelerator. © 2009 by Taylor and Francis Group, LLC
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FIGURE 14.9 Schematic of the experimental setup. A pulsed beam of OH radicals is produced via ArF-laser photodissociation of HNO3 seeded in a heavy carrier gas. The molecular beam passes through a skimmer, hexapole, and Stark decelerator into the detection region. State-selective laser-induced fluorescence (LIF) detection is used to measure the arrival time distribution of the OH(J = 3/2) radicals in the detection zone. (From van de Meerakker, S.Y.T. et al., Annu. Rev. Chem., 57, 159–190, 2006. Copyright (2006) Annual Reviews www.annualreviews.org. With permission.)
The 1188-mm-long Stark decelerator consists of an array of 108 equidistant electric field stages, with a center-to-center distance L of adjacent stages of 11 mm. Each stage consists of two parallel 6-mm-diameter polished hardened-steel rods that are centered 10-mm apart, and located symmetrically around the beam axis. Alternating stages are rotated by 90◦ with respect to one another, providing a 4 × 4 mm2 spatial transverse acceptance area. A photograph of the decelerator, with a close-up of the 4 × 4 mm2 opening between the electrodes, is shown in Figure 14.4. The decelerator is operated using a voltage of ±20 kV applied to the opposing electrodes of a field stage, thus creating a maximum electric field strength near the electrodes of 115 kV/cm. The high voltage pulses are applied to the electrodes using fast semiconductor high-voltage switches. The OH radicals are state-selectively detected 21 mm downstream from the last electric field stage (1.307 m from the nozzle) using an off-resonant laser-induced fluorescence (LIF) detection scheme. The performance of the Stark decelerator can be studied by recording the time-offlight (TOF) profile of the OH radicals exiting the decelerator, that is, by scanning the timing of the detection laser relative to the dissociation laser. Alternatively, the phase-space distribution of the molecules can be studied inside the decelerator by propagating a laser beam along the molecular beam axis, and by performing spatially resolved LIF detection from there. The latter strategy has been implemented by Jun Ye and coworkers at JILA, also using the OH radical [33]. Typical TOF profiles of OH radicals at the exit of the decelerator, obtained in our laboratory, are shown in Figure 14.10. © 2009 by Taylor and Francis Group, LLC
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237 m/sec (a)
LIF signal (arb. units)
194 m/sec (b)
142 m/sec (c)
(d)
95 m/sec
2.0
3.0
4.0
5.0
6.0
Time of flight (msec)
FIGURE 14.10 Observed and simulated TOF profiles of a molecular beam of OH radicals exiting the Stark decelerator when the decelerator is operated at a phase angle of 70◦ for a synchronous molecule with an initial velocity of (a) 470 m/sec, (b) 450 m/sec, (c) 430 m/sec, and (d) 417 m/sec. The molecules that are accepted by the decelerator are split off from the molecular beam and arrive in the detection region at later times, and with the indicated final velocities. (From van de Meerakker, S.Y.T. et al., Annu. Rev. Phys. Chem., 57, 159–190, 2006. Copyright (2006) by Annual Reviews www.annualreviews.org. With permission.)
These TOF profiles were obtained using Kr as a carrier gas, producing a molecular beam with the most probable velocity of 460 m/sec. The decelerator was operated at a phase angle φ0 = 70◦ and an initial velocity of the synchronous molecule of 470 m/sec (curve a), 450 m/sec (b), 430 m/sec (c), and 417 m/sec (d). With these settings, the decelerated bunch of molecules exits the decelerator with final velocities of 237, 194, 142, and 95 m/sec, respectively. The gaps in the profiles of the nondecelerated © 2009 by Taylor and Francis Group, LLC
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beams, which result from the removal of the decelerated bunch of the OH radicals, are indicated by the vertical arrows. The width of the arrival-time distribution of the decelerated packet becomes larger for lower values of the final velocity. This is due to the spreading of the molecular beam while in flight from the exit of the decelerator to the LIF detection zone, and due to the spatial extent of the detection laser beam. Both effects are incorporated in the simulated TOF profiles, shown below the observed ones. In curves (c) and (d), a rich oscillatory structure of the TOF profile of the fast beam is observed (its enlargement is shown in the inset). This structure results from the modulation of the phase-space distribution of the nondecelerated beam in the decelerator [30].
14.2.5
LONGITUDINAL FOCUSING OF A STARK DECELERATED MOLECULAR BEAM
In a Stark decelerator, the force acting on a polar molecule is conservative, because it depends, at any given time, solely on the molecule’s position. Consequently, the product of the velocity and position spreads remains constant throughout the deceleration process, in accordance with Liouville’s theorem. Cooling of a packet of molecules while maintaining the packet’s density is therefore fundamentally impossible in a Stark decelerator. However, what is possible is to reduce the velocity spread at the expense of the position spread, and vice versa—as long as the product of the two spreads remains constant. The “swap” can be done using a so-called buncher. A buncher is an additional array of field stages, mounted some distance behind the decelerator. In the buncher, a beam of polar molecules is exposed to a harmonic potential in the forward, longitudinal direction. This results in a uniform rotation of the longitudinal phase-space distribution of the packet of molecules. By switching the buncher on and off at the appropriate times, it can be used to either produce a narrow spatial distribution at a certain position downstream from the buncher, or a narrow velocity distribution. Both possibilities have been experimentally demonstrated; in particular, a beam of Stark-decelerated ND3 molecules with a longitudinal temperature of 250 μK was produced [34–36]. Figure 14.11 shows schematically the principle of bunching. When the Stark decelerator is operated at a phase angle of 70◦ , the longitudinal position spread of the decelerated ammonia molecules is about 1 mm and the longitudinal velocity spread is about 6.5 m/sec. The phase-space distribution of the decelerated molecules is shown in Figure 14.11 underneath the last stage of the decelerator. While in flight from the exit of the decelerator to the buncher, the packet of ammonia molecules spreads out along the molecular beam axis. This results in the elongated and tilted distribution in the longitudinal phase-space, shown in Figure 14.11. The geometry of the buncher is identical to that of the decelerator, except for an overall scaling factor. The fields are switched such that the synchronous molecule spends an equal time on the downward and upward slope of the potential well, and therefore keeps its original velocity. Molecules that are originally slightly ahead of the synchronous molecule (faster molecules), will spend more time on the upward slope than on the downward slope of the potential well, and will, therefore, be decelerated relative to the synchronous molecule. Molecules that are originally slightly behind the synchronous molecule (slower molecules), will spend less time on the upward slope © 2009 by Taylor and Francis Group, LLC
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Decelerator
Buncher
Laser
20 10 5 –2.5 –5
2.5
–10
–5
5
10
–10 20
–20
10
vz (m/sec)
10
–10
–5
5 –10
z (mm)
10
–5
5 –10
–20
FIGURE 14.11 Schematic of the end part of the decelerator, buncher, and detection region. The calculated longitudinal phase-space distributions of the ammonia molecules pertaining to the exit of the Stark decelerator, the entrance and exit of the buncher, and the detection region are shown. The position and velocity are plotted with respect to the synchronous molecule. The solid curves show lines of equal energy in the buncher potential (compare to Figure 14.7). (From Crompvoets, F.M.H. et al., Phys. Rev. Lett., 89, 093004, 2002. Copyright the American Physical Society. With permission.)
than on the downward slope of the potential well, and will, therefore, be accelerated relative to the synchronous molecule. The calculated distribution at the time when the buncher is switched off is also displayed in Figure 14.11, where the contours of equal energy of the ammonia molecules in the potential well are shown relative to the phase-space position of the synchronous molecule. As the potential well is approximately harmonic the phase-space distribution of the molecules undergoes a clockwise uniform rotation. When the buncher is switched off, the slow molecules are speeding ahead while the fast ones are lagging behind, giving rise to a longitudinal spatial focus at some position downstream. The angle by which the packet is rotated, and thus the exact position where the packet comes to a spatial focus, can be controlled by varying the voltage on the buncher electrodes or by varying the time during which the buncher fields are on [35]. In a Stark deceleration beamline, hexapoles and bunchers are used © 2009 by Taylor and Francis Group, LLC
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to map the phase-space distribution of the molecules at the exit of one element onto the entrance of the next in, respectively, the transverse and longitudinal direction.
14.2.6
DECELERATION OF MOLECULES IN HIGH-FIELD SEEKING STATES
It may appear straightforward to apply the above deceleration and bunching techniques to molecules in HFS states. One could simply let the HFS molecules fly out, instead of into, a high-field region. For the longitudinal motion of the molecules, this is indeed all that is required. However, achieving a simultaneous transverse stability is significantly more difficult for high-field seekers than for low-field seekers. This has to do with a prohibition, imposed by Maxwell’s equations, to create a maximum of field strength of a static electric field in free space. As a result, simple means to transversally focus high-field seekers are lacking [37]. Molecules in HFS states have a tendency to crash into the electrodes where the electric fields are the highest. However, this fundamental problem can be overcome by making use of alternating gradient (AG) focusers [38]. Figure 14.12a captures the general features of an AG decelerator. The AG lenses are formed from pairs of cylindrical electrodes to which a voltage difference is applied. Molecules in HFS states are defocused in the plane containing the electrode center lines, and focused in the orthogonal plane. As the molecules move down the beamline, the orientation of the lenses, and thus the focusing and defocusing directions, alternate. In any transverse direction, the defocusing lenses have less effect than the focusing ones, not because they are weaker (which they are not), but because the molecules are closer to the axis while acted upon by a defocusing lens than by a focusing lens. Molecules in HFS states are accelerated when entering the field of an AG lens and decelerated when leaving the field. By simply switching the lenses on and off at the appropriate times, AG focusing and deceleration of polar molecules can be achieved simultaneously. Figure 14.12b shows the potential energy along the z-axis of a single lens for a metastable CO(a3 Π1 , v = 0, J = 1) molecule in the MΩ = +1 HFS state. The molecules enter each lens with the electric fields turned off, so that their speed is not affected. Subsequently, the fields are rapidly turned on, and the HFS molecules are decelerated as they leave the lens and move from a region of high field to a region of low field. This process is repeated until the molecules reach the desired speed. A prototype machine of this type, consisting of 12 AG lenses, has been used to decelerate HFS metastable CO molecules [41]. In the meantime, the technique has been implemented by the group of Ed Hinds at Imperial College, London, and used to decelerate ground-state YbF molecules [42]. See also the chapter of Tarbutt and colleagues (Chapter 15). The transverse acceptance of the AG decelerator was estimated to be about a factor of 100 smaller than that of a decelerator for molecules in LFS states (of the same aperture) [39]. The acceptance can be increased by using a more sophisticated lens design, consisting of four electrodes rather than two. At Imperial College, a new AG decelerator was constructed to bring YbF molecules to a standstill. At the Fritz-Haber-Institut, AG focusing and deceleration has been applied to larger molecules such as benzonitrile [40]. We note that larger molecules may even lack any LFS states, in which case AG deceleration is the only option to slow them down. © 2009 by Taylor and Francis Group, LLC
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(a)
Lens 3 Lens 2
y z x
Lens 1
Molecular beam HV on
(b)
HV off
Stark shift, W (cm–1)
Leff 0 –0.5 –1 –1.5 –30
Wmax W –20
–10 Position, z (mm)
0
10
FIGURE 14.12 (a) Layout of an AG decelerator for polar molecules showing the first four deceleration stages. Each electrode pair both focuses and decelerates the molecules. (b) Crosssection of a single lens formed from two 20-mm-long rods with hemispherical end-caps, 6 mm in diameter and spaced 2 mm apart. The potential energy is shown along the longitudinal axis z for metastable CO molecules in the a3 Π1 , J = 1, MΩ = +1 state, when the potential difference between the electrodes is 20 kV. The procedure of switching the high voltages is indicated: the voltages are turned on when the bunch of molecules reaches the “HV on” position, and are turned off once they reach the “HV off” position. (From Bethlem, H.L. et al., J. Phys. B, 39, R263–R291, 2006. Copyright IOP Publishing Ltd. With permission.)
14.3 THE ZEEMAN, RYDBERG, AND OPTICAL DECELERATOR Inspired in part by the manipulation of polar molecules by electric fields, a magnetic analog of the Stark decelerator has recently been developed. Deceleration based on the magnetic interaction allows the manipulation of a wide range of atoms and molecules to which the Stark deceleration technique cannot be applied. The requisite rapid switching of the magnetic fields posed a considerable experimental challenge. © 2009 by Taylor and Francis Group, LLC
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The Zeeman deceleration technique was first experimentally demonstrated by slowing down ground-state H and D atoms, initially using six [43,44] and later twelve [45] pulsed magnetic field stages. The deceleration stages consist of 7.8-mm-long copperwire solenoids in which magnetic fields of up to 1.5 T are achieved. The coil design provides a cylindrically symmetric transverse restoring force that keeps the molecular beam focused on the longitudinal axis. When the current through the coils is switched, rise and fall times of the magnetic pulses as short as 5 μsec are achieved. These experiments have demonstrated that magnetic fields can be switched fast enough to allow for Zeeman deceleration of paramagnetic atoms and that the deceleration proceeds under the conditions of phase stability. The Zeeman deceleration technique has also been applied to decelerate metastable Ne atoms [46] in an 18-stage decelerator. Using electromagnetic coils that are encased in magnetic-steel shells with Permendur discs, even higher magnetic field strengths, of 3.6 T were achieved. Recently, the deceleration of metastable Ne atoms [47] and oxygen molecules [48] to velocities as low as 50 m/sec using 64 field stages has been reported. Compared to polar molecules, atoms or molecules in Rydberg states offer a much larger electric dipole moment. Hence, these particles can be manipulated using only modest electric field strengths and either a single or just a few field stages. Level crossings in the dense Rydberg manifolds limit the magnitude of the electric field strength that can be applied. The electric field manipulation of atoms and molecules in high Rydberg states has been pioneered using H2 molecules [49] and Ar atoms [50]. Using a Rydberg decelerator, H atoms could be stopped and electrostatically trapped in two [51] or three [52] dimensions. The short lifetimes of Rydberg states inherently limit the time that is available to store and study such species in a trap. However, if fluorescence to the ground state is the dominant decay process, cold samples of ground-state atoms or molecules can thus be produced. As all atoms and molecules possess Rydberg states, this may provide a versatile route to cold samples of atoms and molecules. Optical fields provide another general means to manipulate the motion of neutral particles. An intense optical field polarizes and aligns molecules [53]. In a laser focus, the molecules are subject to a force that is proportional to the gradient of the laser intensity. This force can be used to focus and trap the molecules. Optical manipulation of molecules was experimentally demonstrated by focusing [54] or deflecting [55] a beam of CS2 using a high-intensity pulsed laser beam. Optical forces have also been used to reduce the translational energy of molecular beams [56]. Benzene molecules were decelerated from 320 to 295 m/sec, while at the same time the xenon carrier gas was decelerated from 320 to 310 m/sec, illustrating the scope of this method. Rather than using a single laser beam, much higher forces can be generated by using two nearly counterpropagating laser beams. The two laser beams interfere and give rise to an optical lattice, that is, a periodic array of potential wells for polarizable atoms and molecules. The lattice can be set in motion by using laser beams with slightly different frequencies. By carefully controlling the frequency difference, the lattice can be moved at the same velocity as the molecules in the molecular beam. By lowering the velocity of the lattice, the molecules can be decelerated to any given velocity [57]. In this so-called optical Stark decelerator, the chirp of the laser beams needs to be very well controlled. In combination with the high intensities required, © 2009 by Taylor and Francis Group, LLC
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this is an experimental challenge. A simpler scheme was recently implemented using a constant-frequency offset between the two laser beams, such that the lattice moved with a velocity slightly below that of a molecular beam [58]. For a suitable choice of the parameters, the molecules make exactly half an oscillation within the optical potential. In this way, NO molecules were decelerated from 400 to 270 m/sec.
14.4 TRAPPING NEUTRAL POLAR MOLECULES 14.4.1
DC TRAPPING OF MOLECULES IN LOW-FIELD SEEKING STATES
The first electrostatic trap for polar molecules was demonstrated in 2000 using Starkdecelerated ND3 molecules [26]. This trap, consisting of a ring electrode and two hyperbolic end-caps in a quadrupole geometry, was originally proposed by Wing for Rydberg atoms [59]. Such a quadrupole trap can be mounted directly behind a Stark decelerator, and, through holes in the end-caps, a slow packet of molecules can be loaded into the trap using a procedure that is shown schematically in Figure 14.13. The specific values of the parameters referred to in the figure apply to the OH trapping experiments [60]. In this case, the Stark decelerator is programmed to produce a packet of molecules with a velocity of approximately 20 m/sec. The slow beam of OH radicals is loaded into the electrostatic trap with voltages of 7, 15, and −15 kV on the first end-cap, the ring electrode, and the second end-cap, respectively. This “loading configuration” of the trap is shown on the left side in Figure 14.13. In the loading geometry, a potential hill in the trap is created that exceeds the remaining kinetic energy of the molecules. The incoming OH radicals therefore come to a standstill near the center of the trap. Subsequently, the trap is switched into the “trapping configuration,” shown on the right side in Figure 14.13; the first end-cap is switched from 7 to −15 kV to create a (nearly) symmetric 500 mK deep potential well. A typical TOF profile, obtained in an OH deceleration and trapping experiment, is shown in Figure 14.14. In this experiment, the fluorescence signal of the OH radicals in the trap is detected through the hole in the end-cap. The TOF profile was recorded with Kr as carrier gas, and is therefore complementary to the series of TOF profiles shown in Figure 14.10. The gap in the TOF profile of the fast beam, which is due to the removal of the decelerated OH radicals, is indicated by a vertical arrow. The decelerated OH radicals come to a standstill about 7.4 msec after their production. At that time, indicated by another arrow, the voltages on the trap electrodes are switched to the trapping configuration. After some initial oscillations, a steady LIF signal is observed from the OH radicals in the trap, attesting to the fact that the molecules are confined. In the inset, the signal of the trapped OH radicals is shown on a 10 sec timescale, from which a 1/e trap lifetime of 1.6 sec can be deduced. We note that when Xe is used as a carrier gas, the most intense part of the molecular beam pulse can be selected, decelerated, and trapped [28]. The efficiency of the trap loading process could be increased by 40% by feedback control optimization using evolutionary strategies [61]. Electrostatic traps with other electrode geometries have been developed and tested as well. A four-electrode trap geometry that combines a dipole, quadrupole, and hexapole fields has been tested using decelerated ND3 molecules. By applying © 2009 by Taylor and Francis Group, LLC
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Loading geometry
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FIGURE 14.13 Schematic of the loading procedure of the electrostatic quadrupole trap. In the “loading configuration.” the voltages on the trap electrodes are set such that a potential hill is created in the trap that is higher than the remaining kinetic energy of the incoming molecules. At the time the molecules come to a standstill in the center of the trap, the trap is switched into the “trapping configuration.” In this geometry a (nearly) symmetric 500 mK deep potential well is created, which confines the molecules. (From van de Meerakker, S.Y.T. et al., Annu. Rev. Phys. Chem., 57, 159–190, 2006. Copyright (2006) by Annual Reviews www.annualreviews.org. With permission.)
different voltages to the electrodes, a double-well or donut trapping potential can be generated. Rapid switching between the various trap potentials offers good prospects for studying collisions as a function of collision energy at low temperatures [62]. Confinement of Stark decelerated molecules in combined magnetic and electric fields has recently been demonstrated in the JILA group with the OH radical. An electric field was superimposed on a magnetic field to create a combined magnetoelectric trapping potential [63]. The adjustable electric field might be advantageous in the study of low-energy dipolar interactions. Rather then actively manipulating fast molecules to generate slow ones, a mere velocity selection of slow molecules from an effusive source has also been successfully used; the velocity selection relied on a bent quadrupole guide with a longitudinal curvature such that only slow molecules could follow [13,64]. © 2009 by Taylor and Francis Group, LLC
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FIGURE 14.14 Measured TOF profile of OH( J = 3/2) radicals for a trapping experiment. The time at which the trap is switched on is indicated. In the inset, the signal of the trapped OH radicals is shown on a 10 sec timescale. (Figure partially reproduced from van de Meerakker, S.Y.T. et al., Phys. Rev. Lett., 94, 023004, 2005. Copyright the American Physical Society. With permission.)
14.4.2
STORAGE RING AND MOLECULAR SYNCHROTRON
In its simplest form, a storage ring is a trap in which the particles are subject to a minimum potential energy on a circle rather than at a point. The advantage of a storage ring over a trap is that it can confine packets of particles with a nonzero longitudinal velocity. While circling the ring, these particles can be made to interact repeatedly, at well-defined times and at distinct positions, with electromagnetic fields and/or other particles. Figure 14.15a shows the experimental setup used in the demonstration of an electrostatic storage ring for neutral molecules [65]. A beam of ammonia molecules was decelerated to a velocity of 92 m/sec, longitudinally cooled to a temperature of 300 μK by a buncher, and focused into a 25 cm diameter hexapole ring. The density of ammonia molecules inside the storage ring was probed via a laser-ionization detection scheme. In Figure 14.15b, the ion signal is shown as a function of the storage time in the ring; the time axis originates at the time when the high voltages on the ring are switched on. Peaks are observed when the packet of molecules passes the detection zone. Upon making successive round trips, the packet of molecules gradually spreads out as a result of the residual velocity spread, until it fills the entire ring. In order to counteract the spreading of the packet in the ring, we have constructed a storage ring consisting of two half-rings separated by a 2 mm gap, schematically shown in Figure 14.16a. By appropriately switching the voltages applied to the electrodes as the molecules pass through the gaps between the two half-rings, molecules can be accelerated, decelerated, or bunched. This device is the neutralparticle analog of a synchrotron for charged particles. Figure 14.16b shows again the density of molecules stored in the ring as a function of storage time. It is seen that, after an initial decrease during the first 25 round trips, the width of the stored packet remains constant. Bunching not only ensures a high density of the © 2009 by Taylor and Francis Group, LLC
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Ion detector Pulsed UV laser
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Dv = 0.83 ± 0.13 m/sec T = 300 ± 100 mK
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FIGURE 14.15 (a) Schematic of an experimental setup with a storage ring. A pulsed beam of ND3 molecules is Stark-decelerated to 92 m/sec, cooled to 300 μK using a buncher, and focused into a hexapole torus storage ring. (b) Density of ammonia molecules at the detection zone inside the ring as a function of storage time up to the 33rd round trip. Due to the spreading of the packet, the peak density decreases as 1/t (shown as the dashed line). In the inset, a measurement of the ammonia density after 49 to 51 round trips is shown, together with a multipeak Gaussian fit. (From Crompvoets, F.M.H. et al., Phys. Rev. A, 69, 063406, 2004. Copyright the American Physical Society. With permission.)
stored molecules, but, in addition, it makes it possible to inject multiple co- or counterpropagating packets into the ring, without affecting the packet(s) already stored [66].
14.4.3 AC TRAPPING OF MOLECULES IN HIGH-FIELD SEEKING STATES Trapping molecules in HFS states is of particular interest for two reasons: (1) The ground state of a system is always lowered by an external perturbation. Therefore, the ground state of any molecule is HFS. In the ground state, trap loss due to inelastic collisions is absent, making it possible to cool such molecules further using evaporative or sympathetic cooling. This is particularly relevant as the dipole–dipole interaction is expected to lead to large cross-sections for inelastic collisions of polar molecules in excited rovibrational states [67]. (2) Molecules composed of heavy atoms or many light atoms, such as polycyclic hydrocarbons, have small rotational constants. Consequently, all states of these molecules become HFS in relatively weak magnetic or electric fields. © 2009 by Taylor and Francis Group, LLC
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FIGURE 14.16 (a) Schematic of a molecular synchrotron. The synchrotron consists of two hexapole half-rings with a 12.5 cm radius, separated by a 2 mm gap. (b) Density of ammonia molecules at the detection zone inside the synchrotron as a function of storage time for up to the 40th round trip. Expanded views of two TOF profiles are shown in the insets, illustrating the narrow widths of these peaks. (From Heiner, C.E., Nature Phys., 3, 115–118, 2007. Copyright MacMillan Publishers Ltd. With permission.)
The problem of trapping molecules in HFS states is essentially the same as that of decelerating molecules in HFS states (discussed in Section 14.2.6). Ideally, one would like to have an electrode geometry that creates a maximum electric field strength at some position away from the electrodes; however, this is prohibited by Maxwell’s equations [37]. Although it is not possible to generate a maximum of the electric field in free space, it is possible to generate an electric field that has a saddle point. This is accomplished by superimposing an inhomogeneous electric field and a homogeneous electric field. In such a field, molecules are focused along one direction, while being defocused along the other. By reversing the direction of the inhomogeneous electric field, the focusing and defocusing directions can be interchanged. If this is done periodically, molecules will be further away from the saddle point along the focusing direction and closer to the saddle point along the defocusing direction, leading to a net time-averaged focusing force in any direction. Such a trap works both for molecules in HFS and LFS states. There are three possible electrode geometries that can be used © 2009 by Taylor and Francis Group, LLC
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to create the desired electric three-dimensional trapping field [68]. By now, all these geometries have been demonstrated for atoms and/or molecules. Figure 14.17a shows schematically the electrode geometry of our cylindrical ac trap that has been used to trap both ND3 molecules [68,69] and Rb atoms [70]. The trap has a hexapole symmetry. Consequently, when positive and negative voltages are applied alternately to the four electrodes of the trap, a perfect hexapole field is obtained. In order to create a saddle point, the voltage applied to one of the end-caps is increased, while the voltage on the other end-cap is decreased. This adds a dipole term to the electric field. If the direction of the hexapole term is reversed, the focusing and defocusing directions are interchanged. Figure 14.17b,c shows the strength of the electric field along the symmetry axis z and along the radial distance r from the z-axis, when the field is either focusing along z or r. 4.6 mm
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FIGURE 14.17 (a) Schematic of a cylindrically symmetric ac trap. (b) Electric field strength along the symmetry axis z. (c) The electric field as a function of the radial distance r from the z-axis. When voltages of 5, 7.5, −7.5, and −5 kV are applied to the electrodes, molecules in HFS states are focused toward the center along the z-axis and defocused in the radial direction r. When voltages of 11, 1.6, −1.6, and −11 kV are applied to the electrodes, molecules in HFS states are radially focused along r and defocused along the z-axis. (From Bethlem, H.L. et al., Phys. Rev. A, 74, 063403, 2006. Copyright the American Physical Society. With permission.)
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ND3 molecules were loaded into the ac trap by Stark-decelerating them to a standstill using their LFS state, and subsequently pumping about 20% of the molecules, with a microwave pulse, to a HFS state. Figure 14.18 shows the density of ND3 molecules at the center of the ac trap as a function of the switching frequency for molecules in either the HFS or LFS component of the ground state of para-ammonia [68]. For clarity, the signal for molecules in LFS states has been vertically offset. The signal for the HFS is scaled up by a factor of five, to correct for the smaller initial density of HFS. At low frequencies of the applied voltages, the trajectories of the molecules in the ac trap are unstable and no signal is observed. Above a frequency of about 900 Hz, the trap becomes abruptly stable. Maximum signal is observed at 1100 Hz. When the frequency is increased further, the molecules have less time to move between switching times, and the net time-averaged force on the molecules decreases. As a result the depth of the trap decreases. Higher-order terms in the trapping potential give rise to a (frequency-independent) potential that reduces the trap depth for molecules in HFS states and increases the trap depth for molecules in LFS states. Therefore, the signal of the HFS drops faster with increasing frequency than the signal of the LFS.
ND3 density (arb. units)
LFS
HFS ×5
500
1000
1500 Wdriven/2p (Hz)
2000
2500
FIGURE 14.18 Density of 15 ND3 molecules in LFS and HFS states of the |J, K = |1, 1 level at the center of the trap as a function of the switching frequency Ωdriven /2π. The measurements were performed 80 msec after the molecules had been loaded into the trap. The signal of the HFS is scaled up by a factor of 5 to compensate for the fact that the initial density of HFS is only about 20% of that of the LFS. (From Bethlem, H.L. et al., Phys. Rev. A, 74, 063403, 2006. Copyright the American Physical Society. With permission.)
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It is instructive to compare the ac trap with the dc trap, discussed in Section 14.4.1. The depth of the electrostatic traps is on the order of 1 K (depending on the molecular species and details of the trap design), and their volume is typically 1 cm3 . An ac electric trap has a depth of about 1 to 10 mK and a volume of about 10−2 cm3 [68].
14.4.4 TRAP LIFETIME LIMITATIONS Returning to Figure 14.14, we see that the time during which the OH radicals were stored in our electrostatic trap was on the order of a few seconds. Molecules can leave the trap via several distinct mechanisms. As there is a zero electric field at the center of the dc quadrupole or hexapole traps, the molecules can undergo transitions to degenerate quantum states that cannot be trapped. In practice, however, such Majorana transitions have not (yet) become a limitation of the lifetime, both because the volume in which these transitions could occur is very small and because, when hyperfine structure is included, molecules are often in states that are exclusively LFS; that is, there is no degeneracy [71]. Molecules can collide with particles in the residual gas in the vacuum chamber, leading to a kinetic energy transfer that results in a direct loss of trapped molecules from the rather shallow trap. Molecules in the trap can also collide with one another. In this case, inelastic collision processes can transfer the molecules from the trapped state to a different quantum state that is either antitrapped (HFS) or that is subjected to a lower trapping potential. Finally, the molecules can absorb black-body radiation from the (room-temperature) environment, leading to a change of the internal quantum state of the molecule. For future studies of collisions between trapped molecules, a quantitative understanding of all trap-loss mechanisms is essential. The trap losses due to optical pumping by black-body radiation and due to collisions with the background gas have been studied by monitoring the population decay of OH and OD radicals in a room-temperature electrostatic trap [72]. By comparing two isotopes of the same molecular species under otherwise identical conditions, the trap-loss mechanisms could be disentangled and quantified. The optical pumping rate by room-temperature black-body radiation was determined as 0.49 sec−1 for the OH radical and 0.16 sec−1 for the OD radical. Trap loss due to black-body radiation is thus a major limitation for the room-temperature trapping of OH radicals. The trapped molecules would have to be shielded from thermal radiation if longer trapping times were required. Most polar molecules exhibit strong electric dipole-allowed rovibrational transitions within the room-temperature black-body spectral region. In Table 14.1, the calculated blackbody pumping rates out of a specified initial quantum state are given for a number of polar molecules for which trapping is being pursued using the currently available techniques.
14.5 APPLICATIONS OF DECELERATED BEAMS AND TRAPPED MOLECULES As already mentioned in the Introduction, the three-dimensional focused packets of decelerated molecules with their tunable velocity and their narrow velocity spread can, for instance, be used for high-resolution spectroscopy and metrology, © 2009 by Taylor and Francis Group, LLC
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TABLE 14.1 Pumping Rates Due to Black-body Radiation at Two Different Temperatures, Out of a Specified Initial State, for a Number of Polar Molecules Pumping Rate (sec−1 ) System OH/OD NH/ND NH/ND NH3 /ND3 SO 6 LiH/6 LiD CaH/CaD RbCs KRb CO
Initial State X2Π
3 3/2 , J = 2
a1 Δ, J = 2 X 3 Σ− , N = 0, J = 1 X˜ 1 A1 , J = 1, |K| = 1 X 3 Σ− , N = 0, J = 1 X 1 Σ+ , J = 1 X 2 Σ+ , N = 0, J = 21 X 1 Σ+ , J = 0 X 1 Σ+ , J = 0 a3 Π1,2 , J = 1, 2
295 K 0.49/0.16 0.36/0.12 0.12/0.036 0.23/0.14 0.01 1.64/0.81 0.048/0.063 <10−3 <10−3 0.014/0.014
77 K 0.058/0.027 0.042/0.021 0.025/0.083 0.019/0.0063 <10−3 0.31/0.11 0.0032/<10−3 <10−3 <10−3 <10−3 /<10−3
Source: Reproduced from Hoekstra, S. et al., Phys. Rev. Lett., 98(13), 133001, 2007. With permission.
taking advantage of the increased interaction times. These beams also enable novel molecular-beam collision studies, in particular the study of (in)elastic collision and scattering processes as a function of collision energy. Trapped samples of neutral molecules can be used to measure lifetimes of metastable states, which is of particular relevance for free radicals, for which no other reliable methods exist to obtain this information. In the following sections, examples of these three applications are given and future prospects are discussed.
14.5.1
HIGH-RESOLUTION SPECTROSCOPY AND METROLOGY
Ultimately, the precision of any spectroscopic measurement is limited by the interaction time of the particle under investigation with the radiation field. The interaction time that can be obtained in a normal molecular beam setup is, at best, in the millisecond range. By decreasing the velocity of the molecular beam, the interaction time and thus the obtainable resolution, can be increased by orders of magnitude. We demonstrate this here, using proof-of-principle microwave experiments with a slow beam of ammonia molecules [20]. Similar measurements were also performed with decelerated OH radicals [21]. The inversion spectrum of the |J, K = |1, 1 level of 15 ND3 consists of 72 hyperfine transitions in a frequency interval of about 300 kHz. Due to this spectral congestion, the hyperfine structure could not be resolved in an earlier molecular beam experiment [71]. We have therefore carried out this experiment with a Starkdecelerated molecular beam, using the setup shown schematically in Figure 14.19a. A beam of ammonia molecules is decelerated from 280 m/sec to either 100 or 50 m/sec, and focused into a microwave zone. The microwave zone provides a nearly rectangular © 2009 by Taylor and Francis Group, LLC
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(a)
TOF mass spectrometer
Hexapole Stark decelerator Microwave strip line
UV laser
Hexapole
(b)
Ion signal (arb. units)
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100 m/sec
25 .5
20 .5
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.5
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FIGURE 14.19 (a) Experimental setup used to demonstrate reduction of transit-time broadening. A beam of 15 ND3 molecules is decelerated from 280 m/sec to either 100 or 50 m/sec, sent through a microwave zone, and subsequently detected. (b) Upon reducing the beam velocity, the transit-time-limited linewidth of the microwave transitions decreases to about 1 kHz at 50 m/sec. (From van Veldhoven, J., Eur. Phys. J. D, 31, 337–349, 2004. Copyright EDP Sciences. With permission.)
field strength distribution along the molecular beam axis with a width of about 65 mm. The microwave field drives the molecules, initially in the upper inversion level, into the lower inversion level. Shortly behind the microwave zone, molecules in the lower inversion level of the |J, K = |1, 1 state are detected. Figure 14.19b shows a small fraction of the recorded inversion spectrum, obtained with a beam of 280 m/sec (upper trace), 100 m/sec (middle trace), or 50 m/sec (lower trace). The broad line observed in the measurements taken at 280 m/sec is fully resolved in the measurements done on the decelerated beam. At 50 m/sec, the transit-time-limited linewidth is about 1 kHz, sufficient to resolve all individual hyperfine transitions. In this way we were able to determine the absolute energy of all 22 hyperfine levels in this system with an accuracy of better than 100 Hz [20]. The accuracy can be further increased by using a longer microwave zone, slower molecules, or both. At some point, however, beam deflection due to Earth’s gravitational pull becomes an issue. A way to deal with it is to use a vertical setup, for example, a molecular fountain. Such a fountain is currently being set up at the Laser Centre Vrije Universiteit in Amsterdam, in collaboration with the Fritz-Haber-Institut. In this fountain, ammonia molecules are decelerated, cooled, and subsequently © 2009 by Taylor and Francis Group, LLC
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launched. The molecules fly upwards some 10 to 50 cm before falling back under gravity, thereby passing a microwave cavity twice. The effective interrogation time in such a Ramsey-type measurement scheme includes the entire flight time between the two passages through the driving field, which can be up to a second. This should make it possible to obtain a precision of 10−12 to 10−14 . In the near future, it is unlikely that the precision of molecular frequency standards will compete with that obtained with the atomic counterparts. Nevertheless, precision measurements on molecules can be used as stringent tests of fundamental theories in physics. The sensitivity of any experiment on looking for a frequency shift due to a certain physical phenomenon depends both on the size of the shift, that is, the inherent sensitivity of the atom or molecule, and on the ability to measure this shift. Although the precision obtained for atoms is far better than the precision obtained for molecules, in a number of cases this is compensated for by the fact that the structure (and symmetry) makes molecules inherently more sensitive. For instance, in certain molecules such as YbF and PbO, time-reversal violating interactions leading to a permanent electric dipole moment (EDM) of the electron are three orders of magnitude stronger than in atom [73]. The possibility of improving the resolution of the EDM experiment by the use of cold molecules, has been an important motivation for the work on AG deceleration [42]; see also the chapter by Tarbutt and colleagues (Chapter 15). Molecules are also used in the search for a difference in the transition frequencies for chiral molecules (i.e., molecules that are each other’s mirror image) [74], and also in this case the use of slow molecules offers a great advantage. Ammonia, in its various isotopomers, might be ideally suited to test the time-variation of the proton-to-electron mass ratio; the inversion frequency in ammonia is determined by the tunneling rate of the protons through the barrier between the two equivalent configurations of the molecule, and is exponentially dependent on the reduced mass, which is closely linked to the proton-to-electron mass ratio. See also the chapter by Flambaum and Kozlov (Chapter 16).
14.5.2
COLLISION STUDIES AT A TUNABLE COLLISION ENERGY
Generally, a Stark decelerator offers new possibilities in all those experiments in which the velocity (distribution) of the beam molecules is an important parameter. Arguably, one of the most interesting applications of Stark-decelerated beams is in scattering experiments, where the tunability of the velocity of the beam molecules can be exploited to accurately measure, for example, scattering resonances. Studies of this kind have thus far been performed by molecular beams crossing under a variable angle [75]. With Stark-decelerated beams, these experiments can be performed with an unprecedented energy resolution in a fixed-angle experimental geometry. As the deceleration process is quantum-state specific, the bunches of slow molecules that emerge from the decelerator are pure, which is of particular importance for studies of inelastic collisions. In addition, the decelerated molecules are all naturally spatially oriented, allowing the steric effect to be studied. The first crossed-beam scattering experiments using a Stark-decelerated beam were performed with the OH–Xe system [22]. By varying the velocity of the OH radicals from 33 to 700 m/sec, with the velocity of the Xe atoms fixed at 320 m/sec, the total © 2009 by Taylor and Francis Group, LLC
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center-of-mass collision energy of the system could be tuned from 50 to 400 cm−1 . The collision energy was thereby scanned over the energetic thresholds for inelastic scattering into the first excited rotational levels of the OH radical. In Figure 14.20, the measured relative inelastic cross-sections are shown as a function of the center-of-mass collision energy for scattering into four different inelastic channels, indicated by arrows in the energy-level diagram. The largest cross-section is observed for scattering into the (X 2 Π3/2 , J = 3/2, e) state. This Λ-doublet-changing collision is the only exoergic channel, and the relative cross-section for this channel therefore approaches 100% at low collision energies. The other channels show a clear threshold behavior. These measurements provide a very sensitive probe of the theoretical potential energy surfaces, from which a detailed understanding of the collision dynamics can be obtained. The solid curves shown in Figure 14.20 are the result of quantum scattering calculations, which are seen to be in excellent agreement with the experiment. In the OH–Xe experiment, the energy resolution of (only) about 13 cm−1 is almost exclusively determined by the relatively large longitudinal velocity spread in the Xe beam. A major improvement in the energy resolution can be expected to come about if both scattering partners are fully under motional control. For this, a new crossed-beam scattering machine, consisting of two Stark decelerators at a 90◦ crossing angle, is currently being built at the Fritz-Haber-Institut. An artist’s rendition of this machine is shown in Figure 14.21. Each decelerator consists of 300 electric field stages. With this relatively large number of stages, the efficiency of the deceleration process can be optimized so that particle densities required for scattering studies would be obtained in the interaction region. cm–1 250
J = 3/2, e J = 5/2, e J = 5/2, f J =1/2, e
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F1(3/2f) F1(3/2e)
50 100 150 200 250 300 350 400 Ecoll (cm–1)
FIGURE 14.20 Relative inelastic cross-sections for scattering of OH (X 2 Π3/2 , J = 3/2, f ) radicals by Xe atoms as a function of the center-of-mass collision energy. As depicted in the energy level diagram, collisions populating the (X 2 Π3/2 , J = 3/2, e), (X 2 Π3/2 , J = 5/2, e), (X 2 Π3/2 , J = 5/2, f ), and (X 2 Π1/2 , J = 1/2, e) states have been studied. The measurements are indicated by points, whereas the solid lines are the theoretically predicted curves. (Partially reproduced from Gilijamse, J. J. et al., Science, 313, 1617–1620, 2006. Copyright AAAS. With permission.)
© 2009 by Taylor and Francis Group, LLC
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FIGURE 14.21 Schematic of the crossed-beam scattering machine. Each decelerator has a modular design and consists of a total of 300 electric field stages. As shown in the inset, the decelerators are designed such that their exits come very close to the collision zone, while simultaneously providing excellent optical access for detection of the scattered products (design: Henrik Haak).
In this machine, molecular inelastic or reactive scattering can be studied between combinations of neutral polar molecules such as OH, NH(a1 Δ), CO(a3 Π), ND3 , SO2 , and H2 CO. The collision energy can be continuously varied in the 1 to 500 cm−1 range, with an overall energy resolution better than 1 cm−1 . This should enable an accurate mapping of, for example, scattering resonances, thereby providing sensitive tests of the molecular potential energy surfaces. In the crossed-beam geometry, the particles encounter each other only once. Alternatively, the decelerated beams can be loaded into a molecular synchrotron, located at the intersection point of the two molecular beams. In a synchrotron containing 20 counterpropagating packets, a packet that completed 100 round trips would have undergone 4000 encounters. We are currently constructing a molecular synchrotron as a collider for neutral polar molecules, which is designed to be combined with the crossed-beam apparatus.
14.5.3
DIRECT LIFETIME MEASUREMENTS OF METASTABLE STATES
The long observation time afforded by a trap can be exploited to directly measure, with high precision, the radiative lifetime of excited rovibrational or electronic states. To demonstrate this, we have electrostatically trapped OH radicals in the first vibrationally excited X 2 Π3/2 , v = 1, J = 3/2 state, and monitored the temporal decay of the trapped molecules [27]. The measured population in the vibrationally excited state as a function of the storage time in the trap is shown in Figure 14.22a. From the observed exponential decay a radiative lifetime of this state of 59.0 ± 2.0 msec was deduced, in good agreement with the calculated value of 58.3 msec [76]. This experiment thus provided a benchmark value for the Einstein A coefficients of the important Meinel system of OH. The same experimental approach has also been applied to accurately measure the lifetime of the metastable a 3 Π electronic state of CO molecules. CO molecules in this state can only decay to the X 1 Σ+ electronic ground state (giving rise to the Cameron bands), via a spin-forbidden transition, weakly allowed because of spin–orbit mixing of the a3 Π state with 1 Π states [78]. The spin–orbit mixing © 2009 by Taylor and Francis Group, LLC
543
(b) Intensity (arb. units)
OH ( X2P3/2, v = 1, J = 3/2)
0
40
80
120
160
Trapping time (msec)
(c) Intensity (arb. units)
(a) Intensity (arb. units)
Slowing, Trapping, and Storing of Polar Molecules
CO (a 3P1)
15
20
25
Trapping time (msec)
CO (a 3P2)
0
100
200
300
400
Trapping time (msec)
FIGURE 14.22 Measured populations of trapped OH (X 2 Π3/2 , v = 1, J = 3/2) radicals (a), CO (a 3 Π1 (v = 0, J = 1)) molecules (b) and CO (a 3 Π2 (v = 0, J = 2)) molecules (c) as a function of the trapping time. The trapping potential is switched on at the zero of the timescale. (Figure partially reproduced from van de Meerakker, S.Y.T. et al., Phys. Rev. Lett., 95(1), 013003, 2005; Gilijamse, J.J. et al., J. Chem. Phys., 127, 221002, 2007. Copyright the American Physical Society. With permission.)
makes the lifetime of the a3 Π state strongly quantum-state-dependent. To measure these quantum-state-specific lifetimes, the CO molecules are laser-prepared in either the a 3 Π1 (v = 0, J = 1) state or the a 3 Π2 (v = 0, J = 2) state, both of which can be decelerated and trapped. The measured trap decay curves are shown in Figure 14.22b and 14.22c, yielding lifetimes of 2.63 ± 0.03 msec and 143 ± 4 msec for the a 3 Π1 (v = 0, J = 1) and the a3 Π2 (v = 0, J = 2) states, respectively. Although the absolute values of these lifetimes were not accurately known prior to these measurements, the ratio of these lifetimes follows from the known energies of the levels of the a3 Π state, with spectroscopic precision, as 1:54.7. The independently determined lifetimes of the two different quantum states are seen to agree perfectly with this ratio [77].
14.6
CONCLUSIONS AND OUTLOOK
The merging of molecular beam methods with those of accelerator physics has yielded new tools to manipulate the motion of molecules. Over the last few years, decelerators, lenses, bunchers, traps, storage rings, and a synchrotron for neutral molecules have been demonstrated. Molecular beams with a tunable velocity and with a tunable width of the velocity distribution can now be produced. We expect this array of new molecular-beam technology to become a valuable tool in a variety of chemical physics experiments, ranging from ultrahigh-resolution spectroscopy to crossed-beam (reactive) scattering experiments. The decelerated beams have enabled the loading of molecules into a variety of traps. In these traps, electric fields are used to keep the molecules confined in a region of space where they can be studied in complete isolation from the environment. This enables the investigation of molecular properties in unprecedented detail. The Stark-deceleration and trapping techniques can be applied to any polar molecule, provided it has a sufficiently large Stark shift in experimentally attainable electric fields. In Tables 14.2 and 14.3, selections of polar molecules with their relevant © 2009 by Taylor and Francis Group, LLC
544
TABLE 14.2 Selection of Polar Molecules Suited for Deceleration and Trapping Experiments and Their Relevant Propertiesa
Molecule
State
Stark Decelaration √ ( ) and Trapped (†)
Multiplicity of Hyperfine Levels
Stark Shift at 200 kV/cm (cm−1 )
No. of Stages Required in Present Setup
Dipole Moment (D)
4 4
1.54 1.88
90 71
1.46
CF
|X 2 Π1/2 , J = 1/2, MΩ = −1/4 |X 2 Π1/2 , J = 3/2, MΩ = −3/4
4 4
0.44 0.32
533 845
0.65
CH2 F2
|Jτ M = |2−2 0
1.52∗
168
1.96
CH3 F
|JKM = |100 |JKM = |1 ± 1 ∓ 1
0.54∗
1.05∗
217 99
1.86
CO
|a 3 ΠΩ=1 , J = 1, MΩ = −1 |a 3 ΠΩ=2 , J = 2, MΩ = −4
2 2
1.71 2.87
89 63
1.37
H2 CO D2 CO
|Jτ M = |11 1 |Jτ M = |11 1
6 6
1.44∗ 1.11∗
130 155
2.34 2.34
H2 O D2 O HDO
|Jτ M = |11 1 |Jτ M = |11 1 |Jτ M = |11 1
6 6 12
0.45 0.72 0.73
1081 667 558
1.82 1.85 1.85
HCN
|(v1 , v2l , v3 ), J, M = |(0, 00 , 0), 1, 0
6
0.92∗
177
3.01
12
1.80∗
79
12
2.58
150
12
1.66∗
109
8
3.11
45
|(v1 , v2l , v3 ), J, Ml = |(0, 11 , 0), 1, −1 |(v1 , v2l , v3 ), J, Ml = |(0, 11 , 0), 2, −1 |(v1 , v2l , v3 ), J, Ml = |(0, 11 , 0), 2, −2 LiH
|X 1 Σ+ , J = 1, M = 0
© 2009 by Taylor and Francis Group, LLC
√ [12] † [77] √ [77] † [77] √ [79]
5.88
Cold Molecules: Theory, Experiment, Applications
CH
|X 2 Π1/2 , J = 1/2, MΩ = −1/4 |X 2 Π3/2 , J = 3/2, MΩ = −9/4
|X 1 Σ+ , J = 1, M = 0
NH
|a 1 Δ, J = 2, M = 2
14 NH
3 15 NH 3 14 ND 3 15 ND 3
|JKM = |1 ± 1 ∓ 1 |JKM = |1 ± 1 ∓ 1 |JKM = |1 ± 1 ∓ 1 |JKM = |1 ± 1 ∓ 1
NO
|X 2 Π1/2 , J = 1/2, MΩ = −1/4
N2 O
OH OD
|(v1 , v2l , v3 ), J, Ml = |(0, 11 , 0), 1, −1 |(v1 , v2l , v3 ), J, M = |(0, 00 , 0), 1, 0 |(v1 , v2l , v3 ), J, M = |(0, 00 , 0), 2, 0 |(v1 , v2l , v3 ), J, Ml = |(0, 11 , 0), 1, −1 |(v1 , v2l , v3 ), J, Ml = |(0, 11 , 0), 2, −1 |(v1 , v2l , v3 ), J, Ml = |(0, 11 , 0), 2, −2 |X 2 Π3/2 , J = 3/2, MΩ = −9/4 |X 2 Π3/2 , J = 3/2, MΩ = −9/4
SH SD
|X |X
SO2
|Jτ M = |10 0
OCS
2Π 3/2 , J = 3/2, MΩ = −9/4 2Π 3/2 , J = 3/2, MΩ = −9/4
√
12 [80] † [81] √ [29]
√ [26] † [26] √ [29] † [29,62]
√ [82] † [60,63] √ [72] † [72]
√
[83]
2.67
34
12
3.34
48
1.49
48 32
2.11 2.11 2.29 2.29
79 84 65 68
1.47 1.47 1.50 1.50
6
0.17
1179
0.16
18
0.26
1041
0.17
1
0.13∗
1172
0.72
1407
1
0.43
2
0.25∗
647
2
0.56
759
2
0.24∗
779
4 6
3.22 3.22
56 58
1.67 1.65
4 6
1.51 1.49
227 235
0.76 0.76
1
1.47
343
1.59
0.70
Slowing, Trapping, and Storing of Polar Molecules
LiD
a See also Ref. [29]. A Stark shift arising at 200 kV/cm is indicated. If the Stark energy reaches a maximum at smaller field Strengths, indicated by the symbol ∗ , the maximum Stark shift is given.
545
© 2009 by Taylor and Francis Group, LLC
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TABLE 14.3 Selection of Molecules Suited for AG Deceleration and AC Trapping and their Relevant Propertiesa
|J = 1, MΩ = +1
CaF
|J = 1/2, MΩ = +1/4
YbF X 2 Σ+
State
|J = 3/2, MΩ = +9/4
|J = 1/2, MΩ = +1/4
15 ND 3
|J = 1, MK = +1
Pyridazine (C4 H5 N)
|JKa Kc |M| = |000 0
Benzonitrile (C7 H5 N)
|JKa Kc |M| = |000 0
Tryptophan I (C11 H12 N2 O2 ) II III IV V VI
|JKa Kc |M| = |000 0
a See also Ref. [39].
© 2009 by Taylor and Francis Group, LLC
Effective dipole (cm−1 /kV/cm) at 100 kV/cm
−1.25
0.0135
–/1.68/–
28
−1.62
0.0191
–/18.52/–
17
[84]
−3.43
0.0420
–/0.34/–
59
[42]
−4.91
0.0569
–/0.24/–
193
† [69,85]
−1.27
0.0134
–/5.14/3.12
20
−5.59
0.0624
0.21/0.20/0.10
80
−6.71
0.0711
0.19/0.051/0.040
103
−6.25 −4.72 −1.71 −11.68 −12.28 −11.37
0.0646 0.0494 0.0183 0.120 0.126 0.116
0.041/0.013/0.012 0.039/0.014/0.012 0.033/0.017/0.013 0.032/0.016/0.013 0.043/0.011/0.0096 0.045/0.011/0.0095
216
√
[41]
√ √ √
√
[40]
Rotational Constants (cm−1 ) A/B/C
Mass (amu)
Cold Molecules: Theory, Experiment, Applications
Molecule CO a 3 Π1 OH X 2 Π3/2
Stark shift (cm−1 ) at 100 kV/cm
AG Decelaration √ ( ) or Trapped (†)
Slowing, Trapping, and Storing of Polar Molecules
547
properties are listed that are suited for deceleration and trapping in LFS and HFS states, respectively. Both the production in a molecular beam and the detection methods for these molecules are well known. Molecules that have already been Stark decelerated and/or trapped are indicated, and the reference to the original experiments is given. The trapping of neutral polar molecules holds great promise in the investigation of molecular interactions and quantum collective effects at ultralow temperatures. For this, the phase-space density needs to be further increased, that is, the number density needs to be made higher and/or the temperature needs to be reduced. The most straightforward way to increase the number density of trapped molecules would be to accumulate several packets of molecules in a trap. Simply reloading the trap, however, requires opening up the trapping potential, thereby losing or heating the molecules that have already been stored. Two different schemes that work specifically for the NH radical [86] and the SO molecule [83] have been proposed to circumvent this fundamental obstacle. The phase-space density of the trapped gas can also be increased by decreasing the temperature of the molecules. Various cooling schemes have been proposed to achieve temperatures below 1 mK. The most promising scheme is sympathetic cooling, in which cold molecules are brought into contact with an ultracold atomic gas and equilibrated with it via elastic collisions. Most suited for sympathetic cooling are HFS molecules confined in an ac trap, eliminating possible inelastic collision channels that lead to trap loss. Experiments to spatially overlap magnetically trapped atoms with molecules in an ac trap are under way [70]. When a further cooling of trapped molecules is achieved, a host of new experiments will become possible. At sufficiently low temperatures, the de Broglie wavelength of the molecules becomes comparable to, or even larger than, the interparticle separation. In this exotic regime, quantum-degenerate effects dominate the dynamics of the particles, and a Bose–Einstein condensate can be formed. Particularly interesting for these experiments is the presence of a permanent electric dipole moment in the molecules. The anisotropic, long-range, dipole–dipole interaction is predicted to give rise to new and rich physics in these cold dipolar gases [87]. The different tools that were developed in the past to transversally manipulate molecular beams, dating back to the Stern and Rabi eras, have proven to be crucial for developments that reach beyond molecular physics proper. The complete control over the full three-dimensional motion of molecules now available adds a new dimension to the long and rich history of the manipulation of molecules by electric fields [88].
ACKNOWLEDGMENTS The experiments described in this chapter are a result of over a decade of research by a large group of people. This research has been launched at the University of Nijmegen, the Netherlands. In 2000, the research group moved to the FOM Institute for Plasmaphysics “Rijnhuizen” in Nieuwegein, the Netherlands. In 2003, it moved yet again to its present home at the Fritz-Haber-Institut der Max-Planck-Gesellschaft in Berlin, Germany. We are greatly indebted to the technical and scientific staff at all three institutions. In particular, we thank all the students, postdocs, senior scientists, and research technicians who have been involved in this work, without whom these experiments would not have been possible. © 2009 by Taylor and Francis Group, LLC
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65. Crompvoets, F.M.H., Bethlem, H.L., Küpper, J., van Roij, A.J.A., and Meijer, G., Dynamics of neutral molecules stored in a ring, Phys. Rev. A, 69, 063406, 2004. 66. Heiner, C.E., Carty, D., Meijer, G., and Bethlem, H.L., A molecular synchrotron, Nature Phys., 3, 115–118, 2007. 67. Bohn, J.L., Inelastic collisions of ultracold polar molecules, Phys. Rev. A, 63, 052714, 2001. 68. Bethlem, H.L., van Veldhoven, J., Schnell, M., and Meijer, G., Trapping polar molecules in an ac trap, Phys. Rev. A, 74, 063403, 2006. 69. van Veldhoven, J., Bethlem, H.L., and Meijer, G., AC electric trap for ground-state molecules, Phys. Rev. Lett., 94, 083001, 2005. 70. Schlunk, S., Marian, A., Geng, P., Mosk, A.P., Meijer, G., and Schöllkopf, W., Trapping of Rb atoms by ac electric fields, Phys. Rev. Lett., 98, 223002, 2007. 71. van Veldhoven, J., Jongma, R.T., Sartakov, B., Bongers, W.A., and G. Meijer, Hyperfine structure of ND3 , Phys. Rev. A, 66, 032501, 2002. 72. Hoekstra, S., Gilijamse, J.J., Sartakov, B., Vanhaecke, N., Scharfenberg, L., van de Meerakker, S.Y.T., and Meijer, G., Optical pumping of trapped neutral molecules by blackbody radiation, Phys. Rev. Lett., 98, 133001, 2007. 73. Hudson, J.J., Sauer, B.E., Tarbutt, M.R., and Hinds, E.A., Measurement of the electron electric dipole moment using YbF molecules, Phys. Rev. Lett., 89, 023003, 2002. 74. Daussy, C., Marrel, T., Amy-Klein, A., Nguyen, C.T., Bordé, C.J., and Chardonnet, C., Limit on the parity nonconserving energy difference between the enantiomers of a chiral molecule by laser spectroscopy, Phys. Rev. Lett., 83, 1554–1557, 1999. 75. Macdonald, R. and Liu, K., State-to-state integral cross sections for the inelastic scattering of CH(X 2 Π) + He: Rotational rainbow and orbital alignment, J. Chem. Phys., 91, 821–838, 1989. 76. van der Loo, M. and Groenenboom, G., Theoretical transition probabilities for the OH Meinel system, J. Chem. Phys., 126, 114314, 2007. 77. Gilijamse, J.J., Hoekstra, S., Meek, S.A., Metsälä, M., van de Meerakker, S.Y.T., Meijer, G., and Groenenboom, G.C., The radiative lifetime of metastable CO (a3 Π, v = 0), J. Chem. Phys., 127, 221102, 2007. 78. James, T., Transition moments, Franck-Condon factors, and lifetimes of forbidden transitions. Calculation of the intensity of the Cameron system of CO, J. Chem. Phys., 55, 4118–4124, 1971. 79. Hudson, E.R., Ticknor, C., Sawyer, B.C., Taatjes, C.A., Lewandowski, H.J., Bochinski, J.R., Bohn, J.L., and Ye, J., Production of cold formaldehyde molecules for study and control of chemical reaction dynamics with hydroxyl radicals, Phys. Rev. A, 73, 063404, 2006. 80. van de Meerakker, S.Y.T., Labazan, I., Hoekstra, S., Küpper, J., and Meijer, G., Production and deceleration of a pulsed beam of metastable NH (a1 Δ) radicals, J. Phys. B, 39, S1077–S1084, 2006. 81. Hoekstra, S., Metsälä, M., Zieger, P.C., Scharfenberg, L., Gilijamse, J.J., Meijer, G., S.Y.T. and van de Meerakker, Electrostatic trapping of metastable NH molecules, Phys. Rev. A, 76, 063408, 2007. 82. Bochinski, J.R., Hudson, E.R., Lewandowski, H.J., Meijer, G., and Ye, J., Phase space manipulation of cold free radical OH molecules, Phys. Rev. Lett., 91, 243001, 2003. 83. Jung, S., Tiemann, E., and Lisdat, C., Cold atoms and molecules from fragmentation of decelerated SO2 , Phys. Rev. A, 74, 040701, 2006. 84. Tarbutt, M. R., private communications.
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85. Schnell, M., Lützow, P., van Veldhoven, J., Bethlem, H., Küpper, J., Friedrich, B., Schleier-Smith, M., Haak, H., and Meijer, G., A linear AC trap for polar molecules in their ground state, J. Phys. Chem. A, 111, 7411–7419, 2007. 86. van de Meerakker, S.Y.T., Jongma, R.T., Bethlem, H.L., and Meijer, G., Accumulating NH radicals in a magnetic trap, Phys. Rev. A, 64, 041401, 2001. 87. Baranov, M., Dobrek, L., Góral, K., Santos, L., and Lewenstein, M., Ultracold dipolar gases—a challenge for experiments and theory, Phys. Scr., T102, 74 –81, 2002. 88. van de Meerakker, S.Y.T., Bethlem, H.L., and Meijer, G., Taming molecular beams, Nature Physics, 4, 595, 2008.
© 2009 by Taylor and Francis Group, LLC
Part V Tests of Fundamental Laws
© 2009 by Taylor and Francis Group, LLC
Preparation and 15 Manipulation of Molecules for Fundamental Physics Tests Michael R. Tarbutt, Jony J. Hudson, Ben E. Sauer, and Edward A. Hinds CONTENTS 15.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Testing Invariance Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Do Fundamental Constants Vary in Time? . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Testing Fundamental Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Beams of Cold Polar Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Translational Temperature and Source Size . . . . . . . . . . . . . . . . . . . . . . . 15.3.3 Molecular Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.4 Rotational Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.5 Source Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Coherent Manipulation of Internal States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 Stark and Zeeman Shifts of the Hyperfine States . . . . . . . . . . . . . . . . . 15.4.2 Two-Pulse Interferometry of a Three-Level System . . . . . . . . . . . . . . 15.4.3 Experiments with Single Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.4 Experiments with Double Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Alternating Gradient Deceleration of Polar Molecules . . . . . . . . . . . . . . . . . . . . 15.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.2 A Model Alternating Gradient Decelerator . . . . . . . . . . . . . . . . . . . . . . . . 15.5.3 Axial Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.4 Transverse Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.5 Beyond the Ideal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Cold Molecules: Theory, Experiment, Applications
INTRODUCTION
Atoms and atomic ions have long held a place at the very heart of precision measurement and metrology. For example, the second is defined in terms of the Cs hyperfine structure interval, the Rydberg constant is measured by spectroscopy of atomic hydrogen, and the ratio of electron-to-proton mass is known from the oscillation frequencies of trapped atomic ions. The importance of atoms in this field lies partly in the detailed understanding that we have of atomic structure but also in the technical capabilities that exist for preparing and manipulating atoms and ions. In recent years, both the computational methods for understanding molecules more fully and the experimental methods for producing and controlling them have advanced enormously. This has led to a surge of interest in using molecules for precision measurements, especially where they offer new properties that are not available from atoms and atomic ions. For example, the rotational, vibrational, and electronic structures within a molecule offer a wider range of coexisting frequencies than one finds in atomic systems. Moreover, polar diatomic molecules have a built-in cylindrical symmetry, whilst more complex molecules can have a handedness—structural conformations that atoms cannot offer. In this chapter, we discuss some applications of molecules to current problems in precision measurement and we outline recent technical advances that make some of these applications possible.
15.2 TESTING INVARIANCE PRINCIPLES 15.2.1
DO FUNDAMENTAL CONSTANTS VARY IN TIME?
Recent measurements have shown that the expansion of the universe is accelerating, requiring the Einstein equations of cosmology to have a “dark energy” term, which was previously assumed to be zero [1]. This surprise, together with the ongoing search for a quantum theory of gravity, has led theorists to question some of the most basic assumptions in our physical model of the universe, including the assumption that the constants of nature are indeed constant over time. In atomic and molecular physics, two of these constants are particularly important. They are the fine structure constant α, and the electron-to-proton mass ratio μ = me /mp . The Rydberg energy Ry sets the gross scale of electronic binding energy. Relative to this, the fine structure splittings are characteristically smaller by the factor α2 and the hyperfine structure is smaller again by a further factor of order μ, because μ relates the magnetic moment of the nucleus to that of the electron. Consequently, it is possible to search for a variation of α by comparing fine and gross structure at two different times. Similarly the variation of μ can be deduced from a further comparison of the hyperfine structure with one of the other energy scales. Molecules bring an important new dimension √ to this search [2] by adding two more energy scales: vibrational energy of order Ry me /M and rotational energy of order Ry(me /M), where M is the reduced nuclear mass. Uzan [3] has recently reviewed both the theoretical framework and the experimental tests for variable constants. So far, the most productive experimental method has been to study astronomical spectra, which permit measurements of Δα/α and Δμ/μ, typically with a precision of 1 part in 105 , over enormous time intervals of order 10 Gyr, giving uncertainties of © 2009 by Taylor and Francis Group, LLC
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order 10−15 /year in the average value of α/α ˙ or μ/μ. ˙ At this level, some observations seem to hint at a variation [4,5], while others do not [6,7]. In order to interpret the astronomical data one needs to know the sensitivity of each transition to α and μ, which requires input both from laboratory data and from numerical modeling. Using H2 in this way, Reinhold and colleagues [5] have found an average variation over 12 Gyr of μ/μ ˙ = (1.7 ± 0.5) × 10−15 /year, which differs from zero by 3.4σ. The ground-state Λ-doublet of OH has also been observed at large redshift. The transition frequency for this interval relative to its hyperfine structure yields information on both α and μ [7]. When compared with the structure of OH today, these astrophysical measurements looking back 6.5 Gyr indicate with 2σ confidence that μ/μ ˙ < 2.1 × 10−15 /year on average and is consistent with zero. A laboratory experiment using a molecular decelerator to make slow OH [8] (there is more on deceleration in Section 15.5) has recently improved our knowledge of these transition frequencies in the current epoch. This will allow further improvement in the astronomical data to yield an even better determination of μ ˙ and α˙ in the near future. Another frequency with high sensitivity to μ is the inversion splitting of NH3 due to tunneling. Analysis of quasar absorption spectra, comparing the inversion line to rotational transitions, has placed a limit μ/μ ˙ = (−1 ± 3) × 10−16 yr −1 [10] on the variation of μ. This leaves the astronomical measurements in an intriguing but uncertain state. Are there systematic errors yet to be uncovered? Do the constants vary or not, and if they do, is the variation irregular in time or perhaps nonuniform in space? There is a good prospect that laboratory measurements can help to clarify the situation. Although the time intervals available for comparison are short—years rather than gigayears—this method provides a promising alternative because recently developed frequency comb methods can link optical frequencies directly to the Cs standard, giving an absolute frequency accuracy below one part in 1015 [11]. Even higher accuracy, approaching one part in 1017 is becoming available in measuring relative frequencies [12]. Several molecular experiments of this type are now under way. It has been + suggested [13] that the cold, trapped molecular ions H+ 2 or HD could be used to 15 measure changes in μ to a precision of 1 part in 10 . A second laboratory-based proposal [14] is to compare the inversion splitting measured in a slow, cold fountain of ND3 to an atomic reference. A third project is in progress to measure vibrational transitions in SF6 [15].
15.2.2 TESTING FUNDAMENTAL SYMMETRIES The electromagnetic forces that bind atoms and molecules together obey Maxwell’s equations and the Dirac equation, as synthesised in quantum electrodynamics. This field theory has three important symmetry properties: it is invariant under space inversion (parity, P), charge conjugation (exchange of particles and antiparticles, C), and time reversal (T). These symmetries have profound experimental consequences. For example, the eigenstates of atoms and molecules have definite parity (unless there are degenerate conformations), leading to selection rules for radiative transitions. For similar reasons, atoms and molecules cannot have a permanent electric dipole moment (EDM) (barring degenerate conformations). For example, the well-known EDM of ammonia is not permanent because the nitrogen atom tunnels back and forth at a © 2009 by Taylor and Francis Group, LLC
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frequency set by the splitting of the two opposite-parity field-free states. Of course, this splitting is small, and therefore it takes only a modest electric field to induce a dipole moment. For some years it was surmised that all interactions possess these symmetries, but an experiment in 1956 showed that weak interactions violate parity, as seen by the fact that radioactive decay particles are emitted with a large left–right asymmetry [16]. Within a decade, an experiment on the decay of kaon particles showed that strong interactions are also not symmetrical, having a small asymmetry under the combined operation CP [17]. There is a theorem for the type of theories used to describe particle interactions (local, Lorenz-invariant field theories) that they must be invariant under the triple reflection CPT. Since CP symmetry is broken, this theorem seems to indicate that T symmetry is also broken at the same level. These symmetry (and asymmetry) properties have played an important role in developing the standard model of particle interactions, which describes electromagnetic, weak, and strong interactions. The discovery of CP-violation led to a fascinating question. Is it possible that particles, atoms, and molecules do have permanent electric dipole moments after all? This would require interactions that violate both P and T, but we know that the weak and strong interactions together can do that. As it happens, the standard model, with its standard P and T violation, predicts exceedingly small EDM values. This is the result of a fortuitous cancellation, which comes about from the simplicity of the standard model. The pressing issue today is to discover what lies beyond the standard model. In order to understand more clearly the origin of mass and in order to accommodate a quantum theory of gravity, it seems very likely that there are more particles than the standard collection. Such an increase in complexity immediately leads to the prediction that particles, atoms, and molecules have EDMs much larger than the standard model values. They are still very small, but they are no longer too small to measure. Therefore, the search for a permanent electric dipole moment of an atom or molecule is really the search for particle physics beyond the standard model. Molecules are beginning to play a central part in this search. The EDM de,p,n σ of an electron, proton, or neutron is neccessarily aligned along the spin direction σ of the particle. In essence, an EDM measurement in an atom or molecule involves polarizing the system with an applied external electric field and between the electronic or nuclear EDM and the searching for the interaction ηdx σ · E polarized atom/molecule. Schiff’s theorem [18] states that η = 0 if the atom/molecule is made of point particles bound by electrostatic forces. In other words, the electronic or nuclear EDM does not see the applied field because it is shielded out by the other charged particles. This theorem is important for its loopholes: nuclei are not point particles and the electric dipole interaction is not screened when the electrons are relativistic. Consequently, η is not zero if the atom/molecule is well chosen [19,20]. For example the best measurement of the proton EDM comes from a measurement on TlF molecules [21], where the large size of the Tl nucleus ends up giving η ∼ 1 for the nuclear spin EDM interaction. The upper limit on the neutron EDM is known both directly, from measurements on free neutrons [22], and indirectly from nuclear spin measurements on Hg atoms [23]. For electron EDM measurements, as opposed to neutron or proton measurements, η can be much larger than 1 if the electron moves relativistically within the atom or © 2009 by Taylor and Francis Group, LLC
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molecule. For example, the Tl atom is very sensitive to the electron EDM de , with η = −585. Currently, the best limit on the electron EDM is derived from such an is linear in E because experiment on Tl atoms [24]. The effective interaction ηde σ · E the polarization of the atom is proportional to the applied electric field. This polarization is small for atoms in laboratory-strength fields because it derives from the mixing of higher electronic states induced by the field and these are typically 1015 Hz higher in energy. By contrast, a heavy polar molecule is polarized by mixing rotational states, which are typically only 10 GHz away, giving five orders of magnitude more polarizability and a correspondingly larger η. The polarization due to rotational mixing stops approaches increasing once the molecule is largely aligned with the field and then ηE a saturated value. For the YbF molecule, this value is an enormous 26 GV/cm [25]. A group at Imperial College, including the authors of this article, are currently in the process of measuring de using the YbF molecule [26]. In comparison with the Tl atom, this molecule gives roughly 500 times more EDM interaction energy, whereas the interaction with magnetic fields is essentially the same. Because stray magnetic fields constitute the primary source of systematic errors, this is a significant advantage. However, Tl beams have much better statistical noise because they are much more intense than YbF beams. At present the gain in sensitivity is roughly offset by the loss in signal, and the YbF experiment is taking data at a level of precision similar to the Tl experiment [26]. In the new era of high-precision molecular beam measurements that we are discussing here, the need for brighter, colder, slower sources is a recurring theme that we address again in the next section. Other molecular approaches to EDM measurement are also being pursued. The group of DeMille is aiming to measure the electron EDM in a metastable Ω-doublet of PbO using vapor in a cell [27]. The group of Cornell is investigating the possibility of using trapped molecular ions [28], among which HfF+ appears to be a promising candidate. There is also a proposal to make a dense sample of some radical such as YbF and to measure the magnetization induced by aligning the electron EDMs in an applied electric field [29]. We turn now to parity violation without T violation. This is very well understood in the standard model of particle physics as a normal feature of weak interactions. It is also well established in atomic physics through the measurements on Cs atoms [30]. However, it remains a fascinating topic in the context of molecular physics, partly because it has not been observed in molecules but mainly because chirality plays such an important role in chemistry. The weak interaction is predicted to alter the energy spectrum between enantiomers of chiral molecules. Indeed, it is still a subject of debate whether the parity violation of weak interactions has played any role in establishing the chirality of the biochemistry in living organisms [31,32]. The most intensively studied species are the methyl halides, CH–XYZ, where X, Y, Z are three different halogens [33]. The largest effect is predicted to be a 50.8 mHz shift in the C–F stretching mode between left- and right-handed versions of CHFBrI. By contrast, the best experimental results [34] reach a precision of 50 Hz. It appears that cold trapped molecules will be necessary to measure weak interactions in these systems. Much larger enantiomer shifts, in the range of several Hz, have recently been predicted for some rhenium and osmium complexes [35] and these may be observable using supersonic beams. © 2009 by Taylor and Francis Group, LLC
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There is also nuclear physics interest in parity violation because it plays a role in nuclear structure. A particularly interesting possibility is that weak interactions in nuclei can induce an anapole moment, a P-odd multipole that produces no external field and corresponds in lowest order to a toroidal flow of current within the nucleus. An experiment is under way at Yale University to measure the anapole moment of the 137 Ba nucleus using BaF molecules [36]. The last symmetry we mention here is Lorentz invariance, which has been a central plank of twentieth-century physics and has so far shown no indication of being violated in nature. Even so, it is possible that new physics, associated with quantum gravity at the Planck energy scale, could lead to very small violations of Lorentz invariance in the laboratory [37]. For example, there might be a change in the energy of an atom depending on the orientation of its spin relative to some preferred direction, such as its velocity in the rest frame of the universe. Very sensitive experiments of this sort have already been performed using a variety of atomic clocks [38]. More recently, it has been pointed out that diatomic molecules provide a new way to investigate Lorentz invariance by orienting the internuclear axis of the molecule relative to the proposed preferred direction. The symmetry violation could then be read out as a shift in the energy, bond length, vibration frequency, or rotation frequency [39]. Sensitivities for H2 and HD and their cations have been calculated in Ref. [39]. The authors conclude that an experiment to measure the ground-state rovibrational transitions can improve limits on some elements of the Lorentz tensor cμν by an order of magnitude [37,40]. It is very likely that other more polar molecules could be convenient to use and that this area will develop as techniques for preparing, cooling, and trapping molecules progress further. In the rest of this chapter, we discuss the production and detection of intense molecular beams, particularly of heavy polar radicals. We also describe methods of using pulsed molecular beams to map fields in the beamline and to detect interactions through the quantum coherences. Finally, we describe advances in slowing heavy polar molecules with a view to trapping them. These rather practical issues are the key points to be addressed if molecules are to fulfil the fantastic promise that we have just outlined for elucidating exotic fundamental physics.
15.3
BEAMS OF COLD POLAR RADICALS
All molecular beams must begin with a source. A good source will provide a large number of molecules in the quantum state of interest for the experiment. In most cases, the molecules need to be prepared in a single low-lying rotational state, and so the temperature of the molecules should ideally be smaller than the rotational energy spacing, typically 1 K or less. Pulsed sources of cold molecules, very narrowly distributed in both position and velocity, offer many advantages. They can be prepared with very high intensity without imposing an excessive gas load on the vacuum system, they can be used to map the electromagnetic field along the beamline with high resolution and precision, they allow quantum coherences to be prepared and manipulated with great accuracy, and they can be slowed down in a Stark decelerator to increase the coherence times. In this section, we concentrate on the formation and detection of cold, pulsed beams of the radical © 2009 by Taylor and Francis Group, LLC
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molecules that are typically required for measuring the P and T violating interactions discussed above. Supersonic expansion is a very common technique for producing cold molecular beams [41,42]. A high-pressure gas expands through a nozzle into a vacuum chamber, acquiring a high center-of-mass velocity but very narrow velocity distribution. The translational degrees of freedom are cooled, as are the internal degrees of freedom of the molecules. While the first beams were continuous, the method was later extended by using pulsed valves with short opening times [43,44]. The individual pulses could then be made very intense, without the gas load becoming excessive. Often, the molecules of interest have low vapor pressure, and need to be formed by laser ablation or electric discharge techniques. This is usually done immediately outside the nozzle of the pulsed valve, or inside an extended nozzle, where the density of carrier gas is high enough to entrain a useful fraction of the molecules produced [45]. In our laboratory, we have used laser ablation to produce cold beams of YbF [46], CaF and LiH molecules [47]. In all cases, we detect our pulsed beams using time-resolved, Doppler-free, laser-induced fluorescence (LIF). This detection method is very well suited for precision measurements where high-sensitivity, high-frequency resolution and good beam diagnostics are all required.
15.3.1 APPARATUS The typical experimental setup is illustrated in Figure 15.1. A solenoid valve emits short pulses of the carrier gas [48], which is usually Ar, Kr, or Xe, into a vacuum chamber maintained at pressures below 10−4 mbar. The central part of the gas pulse passes through a skimmer, of diameter 1 to 2 mm, placed 50 to 100 mm downstream of the source, and into the high-vacuum region where the pressure is below 10−7 mbar. A fast ionization gauge placed on the axis of the beamline can be used to ensure good alignment. Using two such gauges, one immediately outside the nozzle and the other much further downstream, the initial width, the speed, and the translational Plane of photodetector x ƒz y Ablation laser
Light-collecting lenses
Skimmer Solenoid valve Target
Light-collecting mirror
Probe laser Molecular beam
FIGURE 15.1 Typical experimental setup for producing pulsed beams of cold polar radicals, and detecting them by laser induced fluorescence. (Not to scale.)
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temperature of the gas pulses can all be measured. The shortest gas pulses we obtain with this valve have a full-width at half-maximum (FWHM) of 81 μs [46]. In an ideal supersonic expansion from a reservoir at temperature T0 , the carrier gas approaches a terminal velocity vT =
(2kB T0 /m)γ/(γ − 1),
(15.1)
where γ is the specific heat ratio, 5/3 for an ideal monatomic gas, and m is the mass of an atom of the carrier gas. Our measurements using a range of carrier gases and temperatures show that the true speed of the carrier gas is 5 to 15% faster than vT . A suitable target, placed outside the nozzle of the valve, is ablated using light from a Q-switched laser. To ensure minimal disruption of the gas pulse, the target is thin in the direction of the gas-jet, typically 2 to 5 mm. The resulting ablation plume can usually be observed by eye. It is forward peaked along the normal to the target, which is usually perpendicular to the direction of the beamline. Some of the atoms and molecules in the plume become entrained in the high-density carrier gas, thereby cooling to temperatures and speeds that approach those of the carrier gas. The target may either contain all the precursors needed to form the molecules of interest, or some of the precursors can be added to the carrier gas. For example, we have produced beams of YbF either by ablating pure Yb and mixing SF6 into the carrier gas, or by ablating a pressed target containing a mixture of Yb and AlF3 powders. The former method is slightly simpler to realize, but both methods work equally well. We typically use ablation pulses of 5 to 10 ns duration, focused to a spot size of 1 to 2 mm. Under these conditions, the optimal ablation energy is in the range 10 to 50 mJ. A photomultiplier, placed some distance (typically 10 to 150 cm) downstream, detects the molecules by means of cw laser-induced fluorescence. The laser is directed at right angles to the molecular beam in order to minimize the Doppler shift and is tuned to a strong molecular transition. The transit time of the molecules through the laser beam is typically 5 to 10 μs, giving this method high temporal resolution. The width of the fluorescence excitation spectrum is usually greater than the transit time limit, being typically 20 to 50 MHz. This comes from a combination of the residual Doppler width, due to the angular distribution of the molecules, and the natural lifetime of the excited state. Such high spectral resolution is typically needed in order to resolve the hyperfine structure and hence to permit coherent state manipulation and readout at radio frequencies. The fluorescence detection method is very sensitive. If the detection efficiency is (assumed 1) and there are N molecules per shot passing through the detector area in a time interval w , then √ the shot-noise-limited signal-to-noise ratio for a single shot will be s : n = N/ bw + N, where b is the background rate of detected photons. In a good detection setup, bw ≈ 1 and is in the range 1 to 10%. Taking = 0.02 as a typical value, we find that the signal-to-noise ratio per shot is 1 when N = 81 molecules.
15.3.2 TRANSLATIONAL TEMPERATURE AND SOURCE SIZE Figure 15.2 shows the time-of-flight (ToF) profiles of ground-state YbF molecules recorded by two separate LIF detectors placed 340 and 1300 mm from the pulsed © 2009 by Taylor and Francis Group, LLC
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6 5
Signal
4 3 2 1 0
0.5
1
1.5
2
2.5
Time of flight (ms)
FIGURE 15.2 Time of flight profiles of ground-state YbF recorded by two laser-induced fluorescence detectors, situated 340 and 1300 mm from the source. The lines are Gaussian fits to the two datasets.
source. The background laser scatter is lower at the downstream detector. In this experiment, the molecules were produced by ablating pure Yb just outside the nozzle of the solenoid valve, which was pressurized to 4 bar with a mixture of Ar (98%) and SF6 (2%). The probe laser was tuned to excite the F = 1 hyperfine component of the X 2 Σ+ (v = 0) − A2 Π1/2 (v = 0) Q(0) transition. There are four contributions to the spread of arrival times: (1) the temporal spread of the molecules produced at the source, (2) their spatial spread at the source, (3) the forward velocity distribution in the pulse, and (4) the temporal resolution of the detector. The last of these is usually small enough to be neglected. The flux of molecules with speeds in the interval from v to v + dv is usually taken to be f (v) dv = Av 3 exp(−M(v − v0 )2 /2kB T ) dv, where M is the mass, T is the translational temperature, v0 is the central velocity, and A is a normalizing constant. Consider those molecules born in the source at time ts with initial position s along the beam axis. In the LIF detector, placed a distance L away from the source, these molecules produce a time-dependent signal −Mv02 (t0 − s/v0 − t + ts )2 A(L − s)4 h(t, ts , s) = , exp 2kB T (t − ts )5 (t − ts )2
(15.2)
where t0 = L/v0 . As we shall soon see, both the temporal width and spatial width of the source are small, ts t0 and s L. Furthermore, at typical detection distances, the range of arrival times is much smaller than the mean arrival time, and it is valid to set t ≈ t0 everywhere except in the numerator of the exponent. With these approximations, Equation 15.2 simplifies to L4 −4 ln 2(t − t0 − ts − ρ)2 h(t, ts , ρ) ≈ A 5 exp , w2 t0 © 2009 by Taylor and Francis Group, LLC
(15.3)
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1/2 where ρ = −s/v0 , and w = 8 ln 2 kB Tt02 / Mv02 is the temporal width (FWHM) of the pulse due to the thermal spread of forward velocities. The signal at the detector is obtained by integrating over the temporal and spatial distributions present in the source. We do not know these distributions, but we can hope to obtain a measure of their characteristic widths. In this spirit, we assume that the source emits the normalized distribution −4 ln 2 ρ2 4 ln 2 −4 ln 2 ts2 g(ts , ρ) = exp . (15.4) exp πΔts Δρ Δ2ρ Δ2ts Then, the detector records the signal h(t) =
L4 w (t − t0 )2 h(t, ts , ρ) g(ts , ρ) dts dρ = A 5 exp −4 ln 2 , w2 t0 w
(15.5)
where the pulse width w2 = w2 + Δ2ts + Δ2ρ includes the broadening due to the distribution of positions and times where molecules are first formed. As shown by the Gaussian fits in Figure 15.2, the recorded profiles are well described by the model except in the high-velocity tails. The long tail, indicating a hotter component in the beam, is a common feature of our source when optimized for maximum signal and minimum shot-to-shot fluctuation; it can be removed by reoptimizing for low temperature. By using two well-separated detectors, the translational temperature can be obtained: T=
Mv02 w22 − w12 , 8 ln 2 kB t22 − t12
(15.6)
where w1 , w2 , t1 , and t2 are the widths (FWHM) and central arrival times obtained from Gaussian fits to the downstream and upstream data. For the data shown, the speed is v0 = 586 m/s and the temperature is T = 4.8 K. The use of two detectors allows an unambiguous determination of the temperature, whereas a single detector alone sets only an upper limit. However, if the detector is far from the source, that upper limit can be very close to the true temperature. For this data, the downstream detector alone provides an upper limit that is just 1.3% higher than the measured 4.8 K. Having measured the temperature, we can also extract from the data upper limits on the initial temporal and spatial spreads at the source. For the data shown, these are Δts < 14.5 μs and v0 Δρ < 8.5 mm. In separate experiments, using a dual ablation technique, we have measured Δts ≈ 5 μs [47].
15.3.3
MOLECULAR FLUX
We consider next how to determine the absolute flux of molecules from the LIF signal. The detector counts the flux of photons, and we wish to convert this to a flux of molecules by knowing the mean number of fluorescent photons emitted from each © 2009 by Taylor and Francis Group, LLC
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molecule. To find this, we model the molecule as a three-level system and use rate equations. Every molecule starts out in level 1 and passes through the laser, which excites the resonance between levels 1 and 2. The excitation rate is R, as is the rate of stimulated emission. It is directly proportional to the laser intensity I, and depends on the detuning δ = ωL − ω12 of the laser angular frequency, ωL , from the molecular resonance frequency, ω12 . Taking the optical Bloch equations in the limit where the coherences have reached a steady state, one finds that R=
Γ/2 I , 2 2 (1 + 4δ /Γ ) Is
(15.7)
where Γ is the spontaneous decay rate of level 2, Is = 0 c2 (Γ/2)2 /D2 is the saturation intensity, and D is the matrix element of the dipole operator connecting levels 1 and 2. Level 1 is stable, while level 2 decays with rate rΓ to level 1, and rate (1 − r)Γ to level 3, which represents all the other states in the molecule. The rates for excitation or decay out of level 3 are negligible. We solve the rate equations to find the number of molecules in level 2 as a function of time, N2 (t). Integrating ΓN2 (t) over the laser–molecule interaction time τ, we find the number of fluorescent photons emitted per molecule to be RΓ np = R+ − R −
e−R+ τ − 1 e−R− τ − 1 , − R+ R−
where R± = R + Γ/2 ±
R2 + r R Γ + Γ2 /4.
(15.8)
(15.9)
In the limit where R+ τ 1 and R− τ 1, np acquires its asymptotic value n4 p,max = 1/(1 − r); this is simply the sum of the obvious geometric series, np,max = ∞ N N=0 r . For good detection efficiency, we would like to ensure that np reaches its maximum possible value, and so we should examine how easily this limit is reached. If, as is usual, level 2 is an electronically excited state with an allowed electric dipole transition, Γ will typically exceed 107 s−1 . The interaction time τ is usually greater than 1 μs, and so we are in the limit Γτ 1. If we suppose that the laser excitation is weak, in the sense that R Γ (or, equivalently, I Is ), we get np =
1 − e−R(1−r)τ 1−r
(Γτ 1, R Γ).
(15.10)
The limit np = 1/(1 − r) is reached once R(1 − r)τ 1, at which point the fluorescence signal saturates. Note that it is usual for Γτ to be greater than 100, and that for molecules, it is rare to have r close to 1. It follows that the above “saturation” condition can easily be met, even when I Is —the fluorescence signal saturates for laser intensities well below Is because the interaction time is very long compared to the time required for the molecule to reach level 3, the dark state. Figure 15.3 shows the value of np as a function of I/Is for four different values of Γτ with r = 0.5. The maximum value of np is 2, and in the case of Γτ = 100, np is within © 2009 by Taylor and Francis Group, LLC
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Photons per molecule
2.0
100 50 20
1.5
2
10 100
1.0 1
50 20
0.5
10 –4
0.0
0.0
0.5
–2
1.0 I/Is
2
1.5
4
d/G
2.0
FIGURE 15.3 Number of fluorescent photons per molecule, np , when the branching ratio for returning to the initial state is r = 0.5. In the main graph, np is plotted vs. the intensity I/Is for Γ τ = 10, 20, 50, 100. The inset shows for each case how np varies with the laser detuning δ/Γ, when I = Is /3.
1% of this limit at I = Is /4. At lower values of Γτ, the asymptotic value is lowered and more intensity is required to reach it. For example, when Γτ = 10, np has an asymptotic value of 1.84 and is 78% of this value when I = Is . The saturation process described here also leads to a type of power broadening of the spectral line when the laser frequency is scanned. An increase in laser intensity increases the fluorescence in the wings of the resonance more than at the center of the resonance, and so the line is broadened. This is seen in the inset to Figure 15.3, where the lineshape is plotted for the various values of Γτ, in the case where r = 0.5 and I = Is /3. If the decay rate of the excited state happens to be small, it is possible that the system will be in the opposite limit, Γτ 1. In that case, even if R Γ, the mean number of photons scattered by each molecule will be much smaller than 1, and independent of r. We cannot then use Equation 15.8 because the damping has no time to act. However, Equation 15.20, which has no damping, can be integrated to give √ Γτ sin(Γτ I/(2Is )) np = 1− (Γτ 1, R Γ). (15.11) √ 2 Γτ I/(2Is ) The fluorescence saturates to the value Γτ/2 at high laser intensities. Finally, it is interesting to consider the case where the interaction time is long (Rτ 1, Γτ 1) and r is very close to 1 so that each molecule has the possibility of scattering a large number of photons. Then, Equation 15.8 reduces to the familiar atomic physics result, np =
RΓτ Γτ I/Is . = 2R + Γ 2 1 + I/Is + 4δ2 /Γ2
(15.12)
The on-resonance fluorescence now saturates to Γτ/2 when the condition I Is is met. © 2009 by Taylor and Francis Group, LLC
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Returning to the determination of the molecular flux, we write the total number of photons detected per shot as 1 N(x, z)(x, z)np (x, z) dx dz, p= 2 (15.13) L where N(x, z), (x, z), and np (x, z) are the number of molecules per steradian in the quantum state of interest, the detection efficiency and the number of fluorescent photons per molecule, all in the interval dx dz around the point (x, z) in the plane of detection. To determine the molecular flux as accurately as possible, the detection area should be defined by placing a small aperture on the molecular beam axis, slightly upstream of the detector, to block all but a small portion of the molecular beam. Similarly, the probe laser should be collimated and prepared with a top-hat intensity distribution in the x direction. Then N and will be constant over the detection area. The value of np will be independent of x, but because of the Doppler shift, it remains a sensitive function of z, particularly if the degree of saturation is not high. Under these conditions, Equation 15.13 reduces to HN p= (15.14) np (θ) dθ, L where θ = z/L and H is the height (along x) of the detection area. The integral is straightforward to calculate by substituting the Doppler shift δ = 2πv0 θ/λ into Equation 15.7 and then using Equations 15.8 and 15.9 (the probe laser, of wavelength λ, is assumed to be on resonance). The calculation requires some knowledge of r, Γ, and Is , but provided the detection is in the saturated regime, rough estimates will suffice because the result becomes rather insensitive to Γ and Is , and also to r if r is small, as is often the case. For the detection setup shown in Figure 15.1, the total detection efficiency is
= (Ωl /4π) qi (1 + R(λi ))Tl (λi )2 χ(λi ) . (15.15) i
Here, qi is the fraction of fluorescent photons in the emission line whose wavelength is λi ; R, Tl , and χ are the wavelength-dependent mirror reflectivity, lens transmission, and photodetector quantum efficiency; Ωl is the solid angle subtended by the light-gathering lens. If windows and filters are present in the setup, their transmissions need to be included too. By measuring the value of p and performing the above calculations, the flux of molecules in the detected quantum state can be determined. With careful measurements, an uncertainty below 50% should be possible. For the cold YbF molecules produced in our laboratory, the flux is measured to be 1.4 × 109 ground-state molecules per steradian per shot, when the carrier gas is argon [46].
15.3.4
ROTATIONAL TEMPERATURE
The rotational temperature of the molecules can be determined by scanning the laser frequency and recording the rotational spectrum. When the rotational temperature is © 2009 by Taylor and Francis Group, LLC
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Tr , the intensity of a rotational line in the spectrum is proportional to N(J)(J)np (J), where N(J) = (2J + 1) exp[−BJ(J + 1)/kB Tr ] is the relative number of molecules in the rotational state J, B is the rotational constant, and (J), np (J) express the J-dependence of the detection efficiency and the number of scattered photons per molecule. Usually, the efficiency is almost independent of J. In many cases this is also true of np , making it easy to extract the rotational temperature from the relative line intensities once a few rotational lines have been measured. A more accurate temperature determination must take into account the variation of the matrix elements with J. For example, in 1 Σ −1 Σ transitions, the MJ -averaged value of D2 in the R-lines is proportional to (J + 1)/(2J + 1), which decreases from 1 to 1/2 as J goes from 0 to ∞. For the P-lines, by contrast, D2 ∝ J/(2J + 1), which increases from 0 to 1/2. Since this variation of the matrix elements generally affects both the excitation rate R and the branching ratio r, it influences the value of np (J) in both the saturated and unsaturated regimes. In the saturated regime, where np ! 1/(1 − r), the variation of np with J is strongest if r happens to be close to 1. In our laboratory, we have measured rotational temperatures of YbF, CaF, and LiH beams. For the first two, the rotational temperatures are usually very close to the translational temperatures (typically in the range 1 to 5 K) [46]. For LiH we measure rotational temperatures considerably higher than the translational temperature [47].
15.3.5
SOURCE NOISE
The flux of molecules obtained from these sources is subject to shot-to-shot fluctuations, as well as having a slow drift (mostly downward). The timescale for the slow drift is typically 104 to 105 shots on a given spot of the target. We usually attach the target to the rim of a large disk, typically 20 cm in diameter, which we rotate incrementally as each target spot becomes exhausted. In this way, the lifetime of a target is of the order of 107 shots. The ability to run the source continuously for long periods of time is very important for precision measurements. A second essential requirement is that the shot-to-shot fluctuations be small, because they can contribute directly to the noise in the experiment. In measuring the electron electric dipole moment using coldYbF molecules, √ our detector records approximately 3000 photons per shot, with a corresponding N photon shot-noise limit of 2%. Source fluctuations should ideally be kept below that level, but this proves rather difficult to achieve; when optimized, the short-timescale fluctuations of our source are typically 2 to 3%.
15.4 15.4.1
COHERENT MANIPULATION OF INTERNAL STATES STARK AND ZEEMAN SHIFTS OF THE HYPERFINE STATES
In Section 15.3.3, we discussed the excitation of higher electronic states using laser light to drive optical dipole transitions. The interaction was strongly damped by spontaneous emission from the upper level; indeed, the molecules were detected by means of the scattered photons. In this section, we consider some ways to manipulate the hyperfine sublevels within the ground state of the molecule. In contrast to optical transitions, the coherences between these ground-state levels are not radiatively damped, © 2009 by Taylor and Francis Group, LLC
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because the spontaneous transition rates are very low for transition frequencies in the sub-gigahertz range. The control of these coherences provides a basis for exceedingly high-precision measurements of electric and magnetic fields and for measurements such as that of the electron EDM. In order to be concrete in our discussion, let us take the simple case illustrated in Figure 15.4, of two hyperfine levels, F = 1 and F = 0, such as one finds in a diatomic X 2 Σ(N = 0) molecule with nuclear spins of 0 and 1/2. Here the total angular momentum F is the sum of the electronic angular momentum J = 1/2 and the nuclear spin I = 1/2. Two examples of such molecules, which have been studied in our laboratory, are 174YbF and CaF. In the absence of any external fields, the two hyperfine levels are separated by the hyperfine splitting A, as illustrated on the left of Figure 15.4, and the three magnetic sublevels of F = 1 are degenerate (A = 2π × 123 MHz in CaF and 2π × 170 MHz in YbF). For a fuller discussion of hyperfine structure in such molecules, see Ref. [49]. When the molecule is subjected to an electric field Ez , the main effect is the rigid rotor Stark shift due to the electric dipole moment μe along the internuclear axis of the molecule, as we discuss more fully in Section15.5. This shift is large in comparison with the hyperfine interaction; for example the N = 0 state of YbF shifts downwards by 20 GHz at 20 kV/cm. To a good approximation, all four hyperfine levels shift together, but there are some small differential shifts as well, due mainly to the tensor part of the hyperfine interaction between J and I, the electronic and nuclear angular momenta. This effect is analyzed in detail in Ref. [49]. Relative to the F = 0 level, the state (F, MF ) = (1, 0) shifts up by Δ0 whereas states (1, ±1) shift up by Δ1 , as illustrated in the center of Figure 15.4. These shifts are plotted in Figure 15.5 versus electric field for the particular case of YbF. The lines are a calculation following the theory detailed in Ref. [49], which has been confirmed by experiment. One sees that these shifts of the hyperfine frequencies within the N = 0 manifold are typically a thousand times smaller than the overall shift of the manifold itself. The general behavior is a quadratic shift at low electric fields, where the characteristic dipole interaction −μe · E is small compared with the
(1, ±1)
F=1
2Dz
(1,0)
A + D0
(1, –1) (1, 0)
A + D1 A
(1, +1)
A + D1 – Dz
(0, 0) F=0
FIGURE 15.4 Hyperfine levels. Left: field-free levels F = 0, 1. Centre: electric-field-induced shift of levels and tensor Stark splitting of the triplet F = 1. Right: Zeeman splitting of the doublet F = 1, MF = ±1.
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185 A + D1 180
175
170
A + D0
0
5
10 15 Electric field (kV/cm)
20
FIGURE 15.5 Stark shift of the ground-state hyperfine intervals in YbF.
rotational constant B, evolving closer to linear behavior at higher fields and ultimately saturating at the highest fields. It is worth noting that the total Hamiltonian (including the coupling to the external electric field) is invariant under time reversal. Since the two states {(1, +1) in field E} and {(1, −1) in field E} are time-reverses of each other and because the Hamiltonian is invariant under time reversal, it follows that the energy levels of (1, ±1) are exactly degenerate at all electric fields. Thus, even though the molecule has an electric dipole moment along its internuclear axis, and has an induced electric dipole moment along the applied field direction z, it does not have an electric dipole moment proportional to Fz . This is a direct consequence of time-reversal symmetry. By contrast, if the electron were to have a permanent EDM de along its spin, this would lift the degeneracy between the two levels—a direct consequence of the fact that such an EDM violates time-reversal symmetry. This spin-dependent Stark shift is what our group measures in searching for an electron EDM using YbF molecules [26]. A more mundane way to split the (1, ±1) levels, as illustrated on the right of Figure 15.4, is with a static magnetic field B through the Zeeman interaction gF μB F · B, where μB is the Bohr magneton, and the g-factor expresses the ratio of magnetic moment to total angular momentum. Provided this interaction is small compared with the tensor Stark splitting Δ1 − Δ0 , the molecule only responds to the component Bz parallel (or antiparallel) to the electric field. This splits the levels by ±Δz = ±gF μB Bz /. The additional contribution to this splitting resulting from a perpendicular field component B⊥ is of order Δz [μB B⊥ /(Δ1 − Δ0 )]2 and is therefore negligible [26]. No violation of time-reversal symmetry is implied here because the state {(1, −1) in field B} is not the time-reverse of {(1, +1) in field B}: time-reversal also reverses the sign of B. There is no shift (to first order) of the states (F, MF ) = (0, 0) and (1, 0) because they have < Fz > = 0. To summarize, the mean shift of the two levels (1, ±1) relative to level (0, 0) is a measure of the electric field strength, whereas the splitting between the two measures the parallel component of the magnetic field (and a possible very small contribution from the electron EDM). Although we have chosen to illustrate this with a simple © 2009 by Taylor and Francis Group, LLC
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hyperfine system having F = 0 and F = 1, the same behavior applies more generally, namely that any two levels F, ±MF are equally shifted by the Stark interaction with electric field Ez , whereas they are split apart by a magnetic field along z (and by an EDM). In the coherent manipulations we discuss below, the (1, 0) level plays no significant role and we therefore do not consider it further. The remaining three levels can now be conveniently abbreviated to (0), (+1), and (−1), with energies 0, A + Δ1 + Δz , and A + Δ1 − Δz , respectively. In the next section, we will find it convenient to redefine the zero of energy.
15.4.2 TWO-PULSE INTERFEROMETRY OF A THREE-LEVEL SYSTEM Let us write the amplitudes of these three states as a column vector ⎛
⎞ a0 (t) az (t) = ⎝ a+1 (t) ⎠ , a−1 (t) z
(15.16)
where the subscript z indicates that the quantization axis is taken to be along the electric field. The free evolution of these amplitudes from time t1 to time t1 + τ in the presence of the static electric and magnetic fields is given by the propagator ⎛
Ω
ei 2 τ ⎜ Π0 (t1 , τ) = ⎝ 0 0
⎞ 0 ⎟ ⎠ , 0 Ω −i( 2 −Δz )τ e z
0 Ω e−i( 2 +Δz )τ 0
(15.17)
such that az (t1 + τ) = Π0 (t1 , τ)az (t1 ). Here the Stark-shifted hyperfine interval A + Δ1 has been replaced by the symbol Ω, and we have moved the zero of energy to the center of that interval in order to simplify the algebra that follows. Let us suppose that the molecules are prepared in state (0), then subjected to a radiofrequency magnetic field βx cos(ωt + φ) along x in order to drive transitions to states (+1) and (−1). This field excites the coherent superposition state (c) = √1 [(+1) + 2
(−1)] and does not couple at all to the orthogonal superposition (u) = √1 [(+1) − 2 (−1)]. (The converse is true for a radio-frequency (rf) field along y.) This suggests a new basis with quantization along the x-axis, in which the new state amplitudes are given by ⎛ ⎞ 1 a0 ⎜ ax = ⎝ ac ⎠ = Uaz = ⎝ 0 au x 0 ⎛
0
0
√1 2 √1 2
√1 2 − √1 2
⎞⎛
⎞ a0 ⎟⎝ a+1 ⎠ . ⎠ a−1 z
(15.18)
Note that the transformation U is its own inverse: U = U −1 . In the x-basis, only the states (0) and (c) are coupled by the rf field. With this reduction of the problem to a two-level problem, we can write down how the amplitudes evolve in the x-basis under the influence of the rf magnetic field applied from time t1 to time t1 + τ. © 2009 by Taylor and Francis Group, LLC
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Following the standard derivation leading to Equation V.7 of Ramsey’s book [50], we find that ⎞ ⎛ ω ω Zei 2 τ Wei 2 τ ei(ωt1 +φ) 0 ω ω ⎟ ⎜ 0 (15.19) ΠRF (t1 , τ) = ⎝ We−i 2 τ e−i(ωt1 +φ) Z ∗ e−i 2 τ ⎠ , Ω −i 2 τ 0 0 e x where Z = i cos Θ sin W = i sin Θ sin
aτ 2 aτ
+ cos
aτ 2
2 a = (Ω − ω)2 + 4b2
Ω−ω a 2b sin Θ = − a b = 0| − μx βx |c .
cos Θ =
Although the phase of the field φ is not important when a single pulse is applied, we keep it in this formula because it becomes relevant when we consider double pulses. As the state (c) is excited, the effect of any static magnetic field Bz will be to rotate it into the third state (u) at the Larmor frequency Δz . This has been ignored in deriving Equation 15.19, under the assumption that the rf excitation will be performed quickly in comparison with the Larmor precession. Taken together, Equations 15.17, 15.18, and 15.19 provide us with all the tools we need to investigate the evolution of this three-level system under any sequence of short rf pulses. When molecules in state (0) are subjected to a single rf pulse, the probability of excitation to state (c) is given by Equation 15.19 as P1 pulse (0 → c) = |W |2 =
4b2 2 τ 2 + 4b2 . sin (Ω − ω) 2 (Ω − ω)2 + 4b2 (15.20)
This is the usual magnetic resonance lineshape for transitions in a two-level system without damping. At resonance the population oscillates sinusoidally between the two states (this is known as Rabi oscillation). A “π-pulse” is an on-resonance pulse with 2bτ = π, which transfers all the population from state (0) to state (c). In Section 15.4.3 we will discuss how this can be used in a molecular beam to map out the fields along the beamline. An on-resonance “π/2-pulse” (2bτ = π/2) drives the transition only half-way, creating an equal superposition of states (0) and (c) with a definite relative phase. The density matrix element describing this coherence at the end of © 2009 by Taylor and Francis Group, LLC
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the pulse (at time t1 + τ) is (a0 ac ∗ )x = 21 i exp{i[ω(t1 + τ) + φ]}, the phase of which (apart from the fixed factor of i) is just the final phase of the rf field. In conventional Ramsey spectroscopy of a two-level system [50], two short π/2 pulses are applied in succession. If the second pulse comes immediately after the first, the transition is completed and all the population is excited. If instead there is a delay time T between the two pulses, which is long compared to the pulse duration τ, the transition probability becomes sensitive to small differences between the radiofrequency frequency and the molecular transition frequency. The internal coherence evolving at the transition frequency accumulates a phase between pulses of ΩT , whereas the rf field evolves a phase ωT . When the difference between these two reaches π, the second pulse reverses the effect of the first, returning all the population to the initial state. More generally, the probability that a molecule will end up in the excited state is PRamsey (0 → 1) =
1 {1 + cos([Ω − ω]T − δφ)} , 2
(15.21)
where we have allowed for the useful possibility of giving the second pulse a phase φ + δφ when the first has phase φ. These oscillations of the population, resulting from a beat between the coherence and the driving field, are known as Ramsey fringes. They are important because a long waiting time T allows a small difference Ω − ω to be measured, giving rise to very precise spectroscopy. We have generalized this idea to our system of three hyperfine levels. The result, derived from Equations 15.17, 15.18, and 15.19 is P2×pulse (0 → 1) = 1 − cos2 (bτ)1 cos2 (bτ)2 − cos2 (Δz T )sin2 (bτ)1 sin2 (bτ)2 +
1 cos([Ω − ω]T − δφ)cos(Δz T )sin(2bτ)1 sin(2bτ)2 . 2 (15.22)
Here we allow for the possibility that the values of bτ for the two pulses, (bτ)1 and (bτ)2 , are not equal. In the “Ramsey” case of two π/2-pulses, this simplifies to P2× π2 −pulse (0 → 1) =
1 3 − cos2 (Δz T ) + 2 cos([Ω − ω]T − δφ) cos(Δz T ) . 4 (15.23)
If we set Δz to zero, Equation 15.23 reduces to the standard two-level Ramsey result of Equation 15.21 because the third state (u) plays no role when the two levels (+1) and (−1) are degenerate. When Δz = 0 it can be useful to pick out the Ramsey interference by switching δφ between 0 and π and taking the difference, which is cos([Ω − ω]T ) cos(Δz T ). As the frequency of the oscillator is swept, the amplitude of these fringes provides information about the Zeeman shift Δz , while the phase of the fringe pattern reveals the precise value of the splitting Ω with a precision controlled by the choice of T . In Section 15.4.4 we give an example of how this can be used to look for small changes in a large electric field. © 2009 by Taylor and Francis Group, LLC
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With three levels, it becomes possible to do interferometry using π-pulses as well. Taking 2bt = π, Equation 15.22 becomes P2×π−pulse (0 → 1) = sin2 (Δz T ) .
(15.24)
This has a simple interpretation. Molecules are excited by the first pulse to state (c). This state subsequently evolves into cos(Δz T )(c) + i sin(Δz T )(u) because of the splitting between states (+1) and (−1). The second pulse then drives the state-(c) part of the population back to (0), leaving those in state (u) alone. As the magnetic field is scanned, this produces fringes in the final state-(0) population, with a spacing that is inversely proportional to T . These interference fringes can be used for sensitive magnetometry and in searching for an electron EDM.
15.4.3
EXPERIMENTS WITH SINGLE PULSES
Precision measurements usually require careful control and monitoring of stray and applied fields, both electric and magnetic, throughout the interaction region of the apparatus. The small spatial and temporal extent of molecular beam pulses make it possible to do so with high spatial resolution [51]. Figure 15.6 shows an apparatus to demonstrate this using YbF molecules and pulsed rf fields. A pump laser and an interaction region have been added in between the source and detector already shown in Figure 15.1. Although the beam is cold, both hyperfine levels are occupied because the splitting A = 170 MHz is very much less than kT . The pump laser excites the A2 Π1/2 − X 2 Σ+ Q(0) transition at 552 nm, for which the Doppler width is (almost) eliminated by pointing the laser beam perpendicular to the molecular beam. This makes the excitation spectrum narrow enough (∼20 MHz) to excite just the F = 1 population so that it becomes selectively depleted. The remaining N = 0 molecules are virtually all in the F = 0 state, which serves as the initial state for subsequent manipulation by rf pulses. The interaction region starts 450 mm from the skimmer and is 790 mm long. It is magnetically shielded to reduce the ambient field, whilst current-carrying wires
x ƒz y
Probe
Pump RF in
Bƒ Eƒ
RF out Photomultiplier
Source
Interaction region
Detector
FIGURE 15.6 Schematic diagram of the beam machine. Bunches of molecules issue from the source (in the y-direction) and are skimmed before being optically pumped into a single hyperfine state. The molecules enter the magnetically shielded interaction region and fly through a high-voltage capacitor where electric and magnetic fields can be applied along z. This doubles as a rf transmission-line where the rf magnetic field is along x. Finally, the molecules are detected by laser-induced fluorescence.
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inside the shields can generate a magnetic field if required. Within this region there is a pair of electric field plates, 750 mm long and 70 mm wide, with a 12 mm spacing, constant to better than 200 μm over the full length. These are machined from cast aluminum, then gold-coated to improve the uniformity of the surface potential (using a nonmagnetic, nickel-free process). The whole assembly is nonmagnetic. With a field across the gap of 15 kV/cm the leakage current is less than 1 nA. The same plate structure also serves as a 34 Ω transmission line for the radiofrequency field, transporting it as a 170 MHz TEM wave traveling parallel or antiparallel to the beam direction. This is described more fully in Ref. [51]. As a bunch of molecules travels along the interaction region, a hyperfine transition can be induced at any desired position by pulsing the rf field on for a short time. This repopulates the F = 1 state and therefore produces an increase in the fluorescence signal at the detector, as shown by the rf scans in Figure 15.7. Being a TEM wave, the rf magnetic field between the plates is accurately perpendicular to the static electric field and we choose to define its direction as the x-axis. This field therefore drives the transition (0) − (c) discussed in the last section. Figure 15.7 shows three excitation spectra, obtained by applying π-pulses of three different durations (18, 36, and 72 μs) to molecules near the middle of the interaction region. Superimposed on the data points are solid lines corresponding to Equation 15.20, which does a good job of describing the lineshapes, including the positions and relative heights of the sidebands. When we fix the value of 2bτ at some value θ (equal to√π in this case), these lineshapes can be re-written as a universal function sin2 ( 2θ 1 + x 2 )/(1 + x 2 ), where x = (Ω − ω)τ/θ. Thus, the width of the line is inversely related to the duration of the pulse, becoming wider as the pulse is made shorter, as one would expect from the usual Fourier relation between pulse duration and spectral width. For a π-pulse, the full-width at half-maximum is δωFWHM = 5.0/τ.
173.40
173.50 173.60 Frequency w/2p (MHz)
173.70
FIGURE 15.7 Excitation spectra for the ground state F = 0 → F = 1 transition in YbF using π-pulses of three different pulse lengths (upper, τ = 18 μs; middle, τ = 36 μs; lower, τ = 72 μs). A static electric field of 12.5 kV/cm is applied. Symbols: experimental data; lines: plots of Equation 15.20.
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173.40
173.50
173.60
173.70
Frequency w/2p (MHz)
FIGURE 15.8 Upper curve: the hyperfine transition moves to lower frequency when the electric field is reduced. The dashed line marks the line-centre of Figure 15.7 at higher electric field. Lower curve: The line splits when a magnetic field is applied.
The transition probabilities in Figure 15.7 peak at 173.513 MHz, not at the fieldfree frequency of 170.254 MHz, because these spectra were measured with a potential difference of 15 kV across the 12 mm gap of the plates and are therefore Stark-shifted by Δ1 , as illustrated in Figure 15.4. The magnitude of this shift is a measure of the electric field strength at the place occupied by the molecules when the rf pulse was applied. By varying the timing of the rf pulse, it is possible to map out the electric field as a function of position along the beamline [51]. Note that the Doppler shift of the rf transition is only a few hundred Hz and is therefore insignificant at this level of accuracy. In Figure 15.8 we show how the resonance frequency moves down (by 40.246 kHz) when the applied potential difference is reduced (by 203 V). The spectral resolution afforded by the 18 μs π-pulses is quite sufficient to see this shift clearly. Figure 15.8 also shows how the line splits by 2Δz (see Figure 15.4) when a dc magnetic field of 0.8 μT is applied. In order to resolve the 22 kHz splitting fully, the linewidth was narrowed by increasing the pulse duration to 72 μs. Over this time, a YbF molecule travelling at 590 m/s covers 42 mm, so it has been necessary to give up some spatial resolution in order to achieve the higher spectral resolution. The use of long rf pulses for high precision spectroscopy is not ideal because the static field average that is measured can be affected in a complicated way by variations in the strength and polarization of the rf field, either in space or in time. As was first pointed out by Ramsey, it can be more satisfactory to use a pair of rf pulses, and that is what we discuss next.
15.4.4
EXPERIMENTS WITH DOUBLE PULSES
In Equation 15.23, we found an expression for the lineshape using two short π/2pulses separated by a time T . In order to pick out the Ramsey interference, we suggested introducing a phase shift φ between the first and second pulse and taking the difference between φ = π and φ = 0. This is expected to give fringes of the © 2009 by Taylor and Francis Group, LLC
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form cos(Δz T ) cos([Ω − ω]T ), which we have plotted in Figure 15.9a versus rf and magnetic field for the case when T = 900 μs. In Figure 15.9b we display the result of an experiment to test this formula, in which we scanned the rf ω many times through a region close to ω = Ω, stepping the applied magnetic field Bz in order to vary the Zeeman splitting 2Δz . In this experiment, the molecules perform exactly as predicted by Equation 15.23. In Figure 15.10, we scan a much wider frequency range, covering the central 40 Ramsey fringes with Bz fixed close to zero. Here we begin to see a departure from Equation 15.23 in the amplitude of the fringe pattern, which shows a clear decrease on either side of the center. This happens because the individual pulses are starting to become appreciably detuned, and therefore have reduced amplitude. √ Indeed, when the detuning reaches ± 15/(4τ) = ±54 kHz, the 18 μs single pulse transition probabilities go to zero and the Ramsey interference has no amplitude at all. (a)
1000
z)
0
(H
–50
et
0
f fs
Signal (arb. units)
50
nc ue
20 c fie ld (n 40 T)
neti
–1000
Fre q
Mag
yo
0
(b)
0
1000
(H
z)
–50
et
0
f fs
Signal (arb. units)
50
nc –1000
ue
20 c fie ld (n 40 T)
neti
Fre q
Mag
yo
0
FIGURE 15.9 Ramsey interference signals using two π/2-pulses separated by a time T = 900 μs. The signals are plotted versus the rf ω and also as a function of magnetic field in nT. (a) Theoretical signal, scaled to have the same amplitude as the experimental result. (b) Measured Ramsey interference, in excellent agreement with the theory.
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Signal (arb. units)
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20 10 0 –10 –20 –20
–10 0 10 Relative RF magnetic field frequency (w/2p – 173,565 kHz) (kHz)
20
FIGURE 15.10 The central 40 fringes of a Ramsey pattern obtained using two π/2-pulses of 18 μs duration, separated by T = 800 μs. The magnetic field is set very close to zero. Symbols: experimental points; line: theoretical fringes with amplitude adjusted to fit the data.
For a two-level system, the envelope of the interference pattern is just the single-pulse lineshape, as discussed by Ramsey [50]. However, the spectrum of Figure 15.10 involves the third level (u), which makes it sensitive to very small magnetic fields and complicates the shape of the envelope. This detuning effect is not included in Equation 15.22, which assumes that the detuning is small compared with 1/τ. At present we do not have an analytical formula for this case, so the varying amplitude of the −cos([Ω − ω]T ) curve drawn through our data in Figure 15.10 is just a smooth fit to the measured amplitude of the oscillations. The curve in Figure 15.11 shows a blow-up of the central Ramsey fringe (solid line), together with the fringe obtained when the high-voltage leads were reversed (dashed line). The small shift in the phase of the fringe pattern shows that the 12.5 kV/cm electric field decreased in magnitude by 455 ± 11 mV/cm when we attempted to reverse it. This experiment demonstrates that Ramsey interferometry of the hyperfine states can provide very sensitive monitoring of high electric fields in a molecular beam apparatus.
Signal (arb. units)
2 1 0 –1 –2 –1000
–500 0 500 1000 Relative frequency (w/2p – 173,565 kHz) (Hz)
FIGURE 15.11 The central fringe of Figure 15.10. Solid line: fit to data taken with electric field as in Figure 15.10. Dashed line: fit to data taken with high-voltage leads reversed. The evident Stark shift between the two indicates a change in the magnitude of the electric field.
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Signal (arb. units)
We turn now to interferometry using pairs of π-pulses. Whereas the π/2-pulses of Ramsey interferometry prepare and interrogate an rf coherence between the states (0) and (c), the π pulses make and probe the superposition of the states (+1) and (−1), which are almost degenerate. As expressed in Equation 15.24, this superposition evolves at the (±1) splitting frequency, leading to fringes of the form sin2 (Δz T ), where Δz = gF μB Bz . Figure 15.12 shows the interferometer fringes measured in an electric field of 10 kV/cm by scanning the applied magnetic field over 160 nT, with a separation of T = 800 μs between the two π-pulses. The solid line is a fit over the two cental fringes to the lineshape A sin2 (Δz T ) + C, where C represents the background due to unpumped F = 1 molecules and to scattered light. Within the 2% uncertainty of the applied magnetic field calibration, the fringe spacing is found to be 44 nT, corresponding to a g-factor of gF = 1, as one would expect for the F = 1 hyperfine level of this 2 Σ state. The total field Bz , which determines Δz , is taken in our fit to have an adjustable offset in addition to the applied field. This is found to be 1.7 nT in Figure 15.12, which is typical of the field that leaks through the magnetic shielding from outside. While this simple function fits the central two fringes well, the next fringe on either side clearly has less amplitude. This is because the magnetic field detunes the two π-pulse transitions from resonance. Adjusting the magnetic field to reach a steep part of the fringe pattern, the interferometer becomes sensitive to any small change in the splitting 2Δz between the levels (±1). This can be used to monitor variations of the magnetic field in the apparatus. In addition, if the electron has a dipole moment de , then the interaction with the applied electric field E also contributes to Δz , the additional amount being ±de ηE/. Here η is the enhancement factor discussed in Section 15.2.2. If the electric field is reversed, this shift changes sign, causing the fringe pattern to be modulated from side to side as the electric field is flipped back and forth. In this way, a measurement of the signal on the side of a fringe can become very sensitive to the presence of a small electron EDM.
300
200
100
–50
0 Applied magnetic field (nT)
50
FIGURE 15.12 Interferometer fringes obtained by scanning the applied magnetic field, using a pair of π-pulses, separated in time by 800 μs. Symbols: experimental data; line: a fit of the central two fringes to the form A sin2 (Δz T ) + C.
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15.5 ALTERNATING GRADIENT DECELERATION OF POLAR MOLECULES 15.5.1
INTRODUCTION
We have seen in Section 15.4 how spectral resolution can be improved by increasing the time T available for coherent evolution. In a molecular beam with a velocity of a few hundred m/s and an interaction region 1 m long, this time is a few ms, corresponding to minimum linewidths of several hundred Hz. For this reason, the prospect of decelerating and trapping molecules offers significant improvements for some precision measurements, provided the beam flux can remain high and the inhomogeneous broadening due to trapping fields can be kept under control. The basic idea of deceleration is to manipulate polar molecules in an electric field gradient using the force due to the Stark effect. After Stark deceleration was first demonstrated [52], the new technology was rapidly applied to make new measurements. Using a beam of 15 ND molecules decelerated to 52 m/s, the energies of all 22 hyperfine levels of the 3 (J, K) = (1, 1) state were measured with accuracies better than 100 Hz [14]. Using Stark-decelerated OH radicals, greatly improved measurements of the ground-state Λ-doublet microwave transitions were made, thus contributing to the constraint on the evolution of the fine-structure constant over cosmological time [8]. Once molecules could be trapped, it became possible to measure directly the lifetimes of long-lived molecular states [53], and even to measure the optical pumping of molecules by roomtemperature black-body radiation [54], which typically occurs on a timescale of many seconds. Many of the precision measurements discussed in this chapter make use of heavy polar molecules. Stark deceleration of these heavy species is considerably more challenging because (1) the kinetic energy to be removed is proportional to the molecular mass and (2) the low-lying energy states are all high-field seeking whereas the Stark deceleration method works best for low-field seekers. The first difficulty arises because the molecules formed in a supersonic expansion acquire the speed of the carrier gas into which they are seeded; thus their speed is independent of their mass. This difficulty could be mitigated by using a low-temperature effusive source such as the buffer-gas sources recently demonstrated [55,56]. The second difficulty results from the closely packed rotational energy levels of a heavy molecule, which causes all the low-lying states to be high-field seeking HFS when the electric field is strong. This problem is best illustrated by considering the Stark shift of a rigid rotating molecule of reduced mass m , bond length R, and where B = 2 /(2m R2 ) is · E, dipole moment μ. The Hamiltonian is H = BJ 2 − μ is the applied electric the rotational constant, J is the angular momentum vector, and E field. Figure 15.13 shows the first 16 energy eigenvalues, in units of B, as a function of applied electric field, in units of B/μ. The field mixes states having different values of J but the same value of M, the projection of the angular momentum onto the electric field axis. Each energy level is labeled according to the quantum numbers (J, M) that the state evolves into when the electric field is adiabatically reduced to zero. Note that the states (J, M) and (J, −M) are degenerate for all electric fields, a consequence of time-reversal symmetry. The important point to note from Figure 15.13 is that all the weak-field-seeking states have turning points, becoming strong-field-seekers © 2009 by Taylor and Francis Group, LLC
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(3,0) 10
(3,1) (3,2) (2,0) (3,3) (2,1)
Energy (B)
0 –10
(2,2) (1,0)
–20
(1,1)
–30
(0,0)
–40 0
10
20 30 Electric field (B/m)
40
50
FIGURE 15.13 The Stark shift of the low-lying energy levels of a rigid rotating molecule. The electric field is expressed in units of B/μ and the energy in units of B. States are labeled by the quantum numbers (J, M).
at high field. For example, the lowest-lying weak-field-seeking state (1, 0) has its turning point at an electric field of 4.9B/μ, at which point the Stark shift is 0.64B. Taking YbF as an example of a molecule with a small rotational constant, the electric field at the turning point is only 18 kV/cm and the Stark shift only 0.15 cm−1 . This amount of energy, which can be removed from the molecule in a single stage of deceleration, is to be compared with the 682 cm−1 of kinetic energy possessed by a YbF molecule formed in a supersonic expansion at 290 m/s [46]. Clearly a very large number of stages would be needed to decelerate in this way. By contrast, the strong-field-seeking ground state of YbF has a Stark shift of 10.7 cm−1 at a field of 200 kV/cm, and so Stark deceleration in this state seems feasible. For more complex molecules, such as those of biological interest, the situation is even more extremely weighted in favor of the high-field seekers, as discussed in more detail in Ref. [57]. Unlike weak-field-seekers, which are naturally focused onto the axis of the Stark decelerator, strong-field-seeking molecules cannot be focused using static fields. A dynamic focusing scheme needs to be used to prevent the molecules being pulled toward the surfaces of the electrodes, where the field is strongest. The alternating gradient focusing technique solves this problem of transverse confinement. The molecules travel through a sequence of electrostatic lenses, each of which focuses the molecules in one of the two transverse directions and defocuses them in the other. The focusing and defocusing planes alternate between one lens and the next. For a subset of the molecules that enter the decelerator, namely those that lie within the transverse phase-space acceptance of the lens array, the net effect is to focus in both transverse planes. Ideally, the focusing and defocusing forces are linear in the off-axis displacements, and the trajectories of the accepted molecules take them far from the axis inside the focusing lenses, but close to the axis inside the defocusing lenses. Thus, the overall focusing is a direct result of the motion of the molecules, and can operate even when the defocusing power is stronger than the focusing. © 2009 by Taylor and Francis Group, LLC
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The first experiment to demonstrate alternating gradient deceleration of polar molecules used an array of 12 lenses to decelerate high-field seeking metastable CO molecules from 275 to 260 m/s [58]. Ground-state YbF molecules were decelerated from 287 to 277 m/s using a similar machine [59]. The transverse focusing properties of the alternating gradient have been demonstrated [57] by imaging metastable CO molecules exiting from the decelerator. Longer machines, with more sophisticated electrode designs, should be capable of decelerating heavy polar molecules to rest.
15.5.2 A MODEL ALTERNATING GRADIENT DECELERATOR We consider a decelerator consisting of a series of electrostatic lenses whose focusing and defocusing planes alternate. A typical electrode structure is shown in Figure 15.14a. Here, each lens is formed by applying a large potential difference between a pair of rods whose axes lie parallel to the beamline. The forward velocity of the molecules only changes when they pass through the fringe fields formed in the gap that separates one lens from the next. In operation, the decelerator is switched between three states: (1) odd lenses at high voltage, even lenses grounded, (2) even lenses at high voltage, odd lenses grounded, and (3) all lenses grounded. We would like to work out the dynamics of molecules in such a decelerator. We therefore first generate a map of the electric field in the decelerator using an electrostatics solver, and then construct the interaction potential, using the electric field dependence of the Stark shift such as is plotted in Figure 15.13. We emphasize that this is not the electrostatic potential, but rather the Stark potential in which the molecules move. The electrode geometry shown in Figure 15.14a is one of many possible geometries, some of which have been discussed in detail in Ref. [57]. The same reference discusses the Stark shift and shows that it varies linearly with the electric field magnitude for most heavy molecules in the electric fields of a typical decelerator. The potential in which the molecules move is similar in form for all the electrode geometries and molecular states considered, although they vary in their details. Here, we do not specialize to a particular geometry or molecular state, but instead elucidate the dynamics using an analytical form for the potential that approximates the true potential produced in most cases. We take the potential in switch state (1) to be W (x, y, z) = W0 (1 + b((x/r0 )2 − (y/r0 )2 ))f (z),
(15.25)
tanh(z /d + L/2d) − tanh(z /d − L/2d) 2 tanh(L/2d)
(15.26)
where f (z) =
provides a good phenomenological approximation to the actual z-dependence. In Equation 15.25, W0 is the electric field at the origin, r0 is a measure of the transverse aperture of the decelerator, and b measures the transverse curvature of the potential. In Equation 15.26, z = mod(z − D, 2D) − D, where mod(m, n) is the remainder on division of m by n, D is the lens-to-lens spacing, L measures the length of a lens, and © 2009 by Taylor and Francis Group, LLC
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Lens 4
Lens 3 Lens 2
(a)
y z
x Lens 1
Molecular beam (b) –0.0
W/|W0|
–0.2 –0.4 –0.6
C
C A
–0.8 –1.0 0.0
A B
B 0.5
1.0
1.5
2.0
0.5
1.0
z/D (c)
W/|W0|
–0.9
–1.0
–1.1 –1.0
–0.5
0.0 x/r0 or y/r0
FIGURE 15.14 (a) Schematic of a typical electrode structure in an alternating gradient decelerator. (b) Potential in which the molecules move, approximated using an analytical form (see text). Solid and dashed lines represent respectively the switch states (1) and (2) of the decelerator. Positions A, B, and C are referred to in the text. (c) At the center of the first lens, the potential along x is shown by the solid line and that along y by the dashed line.
d measures how rapidly the potential changes at the exit of the lens. The potential in switch state (2) is simply W ( y, x, z − D). Figure 15.14b,c shows the potential for the case where L = 2/3D, d = 1/15D, and b = 0.15. We shall use these parameters throughout. Part (b) is a plot of the potential along the beamline, W (0, 0, z), for both switch state (1) (solid line) and switch state © 2009 by Taylor and Francis Group, LLC
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(2) (dashed line). Part (c) of the figure shows the transverse potentials in the center of a lens, W (x, 0, 0) (solid line) and W (0, y, 0) (dotted line). In our ideal lens, these are everywhere equal and opposite.
15.5.3 AXIAL MOTION Consider molecules traveling down the axis of the decelerator. To calculate when we should switch the potentials we introduce the concept of a synchronous molecule, which enters the decelerator with speed u0 . We design the switching sequence so that this molecule is always at the same relative position in the periodic array every time the field is turned on (e.g., position A in Figure 15.14b) and every time the field is turned off (e.g., position B in Figure 15.14b). We refer to these fixed positions as zon and zoff . Because this molecule is always climbing potential hills, it decelerates as it moves along the beamline, and the time intervals between successive switches must be chosen to increase in correspondence. The required time sequence can be constructed using a simple algorithm. Consider the energy conservation equation that applies between the turn-on and turn-off points, zon and zoff , 2 1 2 Mun−1
+ W (zon ) = 21 M(dz/dt)2 + W (z).
(15.27)
Here, z is the position of the molecule at time t. Rearranging and then integrating gives us the relationship between the nth turn-off time, toff,n , and the nth turn-on time, ton,n , toff,n = ton,n +
zoff zon
dz 2 un−1 + 2(W (zon ) − W (z))/M
,
(15.28a)
where un is the speed of the synchronous molecule immediately after the nth turn-off. The (n + 1)th turn-on is now found using un =
2 un−1 + 2(W (zon ) − W (zoff ))/M,
ton,n+1 = toff,n +
D − (zoff − zon ) . un
(15.28b)
(15.28c)
Because we now have the potential seen by the molecules as a function of both position and time, we can solve the equation of motion numerically for any molecule, synchronous or not. Let us define N to be the minimum number of stages required to stop the synchronous molecule of mass M and initial speed u0 , through the relation |W0 | = Mu02 /(2N). Figure 15.15 shows the result of such a calculation for the case where L = 2/3D, d = 1/15D, and N = 80, and the turn-on and turn-off points are A and B in Figure 15.14b. The thick red line shows the speed of the synchronous molecule versus time, while the thin blue line shows how the speed changes with time for a molecule that has the same initial speed as the synchronous molecule but starts out ahead by D/15. The deceleration of the synchronous molecule appears to © 2009 by Taylor and Francis Group, LLC
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1.0
Speed (u0)
0.8 0.6 0.4 0.2 0.0 0
50
100
150 200 Time (D/u0)
250
300
FIGURE 15.15 The speed of a synchronous (grey line) and a nonsynchronous molecule (black line) as a function of time, using the model potential and the turn-on and turn-off points A and B indicated in Figure 15.14b. The inset reveals the steplike structure of the deceleration.
be uniform, though when magnified, as in the inset, is actually seen to be a series of small steps reflecting the shape of the potential. These steps can also be seen in the main figure when the speed is low. The speed of the nonsynchronous molecule oscillates around that of the synchronous one and also consists of many small steps when examined in detail. If we take snapshots of the position and velocity of the molecules at each of the many turn-off times, we would no longer be able to see the fine structure of the motion. Between one snapshot and the next, the change in kinetic energy of the synchronous molecule is ΔK = MuΔu = MDuΔu/Δzs = W (zon ) − W (zoff ), where we have introduced the quantity Δzs to represent the change in position of the synchronous molecule between one snapshot and the next, and have used the fact that Δzs = D always. Since we have discarded the information about the fine structure of the motion, the deceleration of the synchronous molecule appears to be constant, and we can convert to continuous variables. The above equation then becomes MD u du/dzs = MD d2 zs /dt 2 = W (zon ) − W (zoff ). Applying a similar reasoning to any other molecule, described by the position and velocity coordinates z and v, we obtain MD d2 z/dt 2 = W (zon + z˜ ) − W (zoff + z˜ ), where z˜ = z − zs , and we have made the additional approximation that v − u u so that the distance moved by the general molecule between switching times is also very close to D. Subtracting the equation for the general molecule from that for the synchronous molecule we obtain an equation of motion for the relative coordinate, allowing us to define an effective force, Feff : d2 z˜ W (zon + z˜ ) − W (zon ) − W (zoff + z˜ ) + W (zoff ) = MD dt 2 = Feff /M.
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(15.29)
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Introducing the relative velocity v˜ = v − u = d˜z/dt, the left-hand side of the above equation can be written as v˜ dv/d˜ ˜ z. Integrating, we then obtain 1/2M v˜ 2 + (V (˜z) − V (0)) = E0 , where E0 is a constant and
(15.30)
V (˜z) = −
Feff d˜z
(15.31)
is an effective potential for the relative motion between nonsynchronous and synchronous molecules. Figure 15.16a shows the effective potential obtained from Equations 15.25, 15.26, and 15.31. The solid line corresponds to zon = −D/10, zoff = D/3, while the dashed line has zoff = 11D/30 corresponding to greater deceleration, but a shallower effective potential. Because the effective potential is confining, nonsynchronous molecules with insufficient energy to reach the top of the potential must oscillate about the synchronous molecule. We can use the effective force to solve for the relative motion of nonsynchronous molecules. Figure 15.16b and c show trajectories in phase space obtained in this way, for the solid and dashed effective potentials shown in part (a). In each case, the thicker line separates bounded and unbounded motion and is called the separatrix. All molecules inside the separatrix will remain close to the synchronous molecule throughout the deceleration process, and the area bounded by the separatrix is the axial phase-space acceptance. Comparing parts (b) and (c) of the figure, we see that, as expected from the shallower potential, the acceptance is smaller when the deceleration is greater. We note that the same phase-space plots can be generated without making use of the effective potential, by numerically integrating the complete equation of motion as was done in generating Figure 15.15. The trajectories in phase space then acquire the detailed structure shown in the inset of that figure, but are otherwise found to be identical to those obtained (much more rapidly) from the effective potential. For small-amplitude oscillations about the synchronous molecule, the motion is harmonic. Expanding the right-hand side of Equation 15.29 in a Taylor series about z˜ = 0, gives d2 z˜ /dt 2 − (W (zon ) − W (zoff ))˜z/(MD) = 0 where W (a) = dW /dz evaluated at z = a. The angular oscillation frequency for small-amplitude axial oscillations is therefore W (zoff ) − W (zon ) ωz = . (15.32) MD For our model potential, the solid line in Figure 15.16a, the frequency can be con√ veniently expressed as ωz /2π = 0.309 u0 /( ND). For example, if D = 30 mm, and N = 80 when u0 = 300 m/s, we find ωz /2π = 345 Hz for turn-off at position B.
15.5.4 TRANSVERSE MOTION Our model potential is harmonic in the two transverse directions, with a curvature that varies with z. As a molecule travels through a lens, the curvature is very nearly
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Effective potential (Mu2)
(a) 0.003
0.002
0.001
–0.4
–0.2
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Relative position (D) Relative speed (u)
(b) 0.04
0.02
–0.3
–0.2
–0.1
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Relative position, D –0.02
–0.04
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(c) 0.04
0.02
–0.3
–0.2
0.1 0.2 0.3 Relative position (D)
–0.1
0.4
–0.02
–0.04
FIGURE 15.16 (a) The effective potential, Equation 15.31, as a function of relative position. The solid line corresponds to moderate deceleration (turn-off-position B), while the shallower potential shown by the dashed line is for strong deceleration (turn-off position C). The trajectories in phase space calculated using these effective potentials are shown in (b) and (c) for the deep and shallow potentials, respectively. The thick outer boundaries in these plots are the separatrices. © 2009 by Taylor and Francis Group, LLC
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constant until it reaches the fringe-field of the lens where the curvature drops rapidly to zero. To simplify the analysis, we make the approximation that the curvature has the constant value, W0 b/r02 over the lens length L, is zero in the drift space of length S = D − L, and changes abruptly between these values. As the decelerator structure is periodic in z, it is natural to write the equation of motion with independent variable z rather than time. For a molecule with forward speed u, the equation of motion is d2 x/dz2 + κ2 Q(z)x = 0,
(15.33)
where Q(z) = 1 inside a focusing lens, −1 in a defocusing lens, and 0 in a drift region, and the spatial frequency is κ=
2W0 b = Mu2 r02
b . Nr02
(15.34)
The angular frequency of the transverse oscillation inside a focusing lens is independent of the beam velocity u and is related to κ by Ω = κu. In moving through a region of length l, from an initial axial position z0 , the transverse position and velocity coordinates of a molecule change according to
x/r0 vx /Ωr0
z0 +l
= M(z0 + l|z0 )
x/r0 vx /Ωr0
.
(15.35)
z0
Here, the dimensionless transfer matrix denoted by M(z0 + l|z0 ) takes the values F(l) inside a focusing lens, D(l) inside a defocusing lens and O(l) in a drift region, with
cos(κl) sin(κl) , − sin(κl) cos(κl) cosh(κl) sinh(κl) D(l) = , sinh(κl) cosh(κl) 1 κl O(l) = . 0 1 F(l) =
(15.36a) (15.36b) (15.36c)
If, in moving from left to right, the molecule first travels through a region described by the matrix M1 , followed by a region described by M2 , the matrix in Equation 15.35 is simply the product, M = M2 · M1 . In this way, one complete unit of our alternating gradient array is described by M = F(L) · O(S) · D(L) · O(S), and a sequence of N such units is described (in a more compact but obvious notation) by (FODO)N . It can be shown that the molecular trajectories are stable if the well-known condition −2 < Tr(FODO) < 2 is satisfied (e.g., Ref. [60]). We would like to know whether a molecule that enters the array reaches the exit. For a long decelerator, the above stability condition is a necessary but not sufficient one, because a molecule can be on a stable trajectory that takes it so far from the axis that it crashes into one of the electrodes. Rather than constructing trajectories © 2009 by Taylor and Francis Group, LLC
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piecewise using Equation 15.36, we follow the approach first used by Courant and Snyder in the context of the alternating gradient synchrotron [61]. We look for a solution to Equation 15.33 of the general form i β(z) cos(ψ(z) + δi ) = A1 β(z) cos ψ(z) + A2 β(z) sin ψ(z),
x(z) =
(15.37)
where β is a z-dependent amplitude function that has the same periodicity as the AG array, ψ is a z-dependent phase, and i , δi , A1 , and A2 are defined by the initial conditions. By substitution into Equation 15.33 we find that Equation 15.37 is a valid solution provided z 1 ψ(z) = κ (15.38) dz ) β(z 0 and 1 1 − β2 + ββ + κ2 Q(z)β2 = κ2 . 4 2
(15.39)
To find β, we need to make the connection to the piecewise solution already given in Equation 15.36. From Equation 15.37 we have A1 κ A2 κ x (z) = √ (−α cos ψ − sin ψ) + √ (−α sin ψ + cos ψ) β β
(15.40)
where α=−
1 β. 2κ
(15.41)
Using Equations 15.37 and 15.40 we find the relationship between the coordinates x, vx /Ω at position z and those at position z + lcell , lcell = 2D being the periodicity of the array. We make use of the periodicity constraint on β, β(z + lcell ) = β(z) and thus obtain cos Φ + α sin Φ β sin Φ M(z + lcell |z) = , (15.42) −γ sin Φ cos Φ − α sin Φ where γ = (1 + α2 )/β and Φ = ψ(lcell ) is the phase advance per unit cell. Because the integral in Equation 15.38 is taken over a full period, Φ is independent of z. The matrix in Equation 15.42 is known as the Courant–Snyder matrix. We can now equate this matrix to the explicit form for the transfer matrix for one lattice unit, and so obtain β(z). For example, at a distance z beyond the start of a focusing lens, M(z + lcell |z) = F(z) · O(S) · D(L) · O(S) · F(L − z). We then obtain Φ using the relation cos Φ = Tr(M)/2, and then find β by equating the upper right-hand element of M to that of the Courant–Snyder matrix. Figure 15.17a shows a few trajectories (dashed lines) calculated using Equation 15.37 with κL = 1, κS = 0.5, and arbitrarily chosen values for i and δi . The © 2009 by Taylor and Francis Group, LLC
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F O D O
1.0
x/r0
0.5
2
4
6
8
10
12
14 kz
–0.5 –1.0 (b) F Start
F Center
0.5
0.5
0
0
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–0.5
–0.5
–0.5
vx/(Wr0)
0.5
–1 0.5
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–1
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–0.5
–0.5
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0
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–1
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F End
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1
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FIGURE 15.17 (a) Dashed lines are typical trajectories through a section of the alternating gradient array. Solid lines denote the envelope that bounds the trajectories of all transmitted molecules. The positions in the array of the focusing (F), defocusing (D), and drift regions (O) are indicated by the dotted lines. The figure shows that the beam envelope is largest at the center of the focusing lens, and smallest at the center of the defocusing lens. (b) Evolution of the phase-space ellipse through a unit of the array.
motion is a product of two periodic functions, one of wavelength lcell and the other of longer wavelength 2πlcell /Φ. It is often the case that Φ 2π, in which case the modulation with wavelength lcell has a small amplitude and is called the micromotion, while the much longer wavelength motion is called the macromotion. For the case shown in the figure, Φ = 0.38π and the separation into a micromotion and a macromotion is evident. If we consider a large collection of molecules, all having different values of δi and i , the only constraint being √ that every |i | < , then all the trajectories will be bounded by the envelope ± β. The bold lines in the figure show this envelope. The figure shows that the beam size modulates with © 2009 by Taylor and Francis Group, LLC
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the period of the array, reaching its maximum size in the center of every focusing lens, and its minimum size in the center of every defocusing lens. Because the confining and deconfining forces are linear in the off-axis displacements, the defocusing lenses have less effect on the beam than the focusing lenses, this being the key to the stability of the alternating gradient array. As the power of the lenses increases, so too does the depth of modulation of the envelope until, at the stability boundary where Φ = π, the beam size becomes zero at the center of the defocusing lens. Using the first line of Equation 15.37 and its derivative to form the quantity x 2 + (αx + βvx /Ω)2 , we find the invariant γx 2 + 2αxvx /Ω + β(vx /Ω)2 = i .
(15.43)
This equation defines an ellipse in the phase space with coordinates x and vx /Ω. The coordinates of all molecules with the same value of i but different values of δi lie on this ellipse. Replacing i by gives us the ellipse that bounds the entire set of molecules in the collection discussed above. The shape of this ellipse changes periodically with z, but its area is a constant, π. Figure 15.17b shows the phase-space ellipse at various positions in the array. The beam is diverging as it enters the focusing lens. Inside this lens the ellipse rotates, reaching its maximal spatial extent at the lens center, where its principal axes are parallel to the coordinate axes. The ellipse continues to rotate so that it is converging at the exit of the focusing lens, and still converging as it enters the defocusing lens. At the center of the defocusing lens, the spatial extent is minimized and the principal axes of the ellipse are again along the coordinate axes. The beam then starts to diverge again. Molecules will only be transmitted if their trajectories do not take them outside the natural boundaries formed by the electrodes. The characteristic size scale in the transverse direction is r0 and so to calculate the transverse phase space acceptance, we assume that the electrodes impose an aperture of size 2r0 in√each of the transverse directions. The beam characterized by will be transmitted if β < r0 everywhere. In particular, the condition must be satisfied at the point where β has its maximum value, βmax , which we already know is at the center of every focusing lens. The phase space acceptance in (x, vx ) space is thus found to be πr02 Ω/βmax . Figure 15.18 is a density plot of the phase-space acceptance in either transverse direction as a function of the two dimensionless parameters that define the array, κL and κS. The region of highest acceptance is found near κL ∼ 1, S L, and the maximum value is 0.744r02 Ω obtained at κL = 1.254, κS = 0. The requirement that κL ∼ 1 for high acceptance constrains the aspect ratio of the lenses, that√is, the ratio L/r0 . Using Equation 15.34 and setting κL = 1, we obtain L/r0 = N/b. Typical values for a decelerator with a maximum field of 200 kV/cm are N = 80 and b = 0.15, giving an aspect ratio of L/r0 = 23.1. Since κ is inversely proportional to u, it will increase as the molecules slow down. To preserve the transverse acceptance, the lenses can be made progressively shorter so as to maintain κL ∼ 1. Alternatively, the alternating gradient array can have the structure (FO)n (DO)n , with the value of n decreasing down the beamline. © 2009 by Taylor and Francis Group, LLC
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kS
3
2
p /3 2p /2 p 3 p/
1 6
p/
0 0
0
0.5
1 kL
1.5
2
FIGURE 15.18 Phase-space acceptance in one transverse direction, as a function of κL and κS. The acceptance is shown in units of Ωr02 .
15.5.5
BEYOND THE IDEAL MODEL
In the previous section, we have used an idealized potential because it allows us to understand the main aspects of the dynamics in a straightforward way. This theoretical model misses some largely undesirable effects that are present in reality. In the transverse directions, nonlinear forces are necessarily present and these reduce the transverse acceptance [57,62,63]. The linear part of the force changes sign between one lens and the next, leading to the dynamical stability discussed above, but the leading order nonlinear terms in the transverse force do not change sign between the focusing and defocusing lenses and so tend to upset the dynamical stability. Even when small compared to the linear terms, the nonlinear terms can significantly reduce the transverse acceptance. Calculations for some typical electrode geometries are presented in Ref. [57]. In our idealized model, the axial and transverse potentials are completely decoupled. This cannot be achieved in any real decelerator because axial gradients of the electric-field-needed for deceleration, change the dependence of the electric field on the transverse coordinates. In particular, the fringe fields at the ends of the lenses tend to increase the force constant in the defocusing direction relative to the one in the focusing direction, leading to further beam loss [57]. A realistic simulation of the transmission of molecules through an alternating gradient decelerator should use a three-dimensional map of the electric field magnitude produced by the electrode geometry of the machine, and the full electric-field dependence of the Stark shift. Simulations of this kind show that the true molecular trajectories in phase space are similar to those calculated with our simplified model, although the acceptance volume in phase space may be considerably smaller. © 2009 by Taylor and Francis Group, LLC
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CONCLUDING REMARKS
In this chapter we have summarized how precise measurements on molecules are able to address important questions about the constancy of physical laws and the structure of fundamental interactions. This is a relatively new direction, which has emerged from a growing ability to prepare and manipulate molecules in pure quantum states. We have discussed the preparation of pulsed supersonic molecular beams and have explored how coherent control of the hyperfine levels can provide exquisite sensitivity to electric and magnetic fields. These same methods also provide an opportunity to search within molecules for more interesting effects, such as symmetry violations or variation of fundamental constants, which can be related to new physics on high energy scales. Deceleration, and trapping are important for improving these experiments because they can increase the time available for coherent interaction with molecules from milliseconds to seconds. Although we have focused on decelerating heavy polar molecules using strong electric field seekers, the Stark deceleration method can also be applied to weak-field-seeking polar molecules and to Rydberg states of atoms and molecules. Alternatively, it is possible to decelerate, and trap molecules using optical dipole forces. As methods for cooling, deceleration, and trapping molecules reach ever lower temperatures and higher densities, and as they are extended to heavier and more diverse species, molecules will surely play an increasingly important role in testing and understanding fundamental physics.
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9. Brian C. Sawyer, Benjamin L. Lev, Eric R. Hudson, Benjamin K. Stuhl, Manuel Lara, John L. Bohn, and JunYe, Magnetoelectrostatic trapping of ground state OH molecules, Phys. Rev. Lett., 98, 253002, 2007. 10. Flambaum, V.V. and Kozlov, M.G., Limit on the cosmological variation of mp /me from the inversion spectrum of ammonia, Phys. Rev. Lett., 98, 240801, 2007. 11. Holzwarth, R., Udem, Th., Hänsch, T.W., Knight, J.C., Wadsworth, W.J., and Russell, P.St.J., Optical frequency synthesizer for precision spectroscopy, Phys. Rev. Lett., 85, 2264, 2000. 12. Daussy, C., et al., Long-distance frequency dissemination with a resolution of 10−17 , Phys. Rev. Lett., 94, 203904, 2005. 13. Schiller, S. and Korobov, V., Tests of time independence of the electron and nuclear masses with ultracold molecules, Phys. Rev. A, 71, 032505, 2005. 14. van Veldhoven, J., Küpper, J., Bethlem, H.L., Sartakov, B., van Roij, A.J.A., and Meijer, G., Decelerated molecular beams for high-resolution spectroscopy, Eur. Phys. J. D, 31, 337, 2004. 15. AnneAmy-Klein,Andrei Goncharov, Mickal Guinet, Christophe Daussy, Olivier Lopez, Alexander Shelkovnikov, and Christian Chardonnet, Absolute frequency measurement of a SF6 two-photon line by use of a femtosecond optical comb and sum-frequency generation, Opt. Lett., 30, 3320, 2005. 16. Wu, C.S., Ambler, E., Hayward, R.W., Hoppes, D.D., and Hudson, R.P., Experimental test of parity conservation in beta decay, Phys. Rev., 105, 1413, 1957. 17. Christenson, J.H., Cronin, J.W., Fitch, V.L., and Turlay, R., Evidence for the 2π decay of the K20 meson, Phys. Rev. Lett., 13, 138, 1964. 18. Schiff, L.I., Measureability of nuclear electric dipole moments, Phys. Rev., 132, 2194, 1963; Eugene, D., Commins, J., David Jackson, David P. DeMille, The electric dipole moment of the electron: An intuitive explanation for the evasion of Schiff’s theorem, Am. J. Phys., 75, 532, 2007. 19. Sandars, P.G.H., The electric dipole moment of an atom. Phys. Lett., 14, 194, 1965. 20. Sandars, P.G.H., The search for violation of P or T invariance in atoms or molecules, in Atomic Physics 4, zu Putlitz, G., Ed. Plenum, 1975, p. 71; Sushkov, O.P. and Flambaum, V.V., Parity breaking effects in diatomic molecules, Zh. Eksp. Teor. Fiz., 75, 1208, 1978. 21. Cho, D., Sangster, K., and Hinds, E.A., Tenfold improvement of limits on T violation in thallium fluoride, Phys. Rev. Lett., 63, 2559, 1989; Cho, D., Sangster, K., and Hinds, E.A., Search for time-reversal-symmetry violation in thallium fluoride using a jet source, Phys. Rev. A, 44, 2783, 1991. 22. Baker, C.A. et al., Improved experimental limit on the electric dipole moment of the neutron, Phys. Rev. Lett., 97, 131801, 2006. 23. Romalis, M.V., Griffith, W.C., Jacobs, J.P., and Fortson, E.N., New limit on the permanent electric dipole moment of 199 Hg, Phys. Rev. Lett., 86, 002505, 2001; Griffith, W.C., Swallows, M.D., Loftus, T.H., Romalis, M.V., Heckel, B.R., Fortson, E.N., Improved limit on the permanent electric dipole moment of 199 Hg, arXiv:0901.2328. 24. Regan, B.C. et al., New limit on the electron electric dipole moment, Phys. Rev. Lett., 88, 071805, 2002. 25. Kozlov, M.G. and Ezhov, V.F., Enhancement of the electric dipole moment of the electron in the YbF molecule, Phys. Rev. A, 49, 4502, 1994; Kozlov, M.G., Enhancement of the electric dipole moment of the electron in the YbF molecule, J. Phys. B, 30, L607, 1997; Titov, A.V., Mosyagin, N.S., Ezhov, V.F., P,T-odd spin-rotational Hamiltonian for YbF Molecule, Phys. Rev. Lett., 77, 5346, 1996; Quiney, H.M., Skaane, H., Grant, I.P., Hyperfine and PT-odd effects in YbF 2 Σ, J. Phys. B, 31, L85, 1998 (after correcting for the trivial factor of 2 between s and σ their result becomes 26 GV/cm); Parpia, F.A.,
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Variation of Fundamental 16 Constants as Revealed by Molecules: Astrophysical Observations and Laboratory Experiments Victor V. Flambaum and Mikhail G. Kozlov CONTENTS 16.1 16.2 16.3 16.4 16.5
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dependence of Atomic and Molecular Spectra on α and μ . . . . . . . . . . . . . . . Astrophysical Observations of H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Astrophysical Observations of Microwave Molecular Spectra . . . . . . . . . . . . 16.5.1 Rotational Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.2 The 18 cm Transitions in OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Limit on the Time Variation of μ from the Inversion Spectrum of Ammonia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7 Experiment with SF6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8 Narrow Close-Lying Levels of Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . 16.8.1 Molecules with Quasidegenerate Hyperfine and Rotational Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8.2 Molecules with Quasidegenerate Fine-Structure and Vibrational Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8.3 The Molecular Ion HfF+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8.4 Estimate of the Natural Widths of the Quasidegenerate States . . . 16.9 Proposed Experiments with Cs2 and Sr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . + 16.10 Experiments with Hydrogen Molecular Ions H+ 2 and HD . . . . . . . . . . . . . . . 16.11 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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INTRODUCTION
In this chapter, we describe the application of precision molecular spectroscopy to the study of a possible temporal and spatial variation of the fundamental constants. As we will show below, molecular spectra are mostly sensitive to two such dimensionless constants, namely the fine-structure constant α = e2 /c and the electron-to-proton mass ratio μ = me /mp (note that some authors define μ as an inverse value, i.e., the proton-to-electron mass ratio). At present, NIST lists the following values of these constants [1]: α−1 = 137.035999679(94) and μ−1 = 1836.15267247(80). The fine-structure constant α determines the strength of the electromagnetic (and, more generally, electroweak) interactions. In principle, there is a similar coupling constant αs for quantum chromodynamics (QCD). However, because of the highly nonlinear character of the strong interaction, this constant is not well defined. Therefore, the strength of the strong interaction is usually characterized by the parameter ΛQCD , which has the dimension of mass and is defined as the position of the Landau pole in the logarithm for the strong coupling constant, αs (r) = const./ ln (rΛQCD /c), where r is the distance between the interacting particles. In the Standard Model (SM), there is another fundamental parameter with the dimension of mass—the Higgs vacuum expectation value (VEV), which determines the electroweak unification scale. The electron mass me and quark masses mq are proportional to the Higgs VEV. Consequently, the dimensionless parameters Xe = me /ΛQCD and Xq = mq /ΛQCD link the electroweak unification scale with the strong scale. For the light u and d quarks, Xq 1. As a result, the proton mass mp is proportional to ΛQCD and hence Xe is proportional to μ. In what follows, we use μ instead of Xe as it is more directly linked to the experimentally accessible atomic and molecular observables. Below we show that a huge enhancement of the relative variation takes place in transitions between close-lying atomic, molecular, and nuclear energy levels. Recently, several new systems have been found where the levels are both very close to one another and narrow. Also, a large enhancement of the variation effects can be expected to occur in cold collisions of atoms and molecules near Feshbach resonances. We begin by reviewing the state of the art of the search for the variation of α and μ. Next, we discuss, in some detail, the results that follow from the astrophysical observations of the optical and microwave spectra of molecules. Finally, we describe possible laboratory experiments with molecules on time variation. Such experiments are a novelty, and the accuracy of the laboratory results cannot yet compete with that of the astrophysical observations (see, nevertheless, Section 16.7). However, there are proposals for significant improvements, and several groups have already started implementing them. The analysis of the data from the Big Bang nucleosynthesis [2], quasar absorption spectra, and the Oklo natural nuclear reactor yields the space–time variation of the constants on the timescale of the lifetime of the Universe, that is, from a few billion to more than ten billion years. In comparison, the frequencies of various atomic and molecular transitions in the laboratory experiments yield a time variation on the timescale from a few months to a few years. There is no model-independent connection between the variations on such different timescales. However, in order to
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compare the astrophysical and laboratory results, we will often assume a linear time dependence of the constants. In this way, we can interpret all the results in terms of the time derivatives of the fundamental constants. Within this assumption, we can use the quasar absorption spectra to obtain the best current limit on the variation of the mass ratio μ and of Xe [3], μ/μ ˙ = X˙e /Xe = (1 ± 3) × 10−16 yr −1 .
(16.1)
Combining this result with the atomic clock results [4] yields the best upper bound on the variation of α [5–7]: α/α ˙ = (−0.8 ± 0.8) × 10−16 yr −1 .
(16.2)
The measurements at the Oklo natural reactor provide the best bound on the variation of Xs = ms /ΛQCD , where ms is the strange quark mass [8–10], |X˙s /Xs | < 10−18 yr −1 .
(16.3)
Note that the Oklo data cannot yield any bound on the variation of α since the effect of α there is much smaller than the effect of Xs and should be neglected within the accuracy of the present theory [10]. In addition to the time variation, one may also consider the spatial variation of the constants. Massive bodies (stars or galaxies) may also affect the physical constants. In other words, the fundamental constants may depend on the gravitational potential, for example, δα/α = kα δ(GM/rc2 ),
(16.4)
where G is the gravitational constant and r is the distance from the mass M. The most stringent limit on such a variation, kα + 0.17kμ = (−3.5 ± 6) × 10−7 ,
(16.5)
is obtained in Ref. [6] from the measurements of the dependence of the atomic frequencies on the distance from the Sun due to the ellipticity of the Earth’s orbit [4,11] (the parameter kμ is defined by analogy with Equation 16.4). Below we also discuss some other results, including those that indicate nonzero variation of the fundamental constants.
16.2 THEORETICAL MOTIVATION How may the variation of the physical constants and the violation of the local position invariance come about? Light scalar fields very naturally appear in modern cosmological models, affecting parameters of the SM including α and μ (for the whole list of SM parameters see Ref. [12]). Cosmological variations of these scalar fields are, in turn, expected to arise because of the drastic changes in the composition of the Universe during its evolution. © 2009 by Taylor and Francis Group, LLC
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Theories unifying gravity and other interactions suggest the possibility of spatial and temporal variation of physical “constants” in the Universe [13]. Moreover, there exists a mechanism for making all coupling constants and masses of elementary particles both space- and time-dependent, and influenced by the local environment (see review in Ref. [14]). Variation of the coupling constants could be nonmonotonic, such as, for example, damped oscillations. These variations are usually associated with the effect of massless (or very light) scalar fields. One such field is the dilaton: a scalar that appears in string theories together with the graviton, in a massless multiplet of closed-string excitations. Other scalars naturally appear in those cosmological models in which our Universe is a “brane” floating in a space of a larger dimension. The scalars are simply brane coordinates in extra dimensions. However, the only relevant scalar field recently discovered, the cosmological dark energy, has not shown visible variations so far. Observational limits on the variation of the physical constants given in Section 16.1 are quite stringent, allowing only for scalar couplings, which are tiny in comparison with gravity. A possible explanation was suggested by Damour and colleagues [15,16], who pointed out that cosmological evolution of scalars naturally leads to their selfdecoupling. Damour and Polyakov [16] have further suggested that the variations should take place when the scalars become excited by some physical change in the Universe, such as phase transitions, or by other drastic changes in the equation of state of the Universe. They considered several phase transitions, but since the publication of their paper a new transition has been discovered, one from a matter dominated (decelerating) era to a dark-energy-dominated (accelerating) era. This transition is a relatively recent event, corresponding to a cosmological redshift z ≈ 0.5, or a look-back time of approximately 5 billion years. The time dependence of the perturbation related to the transition from the decelerating to the accelerating era could be calculated [17,18]. The calculation shows that the self-decoupling process is effective enough to explain why after this transition the variation of the constants is as small as observed in the present-time laboratory experiments. However, the calculated time dependence is also consistent with the observations of the variation of the electromagnetic fine-structure constant at z 1 [19–21].
16.3
DEPENDENCE OF ATOMIC AND MOLECULAR SPECTRA ON α AND μ
Atomic and molecular spectra are most naturally described in atomic units ( = 4 me = e = 1), with the energy measured in Hartrees (1 Hartree = e m2 e = 2 Ry = 219474.6313705(15) cm−1 ). One could argue that the atomic energy unit itself depends on α as it can be expressed as α2 me c2 , with me c2 the rest energy of a free electron. However, an experimental search for a possible variation of the fundamental constants relies on the observation of the time variation of the ratios of different transition frequencies to one another. In such ratios, the dependence of the units on the fundamental constants cancels out. Below we will use atomic units unless stated otherwise. © 2009 by Taylor and Francis Group, LLC
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In the atomic units, the nonrelativistic Schrödinger equation for an atom with an infinitely heavy, pointlike nucleus does not include any dimensional parameters. The dependence of the spectrum on α enters only through relativistic corrections, which describe the fine-structure, the Lamb shift, and so on. The dependence of atomic energies on μ is known as the isotope effect and is caused by a finite nuclear mass and volume. There are even smaller corrections to atomic energies, which depend on both α and μ and are known as the hyperfine structure. The relativistic corrections to the binding energies of the atomic valence electrons are on the order of α2 Z 2 , where Z is the atomic number, and become quite large for heavy elements. For our purposes, it is convenient to consider the dependence of the atomic transition frequencies on α2 in the form ω = ω0 + qx,
(16.6)
where x = ( αα0 )2 − 1 ≈ 2δα α and ω0 is a transition frequency for α = α0 . Rough estimates of the q-factors can be obtained from simple one-particle models, but in order to obtain accurate values, one has to account for electronic correlations via large-scale numerical calculations. Recently, such calculations have been carried out for many atoms and ions [22–29]. Isotope effects in atoms are on the order of μ ∼ 10−3 and the magnetic hyperfine structure scales roughly as α2 μZgnuc ≈ 10−7 Zgnuc , where gnuc is the nuclear g-factor. One has to keep in mind that gnuc also depends on μ and the quark parameters Xq . This dependence has to be considered when comparing, for example, the frequency of the hyperfine transition in 133 Cs (Cs frequency standard) [5] or the hydrogen 21 cm hyperfine line [30,31] with various optical transitions [5]. At present, there are many accurate experiments comparing different optical and microwave atomic clocks [4,32–39]. These experiments put stringent limits on the time variation of the different combinations of α, μ, and gnuc . As we mentioned above, the limit on the α-variation (16.2) follows from the experiment of Ref. [4] and the limit (16.1) from the assumption of a linear time-dependence of all constants. A detailed discussion of the atomic experiments can be found in recent reviews [40,41]. A comparison of the hyperfine transition in atomic hydrogen with optical transitions in ions on a cosmological timescale was described in Refs. [30] and [31]. This allows one to study the time variation of the parameter F = α2 gp μ, where gp is the proton g-factor. The analysis of the absorbtion spectra of nine quasars with redshifts 0.23 ≤ z ≤ 2.35 yielded δF/F = (6.3 ± 9.9) × 10−6 , ˙ F/F = (−6 ± 12) × 10−16 yr −1 ,
(16.7) (16.8)
which is consistent with a zero variation of μ and α. Molecular spectroscopy opens additional possibilities for studying the variation of the fundamental constants. It is known that μ defines the scales of electronic, vibrational, and rotational intervals in molecular spectra, Eel : Evib : Erot ∼ 1 : μ1/2 : μ. In addition, molecules have fine and hyperfine structure, Λ-doubling, hindered rotation, © 2009 by Taylor and Francis Group, LLC
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and so on. All these effects have different dependencies on the fundamental constants. Obviously, a comparison of these effects allows one to study different combinations of the fundamental constants. The sensitivity to temporal variation of the fundamental constants may be strongly enhanced in transitions between narrow close-lying levels of different types. Huge enhancement of the relative variation δω/ω can be expected in transitions between nearly degenerate levels of atoms [22,24,25,42,43], molecules [3,44–46,49], and nuclei [50,51]. An interesting case of the enhancement of the effect of the variation of fundamental constants arises in collisions of ultracold atoms and molecules near Feshbach resonances [52]. The scattering length A near a resonance is extremely sensitive to the μ-variation: δA δμ =K , A μ
(16.9)
because the enhancement factor K can be quite large; for example, for Cs–Cs collisions, K ∼ 400 [52]. The enhancement can be further increased by tuning the resonance using external fields. Near a narrow magnetic or optical Feshbach resonance, the enhancement factor K may be increased by many orders of magnitude. The calculation of the factor K in Ref. [52] is based on the analytic formula for the scattering length derived in Ref. [53]. This formula is valid for an arbitrary interatomic potential with an inverse-power long-range tale (1/r 6 for neutral atoms); that is, it includes all anharmonic corrections. To the best of our knowledge, it is the only suggested experiment on time variation where the observable is not a frequency. However, another parameter L with the dimension of length is needed to compare with A, and thus render it dimensionless. In Ref. [52], the scattering length was defined in atomic units (aB ). However, it is important, because of the large enhancement in Equation 16.9, that the possible dependence of L on μ becomes irrelevant. For example, if we measure A in conventional units, meters, which are linked to the Cs standard, then δL/L = −δμ/μ, and δ(A/L) δμ = (K + 1) . (A/L) μ
(16.10)
As long as K 1, the dependence of the units used on the fundamental constants can be neglected. Below, we discuss several other experiments with huge enhancement factors, where this argument can be also applied.
16.4 ASTROPHYSICAL OBSERVATIONS OF H2 H2 is the most abundant molecule in the universe and its UV spectrum has been used in the studies of the possible μ-variation for a long time. For a given electronic transition, the frequency of each rovibrational line has different dependence on μ [54,55]. Therefore, comparing rovibrational frequencies from astrophysical observations with laboratory data can provide information on μ. © 2009 by Taylor and Francis Group, LLC
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In the adiabatic approximation, the rovibrational levels of the electronic state Λ with vibrational and rotational quantum numbers v and J are given by the Dunham expansion [56]: l
1 k J(J + 1) − Λ2 , Yk,l v + (16.11) E(v, J) = 2 k,l≥0
where each term depends on μ in the following way: Yk,l ∝ μl+k/2 .
(16.12)
Because of the smallness of the parameter μ, the Yk,l coefficients rapidly decrease with both k and l, and for small v and J, one ends up with the usual vibrational (k = 1) and rotational (l = 1) terms. The zeroth term of the Dunham expansion (k = l = 0) corresponds to the electronic energy. One can define the sensitivity coefficient Ki for each rovibrational transition i of a given electronic band e − g as [55]: 8 dμ dνi Ki ≡ νi μ dEg dEe μ = − , (16.13) Ee − Eg dμ dμ where both energies, Eg and Ee , are given by expansion 16.11. The sign of Ki depends on the rovibrational energies of the excited (e) and ground (g) states (in the absorbtion spectra of the quasars, only transitions from the ground electronic state are seen). The electronic energy, represented by the term Y0,0 , dominates the expansion and the coefficients Ki are rather small. Typically they are on the order 10−2 , but can reach 0.05 for large values of the quantum numbers v and J. The Dunham coefficients of Equation 16.11 are determined by fitting the measured spectra. The sensitivity coefficients Ki can be found by making use of Equations 16.12 and 16.13. Some rovibrational levels of different electronic excited states lie very close to each other. For such levels, additional nonadiabatic corrections can be included within the two-level approximation [57]. A purported μ-variation, δμ, leads to a difference in the observed redshifts zi for different lines ζi ≡
zi − zq,abs δμ = − Ki , 1 + zq,abs μ
(16.14)
where zq,abs represents the redshifts of a quasar absorption system. One can estimate δμ/μ by plotting the reduced redshifts ζi against the sensitivity coefficients Ki . The most recent study [20] of the possible μ-variation using astrophysical data on H2 was based on the observation of the two redshifts zq,abs = 3.02 and 2.59. An analysis of 76 lines from two UV bands of H2 gave the following result: δμ = (−20 ± 6) × 10−6 . μ © 2009 by Taylor and Francis Group, LLC
(16.15)
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This result indicates, at a 3.5σ confidence level, that μ has increased during the past 12 billion years. Assuming a linear time dependence, we can rewrite Equation 16.15 as μ ˙ = (17 ± 5) × 10−16 yr −1 . μ
(16.16)
This has to be compared with the ammonia result of Equation 16.1, which corresponds to a timescale of about 6.5 billion years and is discussed in more detail in Section 16.6.
16.5 ASTROPHYSICAL OBSERVATIONS OF MICROWAVE MOLECULAR SPECTRA In the previous section, we discussed astrophysical observations of UV spectra of H2 . The corresponding absorbtion bands are very strong and can be observed even for objects with very high redshifts. On the other hand, as we have seen, the sensitivity coefficients Ki in Equation 16.13 are rather small. This is because of the smallness of the rovibrational energy compared with the total transition energy. Therefore, it is expedient to study the microwave spectra of molecules, where the relative frequency changes due to a possible variation of the constants are larger.
16.5.1
ROTATIONAL SPECTRA
In 1996, Varshalovich and Potekhin [58] compared redshifts for the microwave rotational transitions (J = 3 → J = 2) and (J = 2 → J = 1) in the CO molecule with redshifts of optical lines of light atomic ions from the same astrophysical objects at redshifts z = 2.286 and z = 1.944. Because the atomic frequencies are independent of μ while the rotational transition frequencies are proportional to μ, this comparison made it possible to set the following limits on the variation of μ: δμ = (−0.6 ± 3.7) × 10−4 , μ
at z = 2.286,
(16.17a)
δμ = (−0.7 ± 1.0) × 10−4 , μ
at z = 1.944.
(16.17b)
In the same paper [58], the authors compared the (J = 0 → J = 1) CO absorbtion line with the 21 cm hydrogen line for an object with z = 0.2467. They did not find any significant difference in the respective redshifts, and interpreted this result as yet another limit on the variation of μ. However, as noted above, the frequency of the hydrogen hyperfine line is proportional to α2 μgp , and so this result actually puts a limit on the variation of the parameter F = α2 gp [59]. Recently, a similar analysis was undertaken by Murphy and colleagues [60] using more accurate data for the same object at z = 0.247 and for a more distant object at z = 0.6847, which led to © 2009 by Taylor and Francis Group, LLC
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the following limits: δF = (−2.0 ± 4.4) × 10−6 , F δF = (−1.6 ± 5.4) × 10−6 , F
at z = 0.2467,
(16.18a)
at z = 0.6847.
(16.18b)
The object at z = 0.6847 is associated with the gravitational lens toward quasar B0218+357 and corresponds to the look-back time of ∼6.5 Gyr. This object was also used by other authors, as will be discussed in Sections 16.5.2 and 16.6.
16.5.2 THE 18 CM TRANSITIONS IN OH Here we consider transitions between the hyperfine substates of the 2 Π3/2 groundstate Λ-doublet of the OH molecule [61–63]. The Λ-doubling for 2 Π3/2 states appears in the third order of the Coriolis interaction and the corresponding energy interval is inversely proportional to the spin–orbit splitting between the 2 Π3/2 and 2 Π1/2 states, that is, it scales as μ3 α−2 , while the hyperfine structure intervals scale as α2 μgnuc . Therefore, the ratio of the hyperfine interval to the Λ-doubling interval depends on the combination F˜ = α4 μ−2 gnuc . Higher-order corrections modify this parameter to F˜ = α3.14 μ−1.57 gnuc [64]. The hyperfine structure splitting for the OH molecule is approximately 50 MHz and is much smaller than the Λ-doubling interval, which is about 1700 MHz. Because of that, it is actually easier to compare the Λ-doubling transitions in OH to the 21 cm hydrogen line, or to rotational lines of the HCO+ molecule [61–64]. The most stringent limit on the variation of F˜ was obtained from observations of the z = 0.765 absorber and the z = 0.685 gravitational lens [64], ˜ F˜ = 0.44 ± 0.36stat ± 1.0syst × 10−5 , δF/
(16.19)
where the systematic error mostly accounts for the possible Doppler noise, due to the difference in the velocities of the molecules in a molecular cloud. The frequencies of the OH Λ-doublet were recently remeasured in the laboratory with a higher precision using cold molecules produced by a Stark decelerator [65]. That may be of relevance to more accurate future astrophysical observations.
16.6
LIMIT ON THE TIME VARIATION OF μ FROM THE INVERSION SPECTRUM OF AMMONIA
In 2004, van Veldhoven and colleagues suggested using a decelerated molecular beam of ND3 to search for the variation of μ in a laboratory experiment [46]. The ammonia molecule has a pyramidal structure with an inversion frequency depending on the exponentially small probability of tunneling of the three hydrogen (or deuterium) atoms through the potential barrier [66]. Therefore, it is very sensitive to any changes of the parameters of the system, particularly to the reduced mass entering the vibrational inversion mode. Van Veldhoven and colleagues found that for the ND3 © 2009 by Taylor and Francis Group, LLC
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molecule, δω/ω = 5.6 δμ/μ. Therefore, the inversion frequency of ND3 is nearly an order of magnitude more sensitive to μ-variation than typical molecular vibrational frequencies (note that Ref. [46] contains a misprint in the sign of the effect). However, even such enhanced sensitivity is insufficient to make a laboratory experiment on the time variation of μ using conventional or Stark-decelerated molecular beams competitive. A molecular fountain seems to be necessary to increase the sensitivity by the several orders of magnitude needed for a competitive experiment. Work on the molecular fountain is in progress [67]. On the other hand, an only slightly smaller enhancement exists for the inversion spectrum of NH3 , often observed in astrophysics even for high z objects. This spectrum was used in Ref. [3] to obtain limit 16.1, which we will now discuss in more detail. There are two bound vibrational states pertaining to the inversion vibration of ammonia. These two levels are split by tunneling through the barrier into inversion doublets. The splitting of the lower doublet corresponds to a wavelength λ ≈ 1.25 cm and is used in ammonia masers. Molecular rotation leads to a centrifugal distortion of the potential energy curve, and so the inversion splitting depends on the rotational angular momentum J and its projection K on the molecular symmetry axis: ωinv (J, K) = ω0inv − c1 [J(J + 1) − K 2 ] + c2 K 2 + · · · ,
(16.20)
where we omitted terms with higher powers of J and K. The measured values of the parameters are ω0inv ≈ 23.787 GHz, c1 ≈ 151.3 MHz, and c2 ≈ 59.7 MHz [68]. In addition to the rotational structure 16.20, the inversion spectrum has a hyperfine structure. For the main nitrogen isotope 14 N, the hyperfine structure is dominated by the electric quadrupole interaction (∼1 MHz) [69]. Because of the dipole selection rule, ΔK = 0 and the levels with J = K are metastable. In beam experiments, the width of the corresponding inversion lines is usually determined by collisional broadening. In astrophysical observations, the lines with J = K are also narrower and stronger than others, but the hyperfine structure of the spectra with high redshifts is unresolved. For our purposes, it is important to know how the parameters in Equation 16.20 depend on the fundamental constants. The molecular electrostatic potential in atomic units does not depend on the fundamental constants (here we neglect small relativistic corrections, which give a weak α dependence). Therefore, the inversion frequency ω0inv and the constants c1,2 are functions of μ only. Note that the coefficients ci depend on μ through the reduced mass of the inversion mode and because they are inversely proportional to the molecular moments of inertia. This implies a different scaling of ω0inv and ci with μ. The inversion spectrum of Equation 16.20 can be approximately described by the Hamiltonian Hinv = −
1 2 1 1 ∂x + U(x) + J(J + 1) − K 2 + K 2, 2M1 I1 (x) I2 (x)
(16.21)
where x is the distance from the N nucleus to the H-plane, I1 and I2 are the moments of inertia perpendicular and parallel to the molecular axis, respectively, and M1 is the © 2009 by Taylor and Francis Group, LLC
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reduced mass for the inversion mode. If we assume that the length d of the N–H bond does not change during the inversion, then M1 = 2.54 mp and 3 mp d 2 [1 + 0.2(x/d)2 ], 2
I1 (x) ≈
I2 (x) ≈ 3mp d 2 [1 − (x/d)2 ].
(16.22) (16.23)
The dependence of I1,2 on x generates a correction to the potential energy of the form C(J, K) x 2 μ. This changes the vibrational frequency and the effective height of the potential barrier, thereby changing the inversion frequency ωinv given by Equation 16.20. Following Ref. [70], we can write the potential U(x) in Equation 16.21 as U(x) =
1 2 kx + b exp −cx 2 . 2
(16.24)
Fitting the vibrational frequencies for NH3 and ND3 gives k ≈ 0.7598 au, b ≈ 0.05684 au, and c ≈ 1.3696 au. A numerical integration of the Schrödinger equation with the potential 16.24 for different values of μ gives the following result: δω0inv ω0inv
≈ 4.46
δμ . μ
(16.25)
It is instructive to reproduce this result analytically. Within the Wentzel–Kramers– Brillouin (WKB) approximation, the inversion frequency is estimated as [71]: ωvib exp(−S) π ωvib 1 a = 2M1 (U(x) − E) dx , exp − π −a
ω0inv =
(16.26a) (16.26b)
where ωvib is the vibrational frequency of the inversion mode, S is the action in units of , x = ±a are the classical turning points at energy E; for the lowest vibrational state, E = Umin + 21 ωvib . Using the experimental values ωvib = 950 cm−1 and ωinv = 0.8 cm−1 , we obtain S ≈ 5.9. Expression 16.26b allows one to calculate the dependence of ω0inv on the mass ratio √ a μ. Let us write the action as S = Aμ−1/2 −a U(x) − E dx, where A is a numerical constant and the square root depends on μ via E. Then dω0inv 1 dS 0 = ωinv − dμ 2μ dμ 1 ∂S ∂S ∂E 0 = ωinv − − . 2μ ∂μ ∂E ∂μ
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(16.27a) (16.27b)
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It is easy to see that ∂S/∂μ = −S/2μ. The value of the third term in Equation 16.27b depends on the form of the potential barrier q S ∂S =− , ∂E 4 Umax − E
(16.28)
where for a square barrier q = 1, and for a triangular barrier q = 3. For a more realistic barrier shape, q ≈ 2. Using the parametrization 16.24 to determine Umax we obtain δω0inv ω0inv
≈
δμ S ωvib δμ 1+S+ = 4.4 , 2μ 2 Umax − E μ
(16.29)
which is close to the numerical result of Equation 16.25. We see that the inversion frequency of NH3 is an order of magnitude more sensitive to the change of μ than typical vibrational frequencies. The reason for this is clear from Equation 16.29: it is the large value of the action S for the tunneling process. Using Equations 16.21 to 16.23, one can also find the dependence of the constants c1,2 in Equation 16.20 on μ [3] δc1,2 δμ = 5.1 . c1,2 μ
(16.30)
It is clear that the above consideration is directly applicable to ND3 , where the inversion frequency is 15 times smaller and Equation 16.26b gives S ≈ 8.4. According to Equation 16.29, this leads to a somewhat higher sensitivity of the inversion frequency to μ, in agreement with Ref. [46],
ND3 :
⎧ δω δμ inv ⎪ ⎪ ⎨ ωinv ≈ 5.7 μ
(16.31)
⎪ ⎪ ⎩ δc2 ≈ 6.2 δμ . c2 μ
We see from Equations 16.25 and 16.30 that the inversion frequency ω0inv and the rotational intervals ωinv (J1 , K1 ) − ωinv (J2 , K2 ) have different dependencies on μ. In principle, this allows one to study time variation of μ by comparing different intervals in the inversion spectrum of ammonia. For example, if we compare the rotational interval to the inversion frequency, then Equations 16.25 and 16.30 give δ{[ωinv (J1 , K1 ) − ωinv (J2 , K2 )]/ω0inv } [ωinv (J1 , K1 ) − ωinv (J2 , K2 )]/ω0inv
= 0.6
δμ . μ
(16.32)
The relative effects are substantially larger if we compare the inversion transitions with the transitions between the quadrupole and magnetic hyperfine components. However, in practice, this method does not work because the hyperfine splitting is much smaller than typical linewidths in astrophysical spectra. © 2009 by Taylor and Francis Group, LLC
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Again, as in the case of Λ-doubling in the OH molecule, it is more promising to compare the inversion spectrum of NH3 with the rotational spectra of other molecules, where δμ δωrot = . ωrot μ
(16.33)
In astrophysical observations, any frequency shift is related to a corresponding apparent redshift δz δω =− . ω 1+z
(16.34)
According to Equations 16.25 and 16.33, for a given astrophysical object with z = z0 , variation of μ leads to a change of the apparent redshifts of all rotational lines δzrot = −(1 + z0 )δμ/μ. The corresponding shifts of all inversion lines of ammonia therefore are δzinv = −4.46(1 + z0 )δμ/μ. By comparing the apparent redshift zinv for NH3 with the apparent redshifts zrot for the rotational lines, we find zrot − zinv δμ . = 0.289 μ 1 + z0
(16.35)
High-precision data on the redshifts of NH3 inversion lines exist for the already mentioned object B0218+357 at z ≈ 0.6847 [72]. Comparing them with the redshifts of the rotational lines of CO, HCO+ , and HCN molecules from Ref. [73], one can obtain the following conservative limit from Equation 16.35 δμ = (−0.6 ± 1.9) × 10−6 . μ
(16.36)
Taking into account that the redshift z ≈ 0.68 for the object B0218+357 corresponds to the look-back time of about 6.5 Gyr, this limit translates into the most stringent present limit 16.1 on the variation rate μ/μ. ˙
16.7
EXPERIMENT WITH SF6
Here we discuss ongoing experiments on the time variation using the SF6 molecule. We start with a recent experiment on the two-photon vibrational transition (v = 0, J = 4) → (v = 2, J = 3) in SF6 [47]. This is a Ramsey-type experiment with a supersonic beam of SF6 molecules. The beam velocity u = 400 m/sec and the length of the interaction region D = 1 m correspond to a linewidth of u/2D = 200 Hz. A CO2 laser was used to drive the two-photon transition and its frequency was controlled by a Cs standard [48]. This means that the vibrational frequency ωvib in SF6 was compared with the hyperfine transition frequency ωhfs in Cs. Therefore, the experiment was sensitive to the combination of the fundamental constants F = gnuc μ−1/2 α2.83 . The measurements were in progress for 18 months, and the following result was obtained: ˙ F/F = (1.4 ± 3.2) × 10−14 yr −1 .
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This limit is weaker than the most stringent limit obtained with atomic clocks. On the other hand, it constraints a different combination of the fundamental parameters. Most importantly, in atomic experiments, the parameters gnuc and μ always scale as a product gnuc μ, while here we have the combination gnuc μ−1/2 . That makes it possible to combine the atomic clock results [4,35,37] with the limit 16.37 to obtain the best laboratory limit on μ-variation: μ/μ ˙ = (3.4 ± 6.5) × 10−14 yr −1 .
(16.38)
This limit is significantly weaker than the astrophysical limit 16.1, but it will likely soon be markedly improved.
16.8
NARROW CLOSE-LYING LEVELS OF DIATOMIC MOLECULES
In this section we focus on narrow, close-lying levels of varying nature in diatomic molecules. Such levels may come about due to a quasi-degeneracy of either hyperfine and rotational levels [45], or between the fine and vibrational levels within the molecular electronic ground state [49] (see Figure 16.1). The transitions between the quasi-degenerate levels correspond to microwave frequencies, which are experimentally accessible, and have narrow linewidths, typically ∼10−2 Hz. The sensitivity K of the relative variation can exceed 105 in such cases.
16.8.1
MOLECULES WITH QUASIDEGENERATE HYPERFINE ROTATIONAL LEVELS
AND
Consider a diatomic molecule with a 2 Σ ground state (one unpaired electron). Examples of such molecules include LaS, LaO, LuS, LuO, and YbF [74]. The hyperfine interval Δhfs is proportional to α2 ZFrel (αZ)μgnuc , where Frel is an additional relativistic (Casimir) factor [76]. The rotational interval Δrot ∝ μ is approximately Quasidegenerate levels
Electronic ground state
Fine structure
Vibration
Rotation
Hyperfine structure
FIGURE 16.1 Energy levels within the ground electronic state of a diatomic molecule (not to scale).
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independent of α. For a molecule with Δhfs ≈ Δrot , the splitting ω between the hyperfine and the rotational levels depends on the following combination ω ∝ μ α2 Frel (αZ) gnuc − const. . (16.39) The relative variation is then given by
δω Δhfs δα δgnuc δμ ≈ , + (2 + K) + ω ω α gnuc μ
(16.40)
where the factor K comes from the variation of Frel (αZ); for Z ∼ 50, K ≈ 1. As long as Δhfs /ω 1, we can neglect the last term in Equation 16.40. The data on the hyperfine structure of diatomics are hard to come by and usually not very accurate. Using the data from Ref. [74], one can find that ω = (0.002 ± 0.01) cm−1 for 139 La32 S [45]. Note that for ω = 0.002 cm−1 , the relative frequency shift is δω δα ≈ 600 . ω α
(16.41)
As new data on molecular hyperfine constants become available, it is likely that other molecular candidates with the requisite quasidegeneracy will be found.
16.8.2
MOLECULES WITH QUASIDEGENERATE FINE-STRUCTURE LEVELS
AND VIBRATIONAL
The fine-structure interval ωf increases rapidly with the nuclear charge Z: ωf ∼ Z 2 α2 ,
(16.42)
In contrast, the vibrational energy quantum decreases with the atomic mass ωvib ∼ Mr−1/2 μ1/2 ,
(16.43)
where the reduced mass for the molecular vibration is Mr mp . Therefore, we obtain an equation Z = Z(Mr , v) for the spectral lines at fixed Z, Mr where we can expect an approximate cancelation between the fine-structure and vibrational intervals: ω = ωf − vωvib ≈ 0,
v = 1, 2, . . . .
(16.44)
Using Equations 16.42 to 16.44, it is easy to find the dependence of the transition frequency on the fundamental constants δω 1 δα 1 δμ δα v δμ = 2ωf + ωvib ≈K 2 + , (16.45) ω ω α 2 μ α 2 μ ω
where the enhancement factor K = ωf is due to the relative frequency shift for a given change of the fundamental constants. Large values of the factor K are experimentally © 2009 by Taylor and Francis Group, LLC
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favorable, as the correspondingly large relative shifts can be more easily detected. However, large values of K do not always guarantee a more sensitive measurement. For some quasidegenerate levels, this factor may become irrelevant [42]. Thus, it is also important to consider the absolute values of the shifts and compare them with the linewidths of the transitions considered. Because the number of different molecules is finite, we cannot expect to find a molecule with ω = 0. However, for many molecules we do have ω/ωf 1 and thus |K| 1. Moreover, an additional “fine tuning” may be achieved by isotopic substitution and the use of suitable rotational, Ω-doublet, and hyperfine components. Therefore, there are two large manifolds available, one arising from the fine structure and the other from the vibrational structure of the same or of different electronic states. If these manifolds overlap, one may select two or more transitions with different signs of ω. In this case, the expected sign of the |ω|-variation must be different (because the variation δω has the same sign) and so one can eliminate some systematic errors. Such an elimination of the systematic errors was carried out by Budker and colleagues [42, 43,75] for the transitions between the close-lying levels in two dysprosium isotopes. The signs of the energy differences between two levels belonging to different electron configurations were opposite for the 163 Dy and 162 Dy transitions used in that work. In Table 16.1, we present a list of molecules reported in Ref. [74] whose ground state is split into two fine-structure levels such that Equation 16.44 is approximately fulfilled. The molecules Cl+ 2 and SiBr are of particular interest. For both molecules, the frequency ω, as defined by Equation 16.44, is on the order of 1 cm−1 and thus comparable to the rotational constant B. This means that ω can be further reduced by a proper choice of isotopes, the rotational quantum number J, and the hyperfine components. New measurements are needed to determine the exact values of the transition frequencies and to identify the best transitions. The required accuracy of the frequency-shift measurements is easily found: according to Equation 16.45, the expected frequency shift is δω = 2ωf
δα 1 δμ + . α 4 μ
(16.46)
TABLE 16.1 Diatomic Molecules with Quasidegeneracy between the Ground-State Vibrational and Fine-Structure Excitations Molecule Cl+ 2 CuS IrC SiBr
Electronic States
ωf
ωvib
K
2Π 3/2,1/2 2Π
645 433.4 3200 423.1
645.6 415 1060 424.3
1600 24 160 350
2Δ 5/2,3/2 2Π 1/2,3/2
Source: Data are taken from Huber, K.P. and Herzberg, G., Constants of Diatomic Molecules, Van Nostrand, New York, 1979. Note: All frequencies are in cm−1 . Enhancement factor K is estimated using Equation 16.45.
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Assuming δα/α ∼ 10−15 and ωf ∼ 500 cm−1 , we obtain δω ∼ 10−12 cm−1 ∼ 3 × 10−2 Hz. In order to obtain a similar sensitivity from a comparison of the hyperfine transition frequencies of Cs or Rb, one would have to measure the shifts with an accuracy of ∼10−5 Hz.
16.8.3 THE MOLECULAR ION HfF+ The list of molecules in Table 16.1 is incomplete due to a lack of data in Ref. [74]. Let us briefly discuss one interesting case, which has been recently brought to our attention. The HfF+ ion and similar ions are being considered by Cornell and colleagues as candidates for a search for the EDM of the electron [77,78]. The ions are to be trapped in a quadrupole radio-frequency trap, to achieve long coherence times. A similar experimental setup can be used to study a possible time variation of the fundamental constants. A recent calculation by Petrov and colleagues [79] suggests that the ground state of HfF+ is a 1Σ+ state and that the first excited state, 3 Δ1 , lies only 1633 cm−1 higher in energy. The calculated vibrational frequencies for these two states are 790 and 746 cm−1 , respectively. For these parameters the vibrational level v = 3 of the ground state is separated by only 10 cm−1 from the v = 1 level of the 3 Δ1 state. Thus, instead of Equation 16.44, we now have 3 (1) 7 (0) ω = ωel + ωvib − ωvib ≈ 0 , 2 2
(16.47)
where the superscripts 0 and 1 correspond to the ground and excited electronic states, respectively. The electronic transition with the frequency ωel is not a finestructure transition, and so Equation 16.42 is not applicable. Instead, by analogy with Equation 16.6, we can write ωel = ωel,0 + qx,
x = α2 /α20 − 1 .
(16.48)
In order to calculate the q-factor for the HfF+ ion, one needs to perform a relativistic molecular calculation for several values of α, which has not yet been done. However, we can make an order-of-magnitude estimate using an atomic calculation for the 3 Yb+ ion [24]. According to Ref. [79], the 1Σ+ 1 − Δ1 transition is the same, in the first approximation, as the 6s–5d transition of the hafnium ion. It is well known that valence s- and d-orbitals of heavy atoms have a different dependence on α: while the binding energy of the s-electrons increases with α, the binding energy of the d-electrons decreases [22–25]. For the same transition of the Yb+ ion, Ref. [24] gives qsd = 10,000 cm−1 . Using this value as an estimate, we can write by analogy with Equation 16.45: 2q δα ωel δμ δω δα δμ ≈ + ≈ 2000 + 80 (16.49) ω ω α 2ω μ α μ δω ≈ 20,000 cm−1 (δα/α + 0.04δμ/μ). Assuming δα/α ∼ 10−15 we obtain δω ∼ 0.6 Hz. © 2009 by Taylor and Francis Group, LLC
(16.50)
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Cold Molecules: Theory, Experiment, Applications
ESTIMATE OF THE NATURAL WIDTHS OF THE QUASIDEGENERATE STATES
As pointed out above, it is important to compare the frequency shifts caused by a possible time variation of the fundamental constants with the linewidths of the corresponding transitions. Let us first estimate the natural width Γv of the vibrational level v: Γv =
4ω3vib ˆ − 1|2 . |v|D|v 3c3
(16.51)
In order to estimate the magnitude of the dipole matrix element, we can write ∂D(R) D0 ˆ = D (R − R0 ) ∼ (R − R0 ) , (16.52) ∂R R=R0 R0 where D0 is the dipole moment of the molecule at the equilibrium internuclear distance R0 . Using the result for the harmonic oscillator, v|x|v − 1 = (v/2mω)1/2 , we obtain Γv =
2ω2vib D02 v 3c3 Mr mp R02
.
(16.53)
For the homonuclear molecule Cl+ 2 , D0 = 0 and Γv vanishes. For the SiBr molecule, −2 Equation 16.53 gives Γ1 ∼ 10 Hz, where we assume D02 /R02 ∼ 0.1 e2 . Let us now estimate the width Γf of the upper state 2 Π3/2 of the fine-structure doublet, 2 Π1/2,3/2 . By analogy with Equation 16.51 we can write Γf =
+2 4ω3f *2 2 Π |D | Π . 3/2 1 1/2 3 3c
(16.54)
This dipole matrix element is written in the molecular frame and includes the summation over all final rotational states. It corresponds to a spin flip and must be zero in the nonrelativistic approximation. The spin–orbit interaction mixes the 2 Π1/2 and 2Σ 1/2 states + + + 2 (16.55) Π1/2 → 2 Π1/2 + ξ 2 Σ1/2 , and the matrix element in Equation 16.54 becomes [80] *
2
+ Π3/2 |D1 |2 Π1/2 ≈ ξ Π|D1 |Σ ∼
α2 Z 2 , 10(EΠ − EΣ )
(16.56)
where EΣ is the energy of the lowest Σ-state. Substituting Equation 16.56 into Equation 16.54 and using energies from Ref. [74], we obtain the following estimate for the SiBr molecule Γf ∼ 10−2 Hz.
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Here we assumed that the unpaired electron in the SiBr molecule is predominantly located on Si (Z = 14) rather then on Br (Z = 35). As a result, the fine-structure splitting in SiBr comes out smaller than that of Cl+ 2 , where Z = 17 (see Table 16.1). the matrix element in Equation 16.54 vanishes For the homonuclear molecule Cl+ 2 due to the g ↔ u selection rule. We conclude that the natural widths of the molecular levels considered here are on the order of 10−2 Hz or less. For comparison, the natural width of the level 2 D5/2 of Hg+ ion (used in an atomic clock experiment [4]) is 12 Hz. The lifetime of levels as narrow as 10−2 Hz may depend on the interaction with the black-body radiation [81]. According to Ref. [81], the lifetimes of the rovibrational levels of polar molecules at room temperature vary from 1 to 100 sec. Again, we can expect that lifetimes for homonuclear molecules are significantly longer. This makes them very promising candidates for superhigh precision experiments [82].
16.9
PROPOSED EXPERIMENTS WITH Cs2 AND Sr2
In this section we discuss two recently proposed experiments with cold diatomic molecules, one with Cs2 at Yale [44,83] and one with Sr2 at JILA [84]. The Yale experiment is based on the idea, described in Ref. [44] of matching the electronic energy with a large number of vibrational quanta. The difference compared with the case described by Equations 16.42 to 16.44 is that here the electronic 3 + transition is between a 1 Σ+ g ground state and a Σu excited state and thus, in the first approximation, its frequency is independent of α. The energy of this transition is about 3300 cm−1 and the number of the vibrational quanta needed to match it is on the order of 100 (see Figure 16.2). For the vibrational quantum number v ∼ 100, the 1000
a3Â+u ESinglet
6s2S1/26s2S1/2
ETriplet
–1200
–1000
–1220 Energy (GHz)
Energy (cm–1)
0
–2000
ETriplet
D ~ 10 GHz
–1240 –1260
ESinglet
–1280
–3000
X1Â+g
–1300 18.0
18.5
19.0
19.5
20.0
20.5
Internuclear separation (au)
5
10
15 20 Internuclear separation (au)
25
1 + FIGURE 16.2 Levels 3Σ+ u and Σg in Cs2 molecule. (Adapted from Sainis, S., Two-color photoassociation spectroscopy of the 3 Σ+ u state of Cs2 , PhD Thesis, Yale University, 2005.)
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density of the levels is high due to the anharmonicity of the potential and hence it is possible to find two nearby levels belonging to two different potential energy curves. This leads to an enhanced sensitivity to variation of μ (cf. Equation 16.44). Cold Cs2 molecules in a particular quantum state can be produced by the photoassociation of Cs atoms in a trap. Let us estimate the sensitivity of this proposed experiment to the variation of α and μ. The electronic transition energy can be determined from Equation 16.48. If we neglect the anharmonicity, we can write the transition frequency between the closely spaced vibrational levels of the two electronic terms as 1 1 ω = ωel,0 + qx + v2 + ωvib,2 − v1 + ωvib,1 , (16.58) 2 2 where v2 v1 . The dependence of this frequency on α and μ is given by δω ≈ 2q
δα ωel,0 δμ − , α 2 μ
(16.59)
where we made use of the inequality ω ωel,0 . For the ground state of atomic Cs, the q-factor is about 1100 cm−1 , which is close to 41 α2 Z 2 ε6s , where ε6s is the ground-state binding energy. If we assume that the same relation holds for the electronic transition in the molecule, we obtain |q| ∼ 41 α2 Z 2 ωel,0 ∼ 120 cm−1 . Using this rough estimate and Equation 16.59 we have δω ≈ −240
δα δμ − 1600 , α μ
(16.60)
where we assumed that the relativistic corrections reduce the dissociation energy of the molecule, as a result of which q is negative. This estimate shows that the experiment with Cs2 is mostly sensitive to the variation of μ. As noted above, for high vibrational states, the potential is highly anharmonic. This significantly decreases the sensitivity as estimated from Equation 16.60. This can be seen either from the WKB approximation [44,83], or from an analytic solution for the Morse potential [84]. The quantization condition for the vibrational spectrum in the WKB approximation R2 1 2M(U(r) − Ev ) dr = v + π (16.61) 2 R1 yields, by differentiation with respect to μ, the following result 1 δμ 2 δEv = , 2ρ(Ev ) μ v+
(16.62)
where ρ(Ev ) ≡ (∂Ev /∂v)−1 ≈ (Ev − Ev−1 )−1 is the level density. For the harmonic part of the potential, ρ is constant, and the shift δEv increases linearly with v, but for vibrational states near the dissociation limit, the level density ρ(E) −→ ∞ and © 2009 by Taylor and Francis Group, LLC
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δEv −→ 0. Consequently, the maximum sensitivity of about 1000 cm−1 is reached at v ≈ 60, and rapidly drops for higher v. TheYale group has found a conveniently close vibrational level of the upper 3Σu state with v = 138. The sensitivity for this level is, however, only about 200 cm−1 [83]. There are still good prospects for finding other close-lying levels with smaller v, for which the sensitivity may be several times higher. The sensitivity as given by Equation 16.60 for the variation of α is in addition reduced by the anharmonicity of the potential. For the highest vibrational levels of the electronic ground state as well as for all levels of the upper (weakly bound) electronic state, the separation between the nuclei is large, R 12 au (see Figure 16.2). Thus, both electronic wavefunctions are close to either symmetric (for 1Σ+ ) or antisymmetric combination (for 3Σ+ ) of the atomic 6s functions, g u 1 Ψg,u (r1 , r2 ) ≈ √ 6sa (r1 )6sb (r2 ) ± 6sb (r1 )6sa (r2 ) . 2
(16.63)
As a result, all the relativistic corrections are (almost) the same for both electronic states. The deleterious effect of the anharmonicity on the sensitivity to the variation of μ and α can be also obtained from the analysis of the Morse potential. Its eigenvalues are given by Ev = ω0 v +
1 2
1 2 ω20 v + 2 − − d, 4d
(16.64)
√ with ω0 = 2πa 2d/M and d the dissociation energy. The last eigenvalue EN is found from the conditions EN+1 ≤ EN and EN−1 ≤ EN . Clearly, EN is very close to zero and is independent of μ and α and thus of their variation. We conclude that the highest absolute sensitivity can be expected for vibrational levels in the middle of the potential energy curve. However, in this part of the spectrum, there are no close-lying vibrational levels pertaining to different electronic states that would allow to maximize the relative sensitivity δω/ω. An option would be to make use of a frequency comb to perform a high-accuracy measurement. This idea has been proposed by Zelevinsky and colleagues [84], who suggested using an optical lattice to trap Sr2 molecules formed in one of the uppermost vibrational levels of the ground electronic state by photoassociation (see Figure 16.3). As we saw above, this level is not sensitive to the variation of μ. In the next step, a Raman transition is proposed to create molecules in one of the most sensitive levels in the middle of the potential well. In this way, it might be feasible to achieve the highest possible absolute sensitivity for a given molecule. Unfortunately, the dissociation energy of Sr2 is only about 1000 cm−1 , which is three times smaller than that of Cs2 . Consequently, the highest sensitivity for the Sr2 molecule occurs at about 270 cm−1 , that is, only slightly higher than for the v = 138 level in Cs2 . Therefore, it may be useful to try to apply this scheme to another molecule with a larger dissociation energy. Finally, we note that the sensitivity to α-variation in the Sr2 experiment is additionally reduced by a factor (38/55)2 ≈ 1/2 because of the smaller Z. © 2009 by Taylor and Francis Group, LLC
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1S
u ¢ = –6
0
+ 3P 1 0u + 1u
u ¢ ~ 40
Photoassociation
1S 0
+ 1S0 X
u = 27, J = 0
Internuclear distance
FIGURE 16.3 The scheme for Raman spectroscopy of Sr2 ground-state vibrational spacings. A two-color photoassociation pulse prepares molecules in the v = vmax − 2 vibrational state (labeled in the figure as v = −3). Subsequently, a Raman pulse couples the v = −3 and v = 27 states via the v ≈ 40 level of the excited 0u+ state. (From Zelevinsky, T. et al., Phys. Rev. Lett., 100, 043201, 2008; arXiv:0708.1806, 2007. With permission.)
16.10
EXPERIMENTS WITH HYDROGEN MOLECULAR + IONS H+ 2 AND HD
The hydrogen molecular ion is very attractive for fundamental studies because of its simplicity and the feasibility of its cooling and trapping (compare with Chapter 18 by + Roth and Schiller). The use of H+ 2 and HD ions for the study of the time variation of the electron-to-proton and the proton-to-deuteron mass ratios μ = me /mp and mp /md has been suggested in Refs. [86] and [87]. Because of the anharmonicity of the ion’s potential, the ratio of the two vibrational transitions with very different vibrational quantum numbers is μ-dependent [86]. As a result, there is no enhancement of the relative sensitivity here, but the lines are very narrow and high-precision measurements are possible using frequency combs. © 2009 by Taylor and Francis Group, LLC
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Recently, HD+ ions have been cooled to 50 mK and trapped in a linear radiofrequency trap [88]. This made it possible to measure the rovibrational transition v, N = 0, 2 → v , N = 4, 3 with an absolute accuracy of 0.5 MHz. By making use of the sensitivity coefficient from Ref. [87], one can see that this accuracy translates into a 5 × 10−9 (5 ppb) accuracy for μ. Note that modern molecular theory of HD+ has achieved a comparable accuracy [89]. Thus, a direct comparison between theory and experiment allows the determination of the absolute value of μ to 5 ppb.
16.11
CONCLUSIONS
Astrophysical observations of the spectra of diatomic and polyatomic molecules can reveal a possible variation of the electron-to-proton mass ratio μ on a timescale from 6 to 12 billion years. However, the astrophysical results obtained so far are inconclusive; see Equations 16.15, 16.19, and 16.36. Much of the same can be said about the astrophysical search for an α-variation. In principle, the astrophysical observations can be explained by a complex evolution of μ and α in space and time. Likely, there are also systematic errors in the measurements that have not been fully understood. Therefore, it is imperative to complement the astrophysical studies with laboratory measurements of the present-day variation of these constants. This work is under way in a number of laboratories. Most use atomic frequency standards and atomic clocks. In this chapter we discussed several recent ideas and proposals on how to increase the sensitivity of the laboratory tests by using molecules instead of atoms. The only molecular experiment [47,48] capable of putting a limit on the time variation of the fundamental constants, Equation 16.37, has made use of a supersonic beam of SF6 . Although this experiment is less sensitive than the best atomic experiments, it probes a different combination of the fundamental constants. In combination with the results of the atomic-clock experiments, [4,35,37], it provides the most stringent laboratory limit, Equation 16.38, on the time variation of μ. The linewidth in this experiment, Γ ≈ 200 Hz, was determined by the time of flight through a 1 m Ramsey interferometer. A similarly broad linewidth has been a liability in an ND3 experiment [46]. The use of cold molecules holds the promise of reducing the linewidth by several orders of magnitude and hence to dramatically enhance the sensitivity of the molecular experiments. We have seen that, for diatomic radicals, such as Cl+ 2 and SiBr, there are narrow levels belonging to different electronic states that are separated by 1 cm−1 , while the natural widths of these levels are on the order of 10−2 Hz. This comes close to what is needed in order to reach the sensitivity of δα/α ∼ 10−15 , similar to that achieved in the best atomic laboratory tests. In the high-precision frequency measurements, the accuracy is typically better than the linewidth by a few orders of magnitude. In order to be able to benefit from such narrow lines, the molecules need to be cooled. In this respect, the Cl+ 2 ion seems particularly promising. An even higher sensitivity to the temporal variation of α is found in HfF+ and similar molecular ions, which are being considered as candidates for the search for the electron EDM at JILA [77–79]. The transition amplitude between the 3 Δ1 and © 2009 by Taylor and Francis Group, LLC
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states of HfF+ is suppressed. The transition width is larger than those of Cl+ 2 and SiBr because of the larger value of Z and the higher frequency ωf . In Ref. [79], the width of the 3 Δ1 state was estimated to be about 2 Hz. This width is of the same order of magnitude as the expected frequency shift for δα/α ∼ 10−15 . At present, not much is known about these molecular ions. More spectroscopic and theoretical data are required in order to estimate the sensitivity to α-variation reliably. We hope that this monograph will further stimulate such studies. An added benefit is the possibility to measure the electron EDM and the α-variation on the same molecule and with a similar experimental setup. A preliminary spectroscopic experiment with the Cs2 molecule has been recently 1 − performed at Yale [83]. The electronic transition between the 3Σ+ u and Σg states of Cs2 is independent, in the first approximation, of α. On the other hand, the sensitivity to the μ-variation may be enhanced because of the large number of vibrational quanta needed to match the electronic transition. However, the anharmonicity of the potential suppresses this enhancement for very high vibrational levels near the dissociation limit. As a result, the sensitivity to the variation of μ for the v = 138 level is about the same as in Equation 16.46. It is possible that there are other close-lying levels with smaller vibrational quantum number that, consequently, allow for a higher sensitivity. Even if such levels are not found, the experiment with the v = 138 level may improve the current limit on the time variation of μ by several orders of magnitude. An experiment with the Sr2 molecule has recently been proposed at JILA [84]. This experiment has a similar sensitivity to the time variation of μ as the experiment with Cs2 ; these experiments are complementary to the experiments with the molecular radicals, which are mostly sensitive to the time variation of α [49]. Finally, we have shown that the inversion spectra of polyatomic molecules, such as NH3 and ND3 , are potentially even more sensitive to the time variation of μ. This has already been used in astrophysical measurements to put the most stringent limit, Equation 16.36, on the time variation of μ on a cosmological timescale. The corresponding laboratory experiments would require slow molecules, and molecular fountains and traps hold the promise of providing them [46]. 1Σ
0
ACKNOWLEDGMENTS We want to thank J. Ye, D. DeMille, and S. Schiller for their extremely useful comments. We are particularly grateful to D. Budker, whose advice significantly improved the whole manuscript.
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Part VI Quantum Computing
© 2009 by Taylor and Francis Group, LLC
Quantum Information 17 Processing with Ultracold Polar Molecules Susanne F. Yelin, Dave DeMille, and Robin Côté CONTENTS 17.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Quantum Computers: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Quantum Information and Entangled States . . . . . . . . . . . . . . . . . . . . . . . 17.2.2 Platforms to Implement Quantum Computers . . . . . . . . . . . . . . . . . . . . . 17.2.3 Wishlist: Properties of Polar Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Scheme with Permanent Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 General Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 Experimental Parameters and Decoherence . . . . . . . . . . . . . . . . . . . . . . . 17.4 Schemes with Switchable Dipoles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.1 General Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2 Decoherence and Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2.1 Dipole–Dipole Interaction Strength. . . . . . . . . . . . . . . . . . . . . . 17.4.2.2 Molecular State Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2.3 Trap-Induced Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Outlook and Other Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5.1 Superconducting Microwave Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5.2 Optical Quantum Computing in Polar Ensembles. . . . . . . . . . . . . . . . . 17.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1
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INTRODUCTION
Historic developments in information technology have had an unprecedented impact on society. The past few decades have witnessed an explosion of computing capacity consistent with Moore’s law, which asserts that processor power doubles every 18 months. This trend has been accompanied by a commensurate increase in processor 629 © 2009 by Taylor and Francis Group, LLC
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density, one that cannot continue unchecked without dramatic changes to the basic circuit elements of modern processors. Indeed, existing circuit elements are reaching the quantum limit, the size scale where quantum phenomena begin to dominate the physics governing their behavior. While this prediction itself motivates the study of “quantum” information and computation, discoveries of the last decade have revealed that quantum phenomena can be exploited to yield a qualitatively new computational paradigm, one that offers provable speedup over its classical counterpart for a wide variety of computational problems. Indeed, efficient quantum algorithms have been designed for problems believed to be intractable for classical computation, like integer factorization or large database search [1,2]. While this promise of a new, more efficient, frontier of computation is exciting, construction of general purpose quantum computing equipment remains a significant engineering challenge. In fact, it is essential to address coherently quantum states in a system and to perform reversible quantum logic operations. This can be achieved only if the qubits can interact via controlled coherent physical processes; preserving coherence is crucial because quantum interference and entanglement are extremely fragile. Many systems are being studied to manipulate quantum information: cold trapped ions, neutral atoms in optical lattices, atoms in crystals, particle spins, photons in cavity QED or nonlinear optical setups, or mesoscopic ensembles. Polar molecules present a promising new platform for quantum computation [3–5], because they incorporate the prime advantages of neutral atoms (scalability to large numbers of bits) and trapped ions (strong interactions), along with the long coherence times characteristic of both. For example, schemes using entanglement of vibrational eigenstates and optimal control have been proposed. Here, we describe schemes based on polar molecules, where the dipolar interaction is crucial to the implementation of quantum gates. The advances in cooling and storing [6] techniques are beginning to make the precise manipulation of single molecules possible. In addition, polar molecules could be integrated into condensed-matter physics architectures, using, for example, molecule-chips or microtraps connected to superconducting wires. In a recent article [5], the experimental implementation of quantum information processing using superconducting stripline resonators has been studied in detail. After a brief review of some basic concepts in quantum information processing, such as entangled states and quantum logic gates, we list various systems considered to implement these ideas. We then focus our attention on the use of polar molecules for these applications. After a brief overview of their properties, we show how the strong interaction between polar molecules can be used to implement universal logic gates, from which any quantum algorithm can be constructed. We describe two main schemes for the crucial two-qubit gates, focusing on their implementation in optical lattices: one based on large permanent dipole moments, and one based on “dipolar switching,” where dipole moments are changed via occupation of different molecular states. We discuss some sources of decoherence and errors and how they can be controlled. Finally, we briefly describe other ideas related to quantum computing with polar molecules, such as a platform based on supercondutcing striplines, using ensembles of polar molecules, or a combination of both. © 2009 by Taylor and Francis Group, LLC
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17.2
QUANTUM COMPUTERS: AN OVERVIEW
In this section, we briefly describe the foundations of quantum information and computing. We list some physical systems that have been explored to create them and show that polar molecules represent a very attractive possibility for the realization of quantum logic gates.
17.2.1
QUANTUM INFORMATION AND ENTANGLED STATES
The individual unit of classical information is the bit: an object that can take either one of two values, say 0 or 1. The corresponding unit of quantum information is the quantum bit or qubit. It describes a state in the simplest possible quantum system [1,2]. The smallest nontrivial Hilbert space is two-dimensional, and we may denote an orthonormal basis for the vector space as |0 and |1. A single bit or qubit can represent at most two numbers, but qubits can be put into infinitely many other states by a superposition: |Q(1) = c0 |0 + c1 |1
(17.1)
with |c0 |2 + |c1 |2 = 1,
ci ∈ C.
If prepared in a general superposition state, quantum registers consisting of N qubits can store 2N bits of information simultaneously, as compared to classical registers where only N bits of information are stored. However, not all the information contained in quantum memories can be accessed by physical measurements. Nevertheless, so-called quantum parallelism makes quantum computers very fast: they can process quantum superpositions of many numbers in one computational step, where each computational step is a unitary transformation of quantum registers. To achieve this, a universal quantum computer should be able to perform an arbitrary unitary transformation on any superposition of states. Like classical computers, quantum computers can be built out of quantum circuits composed of a set of elementary logic gates. Such a set is universal if it can be used to design any possible computation. It is by now known that all sets of universal gates must include both a two-qubit nonlocal interaction (giving rise to “entangled” states) and a local operation (on a single qubit). One example of a nontrivial two-qubit operation is a so-called phase gate. This denotes a unitary operation Uφ that adds a conditional phase on a two-qubit state, for example, Uφ
α|00 + β|01 + γ|10 + δ|11 −→ α|00 + β|01 + γ|10 + eiφ δ|11. For any φ = 0, such a state cannot be written as the product of any single qubit states. Two qubits in such a state are strongly (and nonlocally) correlated. By using a sequence of such phase gates with φ = π, and simple unitary operations on a single qubit, any quantum computation can be completed. © 2009 by Taylor and Francis Group, LLC
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A similar example is the controlled not (CNOT) gate. In this operation, the state of the target qubit |B is flipped (i.e., |0B ↔ |1B ) if the control qubit |A is in the state |1A ; otherwise qubit |B is not affected. Just as for the phase gate, the combination of CNOT gates and single-qubit gates are sufficient to perform any arbitrary computation. The conditional nature of the CNOT gate is transparently similar to the minimal requirements for classical computation, where a single conditional two-bit logic gate (e.g., NAND) is sufficient. To take advantage of parallelism and entanglement, quantum algorithms are being devised. Three of the best known quantum algorithms illustrate the various speedups expected when compared to classical computing times. Notably, Shor’s algorithm for factoring large numbers is the perfect example of a truly exponential speedup: the problem can be solved in polynomial time instead of the exponential time required for classical computers. Although not all algorithms rely on entanglement, most do: for that reason, entanglement is a critical property that quantum computers should possess. The information contained in qubits and entangled states includes the phase: any error in the phase has important implications (e.g., changing an entangled to a product state). To perform complex quantum computations, we need to reliably prepare a delicate superposition of states of a relatively large quantum system, which cannot be perfectly isolated from the environment; hence the superpositions always decay. The decoherence of an entangled state is even faster: the nonlocal correlations are extremely fragile and decay very rapidly. Also, applications of unitary transformations to qubits will not be flawless, and errors can accumulate. Recent theoretical developments in quantum error correction have addressed these points, and it has been shown that quantum computing can be fault-tolerant.
17.2.2
PLATFORMS TO IMPLEMENT QUANTUM COMPUTERS
A burgeoning experimental effort to process coherent quantum information is taking place. To build hardware for a quantum computer, technology that enables the manipulation of qubits is needed. The requirements for that technology are given, for example, by DiVincenzo [7]; the most elusive ones are the following: 1. A set of individually addressable qubits in which coherence can be stored long enough to complete interesting computations. 2. Quantum gates to perform nontrivial two-qubit interactions; this can be achieved only via controlled, coherent interactions among qubits. 3. Reliable and efficient readout methods. Many systems are being studied to manipulate quantum information. Some make use of individual atoms: cold trapped ions, neutral atoms in optical lattices, atoms in crystals. Other involve particle spins or photons in cavity QED or nonlinear optical setups as well as more exotic ones where geometric combinations of elementary excitations are defined as qubits, such as in topological quantum computing [8]. However, none of these systems has yet emerged as a definitive way to build a quantum information processor.A reason for this is that there is an essential dichotomy: we need © 2009 by Taylor and Francis Group, LLC
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weak coupling to the environment to avoid decoherence, but also strong coupling to at least some external modes to manipulate qubits and have controlled interactions.As we will see in the next section, polar molecules represent a good candidate for obtaining quantum logic gates in which the two required features can peacefully coexist. To implement quantum information processing, polar molecules could be stored in a one- or two-dimensional array so that their dipole moments could be aligned by an electric field perpendicular to the array. We assume the full development of the storage and addressing capabilities of two recently proposed architectures (see Figure 17.1). The first is an optical lattice with a lattice spacing of about 1 μm, as suggested in Ref. [3]. Using a dc field for dipole alignment during trapping naturally allows the repulsive dipole–dipole interaction to aid with homogeneous distribution in the lattice. In this case, addressing single qubits can be accomplished by either using the inhomogeneous dc electric fields proposed in Ref. [3] to create individualized transition (a) Electric field Large dipole Excitation lasers
Molecule Small dipole (b)
Molecules A
B h
Superconducting wire
L
FIGURE 17.1 (a) Molecules are individually addressable by lasers and are stored in an optical lattice. (b) Superconducting wires are used to “deliver” the interaction. In both, molecules are selectively excited, and interact only if both are in |e. (From Yelin, S.F. et al., Phys. Rev. A, 74, 050301, 2006; Kuznetsova, E. et al., Phys. Rev. A, 78, 012313, 2008.)
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frequencies, or by individual addressing with light in the visible part of the frequency spectrum. The second architecture is based on a “stripline wire” architecture, as suggested at Yale and Harvard [5]. Here, molecules sit on their own small traps, which also serve for addressing, and are connected via a superconducting transmission line that allows for long-range dipole–dipole interaction, effectively replacing the 1/r 3 term in Equation 17.2 (later) with 1/h2 r, where h is a characteristic scale describing both the distance of the molecules from the transmission line, and the size of the line itself. Here, the fields need to be in the microwave range.
17.2.3 WISHLIST: PROPERTIES OF POLAR MOLECULES Polar molecules have a permanent dipole moment with respect to their molecular axis, but which averages to zero in the laboratory frame. An electric field with a given orientation in the laboratory frame will orient the molecules and align the two frames. This means that rotational states are mixed, leading to observable Stark shifts of the energy levels of the molecules. Now, because the dipole moment in the molecular frame μ(R) is a function of the separation R of the atoms, the dipole moment depends on the exact internal rovibrational state of the molecule. Usually, because μ(R) is small at large separation, the upper rovibrational states have small dipole moments, while the deeper ones have larger dipole moments. To store the information, a variety of internal molecular states can be used. In addition to specific rotational and/or vibrational states, one can also employ hyperfine states by using molecules with nuclear and electronic spins. Overall, we can list five requirements on properties required of polar molecules that would ensure a realistic implementation of quantum gates: 1. Choice of qubits. We need long-lived states to store the information. These should interact as little as possible with the environment to minimize decoherence. Good candidates are hyperfine and rotational states of a ground electronic molecular state. 2. Coupling strength. Fast one- and two-qubit gates require large interaction strengths, while storage needs long lifetimes. So, transitions between the storage states |0 and |1 should have long lifetime. Either direct transitions, or Raman transitions via intermediate states, could be used to perform one-qubit operations. 3. Robust dipole–dipole interaction. For two-qubit gates, strong interactions are required. The interaction strength can be maximized by choosing the smallest possible (effective) distance between qubits, and also by using molecules with large dipole moments. 4. Cooling and trapping. To fulfill the previous requirements, molecules must be cooled to sub-Kelvin temperatures. This prevents the population of undesired rotational states. Also, the molecules must be tightly trapped, since fluctuations in their position leads to errors in gate and readout operations. 5. Decoherence. In order to store the information for long durations, we need to minimize the interaction with the trapping potential, the environment, and © 2009 by Taylor and Francis Group, LLC
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all other fields used to manipulate the qubits. It helps to use states with negligible dipole moments to store the information, because these states interact only weakly with each other, the environment, or other fields present in the setup. (Use of identical dipole moments for storage also has a similar effect, because the only differential interactions affect the coherence.) Switching on and off strong dipoles at the appropriate time is conceptually attractive for controlling decoherence, but the switching itself must be performed carefully lest it become the largest source of decoherence.
17.3
SCHEME WITH PERMANENT DIPOLES
In this section, we discuss a scheme based on permanent dipoles.
17.3.1
GENERAL NOTIONS
A conceptually simple scheme for using polar molecules as qubits in a large-scale quantum computer was first introduced in Ref. [3]. In this chapter, qubit state |0 (|1) was chosen to be the ground (first excited) rotational level (of a simple diatomic rigid rotor), with angular momentum projection m = 0 along an external polarizing electric These polarized states have nonzero expectation values of the electric dipole field E. Over a wide range of electric field values, the moment operator μ along (against) E. value of μ is constant for both states; hence in this range one can think of the qubit states as simply “dipole up” (|0 = | ↑) and “dipole down” (|1 = | ↓). Single-qubit operations can be performed by applying pulses of resonant radiation at frequency ω = ΔE/, where ΔE is the energy required to flip the electric dipole moment of a qubit: ΔE = (μ ↑ − μ ↓ )E. The drive fields could be either in the microwave regime (for direct excitation) or the optical (using two fields with frequency difference ω to drive resonant Raman transitions). A large, regular spatial array of these qubits was proposed to be assembled in an optical lattice. In Ref. [3] this was described as a one-dimensional lattice with strong transverse confinement, but the scheme can be extended in a straightforward manner to a two-dimensional array as well. It was proposed to address individual sites in the lattice spectroscopically, by applying an electric field gradient along the array. This ensures that the qubit energy ΔE depends on its location in the array. Two-qubit CNOT gates can also be performed by spectroscopic control in this arrangement. This is easiest to envision in the case of two isolated qubits (see Figure 17.2). The fringing at the location electric field from dipole a adds or subtracts from the external field E, of dipole b. This in turn means that the energy required to flip dipole b depends on the state of a. By tuning the drive frequency appropriately, then, it is possible to perform a rotation of qubit b if and only if qubit a is in the state |↑. This is exactly the requirement for a CNOT gate. Indeed, this is the same method used for conditional logic operations in NMR-based schemes for quantum computing [1]. This particular gate implementation also makes evident the relationship between coupling strength and gate speed; that is, via the energy–time uncertainty principle, the time needed to distinguish between the two conditions of qubit a, while flipping qubit b, is inversely proportional to the strength of their interaction. © 2009 by Taylor and Francis Group, LLC
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(b) With interaction Æ Æ H' = A m a · m b
11
01
10
00
FIGURE 17.2 CNOT gate implementation with spectroscopic control. The figure shows the energy-level diagram for a two-qubit system, where the level splitting is different for qubits a and b. A CNOT gate with control qubit b and target qubit a can in principle be performed, by applying a resonant π-pulse on qubit a conditioned on the state of bit b. (a) The level diagram in the absence of interactions between the qubits. Here, the resonant frequency to drive the |0 ↔ |1 transition in qubit a is independent of the state of b, so the CNOT gate cannot be performed. (b) The level structure in the presence of the dipole–dipole interaction, with strength parameterized by A. Here the resonant drive frequency for qubit a is different for the two states of qubit b. Thus the CNOT can be performed successfully as long as energy resolution ΔE < A is maintained; this requires a time τ 1/A.
At first glance, it may appear impossible to generalize this scheme to a large number of coupled bits, due to the presence of this type of “always-on” dipole–dipole interaction. However, a scheme for effectively eliminating always-on interactions was also devised for NMR-based quantum computers [1]. This scheme, known as “refocusing” in the NMR community (and “spin echo” more generally), can be understood simply as follows. Suppose one wants to perform a CNOT operation on a particular pair of neighboring qubits. During the (long) duration of the CNOT drive pulse, one can also apply strong resonant pulses to each of the surrounding qubits, effectively flipping their dipole. If each surrounding qubit spends an equal amount of time in the dipole up vs. down state during the CNOT operation on the selected pair, the interactions with the surrounding qubits will average to zero. (This method depends on the fact that the resonant frequency for each individual bit is separated from the frequencies of its neighbors by much more than the interaction energy; this condition is easily met for the proposed system.) It may also appear that the complexity of the refocusing procedure may prevent scaling to large numbers of qubits. Remarkably, however, it has been proven that—at least in principle—this type of refocusing requires resources that grow only polynomially with the number of qubits in the system [9]. Nevertheless, in practice the refocusing operations will be extremely complex © 2009 by Taylor and Francis Group, LLC
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for a large array, and methods to reduce or eliminate the always-on interactions, as discussed later, would have significant benefits. Readout of the final state of the system also requires site-by-site spatial selectivity. In Ref. [3], it was proposed to perform state-selective resonance-enhanced multiphoton ionization (REMPI), followed by electrostatic magnification of the resulting ion pattern and finally imaging with a microchannel plate. Single-site resolution is at the edge of technological feasibility in this scheme. However, the requirement on readout spatial resolution can be relaxed with a “shift register” readout scheme. Here, the qubits can be selectively transferred from, for example, the |1 state to a higher rotational “storage” state |s, only on sites 1, n + 1, 2n + 1, and so on, again using the electric field gradient for spatial selectivity. Then REMPI detection of state |s determines the qubit state on these sites. Next, sites 2, n + 2, 2n + 2, and so on are read out in the same way. After n iterations, the entire array is read out, but the requirement on spatial resolution for each step is relaxed by a factor of n compared to when the entire array is read out at once.
17.3.2
EXPERIMENTAL PARAMETERS AND DECOHERENCE
The use of an optical lattice trap constrains the system in several different ways. For example, current technology limits the depth of such traps to values corresponding to temperatures of mK or lower. This in turn means that the molecules must be at much lower temperature, that is, in the μK range or (preferably) lower. This has not yet been achieved experimentally, although recent experimental progress toward this goal is rapid and encouraging. The choice of lattice wavelength λ is also critical. Since the dipole–dipole interaction scales as 1/r 3 , faster gates can in principle be made available by making the spacing between lattice sites r = λ/2 as small as possible. However, the lattice itself can lead to decoherence in the system, via inelastic (spontaneous Raman) scattering of lattice photons. Such scattering leaves molecules in excited rotational or vibrational states, outside the qubit state space. Moreover, for a given lattice potential depth, this inelastic scattering rate generically scales as 1/λ4 (for sufficiently large detunings of the lattice laser frequency from any resonance). Hence, the ratio of gate speed to lattice-induced decoherence generally leads one to consider longer-wavelength lattices. We next give some typical numerical values for relevant parameters in this type of architecture. For concreteness, we consider the use of SrO molecules; SrO is a good candidate species for direct trapping and cooling [6]. We assume that a sample of SrO is available with sufficiently low temperature (again, in the μK range or lower) and high density (n (λ/2)3 ) to load the optical lattice. The optimal lattice wavelength depends on the electronic transition frequencies of the molecular species; for SrO, λ ∼ 1 μm is near optimal. (For some other species this could be a factor of 2 to 3 smaller.) SrO has an unusually large molecule-fixed dipole moment μ = 8.9 D, yielding an effective dipole moment μeff ≡ μ↑ − μ↓ ≈ 2.6ea0 . SrO has a rotational constant B ≈ 10 GHz, which sets the scale both for the useful range of electric field values (E ≈ 4 to 10 kV/cm) and for the single-bit transition frequencies (ω ≈ 2π × 35 to 60 GHz over the entire range of electric field). Because of the large dipole moment of SrO, the interaction strength ΔEint is substantial. Numerically, © 2009 by Taylor and Francis Group, LLC
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2 /(λ/2)3 ≈ 2π × 100 kHz. This sets the timescale for two-qubit gates: ΔEint = deff τg2 ≈ 2π/ΔEint ≈ 10 μsec. In principle, the time needed for a single-qubit operation, τg1 , is limited only by the need to resolve the transition frequencies between adjacent qubits in the array; that is, with N qubits in the array, the frequency difference between transitions for any two qubits will be δω ≈ (ωmax − ωmin )/N, and τg1 ≈ 1/δω. Even for very large numbers of qubits N 3 × 105 , τg1 < τg2 . In practice, however, the time to perform a single-qubit operation will be limited for another reason. In particular, the gate operations can excite motion of molecules in the lattice. This is easily understood, since, for example, due to the electric field gradient, the location of the trap minimum is slightly shifted for the states |↑ and |↓. This motion can in turn lead to decoherence and/or gate infidelities because of the uncertain or fluctuating value of the electric field at any site. However, motional excitation can be avoided by performing gate operations adiabatically, with respect to the motional oscillation frequency ωL of the molecule in its lattice site. For typical lattice conditions (e.g., trap depth ∼100 μK), ωL ∼ 2π × 100 kHz, which means that in practice τg1 ≈ τg2 for this example. Note that increasing the trap depth can allow for faster one-qubit gates, at the expense of greater decoherence via Raman scattering. Under the conditions outlined here, this decoherence rate would be Rs ∼ 0.3 sec−1 . Several other potential sources of decoherence, due to both technical noise and fundamental processes, were considered in Ref. [3]. Electric field noise couples directly to the qubit transition frequency and must be deeply suppressed. This noise could arise either from voltage fluctuations, or from relative motion of the electrodes relative to the lattice. Fluctuations in the laser intensity couples to the qubit transition frequency via the optical-frequency tensor polarizability of the molecule, which is typically a few times smaller than the scalar polarizability responsible for the trapping itself. It appears that both these effects could in principle be controlled sufficiently, that the Raman scattering process dominates the decoherence rate. However, actually achieving this level of control would clearly present a serious technical challenge. However, the degree of control needed could be reduced by several orders of magnitude, by using a scheme (like that mentioned earlier) to encode qubits into hyperfine or nuclear spin states instead of directly into rotational states.
17.4
SCHEMES WITH SWITCHABLE DIPOLES
In this section, we discuss schemes based on dipole–dipole interactions that can be switched on and off.
17.4.1
GENERAL NOTIONS
We first describe the generic setup to obtain a phase gate, or universal two-qubit operation, in Figure 17.3 [10]. We assume that the molecules are individually addressable by optical or microwave fields. (Alternatively, a frequency-based addressing scheme akin to the one in the previous section could be used.) We choose |0 and |1 as, for example, hyperfine states within a zero-dipole-moment manifold, in a level with a long coherence time; and |e as a metastable state in a large-dipole-moment manifold. © 2009 by Taylor and Francis Group, LLC
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e
e
t1
t1
t2 Zero dipole moments
t2
1
1
0
0 r A
B
FIGURE 17.3 Phase gate: two molecules A and B separated by r are prepared in a superposition of states |0 and |1. At t1 = 0, we excite |1 of both into |e: both interact via dipole– dipole interactions, and acquire a phase φ. At time t2 = τ such that φ = π, we stimulate coherently both |e back to |1. (Adapted from Yelin, S.F. et al., Phys. Rev. A, 74, 050301, 2006; Kuznetsova, E. et al., Phys. Rev. A, 78, 012313, 2008.)
Single-qubit rotations can be accomplished with optical or microwave fields. The initial states of two individual sites a and b can be prepared in a superposition state, for example, using π/2 Raman pulses. To perform a two-qubit gate, a one- or two-photon transition couples |1 and |e coherently, but not |0 and |e. This can always be accomplished by either polarization or frequency selection. The molecules interact via a dipole–dipole interaction only if both are in the |e state; in this case the system acquires a phase φ(t). After a time t = τ such that φ = π, we coherently stimulate the state |e back to |1. This can be summarized by |00 |01 |10 |11
π−pulse
−→ −→ −→ −→
|00 |0e |e0 |ee
dip−dip
−→ −→ −→ −→
|00 |0e |e0 −|ee
π−pulse
−→ −→ −→ −→
|00 |01 . |10 −|11
The resulting transformation corresponds to a phase gate. The π-phase shift produced in the time τ between the exciting and de-exciting π-pulses is given by 1 τ μ φ=π= dτ 3 3 cos2 θ − 1 ρ2e (τ ), (17.2) 0 r where μ and ρe are the dipole moment and fractional population in the excited state, r the distance between molecules a and b, and θ the angle between the dipole moments. © 2009 by Taylor and Francis Group, LLC
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This formulation allows for finite excitation and de-excitation times and imperfect π-pulses. We now describe three possible setups utilizing variations of our switchable phasegate scheme [10]. The first system is based on carbon monoxide (CO).As far as dipolar molecules are concerned, CO is an anomaly; while its electronic ground state X 1 Σ+ has a very small dipole moment (μ ≈ 0.1 D in the vibrational ground state, which is expected to be the easiest to trap), there exists a very long-lived (τlife ≈ 10 to 1000 ms) excited electronic state a3 Π with a large dipole moment, μ ≈ 1.5 D. As “0” and “1,” we choose, for example, the two nuclear spin projection states of X 1 Σ+ , v = 0, N = 0, I = 1/2, F = 1/2 of 13 CO [11]. With a magnetic-field-induced Zeeman splitting, selective excitation from |1 to |e is possible. The transition frequency between X 1 Σ+ and a3 Π is in the UV (about 48,000 cm−1 ), and the optical lattice architecture would be the ideal choice. With a coherence time in an optical lattice of a few seconds [3] and a necessary dipole–dipole interaction time of several milliseconds, there can be about 103 operations. The scheme, however, is very straightforward, the techniques are in place or nearly so, and CO is a very well studied molecule. A more common situation can be found in molecules such as alkali hydrides or mixed alkali dimers, for example, LiH or LiCs. These molecules have large permanent dipole moments μ (as large as 7 D) in their ground electronic state X 1 Σ+ (for |0 and |1), and a metastable electronic state a3 Σ+ (for |e) for which the potential well is located at large nuclear separation and supports at least one bound state; in most cases, these triplet states have permanent dipole moments close to zero. These properties can be used to implement a scheme in all important points similar to the CO scheme, except for three details. First, the phase gate would be “inverted,” that is, |00 → −|00, |01 → |01, |10 → |10, and |11 → |11. Second, it requires the molecules to be stored, with the help of an aligning dc electric field, in the largedipole state, which would most likely lead to seriously shortened coherence times. In addition, the interaction would happen for all molecules, not just the two we wish to be coupled by a phase gate. However, this can be mitigated for example, by switching on an aligning dc field only during interaction times, and for exactly a 2π phase shift, as measured for the lower states. By adding together this 2π phase shift using a dc field and the “negative” π phase shift for the molecules in the |e state, the phase gate is given by |00 |01 |10 |11
exc+dc
−→ −→ −→ −→
|00 |0e |e0 |ee
π
−→ −→ −→ −→
−|00 |0e |e0 |ee
de−exc
−→ −→ −→ −→
−|00 |01 |10 |11
dc
−→ −→ −→ −→
|00 −|01 . −|10 −|11
Note that the scheme described for CO could be adapted for these molecules by using two vibrational, rotational, or hyperfine states of a3 Σ+ for the doublet |0 and |1, and a low-lying vibrational state of X 1 Σ+ as |e. The last setup we propose here is the “rotational scheme.” It utilizes the fact that in the rotational ground state N = 0 the dipole moment μ is, in fact, zero (as for any pure rotational state). We choose for all states the electronic and vibrational ground © 2009 by Taylor and Francis Group, LLC
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Rotational levels
N=2
N=2
N=0 ˙ 0Ò
˙ 1Ò
˙ eÒ
Hyperfine states
FIGURE 17.4 Example of level system for “rotational scheme”: all states are part of the electronic and vibrational ground state. For example, a laser Raman π-pulse can transfer |1 to a storage state |s that is a sublevel of for example, the N = 2 state. Then a microwave π/2 pulse transfers |s to the superposition state |e ∝ |e1 + |e2 , where |e2 = |s and |e1 is a sublevel of the N = 1 manifold. Alternatively, if lasers are not needed for spatial selectivity, |1 could simply be transferred to the superposition state |e ∝ |1 + |e1 , again with |e1 in the N = 1 manifold. (Adapted from Yelin, S.F. et al., Phys. Rev. A, 74, 050301, 2006; Kuznetsova, E. et al., Phys. Rev. A, 78, 012313, 2008.)
state. While |0, |1 are also in the rotational ground state N = 0, |e is the superposition of neighboring rotational states |e = |e1 + |e2 , as shown in Figure 17.4 [12]. (This superposition can, for example, be created using a microwave field.) Because both |0 and |1 are in the absolute ground state with exactly zero dipole moment, this system has several advantages: maximum coherence time, ease of storage, and no residual dipole–dipole interaction. Moreover, any polar molecule can be used with this scheme, as long as it has at least two hyperfine states. Interesting choices would be CaF or NaCl with a dipole moment of up to 5 to 10 D. Given the fact that rotational levels are spaced in the GHz range and thus only lowfrequency photons are required, this scheme is suitable for both the superconducting wire architectures and the optical lattice (in this case, at least one higher-frequency addressing laser might be necessary); for a dipole moment μ = 10 D, r = 10 μm, and h = 0.1 μm, the necessary interaction time is of the order of 3 μsec. With a coherence time of the order of 100 msec to 1 sec, this setup would thus allow for 105 –106 operations. We note here that other scenarios for the generation of entanglement between rotational quantum states of two polar molecules have been proposed in Ref. [13]. In that approach, the entanglement arises from dipole–dipole interaction and is controlled by a sequence of laser pulses simultaneously exciting both molecules. In addition to cold molecules trapped in optical lattices, cold molecules in solid matrices are also considered in Ref. [13]. © 2009 by Taylor and Francis Group, LLC
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If the sites can be addressed individually and the dipole–dipole interactions are very strong, the previous schemes could take advantage of the so-called dipole blockade mechanism. This mechanism has been introduced for quantum information processing with Rydberg atoms, and generalized to mesoscopic ensembles as well as van der Waals interactions. The original dipole blockade proposal [14] relies on a rapid “hopping” of the excitation between the energy levels of two Rydberg atoms, leading to an effective splitting of the doubly-excited state. When this splitting is sufficiently large, the energy levels are shifted far away from the unperturbed atomic resonance, effectively eliminating the transition to this doubly-excited state; one atom can be excited into a Rydberg state, but additional Rydberg excitations are prevented by the large energy shifts. In a similar fashion, the blockade mechanism can be generalized to polar molecules. If the dipole–dipole interaction is strong enough, that is, larger than the bandwidth of the excitation field, the doubly-excited state corresponding to |ee will be shifted out of resonance and never excited. If both sites a and b are addressable individually, the ability to drive a 2π transition in site b depends on whether site a is excited (see Figure 17.5). At t1 , we apply a π pulse to molecule a and populate the state |ea . At t2 we apply a second pulse (2π) to molecule b: if a is already in |ea , the dipole–dipole interaction shifts the state |eb , the photon is off-resonance, hence no transition. If a is not in |ea , b acquires a phase of π after the process. At t3 , we de-excite a with another π pulse; in summary, |00 |01 |10 |11
t1 −→ −→
t2
|00 |01 i|e0 i|e1
−→ −→ x
t3
|00 −|01 i|e0 i|e1
|00 −|01 . −|10 −|11
−→ −→
This scheme is robust with respect to the separation between the molecules; as long as the excitation is blockaded, the exact separation is not important. t1
t3
t2
e
e
e
e
Shifted
e
e Shifted
p
p
2p
1 0 A
1
1
0
0
B
A
B
1
1
0
0
1 0 A
B
FIGURE 17.5 Principle of the dipole blockade (see text). (Adapted from Yelin, S.F. et al., Phys. Rev. A, 74, 050301, 2006; Kuznetsova, E. et al., Phys. Rev. A, 78, 012313, 2008.)
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A key operation at the end of several qubit operations is the readout of the quantum registers. Several approaches could be employed with polar molecules. As mentioned above, selective ionization of one of the states (0 or 1) and the detection of molecular ions can be readily accomplished. However, this is a destructive method, because the molecule is lost after the readout, and the site would need to be refilled. A different method uses a “cycling” fluorescent transition in which the molecules decay after irradiation directly back into the state from which they came. Although this is substantially more difficult for molecules than for atoms because of the large number of molecular levels, it offers the advantage of being “nondestructive.” Another approach based on recent work on evanescent-wave mirrors for polar molecules might yield promising results [15]; while “0” would stick to the wall, “1” could be reflected. Because reflection takes place far away from the surface of the mirror, it might help to minimize decoherence due to shorter-range interactions with the surface. Finally, as discussed below, for molecules trapped near a microwave stripline cavity, the qubit state can be read out by monitoring the dispersive phase shift of photons transmitted through the cavity.
17.4.2
DECOHERENCE AND ERRORS
The schemes described above may suffer from various sources of error. For example, because our schemes rely only on internal molecular states, there is a possibility that some of the molecules may be translationally “hot.” If molecules are not in the motional ground state of the trap potential, there can be considerable uncertainty and variation in the separation between molecules, which can affect the exact phase. For example, during a ∼1 μsec gate time, the motion of RbCs molecules at 10 μK can lead to ∼3% variation in the phase. We can control and reduce such error, in a variety of ways. Lower temperatures are always preferred. Larger separations reduce this type of error at the expense of requiring longer gate times. Similarly, performing gates over times long compared to the motional period of molecules in the trap (i.e., adiabatically with respect to this motion) leads to “motional averaging” which reduces the size of errors. Alternatively, use of larger dipole moments and shorter separations can enable very fast gates. This can reduce errors if the gate times are much shorter than the motional period; this is known as “bang-bang” control. Note that decoherence and uncertainty due to molecular motion can be completely eliminated using dipole blockade, leading to higher fidelity. 17.4.2.1
Dipole–Dipole Interaction Strength
The fidelity of two-qubit gates depends on the separation and orientation of the molecular dipoles. The question therefore is how the error per gate can be reduced below a threshold value at which fault-tolerant quantum computing can be realized. A typically cited desirable error threshold is 0.01%, or 1% with error correction, while errors per gate demonstrated experimentally to date are 3%. We analyze the error in a phase gate, assuming that two molecules are in a state having a large dipole–dipole interaction matrix element. While the interaction is “switched on,” ideally a π phase shift accumulates. © 2009 by Taylor and Francis Group, LLC
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To analyze the phase dependence on the relative distance and orientation of the dipole moments, we assume that each molecule is in a ground translational state of a one- or two-dimensional optical lattice potential. We use a translational ground state wavefunction of the three-dimensional isotropic harmonic potential for a simple estimate of the mean distance a molecule travels in the ground state of an optical 4 lattice potential. We make a√harmonic approximation of the potential V0 k 2 i xi2 , giving the frequency ω = k √ 2V0 /m with the corresponding translational ground state wavefunction width a = /mω for a molecule of mass m. A typical potential depth a molecule experiences in a lattice is V0 = ηER , where ER = h2 /(2mλ2 ) is the molecular recoil energy; the dimensionless parameter η defines the depth of the optical lattice (typically η = 10 to 50). The width a can then be related to the lattice field wavelength λ and the lattice depth η as a = (2η)−1/4 λ/2π. For molecules in neighboring lattice sites R = λ/2 this gives the ratio a/R ≈ 0.1. While with adiabatic excitation this will not result in any uncertainty of the phase (just a renormalization), nonadiabaticity can result in errors of the order of a/R. For the typical case of a/R ≈ 0.1, the error would too big to tolerate, and dipole blockade would have to be used. If static dipoles are aligned by the same (static) electric field, errors in orientation will be negligible. However, for the “rotational” scheme desribed above, the dipole rotates with a frequency given by the rotational energy splitting. Hence, errors in relative timing of the excitation pulses are equivalent to orientation errors, and hence the timing must be tightly controlled. 17.4.2.2
Molecular State Decoherence
If qubits are stored in hyperfine sublevels of the ground rovibrational electronic state the phase is relatively insensitive to local fluctuations of dc and ac electric fields, but more sensitive to magnetic field fluctuations, which should be minimized. The time scale of this minimization should be of the order of the coherence time of the trap. The states used to “switch on” the dipole–dipole interaction have to be longlived to minimize decoherence from spontaneous emission. With gate operation times of ≤100 μsec and metastable excited state lifetimes of several hundred millisecond, decoherence due to spontaneous emission will be small. Interaction of molecules in the large dipole states can give rise to mechanical forces between molecules, leading to motional decoherence if the gates are not performed adiabatically. In the dipole blockade mechanism, however, molecules are never actually transferred to the large dipole moment states simultaneously, thus this source of motional decoherence is minimized. A similar problem could be excitation into higher translational states due to, for example, momentum transfer from laser Raman pulses, or forces from an electric field gradient. Again, however, typical pulse lengths (i.e., gate operation times) are expected to be larger than the translational oscillation period, allowing adiabatic transfers that minimize this decoherence source sufficiently. 17.4.2.3 Trap-Induced Decoherence Lifetimes of ultracold molecules in a far-detuned optical lattice of ∼1 sec have been obtained by minimizing the scattering of lattice photons. Given that the lifetime of a nuclear spin state of a single molecule, isolated from the environment, can be as © 2009 by Taylor and Francis Group, LLC
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long as hours, the lifetime of a molecule in a lattice, that is, Raman scattering into other translational states, will be the major decoherence mechanism in this context. Similar numbers are expected in other trap schemes. Where our scheme relies on the dipole blockade effect, it is not critical for the gate operation to cool molecules to the translational ground state of a lattice. The spatial dependence of driving fields, however, could cause decoherence due to excitation of higher translational states: The finite width of the translational ground state in a lattice potential does result in a spatially varying Rabi frequency of optical pulses experienced by the molecule. In particular, for a Gaussian Raman drive beam of width σ, the variation of the Rabi frequency over the molecular wavefunction can be of the order of a2 /σ2 , where a is the width of the molecular state. Beams, however, usually will not be focused down below the individual trap size (e.g., the size of one lattice site); hence this error will be at most on the order of 1%. In addition, there are ideas that use quantum coherence in order to mitigate this error type [16].
17.5 17.5.1
OUTLOOK AND OTHER IDEAS SUPERCONDUCTING MICROWAVE RESONATORS
As mentioned above, a very promising approach that is at the moment already being explored experimentally is the one involving so-called stripline resonators (see Figure 17.6). Two length scales are bridged by integrating on a chip a novel type of trap for polar molecules (dubbed the “EZ trap”) with superconducting stripline microwave resonators. The electrostatic EZ traps create an electric field near the surface of the chip, where cold polar molecules can be loaded and their internal states used as qubits, similar to the way described above. André and colleagues [5] discuss the example of CaBr, where the two states with rotational angular momentum N = 1 and 2 are chosen as rotational qubit states |0 and |1. In the proposed
v v v
FIGURE 17.6 External electric and classical microwave fields can be used to manipulate polar molecules in EZ traps to encode information in rotational states and perform onequbit operations. Superconducting stripwire resonators can then couple different molecules and carry two-qubit gates. The sites are selected by adjusting the EZ trap voltage appropriately. (From André, A. et al., Nature Phys., 2, 636, 2006; Côté, R., News & Views, Nature Phys., 2, 583, 2006. With permission.)
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hybrid approach, superconducting stripline resonators confine microwave fields to extremely small volumes. Using the appropriate microwave radiation, each tiny resonator allows manipulation of a single molecule close to it. At the same time, the exchange of so-called virtual microwave photons through the stripline resonator can also mediate dipole–dipole coupling between different polar molecules separated by large distances. This is achieved by locally adjusting the detuning of each molecule with the resonator—for instance, by varying the voltage of individual EZ traps. Hence, two qubits can be coupled over large distances by wires on a chip, leading to good scalability properties and enabling gate operations involving two qubits. In addition, ingenious ideas have been developed to cool the molecules via resonator-enhanced spontaneous emission arising from the strong coupling between the molecules and the resonator microwave cavity. That strong coupling is exploited further in a detection scheme in which a state-dependent phase shift of the resonator field leads to a near-perfect nondemolition measurement of the qubit state. The combination of all these properties leads to the interesting prospect of all-electrical control over the polar molecules qubits in this system. This appears very attractive for integration with other solid-state devices.
17.5.2
OPTICAL QUANTUM COMPUTING IN POLAR ENSEMBLES
The idea of optical quantum computing is to treat photons as information carriers, that is, qubits. Of the five DiVincenzo criteria [7] this means that the critical one is the two-qubit gate, because photons do not interact directly with each other. The idea thus is for two photons in one of the |0 and |1 states (where 0 and 1 can mean presence/absence of the photon, polarization, or similar) acquire nonlinear phase shifts upon interaction inside the molecular gate medium. The photons interact with the molecules via dark-state polaritons, that is, coupled light–matter excitations that utilize the excellent coherence properties of the so-called “dark states” of a multilevel system. The nonlinear phase shift then happens via the dipole–dipole interactions between the molecules. Comparison with the previous sections reveals the similarities and differences with this approach. While up to now we equated molecules with qubits and saw photons as mere carriers of the interaction, here the situation is reversed: we treat the photons as qubits and the molecular medium as interaction carrier. In both cases, however, regular arrays of dipoles are useful for addressability and coherence, respectively. (Although regularity of the trap is not essential for this setup, it is advantageous.) Various schemes for hybrid quantum processors based on molecular ensembles as quantum memories and optical interfaces have been proposed. In Ref. [17], a hybrid quantum circuit using ensembles of cold polar molecules with solid-state quantum processors is discussed. As described above, the quantum memory is realized by collective spin states (ensemble qubit), which are coupled to a high-Q stripline cavity via microwave Raman processes. This proposal combines both molecular ensemble and stripline resonator ideas. A variant of this scheme using collective excitations of rotational and spin states of an ensemble of polar molecules prepared in a dipolar © 2009 by Taylor and Francis Group, LLC
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crystalline phase is given in Ref. [18]. In one- and two-dimensional traps, the crystalline structure protects the molecular qubits from detrimental effects of short-range collisions. Yet, in another proposal, a single mesoscopic ensemble of trapped polar molecules would be used for quantum computing [19]. A “holographic quantum register” with hundreds of qubits would be encoded in collective excitations, each uniquely addressable by optical Raman processes with classical optical fields. Again, one- and two-qubit gates and qubit readout would be accomplished by transferring the qubit states to a stripline microwave cavity field and a Cooper pair box governed by classical microwave fields. Many more ideas for hybrid quantum processors are being developed, and polar molecules remain a very promising platform.
17.6
CONCLUSIONS
In summary, polar molecules offer a very diverse platform to optimize various aspects of quantum information processing. With fabrication, cooling, and trapping maturing (as described elsewhere in this book), the tools to implement the ideas laid out here are almost at our fingertips.
ACKNOWLEDGMENTS SFY and RC would like to thank E. Kuznetsova for help and useful discussions and NSF and ARO for funding.
REFERENCES 1. Nielsen, M.A. and Chuang, I.L., Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. 2. David Mermin, N., Quantum Computer Science: An Introduction, Cambridge University Press, Cambridge, 2007. 3. DeMille, D., Quantum computation with trapped polar molecules, Phys. Rev. Lett., 88, 067901, 2002. 4. Lee, C. and Ostrovskaya, E.A., Quantum computation with diatomic bits in optical lattices, Phys. Rev. A, 72, 062321, 2005. 5. André, A., DeMille, D., Doyle, J.M., Lukin, M.D., Maxwell, S.E., Rabl, P., Schoelkopf, R.J., and Zoller, P., A coherent all-electrical interface between polar molecules and mesoscopic superconducting resonators, Nature Phys., 2, 636, 2006; Côté, R., Quantum information processing—Bridge between two lengthscales, News & Views, Nature Phys., 2, 583, 2006. 6. DeMille, D., Glenn, D.R., and Petricka, J., Microwave traps for cold polar molecules, Eur. Phys. J. D, 31, 375–384, 2004. 7. http://www.research.ibm.com/ss_computing 8. Collins, G.P., Computing with Quantum Knots, Scientific American, 2006. 9. Leung, D.W., Chuang, I.L.,Yamaguchi, F., andYamamoto,Y., Efficient implementation of coupled logic gates for quantum computation, Phys. Rev. A, 61, 042310, 2000. 10. Yelin, S.F., Kirby, K., and Côté, R., Schemes for robust quantum computation with polar molecules, Phys. Rev. A, 74, 050301, 2006; Kuznetsova, E., Côté, R., Kirby, K., and
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11. 12. 13. 14.
15.
16. 17.
18. 19.
Cold Molecules: Theory, Experiment, Applications Yelin, S.F., Analysis of experimental feasibility of polar-molecule-based phase gates, Phys. Rev. A, 78, 012313, 2008. Note that, since 12 CO has a zero nuclear spin, isotopic 13 CO would have to be used. A good way to get into this superposition state would be to couple two neighboring rotation states via microwaves, as in Ref. [10]. Charron, E., Milman, P., Keller, A., and Atabek, O., Quantum phase gate and controlled entanglement with polar molecules, Phys. Rev. A, 75, 033414, 2007. Jaksch, D., Cirac, J.I., Zoller, P., Rolston, S.L., Côté, R., and Lukin, M.D., Fast quantum gates for neutral atoms, Phys. Rev. Lett., 85, 2208, 2000; Lukin, M.D., Fleischhauer, M., Côté, R., Duan, L.M., Jaksch, D., Cirac, J.I., and Zoller, P., Dipole blockade and quantum information processing in mesoscopic atomic ensembles, Phys. Rev. Lett., 87, 037901, 2001; Tong, D., Farooqi, S.M., Stanojevic, J., Krishnan, S., Zhang, Y., P., Côté, R., Eyler, E.E., and Gould, P.L., Local blockade of Rydberg excitation in an ultracold gas, Phys. Rev. Lett., 93, 063001, 2004. Kallush, S., Segev, B., and Côté, R., Evanescent-wave mirror for ultracold diatomic polar molecules, Phys. Rev. Lett., 95, 163005, 2005; Kallush, S., Segev, B., and Côté, R., Manipulating atoms and molecules with evanescent-wave mirrors, Eur. Phys. J. D, 35, 3, 2005. Gorshkov, A.V., Jiang, L., Greiner, M., Zoller, P., and Lukin, M.D., Coherent Quantum Optical Control with Subwavelength Resolution, arXiv:quant-ph/0706.3879. Rabl, P., DeMille, D., Doyle, J.M., Lukin, M.D., Schoelkopf, R.J., and Zoller, P., Hybrid quantum processors: Molecular ensembles as quantum memory for solid state circuits, Phys. Rev. Lett., 97, 033003, 2006. Rabl, P. and Zoller, P., Molecular dipolar crystals as high-fidelity quantum memory for hybrid quantum computing, Phys. Rev. A, 76, 042308, 2007. Tordrup, K., Negretti, A., and Mølmer, K., Holographic quantum computing, arXiv:0802.4406v2 [quant-ph] 22 July 2008.
© 2009 by Taylor and Francis Group, LLC
Part VII Cold Molecular Ions
© 2009 by Taylor and Francis Group, LLC
Sympathetically Cooled 18 Molecular Ions: From Principles to First Applications Bernhard Roth and Stephan Schiller CONTENTS 18.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Sympathetic Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Ion Trapping and Production of Cold Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 Radio-Frequency Ion Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.2 Atomic Ion Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.3 Molecular Ion Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Properties of Cold Trapped Coulomb Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.1 Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.2 Collisional Heating of Ion Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.3 Heating Effects in Multispecies Ensembles . . . . . . . . . . . . . . . . . . . . . . . 18.5 Characterization and Manipulation of Multispecies Ensembles . . . . . . . . . . 18.5.1 Crystal Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5.2 Particle Identification: Destructive and Nondestructive . . . . . . . . . . . 18.5.3 Motional Resonance Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5.4 Species-Selective Ion Removal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6 Chemical Reactions and Photofragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6.1 Ion-Neutral Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6.1.1 Reactions of Laser-Cooled Atomic Ions . . . . . . . . . . . . . . . . . 18.6.1.2 Reactions of Molecular Ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6.2 Photofragmentation of Polyatomic Molecules . . . . . . . . . . . . . . . . . . . . 18.7 Rovibrational Spectroscopy of Molecular Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7.1 Rovibrational Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7.2 Molecular Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7.3 High-Resolution Spectroscopy of Molecular Hydrogen Ions . . . . . 18.7.4 Sub-MHz Accuracy Infrared Spectroscopy of HD+ Ions . . . . . . . . .
652 653 654 654 656 656 658 658 664 666 670 670 672 674 676 678 678 679 681 686 688 688 693 695 695 651
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18.8 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699
18.1
INTRODUCTION
The emerging field of cold molecules offers applications ranging from novel studies of light–molecule interactions (e.g., coherences within rotational states) over interactions between dipolar molecules [1], to ultrahigh-resolution spectroscopy, and to the study of complex systems [2]. In spectroscopy, a similar qualitative jump is expected as was obtained in atomic spectroscopy after the technique of laser cooling was introduced. Vibrational and rotational levels of suitable molecules have long lifetimes (milliseconds to days), implying potentially huge transition quality factors. Fundamental physics experiments with molecules requiring extreme resolution and accuracy include tests of quantum electrodynamics (QED), measurement of and search for a space–time variability of the proton-to-electron mass ratio [3], the search for parity violation effects on vibrational transition frequencies [4,5], and tests of the isotropy of space [6]. Finally, a novel application is the investigation of collisions of molecules with atoms or other molecules at very low temperatures [7,8]. This temperature regime represents a unique situation for the study of quantum-mechanical details of collisional processes, preferably with the collision partners in well-defined internal states [9]. All of these topics can be studied both with neutral molecules and with molecular ions (Figure 18.1). Some molecular systems of particular interest, such as the oneelectron molecules, are inherently ions (molecular hydrogen ions). At present, the range of molecular ions that can be produced at very low temperatures is much wider
Cold molecular ions
Fundamental physics Molecular
physics Fundamental constants Biomolecules and their time variation Molecular High-resolution structure and Parity vibrational and potentials violation electronic spectroscopy RF and Isotropy rotational tests spectroscopy for Internal dynamics (infrared heating, astronomy and IVR*, evaporation, astrochemistry …)
Cold ion-neutral interactions
Ion-neutral reaction/ scattering at low/ultralow temperature
Many-particle systems under extreme conditions New many-body systems Ions in condensates Clusters with electric-dipoledipole interactions
Cool internal degrees of freedom to the quantum ground New state phenomena?
Ion-neutral photoassociation
Weak radiative processes
FIGURE 18.1 Overview of potential applications of cold molecular ions. (∗ IVR, intramolecular vibrational-energy redistribution.)
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than for neutral molecules. This situation also makes them interesting for testing methods that may later be extended to neutral molecules as well. The techniques developed for the laser-cooling of neutral and charged atoms cannot be directly applied to molecules, due to the lack of, closed optical transitions. Therefore, new methods for the production of cold molecules are required. Of particular interest are general methods that do not depend on the internal nature of the particles, such as their magnetic or electric moments or their energy level structure, so that they can be applied to a large variety of molecular species, ranging from light diatomics to complex molecules, such as proteins and polymers. Buffer-gas (collision) cooling is such a general approach, and cold molecular ions have been produced using cold helium gas already a long time ago. Translational and internal temperatures of a few Kelvin were reached [10]. In other work, buffergas cooling was combined with multipole linear radio-frequency traps [11]. Here the achieved temperatures are significantly higher, ∼15 K. This approach is used by several groups for studies of spectroscopy and chemical reactions of both small and large molecular ions [12–17].
18.2
SYMPATHETIC COOLING
Temperatures significantly lower than those obtained with cryogenic He buffer gas cooling can be achieved by using laser-cooled atomic ions as a charged “buffer gas.” This method is referred to as sympathetic or interaction cooling. Here, two or more different species are trapped simultaneously in an ion trap. One of them is directly cooled by lasers, and the other species will, at least in part, eventually also cool down through long-range Coulomb interactions. Sympathetic cooling was first observed in laser cooling experiments of atomic ions in Penning traps [18,19] and later in radio-frequency traps [20–22]. One important advantage of sympathetic cooling is that it does not depend on the internal level structure or on the electric or magnetic moment of the particles, but only their mass and charge. Sympathetic cooling of atomic species has been implemented using as atomic coolants laser-cooled 9 Be+ [23], 24 Mg+ [24–26], 40 Ca+ [27], 114 Cd+ [28], and 138 Ba+ [29], in both electrostatic and electrodynamic ion traps. In these works all sympathetically cooled species were singly charged, medium-mass atoms, except in one experiment on highly charged xenon [23]. Sympathetic ion cooling is equally well applicable to molecular ions. Groundbreaking work has been performed by Drewsen and coworkers, who produced MgH+ molecular ions by reactions of H2 with laser-cooled Mg+ , and observed subsequent sympathetic cooling and crystallization [30,31]. Baba and Waki demonstrated sym+ + + pathetic cooling of H3 O+ , NH+ 4 , O2 , and C2 H5 ions, also using Mg [38]; here the temperatures T ∼ 10 K corresponded to the gas phase. These two groups also showed the utility of linear quadrupole ion traps for sympathetic cooling. The second advantage of sympathetic cooling of ions is the notable cooling strength due to the Coulomb interaction. In our work we showed that using just two atomic ion species, sympathetic cooling can be used to cool any (singly charged) atomic and molecular ion species in the wide mass range 1 to 470 amu, as well as much heavier ions (up to mass 12,400 amu) if they are highly charged (Table 18.1). © 2009 by Taylor and Francis Group, LLC
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TABLE 18.1 Molecular Ions Sympathetically Cooled to Date Crystallized Statea
Non-Crystallized Statea
+ + BeH+ [34], MgH+ , O+ 2 , MgO , CaO [30,32], + , D+ , H+ , D+ , H D+ , HD+ , H+ , HD 2 2 2 3 3 2 + BeH+ , BeD+ , NeH+ , NeD+ , N+ 2 , OH , + + + + + + H2 O , O2 , HO2 , ArH , ArD , CO2 , + KrH+ , KrD+ , BaO+ , C4 F+ 8 , AF350 , GAH+ , R6G+ and fragments ([36] and this work), cytochrome-c proteins (Cyt12+ , Cyt17+ ) [37]
+ + NH+ 4 , H2 O , H3 O , + , O+ C2 H+ , COH 5 2 [29,33], C+ 60 [35]
a
18.3 18.3.1
The term “crystallized” refers to a cold ion ensemble with developed shell structure, with temperatures below 200 mK.
ION TRAPPING AND PRODUCTION OF COLD MOLECULES RADIO-FREQUENCY ION TRAPS
Charged particles can be trapped by a combination of static electric and magnetic fields (Penning trap) or by a combination of a static and a radio-frequency electric field (Paul trap). For one review on ion traps, see Ref. [39]. An advantage of radiofrequency traps compared to Penning traps is the absence of magnetic fields, which lead to Zeeman shifts, splittings and broadenings, and the low attainable ion velocities. We consider here linear radio-frequency traps [40], typically consisting of four cylindrical electrodes, each sectioned longitudinally into three parts (Figure 18.2a). Compared to the radio-frequency quadrupole trap, the linear radio-frequency trap offers a line of vanishing radio-frequency field, and thus the possibility of storing a larger number of particles with little micromotion. In addition, it also allows good optical access, which is favorable for laser cooling and for spectroscopic measurements. Radial confinement of charged particles is achieved by applying a radio-frequency voltage Φ0 = V0 − VRF cos(Ωt) to two diagonally opposite electrodes, with the other two electrodes at ground. VRF and Ω are the amplitude and the frequency of the radio-frequency driving field. In the following we consider mainly the case V0 = 0 (a nonzero value is considered in Section 18.5.9). Axial confinement along the symmetry (z) axis of the trap is achieved by electrostatic voltages VEC applied to the eight end sections (end-caps). In Figure 18.2b, an overview of one complete setup is given [41]. The equations of motion of a single ion inside a linear radio-frequency trap are differential equations of the Mathieu type, which lead to stable or unstable solutions with respect to motion in the x–y-plane, depending on the trap parameters. A necessary condition for stable trapping of an ion or an ensemble of noninteracting ions is a (Mathieu) stability parameter, q = 2QVRF /mΩ2 r02 , of <0.9. Here, Q and m are the ion charge and mass, and r0 is the distance from the trap center to the electrodes. For a single trapped ion, its trajectory is a superposition of a fast oscillating motion (micromotion) with frequency Ω and a slow (secular) motion that can enclose a large area in the trap. The amplitude of the micromotion is proportional to q if q 1. From simulations it is known that when many ions are to be stored in the trap, a small q © 2009 by Taylor and Francis Group, LLC
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(a) y
Small Q2/m
X
VEC VRF Z r Imaging optics Octopole ion guide
Ion trap
Magnetic field coil (b)
Electrospray ionization (ESI) Differential vacuum source and quadrupole mass filter and octopole ion guide
UHVchamber with ion trap
To pumps
FIGURE 18.2 (a) The linear radio-frequency trap, showing a schematic of a linear radio2 frequency trap (left), the effective trap potential for different values of Q /m (right), and a + 2 photograph of the UHV chamber of the ESI/Ba apparatus [42]. # = x + y2 and z are the radial and axial trap coordinates, respectively. (b) Apparatus for sympathetic cooling of heavy molecular ions [41].
parameter is favorable because heating effects induced by the radio-frequency field are less pronounced. For small q 1 one can often approximate the interaction of the ions with the trap by neglecting micromotion altogether and introducing an effective, time-independent, harmonic potential (quasipotential, pseudopotential) m (18.1) Utrap (x, y, z) = (ω2r (x 2 + y2 ) + ω2z z2 ), 2 © 2009 by Taylor and Francis Group, LLC
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where x, y are coordinates transverse to the trap axis and the electrode center lines cross the x and y axes. The mass-specific motion of a single particle in such a potential is a superposition of harmonic motions with a transverse (to the z-axis) √ oscillation frequency ωr = (ω20 − ω2z /2)1/2 , with ω0 = QVRF / 2mΩr02 and a longitudinal frequency ωz = (2κQVEC /m)1/2 . Here κ is a trap-geometry constant. An ensemble of (interacting) ions in such a potential has, at sufficiently low (but not too low) average ion energy (i.e., temperature), an approximately constant density and a spheroidal shape [43]. r0 = (4.32, 4.36) mm, κ ≈ (1.5, 3) × 10−3 mm−2 , and Ω = 2π · (14.2, 2.5) MHz for our Be+ and Ba+ traps, respectively.
18.3.2 ATOMIC ION COOLING A variety of atomic ions can be efficiently laser-cooled to mK temperatures and are in principle suitable as coolants. In our work, beryllium and barium ions are used. They are produced by evaporating neutral atoms from an oven and ionizing them in situ in the trap by an electron gun. For laser cooling of Be+ ions, light resonant with the 2 S1/2 (F = 2) → 2 P3/2 transition at 313 nm is used [44]. Population losses by spontaneous emission to the metastable ground state 2 S1/2 (F = 1) are prevented by using repumping light, red detuned by 1.250 GHz. 138 Ba+ ions are laser cooled on the 62 S1/2 → 62 P1/2 transition at 493.4 nm [45]. A repumper laser at 649.8 nm prevents optical pumping to the metastable 52 D3/2 state. The laser-induced atomic fluorescence can be recorded using a photomultiplier tube and a twice charge-coupled device (CCD) camera. The latter takes pictures with typical exposure times of 0.5 to 2 sec. The direction of observation is perpendicular to the trap symmetry axis. Because the ions do not “cast a shadow,” the CCD images are projections of the complete ion ensemble onto a plane parallel to the trap axis. Strong cooling results in a transition from a gaseous to a fluid and then to a crystalline state (Coulomb crystal); see Figure 18.6 later. The precise meaning of “crystalline” will be explained in Section 18.4. The shape of these ensembles is spheroidal if the quasipotential has axial symmetry, Equation 18.1.
18.3.3
MOLECULAR ION PRODUCTION
There are different ways to produce atoms and molecules to be sympathetically cooled. One way is to leak neutral gases into the vacuum chamber and ionize them in situ by an electron beam crossing the trap center [46,47]. The loading rate is controlled by the partial pressure of the neutral gas and the electron beam intensity. Figure 18.3 illustrates the loading of HD+ ions into a Be+ ion ensemble prepared beforehand. The shape of such a mixed-species ensemble can be changed in real time by variation of the ratio of radial to axial frequency via variation of the trap parameters, as shown in Figure 18.4. The transfer of charged large molecules into the gas phase is possible by several methods. One such method is electrospray ionization (ESI), shown in Figure 18.5. This well-developed method is capable of producing (multiply) charged molecular beams starting from molecules in solution, even for very massive molecules (several © 2009 by Taylor and Francis Group, LLC
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Sympathetically Cooled Molecular Ions Be+
(a)
(d)
t = 10 sec
t = 2 sec
(e)
t = 12 sec
t = 6 sec
(f)
t = 14 sec
t = 0 sec
1.3 mm (b)
HD+ (c)
Heavier ions
FIGURE 18.3 Loading sequence of HD+ ions into a cold (≈20 mK) Be+ ion crystal. The presence of cold HD+ ions is obvious from the appearance of a dark (nonfluorescing) crystal core in the initially pure crystal. Heavier atomic and molecular ions were also loaded to the crystal. Because they are less tightly bound by the trap, they are located outside the fluorescing Be+ ensemble and cause its flattening.
10,000 amu). Charged droplets of the solution are ejected by a needle and undergo rapid evaporation into smaller and smaller droplets until a final Coulomb explosion causes a significant fraction of single molecules to be created with a variable number of protons attached, originating from the solvent. Molecules of a desired charge-tomass ratio are then selected using a quadrupole mass filter. Figure 18.5 illustrates the principle of electrospray ionization and shows a mass-to-charge spectrum obtained for a particular protein. Because the molecules are produced outside the ion trap, a radio-frequency octopole ion guide is used for their transport and injection into the trap (Figure 18.2b) [41].
(a)
HD+ (d)
(c)
(b)
Be+
1.3 mm (e)
(f )
(g)
FIGURE 18.4 Variation of a Be+ –HD+ Coulomb crystal shape with trap anisotropy. The static voltage VEC on the end-cap electrodes (determining the axial confinement) was changed in (a–c), whereas the amplitude of the radio-frequency drive VRF (determining the radial confinement) was changed in (d–g). The pictures are part of a sequence recorded over ≈1 min. The symmetry of the trap potential is cylindrical in all cases. The slight asymmetry along the axial direction is due to the laser pressure.
© 2009 by Taylor and Francis Group, LLC
Cold Molecules: Theory, Experiment, Applications
I
Spray
+3 kV
Apomyoglobin 20 + (16,950 amu)
M
M M
M
M
M
M
M M
M
M
M
M
M
M
10+ 15+ 20+ M
M
M
M
M M
M
M
M
M
M M
M M
M
M M M
M
M
M
M
Needle
M
M
Count rate
M
M
Sheath gas
Molecules in solution
658
14+
25+
M
600
800 1000 1200 M/Q [amu/e]
Taylorconus
FIGURE 18.5 Principle of electrospray ionization and a mass-to-charge ratio spectrum of a protein. Values given are protonation charge states [41].
Many molecular ion species cannot be produced by loading and ionization of neutral gases or by using an ESI source. Here chemical reactions can be used for their production [45,48,49], and are described in Section 18.6.
18.4 18.4.1
PROPERTIES OF COLD TRAPPED COULOMB CLUSTERS MOLECULAR DYNAMICS SIMULATIONS
Although an ion ensemble is a classical system (except in the special case of extremely low temperatures, which shall not be considered here), the large number of degrees of freedom, the finite size and the nonlinear ion–ion interaction make analytical treatments very difficult. Fortunately, molecular dynamics (MD) simulations provide an excellent tool for their analysis. They can even be used to analyze novel methods before implementing them experimentally. MD simulations of threedimensional cold trapped charged particle ensembles have been performed for the one-component plasma in a conservative harmonic potential since the mid-1970s and in radio-frequency traps since the first experimental studies [50,51]. In MD simulations, Newton’s equations of motion are solved for all ions in the trap: mi r¨i = Fi , where i runs over all ions, and mi and r¨i are the mass and position of the ion i. The total force Fi acting on each ion arises from individual contributions, some of which depend on ion positions, ion velocities, and explicitly on the time t, trap
Fi = Fi trap
+ FCoulomb + Fstochastic + Flaser i i i
(18.2)
with the trap force Fi , the Coulomb interaction force due to all other ions FCoulomb , i the stochastic force Fstochastic (due to interactions of ions with the environment, such as i collisions with residual gas, scattered light, electric field noise), and the laser cooling force Flaser , which acts on the laser-cooled (LC) ions only and also includes the light i pressure force. In the simulations, individual photon absorption and emission processes usually need not be taken into account, because the recoil energy of, for example, Be+ is kB (11 μK), that is, 1000 times less than typical temperatures [47,52,53]. © 2009 by Taylor and Francis Group, LLC
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The MD simulations are a flexible and efficient tool: the effect of various forces can be studied individually or in conjunction with other forces. In particular, the simulations can be performed with the time-varying radio-frequency potential, or in the pseudopotential approximation and the results compared. For many purposes the simulations need to compute the ion dynamics only for a few tens of milliseconds. Therefore, for not too large ion numbers (∼1000) the time required to obtain useful information using a personal computer is quite reasonable (hours), making the simulations a practical tool. MD simulations can be straightforwardly extended to sympathetically cooled ensembles with an arbitrary number of species. This was first done in order to explain the experimental structures of a Mg+ /Ca+ mixed crystal [54]. MD simulations are performed in such a way that the ensemble possesses a finite kinetic (secular) energy, equivalent to finite temperature. A basic result is the gas– liquid–crystal phase transition, that is, the appearance of structure with decreasing temperature, as in Figure 18.6. In one-species systems, the regimes in which gaseous, fluid, and crystallized states occur can be distinguished by the interaction parameter
(a)
200 mK
(b)
1K
(c)
200 mK
70 mK
500 mK
50 mK
30 mK
10 mK
50 mK
10 mK
Be+
10 mK
138Ba+
25 mK
Be+
135-7Ba+
FIGURE 18.6 Ion ensembles in the fluid phase. MD simulation of ensembles with (a) one species (500 Be+ ions), (b) two species (500 Be+ ions (outer shells) and 100 HD+ ions (inner shells)), and (c) two species (500 Ba+ ions (outer shells) and 100 barium isotope ions (inner shells)) at different translational temperatures. Particle trajectories integrated over 1 msec are displayed. The view is in the (x–y)-plane. Only a short section along z is shown. The shell structure develops around 200 mK. With lowering of the temperature, the diffusion between and within shells decreases until at temperatures of a few mK and below (not shown) the ions are confined to the immediate neighborhood of particular sites, jittering around them.
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Γ = Q2 /4π0 akB T , the ratio between average nearest-neighbor Coulomb energy and thermal energy (a, average particle spacing) [43]. With decreasing temperature, an ensemble turns from gaseous to liquid when Γ ≥ 2. In the liquid state, the system exhibits short-range spatial correlations, that is, the pair-correlation function is nonmonotonic. Crystallization sets in at Γ ! 170; it is accompanied by an anomaly of the specific heat [2]. The crystallized state is characterized by “caging” of the ions: their thermal motion amplitude is less than the interparticle spacing, they are nearly localized [55]. These values of Γ hold for infinite isotropic systems in a conservative potential [56]. In the approximation where the discreteness of the charges is ignored, the ion spacing or density n = a−3 can be derived from the Laplace equation 2 /(mr 4 Ω2 ). For typical values a ≈ 30 μm the phase transition to the as n = 0 QVRF 0 Coulomb crystal is then predicted at ≈3 mK. In finite systems, however, the phase transition occurs at lower temperatures, corresponding to larger values for Γ [2]. Coulomb crystals do not exhibit the same type of long-range order as solid-state crystals, because the trap potential plays a crucial role. Therefore, the name “crystal” is to be understood as a simplification—the description as a cluster would be more appropriate. Also, in typical experimental situations the ions are not “frozen,” but diffuse between crystal sites (see Figure 18.7). In early work, the spatial structures of single-species crystals were investigated [50]. The simulations reproduced the shell structures found experimentally and also showed a regular ordering within shells, a property not directly observable experimentally for large crystals, because so far experiments only image the structures with a single projection image. At temperatures above a few mK, the case usually encountered experimentally, most ions in a three-dimensional ensemble are not confined to particular sites, but diffuse between sites (Figure 18.7) [2]. Because the CCD images show apparently individual spots, the ensembles seem crystallized, but are not. Except for special sites, the individual spots seen on the experimental images are not the positions where a particular single ion is confined, but where the probability to find any ion is high. Strictly speaking it is thus erroneous (although usual, as here) to denote ensembles at such temperatures as crystallized. “Structured liquids” may be a more appropriate description. Nevertheless, in some sites (e.g., the end sites along the trap axis) the ions are well confined, according to the simulations. For these, their spatial distributions and secular velocity distributions can be obtained, as in Figure 18.8. This information Be+
(b) Be+
(a)
y HD+ x
FIGURE 18.7 Ion diffusion in Coulomb crystals. (a) Trajectories of several individual Be+ ions in a cold (10 mK) Be+ –HD+ ion crystal (time-averaged trap potential assumed; duration: 1 msec). Because of diffusion, except for special sites, individual spots in a (simulated) CCD image are not the positions where a particular single ion is confined, but where the probability to find any ion is high. (b) Axial view of a section of the crystal in (a).
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Sympathetically Cooled Molecular Ions (b) Selected HD+ ion
HD+
Be+
Probability (arb. units)
(a)
(c)
0 20 550 570 590 –20 Axial position (mm) Axial velocity (m/sec)
FIGURE 18.8 (a) An individual, localized HD+ ion at 10 mK. The ion is embedded in a 10 mK Be+ crystal. (b, c) Its spatial (z) and velocity (vz ) distribution. The particle trajectories were simulated for 1 msec. Smooth lines are Gaussian fits.
can be of interest, for example, for determination of the preferred laser propagation direction in spectroscopy and for modeling of the lineshape in spectroscopy. A particular processes of interest is radio-frequency heating, the transfer of energy from the trap field to the ion ensemble. In an important early study [51], it was shown that for a crystallized ensemble, even containing many particles, radio-frequency heating is very small. This was confirmed in experiments on Mg+ ions in a ring trap [57], where the Coulomb crystals did not melt even if the cooling light was interrupted for seconds. A MD study of radio-frequency heating in single-species ensembles as a function of trap parameters has been performed in [58,59]. The influence of phase shift errors on the trap electrodes was investigated in Ref. [53] and found not to be important. All these studies indicate that linear quadrupole traps are adequate for cooling and sympathetic cooling of reasonably large ensembles containing thousands of ions; higher-order multipole traps are not a necessity. This statement does not imply that micromotion is irrelevant or radio-frequency heating always negligible. In fact, the micromotion kinetic energy is typically a few orders of magnitude larger than the secular energy. Figure 18.9 shows crystals with various sympathetically cooled (SC) atomic and molecular species. The most important overall feature of SC and crystallized ensembles is a radial separation of the species due to their different pseudopotential strength: Utrap scales as Q2 /m, as in Figure 18.2a, right panel. In addition, the interspecies interaction is ∼ Q1 Q2 . Thus, for equal charge of all ions, the total potential energy will usually be minimized if the lighter ions are closer to the axis [30,54]. A radial gap between the species develops. For arbitrary charge ratio, in the limit of cylindrical symmetry (very prolate ensembles), the ratio of outer radius r1 of the lower mass-tocharge ratio (m1 /Q1 ) ensemble and inner radius r2 of the higher mass-to-charge ratio (m2 /Q2 ) ensemble is given by r1 /r2 = (Q2 m1 /Q1 m2 )1/2 [60]. Figure 18.11 shows an example. The interaction between the atomic coolants (LC) and the SC ions will be strong if the two species are close, rLC /rSC ! 1. Most favorable is the case (for equal charge states) in which the coolants are slightly heavier than the SC ions (MLC /QLC ≥ © 2009 by Taylor and Francis Group, LLC
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(a)
Cold Molecules: Theory, Experiment, Applications
BeH+/BeD+
Be+
He+
1.60 mm (b) 135-7Ba+/
138Ba+
+
N+
(c)Be+
(d)
135-7Ba+
1.85 mm
AF+
138Ba+
AF+
MD simulation
Ar+/ CO2
1.66 mm (e) 1.33 mm
138Ba+
135-7Ba+
FIGURE 18.9 Examples of mixed-species ion crystals. (a) 900 Be+ , 1200 BeH+ , and BeD+ (in equal amounts) at 15 mK [48]. (b) 300 138 Ba+ , 150 135−137 Ba+ , 240 BaO+ , and ≈200 Ar+ and CO+ at ≈20 mK. [45] (c) 500 Be+ , 1500 He+ , and 800 N+ at ≈12 mK. (d) 830 138 Ba+ , 420 135−137 Ba+ , and 200 protonated Alexafluor ions (AF+ , mass 410 amu). The temperature of the Ba+ including SC isotopes is at 25 mK, the AF+ are at ≈88 mK. (e) MD simulation of the crystal in (d). (From Roth, B. et al., Phys. Rev. A, 73, 042712, 2006; Roth, B. et al., J. Phys. B: At. Mol. Opt. Phys., 38, 3673, 2005; Ostendorf, A. et al. Phys. Rev. Lett., 97, 243005, 2006; Phys. Rev. Lett., 100, 019904(E), 2008. With permission.)
MSC /QSC ), effectively “caging” them (rLC ≥ rSC ). This is the reason for the choice of a light atomic coolant (Be+ being the most suitable) for cooling molecular hydrogen ions and for the choice of a heavy atomic ion for heavy molecular ions. Because atomic ion masses are limited to about 200 amu, the cooling of very heavy molecular ions is not favored. However, if those ions carry a large charge, cooling is facilitated (see Section 18.8). Examples of crystals with radial separation can be seen in Figures 18.3, 18.9 through 18.11. Particularly interesting is the “tubular” structure that ensembles with three or more species can exhibit (Figure 18.9c). Because the radiation pressure of the cooling laser does not act on the SC ions, an asymmetry in axial direction occurs if a single axial cooling laser is employed. The SC ions are located closer to the laser (Figures 18.9b, 18.12a). This feature is
(b)
AF+
(a)
AF+
y
Ba+ x
FIGURE 18.10 Spatial separation of species. (a) Simulation of an ion crystal containing 50 laser-cooled (LC) Ba+ ions and 50 sympathetically cooled (SC) AF+ ions at 15 and 18 mK, respectively. Here, radio-frequency micromotion is included. (b) Cross-section of the crystal in (a). The micromotion is directed toward the electrodes (located on the x and y axes) and leads to a (slight) blurring of the AF+ ions in the radial direction, in addition to the blurring caused by secular motion and found also when the simulations are performed in a time-averaged pseudopotential. (From Zhang, C.B. et al., Phys. Rev. A, 76, 012719, 2007. With permission.)
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Charge Q = 10 e
Charge Q = 20 e
FIGURE 18.11 MD simulation of a Coulomb crystal containing 1000 Ba+ ions and 20 highly charged molecular ions (16,000 amu), of charge QSC = 10 e (top) and 20 e (bottom).
HD+
Be+
(a) (b)
20 mK
(c)
12 mK
(d)
8 mK
(e)
6 mK 1mm
Normalized fluorescence (arb. units)
particularly apparent when the number of SC ions is significantly smaller than the number of atomic coolants. One important application of the simulations to experimental observations is the determination of ion numbers and translational temperatures. Figure 18.12 is a simple example. Simulations are performed for various temperature values and various ion numbers and the best match to the experimental image yields the respective values. For
1.0 0.8
(f)
Experiment Voigt profile fit
0.6 0.4 0.2 0.0 –140 –130 –120 –110 –100 –90 –80 –70 –60
AOM frequency (MHz)
FIGURE 18.12 Determination of translational temperatures of LC and SC ions [53]. CCD image (a) and simulated images (b–e) of a two-species ion crystal are shown on the left. Ion numbers (690 Be+ and 12 HD+ ions) and translational temperatures (10 mK) for LC and SC ions are obtained by visual comparison between experimental and MD images [47]. The asymmetry of shape in the axial direction is caused by light pressure forces. (f) Determination of the translational temperature of a pure Be+ ion crystal by fitting a Voigt profile to the fluorescence line shape, yielding a temperature of 5 ± 5 mK. The fluorescence line shape was obtained by scanning the cooling laser frequency over the Be+ resonance using an acousto-optical modulator (AOM). Note, that the AOM frequency values have an arbitrary offset compared to the atomic resonance. (From Blythe, P. et al., Phys. Rev. Lett., 95, 183002, 2005; Zhang, C.B. et al., Phys. Rev. A, 76, 012719, 2007. With permission.)
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medium-size crystals (<500 ions) containing mostly LC ions, the inaccuracy in the number determination is a few percent only (Figure 18.13) [53]. The resolution of the temperature determination via MD simulation was higher than a direct measurement via the atomic lineshape (Figure 18.12).
18.4.2
COLLISIONAL HEATING OF ION CRYSTALS
An interesting aspect is the stability of ion crystals under (elastic) collisions with a neutral gas. Such collisions can lead to heating, which can be substantial. Here, the most important ion-neutral interaction arises from an induced dipole attraction with potential ϕ = −(α/2)(e/(4π0 r 2 ))2 for a singly charged ion [61]. α is the polarizability of the neutral atom or molecule, e the electron charge, and r the radial separation. Using classical collision theory, one can derive expressions for the ion-neutral collision heating (or cooling) rate hcoll and the momentum transfer collision rate γelastic , respectively [61]: hcoll
3 · 2.21 ekB √ Tn − T c = nn αμ , 4 0 mn + m c
γelastic
2.21 e = nn 4 0
α , μ
(18.3)
where nn is the particle density of the neutral gas, mn (mc ) and Tn (Tc ) are the masses and temperatures of the neutral (charged) particles, μ = mn mc /(mn + mc ) is the reduced mass. For example, in a background gas of N2 at 300 K and a pressure of 1 × 10−9 mbar the average Ba+ –N2 elastic collision rate is γelastic ≈ 0.017 sec−1 per ion. In each collision the average energy transfer is ≈ kB (128 K), leading to a heating rate hcoll = kB (2.2 K/sec) per ion, which increases linearly with the residual
Experiment
Agreement
435 Be+
Too short
425 Be+
Too long
445 Be+
FIGURE 18.13 Determination of ion numbers. The experimental image of a Be+ ion crystal is compared with simulated images of ensembles with different ion numbers. The best fit is achieved for a number of 435 ions. (From Zhang, C.B. et al., Phys. Rev. A, 76, 012719, 2007. With permission.) © 2009 by Taylor and Francis Group, LLC
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TABLE 18.2 Summary of Some Properties of the Multispecies System (Figure 18.9e) Species, j
Number, Temperature, Laser Cooling Rate, Heating Rate, Sympathetic Rate, Nj Tj (mK) cj (kB K/sec) hj (kB K/sec) −(cj + hj ) (kB K/sec)
138 Ba+
830
25
−19
9.9
(LC) 410AF+ (SC1)
200
88
0
15.9
420
37
0
9.9
135−137 Ba+
(SC2)
9.1 (due to SC1, SC2) −15.9 (due to LC, SC2) −9.9 (due to LC, SC1)
Note: Rates are normalized to the species’ particle number.
gas pressure. This should be compared with the heating rates found for our traps and ensembles (Table 18.2). The consequences of the collision between an ion and a residual gas atom or molecule can be simulated. Figure 18.14a shows the kinetic energies of an ion ensemble of 1249 barium ions and of a single ion after the latter has suffered a head-on collision with a helium atom of 23 kB (300 K) kinetic energy. The colliding barium ion suddenly gains a large velocity (76.8 m/sec) leaving the ion crystal. It starts to oscillate in the trap. Its kinetic energy is periodically transformed into potential energy and back. Each time the ion passes through the ion cluster, it transfers some energy to it, with the energy loss rate being lower, the faster the ion is [62].
(b) 0.020
35 30
0.015
25 20
0.010
15 0.005
10 5 0
0.000 0.00
0.05
0.10
0.15
Time (msec)
0.20
Kinetic energy per ion (3/2 kB K) Heating rate per ion (3/2 kB K/sec)
Kinetic energy of a kicked ion (3/2 kBK)
(a)
60 50 ª 88 mK 40 ª 17 mK 30 20
ª 6 mK
10 0 1E–9
1E–8 N2 Pressure (mbar)
FIGURE 18.14 (a) Ion-neutral head-on collision.After its collision with a neutral helium atom a barium ion gradually transfers its gained kinetic energy (left scale) to the whole ion ensemble, whereupon the kinetic energy per ion (right scale) increases (simulation was performed without micromotion). (b) Collision heating of a Ba+ ion crystal as a function of nitrogen pressure. Line: Equation 18.3; symbols: heating rates deduced from the experiment using MD simulations. (From Zhang, C.B. et al., Phys. Rev. A, 76, 012719, 2007. With permission.)
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Finally, the initial kinetic energy is distributed to all ions and the whole ensemble reaches an equilibrium state with increased potential and kinetic energies per ion. For a typical crystal considered here, the thermalization time is τ ∼ 0.2 msec. Simulations including micromotion indicate that also some micromotion energy is converted into secular energy, increasing the heating rate above the value given by Equation 18.3. The collision heating of an ensemble of N ions will be essentially continuous if the time between two collisions in the whole ensemble, 1/Nγelastic , is smaller than the thermalization time τ, that is, for ion numbers N > 1/(γelastic τ). For a typical value of γelastic = 0.002 sec−1 per ion at 1 × 10−10 mbar, this condition is not fulfilled for typical ensembles, but for ensembles with N > 2,500,000 ions. In fact, if the ensemble is small and observed for a time much shorter than the mean interval between collisions, 1/(Nγelastic ), the ensemble may appear colder than the time-averaged temperature. With a typical CCD exposure time of 2 sec this is the case for N < 250 ions. Indeed, our experience is that small clusters show a lower temperature (as found from the CCD images) than larger ones. For large ion ensembles (N > 2000) and sufficiently long CCD camera exposure times there is a sufficiently large total number of collisions in the ensemble that their effect can be modeled in a very simple way. Frequent velocity kicks per ion, but with very low velocity kick magnitudes are implemented. In this way computing time can be saved because equilibrium states are reached much faster (within a few milliseconds of simulated time). The heating of an ensemble of Ba+ ions at different N2 pressures was studied experimentally (Figure 18.14).Assuming a realistic cooling rate β = 2 × 10−22 kg/sec [53], at each pressure the heating rate hcoll that yields the best agreement of simulated and experimental images was fitted. Here, Tn − Tc (Equation 18.3) was set to 300 K. For pressures above 1 × 10−9 mbar, the experimental values agree well with the theoretical heating rate of Equation 18.3, with αN2 = 4π0 (1.76 × 10−24 cm3 ). At the lowest pressures the heating appears to reach a pressure-independent level. This may in part be an artefact due to the finite spatial resolution of our imaging system (which leads to blurring of the crystal image). There may also be heating sources not taken into account, such as electric field noise. As they are hard to quantify, they are not implemented in our simulations directly, but their effect can be included ad hoc in the velocity kick model as a pressure-independent contribution to the kick magnitude [53].
18.4.3
HEATING EFFECTS IN MULTISPECIES ENSEMBLES
The sympathetic cooling is exerted by the LC particles via the Coulomb interaction and depends on the spatial distributions, the temperatures, and the ion numbers of the LC and SC species. Heating of the LC particles competes with sympathetic cooling. A heating rate hj = dEj /dt characterizes the rate of increase of total energy Nj Ej (potential, secular kinetic, and micromotion kinetic) of a subensemble j due to the combined effects of trap noise, collisions with background gas, and radio-frequency heating. It depends on the spatial distribution of the ion subensemble and is expected to increase with increasing distance from the trap axis, because the micromotion © 2009 by Taylor and Francis Group, LLC
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Sympathetically Cooled Molecular Ions
amplitude increases linearly with radial distance. Assigning a single heating rate to a subensemble is an approximation, because the subensemble typically has a finite radial extent; however, the energy exchange within a subensemble is fast and if slow processes are to be described, we may consider the average heating rate only. Note that it would not be accurate (except in special situations) to assume hj = dEj,secular /dt; as a species is heated, its kinetic energy increases, but so does its spatial extension and therefore its potential energy [53]. Therefore not all heating goes into temperature increase. For a given LC ion temperature, the SC ion temperature adjusts such that, for this subensemble, heating and cooling rates are equal in magnitude. In the simulations, it is convenient not to use the LC ion temperature as an input parameter, but instead a LC heating rate hLC and the LC coefficient β. The laser cooling is modeled by a viscous LC friction force, Flaser = (−βz˙i + const)ez , so that the LC rate is itself temperature i dependent. In the approximation where the micromotion is negligible, the cooling rate cLC is proportional to the secular kinetic energy and thus the temperature: cLC =
dELC dt
!− lasercooling
β kB TLC . mLC
(18.4)
Laser cooling strength β, coolant heating hLC and the sympathetic heating exerted by the SC ions on the LC ions determine the LC ion temperature. Consider first the ensemble of Figure 18.12, where a simplified situation occurs. Because the number of SC ions is very small and they are embedded, as a first approximation, a common SC and LC heating rate is set, and a realistic laser cooling coefficient. Three parameters, heating rate and two ion numbers, are varied until the simulated image of the LC ion ensemble agrees with observation. This gives an LC temperature of 10 mK. Then the SC heating rate is varied independently, allowing its effect on the simulated structure to be studied. When its value is sufficiently high, the sympathetic heating causes the shell structure of the LC ensemble to disappear. This occurs when the SC temperature is at 50 mK. By requiring that the structure of the neighborhood of the SC ions agrees with observation, an upper limit for the SC heating rate is found, with a corresponding upper limit of the SC ion temperature, TSC,max ! 20 mK. The accuracy of the SC temperature determination is of course limited by the quality of the experimental image due to the finite spatial resolution of the imaging optics. It is likely that the SC temperature is nearly equal to the LC temperature. An extreme situation in sympathetic cooling occurs when the SC particles are significantly heavier than the LC particles (and of equal or larger charge). Then the large spatial separation reduces the interaction strength and thus the cooling power. A significant difference in the temperatures of the two species can arise. This is unlike the above case where the SC particles are lighter than the LC ions and thus well-embedded, leading to an efficient coupling. From an experimental point of view this situation is difficult to characterize in a straightforward way via the CCD images, because for a large spatial separation and a small SC ion number, their effect on the shape of the LC ensemble is not easily observable. For such a situation the simulations are an important tool. As one example, the sympathetic cooling of protonated © 2009 by Taylor and Francis Group, LLC
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Alexafluor ions (AF+ , mass 410 amu) by 138 Ba+ (Figure 18.9d) could be characterized with the help of the simulation (Figure 18.9e). The ensemble actually contains five species, of which one species is laser-cooled and the other four are sympathetically cooled. Three of the SC species are Ba+ isotopes different from 138 Ba+ ; they are grouped together as a single SC species (SC2). In order to model a three-species system, we must introduce three heating rates hLC , hSC1 , hSC2 . In the steady state, energy conservation requires that the heating rates and 138 Ba+ temperature TLC are related by dEtot /dt = (−βkB TLC /mLC + hLC )NLC + hSC1 NSC1 + hSC2 NSC2 = 0 .
(18.5)
To obtain insight into the effect of the Coulomb interactions between the species, Figure 18.15 shows a MD simulation in which the various heating and cooling interactions are “switched on” in time to clarify their effect. The middle section of the plot indicates that in the absence of heating of the two SC species their temperature is equal to that of the LC particles. The unavoidable heating of the two SC species, turned on in the right section of the plot, increases the temperatures and leads to different 0.14 0.12
Heating only for LC Ba+
hLC = 11.55 kB K/sec hSC2 = 0 kB K/sec 0.10 hSC1 = 0 kB K/sec
Temperature (K)
All heating on
Laser cooling on
b/m = 0/sec
0.08 0.06
LC 138Ba+ SC barium isotopes SC AF+ hLC = 11.55 kB K/sec hSC2 = 0 kB K/sec hSC1 = 0 kB K/sec b/m = 866.4/sec
hLC = 11.55 kB K/sec hSC2 = 11.46 kB K/sec hSC1 = 25.14 kB K/sec b/m = 866.4/sec
0.04 0.02 CLC = –22.1kB K/sec @ 26 mK
0.00 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
Time (sec)
FIGURE 18.15 MD simulation of sympathetic heating and cooling in a multispecies ion ensemble. At t = 0 all species were set to a secular temperature of 0 K. Left: heating is initially implemented only for the 138 Ba+ ions (heating rate hLC = 11.55kB K/sec). The other ions are sympathetically heated. Center: at t = 0.04 sec laser cooling of 138 Ba+ is switched on (cooling coefficient β/mLC = 866.4/sec), and the 138 Ba+ temperature decreases. The 138 Ba+ ions now sympathetically cool the other ions, and the ensemble reaches an equilibrium state at nonzero temperature. Right: at t = 0.1 sec, heating is also turned on for the SC barium isotopes (heating rate hSC2 = 11.46kB K/sec) and the SC 410AF+ ions (heating rate hSC1 = 25.14kB K/sec), which increases the temperatures of all species until they reach the equilibrium state. (From Zhang, C.B. et al., Phys. Rev. A, 76, 012719, 2007. With permission.)
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temperatures for the three species, due to the differing interaction strengths caused by the different spatial distributions. The simulation also shows that the response time of the subensembles is a few msec. Returning to the characterization of the experimentally obtained ensemble, the main goal was the determination of the temperature of the heavy SC species. The experimental data consisted of the Ba+ cooling coefficient β/mBa ≈ 760/sec, the CCD image of the (effective) three-species ensemble, the number of heavy SC1 ions, and the CCD image of the (effective) two-species ensemble, where the latter two data were obtained by removing the heavy SC ions from the trap. The two CCD images were rather similar, which indicated a relatively weak LC–SC interaction compared to the LC heating effects. With these inputs, the MD simulations were used to determine four quantities: the number of LC and SC2 ions, the LC heating rate (with the reasonable assumption that the heating rate for the SC2 Ba+ isotopes has the same value), and the SC1 heating rate for the heavy ions. In fact, the two-species simulation fit of the CCD images of the system without AF+ easily yields a temperature TLC,0 and the ion numbers NLC,0 , NSC2,0 , from which by Equation 18.5, the common heating rate hLC = (β/mLC )kB TLC,0 NLC,0 /(NLC,0 + NSC2,0 ) is obtained. The ion numbers NLC,0 , NSC2,0 could be obtained rather directly, because their determination is not very sensitive to the SC1 heating rate. Finally, the three-species simulation was then performed assuming the heating value hLC remains unchanged in the presence of AF+ . This assumption is reasonable, because the change in size of the LC/SC2 ensemble was small. Thus, only the heating rate hSC1 was varied, until the CCD image was reproduced. It is in this step that the interspecies interactions are crucial: the heating of the relatively distant heavy ions affects the laser-cooled ions, heating them to a temperature TLC given by βkB TLC /mLC = hLC (1 + NSC2 /NLC ) + hSC1 NSC1 /NLC . The temperature value TSC1 then follows from the simulation result. In order to estimate its uncertainty, the uncertainty in the CCD image match and in the value of β (heating rates and temperatures scale with β) were considered, leading to a factor of 3 between upper and lower limits for TSC1 (i.e., TAF + ), the former being 138 mK. The interesting sympathetic cooling rate of the heavy ions SC1 by all other ions (here LC and SC2) is just equal in magnitude to the heating rate hSC1 , because of equilibrium. Table 18.2 summarizes the various rates for this particular experiment (for a value β/mLC = 760/sec). Energy conservation, Equation 18.5, is not exactly fulfilled because the value TLC is obtained from the simulations with a certain inaccuracy. Thus, the simulation shows that approximately 20% of the available total laser cooling “power” cLC NLC acts on the AF+ ensemble via the Coulomb interaction, and approximately 46% acts on all SC ions together. The comparison between observed and simulated ion crystals allows or requires also the determination of the trap parameters, pseudopotential frequencies and (spurious or deliberately applied) offset potentials. An example is shown in Figure 18.16. This mixed-species ensemble is not axially symmetric because of static potentials on the electrodes. It also contains several species. Even in such complicated cases, MD simulations can provide a good explanation, after the ion numbers and trap parameters have been fitted by comparison with the CCD image—which shows only one species. © 2009 by Taylor and Francis Group, LLC
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(a) Experiment 1 mm (b) Simulations Be+ (c)
(d)
BeH+, BeD+
HD+, H2D+, D2H+
(e)
Observation direction
FIGURE 18.16 MD analysis of an asymmetric cold multispecies ion crystal containing Be+ ions and various SC molecular ion species. (a) CCD image. The view is in the (z–y)-plane (as indicated by the arrow in the right panel in (e)). (b–e) MD simulations of the crystal in (a). (b) The Be+ only. (c–e) All species in the crystal. The images show projections of the crystal onto the (z–y)-plane (b,c) and left panel in (e), the (z–x)-plane (d), and the (x–y)-plane, right panel in (e), respectively.
18.5 18.5.1
CHARACTERIZATION AND MANIPULATION OF MULTISPECIES ENSEMBLES CRYSTAL SHAPES
The overall shape of a cold ion plasma depends strongly on the symmetry and “shape” of the trapping potential. For axial trap potential symmetry, the ensembles are spheroidal, and have been characterized both in linear radio-frequency and Penning traps [32,63]. In the absence of axial symmetry the shape of the plasma has been predicted to be that of an ellipsoid [64]. Ellipsoidal plasmas were first observed and studied in Penning traps [65]. The first demonstration in a linear radio-frequency trap was given with our Be+ apparatus, by adding a static quadrupole potential to the trap electrodes, resulting in an fully anisotropic quasipotential [66]. Such a situation can be produced with crystals containing two or more different ion species (Figure 18.17). A comparison of experimental shapes to theoretical predictions from a simple cold fluid plasma model shows agreement for small anisotropy, while deviations observed for larger anisotropy can be explained by the presence of the SC particles causing space charge effects. © 2009 by Taylor and Francis Group, LLC
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310 mm
950 mm
(a)
(e)
(b)
(f )
(c)
(g)
(d)
(h)
FIGURE 18.17 Ellipsoidal deformation of mixed-species crystals by a static quadrupole voltage Vdc applied to the central electrodes. For Vdc = 0 the crystals are spheroidal. Left: small crystal containing !20 Be+ and a smaller number of SC impurity ions at different values Vdc : 0 V (a), 2.8 V (b), 3.6 V (c), 4.2 V (d). Right: medium-sized crystal containing !500 Be+ ions. Vdc is set to 0 V (e), 1.4 V (f), 2.8 V (g), 4.0 V (h). Asymmetries in (b)–(d) and (f)–(h) are due to stray electric fields. (From Fröhlich, U. et al., Phys. Plasmas, 12, 073506, 2005. With permission.)
The ability to reversibly deform cold multicomponent crystals by static quadrupole potentials is interesting for several reasons. It allows for (1) a controlled ejection of heavier ion species from the trap (see below), (2) a complete radial separation of lower-mass SC ions from the LC ions, and (3) opening up the possibility of studying trap modes of oscillation of ellipsoidal crystals, in particular of multispecies crystals. Conversely, a precise measurement of the trap modes of oscillation of cold ion crystals allows for the identification of even small anisotropies of the effective trap potential, which is important for precision measurement applications and the characterization of systematic effects, such as offset potentials [45]. Furthermore, by applying static offset potentials to individual trap electrodes a spatial manipulation of the ion crystals is possible. For example, by carefully aligning © 2009 by Taylor and Francis Group, LLC
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the atomic coolant ions around the dark core containing SC ions (which is possible with an inaccuracy below 20 μm, while observing the crystal using the CCD camera) trap imperfections can be minimized more efficiently compared to the commonly used fluorescence correlation measurements [45,67].
18.5.2
PARTICLE IDENTIFICATION: DESTRUCTIVE AND NONDESTRUCTIVE
Because fluorescence detection by repeated absorption–emission cycles is not applicable to trapped molecular ions in UHV, that is, in the absence of collisions with a buffer gas [68], different techniques are required for their reliable identification. A commonly used destructive technique for molecular ions is time-of-flight (TOF) mass spectroscopy. We have used a simplified variant in the Ba+ apparatus. The trapped ions are extracted from the trap by reducing the radio-frequency amplitude, in the presence of a finite dc quadrupole potential V0 , which causes the ion trajectories to become unstable (the Mathieu q-parameter enters the instability region). Heavy and hot ions escape first. Upon leaving the trap, the ions are guided to and attracted by the cathode of a channel electron multiplier (CEM) and counted. Figure 18.18 shows an example of a mass spectrum obtained from a multispecies ensemble. The ion count peaks occur at clearly separate radio-frequency amplitudes, allowing for identification whether LC and SC ions are present. Assuming equal detection efficiencies, the ion signal sizes are used to determine the ratio of the numbers of LC and SC ions. In addition, the spectrum also provides evidence that the 120
400
(a)
300
+
Ba
Ba+
200
AF+
100
100
0
900
0
50
(b)
800
100 150 200 250 300 350 400
Ba+
700
Ion counts
Ion counts
80
60
500 400 300 200
40
AF+
100
AF+
20
0
600
0
0
50
100 150 200 250 300 350 400
Radio-frequency trap drive amplitude (V)
0
50
100 150 200 250 300 350 Radio-frequency trap drive amplitude (V)
400
FIGURE 18.18 Extraction of ions (of crystal in Figure 18.9d) from the trap by reduction of the radio-frequency drive amplitude. AF+ ions (mass 410 amu) are ejected first. Inset (a), extraction of a different sample of nonlaser-cooled barium and AF+ ions at ∼300 K; the small + + left-hand peak is due to SC CO+ 2 impurities. Inset (b), extraction of a laser-cooled Ba /AF ion cloud at a temperature of a few hundred mK (fluid state). (From Ostendorf, A. et al., Phys. Rev. Lett., 97, 243005, 2006; Phys. Rev. Lett., 100, 019904(E), 2008. With permission.)
© 2009 by Taylor and Francis Group, LLC
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Be+ fluorescence (arb. units)
heavier molecules are sympathetically cooled by the barium ions, as can be seen by comparing the two insets in Figure 18.18. Figure 18.18b shows the mass spectrum of an ensemble when the Ba+ ions have been laser-cooled. The ion count peaks of both LC and SC ions are narrower, indicating a narrowed energy distribution. An accurate temperature determination is not feasible, however. A nondestructive detection technique is often desirable. One such technique is based on the excitation of motional resonances of trapped species. The information about the species is encoded in the frequency of the resonance. This technique has been used in the past for mass spectrometry of ion clouds in the gas or fluid state [29,69]. For a “two-ion crystal” containing a single molecular ion sympathetically cooled by a single atomic ion, a high mass resolution has been demonstrated [31]. Atomic and molecular species identification via motional resonance-based detection is also possible in more complex systems, namely in large multispecies ion crystals of various size, shape, and symmetry [47,52,70] (Figure 18.19). The basic principle of the method is as follows. The radial motion of the ions in the trap is excited using an oscillating electric field of variable frequency applied either to an external plate electrode or to the central trap electrodes. When the excitation field is resonant with the oscillation mode of one species in the crystal, energy is pumped into the motion of that species. Some of this energy is distributed through the crystal, via the Coulomb interaction. This, in turn, leads to an increased temperature of the atomic coolants and modifies their fluorescence intensity, which can be detected.
9.0 2 8.5 Be+
1
Ar+
8.0 7.5
N2+
Ar++
7.0
2
6.5 1 6.0
40
60
80 100 120 140 160 180 200
Excitation frequency, w/2p(kHz)
FIGURE 18.19 Motional frequency spectrum of a crystal containing Be+ , Ar+ , N+ 2 , and Ar2+ at 20 mK. Excitation of a particular (mass-specific) trap oscillation mode pumps energy into the motion of that species, which also leads to heating of the atomic coolants. This increases the atomic fluorescence level, which is simultaneously recorded using a photomultiplier tube (graph) and a CCD camera (images 1, 2). Images were taken at the excitation frequencies indicated (1, 2) in the spectrum. Excitation of a particular mode leads to blurring of the crystal structure, as observed with the camera (2), whereas when the excitation field is not resonant with a particular mode the structures are more pronounced (1); see also Figure 18.22. (From Roth, B. et al., J. Phys. B: At. Mol. Opt. Phys., 39, S1241, 2006. With permission.)
© 2009 by Taylor and Francis Group, LLC
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18.5.3
Cold Molecules: Theory, Experiment, Applications
MOTIONAL RESONANCE COUPLING
According to the quasipotential approximation, the radial oscillation frequency of noninteracting trapped ions ωr ! ω0 ∝ Q/m (Equation 18.1) when the axial potential is weak; that is, the ensembles are strongly prolate. Then, the ratios of motional frequencies of different singly-charged species are equal to the mass ratios. This is indeed what is observed when the ensemble is in the gas state, where the density is low and therefore the ions are weakly interacting. Interactions between the different ion species, especially in the crystalline state, can shift the observed motional frequencies by a significant amount. This complicates the analysis of the experimental spectra, in particular, for mixed-species ion crystals with SC ions having comparable mass-tocharge ratios. Examples of spectra obtained for such an ion crystal are displayed in Figure 18.20. Figure 18.21 shows examples of multispecies ion crystals and their (radial) motional resonance spectra. Figure 18.21b shows the spectrum of a cold beryllium + + ion crystal containing Be+ , H+ 3 , H2 , and H ions at 15 mK. Even though the measured motional frequencies are shifted compared to their calculated single-particle frequencies, particle identification is possible. Figure 18.21d shows a spectrum of a beryllium ion crystal containing C4 F+ 8 ions and various fragment ions. Usually, the observed resonance frequencies are determined by a superposition of several, sometimes opposing, line shifting effects. For example, the motional frequencies depend on the fractions of particles contained in the crystal. For strong coupling between the ion species, this can lead to a significant shift and broadening of individual features in the spectrum, so that they cannot be resolved experimentally (Figure 18.20a). However, even for weaker coupling (Figure 18.20b), various, sometimes more subtle, line-shifting effects can be present, caused, for example, by space charge effects, trap anisotropies, or the finite amplitude of the excitation field. The observed frequency positions also depend on the sweep direction of the excitation field and the plasma temperature. Finally, the state of the ion plasma, crystalline or gaseous/fluid, affects the motional resonances measured (Figure 18.20c). Therefore, an unambiguous identification of particles embedded in large, mixedspecies ion crystals is often not possible based on the experimental measurements only. Comparison with MD simulations leads to an improved interpretation, finally enabling a more accurate interpretation of spectroscopic and chemical experiments. As an example, in Figure 18.22 the measured motional frequency spectrum for a mixed-species ion crystal is compared to the simulated spectrum. First, the number of ions is determined from a MD fit to the CCD images. Then the motional spectrum is obtained by starting with the equilibrium state of the ensemble, shifting the position of the SC particles radially and then evolving the system. The x coordinate values are then Fourier transformed. The measured spectrum shows a fairly complicated structure with features at 58, 82, 122, and 166 kHz. The feature at 82 kHz is due to sympathetically crystallized Ar+ , whereas the features at 122 and 166 kHz are 2+ ions, respectively. The calculated single-particle secular freattributed to N+ 2 andAr + + quencies for Ar , N2 , and Ar2+ are 63, 90, and 126 kHz. This shows how the Coulomb coupling between the ions produces significant shifts compared to the single-particle
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6.5
Be+ H3+ / H2+
H3+ / H2+ 1.8 mm
6.0
5.5 400
600
800
1000
1200
Be+ fluorescence (arb. units)
(b) 4.0
1400
1600
Excitation amplitude: 2 V Be+
H3+
H3+ / H2+ 1.8 mm
3.5 H2+
3.0
400
600
800
1000
1200
1400
1600
(c) 2.5 Excitation amplitude: 1 V H3+
H2+
2.0
1.5
1.0 400
600
800 1000 1200 Excitation frequency, w/2p (kHz)
1400
1600
FIGURE 18.20 Interaction-induced strong (a, b) and weak (c) coupling between radial modes. Motional frequency spectrum of (a) an ion crystal (inset) containing ≈ 1400 Be+ and ≈ 1300 + + SC ions (≈ 700 H+ 2 and H3 , and ≈ 600 BeH ) at ≈ 20 mK. (b) Ion crystal in (a) after partial removal of lighter SC ions (reduced size of the core), containing ≈ 1350 Be+ and ≈ 1200 SC + + + ions: ≈ 450 H+ 3 , ≈ 100 H2 , and ≈ 650 BeH . (c) Spectrum of an ion crystal containing Be , + + H2 , and H3 ions in the gaseous/fluid state. Calculated single-particle secular frequencies + (arrows): 840 kHz (H+ 3 ) and 1260 kHz (H2 ). The feature at ≈ 580 kHz is attributed to the + second harmonic of the Be radial mode (at 280 kHz). (From Roth, B. et al., Phys. Rev. A, 75, 023402, 2007. With permission.) © 2009 by Taylor and Francis Group, LLC
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Fluorescence (arb. units)
10
H3+
H2+
(b)
9 8 H+
7
800 1200 1600 2000 2400 2800 Excitation frequency, w/2p (kHz)
C4F8+ and fragments
(c) Be+
Fluorescence (arb. units)
H3+/H2+/H+
(a) Be+
6.00
Fragments + 5.75 of C4F8 5.50
(d)
C4F8+
5.25 5.00 4.75
0 20 40 60 80 100 120 140 160 180 Excitation frequency, w/2p (kHz)
FIGURE 18.21 Motional frequency spectra of mixed-species crystals. (a) Crystal containing + + Be+ , H+ 3 , H2 , and H ions at ≈15 mK. (b) Spectrum (low-mass range) of the crystal in (a). Measured frequencies are shifted compared to the calculated single-particle frequencies (840 + + + kHz for H+ 3 , 1260 kHz for H2 , and 2520 kHz for H ; the Be frequency is at 280 kHz), due to Coulomb coupling between LC and SC ions. (c) Crystal containing Be+ , C4 F+ 8 , and various fragments of C4 F+ at ≈20 mK. (d) Spectrum (high-mass range) of the crystal in (c). The 8 calculated single-particle frequency for C4 F+ (13 kHz) agrees well with the measured value 8 (15 kHz), due to smaller Coulomb coupling to all other species. The fragment ions act as a conducting layer and ensure efficient sympathetic cooling of the C4 F+ 8 [52]. (From Ostendorf, A. et al., Phys. Rev. Lett., 97, 243005, 2006. With permission.)
frequencies. The experimental feature at 58 kHz is not well reproduced, but can be attributed to the excitation of the axial ωz Be+ mode, which in the simulations appears as a double peak, due to small anisotropies of the trap potential.
18.5.4
SPECIES-SELECTIVE ION REMOVAL
During loading and ionization of neutral gases in the trap along with the species under study, various impurity ions are often produced by chemical reactions between LC or SC ions and the neutral gases. Such impurities can complicate precise measurements of motional resonances or systematic studies on pure or few-species ion plasmas. In particular, for the applications described in Sections 18.6 and 18.7, crystals with a single species of SC particles are required. It is therefore necessary to remove unwanted species from the crystal and leave all other species as intact as possible. Ions with a mass-to-charge ratio larger than that of the atomic coolants are located outside the LC ion shells and can be removed from the trap selectively by adding a static quadrupole potential Vdc to the electrodes. For appropriate strength, ion motion becomes unstable along one radial direction, leading to the ejection of that particular species from the trap [32,66]. © 2009 by Taylor and Francis Group, LLC
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Motional amplitude (arb. units)
(a) 9.0 Ar+
8.5 8.0
N2+
7.5
Ar2+
7.0 6.5 0
25
50
75 100 125 150 Excitation frequency, w/2p (kHz)
175
200
225
200
225
Motional amplitude (arb. units)
(b) 1.0 0.8 0.6 Ar+ N2+
0.4
Ar2+
0.2 0.0 0
25
50
75 100 125 150 Excitation frequency, w/2p (kHz)
175
FIGURE 18.22 Measured (a), as in Figure 18.19a, and simulated (b) motional frequency + 2+ spectrum of the cold (<20 mK) multispecies ion crystal containing Be+ , N+ 2 , Ar , and Ar shown in Figure 18.19. Arrows: single-particle frequencies. (From Roth, B. et al., J. Phys. B: At. Mol. Opt. Phys., 39, S1241, 2006. With permission.)
After switching off the static quadrupole potential again, the absence of the removed species causes a change of shape of the crystal. This procedure is illus+ + trated in Figure 18.23 (left), where N+ 2 , N , and BeH ions were removed from a cold (≈20 mK) beryllium ion crystal. The crystal changed from approximately cylindrical to ellipsoidal. Note that the dark core of the crystal containing lighter SC ions + (hydrogen molecular ions, H+ 3 and H2 ) was not affected. Particles with a mass-to-charge ratio smaller than that of the atomic coolants, and therefore located closer to the trap axis, can be ejected from the crystal in a different way; see Figure 18.23 (right). By detuning the cooling laser far from resonance, the ion crystal in Figure 18.23d undergoes a phase transition to a disordered (gaseous) state. In this situation, the coupling between different ion species is much weaker than in the crystalline state, and the secular motion of the unwanted species can be strongly excited, ejecting those ions from the trap, with almost negligible © 2009 by Taylor and Francis Group, LLC
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VDC = 0V
1.5mm
N2+/N+/BeH+ H3+/H2+
(b) Be+ VDC = 4.2V
H3+/H2+
(c) Be+ VDC = 0V
(d) Be+ fluorescence (arb. units)
H3 + /H2+
(a) Be+
(e)
D3+/D2+ D3+ D2+
D+
(f )
(g)
400 600 800 1000 1200 1400 1600 Excitation frequency, w/2p (kHz)
FIGURE 18.23 Left: Species-selective removal of heavy SC ions from an ion crystal, by applying a static quadrupole potential Vdc to the trap electrodes. CCD images are taken before (a), during (b), and after (c) removal of the particles. After removal a reordering of the Be+ shells is observed. The temperature of the crystal is slightly reduced, because heating effects for particles closer to the trap axis are smaller. Right: Removal of light SC ions. CCD images and spectra before (d, f) and after (e, g) the ejection of D+ ions [47]. In (f), the individual + contributions of D+ 3 and D2 cannot be resolved, due to strong Coulomb coupling between the SC species [in contrast to (g)]. The spectra were taken with different amplitudes of the excitation field, due to different ion numbers involved. (From Blythe, P. et al., Phys. Rev. Lett., 95, 183002, 2005. With permission.)
effect on other species. The remaining ions can then be recrystallized by retuning the cooling laser back close to resonance (Figure 18.23e). This procedure can be repeated to remove as many different species as required. This and the previous procedure can be performed in sequence and thus pure two-component crystals can be obtained.
18.6 18.6.1
CHEMICAL REACTIONS AND PHOTOFRAGMENTATION ION-NEUTRAL CHEMICAL REACTIONS
Reactive and nonreactive collisions of ions with neutrals are of general interest in chemistry [71]. Ideally, the reactions would be studied as a function of collision energy, spanning the range from μeV to eV. Because of the experimental challenges, studies of ion-neutral reactions at low temperatures are still very few to date. For example, using multipole ion traps and cold He buffer-gas cooling, reaction rates and branching ratios of various chemical reactions could be deduced [7,11]. In a quadrupole ion trap, sympathetically cooled molecular ions were used, in order to study the reaction H3 O+ + NH3 → NH+ 4 + H2 O at temperatures of ≈ 10 K [38]. The study of such reactions at even lower temperatures could improve the understanding of ion-neutral reactions occurring in interstellar clouds [8,12,72,73]. Samples of sympathetically crystallized cold ions open up the possibility to investigate these processes with a good accuracy (because the ion density can be determined) © 2009 by Taylor and Francis Group, LLC
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and eventually with resolution of individual quantum states. A first, and simplest step, is the study of reactions with neutral gas at 300 K. This situation implies collision energies (in the center-of-mass frame) below room temperature if the neutral particles are lighter than the ions. The study of this regime is useful in itself but also for preparing future work on reactions at ultralow energies, for example, between cold trapped ions and ultracold neutral atomic or molecular gases. We note that nonreactive, but vibration-rotation-deactivating collisions using cold atomic gases might be useful for achieving internal cooling of translationally cold molecules. Eventually, a further extension will be collisions with quantum-state resolution. This implies the necessity to prepare the reactants not only with welldefined collision energy (inaccuracy smaller than rotational level spacing), but where the partners are in particular quantum states and the quantum states of the products are also detected. Many ion-neutral reactions, such as XY+ + A → XA+ + Y
(18.6)
XY+ + BC → XYB+ + C
(18.7)
are exothermic and proceed without an activation barrier. For such reactions, the Langevin theory predicts a temperature-independent rate coefficient, for suitably low temperatures [74,75]: πα , (18.8) kL = Q 0 μ where Q is the charge of the ion, α is the polarizability of the neutral reactant, and μ is the reduced mass of the particle pair. 18.6.1.1
Reactions of Laser-Cooled Atomic Ions
The cold ion-neutral reactions most easily studied are those involving the laser-cooled atomic ions. The first examples studied were the formation of cold trapped CaO+ ions by the reaction between laser-cooled Ca+ ions and neutral O2 [32,76]. Using the CaO+ ions formed, the back-reaction CaO+ + CO → Ca+ + CO2 was observed [32]. Furthermore, the formation of cold trapped MgH+ by the reaction between laser-cooled Mg+ ions and neutral H2 was also observed [30]. Reaction rates and branching ratios were deduced. A reaction requiring photoactivation is between Be+ and neutral molecular hydrogen gas, shown in Figure 18.24 [48]. This reaction does not proceed with the beryllium ion in its ground electronic state. But when laser-cooled Be+ ions are excited to the 2P 3/2 state, reaction occurs with a rate comparable to the Langevin rate: (Be+ )∗ + HD → BeH+ + D +
→ BeD + H.
(18.9) (18.10)
The product ions were sympathetically crystallized. The reaction could be followed until the last few Be+ ions reacted away, that is, with a particle number resolution down to the single-particle level.
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9 Be+ fluorescence (arb. units)
8 7 6 5 4 3 Background 2 0
200
400 Time (sec)
600
800
(b) 6 Be+(2P3/2) + D2 Be+ fluorescence (arb. units)
5
4
3
2
1
0
50
100
150
200
250
Time (sec)
FIGURE 18.24 Chemical reaction between Be+ and room-temperature molecular hydrogen gases. (a) Decay of a small Be+ crystal (≈160 Be+ at ≈5 mK) after exposure to HD. The smooth line is an exponential fit to the data. The reaction coefficient is k ≈ 1.1 × 10−9 cm3 /sec. (b) Decay of a crystal with ≈45 Be+ ions after exposure to D2 . Discrete steps in the fluorescence toward the end of the process are due to the formation of a single BeD+ and two BeD+ , respectively. Note how the right end of the crystal almost does not shift. The SC product ions cluster on the left side of the crystal because they do not experience the light pressure from the cooling laser, which propagates to the right. (From Roth, B. et al., Phys. Rev. A, 73, 042712, 2006. With permission.) © 2009 by Taylor and Francis Group, LLC
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A reaction that does not rely on the atomic ion being laser-excited is the production of 138 BaO+ molecular ions via background CO2 molecules [45]: Ba+ + CO2 → BaO+ + CO.
(18.11)
Such reactions are relatively simple to characterize because (1) sympathetic cooling of the product ions permits their identification by mass spectrometry, and (2) reaction rates can be deduced from the time evolution of the laser-cooled ion number (observed using a CCD camera), or from the observation of the atomic fluorescence rate (using a PMT). An interesting issue is the effectiveness with which the reaction product ions are sympathetically cooled. This can be seen in Figure 18.25. The MD analysis indicates that nearly all product ions were sympathetically crystallized. A reason for this might be that the neutral product (H or D) is much lighter than the ion product and thus carries away most of the kinetic energy set free in the exothermic reaction. 18.6.1.2
Reactions of Molecular Ions
Perhaps the most fundamental ion-neutral reaction is the astrophysically important, exothermic chemical reaction of H+ 2 ions and neutral H2 gas [72], H2+ + H2 → H3+ + H,
(18.12)
which involves two of the most fundamental molecular ions [77]. This reaction was studied after producing cold H+ 2 ions by electron impact ionization of neutral H2 and sympathetic cooling using laser-cooled Be+ ions and subsequent exposure to H2 gas (Figure 18.26). Because the sympathetic cooling of the product ions is so efficient, the number of molecular ions does not change appreciably (a)
9Be+
t = 75 sec (c)
t = 1672 sec
Cooling laser (b)
BeH+/BeD+ t = 536 sec
(d) MD simulation BeH+
9Be+
BeD+
FIGURE 18.25 CCD camera image of an initially pure large Be+ crystal following exposure to HD gas, leading to the formation of cold BeH+ and BeD+ via chemical reactions. The ellipse is a fit to the initial Be+ crystal boundary. The molecular ions formed are located in the region enclosed by the ellipse. Ion numbers (determined via MD simulations) are (a) 2100 Be+ , (b) 900 Be+ , 1200 BeH+ and BeD+ (in approximately equal amounts), (c) 150 Be+ , 1700 BeH+ and BeD+ . (d) MD simulation of the crystal in (b), with 900 Be+ , 600 BeH+ , and 600 BeD+ ions at approximately 15 mK. (From Roth, B. et al., Phys. Rev. A, 73, 042712, 2006. With permission.)
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Be+ fluorescence (arb. units)
(a)
H2+ + H2
16
H3+ + H
GH + = 0.004 sec–1 3
14
Be+
(b) H3+/H2+
12
GH2+ = –0.007 sec–1
10
Be+
(c) H3
+
8 30
40
50
60
70
80
90
Time (sec)
FIGURE 18.26 Chemical reaction between cold H+ 2 and room-temperature H2 . (a) Motional + frequency spectrum of a cold multi-species ion crystal containing Be+ , H+ 3 , and H2 ions, + + during exposure to neutral H2 gas. H2 and H3 resonances are denoted as circles and squares, respectively. CCD images taken before (b) and after (c) exposure to H2 . No significant loss of Be+ ions occurred [78]. (Adapted from Roth, B., Production, manipulation and spectroscopy of cold trapped molecular ions, habilitation thesis, Heinrich-Heine University, Düsseldorf, 2007.)
and almost no change in the Coulomb crystal structure occurs. The product ions can be identified via secular excitation mass spectrometry (Figure 18.26a). The heights of the peaks in the secular spectrum are a measure of the ion numbers of SC ions in the crystal. During exposure of the cold (≈10 mK) H+ 2 ions to room-temperature H2 gas, in the motional resonance spectrum a decrease of the H+ 2 ion number is observed, + while the H3 ion number increases from an initially present amount. Fitting the peak + maxima decay or increase yields H+ 2 loss and H3 formation rates (Γ) for both molecular ion species. The difference between the two values found is probably due to nonlinear dependence of the signal on ion number due to mode coupling effects. We expect that a detailed understanding of the motional resonance spectrum could be obtained from MD simulations. After cleaning of BeH+ ions generated in addition to Equation 18.12 using the + procedure mentioned in Section 18.5.4, only H+ 3 and Be ions remain Figure 18.27a. Thus, this is an example of the production of a sympathetically cooled molecular ion sample that contains only one molecular species. The ability to produce chemical reactions with a high degree of accuracy and control was used to implement a fast and efficient method for molecule-to-molecule conversion. Cold H+ 3 ions were exposed to room-temperature HD gas, leading to the formation of H2 D+ ions, a species of astrophysical interest [14], via the reaction H3+ + HD → H2 D+ + H2 ,
(18.13)
(see Figure 18.27b). This reaction is exothermic by about 232 K [7]. After subsequent exposure to room-temperature H2 , the ions were converted back to H+ 3 via H2 D+ + H2 → H3+ + HD. © 2009 by Taylor and Francis Group, LLC
(18.14)
683
Sympathetically Cooled Molecular Ions (a) 6.4 H+3 6.0
5.6
5.2 400 Be+ fluorescence (arb. units)
(b)
600
800
1000
1200
1400
800
1000
1200
1400
1200
1400
0 H2D+ 6
2
400
600
(c) H2D+
H3+
4.4
4.2
4.0 400
600
800 1000 Excitation frequency, w/2p (kHz)
FIGURE 18.27 Molecule-to-molecule conversion using SC molecular ions. (a) Motional (secular) frequency spectrum of a Be+ crystal containing cold H+ 3 . (b) After exposure to HD + . (c) Subsequent exposure to H gas leads to partial gas, most H+ ions are converted into H D 2 2 3 back-conversion of H2 D+ to H+ . According to the CCD images, loss of hydrogen molecular 3 ions occurs, while Be+ loss is negligible.
The barrier for this back-reaction is overcome by the thermal energy of the H2 molecules or internal energy of the cold H2 D+ ions. Depending on the exposure time to the neutral H2 gas, back-conversion efficiencies close to unity can be reached. Loss of hydrogen molecular ions (seen as decrease in size of the dark crystal core) is mainly due to reactions with nitrogen molecules present in the background gas, leading to the formation of heavier SC ions embedded outside the Be+ ensemble [49]. The presence of the reaction products is obvious from the increasingly flattened shape of the ion crystal in Figure 18.27 [going from (a) to (b)]. © 2009 by Taylor and Francis Group, LLC
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Cold Molecules: Theory, Experiment, Applications
Heteronuclear diatomic ions with large vibrational and rotational frequencies are promising systems for high-precision laser spectroscopy and fundamental studies, such as tests of time independence of the electron-to-proton mass ratio. They can also serve as model systems for the implementation of schemes for internal state manipulation [79,80]. Molecular hydrides, such as ArH+ and ArD+ , are interesting examples, with the advantage of a relatively simple hyperfine structure of the rovibrational transitions [79,81]. These hydrides were formed by the ion-neutral reactions [49] (Figure 18.28a through c) Ar + + H2 → ArH+ + H.
(18.15)
When the exposure to H2 gas continues, the ArH+ ions are converted to H+ 3, ArH+ + H2 → H3+ + Ar,
(18.16)
leading to a single-species molecular ion ensemble (Figure 18.28d, e). The above is one of two possible reaction paths for the formation of cold H+ 3 ions; the second is Ar + + H2 → Ar + H2+
(18.17)
⇒ H2+ + H2 → H3+ + H.
(18.18)
For both paths (Equations 18.15 and 18.16 and Equations 18.17 and 18.18, respectively) all reactions are exothermic and are expected to proceed with a temperatureindependent Langevin reaction rate constant; see Ref. [49] and references therein for details. 9Be+
(a)
2+
40Ar
10 mK (d)
Cooling laser (b)
2+
40Ar , and small admixtures of
14
+
2+
H3
+
9Be
20 mK
(e) MD simulation
9Be+ 20 mK
+
9Be+ 20 mK
N2 , 40Ar
+
few H3 ions, formed by chemical reactions with residual gas
(c)
40Ar2+, 40ArH2+and small admixtures of 14N2+, 40Ar2+ +
+
H3
2+
40Ar
9Be+ 20 mK
+
H2 and H3
FIGURE 18.28 Production of cold molecular ions using sequential chemical reactions. (a) CCD image of a pure Be+ ion crystal, (b) after loading with Ar+ ions, (c) after H2 inlet, + leading to the formation of (mainly) H+ 3 ions and a smaller fraction of H2 ions (as found by secular excitation mass spectroscopy), (d) after removal of Ar+ , ArH+ , and heavier contami+ 2+ ions were deliberately not removed. (e) nants, and full conversion of H+ 2 ions into H3 . Ar Ion numbers and temperatures for the crystal in (d) are obtained from MD simulations: ≈1150 2+ ions at ≈20 mK. (From Roth, B. et al., J. Phys. B: Be+ ions, ≈100 H+ 3 ions, and ≈30 Ar At. Mol. Opt. Phys., 39, S1241, 2006. With permission.)
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The H+ 3 ion crystals produced in this way could be useful systems for exploring + the chemistry of H+ 3 . In particular, the study of state-specific reactions of H3 via high-resolution infrared spectroscopy could provide valuable input for theories of ion-molecule gas-phase chemistry and precise calculations of molecular transition frequencies of this two-electron molecule. A second example of the production of molecular ions via a sequence of two ionneutral reactions with different neutral reactants is the production of HO+ 2 . The first reaction is Equation 18.12, followed by H3+ + O2 → HO+ 2 + H2 .
(18.19)
This reaction is nearly thermoneutral [82,83]. Because the product ions are heavier than the atomic coolant ions, they crystallize on the outside of the atomic ion ensemble. The dark core of H+ 3 ions therefore decreases and the reaction can be directly followed from the CCD images (Figure 18.29) [48]. In summary: 1. Reactions can be produced in which the laser-cooled and crystallized atomic ions are “spectators” only, opening up the study of a large variety of reactions. 2. In particular, it is possible to study reactions between simple molecules (diatomics, hydrogen molecules), which are of importance for astrochemistry. 3. Reactions can be studied on small ensembles; those are more likely to be prepared in well-defined quantum states in the future. 4. The rates of exothermic barrierless reactions can be set (by choice of neutral gas pressure) such that the reaction can be slow, permitting the application of nondestructive detection, and, in the future, laser manipulation of internal states of the molecules and atoms. (a)
9Be+
+
t = 0 sec
H3
1.3 mm
9Be+
H3
t = 20 sec (e) MD simulation
+
H3
+
+
HO2
(c)
9Be+
+
Cooling laser (b)
(d) MD simulation
9Be+
H3
9Be+ HO2 +
t = 30 sec +
HO2
FIGURE 18.29 (a)–(c) Images of an initially pure Be+ –H+ 3 ion crystal exposed to neutral O2 at 3 × 10−10 mbar. H+ disappears from the crystal core as HO+ 3 2 molecules are formed and embedded in the outer crystal region. Their presence leads to a slight flattening of the upper and lower edge of the crystal in (c). (d) MD simulation of the crystal in (a) containing 1275 + Be+ and 80 H+ 3 ions at ≈30 mK. (e) MD simulation of the crystal in (c) containing 1275 Be , + + 3 H3 , and 75 HO2 ions at ≈30 mK. (From Roth, B. et al., Phys. Rev. A, 73, 042712, 2006. With permission.)
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18.6.2
PHOTOFRAGMENTATION OF POLYATOMIC MOLECULES
Studies of laser-induced fragmentation of molecules are an important topic in chemical physics and can be useful, among others, for the following: 1. The development of techniques to measure the internal state distribution of cold trapped molecules, 2. The measurement and manipulation of branching ratios of dissociating channels in simple molecular systems. 3. The development of theoretical photofragmentation models based on first principles in ab initio computational quantum chemistry (compare, e.g., Ref. [84]), 4. The study of photofragmentation and of conformational dynamics of complex polyatomic molecules, such as proteins and polymers, in the gas phase. 5. As a (destructive) means to probe the change in population of vibrational and rotation levels upon rovibrational excitation [15]. In traps using buffer-gas cooling, photofragmentation is an established method [15], where it is applied to polyatomic ions [15,16,85,86]. In connection with sympathetic cooling of MgH+ molecular ions, two-photon dissociation was demonstrated and the branching ratio of the two possible dissociation channels, Mg + H+ and Mg+ + H, was investigated [76]. In all of these experiments, pulsed lasers were employed. Photodissociation of HD+ is described in the following section. One of the advantages of cold molecular ion ensembles is the long lifetime. Therefore photofragmentation studies can be performed with continuous-wave radiation. This appears as an interesting regime, because multiphoton processes can be avoided, and measured fragmentation rates can therefore be more easily compared with theoretical results. The absorption spectra of Rhodamine 6G ions (R6G+ , mass 479 amu) and Glycerrhetinic acid ions (GAH+ , mass 470 amu) in solution are shown in Figure 18.30.
Absorption (arb. units)
100
GA
Glycyrrhetinic acid (GA), mass 470 amu
R6G
H3C COOH
80 O H3C C3H
60
C3H
HO H3C C3H
40 20
C3H
Barium cooling laser@ 493 nm
Rhodamine 6G, mass 479 amu CH3CH2NH
0 200 250 300 350 400 450 500 550 600 650 Wavelength (nm)
H3C
O
–
NHCH2CH3 – CI CH3 C OCH2CH3 O
FIGURE 18.30 Absorption spectra of solvated GA and Rhodamine (R6G) ions and their structure.
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Sympathetically Cooled Molecular Ions
They indicate that R6G+ will absorb cooling laser light and might fragment as a consequence, while GAH+ will not. This is indeed observed on the cold ions. Figures 18.31 and 18.32 show photofragmentation of cold (≈0.1 K) trapped R6G+ and GAH+ ions. Rhodamine 101 ions were also found to photodissociate in the presence of the cooling laser.
(b)
138Ba+
138Ba+
R6G+
135-7Ba+
135-7Ba+
600 Ion counts (0.5 sec)
(a)
500
138
Ba+ R6G+
135-7 Ba+
R6G+ fragments
(d)
R6G+ fragments
300 200 100
400 Ion counts (0.5 sec)
(c)
Ba+
400
0
R6G+ fragments
Co+2
300
0
50 100 150 200 250 300 350 400 Co+2
Ba+
(e)
R6G+
200 100 0
0 50 100 150 200 250 300 350 400 Trap radio-frequency amplitide Vpp (V)
FIGURE 18.31 Photodissociation of cold, singly charged (protonated) Rhodamine 6G ions by the 493 nm Ba+ cooling laser. Left: Cold Ba+ ion crystal before (a) and after (b) loading of R6G+ . During loading the R6G+ ions (located outside the Ba+ ion ensemble) are exposed to 493 nm radiation and R6G+ fragments are produced. Fragments lighter than Ba+ are embedded inside the Ba+ ion ensemble and lead to the dark core in (b). Fragments heavier than Ba+ are located in between the Ba+ and R6G+ ion ensembles and can also be dissociated by the cooling light. Typically, ions heavier than Ba+ lead to a deformation of shape of the Ba+ ion crystal (depending in size on their number and mass-to-charge ratio) which is, however, not noticeable here. (c) CCD image taken after 60 sec of exposure of the R6G+ to the Ba+ cooling laser (493 nm). The number of fragments lighter than Ba+ and, thus, the size of the dark core have increased. Right: (d) Mass spectrum of an ion crystal similar to the one in (c) containing cold + Ba+ , CO+ 2 (impurities; narrow left-hand peak), and various R6G fragments heavier than + + + Ba (broad right-hand peak). R6G fragments lighter than Ba are probably also contained in the crystal, hidden in the spectrum under the pronounced CO+ 2 peak. The spectrum was obtained by extraction and counting of the ions. (e) Mass spectrum of an ion ensemble that was not laser-cooled and where R6G+ fragments were therefore not formed. The peaks in the spectrum in (e) are broader than the (corresponding) peaks in (d), indicating translational + temperatures for Ba+ , CO+ 2 , and R6G ions above 300 K [36,78]. (Adapted from Offenberg, D. et al., J. Phys. B: At. Mol. Opt. Phys. 42, 035101, 2009; Zhang, C., Ph.D. thesis, HeinrichHeine University Düsseldorf, 2008; Roth, B., Production, manipulation and spectroscopy of cold trapped molecular ions, habilitation thesis, Heinrich-Heine University, Düsseldorf, 2007.)
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Cold Molecules: Theory, Experiment, Applications
138Ba +
GAH +/Bao +
+ + + (b) 138Ba + GAH /GAH fragments / BaO
GAH + fragments
(c)
+ + + Ba + GAH /GAH fragments / BaO
138
GAH + fragments (d)
138Ba +
GAH + fragments / BaO +
GAH + fragments
FIGURE 18.32 Photodissociation of cold, singly charged (protonated) Glycyrrhetinic (GA) acid ions by cw 266 nm UV radiation. Laser-induced fragmentation of the cold (≈100 mK) GAH+ ions leads to the appearance of a dark crystal core containing GAH+ fragments. In addition, GAH+ fragments heavier than Ba+ have probably been formed. Images were taken before (a), during (b, c) and after (d) exposure to the UV radiation for several minutes. (From Offenberg, D. et al., J. Phys. B: At. Mol. Opt. Phys. 42, 035101, 2009; Zhang, C., PhD thesis, Heinrich-Heine University, Düsseldorf, 2008. With permission.)
At least three different fragmentation detection techniques are possible: (1) observation of the change of shape of the cold ion crystal as fragments (lighter than the atomic coolant) are sympathetically cooled into the center of the crystal, (2) extraction and counting of the parent ions and of the fragmentation products, (3) motional resonance mass spectroscopy. The first two methods are shown in Figures 18.31 and 18.32. The third method is described in detail in Ref. [36]. This latter method is important because the first two methods have inherent limitations: for certain systems, the interpretation of the structures in the first method appears too difficult, because of the large number of different product species. The second method is destructive and therefore systematic measurements, for example, as a function of fragmentation laser intensity, will be very time consuming. For such systems, the third method can be an alternative. A particular advantage of these techniques are the long storage times of many minutes (up to hours) in the well-defined and nearly collisionless environment of an ion trap in an ultrahigh vacuum chamber. This allows for the investigation of slow destruction processes such as the photodissociation of large biomolecules. The methods could also be applied for a (systematic) study of highly resolved photodissociation spectra using low-intensity, tunable continuous-wave lasers with narrow linewidths. In order to reduce the spectral congestion due to the presence of different conformers, a cooling of the internal degrees of freedom would be advantageous. This could, for example, be implemented by radiative cooling in a cryogenic environment [36].
18.7 18.7.1
ROVIBRATIONAL SPECTROSCOPY OF MOLECULAR IONS ROVIBRATIONAL SPECTROSCOPY
Traditionally, measurements on molecular ions were performed in discharges [87], in ion beams [88] or in traps equipped with buffer gas cooling (>10 K) [13]. For SC © 2009 by Taylor and Francis Group, LLC
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Sympathetically Cooled Molecular Ions
trapped molecular ions, one wide-ranging application is rovibrational spectroscopy with significantly enhanced resolution and accuracy. This is enabled by the strong suppression of usual line-shifting and broadening effects due to collisions, high thermal velocities, and finite transit time. Spectroscopy of rovibrational transitions in the electronic ground state can take advantage of these special conditions. The low translational temperature of the molecules increases the absorption rate significantly, and efficient excitation is possible even on weak overtone transitions [89]. This allows one to use simpler laser sources, which also simplifies experimentation. Due to the relatively long lifetime of vibrational levels of molecules (∼msec to days), the potential line resolution can be huge. Spectroscopy of vibrational transitions on cold and localized samples of molecular ions was first achieved using HD+ ions [90]. One interesting aspect is the dependence of vibrational and rotational transition frequencies on the ratio of electron mass to the mass of the nuclei [81,91]. In the simplest case, a diatomic molecule, the fundamental vibrational and rotational transition frequencies scale approximately as: υvib ∼
me /μ R∞ ,
υro ∼ me /μ R∞ .
(18.20)
Here, μ is the reduced mass of the two nuclei and R∞ is the Rydberg energy. These dependencies offer two opportunities: (1) the determination of the ratios me /mp , me /md , me /mt m, mp /md , mp /mt (mt is the mass of the tritium nucleus) by measure+ ment of transition frequencies of the one-electron hydrogen molecular ions H+ 2 , D2 , + + HD , HT combined with high-precision ab initio theory, and (2) the search for a time-dependence of the ratio of electron to nuclear mass [3]. The latter option is not restricted to diatomics and therefore offers the option of choice of molecular systems with suitably low systematic shifts. Furthermore, it has been suggested that a possible variation of me /mp (and of the quark masses and the strong interaction constant) over time might be larger than a possible time variation of the fine structure constant α [92–94]; see also [95–97]. Recently, an indication of a variation of me /mp over billion-year timescales was reported in Ref. [98]. This was based on the comparison of laboratory measurements of Lyman bands of neutral H2 with H2 spectral lines observed in quasars, indicating that me /mp could have decreased over the near age of the universe. Currently, the above constants are determined by Penning ion trap mass spectrometry and spin resonance. Their relative accuracies are 2 × 10−10 for mp /md [99], 2 × 10−10 for mp /mt , and 4.6 × 10−10 for me /mp [99,100], respectively. Ab initio calculations of energy levels in molecular hydrogen ions are approaching the limits set by the uncertainties in the values of those fundamental constants entering the calculations, the largest contribution originating from the uncertainty in the value for the electron-to-proton mass ratio me /mp . Therefore, combining molecular hydrogen ion spectroscopy and theory has the potential to eventually yield improved values for the mass ratios. Rovibrational spectroscopy via fluorescence detection will not be feasible for a large class of molecular ions. In particular, it is not feasible in HD+ , because there is no stable excited electronic state. Therefore the fluorescence would be between © 2009 by Taylor and Francis Group, LLC
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vibrational levels. The corresponding low fluorescence rates would require sophisticated photon counting systems in the mid-to far-infrared, which is not practical. In such cases, a destructive technique can be used instead. In the simplest case, a (1 + 1 ) resonance enhanced multiphoton dissociation (REMPD) can be applied. The molecules are excited by an infrared (IR) laser and then selectively photodissociated from the upper vibrational state by a second, fixed-wavelength ultraviolet (UV) laser; see Figure 18.33. The remaining number of molecular ions is measured as a function of the frequency of the excitation laser, see Figure 18.34. The molecular samples used are small (typically 40 to 100 ions). The spectroscopy requires the spectra to be obtained by repeated molecular ion production and interrogation cycles. The loss of HD+ ions not only depends on the REMPD process, but also on transitions induced by black-body radiation (BBR); see Figure 18.35. The respective rates are below 1/sec, but their effects are clearly seen in the experiment: they are responsible for the dissociation of all HD+ ions, even though the laser only excites ions from a particular rotational level. For a comprehensive description of the process, the loss of HD+ ions was modeled by solving the rate equations for the populations of all relevant (v, J) levels including interaction with the IR and UV lasers and the BBR radiation at 300 K; see Figure 18.36. The rate equation model reveals two different timescales at which the HD+ number declines: a first, fast (<1 sec) decay occurs when the IR laser selectively excites ions from a specific rovibrational level (here v = 0, J = 2) which are subsequently photodissociated. The magnitude of this decay depends on the laser intensities (it is small in the case of Figure 18.35). A fraction of the excited ions cascade back to the ground vibrational state, but into
–0.40
Energy (Hartree)
266 nm
ydiss
–0.45 2 ps
1ss
H(1s) + D+ –0.50 D(1s) + H+ 1.4 mm
–0.55
y(u¢ = 4, N¢u) y(u = 0, Nu)
–0.60 1
2
3
4 5 Internuclear distance (a0)
6
7
8
FIGURE 18.33 Principle of (1 + 1 ) REMPD spectroscopy of HD+ ions. A tunable IR diode laser excites a rovibrational overtone transition. The excited HD+ ions are dissociated using a cw 266 nm laser: HD+ (v = 4) + hν → H + D+ or H+ + D. (From Roth, B. et al., Phys. Rev. A, 74, 040501(R), 2006. With permission.)
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Sympathetically Cooled Molecular Ions HD+/D+
Be+
(a)
1.4 mm
0
20
40
60
REMPD+ (d) background loss
(b)
1.4 mm + 266 nm + D lasers on
Be+ fluorescence
Be+ fluorescence
HD+
1.4 mm + 266 nm lasers on
80 100 120 140 160 180 200 Time (sec) Be+
D+
(c)
Background loss only
Secular excitation
5
0
1.4 mm + 266 nm lasers on
10
15
20
25
Time (sec)
FIGURE 18.34 (1 + 1 ) REMPD spectroscopy of HD+ ions. (a) Initial crystal: ≈1100 Be+ , ≈100 HD+ , and ≈20 D+ at ≈20 mK. (b) Repeated secular excitation of the crystal in (a). The excitation frequency was swept between 500 and 1500 kHz every 4 sec. The IR laser was tuned to the maximum of the (v = 4, J = 1) ← (v = 0, J = 2) line. The curve is an exponential fit with a decay constant of 0.04 sec−1 . (c) Crystal after dissociation of all HD+ : ≈1100 Be+ and ≈50 D+ at ≈20 mK. (d) Measurement cycle consisting of repeated probing of the HD+ number before and after exposure to the spectroscopy lasers. (From Roth, B. et al., Phys. Rev. A, 74, 040501(R), 2006. With permission.)
other rotational levels, v = 0, J = 2. The lower level (v = 0, J = 2) is repopulated by BBR and by spontaneous emission at a slower rate and thus the molecular ions are then dissociated at a slower rate [101]. The spontaneous decay rates between vibrational levels differing by Δv = 1 are ≈100/sec and thus the vibrational relaxation is very fast compared to the rotational dynamics in v = 0. Due to the weak coupling between external and internal degrees of freedom, the internal (rotational and vibrational) temperature of the HD+ ions (see Section 18.7.2) is at ≈300 K, in thermal equilibrium with the vacuum chamber, with a significant (>5%) population for rotational levels up to J = 6. Indeed, 12 transitions between 1391 and 1471 nm, from lower rotational levels J = 0 to J = 6 were observed using diode laser spectroscopy. A telecom-type diode laser with a linewidth of ∼5 MHz on 0.74/s
0.9 s
0.55/s Up rate (BB absorption) 0.086/s
1.8 s
0.37/s
0.20/s
4.4 s u=0
15.6 s
149 s 0.029/s J=0
0.12/s 1
0.26/s 2
kB (300 K) Down rate 0.61/s (BB stim. em.)
0.43/s 3
4
5
FIGURE 18.35 Black-body (BB)-induced and spontaneous emission rates within the v = 0 rotational manifold of HD+ . Time values are the natural lifetimes. Rate values are for absorption of BB radiation and stimulated emission by BB radiation. BB temperature is 300 K.
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a b c d e f g
0.5
Relative population
0.4
b
0.3
(u , J ) = 0, 0 (u , J ) = 0, 1 (u , J ) = 0, 2 (u , J ) = 0, 3 (u , J ) = 0, 4 (u , J ) = 0, 5 0, 5* (total molecule number)
g 0.2
d c a
0.1
e f 0
10
20
30
40
50
Time (sec)
FIGURE 18.36 Rotational population dynamics for a translationally cold (20 mK), internally warm (300 K) collision-free ensemble of HD+ ions during REMPD. The IR laser is set to the maximum of the (v = 4, J = 1) ← (v = 0, J = 2) transition (1430 nm). The intensities of the lasers are I(νIR ) = 0.32 W/cm2 , I(266 nm) = 0.57 W/cm2 . J is the rotational quantum number. Hyperfine structure is neglected. At an intensity I(266 nm) = 10 W/cm2 the time at which the ion number has decreased to 50% is reduced to 3.5 sec.
a one-second timescale was used, and its frequency was calibrated with an accuracy of 40 MHz by absorption spectroscopy in a water vapor cell. Detailed measurements for two rovibrational transitions, (v = 4, J = 1) ← (v = 0, J = 2) and (v = 4, J = 3) ← (v = 0, J = 2), are shown in Figure 18.37. (b)
Fractional HD+ loss
Fractional HD+ loss
(a) 0.06 0.04 0.02 0.00 –100 –50 0 50 100 n –2.095882 ¥ 108 (MHz)
0.25 0.20 0.15 0.10 0.05 0.00 –100
–50
0
50
100
n –2.149785 ¥ 108 (MHz)
FIGURE 18.37 (a) The (v = 4, J = 1) ← (v = 0, J = 2) transition at 1430 nm. (b) The (v = 4, J = 3) ← (v = 0, J = 2) transition at 1395 nm. The curves are fits to the data (•), where the theoretical “stick” spectra vertical lines were broadened by ≈40 MHz. The ordinate values are the molecular ion dissociation probabilities for a 5 sec irradiation of 0.65 W/cm2 IR and 10 W/cm2 UV light. The insets show typical error bars. (From Roth, B. et al., Phys. Rev. A, 74, 040501(R), 2006. With permission.)
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Sympathetically Cooled Molecular Ions
693
The measured data show partial resolution of the complicated hyperfine spectrum due to the coupling of four angular momenta (spins of electron, proton, deuteron, and rotational angular momentum). A linewidth of ≈40 MHz [90] was found and was attributed to Doppler broadening, mainly due to a parasitic axial micromotion of the ions.
18.7.2
MOLECULAR THERMOMETRY
One application of rovibrational spectroscopy is molecular thermometry. This type of thermometry is a well-known diagnostic tool and used, for example, in combustion studies, where the rotational and vibrational temperatures are measured, often with coherent Raman spectroscopy, in order to determine their temperature or that of the surrounding medium. An important question is to what extent the rotational motion of sympathetically cooled molecules couples to their translational motion, that is, whether the internal state population can be manipulated by cooling the external degrees of freedom of molecules. The internal temperature is expected to reach a stationary value due to the competition between the (long-range) collisions between charged particles, interaction with black-body radiation, and collisions with residual gas molecules. The cross-section for collisions between a molecular ion and another charged particle (which in the ensembles discussed here can be another molecular ion or a laser-cooled atomic ion) that induce transitions between the rotational or vibrational states has been discussed in Ref. [102]. The transition probability between two (rovibrational) states n and n is modeled by P(n → n ) = 4π2 |n |m(R)|n|2 |Eω (p)ω=ωnn |2 ,
(18.21)
where n |m(R)|n and ωnn (= |En − En |) are the matrix element of the electric dipole moment m(R) and the energy difference between the initial and the final states, respectively. R is the internuclear distance for the molecular ion and Eω (p) is the Fourier component of the electric field strength produced at the molecular ion by the incident charge with a given impact parameter p. In this model, the ion–ion interaction leads to internal heating, similar to the effect of the BBR. At low relative collision energies (large p) in the cold ensembles, the electric field remains small, because the particles do not approach each other more than a few micrometers, and the rate of field strength change is small. The cross-section and excitation/deexcitation probability drop with a high power of the relative energy, and are negligible compared to other effects. Similarly, the influence of trap or noise fields on the rotational distribution is expected to be negligible. In an experiment on sympathetically cooled MgH+ , data from rotational REMPD were compared with results from theoretical simulations [103]. From the measurements it was concluded that the rotational temperature of the MgH+ ions was higher than 120 K. However, the technique was not suited for measuring the rotational temperature of the ions accurately. © 2009 by Taylor and Francis Group, LLC
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With the rotational-state resolved REMPD spectroscopy the rotational distribution can be measured directly. A study was performed on an ensemble of HD+ at 10 mK translational temperature [104]. The HD+ loss rate was measured for rovibrational transitions starting from lower rotational levels J = 0 to J = 6. For this purpose, the IR laser interrogating the ions was tuned to the maximum of individual rovibrational transitions and broadened to approximately 200 MHz, in order to cover their entire hyperfine spectrum. The fractional state population was deduced by fitting the data taking into account population redistribution due to coupling to the BBR. Under the experimental conditions given, the internal (rotational) degrees of freedom were found to be independent of the translational degrees of freedom. The effective rotational temperature found was close to room temperature (335 K), within the 11% measurement accuracy (Figure 18.38). The method used is quite general and can be applied to other molecular species. Furthermore, in the near absence of background gas collisions it allows one to directly relate the rotational temperature of the ions to the temperature of the ambient black-body radiation. This feature (among others, see Refs. [101,105]) suggests the use of molecular ions, such as HD+ or CO+ , for BBR thermometry with possible applications in frequency metrology; that is, it may help to improve the accuracy of frequency standards based on trapped ions [104]. One far-reaching perspective is to perform experiments on translationally localized molecules prepared in specific quantum states. Several methods were proposed to cool the internal degrees of freedom of the molecules, using lasers or lamps [80],
0.4 Fit 335(36) K 125 K
Fractional population
0.3
200 K 300 K 400 K
0.2
500 K
0.1
0.0 0
1
2
4 5 3 Rotational quantum number, J
6
7
FIGURE 18.38 Rotational distribution of cold trapped HD+ ions. (From Koelemeij, J.C.J. et al., Phys. Rev. A, 76, 023413, 2007. With permission.)
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cryogenic techniques, or collisions with cold (μK-regime) neutral atoms. The internal temperature measurement method described could be a useful tool for future studies in this direction and for investigations of collisions in general.
18.7.3
HIGH-RESOLUTION SPECTROSCOPY OF MOLECULAR HYDROGEN IONS
Among all molecular ions, the hydrogen ions deserve particular attention [91]. Molecular hydrogen ions are the simplest molecules in nature, containing only two nuclei and a single electron. Therefore, they have played an important role in the quantum theory of molecules since the birth of the field of molecular physics. More than 800 publications (mostly theoretical) have been written on these molecules over the last 35 years [106]. Molecular hydrogen ions are of fundamental importance in various applications: 1. As benchmark systems for the test of advanced ab initio molecular calculations (in particular, QED contributions) 2. For the measurement of me /mp , mp /md , and mp /mt 3. For the measurement of the deuteron quadrupole moment [107] 4. For tests of Lorentz invariance and time invariance of fundamental constants [3,108] 5. For tests of concepts for the internal state manipulation of molecules 6. For black-body thermometry [104] 7. For the study of collision processes and ion-neutral reactions 8. As model systems for the investigation of radiation-molecule reactions Several spectroscopic investigations on trapped room-temperature molecular hydrogen ions and on ion beams have been performed in the past (Figure 18.39) [88, 109–111]. The lowest spectroscopic relative uncertainties reported so far were achieved in the experiments of Jefferts and of Wing and colleagues [88,112], !1 × 10−6 and 4 × 10−7 (in relative units), respectively. Dissociation energies were obtained with inaccuracy as low as ! 6.5 × 10−7 [113]. Over the years the theoretical accuracy of the rovibrational energies has steadily increased. They have recently been calculated by V. Korobov ab initio with an relative uncertainty below 1 × 10−9 (70 kHz), including QED contributions [114]. Improved hyperfine structure calculations were also reported recently [115], with an estimated uncertainty of about 50 kHz. The different contributions to the calculated transition frequency for a particular rovibrational transition in HD+ are displayed in Figure 18.40.
18.7.4
SUB-MHz ACCURACY INFRARED SPECTROSCOPY OF HD+ IONS
Based on the REMPD spectroscopic method developed for cold molecular ions, the transition frequency for the rovibrational transition (v = 4, J = 3) ← (v = 0, J = 2) at 1395 nm was measured using a novel narrowband grating-enhanced diode laser with resonant optical feedback locked to a femtosecond frequency synthesizer [116]. The latter was stabilized to a maser referenced to the Global Positioning System for long-term stability [117]. © 2009 by Taylor and Francis Group, LLC
696
2
10–4 10–5 10–6 10–7 10–8
Je hf ffer s - ts H + (19 2 69 )
Relative uncertainty
10–3
H di erz ss b e Ca . e rg W ne & m rri icr ng rg T ro ing ie ak vi e ow to s - ez b t n av e r. al H + aw . er ta ( H 1 2, ov l. ( D + 97 H a (1 ib 19 6 D + 97 ,1 2) Ba r. - 93) 97 7) di lak D2 + ,H ss r i . e sh ne na D + rg n ie (1 s - 99 H + 1) 2
10–2
10–9
m ico
2) 97 (1 a w za ke + Ta D g &+ , H 2 er zb . - H r r e H vib ro
R h f icha s, rd Je ph s o ff h f er ot n s - ts o d (1 H + (19 i s s 96 2 68 . - 8) ) H+
10–1
r w Ca av rri e r ng ov ton di ss ib C th .e r. (19 r i i m -H 9 t ne ro s w i c r chl + 6 rg vi o Z a 2, ) ow y ie br rk h D+ s .av (20 - H ang 2, H H e 0 + et D+ ro 1) D+ 2, al H t . D + .( -H 2 + , D 00 2 + 4 2 )
Cold Molecules: Theory, Experiment, Applications
hfs:hyperfine transitions
10–10 1960
1970
1980
1990
2000
2010
Year + + FIGURE 18.39 Experimental uncertainties for various energies of H+ 2 , HD , and D2 .
The spectrum (Figure 18.41) exhibits two peaks of about 40 MHz width. A major contribution to this is excess micromotion along the trap axis. The data allowed a fit to the theoretical hyperfine spectrum with an uncertainty of 0.45 MHz, limited by measurement noise. Consideration of systematic effects led to a slightly increased nonrelativistic frequency: 214 976 047 322 524 accuracy achieved O(a2): ~50
typ. hyperfine contribution (Breit-Pauli)
O(a2): 3411.24
relativ. corrections (Breit-Pauli)
O(a3): –891.61 O(a4): –6.27 QED contributions
O(a5): 0.39 (partial) Unevaluated (estimate): 0.15 10–10
10–9
10–8
10–7
10–6 10–5 10–4 Relative contribution
10–3
10–2
10–1
100
FIGURE 18.40 Contributions to the (v = 4, J = 3) ← (v = 0, J = 2) transition frequency in HD+ (in MHz) [78,111]. (Adapted from Roth, B., Production, manipulation and spectroscopy of cold trapped molecular ions, habilitation thesis, Heinrich-Heine University, Düsseldorf, 2007; Critchley, A.D.J. et al., Direct measurement of a pure rotation transition in H+ 2, Phys. Rev. Lett., 86, 1725, 2001.)
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Fractional HD+ loss
0.25 0.20 0.15 0.10 0.05 0.00 –75
–50
–25 0 25 n – 214 978560.6 (MHz)
50
75
FIGURE 18.41 Spectrum of the (v = 4, J = 3) ← (v = 0, J = 2) transition in HD+ obtained using a precise laser spectrometer. The offset frequency in the abscissa label is the measured result for the unperturbed transition frequency and is accurate to within 0.5 MHz. The smooth curve is a fit to the data. The theoretical hyperfine line spectrum is shown underneath the fit (see Ref. [116] for details). (From Koelemeij, J. et al., Phys. Rev. Lett., 98, 173002, 2007. With permission.)
uncertainty of 0.50 MHz in the determination of the unperturbed (i.e., excluding hyperfine contributions) rovibrational transition frequency. The relative accuracy of 2.3 ppb is 165 times more accurate than the best previous results. The value for the unperturbed transition frequency deduced agrees with the calculated value from V.I. Korobov [114] (theoretical uncertainty 0.3 ppb) to within measurement accuracy. From the measured value the electron-to-proton mass ratio me /mp could be deduced with an accuracy of 5 ppb. The value is in good agreement with the 2002 CODATA value that has a relative uncertainty of 0.46 ppb [99]. It thus appears that the experimental method demonstrated for HD+ provides a new approach for determination of the electron-to-proton mass ratio me /mp . If the experimental accuracy improves by more than one order of magnitude and the uncertainty in the theoretical rovibrational energies improves by a factor of two, a competitive value for me /mp could be obtained. On the experimental side, this requires a trap and a spectroscopy laser that enables Doppler-free spectroscopy.
18.8
SUMMARY AND OUTLOOK
In summary, we note a few salient results obtained on sympathetically cooled molecular ions. It is possible to produce a variety of cold molecular ions, with masses between 2 and 12,400 amu at present. This range can be cooled with just two atomic coolant species. Experimental techniques for destructive and nondestructive characterization and analysis of the content of sympathetically cooled ensembles are available. Molecular © 2009 by Taylor and Francis Group, LLC
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dynamics simulations have proven useful to deduce properties of the ensembles. Chemical reactions between cold trapped atomic and molecular ions and neutral gases can be observed and used to determine reaction rates or to produce a variety of molecular ions in situ. At least for simple molecules, high-resolution rovibrational spectroscopy is feasible even if the internal temperature of the ions is room temperature. A high spectroscopic accuracy can be achieved; as an example, the frequency of a rovibrational transition in cold, trapped HD+ ions was measured with a relative uncertainty of 2.3 × 10−9 . Based on the results obtained so far in the field of cold molecular ions, it appears both important and interesting to extend the toolbox of methods further. Some of the goals are as follows: 1. to develop nondestructive spectroscopic methods (quantum-logic-enabled spectroscopy, state-selective optical dipole force) both for simple and complex molecules 2. to extend high-resolution vibrational spectroscopy to other molecular ions of + + special interest, for example, H+ 3 , HeH , H2 3. to push the accuracy and resolution of rovibrational spectroscopy further by achieving the Lamb–Dicke regime 4. to identify, by theoretical and experimental studies, molecular ions with systematic shifts so low that precise tests of fundamental laws can be performed 5. to demonstrate radio-frequency spectroscopy and two-photon spectroscopy 6. to develop practical methods for internal cooling (e.g., cold shields, optical pumping) 7. to use molecular ions as targets for studies of ion-neutral interactions (elastic and inelastic collisions, charge transfer, rotational and vibrational deactivation) at low temperature 8. to develop methods to study slow processes in polyatomic molecules, for example, triplet–singlet decay rates, taking advantage of the near-collisionfree environment and/or long observation times It is felt that significant progress can be made toward these goals in the near future, so that the challenging scientific topics outlined in the introduction can be addressed.
ACKNOWLEDGMENTS It is a pleasure to acknowledge the colleagues who in the course of the years contributed to the results described here, in particular D. Offenberg, C.B. Zhang, A. Ostendorf, U. Fröhlich, A. Wilson, P. Blythe, J. Koelemeij, H. Wenz, H. Daerr, Th. Fritsch, Ch. Wellers, V. Korobov, D. Bakalov, S. Jorgensen, M. Okhapkin, and A. Nevsky. Technical support was provided by P. Dutkiewicz, R. Gusek, J. Bremer, and H. Hoffmann. We are grateful for financial support from the German Science Foundation, the EC network “Cold Molecules,” the Alexander-von-Humboldt Foundation, the Düsseldorf Entrepreneurs Foundation, the DAAD, and the Studienstiftung des Deutschen Volkes. © 2009 by Taylor and Francis Group, LLC
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