Nonequilibrium Quantum Transport Physics In Nanosystems: Foundation Of Computational Nonequilibrium Physics In Nanoscience And Nanotechnology

NONEQUILIBRIUM QUANTUM TRANSPORT PHYSICS IN NANOSYSTEMS Foundation of Computational Nonequilibrium Physics in Nanoscien...

Author: Felix A. Buot

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NONEQUILIBRIUM QUANTUM TRANSPORT PHYSICS IN NANOSYSTEMS Foundation of Computational Nonequilibrium Physics in Nanoscience and Nanotechnology

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NONEQUILIBRIUM QUANTUM TRANSPORT PHYSICS IN NANOSYSTEMS Foundation of Computational Nonequilibrium Physics in Nanoscience and Nanotechnology

Felix A. Buot

George Mason University, USA

World Scientific NEW JERSEY

•

LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

•

TA I P E I

•

CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

NONEQUILIBRIUM QUANTUM TRANSPORT PHYSICS IN NANOSYSTEMS Foundation of Computational Nonequilibrium Physics in Nanoscience and Nanotechnology Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-256-679-9 ISBN-10 981-256-679-1

Printed in Singapore.

Lakshmi - Nonequilirbrium Quantum Transport.pmd 1

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To my parents; to my wife Evelina, to Max and Eugene

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Preface

The vigorous developments in nanoscience and nanotechnology in such areas as materials science, nanoelectronics and nanodevices, microfluidics, photonics, nanooptoelectronics, plasmonics, phononics, quantum computing, new phases of matter, and quantum information clearly require fundamental knowledge across all areas of physics, aside from familiarity with other established disciplines such as chemistry and microbiology. For physicists and engineers engaged in the analyses of experimental results, physical modeling, and large-scale numerical simulation of active nanodevices and nonequilibrium properties of nanostructures or open nanosystems, there is a need for a broad knowledge of the theoretical foundation of relevant quantum physics. It is the aim of this book to try to fill this need for computational scientists and engineers, as well as for people from other disciplines who felt the need to ‘cross the isle’. This book provides a comprehensive introduction to two significant and pertinent theoretical developments, namely, (a) quantum superfield theory of nonequilibrium quantum physics in real time, and (b) discrete phase-space quantum mechanics. These two formalisms combine to form the foundation of computational nanoscience of functional nanomaterials and nanotechnology. Although the formulation of discrete phase space quantum mechanics was originally motivated to describe energy-band quantum dynamics of charge-carriers in crystalline solid,1 it now occupies the central idea in the dynamics of qubit or two-state system in quantum computing. The field of mesoscopic physics experienced a growth surge in the nineteen eighties. These studies involve the linear approximation of quantum transport in mesoscopic systems, basically serving as a departure from the statistical character of the well-known Kubo formula for the conductivity of condensed matter. On the other hand, the field of nanoelectronics requires the full power of nonequilibrium quantum physics to characterize the highly nonlinear and ultrafast characteristics of quantum nanodevices, where the quantum distribution function inside a nanodevice is 1 There, the number of lattice points is odd by virtue of the crystal inversion symmetry. The use of multiplicative inverse assumes that the number of lattice points is a prime number, primality is required for the nonzero elements to have multiplicative inverses.

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required to match with the classical distribution function at the connecting leads, possessing a much larger number of degrees of freedom. The purpose of this book is to give a broad analytical tool for attacking problems in quantized nonequilibrium dynamics of particles and fields in open systems, especially when openness is due to boundary interface and current-lead contacts (serving as particle and/or energy source and sink in nanodevices). Thus, a more complete background on quantum mechanical techniques is given in the first part of the book. It is intended to make the book self-contained for studying nonequilibrium quantum physics, to help graduate students and entering researchers pursue independent research on the challenging area of highly nonequilibrium nanoscience and nanotechnology. A deeper understanding of discrete quantum mechanics also requires us to give a brief overview of quantum computing and quantum information from a new point of view, which serves as an introduction to this intriguing area of research by itself. A strong emphasis on the principal role of generalized canonical variables in quantum physics, Hermitian and non-Hermitian, discrete and continuum, is the recurring theme that occupies most of the theoretical discussions in this book. This book consists of eight parts. In Part 1 the foundation and methodology in quantum mechanics of particles and fields is given. Part 2 contains discussions of mesoscopic physics, centered on the Landauer and Landauer-Büttiker quantum conductance formulas. In Part 3 a number of quantum semiconductor quantum devices illustrate some of the novel devices that characterized the developments in nanoelectronics, including a more detailed discussion of the device physics of resonant tunneling nanodevices. A generalized quantum superfield theoretical formulation of nonequilibrium quantum physics in real time is discussed in Part 4. In Part 5 the operator Hilbert-space methods and quantum tomography based on mutually-unbiased bases are given in preparation for Part 6. In Part 6 the phase space formulation of discrete Wigner function on finite field and discrete phase-space quantum tomography is given. In Part 7 the phenomenological treatment of open quantum systems in terms of generalized measurements and evolution superoperator is given to supplement Part 4, to help clarify the various nonequilibrium correlation functions used in quantum superfield theoretical technique. For example, the dissipation, noise and mass kernels are discussed in more detail. An overview of quantum computing and quantum information, explicitly from the discrete phase-space point of view, is found in Part 8. I wish to thank the World Scientific Publishing Co. for inviting me to expand my monograph into a book of Advanced Lecture Series. I am deeply grateful to my wife, Evelina (Belen), for her patience, encouragement, and understanding during the writing of this book.

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Contents

Preface

vii

Overview of Quantum Mechanical Techniques

1

1. Quantum Mechanics: Perspectives

3

1.1

1.2 1.3 1.4 1.5

Wave Mechanics of Particles: Schrödinger Wave Function . . . . . 1.1.1 Some Algebraic Relations of Q and P . . . . . . . . . . . . 1.1.2 Deterministic Schrödinger Wave Equation . . . . . . . . . 1.1.3 Isotopic Wavefunction and Many-Body Wavefunction . . . 1.1.3.1 Decoupling of Isotopic Degrees of Freedom . . . 1.1.3.2 Phenomenological ‘Decoupling’ or Reduction of Many-Body Problems . . . . . . . . . . . . . . . Generator of Position Eigenstates . . . . . . . . . . . . . . . . . . Discrete Phase Space on Finite Fields . . . . . . . . . . . . . . . . Non-Hermitian Canonical Variables . . . . . . . . . . . . . . . . . 1.4.1 Left and Right Eigenvectors of Non-Hermitian Operators . Coherent State Formulation as a Mixed q-p Representation . . . .

2. Quantum Mechanics of Classical Fields 2.1

Quantization of Harmonic Oscillator . . . . . . . . . . . . . . . 2.1.1 The Complex Canonical Variables . . . . . . . . . . . . 2.1.2 Classical Schrödinger-Like Equation for Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Second-quantization of the Schrödinger-Like Equation

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23 23

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25 25

Complex Dynamical Variables . . . . . . . . . . . . . . . . . . . . ix

14 14 18 18 19 21 23

3. The Linear Chain of Atoms Coupled by Harmonic Forces 3.1

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3.1.1

Creation and Annihilation Operator for a Coupled Linear Chain of Atoms . . . . . . . . . . . . . . . . . . . . . . . .

4. Lattice Vibrations in Crystalline Solids: Phonons 4.1 4.2 4.3

4.4 4.5 4.6

30

Elementary Lattice Dynamics: The Linear Chain . . . . 4.1.1 Quantization of the Vibrational Mode: Phonons Lattice Vibrations in Three Dimensions . . . . . . Normal Coordinates in Three Dimensions . . . . . . . . 4.3.1 Acoustic and Optic Modes . . . . . . . . . . . . 4.3.2 Frequency Distribution of Normal Modes . . . . Experimental Probes: Elastic Constants . . . . . . . . . Hamiltonian in Terms of Normal Coordinates . . . . . . Phonons in Three Dimensions . . . . . . . . . . . . . .

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5. Quantization of Electromagnetic Fields 5.1 5.2 5.3 5.4

Maxwell Equations . . . . . . . . . . . . . . . . . . The Electromagnetic Wave Equations . . . . . . . 5.2.1 A Single Electromagnetic Wave Equation . Covariant Formulation of Electrodynamics . . . . Complex Dynamical Variables . . . . . . . . . . .

Wave Function for the Harmonic Oscillator . . Second Quantization of the Classical φ and φ∗ Biorthogonal Bases . . . . . . . . . . . . . . . Coherent State Bases . . . . . . . . . . . . . .

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Non-Orthogonality of Coherent States . . . . . . . . . . . . . . . . Completeness of Coherent States . . . . . . . . . . . . . . . . . . . Generation of Coherent States . . . . . . . . . . . . . . . . . . . . Displacement Operator . . . . . . . . . . . . . . . . . . . . . . . . Linear Dependence of Coherent States . . . . . . . . . . . . . . . . General Completeness Relation for States Generated by the Displacement Operator . . . . . . . . . . . . . . . . . . . . . . . . . . Coordinate Representation of a Coherent State . . . . . . . . . . . The Power of Coherent State Representation and the Virtue of Over-Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . .

8. Density-Matrix Operator and Quasi-Probability Density 8.1 8.2

50 51 52 54 56 62

7. Coherent States Formulation of Quantum Mechanics 7.1 7.2 7.3 7.4 7.5 7.6

30 35 36 37 42 44 46 46 48 50

6. Quantum States of Classical Fields 6.1 6.2 6.3 6.4

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Diagonal Representation of Density-Matrix Operator . . . . . . . . Procedures for Determining σ (α) . . . . . . . . . . . . . . . . . .

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9. Operator Algebra 9.1 9.2 9.3

9.4 9.5

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General Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . Boson Annihilation and Creation Operators, Ordering . . . . . . . 9.2.1 Traces of Function of Boson Operators . . . . . . . . . . . Characteristic Functions and Distribution Functions . . . . . . . . 9.3.1 The Wigner Distribution Function . . . . . . . . . . . . . 9.3.1.1 Q-function and P-Function . . . . . . . . . . . . 9.3.2 The Husimi Distribution Function . . . . . . . . . . . . . . Generalized Coherent States and Squeezing . . . . . . . . . . . . . Algebra and Calculus within Ordered Products . . . . . . . . . . . 9.5.1 Algebra within Ordered Products . . . . . . . . . . . . . . 9.5.1.1 Diﬀerentiation within Ordered Products . . . . . 9.5.2 Integration within Ordered Products in Quantized Classical Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 Evaluation of Integral of Some Important Mapping Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.4 Symplectic Transformation and Symplectic Group . . . . . 9.5.4.1 Quadrature States . . . . . . . . . . . . . . . . . 9.5.5 Complex Form of Symplectic Transformation Matrix . . .

10. Discrete Quantum Mechanics of Bloch Electrons

87 91 95 98 100 104 105 109 113 113 114 114 115 116 119 120 124

10.1 Energy-Band Dynamics of Bloch Electrons . . . . . . . . . 10.1.1 Wannier Function and Bloch Function . . . . . . . 10.1.2 Lattice Weyl-Wigner Formulation of Energy-Band Dynamics . . . . . . . . . . . . . . . . . . . . . . . 10.2 Application to Calculation of Magnetic Susceptibility . . .

. . . . 124 . . . . 124 . . . . 125 . . . . 131

11. The Eﬀective Hamiltonian

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11.1 Two-Body Eﬀective Hamiltonian . . . . . . . . . . . . . . . . 136 11.2 Eﬀective Hamiltonian in Second Quantization . . . . . . . . . . . 137 11.3 Eﬀective Non-Hermitian Hamiltonian in a Magnetic Field . . . . . 141 12. Path Integral Formulation 12.1 Evolution Operator and Sum over Trajectories 12.2 Path Integral in Quantum-Field Theory . . . . 12.2.1 Bose Systems . . . . . . . . . . . . . . 12.2.2 Path Integral for Fermion Systems . .

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13. Gauge Theory and Geometric Phase in Quantum Systems

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146 148 148 149 157

13.1 Directional (Covariant) Derivative on Curve Spaces . . . . . . . . 158 13.2 Parallel Transport in Curvilinear Space . . . . . . . . . . . . . . . 159 13.3 Parallel Transport Around Closed Curve . . . . . . . . . . . . . . 160

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13.4 Generalization to Quantum Mechanics . . . . . . . . . . . . . . . . 162 13.5 Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . . 166 14. Generalizations of Geometric Phase: Fiber Bundles

170

14.1 The Fiber Bundle Concept . . . . . . . . . . . . . . . . . . . . . . 14.2 Generalizations of Berry’s Geometric Phase in Quantum Physics . 14.3 Geometric Phase in Many-Body Systems . . . . . . . . . . . . . . 14.3.1 Localized Disturbances of the Ground State of 2+1-D Many-Body Systems . . . . . . . . . . . . . . . . . . . . . 14.3.2 Reconstructing Statistical Quantum Fields in Many-Body Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2.1 Bosonization . . . . . . . . . . . . . . . . . . . . 15. Geometric Phase in Quantum Field Theories: Standard Model 15.1 15.2 15.3 15.4

Classical Gauge Theory . . . . . . . . . . The Yang-Mills Lagrangian for the Gauge Electrodynamics as a Gauge Theory . . . Quantization of Gauge Theories . . . . .

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Feynman Diagrams . . . . . . . . . The Birth of String Theory . . . . . Need for Extra Dimensions in String Nanoelectronics and String Theory .

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182 186 187 187 189

. . . . . . . . . . Theory . . . . .

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Mesoscopic Physics 17. Mesoscopic Physics 17.1 17.2 17.3 17.4 17.5 17.6

176

182

16. String Theory 16.1 16.2 16.3 16.4

170 173 174

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mesoscopic Quantum Transport . . . . . . . . . . . . . . . . . . . Electrical Resistance Due to a Quantum Scattering Event . . . . . The Multichannel Conductance Formula . . . . . . . . . . . . . . . Quantum Interference in Small-Ring Structures . . . . . . . . . . . Generalized Four-Probe Conductance Formula . . . . . . . . . . . 17.6.1 Two-Probe Conductance Formula . . . . . . . . . . . . . . 17.6.2 Three-Probe Conductance Formula: Model of Inelastic Scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6.3 Weakly-Coupled Voltage Probes: Barrier Point Contacts . 17.6.4 The Landauer Four-Probe Conductance Limit . . . . . . .

18. Model of an Inelastic Scatterer with Complete Randomization

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18.1 Conductance Formula for a Sample Containing an Inelastic Scatterer between Two Elastic Scatterers . . . . . . . . . . . . . . . . . 220 18.2 Quantum Coherence in a Chain of Elastic and Inelastic Scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 19. Other Applications of Landauer-Büttiker Counting Argument 19.1 19.2 19.3 19.4 19.5 19.6

Integral and Fractional Quantum Hall Eﬀect . . Universal Conductance Fluctuations . . . . . . . Persistent Currents in Small Normal-Metal Loop Transport in One-Channel Luttinger Liquid . . . Mesoscopic Thermal Noise and Excess Noise . . High-Frequency Behavior . . . . . . . . . . . . .

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20. “Gated” Schrödinger Waveguide Structures and Ballistic Transport

229 230 230 231 231 232 233

20.1 Phenomena Associated with the Quantization of Charge . . . . . . 233 21. Steady-State Nonlinear Many-Body Quantum Transport 21.1 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . 21.2 Integral Equations of Mesoscopic Physics . . . . . . . . . . . 21.3 Tight-Binding Recursive Technique . . . . . . . . . . . . . . 21.3.1 Tight-Binding Expression for the Current . . . . . . 21.3.2 Multidimensional Current Expression . . . . . . . . . 21.3.3 Mesoscopic Transport Along a Linear Atomic Chain 21.3.4 The Four-Probe Landauer Current Formula . . . . . 21.3.5 Current Formula in the Presence of Real Phonon Scatterings . . . . . . . . . . . . . . . . . . . . . . . . 22. Numerical Matrix-Equation Technique in Steady-State Quantum Transport

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22.1 Kinetic Equation at Low Temperatures . . . . . . . . . . . . . . . 259 22.2 Kinetic Equation at Higher Temperatures and Arbitrary Bias . . . 262 22.3 Relation with Multiple-Probe Büttiker Current Formula . . . . . . 263 23. Alternative Derivation of Büttiker Multiple-Probe Current Formula

Heterostructure Quantum Devices: Nanoelectronics 24. Nanoelectronics

268

271 273

24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 24.2 Nanodevices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 24.3 Vertical vs Lateral Transport in Nanotransistor Designs . . . . . . 280

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24.4 Nanotransistor Designs . . . . . . . . . 24.4.1 Vertical Transport Designs . . . 24.4.2 Lateral Transport Designs . . . 24.4.3 GaAs/AlGaAs MODFET-Based

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25. Nanodevice Physics

281 281 289 292 294

25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2 Time-Dependent Nonequilibrium Green’s Function . . . . . . . . . 25.2.1 Electron-Electron Interaction via Exchange of Phonons . . 25.2.2 Relaxation-Time Approximation . . . . . . . . . . . . . . . 25.3 Intrinsic Bistability of RTD . . . . . . . . . . . . . . . . . . . . . . 25.4 Quantum Inductance and Equivalent Circuit Model for RTD . . . 25.4.1 Transient Switching Behavior and Small-Signal Response of RTD from the QDF Approach . . . . . . . . . . . . . . 25.4.2 High-Frequency Behavior and Small Signal Response of RTD using an Equivalent Circuit Model . . . . . . . . . . 25.4.2.1 Linear Response . . . . . . . . . . . . . . . . . . 25.4.2.2 Nonlinear Response . . . . . . . . . . . . . . . . 26. QDF Approach and Classical Picture of Quantum Tunneling

294 295 300 301 301 305 311 313 313 316 318

26.1 Lattice Wigner Function and Band Structure Eﬀects . . . . . . . . 318 26.2 Coherent and Incoherent Particle Tunneling Trajectories . . . . . 319 27. RTD as a Two-State Memory Device, a Memdiode or a Memristor 27.1 Binary 27.1.1 27.1.2 27.1.3 27.1.4

Information Storage at Zero Bias . . . . . . . . Intrinsic Behavior of Double-Barrier Structures The Physical Picture . . . . . . . . . . . . . . . Analysis of a RTD Memory or Memdiode . . . Two-State I-V and Two Charge States . . . . .

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28. RTD as a Tera-Herz Source 28.1 Type I RTD High-Frequency Operation . . . . . . . . . . . . . . . 28.2 Type II RTD High-Frequency Operation . . . . . . . . . . . . . . 28.3 Regional Block Renormalization: Type-I RTD . . . . . . . . . . . 28.3.1 Estimation of J2c and J1c . . . . . . . . . . . . . . . . . . . 28.3.2 Elimination of Fast-Relaxing Variable for Type-I RTD . . 28.4 Regional Block Renormalization: Type-II RTD . . . . . . . . . . . 28.5 Two Sites Bloch-Equation ‘Instanton’ Approach . . . . . . . . . . 28.5.1 Type-I RTD . . . . . . . . . . . . . . . . . . . . . . . . . . 28.5.1.1 Tunneling Matrix Elements . . . . . . . . . . . . 28.5.1.2 Elimination of Oﬀ-Diagonal Elements of the Density-Matrix . . . . . . . . . . . . . . . . . . .

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28.5.2 Type-II RTD . . . . . 28.6 Stability Analysis . . . . . . . 28.7 Numerical Results . . . . . . . 28.8 Perturbation Theory and Limit

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General Theory of Nonequilibrium Quantum Physics

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29. General Theory of Nonequilibrium Quantum Physics in Real Time

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29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 29.2 Quantum Dynamics in Liouville Space . . . . . . . . . . . . . . . . 363 30. Super-Green’s Functions

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30.1 Connected Diagrams: Correlation Function K . . . . . . . . . . . . 30.2 Self-Consistent Equations for GQDF . . . . . . . . . . . . . . . . . 30.2.1 Schwinger Equation as Generalization of the Kohn-Sham and Gross-Pitaevskii Equations . . . . . . . . . . . . . . . 30.2.2 Closure Problem and Renormalization Procedure . . . . . 30.2.3 Iterative Equations for the Vertex Functions . . . . . . . . 31. Quantum Transport Equations of Particle Systems

379 380 381 381 384 387

31.1 General Quantum Transport Equations . . . . . . . . . . . . . . . 390 31.2 Transport Equations and Lattice Weyl Transformation . . . . . . 392 32. Generalized Bloch Equations 32.1 32.2 32.3 32.4 32.5 32.6

Generalized Bloch Equations in Quantum Optics . The Bloch Vector Representation . . . . . . . . . . Bloch Vector Equations . . . . . . . . . . . . . . . Atomic Energy and Dipole Moment . . . . . . . . Rotating Wave Approximation . . . . . . . . . . . Transformation to Rotating Frame . . . . . . . . . 32.6.1 State Preparation and Adiabatic Following 32.7 Analytical Solutions of the Bloch Equations . . . . 32.7.1 The Rabi Problem . . . . . . . . . . . . . 32.7.2 Response to Light Pulse . . . . . . . . . . 32.7.3 Self-Induced Transparency . . . . . . . . . 33. Generalized Coherent-Wave Theory

396 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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397 401 403 403 406 406 408 408 408 410 411 415

33.1 The Tight-Binding Limit . . . . . . . . . . . . . . . . . . . . . . . 418 33.1.1 Flat Band Case . . . . . . . . . . . . . . . . . . . . . . . . 419 34. Impact Ionization and Zener Eﬀect

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34.1 Coulomb Pair Potential ∆ for Impact Ionization and Auger Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 34.2 Pair Potential ∆ due to Zener Eﬀect . . . . . . . . . . . . . . . . . 424 35. Quantum Transport Equations in Phase Space

426

35.1 Conservation of Particle in Zener Tunneling . . . . . . . . . . . . . 429 35.2 Nanosystem Applications . . . . . . . . . . . . . . . . . . . . . . . 429 35.2.1 Resonant Tunneling Diode (RTD) . . . . . . . . . . . . . . 429 36. QSFT of Second-Quantized Classical Fields: Phonons 36.1 36.2 36.3 36.4 36.5

Liouvillian Space Phonon Dynamics . . . . . . . . . . The Phonon Super-Green’s Function . . . . . . . . . . Transport Equation for the Phonon Super-Correlation Phonon Transport Equations in Phase Space . . . . . The Phonon Boltzmann Equation . . . . . . . . . . .

431 . . . . . . . . . . . . Function . . . . . . . . . . . .

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Operator Space Methods and Quantum Tomography

445

37. Operator Hilbert-Space Methodology in Quantum Physics

447

37.1 The Density Operator in Operator Vector Space . . . . . . . 37.2 Formulation in Terms of Translation Operators . . . . . . . . 37.2.1 Weyl Transform of GPM Operator . . . . . . . . . . 37.2.2 Weyl Transform of the GPM Eigenstate Projector . . 37.3 Point Projector in Terms of Line Projectors . . . . . . . . . . ˘ (p, q) in Terms of Intersecting Lines at Point (p, q) 37.3.1 ∆

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38. The Wigner Function Construction 38.1 The Quasi-Probability Distribution and Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.1.1 The Radon Transform . . . . . . . . . . . . . . . 38.2 Line Eigenstates and Line Projection Operators . . . . . 38.2.1 Density Operator in Terms of Line Projectors . . 38.3 Translational Covariance of the Wigner Function . . . . . 38.4 Transformation Properties of the Radon Transform . . . 38.5 Intersection of Line Projectors: Mutually Unbiased Basis

Discrete Phase Space on Finite Fields 39. Discrete Phase Space on Finite Fields

433 435 438 439 442

447 450 452 454 456 456 460

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39.1.1 Line in Discrete Phase Space: Pure Quantum State . . . . sym . . 39.1.2 Commutation Relation Between Q (λ) and T (q, p) 39.2 Generalized Pauli Matrices . . . . . . . . . . . . . . . . . . . . . . 39.2.1 Commutation Relations and Products of Yq p . . . . . . . 39.2.2 Expansion of Operators: Hamiltonian in Terms of Generalized Pauli Matrices . . . . . . . . . . . . . . . . . . 39.2.3 Pauli Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 39.3 Discrete Fourier Transform and Generalized Hadamard Matrix . . 39.3.1 Eigenfunctions and Eigenvalues of X1 , Z1 , and Y1,1 . . . . 39.3.2 General Quantum State of a Two-Level System: Bloch Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.3.2.1 Range of the Parameters . . . . . . . . . . . . . 39.3.2.2 Bloch Sphere . . . . . . . . . . . . . . . . . . . . 39.3.2.3 Quantum States on Opposite Points of Bloch Sphere . . . . . . . . . . . . . . . . . . . . . . . . 39.3.3 Exponential Map . . . . . . . . . . . . . . . . . . . . . . . 39.3.3.1 Rotation about an Arbitrary Axis in Real 3-D Space . . . . . . . . . . . . . . . . . . . . . . . . 39.3.3.2 Arbitrary Unitary Operator for a Qubit: Quantum Control . . . . . . . . . . . . . . . . . . . . 39.3.4 Density Operator for a Two-Level System: Disordered and Pure States . . . . . . . . . . . . . . . . . . . . . . . . . . 40. Discrete Quantum Mechanics on Finite Fields

41.1 Discrete Wigner Function for a Single Qubit . . . . . . . . 41.2 Discrete Phase Space Structure for Two Qubits . . . . . . . 41.2.1 Striations Construction . . . . . . . . . . . . . . . . 41.2.2 Binary String Encoding of Points in Discrete Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Construction of Dual Field Basis for Two Qubits . 41.2.3.1 Commutation Relation . . . . . . . . . .

483 484 486 487 489 491 491 493 493 497 498 500 501

40.1 Tensor Product of Operators . . . . . . . . . . . . . . . . . . 40.1.1 Entanglement Due to Interactions . . . . . . . . . . . 40.1.2 The No-Cloning Theorem . . . . . . . . . . . . . . . 40.1.2.1 Consequences . . . . . . . . . . . . . . . . . 40.2 Quantum Control . . . . . . . . . . . . . . . . . . . . . . . . 40.2.1 Pauli Operators over Power-of-Prime Finite Fields . 40.2.1.1 Phase Space for a Spin- 12 System or Single Qubit . . . . . . . . . . . . . . . . . . . . . 40.3 Striations and Mutually Unbiased Bases . . . . . . . . . . . . 41. Discrete Wigner Distribution Function Construction

478 479 480 481

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501 505 506 507 507 509

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41.3 Line Projectors for Two Qubit Systems . . . . . . . . . . . 41.3.1 Product Hilbert Space for a Two Qubit System . . 41.3.2 Eigenvectors of Commuting Translation Operators 41.3.3 Vertical Striation Ray and ‘Position’ Basis . . . . . 41.3.4 Horizontal Striation Ray and ‘Momentum’ Basis . . 41.3.5 Diagonal Striation Ray and ‘Y Y ’ Basis . . . . . . . 41.3.6 Low-Slope-Striation Ray and ‘Belle’ Basis . . . . . 41.3.7 High-Slope-Striation Ray and ‘Beau’ Basis . . . . . 41.4 Discrete Wigner Function for Two Qubits . . . . . . . . . . 41.4.1 The Origin in Phase Space, q = 0, p = 0 . . . . . . 41.4.2 The Point (1, 0) in Phase Space Structure . . . . . 41.4.3 The Point (ω, 0) . . . . . . . . . . . . . . . . . . . . 41.4.4 The Point (˘ ω , 0) . . . . . . . . . . . . . . . . . . . . 41.4.5 The Point (0, 1) . . . . . . . . . . . . . . . . . . . . 41.4.6 The Point (0, ω) . . . . . . . . . . . . . . . . . . . . 41.4.7 The Point (0, ω ˘) . . . . . . . . . . . . . . . . . . . . 41.4.8 The Point (1, 1) . . . . . . . . . . . . . . . . . . . . 41.4.9 The Point (ω, ω) . . . . . . . . . . . . . . . . . . . 41.4.10 The Point (˘ ω, ω ˘) . . . . . . . . . . . . . . . . . . . 41.4.11 The Point (ω, 1) . . . . . . . . . . . . . . . . . . . . 41.4.12 The Point (˘ ω , 1) . . . . . . . . . . . . . . . . . . . . 41.4.13 The Point (˘ ω , ω) . . . . . . . . . . . . . . . . . . . 41.4.14 The Point (1, ω) . . . . . . . . . . . . . . . . . . . . 41.4.15 The Point (ω, ω ˘) . . . . . . . . . . . . . . . . . . . 41.4.16 The Point (1, ω ˘) . . . . . . . . . . . . . . . . . . . . 41.5 Examples of Two-Qubit Discrete Wigner Function . . . . . 41.5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . 41.5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . 41.5.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . 41.6 Quantum Nets: Arbitrary Assignment to a ‘Vacuum’ Line . 41.7 Potential Applications . . . . . . . . . . . . . . . . . . . . .

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Phenomenological Superoperator of Open Quantum Systems: Generalized Measurements

567

42. Interference and Measurement 42.1 Projective Measurements . . . . . . . . . . . . . . 42.1.1 Eﬀects of Measurements . . . . . . . . . . 42.1.2 Eﬀects of Measurements on Entanglement 42.1.3 Measurements in Quantum Teleportation . 43. Quantum Operations on Density Operators

535 535 541 541 544 547 550 552 556 556 557 557 557 558 558 558 559 559 559 560 560 560 561 561 561 562 562 562 563 564 565

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43.1 The Kraus Representation Theorem . . 43.2 Examples of Quantum Operations . . . 43.2.1 Unitary Evolution . . . . . . . . 43.2.2 Probabilistic Unitary Evolution 43.2.3 Von Neumann Measurements . 43.2.4 POVMs . . . . . . . . . . . . .

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44. Generalized Measurements

576 576 576 576 576 577 579

44.1 Distinguishing Quantum States . . . . . . . . . . . . . . . . . . . . 583 44.2 Utility of POVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 45. Phenomenological Density Matrix Evolution

586

45.1 Quantum Channels . . . . . . . . . . . . . . . . . . . 45.1.1 Time Evolution . . . . . . . . . . . . . . . . . 45.1.2 Partial Trace . . . . . . . . . . . . . . . . . . 45.2 Depolarizing Channel . . . . . . . . . . . . . . . . . . 45.2.1 Unitary Representation of the Channel . . . . 45.2.2 Kraus Representation of the Channel . . . . . 45.2.3 Relative-State Representation . . . . . . . . . 45.2.4 Bloch Sphere Picture . . . . . . . . . . . . . . 45.2.5 Semigroup Property . . . . . . . . . . . . . . 45.3 Phase Damping Channel . . . . . . . . . . . . . . . . 45.3.1 Unitary Representation for the Whole System 45.3.2 Kraus Operators . . . . . . . . . . . . . . . . 45.4 Amplitude-Damping Channel . . . . . . . . . . . . . . 45.4.1 POVM and Unchanging Environment . . . . .

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46. Master Equation for the Density Operator

598

46.1 The Lindblad Master Equation . . . . . . . . . . . . . . . . . 46.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46.2.1 Spontaneous Emission . . . . . . . . . . . . . . . . . 46.2.2 Bloch Equations in Magnetic Resonance for Spin 1/2 46.3 The Pauli Master Equation . . . . . . . . . . . . . . . . . . . 46.4 Lindblad Equation for a Damped Harmonic Oscillator . . . . 46.5 Lindblad Equation for Phase Damped Harmonic Oscillator . 46.6 Coherent State and Decoherence . . . . . . . . . . . . . . . .

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47. Microscopic Considerations of a Two-Level System Revisited 47.1 Quantized Radiation Field . . . . 47.2 Perturbation Expansion of Density 47.2.1 First-order Contribution . 47.2.2 Resonance Approximation

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588 589 589 589 590 590 591 593 593 593 594 594 595 596

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599 603 603 604 605 606 608 610 612

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47.2.3 Bloch Equation . . . . . . 47.3 Second Order Contribution . . . . 47.4 Master Equation to Second Order 47.4.1 Thermal Reservoir . . . .

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48. Stochastic Meaning of Nonequilibrium Quantum Superfield Theory

622 624 626 629 634

48.1 Kubo-Martin-Schwinger Condition . . . . . . . . . . . . . . . . . . 48.1.1 Mass, Dissipation, and Noise Kernels in Nonequilibrium Quantum Superfield Theory . . . . . . . . . . . . . . . . . 48.2 A Two-State System Interacting with a Heat Bath . . . . . . . . . 48.3 Nonequilibrium Quantum Superfield Theory Correlations . . . . . 48.4 Lamb Shift, Dissipation Kernel, and Noise Kernel . . . . . . . . . 48.4.1 Comparison with the Master Equation of Sec. 47.4 . . . .

636 639 641 644 650 652

Quantum Computing and Quantum Information: Discrete Phase Space Viewpoint 657 49. Discrete Phase Space Viewpoint 49.1 Quantum Teleportation . . . . . . . . . . . . . . . . . . . . . . . 49.1.1 Unified Teleportation Procedure . . . . . . . . . . . . . . 49.2 N-State Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.3 Formal Derivation of Entangled Basis States . . . . . . . . . . . 49.3.1 Bell Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 49.3.2 Three-Qubit Entangled Basis . . . . . . . . . . . . . . . 49.3.3 A Qubit Teleportation Using Three-Particle Entanglement . . . . . . . . . . . . . . . . . . . . . . . . 49.4 Teleportation Using Three-Particle Entanglement and an Ancilla 49.5 Two-Qubit Teleportation Using Three-Particle Entanglement . .

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659 664 665 665 666 670

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50. Superdense Coding

680

50.1 General Dense Coding Scheme . . . . . . . . . . . . . . . . . . . . 684 50.2 Reduced Density Matrices . . . . . . . . . . . . . . . . . . . . . . . 684 50.3 Quantum Channel, Generalized Dense Coding . . . . . . . . . . . 685 51. Quantum Algorithm 51.1 Quantum Fourier Transform . . . . . . . . . . . . . . . . . . . . . 51.1.1 Order-Finding Algorithm . . . . . . . . . . . . . . . . . . . 51.1.2 Phase Estimation Algorithm . . . . . . . . . . . . . . . . . 51.1.3 Connection Between Root Finding and Phase Estimation . 51.2 Quantum Search Algorithm . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Performance of the Search Algorithm . . . . . . . . . . . .

687 687 693 695 699 702 704

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51.3 Discrete Logarithms . . . . . . . . . . . . . . . 51.3.1 Quantum Solution . . . . . . . . . . . 51.4 Hidden Subgroup Problem . . . . . . . . . . . 51.4.1 Quantum Hidden Subgroup Algorithm

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704 705 706 708

Appendix A Commutation Relation between Components of π (x, t) and A (x , t)

711

Appendix B Lattice Weyl Transform of One-Particle Eﬀective Hamiltonian in Magnetic Field

715

Appendix C Second Quantization Operators in Solid-State Band Theory

718

Appendix D Direct Construction of Fermionic Path Integral

724

Appendix E Hot-Electron Green’s Function

730

Appendix F Derivation of Generalized Semiconductor Bloch Equations

732

Appendix G Calculation of Nonequilibrium Self-Energies

741

G.1 Nonequilibrium Self-Energy due to Electron-Electron Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 G.1.1 First-Order Contribution to the Electron Self-Energy . . . 741 G.1.2 Four-Point Vertex Function to Second Order . . . . . . . . 742 Appendix H Radon Transformation of Phase Space Functions

776

Appendix I Introduction to Finite Fields

790

I.1

I.2 I.3 I.4 I.5

Constructing Finite Fields . . . . . . . . . . I.1.1 GF(9) . . . . . . . . . . . . . . . . . I.1.2 GF(8) . . . . . . . . . . . . . . . . . Constructing Bases of Finite Field . . . . . . Trace Operation on Elements of Finite Field Dual Basis . . . . . . . . . . . . . . . . . . . I.4.1 Construction of Dual Basis . . . . . . Transformation of Coordinates . . . . . . . .

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793 793 795 796 798 800 800 802

Bibliography

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Index

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PART 1

Overview of Quantum Mechanical Techniques

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Chapter 1

Quantum Mechanics: Perspectives

Quantum mechanics essentially deals with geometric and statistical concepts, involving information and measurement theories. Geometric concepts are used to build up quantum-state space and its supporting or base manifold1 , with its corresponding metric and algebra, examples are phase space, Hilbert space, etc. Because of its statistical nature, quantum states are no longer represented by a determinate point in phase space in contrast with classical mechanics. Canonical dynamical variables are represented by operators, such operators as the Hamiltonian, position operator, momentum operator, phase-space point projectors, and phase-space line projectors, whose eigenvalues correspond to classical quantities. What is perhaps even more remarkable is that a general quantum-state Hilbert-space vector is represented as a superposition of a complete set of Hilbert-space eigenvectors, basically encompassing all classical possibilities at a given instant of time. Canonically conjugate dynamical variables can no longer be measured simultaneously due to the quantum-mechanical uncertainty principle2 . On the other hand, the informatics aspects of physics is common to classical and quantum physics, and pertains to abstract concepts of the basic guiding principles. Thus, the idea that ‘information is physical’ is still a subject of debate. Information theory deals with the quantification of uncertainty, indeterminism, competing possibilities, and chance, as well as the abstract characterization of a collection of diﬀerent ‘forms’ ( or diﬀerent configurations) through the concept of entropy. However, the concept of measurement does not play a mathematically formal role in classical mechanics. The reason for this is that classical mechanics deals directly with measurable, deterministic, or realizable quantities, corresponding to the eigenvalues in quantum mechanics. In quantum mechanics,operator algebra of measurement is an integral part and occupies a central formal operational concept in terms of quantum operators which operate on suitable abstract space. According to Schwinger, quantum mechanics is a mathematical formalism of physical measurements. In view of its informatics aspects, classical mechanics is a prerequisite for understanding the foundation of quantum mechanics. In fact, the abstract informatics aspect of classical mechanics, such as the maximum entropy principle, ergodicity, 1 One

familiar example is the space-time manifold. their respective quantum states are related by a unitary transformations, specifically by Fourier transformations. 2 Mathematically,

3

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the principle of least action, causality, symmetry and invariance guiding principle3 , are valuable theoretical tools in quantum mechanics. It is not possible to have a good working appreciation of the principles of quantum mechanics without a familiarity of the informatics aspect of classical mechanics. More recently, research in quantum computing and quantum entanglement calls for a deeper re-examination of classical information theory. Indeed, quantum information theory is a very active field of theoretical and experimental investigations and is expected to play an important role in the scientific advances of the 21st century. The beginning of the 20th century was timely for the formal development of quantum mechanics. The Hamilton and Lagrange formulation of classical mechanics provided the mathematical abstraction of classical mechanics in terms of generalized coordinates and canonical dynamical variables4 . Lagrange’s equations looks the same in any coordinate system. Indeed, the classical Hamiltonian (expressed in terms of generalized coordinates and momentum) is the Legendre transform of the classical Langrangian (expressed in terms of generalized coordinates and velocities). In 1746 Maupertuis first formulated the Principle of Least Action (PLA), further developed by the three great mathematicians, Euler, Lagrange, and Hamilton. PLA brought the concept of optimization, considered as the intelligence or supreme guiding principle to classical mechanics. Indeed, the PLA is considered one of the greatest generalizations in physical science, useful in the development of quantum mechanics. PLA describes the tendency of physical changes and processes to take the optimum path if left alone. Almost the whole of physics obeys the geodetic form. Water seeks the steepest descent, and distributes itself so that its surface is as low as possible, the water then has the minimum potential energy in the earth’s gravitational field. Light finds the quickest trajectory through an optical system (Fermat’s principle of Least Time). The path of a body in a gravitational field (i.e. free fall in space time) is a geodesic. Feynman’s formulation of quantum mechanics is based on a PLA, using path integrals. Schwinger’s formulation of quantum field theory makes use of the PLA. Maxwell’s equations can be derived as conditions of least action. Newton’s mechanics is derived from Hamilton’s principle of least action, and also Gauss’s principle of least constraint. Thomson’s theorem states that electrically charged particles configure so as to have the least energy. The Second Law of Thermodynamics requires that thermal systems change along a sequence of configurations, each having a higher probability of occurrence than the preceding configuration. In fact optimization in all aspect of physical, life, and socioeconomic dynamics has been seen to be the rule. Quantum mechanics also embodies these ‘informatics’ elements of classical mechanics, generalized probability theory, and information theory. 3 This guiding principle was notably first used by Einstein in constructing his theory of relativity. Indeed, symmetry considerations for establishing the Lagrangian coupled with PLA has been the modus operandi of major advances in modern theoretical physics. 4 In going over to quantum mechanics, the classical canonical variables becomes noncommuting operators for bosons (whose eigenvalues are c-numbers) and non-anticommuting operators for fermions (whose eigenvalues obey Grassmann algebra). Complex-valued canonical variables readily appeared in classical harmonic oscillator system and its quantization leads to the concept of ladder operators or creation and annihilation operators for bosons.

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Quantum Mechanics: Perspectives

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5

The central idea inherent in quantum mechanics lies in the mathematical concept of measurement, defined as an application of a measurement operator to an abstract quantum state of a physical dynamical system. In quantum mechanics, the state of a physical system is identified with an abstract vector in a complex Hilbert space (Hspace), without any reference to specific chosen basis states. Each abstract vector in H-space is called a “ket”, and written as |Ψ , and every ket has a dual bra, written as Ψ| in the dual H-space. For conservative systems, the bra is simply the conjugate transpose (also called the Hermitian conjugate) of the ket. Each bra corresponds to exactly one ket, and vice versa, hence the use of the same labeling. One may also interpret |Ψ as a column vector under any arbitrary chosen basis states, while Ψ| is the corresponding row vector under the same basis states with components as complex conjugate of the components of |Ψ . The inner product Φ| Ψ is the mapping from H-space to complex numbers, whereas Ψ| Ψ is a special case, called the norm of |Ψ , and maps to real numbers only. |Ψ is in general a superposition of elements from a set of eigenstates which forms complete basis states in H-space, |Ψ =

q

= q

ψ (q) |q , q |Ψ |q ,

(1.1)

where q is the quantum label of a basis state considered and ψ (q) = q |Ψ is the complex component or projection of |Ψ along the basis state |q , the so-called wavefunction or probability wavefunction. The absolute value of q |Ψ , which is 2 given by |ψ (q)| is the probability of finding the quantum state in the state |q . An eigenstate is defined as the state in H-space which yield deterministic value of the measurement operator, Q |q = q |q , where Q is the mathematical symbol for a measurement operator and q is the corresponding deterministic eigenvalue or classical quantity. The projection operator for the state |q in H-space is defined by |q q|. The completeness of the eigenstate vectors as basis states in H-space is expressed by

q

|q q| = 1.

Hermitian operators corresponds to observables such as position, momentum, and energy. The eigenstates of a Hermitian operator are orthonormal, q | q = δ q ,q . Thus the measurement operator Q may be written as Q= q

q |q

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Any arbitrary operator other than Q may be expanded in terms of the q-eigenstates as

P = q q

q | P |q

|q

q |.

In this sense measurement operators are projectors in H-space. The role played by the position operator Q and momentum operator P typifies the role of other canonical operators, Hermitian or non-Hermitian, having either real or complexvalued eigenvalues. The combined state vector of two or more quantum mechanical systems are defined by the tensor product of their respective state vectors, the result can thus be either in ‘entangled’ and/or ‘separable product’ quantum state. Quantum entanglement is the functional basis of quantum computers consisting of quantum bits or ‘qubits’ which has found explosive development in the last decade. In later chapters, measurement operators are generalized to quantum operation on the subsystem of a composite system where the total state is described by quantum-state tensors instead of quantum-state vector. In view of the general description of a quantum state as superposition of eigenstates, quantum mechanics deals with the totality of all possibilities that the whole physical system may evolve and can be realized at a given time. Equation of motion in quantum dynamics deals with the wavefunctions (or Wigner distribution function5 ) in the Schrödinger picture, or deals with the quantum operators in the Heisenberg picture, not with the canonical classical variables directly in stark contrast with the determinism of classical mechanics. For example, if the eigenstate |q represent the eigenstate of the position operator, then ψ (q) in Eq. (1.1) represents a complex wavefunction whose value, |ψ (q)|2 , measure the probability density in real space. This is clearly a generalization6 of positive-valued measure of classical probability theory, a generalization which allows for quantum interference eﬀects in its dynamics. Any measurement process which is intended to gain definite location of the system then collapses the state of the physical system to a specific position eigenstate, i.e., a specific location in real space. This has come to be known as the ‘wavepacket collapse’ in the quantum physics community. This collapse however entails complete lack of knowledge of the canonically conjugate momentum, i.e., somehow a decrease of entropy with respect to the position results in maximum entropy with respect to the momentum since subsequent measurements of momentum in this collapse state will only yield completely unpredictable random values of the momentum. This is another expression of the quantum mechanical uncertainty principle. 5 We shall see later that the Wigner distribution function represents the component of the density operator expanded in terms of the complete set of phase-space point projectors, often denoted by ∆ (p, q). 6 This is reminiscent of the generalization of positive real numbers to complex numbers.

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Wave Mechanics of Particles: Schrödinger Wave Function

The state of the particle in classical mechanics is described by a point in momentum (p)-position (q) space or phase space. The passage to quantum mechanics is to abandon the classical phase space point as representing the state of the particle. The momentum of the particle must now be measured by applying the momentum operator P on the eigenstates of momentum, |p , in Hilbert space. Likewise the position of the particle must be measured by applying the position operator Q on the eigenstate of position, |q , in Hilbert space. In quantum mechanics these two ‘measurements’ do not commute, since the measurement of position abandon any information about the momentum and vice versa. This can be illustrated by simulating the measurement in the limit of zero error, as in classical mechanics, using the position operator Q whose eigenvalue is q and abstract eigenket |q . In this limiting case, the corresponding eigenstate in Hilbert space is the Dirac-delta function in the coordinate representation given by |x, q = x |q = δ (x − q) centered at position q. Thus, we Q |x, q = q |x, q . Since the set of position eigenfunctions, {|q }, forms a complete basis states, we can expand the momentum eigenfunction in terms of the position eigenfunctions of all q values. Thus, in the coordinate representation we can write |x, p = x |p =

q

q| p |x, q .

By virtue of the completeness of the momentum eigenfunctions, the inverse transform then leads to |x, q =

p

p| q |x, p .

Thus the transformation function is the Fourier transformation function, i q| p = exp − p · q , |

(1.2)

with |x, p = C exp |i p · x , where C is the normalization constant, usually taken to be √1V , where V is the volume. Clearly, if the state of the particle after measurement collapses to the position eigenstates, δ (x − q), then all the momentum values are equally probable in this collapse state and hence all information of the momentum is lost and the subsequent application of the momentum operator will yield an indeterminate or random values for the momentum. This embodies the statement that the operators Q and P do not commute. In general, an arbitrary quantum-state vector of the particle, |Ψ is a superposition of the complete basis in Hilbert space spanned by either the abstract momentum or position eigenstates. If we expand |Ψ in terms of the position eigenstates, we have |Ψ =

q

ψ (q) |q = C

dq ψ (q) |q .

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In this coordinate representation, we have Q |Ψ =

dq q ψ (q) |q

q ψ (q) |q = C

q

and P |Ψ =

q

=C

dq ψ (q) p

dq ψ (q)

=C C

dq ψ (q) P |q

ψ (q) P |q = C

dq ψ (q) P |q = C

dq ψ (q)

i p exp − p · q |

| ∇q −i

p

|p

i exp − p · q |

|p ,

| ∇q |q . −i

(1.3)

Upon integrating by parts in the last line, we have P |Ψ = C

dq

| ∇q ψ (q) i

|q .

Note that the momentum operator acting on the basis states is i|∇q , Eq. (1.3), whereas the same momentum operator acting on the probability amplitude ψ (q) is −i|∇q . Now the Schrödinger wavefunction in the coordinate representation is given by ψ (q) = q| Ψ . By virtue of the orthonormality of the basis states, it follows that the momentum and position operators for the Schrödinger wavefunction are defined by the following relations, P ψ (q) ⇒ −i|∇q ψ (q) ,

(1.4)

Qψ (q) ⇒ qψ (q) .

(1.5)

In terms of momentum eigenstates we also have |Ψ =

p

ψ (p) |p = C

dp ψ (p) |p .

Thus, we also have in the momentum representation of the Schrödinger wavefunction, P ψ (p) ⇒ pψ (p) ,

(1.6)

Qψ (p) ⇒ i|∇p ψ (p) ,

(1.7)

Q |p ⇒ −i|∇p |p .

(1.8)

The noncommutation relation between P and Q is therefore expressed as [Q, P ] = i|.

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We can also deduce the noncommutation relation between P and Q using the basis states |q . We have P |q = i|∇q |q . Therefore P |q q| = (i|∇q |q ) q| . Integrating or summing over q and using the closure relation q-eigenstates, we obtain

q

P |q q| =

P q

q

(i|∇q |q ) q| ,

|q q| = P = P = i|

dq (i|∇q |q ) q| dq {∇q (|q q|) − |q (∇q q|)}

= −i|

dq |q (∇q q|) .

The representation for the position operator Q is trivial Q= q

q |q q| .

The commutation of Q and P using the basis vector now reads [Q, P ] = −i|

dq |q ∇q q | + i|

dq q |q q|

dq |q ∇q q |

= −i|

dq dq q |q q| |q ∇q q | + i|

= −i|

dq dq q |q δ (q − q ) ∇q q | + i|

= −i|

dq q |q ∇q q| + i|

dq ∇q q| q |q

= −i|

dq q |q ∇q q| + i|

dq (∇q q|) q |q + i|∇q q

+i|

dq dq ∇q q | q |q q| |q dq dq ∇q q | q |q δ (q − q )

dq q q| ∇q |q .

The first two terms cancel and we are left with [Q, P ] = i|

dq ∇q q |q q| + i|

= i| + i|

dq q |q q|

dq q q| ∇q |q ,

dq q q| ∇q |q

dq |q q|

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where we have used the relation, ∇qi qj = δ ij . We can show that the last term is zero by writing dq q q| ∇q |q =

i

dpe− | p.q |p

dq q q| i|∇q i

=

dqdp q q| e− | p.q p |p

=

dqdp q e− | p.q p q| |p

=

dqdp q e− | p.q pe | p.q

=

dq q

i

i

i

dp p

= 0, since both remaining integrals have odd integrand. The noncommutativity of Q and P signifies that their bases states are mutually unbiased, in the sense that the complete knowledge the eigenstates of P renders all the eigenstates of Q to be equally probable, and vice versa. Note that we have used the same basis states to represent Q and P to evaluate the commutation relation. 1.1.1

Some Algebraic Relations of Q and P

From this commutation relation one can easily prove by mathematical induction the following algebraic relations involving powers of Q and P, [Q, P m ] = i|mP m−1 = i|

∂ m P , ∂P

[P, Qm ] = −i|mQm−1 = −i|

∂ m Q . ∂Q

(1.9)

(1.10)

From these relations, it also follows that if F (P ) and G (Q) are functions that may be expanded in power series, then [Q, F (P )] = i|

∂ F (P ) , ∂P

[P, G (Q)] = −i|

∂ G (Q) . ∂Q

(1.11)

(1.12)

For a function of P and Q, namely, F (P, Q), we have [Q, F (P, Q)] = i|

∂ F (P, Q) , ∂P

[P, F (P, Q)] = −i|

∂ F (P, Q) , ∂Q

(1.13)

(1.14)

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where in the last two relations, the order of the factors involving Q and P in F (P, Q) must be preserved. 1.1.2

Deterministic Schrödinger Wave Equation

The particle Hamiltonian operator, H, acting on the state |Ψ now reads, using the position eigenfunction expansion of |Ψ , i|

∂ |Ψ = H |Ψ = C ∂t

dq

−

|2 2 ∇ + V (q) ψ (q, t) 2m q

|q .

Since H is Hermitian, the presence of i| renders the time evolution as a unitary evolution of the quantum states. Upon multiplying by q | on both side of the equation and using the orthonormality of the position eigenfunction, we obtain the Schrödinger equation in the coordinate representation, i|

∂ ψ (q, t) = Hψ (q, t) ∂t |2 2 ∇ + V (q) ψ (q, t) , = − 2m q

(1.15)

where V (q) is the external potential seen by the particle. The above equation can readily be the be separated into temporal and spatial parts using separation of variables, by writing ψ (q, t) = ψ (q) Φ (t) , which yields −

|2 2 ∇ + V (q) ψ (q) = Eψ (q) 2m q

(1.16)

or Hψ (q) = Eψ (q) , where i

Φ (t) = e− | Et . Similarly, in the momentum representation the Schrödinger equation is given by i|

∂ ψ (p, t) = Hψ (p, t) ∂t p2 + V (i|∇p ) ψ (p, t) . = 2m

Since in general V (q) is a complicated function of q, the momentum representation of the Schrödinger equation is rarely employed and solved directly. However, it should be pointed out that in energy-band dynamics of Bloch electrons in solid, where the energy bands are nondegenerate, each energy band could be a complicated

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function of the crystal momentum, Eλ (p), so that the eﬀective Hamiltonian for each band λ, to be used in Eq. (1.16), becomes7 Heff = Eλ (−i|∇q ) + Vλ (q) .

(1.17)

From the Schrödinger equation, Eq. (1.15), the expression for the expectation (average) value of energy or classical energy value is given by H= E =

dq ψ † (q, t) −

|2 2 ∇ + V (q) ψ (q, t) , 2m q

(1.18)

where the time-dependence drops out for a conservative system. In the above expression, the order of ψ† (q, t) relative to ψ (q, t) is fixed, and no ambiguity should arise since these are classical quantities for now. What we have done so far is to treat the quantum statistical aspects of the momentum and position measurements. The Schrödinger complex wavefunction ψ (q, t) is called the probability amplitude since it is the coeﬃcient in the expansion of a quantum state in terms of the position-eigenfunction basis states. The dynamical aspect is given by the time-dependent Schrödinger equation, which should be viewed as derivable from the Least Action Principle. Indeed, i|ψ† (q) and ψ (q) acquire the status of a canonically conjugate variables. In the so-called second quantization scheme to treat many-particle systems, ψ (q) and ψ† (q) become annihilation and creation operators, respectively, at point q. 1.1.3

Isotopic Wavefunction and Many-Body Wavefunction

‘Isotopic’ or multi-component wavefunction is often associated with a particle or system under consideration which can have more than one internal degrees of freedom. This component wavefunction has the form, ψα (x), α = 1, 2, ...n, where α is an internal parameter, such as spin for electron, for energy level of two-level atoms, or for a system and an environment which are accessible to a quasiparticle. The total wavefunction is written as,   ψ1 (x) Ψ (x) =  ....  . ψn (x)

On the other hand, state-tensor or direct-product wavefunction is often associated with many-body problems, such as a combined system of spin- 12 electron and electromagnetic-field photons or lattice vibrational-mode phonons, where the wavefunction for each part must be specified simultaneously. 7 For narrow-gap semiconductors and semimetals, the eﬀective Hamiltonian is a matrix in the energy-band indices. When the quantum dynamics is limited to the energy-band edge, as in most cases using the k · p method, a two-band Hamiltonian matrix is often similar to the Hamiltonian of the relativistic Dirac electrons, such as the k · p Hamiltonian for bismuth-antimony alloys. A ‘Foldy-Woutheysen type’ of energy-band decoupling procedure would also endow each decoupled energy-band dynamics with eﬀective g-factor for the spin degree of freedom.

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1.1.3.1 Decoupling of Isotopic Degrees of Freedom The reduction of a simple8 isotopic problem, described by a multi-component wavefunction, to a single-component Schrödinger wave equation involves a modification of the diagonal Hamiltonian by the addition of an embedding potential or ‘selfenergy’ terms. For example, for time-independent Schrödinger equation with twocomponent wavefunction, HSS HSE HES HEE

ψS ψE

=E

ψS ψE

or E − HSS −HSE −HES E − HEE

ψS ψE

= 0,

can be written in terms of Schrödinger equation for ψS alone HSS + HSE (E − HEE )−1 HES ψS = EψS . Thus, the elimination of ψE results in the inclusion of an extra energy-dependent term, which we denote by Σ (E) , Σ (E) = HSE (E − HEE )−1 HES , and the self-consistent ψ S is an eigenfunction of the eﬀective Hamiltonian, Hef f = HSS + HSE (E − HEE )−1 HES . For time-dependent problem, we have i|∂ ∂t

ψS ψE

=

HSS HSE HES HEE

ψS ψE

or i|∂ ψ = HSS ψS + HSE ψE , ∂t S

(1.19)

i|∂ − HEE ψE = HES ψS . ∂t

(1.20)

We can formally solve for ψE of Eq. (1.20) using the retarded Green function, G (t − t ), which satisfies i|∂ − HEE G (t − t ) = IˆEE δ (t − t ) . ∂t With HES ψS as a source function in Eq. (1.20), we readily obtain t

ψE (t) = −∞ 8 For

G (t − t ) HES ψS (t ) dt .

the Dirac relativistic Hamiltonian, the decoupling scheme is more involved through the so-called Foldy-Woutheysen transformation, which endow spin for the fermion quasiparticles.

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Upon substituting in Eq. (1.19), we have in terms of ψS alone, t

i|∂ ψ = HSS ψS + ∂t S −∞

HSE G (t − t ) HES ψS (t ) dt .

Note that the energy-dependence of the ‘self-energy’ of the time-independent problem has now emerge as a memory kernel of the time-dependent problem. The decoupling schemes in the above simple examples are pretty straightforward, whereas the decoupling scheme for Dirac relativistic electrons and multi-band dynamics in crystalline solids makes use of the Foldy-Woutheysen type of transformations. Moreover, we shall see in Part 4 of this book that the multicomponent superfield theory of real-time nonequilibrium Green’s function technique is much more involved. Indeed, the theory of nonequilibrium quantum physics involves both multicomponent and many-body quantum states. 1.1.3.2 Phenomenological ‘Decoupling’ or Reduction of Many-Body Problems The reduction of direct-product problem, or many-body systems, has acquired great importance in the area of quantum information and computing. The entanglement of qubit subsystem with the environment, and the need to perform quantum operations on the qubit subsystem have focused attention to phenomenological superoperator and generalized quantum operations. In particular, this has lead to generalization of the Von Neumann measurements with the so-called Kraus representation of a generalized quantum operation. These will be further discussed in Part 7 and 8 of this book. 1.2

Generator of Position Eigenstates

From Eq. (1.3), we have P |q = i|∇q |q . Using Fourier series expansion, one can generate coordinate representation of state |q , namely x |q = δ (x − q) by a displacement operator T (q), x |q = δ (x − q) = T (q) δ (x) = exp {−q · ∇x } δ (x) = exp {q · ∇q } δ (x − q )|q →0 = exp

1 q · (i|∇q ) i|

δ (x − q )|q →0 .

Therefore, we have i |q = exp − q · P |

|0 ,

(1.21)

where the operator P = i|∇q is operating on the basis eigenstates |q , refer to Eq. (1.3), since a negative displacement in the components correspond to positive

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displacement of the eigenstate basis. We also note from Eq. (1.11) that i Q, exp − q · P |

∂ i exp − q · P ∂P | i = q exp − q · P . | = i|

(1.22)

Therefore, i Q exp − q · P |

exp

i = exp − q · P |

i q·P |

i Q + q exp − q · P |

i Q exp − q · P |

,

(1.23)

= Q + q.

(1.24)

Multiplying both sides of Eq. (1.23) by the eigenket with component x |q δ (x − q ) and taking the limit q ⇒ 0, we have i Q exp − q · P |

i |0 = exp − q · P |

i Q |0 + q exp − q · P |

=

|0 ,

this reduces to i Q exp − q · P |

|0

i = q exp − q · P |

|0

,

which verifies that the state exp − |i q · P |0 = |q is an eigenstate of the position operator with displaced eigenvalue by q. However, if the limit q ⇒ 0 is not taken then the state exp − |i q · P |q = |q + q is an eigenstate of the position operator with eigenvalue q + q. We can symmetrize the translation operator by inserting exp |i p · Q in front of |0 which eﬀectively insert unity. We thus have i |q = exp − q · P |

exp

i p · Q |0 . |

By the use of the Campbell-Baker-Hausdorﬀ operator identity, namely, exp (A + B) = exp (A) exp (B) exp − provided that [A, [A, B]] = 0 = [B, [A, B]] ,

[A, B] , 2

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a condition that is easily satisfied concerning P and Q since [Q, P ] = i|, a c-number. We have A = − |i q · P , B = |i p · Q, and obtained i exp − q · P |

exp

i p·Q |

i = exp − (q · P − p · Q) exp |

− |i q · P, |i p · Q 2

i p·q i = exp − (q · P − p · Q) exp − | | 2

.

(1.25)

Therefore we have the symmetric form for the displacement operator generating the state |x, q from |x, 0 given by sym

T (q)

= exp −

i p·q | 2

exp

i (p · Q − q · P ) . |

(1.26)

The displacement operator may also be interpreted as an operator for the preparation of the quantum eigenstate out of the ‘vacuum’, |0 9 , basis eigenstate. The symmetric operator Yp,q = exp = exp

i (p · Q − q · P ) | i p·q i exp − q · P | 2 |

exp

i p·Q , |

(1.27)

is often referred to as the generalized Pauli-matrix operator. This will be made clear when we discuss discrete phase space on finite fields. Similarly, we have for the displacement operator of the momentum eigenket |p x |p = C exp

ip.x |

= T (p) C exp

ip .x |

p ⇒0

= exp {p · ∇p } C exp

ip .x |

= exp {p · ∇p } C exp

ip .x |

= exp |p = exp

p ⇒0

p ⇒0

1 p · (−i|∇p ) C exp −i| i p · Q |p = 0 . |

ip .x |

, p ⇒0

(1.28)

9 Here, the concept of a vacuum state does not have a special meaning since |0 represent arbitrary reference position. It is introduced simply to bring analogy with zero-eigenvalue of non-Hermitian operators in later chapters, there the state |0 has a distinguished position.

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We also note from Eq. (1.12) that ∂ i exp p·Q ∂Q | i p·Q . = p exp |

i p·Q |

P, exp

= −i|

(1.29)

Therefore, P exp

i p·Q |

= exp

i p · Q P + p exp |

i exp − p · Q P exp |

i p·Q |

i p·Q , |

= P + q.

(1.30)

(1.31)

Multiplying both sides of Eq. (1.30) by the eigenket with component |p and taking the limit p ⇒ 0, we have P exp

i p · Q P |p = 0 + p exp |

i p · Q |p = 0 = exp |

i p · Q |p = 0 , |

this reduces to P exp

i p · Q |0 |

= p exp

i p · Q |0 |

,

which verifies that the state exp

i p · Q |0 = |p |

(1.32)

is an eigenstate of the position operator with displaced eigenvalue by p. However, if the limit p ⇒ 0 is not taken then the state exp

i p · Q |p |

= |p + p

is an eigenstate of the position operator with eigenvalue p + p. This translation operator, exp |i p · Q , on the |p states is the one we have used in combination with the |q -state displacement operator to form a symmetric operator. Note that the displacement of the eigenvalues of the eigenstates are for positive displacements. By virtue of the covariance of the coeﬃcient of expansion of arbitrary state in terms of the position and momentum eigenkets, the coeﬃcient of expansion has their eigenfunction parameter shifted by negative values, since Q and P are Hermitian operators. This result is useful, for example, in determining the transformation properties of the Wigner distribution function, which as we shall see later, is the coeﬃcient of expansion of the density operator in terms of the phase-space point projectors.

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Unification of Canonical Variables

Universal Canonical Operators Bosons Fermions Mixed Universal q-p Representation: Coherent States

1.3

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P i|φ† (x) i|ψ † (x) i|a†l

Discrete Phase Space on Finite Fields

The formalism discussed above holds naturally for discrete ‘lattice’ points of the eigenvalues q and p, i.e., discrete phase space over a finite field. Examples of these are (i) discrete finite number of lattice atomic position, q, in a crystalline solid, obeying the Born-von Karman boundary condition, and (ii) the simplest finite field consisting of only of two points in either q or p eigenvalues, governed by modular arithmetic. This describes the quantum mechanics of two-level systems and spin 12 particles. This will be the topic of later chapters. 1.4

Non-Hermitian Canonical Variables

In our discussions above, we have focused on the canonical variables, p and q, whose corresponding quantum operators are P and Q, respectively. The meaning we have attached to these canonical variables, so far, are that q and p are the position and momentum variables, respectively, of a single particle. Thus the operators, P and Q, are Hermitian operators since these are observables. However, in dealing with the quantization of classical fields and Schrödinger wave fields (also known as second quantization10 ), we have to deal with non-Hermitian canonical operators, these are the creation and annihilation field operators for bosons and fermions particles. For classical fields, these canonical variables are basically the normal coordinates labeled by index, l. In dealing with non-Hermitian canonical operators, the operators P and Q assume a universal meaning. Table 1.1 shows the translation of the canonical operators P and Q in the language of the boson and fermion field operators, φ (x) , φ† (x) and ψ (x) , ψ† (x) , respectively, and of the mixed q-p representation of the coherent states formulation al , a†l , where x and l label the field components, Then our previous treatment of the Hermitian operators, P and Q, is universally applicable to these non-Hermitian canonical operators as well, the existence of zero eigenvalue, the respective right eigenvectors of Q and P , the respective left eigenvectors of Q and P , and the universal generation of eigenvectors through appropriate translation operation from the vacuum (zero eigenvalue) eigenstate. In what follows, we will retain the symbols q and p for the eigenvalues of the nonHermitian field operators, with the caveat that for fermions q and p are elements of the Grassmann algebra11 , not ordinary c-numbers. 10 Classical fields, such as vibrational fields and electromagnetic fields, naturally exhibit discrete (‘quantum’) oscillating modes, similar to the Schrödinger wave equation. Hence the name second quantization. 11 Whereas the commutation relation of boson variables readily follows from that of harmonic

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Left and Right Eigenvectors of Non-Hermitian Operators

We have the right eigenvectors for bosons φ (x) |q (x) = q (x) |q (x) ,

i|φ† (x) |p (x) = p (x) |p (x) , and for fermions ψ (x) |q (x) = q (x) |q (x) ,

†

i|ψ (x) |p (x) = p (x) |p (x) . Similarly, we have the left eigenvectors for bosons defined as q (x)| φ (x) = q (x)| q (x) , p (x)| i|φ† (x) = p (x)| p (x) , and for fermions q (x)| ψ (x) = q (x)| q (x) , p (x)| i|ψ † (x) = p (x)| p (x) . Then, considering Eq. (1.21) as a universal relation, we have i |q = exp − q · P |0 | i = exp − q · i|φ† |0 | = exp q · φ† |0 for bosons, and for fermions this becomes i |q = exp − q · P |0 | i = exp − q · i|ψ† |0 | = exp q · ψ† |0 . oscillator problem; the anticommutation property of fermion variables is derived from the Slater determinant formalism of many-body quantum theory of half-integral spin particles. However, we shall see that both statistical properties originate from the geometric phase concept by viewing bosons and fermions as quasiparticles in a vacuum condensate structure, similar to the BornOppenheimer approximation in condensed matter.

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Equation (1.32) becomes for bosons i p · Q |p = 0 | i p · φ |p = 0 , |

|p = exp = exp and for fermions |p = exp = exp

i p · Q |p = 0 | i p · ψ |p = 0 . |

The dot products is understood to be an Einstein summation convention, e.g., q ·φ† = qx φ† (x). It should be pointed out that for Hermitian operators the zerox

eigenvalue state does not have a distinguished position among all the eigenstates, it is just an arbitrary position. However, for non-Hermitian operators, the zero eigenvalue state have a distinguished position and is often referred to as the vacuum state. We note that the left and right eigenvectors form a complete biorthogonal basis states, Q |q = q |q , q | Q |q = q q | |q , q q | |q = q q | |q . Therefore q q | |q − q q | |q = 0, (q − q) q | |q = 0, hence, if q = q, then we conclude that q | |q = 0. Similarly, p | |p = 0 if p = p. These are also complete bases,

|q q| dq = I and

|p p| dq = I. The

transformation function, Eq. (1.2), is now translated into the dot product of complex eigenvalues q and p, by virtue of the non-Hermitian character of the field operators, so that |p = C

q

i exp − p · q |

|q ,

where C is a normalization constant and the summation sign may stand for the functional integral over the space of complex functions, q. We will not discuss here the case for fermions since the Grassmann algebra for the eigenvalues, q and p, complicates the notations12 . 12 A good discussion of Grassmann algebra for fermions is given by J. Schwinger in his book, “Quantum Kinematics and Dynamics” (Addison-Wesley, 1991). See also the construction of path integrals for fermion many-body problems given by F. A. Buot, Phys. Rev. A 33, 2544 (1986).

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Quantum Mechanics: Perspectives

1.5

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Coherent State Formulation as a Mixed q-p Representation

We will focus on bosons, in particular the coherent formulation of harmonic oscillators (to be discussed in more details in later chapters), which as we shall see is really a mixed q-p formulation in the universal/generalized sense. The canonical non-Hermitian operators are the creation and annihilation operators, a ˆ and a ˆ† , respectively. Coherent state formulation is based on the right eigenstate of a ˆ, ˆ† = α| α† . Thus, α and a ˆ |α = α |α coupled with the left eigenvector of a ˆ† , α| a α† correspond to the eigenvalues q ⇒ α and p ⇒ i|α† respectively. Therefore, the transformation function corresponding to Eq. (1.2) becomes i α| α = exp − p · q | i = exp − i|α† · α | = exp |α|2 ,

(1.33)

2 which is sometimes written in the formulation as α| α = co exp |α| where co is chosen depending on the common phase transformation of the q states relative to the p states, usually co = (2π|)−d , here d is the dimension or number of canonical variables, q or p . The generation of the eigenstates from the vacuum follows from the prescription of the generalized Pauli operator of Eq. (1.27)

|α = exp = exp = exp

i p·q i i p · Q |0 exp − q · P exp | 2 | | i p·q i exp − q · P |0 | 2 | i i|α† · α i a† |0 exp − α · i|ˆ | 2 |

= exp −

|α|2 2

exp αˆ a† |0 .

The completeness relation in the mixed q-p representation, by virtue of the coherent state formulation, is given by (|q q| |p p|) dq ∧ dp =

∗

|q ( p| |q )

p| dq ∧ dp

i 1 (|q p|) e | qp dq ∧ dp 2π| ∗ 1 i (|α α|) e | αi|α dα ∧ d (i|α∗ ) ⇒ 2π| 2 1 (|α α|) e−|α| id (αr + iαi ) ∧ d (αr − iαi ) = 2π 2 1 (|α α|) e−|α| dαr dαi = π = 1, (1.34)

=

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where the operation denoted by ‘∧’ is the outer product (cross product) of the complex quantities, dq and dp so as to yield an incremental area in a complex plane13 . The last line is of Eq. (1.34) is often written as 1 π

(|α α|) d2 α = 1, |α|2

which follows by normalizing the coherent states, |α ⇒ e− 2 |α [refer to Eq. (1.33)]. Based on this coherent-state normalization, the transformation function, β| α , becomes a non-orthogonality relation for coherent states, β| α = e−

|β|2 2

e−

|α|2 2

i exp − i|β † · α |

2

= e−|α−β| ,

(1.35)

which becomes approximately orthogonal as |α − β|2 increases. It should be emphasized that through the use of universal viewpoint for nonHermitian canonical variables, we have obtained above the important relations for, and true nature of, coherent states without the aid of occupation-number states of a harmonic oscillator, |n , as is usually done in almost all discussions giving the introduction of coherent states. In fact, as we shall see in this book, the universal nature of the canonical operators, Q and P , and their respective eigenvalues, q and p, permeates all formal aspects of quantum dynamics, either in the continuum phase space or discrete phase space on finite fields. Examples of the later are (a) finite discrete lattice points in crystalline solid obeying the Born-von Karman boundary condition, and (b) two discrete points obeying modular arithmetic for two-level quantum systems, such as a two-level atom or spin 12 particle. These will be further discussed in later chapters.

13 The ‘volume’ (area for simple harmonic oscillator) element in phase space between the canonically transformed phase spaces are related by

J (dq × dp) = dQ × dP Therefore for canonical transformation, J = 1, and the so-called ‘wedge form ’ (diﬀerential geometry terminology) dq × dp = dQ × dP is also invariant. In two variables this means the elementary ‘area’ in phase space is invariant (this also reflects the invariant way of counting of states).

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Chapter 2

Quantum Mechanics of Classical Fields

2.1

Quantization of Harmonic Oscillator

The classical harmonic oscillator has a very important historical role in the development of quantum mechanics of many-body systems. Indeed, the harmonic oscillator quantum physics is a forerunner of the second quantization scheme of many-particle systems, either fermions or bosons. The reason why the harmonic oscillator model is central to quantization of classical fields is that the Hamiltonian of classical vibrational fields and electromagnetic fields can be transformed into sums of collective harmonic oscillator modes. Note that in the coordinate transformation to normal coordinates in coupled harmonic oscillator systems (vibrational fields), the normal coordinates are not the original displacements of particles from the equilibrium positions, but correspond to collective distortion-vibrational modes, with the appropriate dispersion relation which relates the angular frequency ω to the wave number simulating traveling energy packets or quantized ‘particles’ known as phonons. In what follows, we will devote detailed discussion on the harmonic oscillator model.

2.1.1

The Complex Canonical Variables

A classical harmonic oscillator has energy, E, or classical Hamiltonian, H, given by H=E=

p2 1 + κq 2 . 2m 2

The classical Hamilton equation of motion follows from this Hamiltonian as q˙ =

p˙ = −

p ∂H = , ∂p m

∂H = −κq = −mω 2 q, ∂q 23

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where ω = k/m. It is easier to solve the above equation of motion by transforming the variables as follows mω 2|

φ=

q+

ip mω

,

(2.1)

where we use the Planck’s constant | in anticipation of the quantum mechanical treatment that follows (this cancels in the harmonic oscillator energy expression). By scaling the dynamical variables, φ can also be written as φ=

1 (˜ q + i˜ p) , 2

(2.2)

1 where q˜ = mω ˜ = |mω p. Then the classical Hamilton equation of motion | q and p becomes a very simple equation to solve, namely,

φ˙ = −iωφ, yielding the harmonic oscillatory solution φ = φ0 exp {−iωt} .

(2.3)

The harmonic oscillator point in 2-D (p, q) phase space oscillates with frequency ω. The solution for q (t) is given by q (t) =

2| |φ| cos (ωt − arg φ) , mω

√ p (t) = − 2|mω |φ| sin (ωt − arg φ) ,

(2.4)

(2.5)

which describe the motion of classical harmonic oscillator of frequency ω having a well-defined complex amplitude φ in phase space. The harmonic-oscillator orbits in (p q)-phase space is highly ‘squeezed’ along the q-axis. The absolute value of φ is of course a well-defined physical quantity, i.e., proportional to the oscillation amplitude of the oscillator. The expression for the classical Hamiltonian becomes much simpler in the transformed dynamical variables expressed in terms of φ, 1 ∗ (φ |ωφ + φ |ωφ∗ ) 2 1 2 mω2 2 p + q =H = 2m 2 |ω 2 = p˜ + q˜2 . 2

Hφ = φ∗ |ωφ =

(2.6)

We will denote φ as ‘field’ variable since it is a function of points in phase space. It is interesting to note that if we take i|φ∗ as the complex conjugate to φ, then the

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25

Hamilton equation for the transformed Hamiltonian Hφ reads ∂Hφ = −iωφ, φ˙ = i|∂φ∗ ∗

i|φ˙ = −

∂Hφ = −φ∗ |ω, ∂φ

∗ φ˙ = φ∗ iω,

which are the same as before, since the second equation is the complex conjugate to the first equation. The complex conjugate solution is φ∗ = φ∗0 exp {iωt} . 2.1.2

Classical Schrödinger-Like Equation for Harmonic Oscillator

It is quite remarkable that the form of the classical equation of motion of the cnumber φ variable has the form of the quantum Schrödinger equation for a single harmonic oscillator excitation (boson) with energy or Hamiltonian H = |ω, i.e., i|φ˙ = |ωφ.

(2.7)

In fact just like the complex wavefunction of the Schrödinger equation only the absolute value of φ is a measurable physical quantity. The element of self-consistency also comes in since φ is a function of |ω. We will refer to Eq. (2.7) as the ‘classical Schrödinger equation’. However, possible states of classical harmonic oscillator described by the complex amplitudes, φ, which describes orbits of rotating vector in phase space, is continuous. Note that, classically, the magnitude of φ is a measurable quantity and a classical dynamical variable. Hence, this quantity is naturally expected to play a major role in the quantization of the harmonic oscillator as we shall see. 2.1.3

Second-quantization of the Schrödinger-Like Equation

Pedagogically, we will see later that the quantum treatment of the classical harmonic oscillator to account for diﬀerent possible quantized-excitation states (different quantized absolute value, |φ|, of the amplitude states) involves procedure identical to the second quantization of the corresponding Schrödinger equation for a single particle, such as Eqs. (1.15) and (1.18), in order to treat many-particle problems. The scheme is basically based on treating φ and φ∗ as annihilation and creation operators, respectively. Indeed, the situation is practically identical when treating boson excitations in many-body problems.

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Chapter 3

The Linear Chain of Atoms Coupled by Harmonic Forces

A simple example of coupled harmonic oscillators is a linear chain of atoms coupled by harmonic forces with spring modulus κ and chain length L. The kinetic energy is given by L

1 T = 2

ρdx (q˙ (x, t))2 ,

(3.1)

0

where ρ is the mass density (i.e., mass per unit length). The potential energy is given by L

1 V = g 2

dx

∂q (x, t) ∂x

2

,

(3.2)

0

where g = κa is a constant product, with a being the distance between atoms taken in the limit a =⇒ 0. The Lagrangian, T − V , follows from these equations L

1 L= 2

L

1 ρdx (q˙ (x, t)) − g 2 2

0

dx

∂q (x, t) ∂x

2

.

(3.3)

0

Upon the application of the Euler-Lagrange equation, namely, δL d δL − = 0, dt δ q˙ (x) δq (x) we obtain the equation of motion, which is a wave equation along the chain, ρ¨ q (x, t) − g

∂ 2 q (x, t) = 0. ∂x2

(3.4)

Moreover, the Lagrangian enables us to define the canonically conjugate momentum, π (x), as π (x) =

δL = ρq˙ (x) . δ q˙ (x) 26

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The quantum mechanical treatment then allows us to treat q (x) and π (x) as operator obeying the equal-time commutation relations q (x ) π (x) − π (x) q (x ) | δ | δ − q (x) = i|δ (x − x ) , = q (x) i δq (x) i δq (x)

3.1

(3.5)

q (x ) q (x) − q (x) q (x ) = 0,

(3.6)

π (x ) π (x) − π (x) π (x ) = 0.

(3.7)

Complex Dynamical Variables

The general solution to Eq. (3.4) which obey the cyclic boundary condition, q (x + L, t) = q (x, t) with q (x, t) a real classical quantity is given by q (x, t) = k

1 1 Ak √ ei(kx−wk t) + A†k √ e−i(kx−wk t) , L L

(3.8)

1 1 (−iω k ) Ak √ ei(kx−wk t) + (iω k ) A†k √ e−i(kx−wk t) , L L k (3.9) g g where the mode frequency, ωk = ρ k = vk k (vk = ρ is the speed of the mode π (x) = ρq˙ (x) = ρ

propagation along the chain), corresponds to ω = oscillator. 3.1.1

κ m

of the simple harmonic

Creation and Annihilation Operator for a Coupled Linear Chain of Atoms

Our goal is to transform the corresponding classical Hamiltonian to the form of Eq. (2.6), i.e., in terms of complex variables which can then be identified as the creation and annihilation operators, akin to the second quantization procedure, in the quantum treatment of the linear chain of coupled atoms. In analogy with Eqs. (2.4) and (2.5), let us write Ak e−iwk t =

A†k eiwk t =

| ak (t) , 2ρωk

| † a (t) . 2ρωk k

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Thus, we have 1 √ L

q (x, t) = k

i π (x, t) = √ L

| ak (t) eikx + a†k (t) e−ikx 2ρω k

|ωk ρ −ak (t) eikx + a†k (t) e−ikx 2

k

,

(3.10)

.

(3.11)

Note that corresponding to Eq. (2.2), we may also redefine the factor for each mode k to yield the scaled q˜ (x, t) and π ˜ (x, t) as 1 q˜ (x, t) = √ L

i π ˜ (x, t) = √ L

ρω k |

k

1 |ωk ρ

k

| ak (t) eikx + a†k (t) e−ikx 2ρωk

|ωk ρ −ak (t) eikx + a†k (t) e−ikx 2

,

,

which allow us to define the complex variables φ (x, t) =

1 [˜ q (x, t) + i˜ π (x, t)] , 2

which corresponds to Eq. (2.2), where 1 φ (x, t) = √ L 1 φ† (x, t) = √ L

ak (t) eikx , k

a†k (t) e−ik x . k

The Hamiltonian is given by H=

π (x, t) q˙ (x, t) dx − L L

L

g π (x) dx + 2

1 = 2ρ

∂q (x) ∂x

2

0

2

dx.

0

Upon substituting the expressions in Eqs. (3.10) and (3.11) using the orthogonality relations, L

1 L

eikx−ik x dx = δ kk , 0

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29

we arrived at the classical expression for the Hamiltonian in terms of the complex variables ak (t) and a†k (t), H=

1 2

|ω k a†k ak + ak a†k .

(3.12)

k

Note that Eq. (3.12) is still a classical expression for the Hamiltonian, similar to Eq. (2.6). We now apply the commutation relations for π (x) and q (x), Eqs. (3.5) - (3.7), now considered as quantum operators to yield the commutation relation for the corresponding quantum operators ak and a†k , which are the annihilation and creation operators, respectively. We then obtained the following relations, a ˆk , a ˆ†k

= δ kk ,

[ˆ ak , a ˆk ] = 0, a ˆ†k , a ˆ†k

= 0.

Using the above commutation relations, we can now rewrite the quantized Hamiltonian as H=

|ωk a†k ak + k

1 2

,

(3.13)

which is a sum of various modes of vibration labeled by the index k, including the vacuum-fluctuation energy. In terms of the dynamical variables φ (x, t) and φ† (x, t), we have the commutation relations ˆ (x, t) , φ ˆ † (x , t) = δ (x − x ) , φ ˆ (x, t) , φ ˆ (x , t) = 0, φ ˆ † (x, t) , φ ˆ † (x , t) = 0, φ and the corresponding Hamiltonian expression is H=

1 2

ˆ † (x, t) |ω (−i∇x ) φ ˆ (x, t) + φ ˆ (x, t) |ω (i∇x ) φ ˆ † (x, t) , dx φ

which is the Fourier transform of Eq. (3.13), where ωk =

g ρk

(3.14)

and ω (i∇x ) =

ˆ † (x, t) is the creation operator for localized phonon The operator φ wavepacket at position x in the linear chain. g ρ i∇x .

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Chapter 4

Lattice Vibrations in Crystalline Solids: Phonons

In what follows we will make heavy use of the Hamilton equations, namely, p˙ (t) = −

q˙ (t) =

4.1

∂H (p (t) , q (t) , t) , ∂q (t)

∂H (p (t) , q (t) , t) . ∂p (t)

Elementary Lattice Dynamics: The Linear Chain

Many of the features of the lattice dynamics of a crystal can be brought out by considering a simpler system, namely a linear chain of point masses. We take the masses m to be identical with a lattice separation a, and represent the interaction between nearest neighbors by a spring of force constant f . Various boundary conditions are clearly possible, but we shall consider only the cyclic boundary condition where one end of the chain is joined to the other so that the N th mass is also the 0th . The Hamiltonian of the system is then H = T +V N

= l=1

2 1 [p (l)] + g [u (l + 1) − u (l)]2 2m 2

,

where the displacement u (l) is longitudinal, i.e. along the chain. The equilibrium position of the lth mass is Rl = la and its equation of motion is m

∂ 2 u (l) = g {[u (l + 1) − u (l)] + g [u (l − 1) − u (l)]} . ∂t2 30

(4.1)

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We will use the cyclic or periodic boundary condition for a chain of N masses such that u (l) = u (l + N ). The real solution for u (l) is |B (q)| u (l) = √ cos (qRl − ω (q) t + α (q)) , Nm

(4.2)

with amplitude, |B (q)|, and phase, α (q), determined by the initial condition. Substitution in the equation of motion, Eq. (4.1) gives a ‘tight-binding’ expression for ω 2 (q) , ω 2 (q) =

2g (1 − cos qa) . m

The solution can be found somewhat more formally as follows. We make a transformation of coordinates by writing 1 u (l) = √ N

q

Q (q) exp {iqRl } .

(4.3)

For a linear chain, Q (q) is referred to as the normal coordinates1 , this will become clear below. Since u (l) is real2 , Q (−q) = Q∗ (q) . Comparison of Eq. (4.3) with Eq. (4.2), we see that 1 B (q) e−iω(q)t + B ∗ (−q) eiω(q)t , Q (q) = √ 2 m

(4.4)

where B (q) = |B (q)| eiα(q) , |B (−q)| = |B (q)| , α (−q) = α (q) , which indeed gives 1 Q (−q) = √ B (−q) e−iω(−q)t + B ∗ (q) eiω(−q)t 2 m 1 B (−q) e−iω(q)t + B ∗ (q) eiω(q)t = √ 2 m = Q∗ (q) . 1 These are the coordinates of the independent modes of vibrations, labeled by the wavevector q, describing the coupled linear chain. 2 Otherwise we could have written

1 √ 2 Nm

q

Q (q) exp {iqR (l)} + Q∗ (q) exp {−iqR (l)}

1 = √ 2 Nm

q

Q (q) exp {iqR (l)} + Q (−q) exp {−iqR (l)}

u (l) =

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Thus, Eq. (4.3) can be written as 1 u (l) = √ 2 N

q

1 = √ 4 Nm

[Q (q) exp {iqRl } + Q (−q) exp {−iqRl }] , 

 B (q) exp {i (qRl − ω (q) t)}  +B ∗ (−q) exp {i (qRl + ω (q) t)}     +B (−q) exp {−i (qRl + ω (q) t)}  . +B ∗ (q) exp {−i (qRl − ω (q) t)}

q

(4.5)

Here we see the meaning of the amplitude, B (q), namely, B (q) and B ∗ (q) specify a real wave traveling in positive direction, whereas, B ∗ (−q) and B (−q) specify a real wave traveling in the negative direction. In terms of the normal coordinates, the expression for the kinetic energy becomes T =

1 N

1 ˙ Q (q) Q˙ (q ) exp {i (q + q ) Rl } . 2m

q,q

l

We now make use of the result that when N is large

l

exp {iqRl } =

l

exp {iqla}

= N ∆ (q) , where 2πn 2π , .... , etc. a a otherwise.

∆ (q) = 1

for q = 0,

=0 Thus, T =

1 2m

2

Q˙ (q) . q

Similarly one finds that the potential energy is given by V =

1 2

2

q

2

m [ω (q)] |Q (q)| ,

so that in terms of the new normal coordinates,   2   ˙ (q)   Q 1 + m [ω (q)]2 |Q (q)|2 . H=  2 q   m 

From the Lagrangian, L = T − V , the momentum conjugate to Q(q) is mQ˙ ∗ (q) since ∂L = Q˙ (q) ∂mQ˙ ∗ (q)

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so that we can also write H=

1 2m

q

|P (q)|2 + m2 [ω (q)]2 |Q (q)|2 ,

(4.6)

which is of the form of the sum of harmonic oscillators labeled by index q. From the form of this expression, we recognize the Q (q) as normal coordinates of the linear chain, each of which describes an independent mode of vibration of the system labeled by index q. From the Hamilton’s equations we may write the equation of motion of the coordinate Q (q) as ¨ (q) = − Q

∂L = −m [ω (q)]2 |Q (q)| ∂Q∗ (q)

or ¨ (q) + m [ω (q)]2 Q (q) = 0. Q The solution can be expanded in terms of B (q), Eq. (4.4), can be written 1 Q (q) = √ {B (q) exp (−iω (q) t) + B ∗ (−q) exp (iω (q) t)} , 2 m

(4.7)

hence  2 2  |B (q)| + |B ∗ (−q)| 2 2 2 m [ω (q)] |Q (q)| = [ω (q)] +B (q) B (−q) exp (−2iω (q) t) .  4 +B ∗ (−q) B ∗ (q) exp (2iω (q) t)  1

We also have

P (q) = mQ˙ ∗ (q) 1 = − √ imω (q) {B (−q) exp (−iω (q) t) − B ∗ (q) exp (iω (q) t)} 2 m 1 = √ imω (q) {B ∗ (q) exp (iω (q) t) − B (−q) exp (−iω (q) t)} , 2 m hence 2

|P (q)| = [ω (q)]2 Q (q) Q∗ (q) m   2 2  + |B (−q)|  |B (q)| 1 = [ω (q)]2 −B ∗ (q) B ∗ (−q) exp (2iω (q) t) .   4 −B (−q) B (q) exp (−2iω (q) t)

From Eq. (4.5), note that B (q) and B (−q) are independent, whereas Q (q) and Q (−q) are not. There, it was shown that B (q) and B ∗ (q) describe a real wave traveling in positive q direction and B ∗ (−q) and B (−q) specify a real wave traveling in the negative direction.

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From Eq. (4.6), the total energy of the system is H=

1 2

q

[ω (q)]2 |B (q)|2 .

Note that by writing

B (q) exp (−iω (q) t) =

∗

B (q) exp (iω (q) t) =

1 2

2| ω (q)

2| ω (q)

φ (q, t) ,

1 2

φ† (q, t) ,

then we have φ† (q, t) |ω (q) φ (q, t)

H= q

= q

1 φ† (q, t) |ω (q) φ (q, t) + φ (q, t) |ω (q) φ∗ (q, t) , 2

(4.8)

where φ (q, t) = φo (q) exp (−iω (q) t) . Thus, we have the Schrödinger-like equation for each mode q given by i|φ˙ (q, t) = |ω (q) φ (q, t) , with Hamilton equation for φ˙ (q, t) given by φ˙ (q, t) =

∂H . i|∂φ∗ (q, t)

Therefore, i|φ∗ (q, t) is now the canonical-conjugate complex-dynamical variable to φ (q, t). Each mode q in Eq. (4.8) clearly labels each independent harmonic oscillator, using complex canonical fields, within the summation. For a system obeying the laws of classical physics and in thermal equilibrium with its surroundings φ† (q, t) |ω (q) φ (q, t) = kB T, where kB is the Boltzmann constant and T is the temperature.

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4.1.1

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Quantization of the Vibrational Mode: Phonons

From Eq. (4.7), we define annihilation and creation operators, a(q) and a† (q), respectively, by means of the relations, 1 Q (q) = 2

| 2mω (q)

=

1 2

2| mω (q) 1 2

φ (q, t) + φ† (−q, t) φ (q, t) + φ† (−q, t) ,

where we make the following substitutions ( a procedure often referred to as ‘second quantization’) φ (q, t) ⇒ a ˆ(q, t), φ† (−q, t) ⇒ a ˆ† (−q, t). Then we can write the normal coordinates as Q (q) =

| 2mω (q)

P (q) = i

|ω (q) m 2

1 2

a ˆ(q, t) + a ˆ† (−q, t) , 1 2

a ˆ† (q, t) − a ˆ(−q, t) .

From the commutation relation of the normal coordinates, [Q (q) , P (q )] = i|δ (q − q ) , we have 1

1

2 |ω (q) m 2 | ˆ† (q, t) − a ˆ(−q, t) a ˆ(q, t) + a ˆ† (−q, t) a i † − a ˆ (q, t) − a ˆ(−q, t) a ˆ(q, t) + a ˆ† (−q, t) 2mω (q) 2 i| ˆ(q, t)ˆ a(−q , t) + a ˆ† (−q, t)ˆ a† (q , t) − a ˆ† (−q, t)ˆ a(−q , t) a ˆ(q, t)ˆ a† (q , t) − a = † † † ˆ (q, t)ˆ a(q , t) + a ˆ (q, t)ˆ a (−q , t) − a ˆ(−q, t)ˆ a(q , t) − a ˆ(−q, t)ˆ a† (−q , t) 2 − a

i| ˆ† (−q, t)ˆ a(−q , t) a ˆ(q, t)ˆ a† (q , t) − a † ˆ (q, t)ˆ a(q , t) − a ˆ(−q, t)ˆ a† (−q , t) 2 − a i| = ˆ(−q, t), a ˆ† (−q , t) a ˆ(q, t), a ˆ† (q , t) + a 2

=

= i|δ (q − q ) .

Therefore, we must have the equal-time commutation relation, 1 2

ˆ(−q, t), a ˆ† (−q , t) a ˆ(q, t), a ˆ† (q , t) + a

which yields a ˆ(q, t), a ˆ† (q , t) = δ (q − q ) .

= δ (q − q ) ,

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Nonequilibrium Quantum Transport Physics in Nanosystems

We also have ˆ† (q, t), a ˆ† (q , t) = 0. [ˆ a(q, t), a ˆ(q , t)] = a Because of the commutation relation given above, the classical Hamiltonian expression in Eq. (4.8) becomes H=

q

= q

1 |ω (q) a ˆ† (q, t) a ˆ (q, t) + a ˆ (q, t) a ˆ† (q, t) 2 1 |ω (q) 2ˆ a† (q, t) a ˆ (q, t) + 1 2 |ω (q) a ˆ† (q, t) a ˆ (q, t) +

= q

1 2

.

ˆ (q, t) = n ˆ (q) is the number Later on we shall see that the operator product a ˆ† (q, t) a operator for quantized vibrations known as phonons. Thus we may write H= where

4.2

|ω(q) 2

|ω (q) n ˆ (q) + q

1 2

,

is the ‘vacuum’ fluctuation energy due to uncertainty principle.

Lattice Vibrations in Three Dimensions

There is no formal diﬃculty in extending the results of foregoing sections so that they apply to a periodic array of point masses in three dimensions. At the same time we remove the restriction that there is only one mass per unit cell, and identify the diﬀerent masses (atoms) by an index κ = 1, 2...r. The position of the κth atom in the lth cell is Xl,κ = Xl + xκ . A basic assumption which we shall make meantime is that the change in potential energy when the atoms are given small displacements u(l, κ) is ∆Φ(2) =

1 2

uα (l, κ)Φll ,κκ ,αβ uβ (l , κ ), ll ,κκ ,αβ

where α, β, denote components in a Cartesian coordinate system. We have Φll ,κκ ,αβ =

∂2Φ ∂uα (l, κ)∂uβ (l , κ )

which is the analogue of the spring constant. The quantity Φll ,κκ ,αβ uβ (l , κ ) is the vector force in the α-direction on the atom at (l, κ) when the atom at (l , κ ) is

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37

displaced in the β-direction an amount uβ (l , κ ). Since there can be no net force on any atom as a result of a uniform (unit) translation of the crystal, it follows that Φll ,κκ ,αβ = 0, l ,κ

Φll,κκ,αβ = −

Φll ,κκ ,αβ .

The Hamiltonian of the system is    1 mκ u˙ 2α (l, κ) + uα (l, κ)Φll ,κκ ,αβ uβ (l , κ ) H=  2 l,κ,α ll ,κκ ,αβ    p2α (l, κ) 1 + uα (l, κ)Φll ,κκ ,αβ uβ (l , κ ) . =  2 mκ l,κ,α

(4.9)

l =l,κ =k

(4.10)

ll ,κκ ,αβ

The equation of motion of an individual atom is, ¨α (l, κ) = − mκ u

4.3

Φll ,κκ ,αβ uβ (l , κ ).

(4.11)

l ,κ ,β

Normal Coordinates in Three Dimensions

We make the Fourier series transformations,3 1 uα (l, κ) = √ NV 1 pα (l, κ) = √ NV

q

q

Uα (q, κ) exp {−i (q · Xl,κ )} ,

(4.12)

Pα (q, κ) exp {i (q · Xl,κ )} .

(4.13)

The above expansion is similar to the expansion of the ‘Wannier function’ labeled by lattice coordinates, uα (l, κ), in terms of the ‘Bloch function’ labeled by reciprocallattice coordinates , Uα (q, κ), with exp {−i (q · Xl,κ )} playing the role of the transformation function. Since uα (l, κ) is real then we must have Uα (−q, κ) = [Uα (q, κ)]∗

(4.14)

∗ Pα (−q, κ) = [Pα (q, κ)] .

(4.15)

and similarly,

3 There

are two types of normalization that one can use in the expansion: (1) the box normalization scheme in which N is the number of lattice point or number of unit cell and V is the volume of each unit cell, NV is the total normalization-box volume in √ 1 as used here, (2) discrete NV

1 in Fourier transform normalization where √ 1 is simply replaced by √1 which goes to (2π| )3 NV N the large lattice or continuum limit. It does not matter which normalization is used as long as consistent treatment is followed.

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Upon substitution in the Hamiltonian, Eq. (4.10), we have    1 p2α (l, κ) H= + uα (l, κ)Φll ,κκ ,αβ uβ (l , κ )  2 mκ l,κ,α ll ,κκ ,αβ     Pα (q,κ) exp{i(q·Xl,κ )} Pα (q ,κ) exp{i(q ·Xl,κ )}       q   q     m   κ     l,κ,α 1 1 = + Uα (q, κ) exp {−i (q · Xl,κ )} Φll ,κκ ,αβ uβ (l , κ )  , 2 NV      q   ll ,κκ ,αβ         × Uα (q , κ) exp {−i (q · Xl,κ )}     q

   Pα (q,κ)Pα (q ,κ) exp{i((q+q )·Xl,κ )}          qq     mκ   1 1 l,κ,α , H= 2 NV    + Uα (q, κ)Φll ,κκ ,αβ Uβ (q , κ )          ll ,κκ ,αβ qq     × exp {−i (q · Xl,κ + q · Xl ,κ )}

           

Pα (q,κ)Pα (−q,κ) mκ κ,α,q

+ Uα (q, κ) 1 κκ qq   H= 2N V       Φ exp {−i ([q + q ] · X + q · [X − X ])} ×  ll ,κκ ,αβ l,κ l ,κ l,κ       ll ,αβ   ×Uβ (q , κ )

                      

We note that by virtue of translational invariance Φll ,κκ ,αβ depend on the relative distance, [Xl ,κ − Xl,κ ], i.e., Φll ,κκ ,αβ = Φκκ ,αβ (Xl ,κ − Xl,κ ) .

We can now write the summation over l as summation over h = Xl ,κ − Xl,κ , thus lh,αβ

Φκκ ,αβ (h) exp {−i ([q + q ] · Xl,κ + q · h)} = l,αβ

=

exp {−i ([q + q ] · Xl,κ )} NV δ (q + q )

αβ

h

h

Φκκ ,αβ (h) exp {−iq · h}

Φκκ ,αβ (h) exp {−iq · h} .

.

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Thus the Hamiltonian can be written as

H=

          

          

Pα (q,κ)Pα (−q,κ) mκ κ,α,q

+ Uα (q, κ) 1 αβ κκ qq 2NV     Φκκ ,αβ (h) exp {−iq · h} × NV δ (q + q )     h   ×Uβ (q , κ )

H=

          1

Pα (q,κ)Pα (−q,κ) mκ κ,α,q

+

Uα (q, κ) αβ

2    ×      

h

κκ q

Φκκ ,αβ (h) exp {iq · h} ×Uβ (−q, κ )

Using Eqs. (4.14)-(4.15, we end up with the expression,

H=

1 2

q

where

   

                    

Pα (q,κ)Pα∗ (q,κ) mκ κ,α

  +

κκ

αβ

h

Dκκ ,αβ (−q) =

,

.

   

Uα (q, κ)Dκκ ,αβ (q) Uβ∗ (q, κ )   

Dκκ ,αβ (q) =

         

,

Φκκ ,αβ (h) exp {iq · h} ,

Φκκ ,αβ h Dκ∗ κ,βα (q)

(−h) exp {iq · h}

= = Dκκ ,αβ (q) ,

(4.16)

showing that the matrix Dκκ ,αβ (q) is Hermitian. Thus the total Hamiltonian is a sum of Hq , where

Hq =

   1

2  +

Pα (q,κ,t)Pα∗ (q,κ,t) mκ κ,α

αβ

κκ

   

Uα (q, κ)Dκκ ,αβ (q) Uβ∗ (q, κ , t)   

.

(4.17)

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The classical equation of motion for each mode is P˙ α (q, κ, t) = −

∂Hq , ∂Uα (q, κ, t)

¨α∗ (q, κ, t) = − mκ U ¨α (−q, κ, t) = − mκ U ¨α (q, κ, t) = − mκ U

Dκκ ,αβ (q) Uβ∗ (q, κ , t), βκ

Dκκ ,αβ (q) Uβ (−q, κ , t), βκ

Dκκ ,αβ (−q) Uβ (q, κ , t).

(4.18)

βκ

If we write the solution as Uα (q, κ, t) = √

1 α e exp (−iω (q) t) , mκ q,κ

then upon substituting in Eq. (4.18) we obtained mκ −√ [ω (q)]2 eα q,κ exp (−iω (q) t) mκ 1 Dκκ ,αβ (−q) √ eβq,κ exp (−iω (q) t) , =− m κ βκ

βκ

Dκκ ,αβ (q) − [ω (q)]2 δ κκ δ αβ eβq,κ = 0. √ mκ mκ

(4.19)

The last equation is a matrix equation of order 3f × 3f , where f is the number of atoms per unit cell in crystal lattice, i.e., κ = 1, 2, 3, .....f with α (or β) = 1, 2, 3(we use α, β, γ in what follows for the three Cartesian indices). We write explicitly the square Hermitian matrix D (q) with matrix elements Dκκ ,αβ (q) as D (q) 

 D11,αα (q) D11,αβ (q) D11,αγ (q) D12,αα (q) D12,αβ (q) .... D1f,αγ (q)  D11,βα (q) D11,ββ (q) D11,ββ (q) D12,βα (q) D12,ββ (q) .... D1f,βγ (q)  . =   ...... ....... ....... ........ ........ ....... ...... Df 1,γα (q) Df 1,γβ (q) Df 1,γγ (q) Df 2,γα (q) Df 2,γβ (q) ..... Df f,γγ (q)

For a consistent solution, the determinant of the matrix within the curly bracket of Eq. (4.19) must vanish, Dκκ ,αβ (q) − [ω (q)]2 δ κκ δ αβ = 0, √ mκ mκ

for which we must then have [ω (q)]2 as the real roots of the equation of order 3f . 2 We label the roots by the index j i.e., [ω j (q)] where runs from 1, 2, .......3f. For 2 each root [ω (q)]j there will be a corresponding polarization components eβ,j q,κ where

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the column eigenvector for the j th root is  α,j  eq,1  β,j   j  e  eq,1  q,1   γ,j   j  e  e   q,1   q,2  ˇ ejq =  α,j  =  .  e   ...   q,2      j  ...  e q,f   eγ,j q,f

The polarization component eβ,j q,κ specifies the motion of the atoms in the mode q j and frequency ω j (q), i.e., ˇ eq is the eigenvector and [ωj (q)]2 is the corresponding real eigenvalue of the Hermitian matrix D (q). Therefore, the eigenvectors can be chosen to satisfy the orthonormality and closure conditions, namely,

κ,α

α,j e∗α,j e∗j ejq = δ jj , q,κ eq,κ = ˇ q •ˇ ∗β,j eα,j q,κ eq,κ = δ jj δ κκ .

j

Since the system is stable against small displacements, we must have [ωj (q)]2 ≥ ejq = ±ˇ ej∗ 0. Moreover, ω j (q) = ω j (−q), we can consistently choose ˇ −q . Since j∗ ∗ j eq = ˇ e−q . Uα (q, κ) = Uα (−q, κ), we choose the positive convention ˇ We now expand the general solution of Eq. (4.18) for Uα (q, κ, t) in terms of the eigenvectors ˇ ejq . We have ˘ U(q, t) =

Qj (q, t) ˇ ejq ,

(4.20)

˘ Qj (q, t) = ˇ e∗j q • U(q, t),

(4.21)

j

where

which is a complex scalar function defines the normal coordinates along the 3fpolarizations index j, and   ej Qj (q,t) √q,1 m1   j   j  eq,2   Qj (q,t) √m2    ˘ U(q, t) =  j      ...   j eq,f   Qj (q,t) √mf j

= (Uκ (q, t)) ,

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where ej

Qj (q,t) √q,κ mκ

Uκ (q, t) =

,

(4.22)

j

with the time-dependence of the normal coordinates given by, Qj (q,t) = Q0j (q) exp −iωj (q) t. The displacement of the atoms, Eq. (4.12), may thus be written as uα (l, κ, t) = √

1 NV

jq

eαj q,κ Qj (q,t) √ exp {−i (q · Xl,κ )} , mκ

(4.23)

which should be compared with Eq. (4.12) showing that the ‘Bloch function’, Uα (q, κ), there is given by Eq. (4.22). 1 uα (l, κ) = √ NV 4.3.1

q

Uα (q, κ) exp {−i (q · Xl,κ )} .

Acoustic and Optic Modes

From Eq. (4.16), we have at q = 0 Dκκ ,αβ (0) =

Φκκ ,αβ (h) . h

Therefore Dκκ ,αβ (0) = κ

Φκκ ,αβ (h) h,κ

=0 =

Dκκ ,αβ (0) .

κ

The last line comes from Eq. (4.9). Thus, the eigenvalue equation

βκ

Dκκ ,αβ (0) − [ω (0)]2 δ κκ δ αβ eβ0,κ = 0. √ mκ mκ

Clearly for a trivial uniform displacement of the whole lattice, we can assume eβ0,κ ≡ d independent of κ . Then it is easy to see that limq⇒0 ω (q) = 0, since   ∂ 2 Dκκ ,αβ (q)   qq  Dκκ ,αβ (0) +  ∂q∂q q⇒0 − [ω (q)]2 δ κκ δ αβ eβ0,κ = 0. √   mκ mκ  βκ 

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Then

∂ 2 Dκκ ,αβ (q) qq ∂q∂q q⇒0

√

mκ mκ

− [ω (q)]2 δ κκ δ αβ = 0.

The polarization vector must be one of the principal axis of the tensor ∂ 2 Dκκ ,αβ (q) ∂q∂q

q⇒0

qq , which is practically independent of index κ for very small q,

and hence ω (q) is proportional to q for small q. Thus there will be three frequencies ω j (0) = 0. Those branches j of the polarization for which limq⇒0 ω (q) = 0 are called the acoustic branches. The remaining 3f − 3 branches are called optic branches and for these limq⇒0 ωj (q) = 0. Thus if f = 1, that is, there is only one atom per unit lattice cell, only acoustic modes are present. The acoustic branches are made of two transverse branches and one longitudinal branch, with respect to the direction of propagation q. The modes in the longitudinal branch are basically compression waves for which the elastic restoring forces are generally stronger than the restoring forces of the transverse modes which are shear j (q) waves. This translates to higher group velocities, vj (q) = ∂ω∂q , for the longitudinal acoustic branch than for the transverse acoustic branches in major portion of the Brillouin zone of the reciprocal space. The 3f − 3 branches have dispersion relation ω j (q) that are higher than the acoustic branches and does not tend to zero at any point of the Brillouin zone. In the acoustic branch the motion of the atoms in the same lattice cell is always in the same sense. One way of looking at the lattice modes is to think of the unit lattice cell as a molecule containing f atoms. The acoustic modes are displacements of the ‘molecule’ as a whole unit in the general motion of the crystal lattice. On the other hand, the optical modes represents the internal vibrations of the atoms relative to one another in the molecule (localized excitations in a unit cell and therefore at higher frequencies than the acoustic modes), loosely coupled from cell to cell throughout the crystal. If the unit cell is strongly bound internally and interacts only weakly with each neighbor (tight binding), the internal forces will dominate the optical modes; the frequency will be approximately that of the free ‘molecule’ with only small coupling with surrounding lattice (nearly a flat band of frequencies). Each optical branch will thus have a very narrow bandwidth of frequencies, well separated from the acoustic branches. As an example, if we think of the crystal lattice as representing an ionic crystal, with two types of atoms per unit cell, i.e., ions of opposite charge, in the three optic branches the atoms in a single cell move in opposite directions to one another; this type of motion may generate large oscillating moments. The vibration frequencies in real crystals correspond to electromagnetic radiation frequencies in the infrared. Thus, infrared radiation are strongly absorbed because of the large dipole moment associated with the optical vibrations. Hence, the name optical modes.

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Frequency Distribution of Normal Modes

For crystal with N unit cells with f atoms per unit cell, we define the following expression, ωm

g (ω) dω = 3N f, 0

where ωm is the maximum frequency and g (ω) dω is the number of modes of vibration having a frequency between ω and ω + dω. The number 3Nf reflects the 3f branches containing N modes per branch. This is related to the distribution of squared frequencies, G ω2 , through the relation, ωm

ωm

G ω 2 dω 2

g (ω) dω = 0

0 ωm

2ωG ω2 dω

= 0

= 3N f. For a linear chain of atoms, [ω (q)]2 =

K (1 − cos qa) , m

where K is the spring constant and a is the lattice constant of the chain. Then K (1 − cos qa) m K (1 − cos qa) = m = ω m (1 − cos qa) 1 = ω m sin qa. 2

ω (q) =

Therefore the number of frequency is the same as the number N of q and hence 2π a

ωm

g (ω) dω = 0

g (q) dq = N 0 2π a

=

{|∇q ω (q)| g (ω (q))} dq, 0

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where g (q) =

N 2π a

.

(4.24)

Therefore

g (ω (q)) = = =

=

N 2π a 2π a

1 1 aωm cos qa 2 2 N N 1 = 1 1 π ω m cos 12 qa 2 aω m cos 2 qa 1

(∇q ω (q))−1 =

N π ω m N π

N

1 − sin2 21 qa 1

ω 2m −

−1

2π a

=

ω 2m sin2 21 qa

N π

1 ω 2m

− ω (q)

. 2

In three dimension, by extending the result of Eq. (4.24), we have g (q) =

N (2π)3 V

=

NV (2π)3

,

where V is the volume of a unit cell, and g (ω (q)) =

NV (2π)3

d3 q, ω<ω(q)<ω+dω

where the integral is extended over the region of the Brillouin zone for which the frequency ω (q) lies between ω and ω+dω. We have the volume element in reciprocal space between constant frequency surfaces ω and ω + dω given by d3 q =

dS dω , |∇q ω (q)|

where dS is an element of the area on the surface in q-space in which the frequency is equal to ω. By accounting for the diﬀerent branches of the dispersion curve, we end up with g (ω (q)) =

3f

NV (2π)

3 j=1

ω=ωj (q)

dS dω . |∇q ω j (q)|

From the above equation, it is clear that characteristic features of g (ω) should show at frequencies for which ∇q ω j (q) = 0; this occurs at point which are called van Hove singularities. Usually ∇q ω j (q) = 0 for each branch corresponding to q at, and perpendicular to, a surface of the Brillouin zone.

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Experimental Probes: Elastic Constants

The long wavelength acoustic modes of vibration of a crystal describe macroscopic deformations, and are identical with sound waves. The speed of propagation of sound waves therefore determines the slope of the dispersion curves for q ⇒ 0, which is in turn determined by the elastic constants of the crystal. The equation of motion of a macroscopic portion of the crystal is ρ

∂ 2 uα = ∂t2

β

∂Tαβ , ∂Xβ

where ρ is the density and Tαβ the stress tensor given by Tαβ =

1 2

Cαβγδ γδ

∂uδ ∂uγ + ∂Xδ ∂Xγ

,

where Cαβγδ is the elastic constants in tensor notation. For a special case of a plane wave solution u exp {i (q · X − ωt)}, the macroscopic equation of motion becomes ρω 2 uα =

Cαβγδ (qβ qδ uγ + qβ qγ uδ ) . βγδ

This equation corresponds to Eq. (4.11) in the long-wavelenght limit, thus one can relate the elastic constants to the force constants. The experimental technique to measure elastic constants is to propagate acoustic disturbance in a known direction along a crystal specimen and measure the velocity of propagation. Optic modes are experimentally studied using electromagnetic radiation with frequency comparable with that of the lattice vibrations, i.e., in the infrared. The experimental technique is infrared-absorption and Raman-scattering spectroscopy. However, these techniques only probe the long-wavelength optic modes. Light is also inelastically scattered by acoustic modes and the process is known as Brillouin scattering. The use of thermal neutrons provides a direct way to study the energy dispersion because the energy and wave vector of the radiation are similar to those of lattice vibrations. 4.5

Hamiltonian in Terms of Normal Coordinates

The Hamiltonian in Eq. (4.10) can now be expressed in terms of the normal coordinates, in preparation for the second quantization. We have, H = Hq , and q

from Eq. (4.17),

Hq =

   1

2  +

Pα (q,κ,t)Pα∗ (q,κ,t) mκ κ,α

αβ

κκ

   

Uα (q, κ)Dκκ ,αβ (q) Uβ∗ (q, κ , t)   

.

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47

By using the conjugate momentum as ˙ ∗κ (q, t) Pκ (q, t) = mκ U = j

√ −iωj (q) Q∗j (q,t) mκ ejq,κ

,

which can be written in terms of the κ components as   √ m1 Q˙ ∗j (q,t) ejq,1   j   √ j   ∗ ˙ m2 Qj (q,t) eq,2     ˘ P(q, t) =  j    ...     √ j  mf Q˙ ∗ (q,t) e  j

q,f

j



=

We have  1 Hq = 2 =

=

 1

2

∗ Pα (q,κ,t)Pα (q,κ,t) mκ

κ,α

1 √ αj √ ˙∗ ˙ mκ mκ Qj (q, t) eq,κ mκ Qj

jj ακ

[ωj (q)]2 Q∗j (q, t) Qj (q, t) jj

2  +

Qj (q, t) e∗α,j q,κ



j

jj

αj eαj q,κ eq,κ

ακ Dκκ ,αβ (q) β,j √ e q,κ mκ mκ

αβjj κκ

jj 2

   

Q∗j (q, t)   

[ω j (q)]2 Q∗j (q, t) Qj (q, t) δ jj

αβjj κκ

2  +

 (q, t) eαj q,κ 

˘ ˘ ∗ (q, t) +U(q, t) • D (q) • U

which further reduced to    1 Hq = 2  + =

 

˘ ˘ ∗ (q, t)  +U(q, t) • D (q) • U

   1

   1

j

 √ Q˙ ∗j (q, t) mκ ejq,κ  .

,

   

β,j ∗  Qj (q, t) e∗α,j q,κ [ω (q)] δ κκ δ αβ eq,κ Qj (q, t)  

[ω j (q)]2 Q∗j (q, t) Qj (q, t)

   

Qj (q, t) [ωj (q)]2 δ jj Q∗j (q, t)   

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=

=

   1

j

2  +

 1 2

 2  [ω j (q)] Q∗j (q, t) Qj (q, t)   2

j

[ω j (q)] Qj (q, t) Q∗j (q, t)   

[ω j (q)]2 Q∗j (q, t) Qj (q, t) + Qj (q, t) Q∗j (q, t) j

From the relation

  

.

Q˙ j (q,t) Q˙ †j (q,t) = [ωj (q)]2 Q∗j (q, t) Qj (q, t) , we can write in terms of the normal coordinates, Qj (q, t), as    1 Q˙ ∗j (q, t) Q˙ j (q, t) + [ωj (q)]2 Qj (q, t) Q∗j (q, t) , Hq =  2 j

which is a form of a sum of harmonic oscillator of diﬀerent polarization labeled by index, j. A transformation of the type Qj (q, t) =

| ω j (q)

Q∗j (q, t)

| ω j (q)

=

1 2

aj (q, t) , 1 2

a†j (q, t) ,

will yield upon substitution the following expression for the Hamiltonian    1 | [ω j (q)]2 Hq = a†j (q, t) aj (q, t) + aj (q, t) a†j (q, t)  2 j ωj (q)    1 = |ω j (q) a†j (q, t) aj (q, t) + aj (q, t) a†j (q, t) .  2 j

4.6

Phonons in Three Dimensions

So far everything is based on classical arguments. From Eqs. (4.12) and (4.13) the quantum mechanical commutation relations of the dynamical variables in the quantized version are ˆα (q, κ), Pˆβ (q , κ ) = i|δ qq δ αβ δ κκ , U ˆα (q, κ), Q ˆ β (q , κ ) = Pˆα (q, κ), Pˆβ (q , κ ) = 0. U

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From Eq. (4.20), we must also have Q˙ ∗j (q , t) Qj (q, t) = ihδ qq δ κκ δ jj , , mκ √ √ mκ mκ Qj (q, t) , iωQ∗j (q , t) = ihδ qq δ jj , | ωj (q)

1 2

a ˆj (q, t) , iω j (q )

1 2

| ω j (q)

a ˆ†j (q , t) = ihδ qq δ jj ,

a ˆj (q, t) , a ˆ†j (q , t) = δ qq δ jj . Thus, we can now write,    1 |ωj (q) a ˆ†j (q, t) a ˆj (q, t) + a ˆj (q, t) a ˆ†j (q, t) Hq = =  2 j

|ω j (q) a ˆ†j (q, t) a ˆj (q, t) +

=

j

|ω j (q) n ˆ j (q, t) +

= j

1 2

1 , 2

where n ˆ j (q, t) is the phonon number operator whose eigenvalue at thermal equilibrium depends on the temperature through the Maxwell-Boltzmann distribution. The total energy of the whole system is then made up of the sum of all phonon occupation number n ˆ j (q, t) over all branch indices j and all points q in the Brillouin zone.

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Chapter 5

Quantization of Electromagnetic Fields

Before going into the ‘second’ quantization of electromagnetic fields, it is convenient to cast the mathematical description of the free electromagnetic field in terms of canonically conjugate position and momentum variables. Then we will see that the quantization of the free electromagnetic field is similar to the “second quantization” of the chain of harmonically coupled atoms.

5.1

Maxwell Equations

The “microscopic” (i.e., without the use of permittivity and permeability of a material, which are only averaged quantities) classical equations for the free electromagnetic field are given by the microscopic Maxwell equations, given here in CGS units as ∇ × E (r, t) = −

∇ × B (r, t) =

1 ∂ B (r, t) , c ∂t

4π 1 ∂ J (r, t) + E (r, t) , c c ∂t

(5.1)

(5.2)

∇ • E (r, t) = 4πρ (r, t) ,

(5.3)

∇ • B (r, t) = 0,

(5.4)

where c is the speed of light in vacuum, B (r, t) and E (r, t) are the classical electric and magnetic fields at the point (r, t), which are vector fields, εo and µo are the permittivity and permeability of the vacuum, respectively. Equation (5.4) shows that the vector field B (r, t) has a zero divergence, thus there exists a vector field A (r, t) such that B (r, t) = ∇ × A (r, t) . 50

(5.5)

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A (r, t) is called the vector potential. However, the vector field A (r, t) is not uniquely defined since another vector potential, A (r, t), given by A (r, t) = A (r, t) + ∇Λ (r, t) ,

(5.6)

which diﬀers by a gradient of a scalar field, Λ (r, t), gives the same physical magnetic field B (r, t) and hence does not alter the physics. Upon substituting for B (r, t) in terms of the vector potential, Eq. (5.1) becomes ∇ × E (r, t) +

1 ∂ A (r, t) c ∂t

= 0.

Now any vector field that is irrotational or has a zero curl can be expressed as a gradient of a scaler field, therefore we may write E (r, t) = −∇Φ −

1 ∂ A (r, t) . c ∂t

(5.7)

Our choice for the minus sign in front of ∇Φ is dictated by the fact that E (r, t) is a force per unit charge. In order that the electric field is not altered as well when A (r, t) is transformed by Eq. (5.6) to A (r, t), the scalar potential Φ (r, t) must be simultaneously transformed to Φ (r, t) = Φ (r, t) −

1 ∂Λ (r, t) . c ∂t

(5.8)

Note that, from Eqs. (5.6) and (5.8), only the longitudinal part of the vector potential and the scalar potential are aﬀected by the gauge transformation. Moreover, for Coulomb gauge, namely, ∇ · A (r, t) = 0, the vector potential only consist of transverse components. 5.2

The Electromagnetic Wave Equations

Equations (5.2) and (5.3) involving sources, in terms of the vector and scalar potentials yield coupled electromagnetic wave equations for the vector and scalar fields. In terms of the vector and scalar potential, Eq. (5.2) becomes ∇ × ∇ × A (r, t) =

4π 1 ∂ J (r, t) − c c ∂t

1 ∂ A (r, t) + ∇Φ (r, t) . c ∂t

Using the vector product identity ∇ × ∇ × A (r, t) = ∇ ∇ • A (r, t) − (∇ • ∇) A (r, t) , we have ∇2 A (r, t) −

1 ∂ 1 ∂2 4π Φ (r, t) . (5.9) A (r, t) = − J (r, t) + ∇ ∇ • A (r, t) + c2 ∂t2 c c ∂t

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Equation (5.3), also with a source term becomes 1 ∂ A (r, t) − ∇2 Φ (r, t) = 4πρ (r, t) , c ∂t 1 ∂ 2 Φ (r, t) 1 ∂ 2 Φ (r, t) 1 ∂ A (r, t) ∇2 Φ (r, t) − 2 = −4πρ (r, t) − −∇• c ∂t2 c2 ∂t2 c ∂t

−∇ •

= −4πρ (r, t) −

1 ∂ c ∂t

∇ • A (r, t) +

1 ∂Φ (r, t) . c ∂t (5.10)

Note that both Eq. (5.9) and Eq. (5.10) involve the expression ∇ • A (r, t) + Because of the freedom in choosing the vector potential allowed by the gauge transformations, Eqs. (5.6) - (5.8), we have 1 ∂ϕ(r,t) c ∂t .

1 ∂Λ(r,t) 1 ∂ Φ (r, t) − c ∂t 1 ∂Φ (r, t) ⇒ ∇ • A (r, t) + ∇Λ (r, t) + ∇ • A (r, t) + c ∂t c ∂t 1 ∂Φ (r, t) 1 ∂ 2 Λ (r, t) 2 = ∇ • A (r, t) + . + ∇ Λ (r, t) − 2 c ∂t c ∂t2

Therefore choosing A (r, t) such that ∇ • A (r, t) +

1 ∂Φ (r, t) = 0, c ∂t

(5.11)

which is universally known as the Lorentz condition (although Lorenz is credited with originating this supplementary condition 25 years before Lorentz), amounts to fixing the gauge since the scalar field Λ (r, t) must now satisfy a wave equation, 2 ∇2 Λ (r, t) − c12 ∂ Λ(r,t) = 0, which by adopting a “sensible” boundary conditions ∂t2 in both space and time then the only solution to this Λ (r, t) wave equation is Λ (r, t) = 0. We thus obtained the electromagnetic wave equations given by ∇2 A (r, t) −

1 ∂2 4π A (r, t) = − J (r, t) , c2 ∂t2 c

(5.12)

1 ∂ 2 Φ (r, t) = −4πρ (r, t) . c2 ∂t2

(5.13)

∇2 Φ (r, t) −

These are wave equations with sources J (r, t) and ρ (r, t). 5.2.1

A Single Electromagnetic Wave Equation

We can condense the two wave equations, by using the mathematical device similar to the one employed in Eq. (2.1) for the harmonic oscillator. Thus by defining a new quantity for the wave functions given Θ (r, t) = A (r, t) + iΦ (r, t) ,

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and the corresponding new source function given by Ξ (r, t) = J (r, t) + icρ (r, t) . Then we can combine the vector and scalar electromagnetic wave equations into a single four-dimensional wave equation as ∇2 Θ (r, t) −

1 ∂2 4π Θ (r, t) = − Ξ (r, t) . c2 ∂t2 c

(5.14)

Equation (5.14), involved strange quantities, Θ (r, t) and Ξ (r, t). By decomposing A (r, t) into its transverse and longitudinal components, A (r, t) = A⊥ (r, t) + AL (r, t), a complex-valued variable, Z (r, t), can be formed by combining the longitudinal component AL (r, t) with the scalar potential Φ (r, t) , Z (r, t) = AL (r, t) + iΦ (r, t) , thus reducing the number of wave equations from four to three components. It is remarkable to note that only the complex variable, Z (r, t), is aﬀected by the gauge transformations, Eqs. (5.6) and (5.8), Z (r, t) = Z (r, t) + Λ (r, t) ,

(5.15)

1 ∂ (here we used the notation, = ∇+ ic ∂t ), whereas A⊥ (r, t) is invariant under gauge transformation. In the absence of charge particle, Z (r, t) can be eliminated by gauge transformation leaving only the transverse vector potential often referred to as the radiation field. Note that making Φ (r, t) = 0, by means of gauge transformation, is not a gauge invariant way of working since we have to redo the gauge transformation to make Φ (r, t) = 0 in another space-time coordinate frame. In a fully covariant quantization of electrodynamics, the combined eﬀects of the resulting AL (r, t) and Φ (r, t) photons ( in our notation as Z (r, t) photons) give rise to unobservable virtual photons which only physically manifest themselves in the Coulomb forces between charge particles. Clearly, Z (r, t), which is sensitive to gauge transformation, is a gauge field associated with charge particle and their force of interaction. Perhaps, it is not surprising that gauge invariance, and in general gauge symmetry, in quantum mechanics has become a cornerstone of modern gauge theories of elementary particles and forces in nature. Thus, by separating the eﬀects of Z (r, t) and A⊥ (r, t), the non-relativistic Hamiltonian of particles in electromagnetic fields can still be written, for practical use, in the familiar form as

H= j

e 1 p − A⊥ (rj , t) mj c

where A⊥ (rj , t) is the radiation field

2

+ i>,j

ei ej + Hradiation , 4π |ri − rj |

∇ · A⊥ (rj , t) = 0 , and Hradiation is the

Hamiltonian of the pure radiation field. Note that AL (r, t) and Φ (r, t) do not appear in the Hamiltonian but instead we have the Coulomb interaction potential e i ej given by the term, 4π|ri −rj | , involving Coulomb charges. i>,j

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Covariant Formulation of Electrodynamics

The traditional approach of separating the radiation field in the literature is dictated by the requirement of Lorentz covariance in special relativity theory, this is done by expressing the potential and source in terms of 4-vectors, represented as Aµ = A, iΦ ,

Jµ = J, icρ . Thus iΦ and icρ are the time-like components of the 4-vectors, Aµ and Jµ , respectively. Equations (5.12) and (5.13) suggest that our four-dimensional space-time coordinates are x1 = x, x2 = y, x3 = z, x4 = ict, xµ = (x, y, z, ict) . Then the single wave equation, in terms of the 4-vector is 2

Aµ = −4πJµ ,

(5.16)

2

1 ∂ where 2 = ∇2 + (ic) 2 ∂t2 is the space-time Laplacian operator or the D’Alambertian operator, while the Lorentz condition, Eq. (5.11) that goes with it now becomes

∂Aµ = 0. ∂xµ Note that the charge continuity equation for the sources, expressed as ∂Jµ = 0. ∂xµ

(5.17) ∂ρ ∂t

= −∇ · J, is now also (5.18)

The virtue of expressing the electromagnetic equations in terms of Aµ , Jµ , and xµ , is that the resulting equations do not change their forms under the Lorentz transformation of the space-time coordinates, i.e., electrodynamics is given a covariant formulation. Note that in physics, Lorentz covariance is a key property of space-time that follows from the special theory of relativity, where it applies globally. Lorentz covariance has two distinct, but closely related meanings. (1) A physical quantity is said to be Lorentz covariant if it transforms under a given representation of the Lorentz group. According to the representation theory of the Lorentz group, these quantities are built out of scalars, four-vectors, fourtensors, and spinors. In particular, a scalar (e.g. the space-time interval or ‘distance’) remains the same under Lorentz transformations and is said to be a Lorentz invariant.

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(2) An equation is said to be Lorentz covariant if it can be written in terms of Lorentz covariant quantities. This condition is a requirement according to the principle of relativity, i.e. all non-gravitational laws must make the same predictions for identical experiments taking place at the same space-time event in two diﬀerent inertial frames of reference. It is therefore desirable to write the Maxwell equations in covariant form as well. Note that the electric field by itself does not transform as a tensor and the magnetic field by itself does not either. However, the covariant anti-symmetric field-strength tensor, Fµν , defined below, does. We can combined Eqs. (5.5) and (5.7) into an expression for an anti-symmetric second rank tensor, Fµν =

∂Aν ∂Aµ − , ∂xµ ∂xν

a 4-dimensional generalization of the curl operation, which expresses the covariant anti-symmetric field-strength tensor. Explicitly, Fµν is given as   0 Bz −By −iEx  −Bz 0 Bx −iEy   (5.19) Fµν =   By −Bx 0 −iEz  . iEx iEy iEz 0

In terms of the anti-symmetric field-strength tensor, the inhomogeneous Maxwell equations, Eqs. (5.2) and (5.3) can be combined into a covariant equation as 4π ∂Fµν Jµ . = ∂xν c

(5.20)

Similarly, Eqs. (5.1) and (5.4) of the Maxwell equations can be reduced to a covariant form as ∂Fλµ ∂Fνλ ∂Fµν + + = 0, ∂xλ ∂xν ∂xµ

(5.21)

where λ, µ, ν can each takes the value 1, 2, 3, 4. Equations (5.20) and (5.21) form the manifestly covariant form of the Maxwell’s equations, and because it involves measurable quantities these equations are also gauge invariant. Each term in Eq. (5.21) transforms like a 4-tensor of third rank so that Eq. (5.21) is also covariant in form. Note that Eq. (5.18) is consistent with Eq. (5.20) since by taking the 4-gradient of Eq. (5.20) we have 4π ∂Jµ ∂ 2 Fµν = . ∂xµ ∂xν c ∂xµ Since Fµν is anti-symmetric and the second derivative is symmetric, we must have ∂Jµ ∂xµ = 0, which yields the covariant (scalar) continuity equation, Eq. (5.18). Thus, Maxwell’s equations, including the continuity equation, become simply tensor equations.

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5.4

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Complex Dynamical Variables

We will not go into the covariant quantization of the electromagnetic field since it would be more instructive to consider free electromagnetic field, i.e., without the 4-vector current sources. For free electromagnetic field, the Lorentz condition becomes the Coulomb-gauge condition ∇ · A = 0, which signify that we are only dealing with transverse vector potential, A⊥ . Then the electromagnetic wave equation, Eq. (5.16), becomes (dropping the subscript ⊥) 2

A = 0.

In analogy with the Lagrangian of Eq. (3.3) leading to the wave equation for a linear chain of coupled atoms, Eq. (3.4), the free electromagnetic wave equation should be obtainable from a Lagrangian of the form 1 1 1 A˙ 2 − (∇Ax )2 − (∇Ay )2 − (∇Az )2 d3 x. 2c2 2 2 2

L = const

Hence the canonically conjugate momenta are given by δL 1 = const 2 A˙ x , ˙ c δ Ax δL 1 πy = = const 2 A˙ y , ˙ c δ Ay δL 1 πz = = const 2 A˙ z . ˙ c δ Az

πx =

(5.22)

We can determine the const from the well-known expression for the energy of the free electromagnetic field given by H=

1 8π

|E|2 + |B|2 d3 x.

Then we have L= = =

1 ˙ 2 1 A − c2 8π 1 1 const 2 A˙ 2 − c 8π 1 1 const 2 A˙ 2 − c 8π

const

|E|2 + |B|2

d3 x

2 1 ˙2 A + ∇ × A d3 x c2 1 ˙2 A + (∇Ax )2 + (∇Ay )2 + (∇Az )2 c2

where we have made use of Eq. (5.7) and the equivalence relation, 2

∇×A

d3 x =

(∇Ax )2 + (∇Ay )2 + (∇Az )2 d3 x.

d3 x,

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Note that 2

∇×A

(∇Ax )2 + (∇Ay )2 + (∇Az )2 d3 x = −

d3 x −

(∇ · A)2 d3 x ≡ 0

can be proved by assuming that A vanishes at the boundary of the integration region and ∇ · A = 0. Therefore we must have const =

1 4π

to give π=

δL 1 ˙ A = ˙ 4πc2 δA

and L=

1 1 1 A˙ 2 − (∇Ax )2 − (∇Ay )2 − (∇Az )2 d3 x. 2c2 2 2 2

1 4π

In analogy with the linear chain of atoms coupled by harmonic forces, Sec. 2.1.3, 1 where the mass density ρ there is now replaced by 4πc 2 and the parameter g replaced 1 by 4π , we write the Fourier decomposition of A and π as   2 |2πc 1 √ (5.23) ˆεk,j ak,j (t) eikx + a†k,j (t) e−ikx  , A (x, t) = ωk V k,j

where in further analogy with Eq. (4.23), ak,j (t) correspond to the normal modes of the electromagnetic fields. We also have, i π (x, t) = √ V

k,j

|ωk ˆεk,j −ak,j (t) eikx + a†k,j (t) e−ikx 2 (4πc2 )

,

(5.24)

where we have introduced the two orthogonal polarization unit vector ˆεk,j ≡ ˆε−k,j (j = 1, 2, where for convenience in what follows, we let ˆεk,1 = ε and ˆεk,2 = ε ) for each mode k. The linear dispersion relation for the frequency is given by 4πc2 k 4π = ck.

ωk =

Again, from Eq. (5.7), we can rewrite L as L=

1 |E|2 − |B|2 . 8π

Using Eq. (5.19), we can also write L in terms of the field-strength tensor as L =−

1 1 Fµν F µν = |E|2 − |B|2 . 16π 8π

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To proceed with the quantization of the electromagnetic field, we now assume the commutators of the components of the fields, and also those involving only their canonical conjugate momenta, vanish [Ai (x, t) , Aj (x , t)] = 0, [πi (x, t) , π j (x , t)] = 0. The commutation of the canonical variables become i [π i (x, t) , Aj (x , t)] = √ V × −

k,γ

k ,γ

k ,γ

i ×√ V

|ωk ˆεk,γ −ˆ ak,γ (t) eikx +ˆ a†k,γ (t) e−ikx 2 (4πc2 )



√1 V  √1 V k,j

 |2πc2 ˆk ,γ (t) eik x +ˆ ˆεk ,γ a a†k ,γ (t) e−ik x  ωk  |2πc2 ˆk ,γ (t) eik x +ˆ ˆεk ,γ a a†k ,γ (t) e−ik x  ωk

|ω k ˆεk,γ −ˆ ak,γ (t) eikx +ˆ a†k,γ (t) e−ikx 2 (4πc2 )

,

where to avoid proliferation of indices, we have postpone evaluating the corresponding Cartesian components of the right-hand side expression; we will indicate this operation later. We have [π i (x, t) , Aj (x , t)] =

i| 2V

k,γ,k ,γ

ωk ˆεk,γ ˆεk ,γ ωk

−ˆ ak,γ (t) eikx + a ˆ†k,γ (t) e−ikx

× a ˆk ,γ (t) eik x + a ˆ†k ,γ (t) e−ik x ik x

ˆk ,γ (t) e × ˆεk ,γ ˆεk,γ a

+a ˆ†k ,γ

i| 2V

−

k ,j ,k,j

ωk ωk

−ik x

(t) e

ˆ†k,γ (t) e−ikx , × −ˆ ak,γ (t) eikx + a [πi (x, t) , Aj (x , t)] =

i| 2V

k,γ,k ,γ

ωk ˆεk,γ ˆεk ,γ ωk

a ˆk,γ (t) a ˆk ,γ (t) ei(kx+k x )

ˆk ,γ (t) a ˆk,γ (t) ei(k x +kx) + ˆεk ,γ ˆεk,γ a + ˆεk,γ ˆεk ,γ

a ˆ†k,γ (t) a ˆ†k ,γ (t) e−i(kx+k x )

ˆ†k ,γ (t) a ˆ†k,γ (t) e−i(k x +kx) − ˆεk ,γ ˆεk,j a − ˆεk,γ ˆεk ,γ

a ˆk,γ (t) a ˆ†k ,γ (t) ei(kx−k x )

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ˆk ,γ (t) a − ˆεk ,γ ˆεk,γ a ˆ†k,γ (t) ei(k x −kx) + ˆεk,j ˆεk ,j

a ˆ†k,γ (t) a ˆk ,γ (t) e−i(kx−k x )

ˆ†k ,γ (t) a ˆk,γ (t) e−i(k x −kx) , + ˆεk ,γ ˆεk,γ a [πi (x, t) , Aj (x , t)] i| ωk = 2V ωk

(5.25)

k,γ,k ,γ

a ˆk,γ (t) a ˆ†k,γ (t) a ˆ†k ,γ (t) ei(kx−k x ) − a ˆk ,γ (t) e−i(kx−k x )

− ˆεk,γ ˆεk ,γ

ˆk ,γ (t) a ˆ†k ,γ (t) a ˆ†k,γ (t) ei(k x −kx) − a ˆk,γ (t) e−i(k x −kx) . − ˆεk ,γ ˆεk,γ a (5.26) Taking Fourier transform on both sides, we have 1 V

eiκ·x e−iκ ·x dx dx [π i (x, t) , Aj (x , t)] = π †i (κ, t) , Aj (κ , t) i| 2V 2

=

ωk ωk

k,γ,k ,γ



 − ˆεk,γ ˆεk ,γ 

−ˆ a†k,γ

 − ˆεk ,γ ˆεk,γ 

−ˆ a†k ,γ



1 V

ei(κ+k)·x e−i(κ +k )·x dx dx

a ˆk,γ (t) a ˆ†k ,γ (t)

i(κ−k)·x −i(κ −k )·x

(t) a ˆk ,γ (t)

e

e

dx dx

ei(κ−k)·x e−i(κ −k )·x dx dx

a ˆk ,γ (t) a ˆ†k,γ (t)

i(κ+k)·x −i(κ +k )·x

(t) a ˆk,γ (t)

e

e

dx dx

  



 ,

eiκ·x e−iκ ·x dx dx [π i (x, t) , Aj (x , t)] = π †i (κ, t) , Aj (κ , t) =

i| V 2V

γ,γ

ωκ ωκ

× −ˆεk,γ ˆεk ,γ =

i| 2 γ,γ

a ˆ−κ,γ (t) a ˆ†−κ ,γ (t) ˆκ ,γ (t) −ˆ a†κ,γ (t) a

ωκ ˆεk ,γ ˆεk,γ ωκ

− ˆεk ,j ˆεk,j

a ˆκ ,γ (t) a ˆ†κ,γ (t) † −ˆ a−κ ,γ (t) a ˆ−κ,γ (t)

a ˆ†−κ ,γ (t) , a ˆ−κ,γ (t) + a ˆ†κ,γ (t) , a ˆκ ,γ (t)

.

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If we impose the commutation relation for the radiation annihilation and creation operators, ˆ−κ,γ (t) = −δ κκ δ γγ a ˆ†−κ ,γ (t) , a = a ˆ†κ,γ (t) , a ˆκ ,γ (t) , then we are left with the expression π†i (κ, t) , Aj (κ , t) = − =

| i

i| ˆεk ,γ ˆεk,γ {2δ κκ δ γγ } 2 γ,,γ ˆεk,γ ˆεk,γ , γ

where the last sum is over the transverse polarization index, γ. Since the lefthand side expresses the components in the Cartesian frame, the right hand side in terms of transverse polarization directions, must now be expressed in terms of their Cartesian components, a procedure we have postpone till now. The result derived in Appendix A is

,j

ˆεk,j ˆεk,j = δ ij −

κi κj |κ|2

.

Thus, π†i (κ, t) , Aj (κ , t) =

| i

δ ij −

κi κj |κ|2

= ∆⊥ ij (κ, κ ) , where we refer to ∆⊥ ij (κ, κ ) as the ‘transverse delta’ function. ⊥ Hence defining δ ⊥ ij (x − x ) as the inverse Fourier transform of ∆ij (k, k ), we obtain δ⊥ ij (x − x ) =

=

| 1 i (2π)3 | 1 i (2π)3

e−ik·x eik ·x dk dk ∆⊥ ij (k, k ) 



ki kj   eik·(x −x) dk δ ij − 2 . k

To evaluate the right hand side of the last equation, we express ki and kj as diﬀerentiation with respect to xi and xj , respectively, acting on the exponential function

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Fig. 5.1

61

→ − − → Orthogonal unit vectors transverse to the propagating wave direction k are → ε and − ε

in the form





ki kj   eik·(x −x) dk δ ij − 2 k       1  | ∂ ∂ 1 ik·(x −x) e dk = δ ij δ (x − x ) + 3 2  i ∂xi ∂xj   (2π) k 

δ⊥ ij (x − x ) =

=

| 1 i (2π)3

∂ ∂ | δ ij δ (x − x ) + i ∂xi ∂xj

1 1 4π |x − x |

.

Therefore Eq. (5.26) must now read [πi (x, t) , Aj (x , t)] =

| i

δ ij δ (x − x ) +

∂ ∂ ∂xi ∂xj

1 1 4π |x − x |

.

(5.27)

The reason why the right hand side of Eq. (5.27) becomes complicated is due to the constraint imposed by the Coulomb gauge, namely ∇ · A = 0, which implies that the three components of Ai (x) (i = x, y, z) are not independent. However, if we only consider the (ε, ε , k) system of Fig. 5.1, for each mode k, i.e., isolate the truly independent dynamical field variables, then the natural quantization that results follows from | [π ε (x, t) , Aε (x , t)] = δ εε δ (x − x ) , i [π ε (x, t) , Aε (x , t)] =

| δ εε δ (x − x ) . i

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Chapter 6

Quantum States of Classical Fields

6.1

Wave Function for the Harmonic Oscillator

The quantum Hamiltonian of a harmonic oscillator, H, reads H=

1 P2 + mω 2 Q2 . 2m 2

The corresponding Schrödinger eigenvalue equation for the harmonic oscillator is −|2 ∂ 2 1 + mω 2 q 2 ψ (q) = E ψ (q) . 2m ∂q 2 2 The eigensolutions for the energy eigenvalues, or energy eigenstate, are given in terms of the Hermite polynomials, ψn (q) = where Hn

mω | q

mω π|

1 4

1 √ Hn 2n n!

mω q |

exp −

mω 2 q , 2|

(6.1)

is nth Hermite polynomial, with eigenvalue given by En = n +

1 2

|ω,

where n is a positive integer. Note that from Eq. (2.6) the above quantization 2 of energy corresponds to the quantization of the classical amplitude |φ| in phase space.

6.2

Second Quantization of the Classical φ and φ∗

Introduce the following annihilation operator, a ˆ, and creation operator, a ˆ† , in terms of linear combination of the momentum and position operator, in exactly the same manner as changing the classical position, q, and momentum, p, occurring in complex ‘field’ variables φ and φ∗ into Q and P , respectively. This amounts to changing 62

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φ as annihilation operator and φ∗ as creation operator, i.e., φ⇒a ˆ=

mω 2|

Q+

i P mω

1 ˜ + iP˜ , = √ Q 2 ˆ† = φ∗ ⇒ a

mω 2|

(6.2)

Q−

i P mω

1 ˜ − iP˜ , = √ Q 2

(6.3)

where the scaled quantum operator for the harmonic oscillator are ˜= Q

mω Q, |

P˜ =

1 P. |mω

(6.4)

Note that the commutation relation for the scaled operators is ˜ P˜ = i| = i. Q, | Eﬀectively, the classical φ and φ∗ are now viewed as quantum annihilation operator, ˆ, a ˆ† = 1. a ˆ, and creation operator, a ˆ† , respectively, which do not commute, i.e., a From Eqs. (2.4)-(2.5), we may interpret a ˆ as a complex amplitude operator. Note that a ˆ and a ˆ† are not equal hence a ˆ is a non-Hermitian operator, whereas, Q and P are Hermitian operators since these represent observables. Because of the noncommutativity of a ˆ and a ˆ† , the replacement of φ and φ∗ in the classical Hamiltonian † by a ˆ and a ˆ , respectively, must be done using the symmetric expression in Eq. (2.6), i.e., 1 ∗ (φ |ωφ + φ |ωφ∗ ) 2 1 ∗ a |ωˆ a+a ˆ|ωˆ a∗ ) ⇒ H = (ˆ 2 1 ˆ+ = |ω a ˆ∗ a , 2

Hφ =

where the last line is obtained by using the commutation relation of a ˆ and a ˆ† . 1 Therefore the vacuum-state energy, 2 |ω, is due to the uncertainty principle in the momentum and position and hence represent vacuum fluctuation energy. More precisely, as we shall see later, 12 |ω is due to minimum uncertainty principle, which is completely absent in the classical case. Then we have ˆ+ H = |ω a ˆ† a

1 2

.

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The quantum Hamiltonian is written in normal order form in which creation operators stand to the left of annihilation operators. The corresponding normal ordering operation is denoted by putting semicolon symbol on both sides of the function of a ˆ and a ˆ† . Thus we may also write H =: |ω a ˆa ˆ† +

1 2

:,

(6.5)

that is inside the normal ordering signs the Bose creation and annihilation operators behave just like c-numbers. Note that the Hermitian operators Q and P are expressed by the real and imaginary parts of a ˆ, Q=

| a ˆ+a ˆ† = 2mω

| 1 √ a ˆ+a ˆ† 2 mω

,

(6.6)

P =

|mω a ˆ−a ˆ† = 2

| √ mω a ˆ−a ˆ† 2

.

(6.7)

From the Schrödinger equation, we deduce that the new eigenvalue equation ˆ ψ n (q) = n ψ n (q). Hence, the bilinear operator a ˆ† a ˆ, which commutes with the a ˆ† a Hamiltonian H is called the number operator with eigenvalues n. The vacuum state corresponds to n = 0. From the relation, a ˆ, a ˆ† = 1, the commutation of a ˆ and a ˆ† with the Hamiltonian are respectably given by [H, a ˆ] = −|ωˆ a, H, a ˆ† = |ωˆ a† . From the above relations, we can characterize the eigenstates obtained by operating a ˆ on the energy eigenstates |ψn for n = 0. We have ˆ] + a ˆH) |ψn Ha ˆ |ψn = ([H, a = (En − |ω) (ˆ a |ψn ) , Ha ˆ† |ψn =

H, a ˆ† + a ˆ† H |ψn

ˆ† |ψn = (En + |ω) a

.

These last relations implies the following a ˆ |ψn = C ψn−1 , a ˆ† |ψn = C ψn+1 . ˆ |ψn = n |ψn , we must have ψn | a ˆ† a ˆ |ψn = n = |C|2 . Now from the relation a ˆ† a √ † ˆ, a ˆ = 1, we have a ˆa ˆ† |ψn = Therefore C = n. From the commutation relation a √ 2 ˆa ˆ† |ψn = (n + 1) = |C | . Therefore, C = n + 1. Thus, (n + 1) |ψn and ψ n | a

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√ √ a ˆ |ψn = n ψn−1 and a ˆ† |ψn = n + 1 ψn+1 . In order to be consistent, we define a ˆ |ψ0 = 0. Denoting the vacuum energy eigenstate by |ψ0 , then we have a ˆ† n |ψn = √ |ψ0 . n!

(6.8)

The fact that a ˆ and a ˆ† perform discrete shifting operations on the energy eigenstates has found very useful applications in the quantization of classical fields and second quantization of the eﬀective one-particle Schrödinger wavefunction equation. Moreover, the general aspects of the construction of the annihilation and creation operators as a linear combination of the canonically conjugate operators is very intriguing since an eigenstate of any of these operators would be labeled by the classical values of the variables, resembling the system of labeling of the states in classical mechanics. 6.3

Biorthogonal Bases

Because a ˆ and a ˆ† operators are non-Hermitian, one can expect the right and left eigenvectors of a ˆ and a ˆ† , respectively, to form bi-orthogonal bases states1 . This follows from the supposed eigenvector equations for the non-Hermitian operators a ˆ and a ˆ† a ˆ |α = α |α , a ˆ† |α∗ = α∗ |α∗ , where the set of states {|α } and {|β }are in diﬀerent topological function spaces. Then the left eigenvector equation for a ˆ† and a ˆ, respectively are given by ˆ† = α∗ | α∗ , α∗ | a which means that the left eigenvector of a ˆ† is α∗ |, on the other hand α| a ˆ = α| α, shows that the left eigenvector of a ˆ is α|. The eigenvalues corresponding to |α and α| are identical, and likewise for |α∗ and α∗ |. Therefore the label α and α∗ is to remind us that this eigenvectors belong to diﬀerent topological spaces. We can readily show that |α and α| form a biorthogonal system as follows. Consider ˆ |αj = αj β i | αj = αi αi | αj . αi | a Then, we have (αj − αi ) αi | αj = 0, showing that for αj = αi , αi | αj = 0. Likewise similar results can be obtained for the pair |α∗ and α∗ |. 1 Recall

discussions in Sec. 1.5.

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The biorthogonal completeness relation may be written as

α

α∗

|α α| = 1,

(6.9)

|α∗ α∗ | = 1.

(6.10)

The following bi-orthogonal expansion holds |α∗i = α∗i | =

αj

αj

αj | α∗i |αj ,

(6.11)

α∗i | αj

αj | .

(6.12)

α∗j αi

α∗j ,

αi | α∗j

α∗j .

We also have |αi =

αi | =

α∗

α

From these expansions, Eqs. (6.11)-(6.10), we see that either the αj | α∗i ’s or the α∗j αi ’s cannot be made zero or that either the α-state space or α∗ -state space cannot be made orthogonal, since these are transformation functions. These functions satisfy the canonical principle that a collapsed state in one canonical space signifies a random distribution of finding this collapsed state in the other canonical topological space. This is very much like the situation found in Fourier transformation from the momentum eigenfunction space to coordinate eigenfunction space and vice versa. 6.4

Coherent State Bases

The formulation of coherent states is a mixed q-p representation of quantum mechanics since the theoretical framework is based on the use of the α∗i | and |α belonging to diﬀerent, although canonical bases. We shall see however that with proper normalization the following completeness relations can be obtained

α

|α α∗ | = 1.

(6.13)

If we drop the asterisk (∗ ) distinction of the eigenket and eigenbra, then any of the ‘α-space’ eigenvector can be expanded in terms of all the other α-space eigenvectors. This means that the set of coherent state eigenvectors in α-space is overcomplete.

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It is probably because the classical counterpart of the annihilation operator a ˆ, namely, φ, fully describes the classical state of a harmonic oscillator that attention was focused on the eigenvalue and eigenfunction of corresponding annihilation operator a ˆ. Moreover, there are more revealing physical picture and calculational advantages of the coherent state formulation that surprisingly the major theoretical development happens only in the last four decades, i.e., much later in the development of quantum mechanics. This formulation has played a prominent role in quantum optics. Other than the lack of orthogonality, the use of this pair of topological vector spaces has several advantages. We will discuss these in more detail in what follows.

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Chapter 7

Coherent States Formulation of Quantum Mechanics

Coherent state is defined to be the right eigenstate of the annihilation operator a ˆ, a ˆ |α = α |α .

(7.1)

Since a ˆ is non-Hermitian, α = |α| eiθ is complex, |α| and θ are the amplitude and phase of the eigenvalue. The conjugate state α| is the left eigenstate of the creation operator a ˆ† , this follows by taking the Hermitian conjugate of Eq. (7.1). The state |α is the so-called coherent state. The usefulness of coherent states is that they form a basis for the representation of other states. Coherent states can never be made orthogonal, although for well-separated eigenvalues α, they can be made approximately orthogonal. Moreover, the set of coherent states is overcomplete, in the sense that the set of coherent states form a basis but are not linearly independent, i.e., they are expressible in terms of each other. The nice thing is that the complex eigenvalue α is labeled by the classical (average) values of position and momentum in the following sense, α| a ˆ |α = α =

i mω α| Q + P 2| mω

α| Q |α =

α| P |α =

2| Re α, mω

√ 2|mω Im α,

1 ˜ |α , Re α = √ α| Q 2 1 Im α = √ α| P˜ |α , 2 68

|α ,

(7.2)

(7.3)

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˜ and P˜ are the scaled canonical operators given by Eq. (6.4). Indeed, we where Q have 1 ˜ |α + i α| P˜ |α α= √ α| Q 2 = α| a ˆ |α . (7.4) Observe that the counting of α states can be derived from the counting of states in (pc , qc ) phase-space, where qc = α| Q |α and pc = α| P |α , namely, 1 1 dqc dpc = d Re α d Im α 2π| π 1 ⇒ dα2 (abbreviated). π We also note that |α|2 = (Re α)2 + (Im α)2

mω α| P |α α| Q |α 2 + 2| 2|mω mω 2 p2c q + . = 2| c 2|mω

2

=

(7.5)

It is easy to see that 1 mω 2 P2 Q + |α − 2| 2|mω 2 1 1 † = α| a ˆ a ˆ + |α − 2 2 ˆ |α = n , = α| a ˆ† a

2 |α| = α|

since mω 2 1 mω α| Q |α 2 = α| Q |α − , 2| 2| 4 P2 1 α| P |α 2 = α| |α − . 2|mω 2|mω 4 Moreover, coherent states are minimum uncertainty states, i.e., they provide equality in the Heisenberg uncertainty relation. Using the commutation relation of a ˆ and a ˆ† , we have 2 α| (∆Q) |α =

| , 2mω

(7.6)

α| (∆P )2 |α =

|mω , 2

(7.7)

which leads to 2 2 α| (∆Q) |α α| (∆P ) |α =

|2 . 4

(7.8)

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Since these relations are independent of the value of α, the minimum uncertainty in the value of momentum and position has the same value as in the vacuum state, ∆q ∆p = |2 . This nonzero action multiplied by the oscillator frequency ω yields the vacuum fluctuation energy, |ω 2 . Since the energy eigenstates, |ψn , form a complete set, we expand |α in the form |α =

∞ n=0

Cn |ψn ,

where the Cn are complex numbers to be determined. Substituting the expansion of |α in Eq. (7.1), we obtain upon equating coeﬃcients of ψn−1 the recursion formula for Cn α Cn = √ Cn−1 n αn Co , (n!)

= so that |α = Co

∞

αn √ |ψn . n! n=0

We can take Co to be real and positive so that it can be completely determined by the requirement that |α be normalized to unity. Using the orthogonality of the set of |ψn in the equation 2

α| α = |Co |

= |Co |2

∞

∞

α∗n αm √ √ ψn |ψm = 1 n! m! n=0 m=0 ∞

|α|2n = |Co |2 exp |α|2 , n! n=0

we arrived at |Co | = Co = exp −

|α|2 2

.

Hence |α|2 |α = exp − 2

∞

αn √ |ψn n! n=0

(7.9)

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71

and |α| 2

exp

2

|α = =

∞

αn √ |ψ n n! n=0 ∞

rn einθ √ |ψn , n! n=0

therefore ∞

−r2

=e

∞

αn α∗n √ √ |ψn ψn | n! n ! n=0 n =0

|α α| = exp − |α|2 ∞

∞

rn+n ei(n−n )θ √ √ |ψn ψn | . n! n ! n=0 n =0

(7.10)

From Eq. (7.9), we can also obtain the expression of |ψn and ψn | in terms of coherent states 1 ∂n |ψn = √ n! ∂αn n

1 ∂ = √ n! ∂rn

|ψn

2

|α e−inθ exp

√ n! ∂ n = exp n! rn

1 ∂n ψn | = √ n ! ∂α∗n n

1 ∂ = √ n ! ∂rn so that we also have √ √ n! n ! |ψn ψn | = (n + n )!

∂ ∂r

|α| 2

|α exp

α⇒0 2

|α| 2

, α⇒0

2

|α| 2

α| exp

e−inθ |α |α|2 2 2

α| ein θ exp

|α| 2

, r⇒0

α∗ ⇒0

, r⇒0

n+n

|α α| exp |α|2 + i (n − n) θ

.

(7.11)

r⇒0

Equation (7.11)is a very useful relation later on in proving the diagonal coherent state representation of density matrix operator. Note that for α = 0, the coherent state |α = 0 can be regarded as the vacuum state |ψ0 . The probability P (n) of finding n quantized excitations in coherent state |α is thus given by P (n) = | ψn |α |2 =

|α|2n exp − |α|2 . n!

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Noting that |α|2 = α| a ˆ† a ˆ |α = n , reflecting the quantization of vibrational amplitude in phase space, we are lead to P (n) =

nn exp {− n } , n!

which is a Poisonnian distribution of the quantized excitations.

7.1

Non-Orthogonality of Coherent States

Note, however that for α = α , the following inner product cannot be made to vanish, expressing the non-orthogonality and over-completeness of the coherent states,

α | α = exp −

|α|2 2

|α|2 = exp − 2 = exp −

exp −

|α |2 2 2

|α | exp − 2

∞

∞

α ∗n αm √ √ ψn |ψm n! m! n=0 m=0 ∞

(α ∗ α)n n! n=0

2 2 |α| − 2α ∗ α + |α | 2

= exp −

|α − α |2 2

= exp −

|α − α | 2

exp

αα ∗ − α α∗ 2

2

exp {i Im αα ∗ } .

Thus, we have 2

2

| α | α | = exp − |α − α |

,

(7.12)

which can never be made to vanish for any set of values for α and α . Note that for α = 0, we have

0| α = exp −

|α|2 2

,

which states that the excited coherent states are not orthogonal to the vacuum state. Notwithstanding this non-orthogonality, coherent states span the whole Hspace of state vectors and form convenient basis for the representation of other H-space states.

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73

Completeness of Coherent States

The completeness relation can be seen as follows. Employing the proper counting of coherent states, we have 1 π

|α α| d2 α =

1 π

|α α| d (Re α) d (Im α) .

Writing α = |α| eiθ = reiθ , so that 1 π

1 |α α| d α = π 2

=

1 π

∞ 2π

(|α α|) r dr dθ 0 0 ∞ 2π

dr 0

∞

∞

n+m+1 2 r e−r √ √ ei(n−m)θ n! m! n=0 m=0

dθ 0

|n

m| ,

where in the last line we make use of the expansion of coherent states in terms of the number states. Integrating with respect to the angle θ, we obtain  ∞  ∞ 2 1 |n n|  |α α| d2 α = 2 dr e−r r2n+1  π n! n=0 0 ∞  ∞ |n n|  = dz e−z z n  n! n=0 0

= =

∞

|n

n=0 ∞ n=0

n|

n!

|n

n!

(7.13)

n| = 1,

(7.14)

thus showing the completeness of the coherent states. Note however that the completeness of the coherent states was derived in Sec. 1.5 without the aid of the occupation number states. 7.3

Generation of Coherent States

Rewriting Eq. (7.9) in terms of the vacuum state, with the use of Eq. (6.8), we have |α = exp −

|α|2 2

= exp −

|α|2 2

∞ n=0

αˆ a† n!

n

|ψ0

exp αˆ a† |ψ0 .

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Inserting the term exp {−α∗ a ˆ} right in front of |ψ0 , which has the eﬀect of multiplying by unity, we obtain a more symmetric form as |α = exp −

|α|2 2

exp αˆ a† exp {−α∗ a ˆ} |ψ0 .

Since the commutator, a ˆ, a ˆ† , is a c-number (i.e., not an operator), we can readily make use of the Campbell-Baker-Hausdorﬀ operator identity, namely, exp αˆ a† exp {−α∗ a ˆ} = exp αˆ a† − α∗ a ˆ exp −

αˆ a† , α∗ a ˆ 2 2

ˆ exp = exp αˆ a† − α∗ a

|α| 2

.

(7.15)

Therefore, we can write1 |α = exp αˆ a† − α∗ a ˆ |ψ0 = D (α) |ψ0 .

(7.16)

Thus D (α) is the displacement operator that generates the coherent state |α from the vacuum state |ψ0 . Note that D (α) is a unitary operator, D (α)† D (α) = D (α) D (α)† = 1, and D (α)† = D (−α). We should point out that the generation of quantum states by some sort of displacement operator from the vacuum state is ubiquitous in quantum mechanics. Indeed, from the identity i ( α| P |α Q − α| Q |α P ) , |

ˆ= αˆ a† − α∗ a

we can also write the displacement operator in terms of the position and momentum operators as D (α) = exp

i ( α| P |α Q − α| Q |α P ) . |

1 Equation (7.15) readily follows from the symmetric form of translation operator in Eq. (1.25), involving the universal canonical operators

i i exp − q · P exp p·Q | | i p·q i = exp − (q · P − p · Q) exp − | | 2 if one substitute the following relations P = i|ˆ a† p = i|α∗ Q=a ˆ q =α

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2| Indeed, for a real-valued α, α| Q |α = mω Re α = r, Im α = √12 α| P |α = 0; in the q-representation the displacement operator, D (α), acts as an argument shift

i D (α) ψ (q) = exp − ( α| Q |α P ) ψ (q) | i = exp − (qc P ) ψ (q) | = ψ (q − qc ) . 7.4

Displacement Operator

(a) The unitary transformation of any function of a ˆ and a ˆ† , f a ˆ, a ˆ† , having power series expansion, yields D (α)† f a ˆ, a ˆ† D (α) = f a ˆ + α, a ˆ† + α ∗ . (b) The product of two displacement operators is another displacement operator a† − α∗ a ˆ exp α a ˆ† − α ∗ a ˆ D (α) D (α ) = exp αˆ = exp (α + α ) a ˆ† − (α∗ + α ∗ ) a ˆ exp

αα ∗ − α∗ α 2

αα ∗ − α∗ α D (α + α ) 2 = exp {Im αα ∗ } D (α + α ) .

= exp

(c) The trace of D (α) is T rD (α) =

|α|2 1 exp − π 2

d Re α d Im α α | exp αˆ a† exp {−α∗ a ˆ} |α

=

|α|2 1 exp − π 2

d Re α d Im α exp {αα ∗ } exp {−α∗ α }

=

|α|2 1 exp − π 2

d Re α d Im α exp {αα ∗ − α∗ α }

=

|α|2 1 exp − π 2

d Re α d Im α exp {2 Im αα ∗ }

=

|α|2 1 exp − π 2

d Re α d Im α exp {2i Im α Re α ∗ } exp {2i Re α Im α ∗ }

=

|α|2 1 exp − π 2

= πδ 2 (α) .

πδ (Im α) πδ (Re α)

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(d) Two displacement operators are orthogonal, in the sense of the trace T r D (α) D† (β) = T r exp {−2 Im αβ ∗ } D (α − β) = exp −2 Im |α|

2

δ 2 (α − β)

= πδ 2 (α − β) . 7.5

Linear Dependence of Coherent States

Some of the properties of the displacement operator is useful for example in demonstrating the linear dependence of coherent states. For example, we may write |α =

1 π

|α α| α d2 α.

We can evaluate α| α readily from the property of D (α)† D (α ) = D (−α) D (α ) , α| α = 0| D (−α) D (α ) |0 α∗ α − αα ∗ = exp 0| D (α − α) |0 2 α∗ α − αα ∗ = exp 0| α − α 2 = exp {i Im α∗ α } exp −

|α − α|2 2

.

Therefore, we have |α =

1 π

exp {i Im α∗ α } exp −

|α − α|2 2

|α d2 α.

(7.17)

We have therefore obtained an expansion of one of the coherent states in terms of all of them, showing the linear dependence. The set of coherent states are said to be over-complete since the states form a basis in H-space and yet are not linearly independent. Moreover, from Eq. (7.17), we can form the following expansion or resolution of zero, 2

|α − α| 1 exp {i Im α∗ α } exp − π 2

− δ (α − α ) |α d2 α = 0.

We also have the following expression which can easily be proved by writing α = reiθ and integrating over θ, namely, αn |α d2 α = 0. Thus any of these two resolution of zero can be added to the expansion of a quantum state in terms of coherent states, rendering any expansion not unique.

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77

General Completeness Relation for States Generated by the Displacement Operator

One can also show that for a given normalizable quantum states, |ref , referred to here as a fixed reference state, the displaced or prepared states formed by |α, ref = D (α) |ref also form a complete set for that fixed |ref . The proof makes use the completeness ˆ of the coherent states above and the following results arbitrary operator O, ˆ= O α

α

ˆ |α |α α| O

α |.

Upon taking the diagonal element, ˆ |β = β| O

α

α

ˆ |α β |α α| O

α|β .

(7.18)

ˆ is an identity operator, i.e., The results can only be made unity if and only if O ˆ |β = 1 if and only if O ˆ = I. Now we are ready to prove that the set of β| O ˆ be {|α, ref } forms a complete set. Let O ˆ= 1 O π 1 = π

d2 α |α, ref

α, ref |

d2 αD (α) |ref

ref | D† (α) .

(7.19)

ˆ Now take the diagonal elements of O 1 π 1 = π

ˆ |β = β| O

d2 α β| D (α) |ref

ref | D† (α) |β

d2 α 0| D† (β) D (α) |ref

ref | D† (α) D (α) |0 .

We make use of the algebraic property of the displacement operator D (α) and obtain ˆ |β β| O β ∗ α − βα∗ α∗ β − αβ ∗ 1 d2 α 0| exp D (α − β)|ref ref| exp D (β − α)|0 = π 2 2 1 d2 α 0| D† (β − α) |ref ref | D (β − α) |0 = π 1 d2 α ref| D (β − α) |0 0| D† (β − α) |ref . = π

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Changing the variable of integration, we finally end up with ˆ |β = 1 β| O π

d2 α ref | |α α| |ref

1 d2 α |α α| |ref π = ref | ref = 1,

= ref |

(7.20)

since the reference state |ref is normalizable. Thus, from the unity condition for Eq. (7.18) stated above, we have prove the completeness relation for the state |α, ref = D (α) |ref by virtue of Eq. (7.20). The use of reference number state, |α, ref = D (α) |n has been found fruitful in quantum optics. We note that calculating matrix elements between coherent states is greatly facilitated if the operators are arranged in normal order, meaning that the creation operators always standing to the left of the annihilation operators. Then the expectation value of normally-ordered operator is obtained by simply replacing all creation and annihilation operators by their left and right eigenvalues, respectively.

7.7

Coordinate Representation of a Coherent State

The eigenvalue equation for the annihilation operator, Eq. (7.1), can be solved in the coordinate representation. We have q| a ˆ |α = α q| α . Upon expressing the annihilation operator, a ˆ, in terms of the position and momentum operators and noting that P q| α = P ψα (q) = −i|∇q ψα (q), we obtained the following first-order diﬀerential equation for the wavefunction ψα (q), 1 ∂ mωq + | ψα (q) = αψα (q) . 2|mω ∂q The general solution may be written as   mω 2| q− α ψα (q) = A exp −  2| mω = A exp −

mω (q − qc )2 exp 2|

2

  

i pc · (q − qc ) exp |

1 p2 , 2|mω c

where the last line made use of Eqs. (6.4) and (7.4). The normalization constant A is determined from the equation 2 |ψα (q)| dq = 1,

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which yield A=

1 4

mω π|

exp −

1 p2 . 2|mω c

Thus, we have ψ α (q) =

mω π|

1 4

exp −

mω (q − qc )2 exp 2|

i pc · (q − qc ) , |

which reduces to the vacuum energy eigenfunction, ψ0 (q), of Eq. (6.1). The probability distribution in position space is mω mω exp − (q − qc )2 . π| |

|ψα (q)|2 =

(7.21)

This expression represents a Gaussian wavepacket with the center following the classical coordinates, and moreover, the wavepacket does not change its shape. The width of the wavepacket is given by ∆qc =

| . mω

In the momentum representation, similar result is obtained with the width of the corresponding wavepacket about the classical momentum value given by √ ∆pc = |mω. Thus, the uncertainty about the classical value is given by ∆qc ∆pc = | >

| . 2

The time evolution for α is the same as for the classical φ of Eq. (2.3). 7.8

The Power of Coherent State Representation and the Virtue of Over-Completeness

The power of the coherent state representation lies in being able to have a representation of quantum states and operators by entire functions that is unique. This uniqueness of analytic continuation of entire functions is a rather amazing and extremely powerful statement. It says in eﬀect that knowing the value of a complex function in some finite complex domain uniquely determines the value of the function at every other point. It follows that, in the coherent state representation, |Ψ is completely determined by its matrix elements α| Ψ within some arbitrarily small but finite range of α∗ . Consider |Ψ =

1 π

d2 α α| Ψ |α .

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Using the number state expansion of α|, we have

α| Ψ = exp −

|α|2 2

∞

α∗n √ n| Ψ . n! n=0

(7.22)

Since | n| Ψ | ≤ 1, the series on the right hand side is absolutely convergent for all |Ψ and α∗ . Thus, we can represent this series in Eq. (7.22) by an entire analytic function of α∗ ∞

α∗n √ n| Ψ = exp n! n=0

2

|α| 2

α| Ψ

= FΨ (α∗ ) . 2

α| Ψ vanishes for Therefore if the entire analytic function FΨ (α∗ ) = exp |α| 2 ∗ some arbitrarily small but finite range of α , this is possible only for null abstract vector |Ψ , thus completely determining |Ψ throughout all the ranges of α∗ . Let us also consider the corresponding coherent state representation of a traceˆ able, positive definite Hermitian operator O, 1 π

ˆ= O

2

ˆ |α d2 αd2 α |α α| O

α |.

Again, using the number state expansion of α| and |α , we have 2

ˆ |α = exp − |α| α| O 2

exp −

|α |2 2

∞

∞

α∗n α m ˆ |m . √ √ n| O n! m! n=0 m=0

(7.23)

ˆ |m has an upper bound for the double sum to be an We have to show that n| O entire analytic function of two variables, α∗ and α . This can be shown for traceable, ˆ In order to see this, we expand O ˆ in terms of its non-negative definite operator O. own eigenstates |oi . Then we have the expansion ˆ= O i

oi |oi oi | ,

ˆ |m = n| O

i

oi n| oi oi | m

≤

i

oi | n| oi | | oi | m |

≤

i

ˆ oi = T r O,

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ˆ as the upper bound. Thus we can write showing T r O ∞

∞

α∗n α m ˆ |m = FO (α∗ , α ) √ √ n| O n! m! n=0 m=0 = exp −

|α|2 2

exp −

|α |2 2

ˆ |α , α| O

where FO (α∗ , α ) is an entire analytic function of two variables, α∗ and α , and view ˆ By virtue of the function FO (α∗ , α ) as a unique representation of the operator O. ∗ ˆ |α the analyticity of FO (α , α ), it is suﬃcient for the diagonal matrix element α| O ∗ ˆ to be known for some α and α for the entire operator O to be determined. For ˆ |α = 0 for some small range of α∗ and α , then it must vanish for example if α| O ∗ ˆ |α for some small range of α, all range of α and α . Hence if α| Aˆ |α = α| B ∗ ˆ This result has ˆ ˆ then α| A − B |α = 0 for all range of α and α and hence Aˆ = B. no counterpart in any representation based on complete set of states. ˆ in some From Eq. (7.23), we can derive the matrix elements of the operator O ˆ other representation explicitly from the matrix α| O |α . Treating α and α∗ as independent variables we have ∂ n+m ˆ |m = √1 √1 ˆ |α exp |α|2 n| O α| O ∗n ∂αm ∂α n! m!

. α=0 α∗ =0

Thus the diagonal matrix element in the coherent state representation is suﬃcient to ˆ |m and it is only necessary to know α| O ˆ |α in generate all matrix elements n| O ˆ the neighborhood of α = 0. Moreover, once all the n| O |m has been determined, we can substitute in Eq. (7.23) and calculate any oﬀ-diagonal matrix element ˆ |α . α| O

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Density-Matrix Operator and Quasi-Probability Density

The density-matrix operator satisfies the condition for it to be represented by an entire analytic function in the coherent state representation. Let ρ ˆ be the density matrix operator. Rather than writing the longer form ρ ˆ = 1 2 π

d2 αd2 α |α α| ρ ˆ |α

α |, it maybe possible to represent ρ ˆ in terms of co-

herent state projector |α α|, i.e., in terms of its diagonal element ˆρ =

σ (α) |α α| d2 α,

(8.1)

in which σ (α) is a real function of α. This density-matrix operator representation suggests that the state of the harmonic oscillator (or any classical field being quantized) may be regarded as a mixture of coherent states with relative weight σ (α), which is real since ˆρ is Hermitian. Since the coherent state |α is analogous to the classical field of complex amplitude, α, of vibration, the pure state is represented by σ (α) as a delta function. Since ρ ˆ have a unit trace, we have

T rˆ ρ = Tr =

σ (α) |α α| d2 α

σ (α) d2 α = 1.

Thus, σ (α) is the probability density in phase space. Note however, σ (α) does not describe probabilities of mutually exclusive states since the diﬀerent coherent states are not orthogonal. In general σ (α) may take on negative values just like the Wigner distribution function, therefore σ (α) is also called a quasi-probability density. The classical limit is defined to be the case when σ (α) behaved like a true probability density. Although, for certain states of the field, namely, the highly non-classical states, σ (α) can take negative values and become more singular than the delta function, it has one very useful advantage over all the other phase space distribution since it allows immediate determination whether the state of the field has a classical description or not. 82

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Diagonal Representation of Density-Matrix Operator

The diagonal representation of the density-matrix operator has two principal advantages, namely, (a) the behavior of σ (α), whether it behaves like a probability density or has a non-classical behavior in the sense that it becomes negative or more singular than a delta function, allows us to determine the classical and non-classical behavior of the quantized classical field, for example light emitted from lasers has a classical analog, because in this case σ (α) is a probability density; (b) the representation, Eq. (8.1), allows us to calculate expectation values of normally-ordered operators (in which the creation operators always stand to the left of annihilation operators) in a way an average is calculated in classical mechanics. Although, this is not a unique to σ (α), since the Wigner distribution function allows classical calculation of averages, σ (α) has is distinguished from other phase space densities since it coincides with corresponding classical probability density whenever a classical description of the state of the quantized field exists. For classical behavior, σ (α) is non-negative. Hence, we can have a very simple test for recognizing a non-classical state. For example, given a real positive function of α, say f (α), then σ (α) f (α) d2 α can vanish only if σ (α) represents a non-classical state. To be specific, let us consider the probability of n photons, p (n), in the quantum state represented by the density-matrix operator ρ ˆ. Then we have, making use of Eqs. (7.9) and (8.1), p (n) = T r (ˆ ρ |n n|)

2

= Tr

σ (α) n| |α α| |n d2 α

= Tr

σ (α) e−|α|

2

|α|2n 2 d α . n!

2n

Since e−|α| |α|n! > 0, when α = 0, then for p (n) = 0 for any n we must have σ (α) behaving like a probability density. For the pure vacuum quantum state σ (α) = δ (α) and p (0) = 1. Thus, except for the quantum mechanical vacuum state, any state for which p (n) = 0 has no classical analog and is purely quantum mechanical. This lack of classical analog is the reflection of the understanding that the coherent state |α contains contributions from all Fock states (single mode for a harmonic oscillator), |n , so that p (n) = 0 is beyond the realm of classical understanding. Consider any normally ordered function gN a ˆ, a ˆ† of the annihilation and cre† ation operators a ˆ and a ˆ . Let gN a ˆ, a ˆ† =

cn,m a ˆ†n a ˆm . n,m

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Then the expectation value of g N a ˆ, a ˆ† in the state characterized by the density operator ˆρ is given by ˆ, a ˆ† gN a

= Tr ρ ˆg N a ˆ, a ˆ†

.

Using the diagonal representation of ρ ˆ, we have ˆ, a ˆ† gN a

= Tr

σ (α) n,m

=

σ (α) n,m

cn,m |α α| a ˆ†n a ˆm d2 α

cn,m α| a ˆ†n a ˆm |α d2 α.

ˆ, we have Since α| is the left eigenstate of a ˆ† and |α is the right eigenstate of a ˆ, a ˆ† gN a

=

cn,m α†n αm d2 α

σ (α) n,m

=

σ (α) g N α, α† d2 α

= g N α, α†

ensemble

.

Thus, if α is viewed as a random complex variable with probability density σ (α) then the classical expectation value of the function g N α, α† over the ensemble will ˆ, a ˆ† = g N α, α† ensemble , be denoted as g N α, α† ensemble , so that g N a that is the quantum mechanical average is equal to the classical ensemble average. Example of this is the number operator g N a ˆ. We have ˆ, a ˆ† = a ˆ† a ˆ = n ˆ = αα∗ = |α|2 . For coherent state |α , we have the density operaa ˆ† a tor, ρ ˆ = δ (α − α ) |α

α |, and hence n ˆ =

δ (α − α ) α | a ˆ† a ˆ |α d2 α = |α |2 ,

2 i.e., the quantum average equals the ‘classical’ value of |α| in a state |α .

8.2

Procedures for Determining σ (α)

From Eq. (7.14), writing α = reiθ , we note that the integral with respect to θ gives 2π

1 2π

(|α α|) dθ 0 2π

1 = 2π

dθ 0

−r2

=e

∞

∞

∞

2 rn+m e−r √ √ ei(n−m)θ n! m! n=0 m=0

r2n |n n! n=0

n| .

|n

m|

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A more useful expression is obtained from Eq. (7.10) as ∞

∞

rn+m √ √ ei(n−m)θ n! m! n=0 m=0

2

er |α α| =

|n

m| .

Multiplying this by eikθ and integrating over θ, we have 1 2π

2

er eikθ |α α| dθ ∞

∞

∞

∞

rn+m 1 √ √ n! m! 2π n=0 m=0

=

ei(k+n−m)θ dθ

rn+m √ √ δ (k + n − m) n! m! n=0 m=0

=

|n

|n

m|

m| .

Diﬀerentiating both sides with respect to r, we have l

d dr

2

er eikθ |α α| d dr

=

rn+m √ √ δ (k + n − m) n! m! n=0 m=0

|n

m|

rn+m−l (n + m)! √ √ δ (k + n − m) n! m! (n + m − l)! n=0 m=0

|n

∞

=

∞

∞

l

dθ 2π

∞

m| .

On evaluating the above equation at r = 0, the only surviving term in the right-hand side is for n + m − l = 0. Thus, we can write d dr

l

2

er eikθ |α α| =

∞

dθ 2π

r⇒0

∞

1 (n + m)! √ √ δ (n + m − l) δ (k + n − m) (n n! m! + m − l)! n=0 m=0

|n

m| .

From this we have √ √ n! m! (n + m)!

∂ ∂r

n+m

2π 2

(|α α|) er ei(m−n)θ dθ 0

= |n

r⇒0

m| .

Using the following identity for the integral involving the derivatives of delta function, namely, f (x)

∂ ∂x

n

δ (x) = (−1)n

∂ ∂x

n

f (x) δ (x) .

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We may also write as 2π

|n

m| =

2πr dr

√ √ n! m! (|α α|) dθ (n + m)!

0 r2

×

e ei(m−n)θ 2πr

(−1)n+m 

d2 α (|α α|) 

=

×

∂ ∂r

n+m

δ (r)

√ √ 2 n! m! er i(m−n)θ (n+m)! 2πr e ∂ n+m (−1)n+m ∂r δ (r)



.

Now the density matrix operator can be expanded in terms of the number states or Fock representation, ρ ˆ= =

∞

∞

n=0 n =0 ∞ ∞ n=0 n =0

ρ (n, n ) |n n| ρ |n |n

n| n |.

From the relation given in Eq. (7.11), we can further write the last line as ˆρ =

∞

∞

n=0 n =0

=

d2 α

  

n| ρ |n |n ∞

∞

n=0 n =0

× (|α α|) =

n|



n| ρ |n 

×

√ √ 2 n! m! er i(m−n)θ (n+m)! 2πr e ∂ n+m (−1)n+m ∂r δ (r)

   

d2 α σ (α) |α α| ,

clearly showing the density-matrix operator in diagonal form in the coherent state representation, where the diagonal element is given by the expression    √ √ 2 n! m! er i(m−n)θ ∞ ∞  e (n+m)! 2πr  σ (α) = n| ρ |n  . (8.2) ∂ n+m  × (−1)n+m ∂r δ (r)  n=0 n =0

This explicit expression for σ (α) was first derived by Sudarshan1 . The explicit expression for the weight factor σ (α) is in general not a well-behaved function and can be interpreted only in the sense of generalized function theory by virtue of the presence of the derivative of the delta function. It is therefore desirable to to give some other explicit expression for σ (α) which yield well-behaved function whenever possible. First let us make a digression on operator algebra. 1 E.

C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).

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Chapter 9

Operator Algebra

Since quantum mechanics deals with noncommutative operators as the dynamical variables, it is desirable to be able to manipulate quantum operators to solve quantum mechanical problems. Thus, we need to study some operator algebra. 9.1

General Operators

We start here with general operator identities. First we define the concept of function of operators. Any function F (B) of operator B is tacitly assume that F (B) may be expanded in a power series F (B) =

∞

cn B n ,

n=0

with c-number expansion coeﬃcients cn . It follows that in terms of the eigenvalues b of B the corresponding c-number function F (b) =

∞

cn bn must converge to be

n=0

meaningful. The following identities involving operators hold. (1) If O and B are two noncommuting operators, and if the inverse O−1 exists, then OB m O−1 = OBO−1

m

,

OF (B)O−1 = F OBO−1 . The proof makes use of the identity OO−1 = 1 and writing OB m O−1 as product of m factors OBO−1 OBO−1 ...OBO−1−1 = OBO−1

m

.

The second relation follows from the power series expansion of F (B). (2) Let O = exp {ξA} where ξ is a c-number. Then we have exp {ξA} B m exp {−ξA} = (exp {ξA} B exp {−ξA})m ,

exp {ξA} F (B) exp {−ξA} = F (exp {ξA} B exp {−ξA}) . 87

(9.1)

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Example 9.1

Let F (B) = exp B, then we have

exp {ξA} [exp B] exp {−ξA} = exp {exp {ξA} B exp {−ξA}} Example 9.2 Let F (B) = Q and let exp {ξA} = exp |i ξP , where Q is the position operator and P is the momentum operator satisfying [Q, P ] = i|. Then, from Eq. (1.24) we have exp

exp

i ξP |

i ξP |

i Q exp − ξP |

i F (Q) exp − ξP |

=Q+ξ

(9.2)

= F (Q + ξ)

A more general way of deriving the last line makes use of Eq. (1.12). Let f (ξ, P, Q) = exp |i ξP F (Q) exp − |i ξP with f (0, P, Q) = F (Q). Then by diﬀerentiating with respect to ξ we have ∂ i f (ξ, P, Q) = [P f (ξ, P, Q) − f (ξ, P, Q) P ] ∂ξ | i = [P, f (ξ, P, Q)] |

(9.3)

By the use of Eq. (1.12), the right-hand side can be reduced to diﬀerentiation with respect to Q, thus we have the diﬀerential equation for f (ξ, P, Q) given by, ∂ ∂ f (ξ, P, Q) = f (ξ, P, Q) ∂ξ ∂Q with the boundary condition f (0, P, Q) = F (Q). It immediately follows that f (ξ, P, Q) = F (Q + ξ). Note that by iterating Eq. (9.3), using Eq. (1.12) to reduce to diﬀerentation with respect to Q, we have ∂2 f (ξ, P, Q) = ∂ξ 2

2

i [P, [P, f (ξ, P, Q)]] | ∂2 = f (ξ, P, Q) ∂Q2

∂n f (ξ, P, Q) = ∂ξ n

n

i P (1) , P (2) , .... P (n) , f (ξ, P, Q) | ∂n f (ξ, P, Q) = ∂Qn

(9.4)

etc., where the superscripts counts the order of ocurrence of the operator P . Thus, using the Maclaurin series expansion of f (ξ, P, Q), noting that

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∂n ∂ξn f

(ξ, P, Q)

ξ=0

=

i n |

89

P (1) , P (2) , .... P (n) , f (0, P, Q) , we have

ξ2 i f (ξ, P, Q) = F (Q) + ξ [P, f (0, P, Q)] + | 2!

i |

2

i ξ2 f (ξ, P, Q) = F (Q) + ξ [P, F (Q)] + | 2!

i |

2

f (ξ, P, Q) = F (Q) + ξ

[P, [P, f (0, P, Q)]] + ..

[P, [P, F (Q)]] + ..

ξ2 ∂ 2 ∂ F (Q) + F (Q) + .. ∂Q 2! ∂Q2

= F (Q + ξ) as before. (3) If A and B are two noncommuting operators, ξ a c-number and f (ξ, A, B) = exp {ξA} B exp {−ξA}, f (0, A, B) = B, then from Eq. (9.4) and expanding f (ξ, A, B) in a Maclaurin series in powers of ξ, we have f (ξ, A, B) = B + ξ [A, B] +

ξ3 ξ2 [A, [A, B]] + [A, [A, [A, B]]] + .... 2! 3!

Some interesting cases occur when the series terminates, as when we let A = |i P and B = Q. Another case is when we let ξ = |i t, A = V (Q) and B = P = ∂ , then we have −i| ∂Q [A, B] = [V (Q) , P ] = i|

∂ V (Q) , Q

[A, [A, B]] = 0, which leads to 1 1 exp i V (Q) t P exp −i V (Q) t | |

=P −

∂ V (Q) t. Q

We also have for some function of P , say F (P ), the following expression, 1 1 exp i V (Q) t F (P ) exp −i V (Q) t | | t Now if we let ξ = i |m ,A=

P2 2

=F

P−

and B = Q, then we have

[A, B] =

P2 , Q = −i|P, 2

[A, [A, B]] = 0,

∂ V (Q) t . Q

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which leads to exp i

1 P2 | 2m

Q exp −i

1 P2 | 2m

=Q+

P t. m

Similarly, we have exp i

1 P2 | 2m

V (Q) exp −i

1 P2 | 2m

=V

Q+

P t . m

(4) Campbell-Baker-Hausdorﬀ (CBH) operator identity: if A and B are two noncommuting operators that satisfy the conditions, [A, [A, B]] = [B, [A, B]] = 0, then 1 exp {A + B} = exp {A} exp {B} exp − [A, B] 2 1 [A, B] . = exp {B} exp {A} exp 2

(9.5)

That the condition, [A, [A, B]] = [B, [A, B]] = 0, are satisfied by two operators whose commutator is a c-number, such as [Q, P ] = i| and a ˆ, a ˆ† = 1. To see how the CBH operator identity arise, let us consider the following operator function f (ξ, A, B) = exp {ξA} exp {ξB} , where again ξ is a c-number. Diﬀerentiating with respect to ξ, we have ∂ f (ξ, A, B) = A exp {ξA} exp {ξB} + exp {ξA} exp {ξB} B ∂ξ = [A + exp {ξA} B exp {−ξA}] exp {ξA} exp {ξB} = [A + exp {ξA} B exp {−ξA}] f (ξ, A, B) .

(9.6)

Using the condition that the commutator of A and B is a c-number, we have exp {ξA} B exp {−ξA} = B + ξ [A, B] . Thus we can write Eq. (9.6) as ∂ f (ξ, A, B) = {A + B + ξ [A, B]} f (ξ, A, B) . ∂ξ

(9.7)

From the condition, [A, [A, B]] = [B, [A, B]] = 0, we see that [(A + B) , [A, B]] = 0, so that we can treat (A + B) and [A, B] as two ordinary commuting variables. We can thus readily integrate Eq. (9.7) and obtain the solution, obeying the boundary condition f (0, A, B) = 1, as f (ξ, A, B) = exp (A + B) ξ +

ξ2 [A, B] , 2

exp {ξA} exp {ξB} = exp {(A + B) ξ} exp

ξ2 [A, B] . 2

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91

Since this is true for arbitrary ξ, we can take ξ = 1. We therefore have, exp {A} exp {B} = exp {(A + B)} exp

1 [A, B] . 2

Upon multiplying both sides by exp − 12 [A, B] we have 1 exp {(A + B)} = exp {A} exp {B} exp − [A, B] . 2 Upon interchanging the operators A and B, we also have exp {(A + B)} = exp {B} exp {A} exp Example 9.3 have

Let A =

i |

(p − P ) · v and B =

i |

1 [A, B] . 2 q − 12 v − Q · u. Then we

i i 1 (p − P ) · v exp q− v−Q ·u | | 2 i i 1 (p − P ) · v + = exp q− v−Q ·u | | 2 i i 1 (p − P ) · v , × exp q− v−Q ·u | | 2 i i (p − P ) · v + (q − Q) · u = exp | | exp

Thus, we have changed the seemingly asymmetric expression of the first line into a symmetric form of the last line.

9.2

Boson Annihilation and Creation Operators, Ordering

The solutions of quantum problems involving annihilation and creation operators are often made possible by means of the more powerful operator techniques. These techniques involve the concepts of ordered operators, which have a one-to-one correspondence with ordinary c-numbers. Any function f a ˆ, a ˆ† of boson operators a ˆ, a ˆ† = 1 is generally ˆ and a ˆ† with a defined by its power series expansion f a ˆ, a ˆ† =

f (j, k, l, m, ....n) a ˆ†j a ˆk a ˆ†l a ˆ†m ....ˆ an ,

...... j

n

j, k, l, m, ..n = 01, 2, 3, .... A function of operator is in normal ordering if all the a ˆ’s in every term in the expansion are to the right of all a ˆ† ’s. We are of course always free to rearrange the operators to normal order by the use of the commutation relation, a ˆa ˆ† = a ˆ† a ˆ + 1.

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Thus we may write the resulting ordered function, f (n) a ˆ, a ˆ† , as ˆ, a ˆ† = f a ˆ, a ˆ† = f (n) a

(n) †r s fr,s a ˆ a ˆ , r,s

(n)

ˆ and a ˆ† . For examwhere fr,s are expansion coeﬃcients which are independent of a † † † † † † † † ˆ a ˆa ˆ a ˆa ˆ a ˆ a ˆ a ˆ a ˆ, then ˆ a ple, if f a ˆ, a ˆ =a ˆ† a ˆa ˆ† a ˆa ˆ† a ˆ† a ˆ† a ˆ† a ˆ f a ˆ, a ˆ† = a ˆ† a ˆ† a ˆ† a ˆ† a ˆ† a ˆ† a ˆ† a ˆa ˆa ˆ+9 a ˆ† a ˆ† a ˆ† a ˆ† a ˆ† a ˆ† a ˆa ˆ + 16 a ˆ† a ˆ† a ˆ† a ˆ† a ˆ† a ˆ. = f (n) a ˆ, a ˆ† = a ˆ† a Alternatively, a function of operator is in antinormal ordering if all the a ˆ† ’s in every term in the expansion are to the right of all a ˆ’s. We may write the resulting antinormal ordered function, f (a) a ˆ, a ˆ† , as ˆ, a ˆ† = f a ˆ, a ˆ† = f (a) a

(a) r †s fr,s a ˆ a ˆ . r,s

(n)

(a)

ˆ, a ˆ† = f (n) a In general, fr,s = fr,s although f a ˆ, a ˆ† = f (a) a ˆ, a ˆ† . One can see that in all but few trivial cases, the ordering procedure is a very tedious process. In what follows, some very useful techniques for obtaining the ordered operators by indirect means will be developed. These techniques rely on establishing a one-toone correspondence between either f (n) a ˆ, a ˆ† or f (a) a ˆ, a ˆ† and ordinary functions (n) † (a) † α, α or f α, α of a complex variable α. f To see how this one-to-one correspondence becomes extremely useful in indirectly obtaining ordered operators, let us consider a general set of noncommuting operators, say, a ˆ1 , a ˆ2 , a ˆ3 , .........ˆ af in the Schrödinger picture which obey some set of commutation (boson) or anticommutation (fermion) relations. Let Q (ˆ a1 , a ˆ2 , a ˆ3 , .........ˆ af ) be some function of these operators which may be expanded in power series. We may use the commutation or anticommutation relations, as the case may be, to reorder the terms in Q (ˆ a1 , a ˆ2 , a ˆ3 , .........ˆ af ) into some chosen order. Let this chosen order of operators be a ˆ1 , a ˆ2 , a ˆ3 , a ˆ4 , .........ˆ af , Q (ˆ a1 , a ˆ2 , a ˆ3 , .........ˆ af ) = Q(c) (ˆ a1 , a ˆ2 , a ˆ3 , .........ˆ af ) =

r

Q(c) ˆr11 , a ˆr22 , a ˆr33 , a ˆr44 , .........ˆ aff , r1 .....rf a

....... r1

rf

where the superscript (c) in Q(c) (ˆ a1 , a ˆ2 , a ˆ3 , .........ˆ af ) indicates that the function is in (c) chosen order, which is equal to the function in the original order, and Qr1 .....rf is the coeﬃcient in the expansion in terms of the chosen-ordered series expansion. Once we have Q(c) (ˆ a1 , a ˆ2 , a ˆ3 , .........ˆ af ), and by virtue of the one-to-one correspondence with a c-number function, we may define an associated c-number function by means of ¯ (c) (α1 , α2 , α3 , .........αf ) = Q

r

r1 r2 r3 r4 f Q(c) r1 .....rf α1 , α2 , α3 , α4 , .........αf ,

....... r1

rf

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93

where the operator a ˆi is replaced by αi which is real or complex depending on ¯ (c) helps in indicating that this whether a ˆi is Hermitian or not. The bar in Q function is a c-number function. Thus, we may define a linear transformation operator C −1 which transform Q(c) (ˆ a1 , a ˆ2 , a ˆ3 , .........ˆ af ) to an ordinary function ¯ (c) (α1 , α2 , α3 , .........αf ) by replacing a ˆi by αi , Q ¯ (c) (α1 , α2 , α3 , .........αf ) . a1 , a ˆ2 , a ˆ3 , .........ˆ af ) = Q C −1 Q(c) (ˆ Due to the one-to-one correspondence, we also have ¯ (c) (α1 , α2 , α3 , .........αf ) . a1 , a ˆ2 , a ˆ3 , .........ˆ af ) = C Q Q(c) (ˆ Formally, we may write f

¯ (c) (α1 , α2 , α3 , .........αf ) = CQ

......

¯ (c) (α1 , α2 , α3 , .........αf ) Q i=1

δ (αi − a ˆi ) dαi ,

f

where the δ-functions in i=1

δ (αi − a ˆi ) dαi are arranged in the chosen order. We

may now represent the δ-function by 1 ˆi ) = δ (αi − a 2π

∞

eiξi (αi −ˆai ) dξ i .

−∞

ˆi is non-Hermitian and αi is complex, and the If a ˆi is Hermitian and αi is real. If a set of noncommuting operators a ˆ1 , a ˆ2 , a ˆ3 , .........ˆ af reduces to the set a ˆ, a ˆ† , then the δ-function integration is over the entire complex plane and if the chosen order is a ˆ, a ˆ† then we have δ (α − a ˆ) δ α∗ − a ˆ† =

1 π2

∗ ∗ † eiξ(α−ˆa) eiξ (α −ˆa ) d (Re ξ) d (Im ξ) .

First let us treat the case where a ˆi is Hermitian and αi is real. Then we can write a1 , a ˆ2 , a ˆ3 , .........ˆ af ) Q(c) (ˆ 1 2π

= =

......

¯ (c) (α1 , α2 , α3 , .........αf ) dα1..... dαf Q i=1

...... ×

f

f

......

δ (αi − a ˆi )

¯ (c) (α1 , α2 , α3 , .........αf ) dα1..... dαf Q dξ 1..... dξ f eiξ1 (α1 −ˆa1 ) ......eiξf (αf −ˆaf ) .

Interchanging the order of integration of the ξ’s and α’s, we may write a1 , a ˆ2 , a ˆ3 , .........ˆ af ) = Q(c) (ˆ

......

dξ 1..... dξ f e−iξ1 aˆ1 ......e−iξf aˆf F (c) ξ 1..... ξ f (9.8)

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where F (c) ξ 1..... ξ f is just the Fourier transform of the associated c-number func¯ (c) (α1 , α2 , α3 , .........αf ), tion Q F (c) ξ 1..... ξ f =

1 2π

f

¯ (c) (α1 , α2 , α3 , .........αf ) . (9.9) dα1..... dαf ei(ξ1 α1 +....+ξf αf ) Q

......

Therefore, in parallel with Eq. (9.8), we obtain, upon taking the inverse Fourier transform, ¯ (c) (α1 , α2 , α3 , .........αf ) Q =

......

dξ 1..... dξ f e−i(ξ1 α1 +....+ξf αf ) F (c) ξ 1..... ξ f .

(9.10)

Equation (9.8) may be viewed as an expansion of arbitrary operator cast in chosen order in terms of elementary operator basis, similar to a Fourier expansion of c-number functions. For symmetric order, this type of expansion leads to expansion of operators in terms of generalized Pauli-matrix operators, discussed in later ˆ and a ˆ† chapters. For the case of non-Hermitian set of operators, a ˆ and a ˆ† , where a are boson operators and the chosen order is the normal order, we have ¯ (n) (α, α∗ ) ˆ, a ˆ† = C Q Q(n) a = 1 π2 1 = π =

¯ (n) (α, α∗ ) δ α∗ − a dα2 Q ˆ† δ (α − a ˆ) ¯ (n) (α, α∗ ) dα2 Q dξ 2 e−iξ

∗ †

a ˆ

∗ ∗ † dξ 2 eiξ (α −ˆa ) eiξ(α−ˆa)

e−iξˆa F (n) (ξ, ξ ∗ ) ,

(9.11)

where F (n) (ξ, ξ ∗ ) =

1 π

dα2 ei(ξ

∗

α∗ +ξα)

¯ (n) (α, α∗ ) . Q

(9.12)

F (n) (ξ, ξ ∗ ) .

(9.13)

Corresponding to Eq. (9.11), we also have ¯ (n) (α, α∗ ) = Q

dξ 2 e−i(ξ

∗

α∗ +ξα)

By virtue of the equation pair, Eqs. (9.8) and (9.10), the equation pair, Eqs. (9.11) and (9.13), it is desirable to derive the Fourier transforms, F (c) ξ 1..... ξ f ¯ (c) (α1 , α2 , α3 , .........αf ) and and F (n) (ξ, ξ ∗ ), of c-number associated functions, Q (n) ∗ ¯ Q (α, α ) respectively, from the original function of operators. In what follows, we will focus on deriving F (n) (ξ, ξ ∗ ) and F (a) (ξ, ξ ∗ ) for function of boson annihilation and creation operators, a ˆ and a ˆ† . We note that there is a direct relation between the function of operators and its associated c-number function. This makes use of the completeness relation,

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Eq. (7.14). To show this, it is convenient to use the antinormal ordering of any ˆ, a ˆ† . Using its series expansion, we have function f a ˆ, a ˆ† = f (a) a f a ˆ, a ˆ† = f (a) a ˆ, a ˆ† =

(a) r †s fr,s a ˆ a ˆ . r,s

Upon inserting the completeness relation, we have f a ˆ, a ˆ† =

(a) r †s fr,s a ˆ a ˆ = r,s

1 π

d2 α r,s

(a) r fr,s a ˆ |α α| a ˆ†s

1 = π

d2 α |α α|

1 = π

d2 α |α α| f¯(a) (α, α∗ ) .

(a) r ∗s fr,s α α r,s

(9.14)

Thus, we have 1 π = A f¯(a) (α, α∗ ) ,

ˆ, a ˆ† = f a ˆ, a ˆ† = f (a) a

d2 α |α α| f¯(a) (α, α∗ ) (9.15)

where the transformation operator C ⇒ A to specify antinormal ordering of the set a ˆ, a ˆ† . We will also use N to specify normal ordering of this same set of oper∗ ∗ ∗ † ∗ ∗ ators. For example, N e−i(ξ α +ξα) = e−iξ aˆ e−iξˆa , whereas A e−i(ξ α +ξα) = ∗ † e−iξˆa e−iξ aˆ . 9.2.1

Traces of Function of Boson Operators

One of the virtues of associated c-number function lies in the evaluation of traces of function of a ˆ and a ˆ† . Using Eq. (9.14), we immediately get the expression of the † trace of f a ˆ, a ˆ by noting that the T r |α α| = 1, 1 π

Tr f a ˆ, a ˆ† =

d2 α f¯(a) (α, α∗ ) .

One can also show that the trace may be found by integrating the normal associated c-number function, f¯(n) (α, α∗ ), over the complex α-plane 1 f a ˆ, a ˆ† |α α| d2 α π 1 α| f a ˆ, a ˆ† |α d2 α π 1 α| f (n) a ˆ, a ˆ† |α d2 α π 1 f¯(n) (α, α∗ ) d2 α. π

Tr f a ˆ, a ˆ† = T r = = =

(9.16)

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Note that from Eq. (9.12), we can also write Tr Q a ˆ, a ˆ† = F (n) (ξ = 0, ξ ∗ = 0) = F (a) (ξ = 0, ξ ∗ = 0) . The trace of a product of two functions of a ˆ and a ˆ† can be evaluated as an integral over the complex α-plane of the product of their associated c-number functions. Let f a ˆ, a ˆ† and g a ˆ, a ˆ† be two functions of a ˆ and a ˆ† . To evaluate the trace of their product, it is convenient to express them in their equivalent antinormal and normal ordering, f (n) a ˆ, a ˆ† or f (a) a ˆ, a ˆ† and g (n) a ˆ, a ˆ† or g (a) a ˆ, a ˆ† . Thus, if we (a) † (n) † expand the product of g a ˆ, a ˆ f a ˆ, a ˆ , we have Tr

g a ˆ, a ˆ† f a ˆ, a ˆ†

g(a) a ˆ, a ˆ† f (n) a ˆ, a ˆ†

= Tr

(n)

(a) r †s gr,s a ˆ a ˆ

= Tr r,s

fl,m a ˆ†l a ˆm l,m

(a) (n) gr,s fl,m

=

Tr a ˆr a ˆ†s a ˆ†l a ˆm

r,s l,m (n)

(a) gr,s fl,m T r a ˆ†s+l a ˆm+r .

= r,s l,m

From Eq. (9.16), we have Tr

ˆ, a ˆ† f (n) a ˆ, a ˆ† g (a) a

= r,s l,m

1 (a) (n) gr,s fl,m α∗s+l αm+r d2 α π

(a) ∗s r gr,s α α

= r,s

= r,s

Thus, we have Tr

g a ˆ, a ˆ† f a ˆ, a ˆ†

1 (n) fl,m α∗l αm d2 α π l,m  

(a) r ∗s  gr,s α α

l,m

1 (n) fl,m α∗l αm  d2 α. π

=

1 g(a) (α, α∗ ) f (n) (α, α∗ ) d2 α. π

(9.17)

=

1 g(n) (α, α∗ ) f (a) (α, α∗ ) d2 α. π

(9.18)

Similarly, one can also show that Tr

g a ˆ, a ˆ† f a ˆ, a ˆ†

Now from Eq. (9.12), we have 1 π 1 = π

F (n) (ξ, ξ ∗ ) =

dα2 ei(ξ

∗

α∗ +ξα)

dα2 eiξα eiξ

∗

α∗

¯ (n) (α, α∗ ) Q

¯ (n) (α, α∗ ) . Q

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Therefore from Eq. (9.17), we end up with F (n) (ξ, ξ ∗ ) = T r eiξˆa eiξ

∗ †

a ˆ

Q a ˆ, a ˆ†

Similarly, using Eq. (9.18), we have F (a) (ξ, ξ ∗ ) = T r eiξ

∗ †

a ˆ

eiξˆa Q a ˆ, a ˆ†

.

We conclude our operator-ordering discussion by deriving the relation between normal-order and antinormal-order associated functions. If we take the diagonal matrix elements of both sides of Eq. (9.15), we have α |f a ˆ, a ˆ† |α =

1 π

2 d2 α f¯(a) (α, α∗ ) | α |α | .

The left side of the above equation can easily be evaluated by using the normal (n) †r s ordering form of f a ˆ, a ˆ† = fr,s a ˆ a ˆ . Thus we immediately have r,s

α |f a ˆ, a ˆ† |α =

r,s

(n) fr,s α |a ˆ†r a ˆs |α (n) ∗r s fr,s α α

= r,s

= f¯(n) (α , α ∗ ) .

(9.19)

The expression on the right hand side is given by Eq. (7.12), i.e., | α |α |2 = exp − |α − α |2 . Therefore, we have the relation between f¯(a) (α, α∗ ) and f¯(n) (α, α∗ ) given by 1 f¯(n) (α , α ∗ ) = π

2 d2 α f¯(a) (α, α∗ ) exp − |α − α | ,

(9.20)

so that if f¯(a) (α, α∗ ) is known then in principle we may obtain f¯(n) (α, α∗ ). It is remarkable that, as shown by Eq. (9.20), the smoothing of the antinormal-order associated function, f¯(a) (α, α∗ ), by a Gaussian function, exp − |α − α |2 , yields the normal-order associated function, f¯(n) (α , α ∗ ). The importance of Gaussian smoothing will be further dealt with in our discussion of various phase-space distribution functions. For now, let us note that in terms of momentum and position variables, Eq. (7.4), the smoothing Gaussian function in Eq. (9.20) is given by 2

exp − |α − α | = exp −

mω 1 2 2 ( α| Q |α − α| Q |α ) − ( α| P |α − α| P |α ) . 2| 2|mω (9.21)

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9.3

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Characteristic Functions and Distribution Functions

We make use of the generator of coherent states, Eqs. (7.16), and the diagonal representation of the density operator, Eq. (8.1) to cast the density operator as a function of a ˆ and a ˆ† , σ (α) |α α| d2 α

ˆρ =

σ (α) exp αˆ a† − α∗ a ˆ |ψ0 ψ0 | exp α∗ a ˆ − αˆ a† .

=

From Eq. (9.49) below, the vacuum projector |ψ0 ψ0 | may be replaced by its expression in normal order, |ψ0 ψ 0 | =: exp −ˆ a† a ˆ :, where the semicolons on both sides indicate normal ordering. Therefore we have the expression for the density operator given by ρ ˆ=

ˆ : exp −ˆ a† a ˆ : exp α∗ a ˆ − αˆ a† . σ (α) exp αˆ a† − α∗ a

In terms of the density operator function, ρ ˆ, Eqs. (9.12) and (9.13) are closely related to the concept of characteristic functions which is often used to characterize distributions in the classical theory of random processes. There, F (n) (ξ, ξ ∗ ) is the ¯ (n) (α, α∗ ). By characteristic function which determine the associated function, Q virtue of the noncommuting operators in quantum theory, there are three ways to define the characteristic functions C (1,2,3) (ξ, ξ ∗ ) for the associated c-number function ρ ¯(1,2,3) (α, α∗ ) of the density operator. These are the normal, antinormal and Wigner characteristic functions, defined by ∗ †

C (n) (ξ, ξ ∗ ) = ρ(a) (−ξ, −ξ ∗ ) = T r

ρ ˆ eiηξ

C (a) (ξ, ξ ∗ ) = ρ(n) (−ξ, −ξ ∗ ) = T r

ρ ˆ eiηξˆa eiηξ

C (w) (ξ, ξ ∗ ) = ρ(w) (−ξ, −ξ ∗ ) = T r

ρ ˆ eiη(ξˆa+ξ

a ˆ

eiηξˆa , ∗ †

a ˆ

∗ †

a ˆ

(9.22)

,

(9.23)

) ,

(9.24)

respectively, where ξ is a complex parameter and η, a real parameter, is introduced here for convenience in taking the derivatives for obtaining normally-ordered, antinormally-ordered and symmetrically-ordered moments Tr

Tr

Tr

ρ ˆa ˆ†l a ˆm = ˆ†m = ρ ˆa ˆl a

∂ (l+m) ∗ l

∂ (iηξ ) ∂ (iηξ) ∂

m

m

l

=

∂l ∂ (iη)l

, η=0

(l+m)

∂ (iηξ)l ∂ (iηξ ∗ )

ˆ† ρ ˆ ξˆ a + ξ∗a

C (n) (ξ, ξ ∗ ) C (a) (ξ, ξ ∗ )

, η=0

C (w) (ξ, ξ ∗ )

. η=0

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Note that for ξ = 0, all the characteristic function are equal to T r ρ ˆ = 1. Using the Campbell-Baker-Hausdorf formula, Eq. (9.5), we have eiη(ξˆa+ξ

∗ †

a ˆ

) = eiηξ∗ aˆ† eiηξˆa exp − 1 η2 |ξ|2 2

= eiηξˆa eiηξ

∗ †

a ˆ

exp

1 2 2 η |ξ| . 2

Therefore, we may easily obtain all the three characteristic functions if one is known since 1 C (w) (ξ, ξ ∗ ) = exp − η2 |ξ|2 C (n) (ξ, ξ ∗ ) = exp 2

1 2 2 η |ξ| C (a) (ξ, ξ ∗ ) . (9.25) 2

We can further characterize this relationship by writing g (a) (ξ, η) = exp − 12 η2 |ξ|2 , g(n) (ξ, η) = exp 12 η2 |ξ|2 and g (w) (ξ, η) = 1. Then we may rewrite the relation as C (n) (ξ, ξ ∗ ) C (a) (ξ, ξ ∗ ) = , g (a) (ξ, η) g (n) (ξ, η) C (n) (ξ, ξ ∗ ) C (w) (ξ, ξ ∗ ) , = 1 g (a) (ξ, η) C (w) (ξ, ξ ∗ ) C (a) (ξ, ξ ∗ ) . = 1 g(n) (ξ, η)

(9.26)

From Eqs. (9.17) and (9.18) the definitions of the three characteristic functions, Eqs. (9.22)-(9.24), we may also write C (n) (ξ, ξ ∗ ) = T r

ρ ˆ eiηξ

∗ †

a ˆ

eiηξˆa

ρ ¯(a) (α, α∗ ) eiη(ξ

=

∗

α∗ +ξα)

1 2 d α π

= ρ(a) (−ξ, −ξ ∗ ) , C (a) (ξ, ξ ∗ ) = T r =

(9.27)

ρ ˆ eiηξˆa eiηξ

∗ †

a ˆ

ρ ¯(n) (α, α∗ ) eiη(ξ

∗

α∗ +ξα)

1 2 d α π

= ρ(n) (−ξ, −ξ ∗ ) ,

(9.28)

and hence the associated functions are given by ρ ¯(a) (α, α∗ ) = η2 = η2

1 2 d ξ π ∗ ∗ 1 e−iη(ξ α +ξα) C (n) (ξ, ξ ∗ ) d2 ξ, π eiη(ξ

∗

α∗ +ξα) (a)

ρ

(ξ, ξ ∗ )

(9.29)

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ρ ¯(n) (α, α∗ ) = η2 = η2

1 2 d ξ π ∗ ∗ 1 e−iη(ξ α +ξα) C (a) (ξ, ξ ∗ ) d2 ξ. π eiη(ξ

∗

α∗ +ξα) (n)

ρ

(ξ, ξ ∗ )

(9.30)

Note that in the literature the notation F N (α, α∗ ) and F AN (α, α∗ ) is sometimes used, which is tied to the characteristic-function ordering so that ρ ¯(a) (α, α∗ ) ⇒ F N (α, α∗ ) , ¯(n) (α, α∗ ) ⇒ F AN (α, α∗ ) , ρ where ρ ¯(n) (α, α∗ ) or F AN (α, α∗ ) is a special case of the Husimi distribution to be discussed below. However, note that from Eq. (9.19) the normal-order associated function is equal to the diagonal matrix element in the coherent state representation, thus ¯(n) (α, α∗ ) ⇒ F AN (α, α∗ ) = α| ρ a ρ ˆ, a ˆ† |α .

(9.31)

A direct relation between the density operator and its characteristic function may be obtained by making use of Eqs. (9.15) and (9.29), 1 π η2 = 2 π

ρ a ˆ, a ˆ† =

9.3.1

d2 α |α α| ρ ¯(a) (α, α∗ ) C (n) (ξ, ξ ∗ ) d2 ξ

d2 α e−iη(ξ

∗

α∗ +ξα)

|α α| .

(9.32)

The Wigner Distribution Function

The associated c-number function, ρ ¯(w) (α, α∗ ), which we denote by W (α, α∗ ) = (w) ∗ ρ ¯ (α, α ), can be expressed through Eq. (9.10) as W (α, α∗ ) = ρ ¯(w) (α, α∗ ) = η2

e−iη(ξ

∗

α∗ +ξα)

1 C (w) (ξ, ξ ∗ ) d2 ξ, π

W (α, α∗ ) is the Wigner distribution function. As with the other associated cnumber functions for the density operator, integrating W (α, α∗ ) over all α-states gives 1 W (α, α∗ ) d2 α = C (w) (0, 0) = 1. π We now express the Wigner distribution function in terms of the position and momentum eigenvalue notations common in the literature. We make use of Eqs. (6.2)-(6.4) for the transformation from the position and momentum operators to annihilation and creation operators. For this purpose, we make the change of variables given by Eq. (2.1) appropriate for the c-number variables α, 1 α= √ (mωq + ip) , 2|mω

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1 α∗ = √ (mωq − ip) , 2|mω where q = α| Q |α and p = α| P |α . Similarly, we make a change in the c-number complex variable ξ in terms of u and v, ηξ =

1√ u − iv , 2|mω 2| mω

ηξ ∗ =

1√ u + iv . 2|mω 2| mω

Therefore, we have e−iη(ξα+ξ

∗

α∗ )

i

= e− | (qu+pv)

and C (w) (ξ, ξ ∗ , t) ⇒ C (w) (u, v, t)

i

= T rρ (P, Q, t) e | (Qu+P v) .

(9.33)

The ‘metric’ for counting of states are related as (d Re α) (d Im α) 1 d2 α = = π π π 1 dpdq, = 2π|

1 mω dq √ dp 2| 2|mω

η2 1 η2 d2 ξ = (d Re ξ) (d Im ξ) = π π π (2|)2 1 dudv. = 2π|

2| √ du 2|mωdv mω

We define f (w) (p, q, t) in terms of W (α, α∗ , t) so that 1 1 W (α, α∗ ) d2 α = π 2π|

f (w) (p, q, t) dp dq = 1.

Therefore f (w) (p, q, t) = W (α, α∗ , t) i 1 e− | (qu+pv) C (w) (u, v, t) dudv. = (2π|) By redefining the Fourier transform (w) Cww (u, v, t) =

1 2π|

i

e | (qu+pv) f (w) (p, q, t) dp dq,

(9.34)

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then we may rewrite the Wigner distribution function as integral of the scaled characteristic function, i

(w) e− | (qu+pv) Cww (u, v, t) dudv,

f (w) (p, q, t) =

(9.35)

(w)

1 where Cww (u, v, t) = 2π| C (w) (u, v, t). The inverse Fourier transform of Eq. (9.35) will be useful in the discussion of smoothing of the distribution functions (w) Cww (u, v, t) =

1 2π|

i

e | (qu+pv) f (w) (p, q, t) dp dq.

(9.36)

From Eq. (9.33), and using the C-B-H formula, Eq. (9.5), we have i

C (w) (u, v, t) = T r ρ (P, Q, t) e | (Qu+P v) i

i

= T r ρ (P, Q, t) e | Qu e | P v exp −i

i

i

uv [Q, P ] 2|2

i

i

= T r e 2| P v e 2| P v e | Qu e 2| P v e 2| P v exp

iuv ρ (P, Q, t) . (9.37) 2|

Making use of Eq. (9.1), we have −i

i

i

e 2| P v e | Qu e 2| P v

−i i i e 2| P v Qe 2| P v u . |

= exp

On the other hand, from Eq. (9.2), the right hand side can be written as exp

−i i i e 2| P v Qe 2| P v u |

v i Q− u . | 2

= exp

Substituting this in Eq, (9.37), we see that i

i

i

C (w) (u, v, t) = T r e 2| P v e | Qu e 2| P v ρ (P, Q, t) i

i

i

= T r e 2| P v ρ (P, Q, t) e 2| P v e | Qu . We now evaluate the trace in the position (q) representation, C (w) (u, v, t) =

i

i

dq e | q

=

i

i

q | e 2| P v ρ (P, Q, t) e 2| P v e | Qu |q

dq

i

u

i

q | e 2| P v ρ (P, Q, t) e 2| P v |q

.

From Eq. (1.21) the last line can be written as C (w) (u, v, t) = (w) Cww (u, v, t) =

i

dq e | q 1 2π|

u

q + i

dq e | q

u

v v ρ (P, Q, t) q − , 2 2 v v ρ (P, Q, t) q − . q + 2 2

(9.38)

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Substituting this expression in Eq. (9.35) for the expression for the Wigner distribution function, we obtain 1 2π| 1 = 2π| 1 = 2π| 1 = 2π|

i

∞

=

i

dudve− | (qu+pv) T r ρ (P, Q, t) e | (Qu+P v)

f (w) (p, q, t) =

v v ρ (P, Q, t) q − dudv 2 2 −i v v ρ (P, Q, t) q − dudv dq e | (q−q )u q + 2 2 v v ρ (P, Q, t) q − dv2π|δ (q − q ) dq q + 2 2

i

i

e− | (qu+pv) i

e− | pv i

e− | pv i

e− | pv q +

−∞

dq e | q

u

q +

v v ρ (P, Q, t) q − dv. 2 2

Upon changing the sign of the integration variable v ⇒ −v, we end up with f

(w)

∞

i

(p, q, t) = −∞

e | pv q −

v v ρ (P, Q, t) q + dv, 2 2

which is the well-known expression of the phase-space Wigner distribution function as a Weyl transform of the density operator. More general phase-space distribution functions, f (g) (p, q, t), can be obtained from the expression 1 2π| 1 = 2π|

f (g) (p, q, t) =

i

dudve− | (qu+pv)

i

T r ρ (P, Q, t) e | (Qu+P v) g (u, v)

i

dudve− | (qu+pv) C (w) (u, v, t) g (u, v) ,

(9.39)

where for g (u, v) = 1, this general expression yields the Wigner distribution function. The other phase-space distribution functions can be calculated by making use of the relation between symmetric, antinormal and normal characteristics functions, Eq. (9.26). Using the relation mω 2 1 2 2 u2 η |ξ| = + v 2 4|mω 4| and Eq. (9.38), we have the following normal and antinormal characteristic functions given by C (n) (u, v, t) = exp = exp ×

1 2π|

1 1 u2 + (mω)2 v 2 2 2|mω 1 1 u2 + (mω)2 v 2 2 2|mω i

(w) Cww (u, v, t)

e | (qu+pv) f (w) (p, q, t) dp dq,

(9.40)

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where we made use of the inverse transformation of Eq. (9.36) for Cww (u, v, t). Similarly, we have 1 1 (w) u2 + (mω)2 v2 Cww (u, v, t) 2 2|mω 1 1 u2 + (mω)2 v2 = exp − 2 2|mω i 1 × e | (q u+p v) f (w) (p , q , t) dp dq . 2π|

C (a) (u, v, t) = exp −

(9.41)

9.3.1.1 Q-function and P-Function From Eqs. (9.29) - (9.30), and denoting ¯ ρ(n) (α, α∗ , t) ⇒ f (n) (p, q, t) and (a) ∗ (a) ρ ¯ (α, α , t) ⇒ f (p, q, t), we have the normal and antinormal phase-space distributions corresponding to the Wigner distribution function f (w) (p, q, t), 1 2π|

f (n) (p, q, t) = =

1 2π| 1 × 2π|

i

e− | (qu+pv) C (a) (u, v, t) dudv 2 i

e− | (qu+pv) exp − e | (q i

u+p v) (w)

f

1 1 u2 + (mω)2 v 2 2 2|mω

(p , q , t) dp dq dudv.

(9.42)

Note that f (n) (p, q, t) is also known as the Q-function in quantum optics. Similarly, we have ρ ¯(a) (α, α∗ , t) = η2

f (a) (p, q, t) = =

1 2π| 1 2π| 1 × 2π|

e−iη(ξ

∗

α∗ +ξα) (n)

ρ

1 (ξ, ξ ∗ ) d2 ξ, π

i

e− | (qu+pv) C (n) (u, v, t) dudv 2 i

e− | (qu+pv) exp e | (q i

u+p v) (w)

f

1 1 u2 + (mω)2 v2 2 2|mω

(p , q , t) dp dq dudv.

(9.43)

We also point out the f (a) (p, q, t) is also known as P-function in quantum optics. Equation (9.43) contains anti-Gaussian term and shows that in general the P-function is not an analytic function but exist only, by virtue of Eq. (8.2), in the sense of a generalized distribution function, i.e., they are only defined when they 1 appear under the integral sign. Because of the antiGaussian terms: exp 4| mωv2 (w) 1 u2 the integral can only converge provided Cww (u, v, t) in Eq. and exp 4|mω (9.40) decays faster than the anti-Gaussian term grows. However the integral of the P-function or f (a) (p, q, t) always exists as a distribution, as was shown by Sudarshan, Eq. (8.2). In contrast, the Q-function or f (n) (p, q, t) always exists since C (a) (u, v, t) ⇒ 0 when u, v ⇒ ∞. Moreover, due to Gaussian versus anti-Gaussian

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term in the characteristic functions, C (a) (u, v, t) and C (n) (u, v, t), respectively, the inverse Fourier transforms, namely, the Q-function or f (n) (p, q, t) is always broader then the P-function or f (a) (p, q, t). From Eqs. (9.17)-(9.18), the Q-function or f (n) (p, q, t) is used to obtain averages of antinormally-ordered operators, whereas, the P-function or f (a) (p, q, t) is used to obtain averages of normally-ordered operators. A simpler procedure to calculate the associated function will be given in the next subsection. It is worth mentioning that states with nonclassical features such as photon number states or squeezed states have P-functions that exist only in the sense of generalized distribution function. For example there are cases that when the Qfunction is a Gaussian, the corresponding P-function is a delta function. We will not go into the details of these cases but refer the reader to the book by Scheich1 . Indeed, it can be shown that in going from the P-function through the Wigner distribution function to the Q-function, a series of Gaussian smoothing in the sense discussed below occurs. A good reference on Gaussian smoothing among various distribution functions is given in a review paper2 . The Husimi Distribution Function

9.3.2

The Q-function or f (n) (p, q, t) is a special case of the so-called Husimi distribution function, which is a class of positive-definite and smooth distributions. Before discussing Husimi distribution function, we show how the diﬀerent distributions, f (n) (p, q, t), f (a) (p, q, t), and f (w) (p, q, t) are related to each other by smoothing integration. We make use of Eq. (9.39) for the general expression for the diﬀerent phase-space distribution functions. 1 2π| 1 = 2π|

f (g) (p, q, t) =

i

dudve− | (qu+pv)

i

T r ρ (P, Q, t) e | (Qu+P v) g (u, v)

i

dudve− | (qu+pv) C (w) (u, v, t) g (u, v) . i

Making use of the expression for C (w) (u, v, t) = T r ρ (P, Q, t) e | (Qu+P v) (9.38), we have (w) Cww (u, v, t) =

1 2π|

i

dq e | q

u

q +

v v ρ (P, Q, t) q − , 2 2

and therefore f (g) (p, q, t) =

1 2π| × ψ∗

1 See 2 H.

2

i i g (u, v) exp − (q − q ) u exp − pv | | 1 1 dudvdq . q − v ρ ψ q + v 2 2

W. P. Schleich, Quantum Optics (Wiley-VCH, New York, 2001). W. Lee, Phys. Rep. 259, 147 (1994).

in Eq.

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From Eq. (9.34), using Eq. (9.26) we have (w) Cww (u, v, t) =

1 2π|

i

e | (qu+pv) f (w) (p, q, t) dp dq,

(w)

C (a) (u, v, t) =

Cww (u, v, t) 1 1 = (n) g (n) (u, v) g (u, v) 2π|

i

e | (qu+pv) f (w) (p, q, t) dp dq.

From Eq. (9.30), we have f (n) (p, q, t) (w)

1 2π| 1 = 2π|

Cww (u, v, t) dudv g (n) (u, v) i 1 i 1 e− | (qu+pv) e | (q u+p v) f (w) (p , q , t) dp dq (n) 2π| g (u, v) i

e− | (qu+pv)

=

2

1 2π|

=

dp dq du dv e | ((q −q)u+(p −p)v) i

dudv

1 f (w) (p , q , t) . g (n) (u, v)

Integrating over the variables u and v eﬀectively takes the two-dimensional Fourier −1 transform of g (n) (u, v) as 1 2π|

2

1 = 2π|

du dv e | ((q −q)u+(p −p)v) ∞

−∞

=

1

i

g (n) (u, v)

u2 du e (q −q)u exp − 4|mω i |

1 2π| 2

mω 1 (p − p) 2 exp − (q − q) exp − π| | mω|

∞

−∞

dve | (p −p)v exp − i

mω 2 v 4|

.

Therefore, we have (p − p)2 mω 2 (q − q) exp − f (w) (p , q , t) . | mω| (9.44) Thus, f (n) (p, q, t) is obtained by smoothing the Wigner distribution function. Indeed from Eq. (7.21) the smoothing made use of the Gaussian probability of the coherent-state minimum wavepacket. From (9.26), we show that f (w) (p, q, t) can in turn be obtained by Gaussian smoothing of f (a) (p, q, t). We have f (n) (p, q, t)=

1 π|

dp dq exp −

C (w) (u, v, t) = exp −

mω 2 u2 − v C (n) (u, v, t) , 4|mω 4|

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and from Eq. (9.35) we write 1 2π|

f (w) (p, q, t) =

i

e− | (qu+pv) exp −

mω 2 u2 − v C (n) (u, v, t) du dv. 4|mω 4|

Upon substituting by the inverse Fourier transform, Eq. (9.29), of C (n) (u, v, t), C (n) (u, v, t) =

1 2π|

e | (q u+p v) f (a) (p , q , t) dp dq , i

(9.45)

yields f (w) (p, q, t) 2

=

mω 2 u2 1 i − v e− | (qu+pv) exp − 2π| 4|mω 4| i × e | (q u+p v) f (a) (p , q , t) dp dq du dv 1 π|

=

dp dq exp −

(p − p)2 mω 2 (q − q) exp − | mω|

f (a) (p , q , t) .

Therefore, the process of Gaussian smoothing with coherent-state wavepacket probability distribution provides the series of transformation from the P-function or f (a) (p, q, t) to the Wigner distribution function, f (w) (p, q, t), and finally to the Qfunction or f (n) (p, q, t). Thus, clearly the smoothing transformation from the P-function or f (a) (p, q, t) directly to the Q-function or f (n) (p, q, t) requires double smoothing in the following sense. From Eq. (9.25), we have the relation between their characteristic functions as exp −η2 |ξ|2 C (n) (ξ, ξ ∗ ) = C (a) (ξ, ξ ∗ ) . Therefore 1 2π| 1 = 2π|

f (n) (p, q, t) =

i

e− | (qu+pv) C (a) (u, v, t) dudv i

e− | (qu+pv) exp −

mω 2 u2 − v C (n) (u, v, t) dudv. 2|mω 2|

Upon substituting the expression in Eq. (9.45) for C (n) (u, v, t) we have f (n) (p, q, t) 2

=

=

mω 2 u2 1 i − v e− | (qu+pv) exp − 2π| 2|mω 2| i × e | (q u+p v) f (a) (p , q , t) dp dq dudv 1 2π|

dp dq exp −

(p − p)2 mω 2 (q − q) exp − 2| 2|mω

f (a) (p , q , t) ,

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which shows that the double√Gaussian smoothing from f (a) (p, q, t) directly to f (n) (p, q, t) is performed with 2 times Gaussian widths, or two times the smoothing area, ∆q ⇒

∆p ⇒

√ 2

| , 2mω

√

|mω , 2

2

so that for this direct-smoothing Gaussian wavepacket, ∆p∆q = |. Since the smoothing process eliminates fast fluctuations in the distributions, this provide another explanation why when the P-function or f (a) (p, q, t) becomes a delta function the Q-function or f (n) (p, q, t) still behaves regularly. It should be noted that in the process of smoothing, no physical information is lost. The expression for the Q-function or f (n) (p, q, t) or F AN (p, q, t) given by Eq. (9.44) is a special case of the Husimi distribution function. From Eq. (9.31), the Q-function or f (n) (p, q, t) ˆ in the or F AN (p, q, t) is the diagonal matrix element of the density operator ρ coherent-state representation The definition of the Husimi distribution is the generalization of Eq. (9.44) by replacing the harmonic oscillator frequency ω by an arbitrary constant ζ. Thus, denoting the Husimi distribution by F H , we have mζ (p − p)2 2 (q − q) exp − F W (p , q , t) , | |mζ (9.46) where we also change the notation from f (w) (p , q , t) to F W (p , q , t) in Eq. (9.46). Since the only diﬀerence between the definition of Q-function or f (n) (p, q, t) or F AN (p, q, t) and F H (p, q, t) is the replacement of ω by ζ, Eq. (9.46) for F H (p, q, t) may be viewed as the generalized Q-function or f (n) (p, q, t) or F AN (p, q, t) [note: F AN (p, q, t) is a notation which is tied to antinormal-order characteristic function instead of the normal-order density operator, Eqs. (9.17)-(9.18), Eqs. (9.22)-(9.23), and Eq. (9.27)-(9.28)]. It should be pointed out at this juncture that our discussion has moved away from the original harmonic-oscillator coherent-state formulation, the oscillator frequency has been substituted by the abstract quantity, ζ, and the only remaining original parameter of the oscillator is its mass. We may further entirely leave the harmonic oscillator problem by a further generalization, i.e., replacing mζ by new real parameter, κ. The resulting formalism of quantum mechanics may thus be called the coherent-state formulation of quantum mechanics by virtue of its harmonic-oscillator origin, or more appropriately the squeezed coherent-state formulation of quantum mechanics. Squeezing has to do with changes in the respective minimum uncertainty widths, ∆q and ∆p, of the Gaussian-smoothing wavepacket F H (p, q, t) =

1 π|

dp dq exp −

| while the product remains 12 | [i.e., ∆q = 2mζ , ∆p = will become clear in the following discussion.

|mζ 2 ,

∆q∆p = 12 |]. This

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Generalized Coherent States and Squeezing

The replacement of the harmonic oscillator parameter mω to an arbitrary real parameter κ also leads us to redefine the annihilation and creation operator of Eq. (6.2)-(6.3) in a straightforward way as b=

κ i Q+ P 2| κ

,

b† =

κ 2|

.

Q−

i P κ

Compared to the definition of the harmonic-oscillator annihilation and creation operator, Eq. (6.2)-(6.3), it can be shown that the new annihilation operator b can be written in terms of the harmonic-oscillator annihilation and creation operator as b = µˆ a + νˆ a† ,

(9.47)

a† + νˆ a, b† = µˆ

(9.48)

with µ=

1 2

κ + mω

mω κ

,

ν=

1 2

κ − mω

mω κ

,

where µ2 − ν 2 = 1. Equations (9.47) - (9.48) represent the step towards generalized annihilation and creation operators. b=

+ =

=

1 2

mω κ

κ − mω

1 2 +

=

κ + mω

1 2

κ + mω 2| 1 2

mω κ 1 2|κ

κ − mω 2|

1 2|κ

κ (Q) + 2| κ 2|

Q+

i P κ

mω 2|

1 2|κ ,

Q+

i P mω

mω 2|

Q−

Q+

i P mω

Q− (iP )

i P mω

i P mω

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µ2 =

mω 1 κ + +2 , 4 mω κ

ν2 =

mω 1 κ + −2 . 4 mω κ

A further generalization is to extend the parameters to complex values, ζ 1 and ζ 2 , and write the corresponding annihilation and creation operators as 1 c = √ (ζ 1 Q + iζ 2 P ) , 2| 1 c† = √ (ζ ∗1 Q − iζ ∗2 P ) . 2| Corresponding to Eqs. (6.6) - (6.7) for Q and P in terms of annihilation and creation operators, we have 1 ζ ∗2 c + ζ 2 c† = √ [(ζ ∗2 ζ 1 + ζ 2 ζ ∗1 ) Q + i (ζ ∗2 ζ 2 − ζ 2 ζ ∗2 ) P ] 2| 2 Q, = | 1 ζ ∗1 c − ζ 1 c† = √ [(ζ ∗1 ζ 1 − ζ 1 ζ ∗1 ) Q + i (ζ ∗1 ζ 2 + ζ 1 ζ ∗2 ) P ] 2| 2 P. =i | Therefore, | ∗ ζ c + ζ 2 c† , 2 2

Q=

P = −i

| ∗ ζ c + ζ 1 c† . 2 1

We can also write, c = µc a ˆ + νca ˆ† , where µc =

1 2

√ ζ √ 1 + ζ 2 mω , mω

νc =

1 2

√ ζ √ 1 − ζ 2 mω . mω

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Thus, √ mω ζ i √ 1 + ζ 2 mω P Q+ mω 2| mω √ 1 mω ζ i √ 1 − ζ 2 mω + P Q− 2 mω 2| mω 1 mω i ζ √ 1 + ζ2 √ P = Q+ 2 mω 2| 2| 1 mω i ζ √ 1 − ζ2 √ P + Q− 2 mω 2| 2| ζ 1 = √ 1 Q + ζ2 √ (iP ) 2| 2| 1 = √ (ζ 1 Q + iζ 2 P ) . 2|

c=

1 2

Similarly, √ mω ζ∗ i √ 1 + ζ ∗2 mω P Q− mω 2| mω √ 1 mω ζ∗ i √ 1 − ζ ∗2 mω + P Q+ 2 mω 2| mω 1 mω i ζ∗ √ 1 + ζ ∗2 √ P = Q− 2 mω 2| 2| 1 mω i ζ∗ √ 1 − ζ ∗2 √ P + Q+ 2 mω 2| 2| ζ∗ 1 = √ 1 Q + ζ ∗2 √ (−iP ) 2| 2| 1 = √ (ζ ∗1 Q − iζ ∗2 P ) . 2|

c† =

1 2

The commutation relation c, c† = 1, becomes 1 1 √ (ζ 1 Q + iζ 2 P ) , √ (ζ ∗1 Q − iζ ∗2 P ) 2| 2| 1 1 1 1 = √ (ζ 1 Q + iζ 2 P ) √ (ζ ∗1 Q − iζ ∗2 P ) − √ (ζ ∗1 Q − iζ ∗2 P ) √ (ζ 1 Q + iζ 2 P ) 2| 2| 2| 2| 1 = ζ ζ ∗ Q2 + ζ 2 ζ ∗2 P 2 − iζ 1 ζ ∗2 QP + iζ 2 ζ ∗1 P Q 2| 1 1 1 − ζ ζ ∗ Q2 + ζ 2 ζ ∗2 P 2 + iζ ∗1 ζ 2 QP − iζ ∗2 ζ 1 P Q 2| 1 1

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1 1 (−iζ 1 ζ ∗2 QP + iζ 2 ζ ∗1 P Q) − (iζ ∗1 ζ 2 QP − iζ ∗2 ζ 1 P Q) 2| 2| −i ((ζ 1 ζ ∗2 + ζ ∗1 ζ 2 ) QP − (ζ 2 ζ ∗1 + ζ ∗2 ζ 1 ) P Q) = 2| −i −i −i2 (QP − P Q) = [Q, P ] = i| = 1, = 2| | |

=

where we have used the relations, [Q, P ] = i|, (ζ 1 ζ ∗2 + ζ ∗1 ζ 2 ) = (ζ 2 ζ ∗1 + ζ ∗2 ζ 1 ) = 2. We also have, 2 2 |µc | − |ν c | =

√ ζ √ 1 + ζ 2 mω mω

2

√ 1 ζ1 √ + ζ 2 mω 4 mω

2

2

−

√ 1 ζ1 √ − ζ 2 mω , 4 mω

√ ζ∗ √ 1 + ζ ∗2 mω mω √ √ |ζ |2 ζ ζ∗ = 1 + mω |ζ 2 |2 + √ 1 ζ ∗2 mω + ζ 2 mω √ 1 mω mω mω

=

√ ζ √ 1 + ζ 2 mω mω

|ζ 1 |2 + mω |ζ 2 |2 + 2, mω √ √ ζ ζ∗ √ 1 − ζ ∗2 mω = √ 1 − ζ 2 mω mω mω 2 √ √ |ζ | ζ ζ∗ = 1 + mω |ζ 2 |2 − √ 1 ζ ∗2 mω − ζ 2 mω √ 1 mω mω mω =

√ ζ √ 1 − ζ 2 mω mω

2

=

|µc |2 − |ν c |2 =

1 4

|ζ 1 |2 + mω |ζ 2 |2 − 2, mω

1 |ζ 1 |2 + mω |ζ 2 |2 + 2 − mω 4

|ζ 1 |2 + mω |ζ 2 |2 − 2 mω

= 1. The eigenfunction of the annihilation operator c is our new coherent state which we denote by |γ . Thus, c |γ = γ |γ , where γ is a complex number.

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Algebra and Calculus within Ordered Products

9.5.1

Algebra within Ordered Products

If U and V belongs to Fermi field operators, then : UV := − : V U : Otherwise : U V :=: V U : . Ordered products satisfy the rules of addition3 : U.......V : + : W....X :=: (U.....V + W.......X) :, : U....V...X : + : U....W....X :=: U..... (V + W ) .......X) : . If c is a number, then c : U.......V :=: cU.......V : . If A is any operator, and : U.......V :=: W....X :, then : AU.......V :=: AW....X :, : U.......V A :=: W....XA : . An ordered product of an ordered product satisfies the relation : U.......V (: W....X :) :=: (U.....V W.......X) : . An ordered product of canonical variables and their space derivatives can be expressed in terms of their ordinary products by a relation of the form ci fi ,

: U.......V := U.......V + i

where the ci are numbers and fi are products of the canonical variables and their space derivatives obtained from U.......V by leaving out one or more pairs of Bose or Fermi particles. If the canonical variables or their space derivatives transform under a unitary transformation U ⇒ U ,......., V ⇒ V , then the ordered products of these variables transforms as : U.......V :⇒: U .......V : . 3 See

for example, S. N. Gupta, Phys. Rev. 107, 1722 (1957).

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9.5.1.1 Diﬀerentiation within Ordered Products Let us consider the ordered product : f :=: V U : . Then we have : δf :=: V δU : + : δV U :, : δf :=: V δU : ± : U δV :, : δf :=:

∂f ∂f δU : ± : δV :, ∂U ∂V

the plus sign for the Bose operators and the minus sign for the Fermi operators. Similarly, if : f :=: V..........U : denotes an ordered product of any number of creation and annihilation field operators, then we : δf :=:

∂f ∂f δU : + : δV :, ∂U ∂V

where the primed derivative considers Fermi operators and takes into account the number of operators in the right side of V , if this number is odd the minus sign for the ordinary derivative is taken. A consistent quantum field theory can be formulated in terms of ordered products. The virtue of the formalism is that it is free from ambiguities in the order of the field operators. Within the ordered products the properties of the ordered products are remarkably similar to ordinary products in classical mechanics and moreover, the zero point energy of the radiation field is automatically eliminated. 9.5.2

Integration within Ordered Products in Quantized Classical Field

Recently, the algebra and calculus of ordered products has acquired new vigor and significance in the development of quantum optics, i.e., in dealing with Bose annihilation and creation operators. The idea is that within ordered products, the annihilation and creation operators only play the role of parameters in the integration. Now since all operators are projection operators, and since all coherent states can be generated from the vacuum, the starting point for theory of integration within ordered products is the expression of the vacuum projection operator in terms of ordered product. From the completeness relation

∞

n=0

|n =

a ˆ†n √ n!

|0 , a ˆ† a ˆ |n = n |n , and using the following identity √

1 n!n !

d dZ ∗

n

(Z ∗ )n

= δ n,n , Z ∗ ⇒0

|n n| = 1, with

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we have 1=

=

=

=

∞ n ,n=0 ∞

|n n | √

1 n!n !

d dZ ∗

n n

(Z ∗ )

Z ∗ ⇒0

a ˆ†n a ˆn 1 √ |0 0| √ √ n! n! n!n ! n ,n=0 ∞

a ˆn a ˆ†n |0 0| n! n! n ,n=0 ∞

a ˆ†n

n ,n=0

= exp a ˆ†

d n dZ ∗

n! d dZ ∗

d dZ ∗

|0 0|

d dZ ∗

n

(Z ∗ )n Z ∗ ⇒0

n

(Z ∗ )n Z ∗ ⇒0

a ˆn (Z ∗ )n n! Z ∗ ⇒0

|0 0| exp {ˆ aZ ∗ }

. Z ∗ ⇒0

The last line is of the form of normal-ordered products in terms of the Bose annihilation and creation operators, i.e., all creation operators are placed to the left of all annihilation operators. Therefore we can write 1 = : exp a ˆ†

d dZ ∗

|0 0| exp {ˆ aZ ∗ } :

= : exp a ˆ† a ˆ |0 0| :

Z ∗ ⇒0

ˆ : |0 0| :: = : exp a ˆ† a ˆ : W :: = : exp a ˆ† a

The last line shows that the projection operator has a normal product form given by |0 0| = : W : = : exp −ˆ a† a ˆ :.

(9.49)

Thus, we have the following expansion for the vacuum projection operator in terms ˆ =a of the number operator N ˆ† a ˆ ˆ N ˆ −1 − 1N ˆ N ˆ −1 ˆ + 1N |0 0| = 1 − N 2! 3! 9.5.3

ˆ − 2 + ........ N

(9.50)

Evaluation of Integral of Some Important Mapping Operators

We have already indicated that quantum dynamical operators can be expanded in terms of projection operators via the completeness relation of the quantum states. However, the nature of the mapping operator, such as scaling mapping operator or transformation of dynamical variables, involves projection which are not symmetric in the expansion of the set of eigenstates and its dual space. For example, the

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following scaling operator integral ∞

Sq = −∞

dq q √ µ µ

q|

µ>0

is perfectly meaningful since it projects all |q states onto µq states, this is the so-called quantum mechanical squeezing operator. The inverse to Sq is Sq†

∞

= −∞

dq √ |q µ

q µ

µ > 0,

so that Sq† Sq

∞

dq √ µ

=

−∞ ∞

=

1 µ

∞

dq √ |q µ

dq

−∞

q|

−∞ ∞

−∞ ∞

=

q q µ µ

−∞

dq |q

dq |q µ δ (q − q) q|

q| = 1.

Another interesting mapping operator or operator for the transformation-ofvariables is the so-called symplectic transformation. 9.5.4

Symplectic Transformation and Symplectic Group

A motivation for studying symplectic transformation may be traced to our classical harmonic oscillator equation when cast in a form of a vector equation, with a column vector defined by filling the leading rows with coordinates and the remaining rows with momentum. Thus for our classical harmonic oscillator, cast in vector notation, we have q˙ =

p˙ = −

∂H p = , ∂p m

∂H = −κq = −mω 2 q. ∂q

We can write this in vector notation as ∂ ∂t

q p

= =

1 0 m −κ 0

0 1 −1 0

q p ∂ ∂q ∂ ∂p

H.

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The last line is the Hamilton’s canonical equations written using the ‘symplectic metric matrix’ g, g=

0 1 −1 0

,

so that we also write ∂H ∂ V =g , ∂t ∂V

(9.51)

where the column vector V is V =

q p

.

Q . Then the transP formation is canonical if and only if in the new variables the Hamilton’s canonical equations also hold

Now let a transformation M changes the vector V to V =

∂H ∂ V =g . ∂t ∂V This implies that from Eq. (9.51), ∂H ∂ M V = Mg ∂t ∂V ∂H , ∂V

= MgM T

that the transformation is canonical if and only if it leaves the ‘symplectic metric matrix’ g invariant. MgM T = g.

(9.52)

If we write M as M=

ab cd

,

then the condition, Eq. (9.52) yields relation between the matrix elements of M , namely, det M = ad − bc = 1. Since M is a linear transformation M=

∂Q ∂q ∂P ∂q

∂Q ∂p ∂P ∂p

,

MT =

∂Q ∂q ∂Q ∂p

∂P ∂q ∂P ∂p

,

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we have M gM

T

=

∂Q ∂q ∂P ∂q

∂Q ∂p ∂P ∂p

=

∂Q ∂q ∂P ∂q

∂Q ∂p ∂P ∂p

0

= 

=

∂P ∂Q ∂q ∂p

∂Q ∂q ∂Q ∂p

0 1 −1 0

−

∂Q ∂P ∂p ∂p ∂Q ∂P − ∂q ∂q ∂Q ∂P ∂q ∂p ∂Q ∂P ∂q ∂p

0

∂(Q,P ) ∂(q,p)

) − ∂(Q,P ∂(q,p)

0



=

−

∂P ∂q ∂P ∂p

∂Q ∂P ∂p ∂q

0 0 1 −1 0

.

Therefore, for canonical transformation the following must hold ∂Q ∂P ∂Q ∂P ∂ (Q, P ) = − = 1. ∂ (q, p) ∂q ∂p ∂p ∂q But this is the expression for the Jacobian of the transformation, namely, J=

∂Q ∂q ∂P ∂q

∂Q ∂p ∂P ∂p

= 1.

Now the ‘volume’ (area for simple harmonic oscillator) element in phase space between the canonically transformed phase spaces are related by J (dq × dp) = dQ × dP. Therefore for canonical transformation, J = 1, and the so-called ‘wedge form’ (diﬀerential geometry terminology), dq × dp = dQ × dP, is also invariant. In two variables this means the elementary ‘area’ in phase space is invariant (this also reflects the invariant way of counting of states). We can also designate J as the Poisson bracket of Q and P , i.e., [Q, P ]pois = J = 1 = [q, p]pois , which means that symplectic transformation of the variables must render the Poisson bracket, [Q, P ]pois , invariant. In quantum mechanics, the classical Poisson bracket is the leading term, in gradient expansion in the limit | ⇒ 0, in the commuˆ and Pˆ , Q, ˆ Pˆ , in the Weyl transform repretation relation between the operators Q ˆ Pˆ ⇒ i| [Q, P ]. sentation. For linear transformation, M, the Weyl transform of Q, Moreover, the Hamilton’s canonical equations still holds in quantum mechanics. Thus in quantum mechanics, the classical Poisson bracket relation, [Q, P ]pois = 1, ˆ Pˆ = i|. is replaced by the operator commutation relation Q,

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9.5.4.1 Quadrature States Example of M is the following matrix, M=

cos θ sin θ − sin θ cos θ

,

so that we have Q P

= MV =

cos θ sin θ − sin θ cos θ

q p

=

q cos θ + p sin θ q (− sin θ) + p cos θ

=

q cos θ + p sin θ q cos θ + π2 + p sin θ +

π 2

.

(9.53)

The resulting eigenstates of the symplectically-transformed operators are known in quantum optics as quadrature states. It is a simple matter to treat many-particle systems by extending the column vector V to include all q’s in the first n rows followed by the p’s in the next n rows.   q1  q2     .     qn   V ⇒  p1  .    p2     .  pn Then the symplectic metric matrix g becomes g⇒

0 In −In 0

,

where In is a n-dimensional diagonal matrix. The corresponding symplectic transformation matrix, M, becomes M⇒

AB CD

,

and the relations between A, B, C, and D matrices imposed by the condition of Eq. (9.52) are AB T CDT AT C BT D

= BAT , = DC T , = C T A, = DT B.

(9.54)

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ADT − BC T = In , AT D − C T B = In .

(9.55)

The group of linear transformation preserving the classical Poisson bracket, in the case of classical mechanics, and quantum commutation relations, in the case of quantum mechanics, among n (q-p)-pairs of variables (for n-particle systems) is called a symplectic group, denoted by Sp(2n, F ) where F is a real field. Because symplectic matrices have unit determinant, the symplectic group is a subgroup of the special linear group SL(2n, F ), over the field F which can have real or complex values.

9.5.5

Complex Form of Symplectic Transformation Matrix

In dealing with quantized classical fields, we have seen that it is convenient to work ˆ and Pˆ ’s in terms of the annihilation and with complex combinations of the Q’s creation operators, a ˆ=

mω 2|

ˆ + i Pˆ , Q mω

a ˆ† =

mω 2|

ˆ − i Pˆ . Q mω

We can form a column vector with components a ˆ and a ˆ† as Vˆ =

a ˆ a ˆ†

=Z

ˆ Q Pˆ

,

where 

 Z =

mω 2| mω 2|

i −i

1 2|mω 1 2|mω

From Eq. (9.51), we can write ∂H ∂ ZV = ZgZ T , ∂t ∂ Vˆ ∂H ∂ ˆ V = ZgZ T , ∂t ∂ Vˆ



 .

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where the quantum Hamiltonian, H, is given by Eq. (6.5) written in normal-ordered form. Thus, we have ∂H ∂ˆ a ∂H ∂ˆ a†

∂ ∂t

a ˆ a ˆ†

= ZgZ

∂ ∂t

a ˆ a ˆ†

=−

i |

=

−iωˆ a iωˆ a†

T

, ∂H ∂ˆ a ∂H ∂ˆ a†

0 1 −1 0 ,

where ZgZ T = − |i g. Therefore, the requirement for the symplectic transformation a ˆ of Vˆ = is that ZgZ T = − |i g remains invariant, this implies the equation, a ˆ† S −

i |

0 1 −1 0

ST = −

i |

0 1 −1 0

,

(9.56)

in ∂ S ∂t

a ˆ a ˆ†

=S −

i |

0 1 −1 0

∂ ∂t

a ˆ a ˆ†

=S −

i |

0 1 −1 0

∂H ∂ˆ a ∂H ∂ˆ a†

ST

, ∂H ∂ˆ a ∂H ∂ˆ a†

.

(9.57)

We are interested in finding the symplectic transformation S in terms of the matrix elements of the symplectic transformation of the position and momentum variables. Now consider the symplectic transformation on the position and momentum operators, ∂ ∂H M V = Mg , ∂t ∂V ∂ ∂H V = MgM T . ∂t ∂V Applying the annihilation/creation transformation on both sides, we have ∂H ∂ Z (M V ) = ZM g , ∂t ∂V ∂ ∂H Z (V ) = ZM gM T , ∂t ∂V ∂H ∂ ˆ V = ZM gM T Z T . ∂t ∂ Vˆ

(9.58)

Therefore, ZM gM T Z T = − |i g = ZgZ T since M gM T = g so that we have the

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canonical relation in the transformed operators a ˆ a ˆ†

∂ ∂t

=−

i |

∂H ∂ˆ a ∂H ∂ˆ a†

0 1 −1 0

.

To find the symplectic transformation S for the annihilation and creation column vector in Eq. (9.57), we write the last line of Eq. (9.58) as ∂ ˆ V = ZMZ −1 ZgZ T Z T ∂t

−1

M T ZT

∂H ∂ Vˆ

T

M T ZT

∂H ∂ Vˆ

= ZMZ −1 ZgZ T Z −1 = ZMZ −1 −

i |

0 1 −1 0

ZT

−1

M T ZT

∂H . ∂ Vˆ

Thus, from Eq. (9.57), we identify S = ZM Z −1 U R R∗ U ∗

=

.

To study the properties of S, it is more convenient to work with scaled position and momentum operators, Eq. (6.4) so that the annihilation and creation operators have very simple forms, Eqs. (6.2)-(6.3), leading to the following expression Vˆ =

a ˆ a ˆ†

= Zscaled

˜ Q P˜

,

where 1 2

Zscaled =

1 i 1 −i

.

The transformation to annihilation and creation operator, Zscaled , has a very nice † −1 property, namely, Zscaled = Zscaled . Then the symplectic transformation S becomes † . S = Zscaled MZscaled

Upon evaluating this last equation in terms of the matrix elements of the symplectic transformation of the scaled position and momentum operators, we have † S = Zscaled M Zscaled

1 2

= =

1 2

1 1 1 2 −i i a + d + i (c − b) a − d + i (b + c) a − d − i (b + c) a + d + i (b − c) 1 i 1 −i

ab cd

,

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which proves the form U R R∗ U ∗

S=

.

Since S obeys the symplectic condition of Eq. (9.56), the symplectic property of S corresponding to Eqs. (9.54)-(9.55) are, upon generalizing to many modes, given by U RT = RU T = URT R∗ U † = U ∗ R† = R∗ U † U T R∗ = R† U = U T R T

∗

†

T

R U =U R= R U

T

, T

,

∗ T

,

∗ T

,

UU ∗ − R∗ R = In , U T U ∗ − R† R = In .

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Chapter 10

Discrete Quantum Mechanics of Bloch Electrons

In what follows we will develop a formalism of the quantum mechanics of charge carriers appropriate to problems in nanoscience and nanoelectronics. In this formalism, we treat the eigenvalues to be discrete and we will only go to the continuum limit, if appropriate, for simplifying the calculational process. This discreteness pertains to the lattice coordinates and crystal momentum in solids, whose eigenstates are the Wannier function and Bloch function, respectively1 .

10.1

Energy-Band Dynamics of Bloch Electrons

The lattice coordinates and crystal momentum thus form a discrete phase space. This approach to the quantum dynamics of solids is the most logical starting point assuming that the energy bands has been calculated theoretically or known experimentally. Moreover, general discrete and canonical dynamical variables exist, characterized as finite fields in discrete mathematics which has relevance to quantum computing.

10.1.1

Wannier Function and Bloch Function

The fundamental assumption made is that there exists a complete set of localized functions labeled by band index λ and lattice points q, and a complete set of extended functions labeled by band index λ and crystal momentum p. The two sets are connected by a unitary “lattice Fourier transformation” which serve as the transformation function between the two mutually-unbiased Hilbert spaces. Let wλ (x, q) be any localized state labeled by a band index λ and lattice point q. Let bλ (x, p) be the corresponding eigenfunction labeled by the crystal momentum p. Thus we can write bλ (x, p) = N |3

− 12

i

e | p·q wλ (x, q), q

1 We

assumed for simplicity that the band index is a good quantum number. 124

(10.1)

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wλ (x, q) = N |3

− 12

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125

i

e− | p·q bλ (x, p),

(10.2)

q

e− | (p−p )·q = N |3 δ p,p ,

(10.3)

e− | (q−q )·p = N |3 δ q,q ,

(10.4)

i

q i

q

where N is the total number of lattice points2 . For convenience, we introduce the Dirac ket and bra notation for the eigenvectors: bλ (x, p) ⇒ |p, λ = |p ,

(10.5)

wλ (x, q) ⇒ |q, λ = |q ,

(10.6)

where the transformation functions are p, λ q, λ = N |3

−1 − i p·q |

e

δ λ,λ ,

(10.7)

p, λ| p , λ

= δ p,p δ λ,λ ,

(10.8)

q, λ| q , λ

= δ q,q δ λ,λ .

(10.9)

The completeness expressions are given by

10.1.2

|p p| = 1,

(10.10)

p

|q q| = 1.

(10.11)

q

Lattice Weyl-Wigner Formulation of Energy-Band Dynamics

By using the closure relations, Eqs. (10.10) and (10.11), the following identity holds for an arbitrary operator A A= p ,p ,q ,q

|p

p |q

q | A |q

q|p

p |.

(10.12)

Introducing the notation p = p + u,

q = q + v,

p = p − u,

q = q − v,

2 N is necessarily an prime number since the crystal considered has inversion symmetry and each point has an inverse, thus forming a finite field.

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and using Eq. (10.7), we obtain A = N |3

−1

Aλλ (p, q) ∆λλ (p, q) ,

(10.13)

p,q,λ,λ

where 2i e( | )p·v q − v, λ| A q + v, λ ,

Aλλ (p, q) = v

∆λλ (p, q) = u

2i e( | )q·u |p − u, λ

p + u, λ .

(10.14)

(10.15)

By applying Eqs. (10.1) and (10.2) it is easy to obtain the following equivalent expressions for Aλλ (p, q) and ∆λλ (p, q): 2i e( | )q·u p + u, λ| A p − u, λ ,

Aλλ (p, q) = u

∆λλ (p, q) = v

2i e( | )p·v |q + v, λ

q − v, λ .

(10.16)

(10.17)

In Eq. (10.13) the operator nature of A is transferred to ∆λλ (p, q), in other words A is expanded in terms of the phase-space point projector, ∆λλ (p, q). We will refer to Aλλ (p, q) as the lattice Weyl transform of the operator A. [Our definition of Aλλ (p, q) diﬀers from that of the continuous (p, q) formalism by a factor of 2 in the exponential and the absence of 12 inside the ket and bra, and, of course, by the replacement of the integral with the summation]. Thus an alternative formulation of the quantum mechanics of solids is possible using a complete set of functions wλ (x, q) labeled by a band index λ and lattice point q and a complete set of functions bλ (x, p) labeled by a band index λ and crystal momentum p, the two complete sets being connected by a unitary lattice Fourier transformation. Examples of these complete sets are the Wannier functions and Bloch functions, both with and without a magnetic field. Blount’s “mixed representation” [1] and Wannier’s formulations [2] of the dynamics of Bloch electrons in a solid are embryonic forms of a discrete (p, q) version of the statistical formulation of quantum mechanics when considered in terms of the Weyl transform instead of operators and the Wigner function instead of state vectors [3]. The power of this method of doing quantum mechanics in solid-state theory has been first demonstrated [4] in calculating the magnetic susceptibility of bismuth. Several powerful applications of this formalism in solid state physics revolve in the expression for the trace of nth power of an arbitrary operator A, T rAn , particularly the nth power of the Hamiltonian operator H. This has not been given in the continuous (p, q) formalism. From Eq. (10.13) we can see that, because of the way the operator ∆ is defined, the only nontrivial way in which the lattice Weyl transform of ∆ explicitly occurs is when one takes the lattice Weyl transform of at least a product of two operators A and B. Therefore we have as fundamental binary operations the taking of the trace and the lattice Weyl transform of products

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of two operators or, equivalently, the trace and lattice Weyl transform of products of two ∆’s. Let ∆αα (p1 , q1 ) = u1

∆ββ (p2 , q2 ) = u2

2i e( | )q1 ·u1 |p1 − u1 , α p1 + u1 , α | ,

(10.18)

2i e( | )q2 ·u2 |p2 − u2 , β

p2 + u2 , β .

(10.19)

p2 + u2 , β p, γ

(10.20)

Then we have T r ∆αα (p1 , q1 ) ∆ββ (p2 , q2 ) exp

= p,γ u1 ,u2

2i (q1 · u1 + q2 · u2 ) |

× p, γ |p1 − u1 , α p1 + u1 , α | p2 − u2 , β =

exp p,γ u1 ,u2

2i (q1 · u1 + q2 · u2 ) |

× δ p1 −u1 ,p δ p2 +u2 ,p δ p1 +u1 ,p2 −u2 δ γ,α δ α ,β δ β ,γ =

exp p,γ

(10.21)

2i [p · (q2 − q1 ) + q1 · p1 − q2 · p2 ] δ p1 ,p2 δ γ,α δ α ,β δ β ,γ . (10.22) |

The last line is obtained by writing q1 · (p1 − p) + q2 · (p − p2 ) as p · (q2 − q1 ) + q1 · p1 − q2 · p2 . We thus obtain, using Eq. (10.13), T rAB = N |3

−2

Aαα (p1 , q1 ) Bββ (p2 , q2 ) p1 ,q1 ,α,α , p2 ,q2 ,β,β ,p,γ

× exp

2i [p · (q2 − q1 ) + q1 · p1 − q2 · p2 ] |

×δ p1 ,p2 δ γ,α δ α ,β δ β ,γ = N |3

−1

Aγα (p, q) Bαγ (p, q) .

(10.23) (10.24)

p,q,α,γ

In taking the lattice Weyl transform of ∆αα (p1 , q1 ) × ∆ββ (p2 , q2 ) it is convenient to use the expression in Eq. (10.16) applied to the ∆ operator in Eq. (10.15). We

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Nonequilibrium Quantum Transport Physics in Nanosystems

therefore have ∆αα (p1 , q1 ) ∆ββ (p2 , q2 )

γγ

(p, q)

e( )q·u p + u, γ| ∆αα (p1 , q1 ) ∆ββ (p2 , q2 ) |p − u, γ 2i |

= u

exp

= u,u1 ,u2

2i |

(10.25)

(q · u + q1 · u1 + q2 · u2 ) δ p+u,p1 −u1

× δ p1 +u1 ,p2 −u2 δ p2 +u2 ,p−u δ γ,α δ α ,β δ β ,γ 2i exp = (q · u + q1 · u1 + q2 · u2 ) δ p+u,p1 −u1 |

(10.26)

× δ u1 ,p2 −p δ u2 ,p−p1 δ γ,α δ α ,β δ β ,γ 2i exp [(p2 − p) · (q1 − q) + (q2 − q) · (p − p1 )] = |

(10.27)

u,u1 ,u2

u,u1 ,u2

× δ γ,α δ α ,β δ β ,γ ,

(10.28)

where the last line is obtained by writing q · (p1 − p2 ) + q1 · (p2 − p) + q2 · (p − p1 ) as (p2 − p) · (q1 − q) + (q2 − q) · (p − p1 ). Introducing pˆ = (p2 − p) and qˆ = (q2 − q), the lattice Weyl transform of a product of two opera-tors A and B may now be written as (AB)λλ (p, q) = N |3

−2

exp p1 ,q1 ,ˆ p,ˆ q,β

2i [ˆ p · (q1 − q) − qˆ · (p1 − p)] |

×Aλβ (p1 , q1 ) Bβλ (ˆ p + p, qˆ + q) .

(10.29)

Further simplification of the above expression depends on our assumption that there exist two continuous functions of q having an infinite radius of convergence, which are equal to Aλβ (p1 , q1 ) and Bβλ (ˆ p + p, qˆ + q), respectively, at all lattice points [1]. (We need not worry about the p dependence since, in the limit of infinite volume, p becomes continuous.) Therefore we can expand Bβλ (ˆ p + p, qˆ + q) in a Taylor series around Bβλ (p, q) and obtain (AB)λλ (p, q) = N |3

−2

exp p1 ,q1 ,ˆ p,ˆ q,β

×Aλβ (p1 , q1 ) exp pˆ · = N |3

−2

2i [ˆ p · (q1 − q) − qˆ · (p1 − p)] |

∂ ∂ + qˆ · Bβλ (p, q) ∂p ∂q

(10.30)

Aλβ (p1 , q1 ) p1 ,q1 ,ˆ p,ˆ q,β

2i [ˆ p · (q1 − q) − qˆ · (p1 − p)] | ←− ←− | ∂ ∂ ∂ ∂ · − · Bβλ (p, q) , × exp 2i ∂p ∂q ∂q ∂p

× exp

(10.31)

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where the arrows pointing to the left mean diﬀerential operator operating to the left. Summing over pˆ, qˆ, p1 , q1 and using Eq. (10.4) we finally end up with (AB)λλ (p, q) =

exp β

| ∂ (a) ∂ (b) ∂ (a) ∂ (b) · − · 2i ∂p ∂q ∂q ∂p

×Aλβ (p, q) Bβλ (p, q) ,

(10.32)

where the superscript (a) and (b) indicate the objects of the diﬀerential operator. In matrix notation we may write exp

(AB) (p, q) = β

| ∂ (a) ∂ (b) ∂ (a) ∂ (b) · − · 2i ∂p ∂q ∂q ∂p

×A (p, q) B (p, q) .

(10.33)

It is easy to see that the lattice Weyl transform of the anticommutator of A and B is | ∂ (a) ∂ (b) ∂ (a) ∂ (b) · − · 2 ∂p ∂q ∂q ∂p × [A (p, q) B (p, q) + B (p, q) A (p, q)]

{A, B} (p, q) = cos

| ∂ (a) ∂ (b) ∂ (a) ∂ (b) · − · 2 ∂p ∂q ∂q ∂p × [A (p, q) B (p, q) − B (p, q) A (p, q)] . +i sin

(10.34)

Putting A = B in the last expression we obtain A2 (p, q) = cos ×

| 2

∂ (1) ∂ (2) ∂ (1) ∂ (2) · − · ∂p ∂q ∂q ∂p

1 (1) A (p, q) A(2) (p, q) + A(2) (p, q) A(1) (p, q) . 2

(10.35)

By reiterating the binary operation prescribed by Eq. (10.35) one can easily convince oneself that for any power n   | An (p, q) = cos  2 ×

n

j,k=1 j=k

∂ (j) ∂ (k) ∂ (j) ∂ (k)   · − · ∂p ∂q ∂q ∂p 

1 A(1) (p, q) A(2) (p, q) ...A(n−1) (p, q) A(n) (p, q) . (10.36) 2 +A(n) (p, q) A(n−1) (p, q) ....A(2) (p, q) A(1) (p, q)

Equations (10.24) and (10.36) are all we need to evaluate the trace of the nth power of an arbitrary operator A. Thus by using Eq. (10.24) we have T rAn = N|3

−1

An−1 (p, q) A (p, q) ,

T rband p,q

(10.37)

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and with aid of Eq. (10.36) we end up with  T rAn = N |3

−1

T rband

p,q

×

| cos  2

n

j,k=1 j=k



∂ (j) ∂ (k) ∂ (j) ∂ (k)   · − · ∂p ∂q ∂q ∂p 

1 A(1) (p, q) A(2) (p, q) ...A(n−1) (p, q) A(n) (p, q) , 2 +A(n) (p, q) A(n−1) (p, q) ....A(2) (p, q) A(1) (p, q)

(10.38)

where T rband means trace over band indices. To make explicit progress beyond Eq. (10.38), we expand the cosine function to order |2 where we have [4] T rAn

= N|3

−1

T rband p,q

+ O |4 .

                                             1 [A (p, q)]n −  2!                                           

           ∂ 2 A(p,q) ∂ 2 A(p,q)    ∂p∂p ∂q∂q        ×    ∂ 2 A(p,q) ∂ 2 A(p,q)   −    ∂p∂q ∂q∂p       (n−1)(n−2)(2n−3)      6 n−3 [A (p, q)]  +      (n−1)(n−2)     −  2         ∂ 2 A(p,q) ∂A(p,q) ∂A(p,q)    ∂p∂p ∂q ∂q           2 A(p,q)   ∂ ∂A(p,q) ∂A(p,q)   +      ∂q∂q ∂p ∂p    2  ×   2 A(p,q) |   ∂A(p,q) ∂A(p,q)    − ∂ ∂p∂q    ∂q ∂p 2       ∂ 2 A(p,q) ∂A(p,q) ∂A(p,q)    −    ∂q∂p ∂p ∂q      n(n−1)(n−2)       6 n−3 [A (p, q)]   +2     (n−1)(n−2)  −    2       ∂A(p,q) ∂ 2 A(p,q) ∂A(p,q)   · ·    ∂p ∂q∂p ∂q         2 A(p,q) ∂A(p,q) ∂ ∂A(p,q)    + · · ∂q ∂p∂q ∂p       ×  2   ∂A(p,q) ∂ A(p,q) ∂A(p,q)     · ·  −   ∂q ∂p∂p ∂q       2  ∂A(p,q) ∂ A(p,q) ∂A(p,q)  · · − ∂p ∂q∂q ∂p 

(n−1)(n−2) [A (p, q)]n−2 2

(10.39)

{ } is a symmetrized tensor contraction, i.e., ∂ 2 A (p, q) ∂p∂p

∂ 2 A (p, q) ∂q∂q

=

∂ 2 A (p, q) ∂ 2 A (p, q) ∂ 2 A (p, q) ∂ 2 A (p, q) + ∂pi ∂pj ∂qi ∂qj ∂qi ∂qj ∂pi ∂pj

and moreover, the p’s must never have identical indices as well as the q’s where repeated indices are summed over. In evaluating the summation by an integral, N |3

−1

goes over to

1 (2π|)3

d3 p d3 q. Equation (10.38) indeed will give us a

p,q

systematic expansion of T rAn in powers of |2 .

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If A (p, q) is of the functional form A p − ec A (q) , f (p, q) , where f (p, q) ⇒ 0 as

|q| ⇒ ∞, then integration by parts permits the reduction of the [A (p, q)]n−3 term n−2 term plus gradient terms which do not contribute to the form of the [A (p, q)] after integration with respect to p and q. To prove this, one has to make use of the periodicity in p space and also the fact that the result of p integration of an integrand, which is a function of p − ec A (q) only, does not depend on q. The final result can be written   [A (p, q)]n    n−2  1 | 2 n(n−1) [A (p, q)]   − 2! 6    2  1   2 2 3 3 ∂ A(p,q) ∂ A(p,q) d p d q T rAn = T rband  , (10.40)  3 ∂p∂p ∂q∂q   (2π|)    ×   ∂ 2 A(p,q) ∂ 2 A(p,q) −   ∂p∂q ∂q∂p +O |4

and the trace of an arbitrary function of operator A defined in terms of power series can thus be expressed as   F (A (p, q))     − 1 | 2 n(n−1) F (A (p, q))   2! 2 6   ∂ 2 A(p,q) ∂ 2 A(p,q)   1  3 3  T rF (A) = T rband d p d q .  ∂p∂p ∂q∂q    (2π|)3   ×   ∂ 2 A(p,q) ∂ 2 A(p,q) −   ∂p∂q ∂q∂p 4 +O | 10.2

Application to Calculation of Magnetic Susceptibility

The need to evaluate the trace of the nth power of an operator such as the Hamiltonian operator in solid-state theory arises in many instances. One example is the calculation of the magnetic susceptibility of the system from the formulas χ = LimB→0 −

1 ∂2F , V ∂B 2

(10.41)

where F is given by F = Nµ + T rF (H) ,

(10.42)

with F (H) as the partition function F (H) = −kB T ln 1 + e

(µ−H) kB T

.

(10.43)

We wish to expand F (H) in powers of H. However, F (H) has a small radius of convergence which vanishes at T = 0. Thus, it is more convenient to employ the

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Laplace transform of F (H) since the term in the Laplace transform involving H is esH which has infinite radius of convergence. Therefore, let us write c+i∞

φ (s) esH ds,

F (H) = c−i∞

esH = n

sn n H . n!

(10.44)

In calculating the magnetic susceptibility of Bloch electrons in a uniform magnetic field, one only needs to evaluate T rHn up to the second order in the magnetic field strength. This was done by Wannier and Upadhyaya [5]. Here, we have given a generalization of their ad hoc method of evaluating T rHn . Another problem where T rHn is needed which made impact on its resolution occurs in inhomogeneous systems, namely, the theory of the magnetic susceptibility of alloys, which has attracted many theorists [6]. Only perturbative solutions in powers of the strength of the impurity potential existed and only in the special case of a free-electron band does the problem seem to have been formulated. We wish to point out that T rHn in calculating the magnetic susceptibility of Bloch electrons in a uniform magnetic field exactly follows from Eq. (10.40), where the lattice Weyl transform H (p, q) of the Hamiltonian is H (p, q) = Wλ p − ec A (q) ; B δ λλ , W being referred to as the renormalized energy band function in the presence of magnetic field. (For the case where spin-orbit coupling is included, for each band index λ, W is a 2 × 2 matrix, and this is given by Roth [7]. Because of the particular combination of p and q in Wλ p − ec A (q) , q did not explicitly appear in the problem for uniform systems. In the presence of a substitutional impurity potential, q will explicitly appear in the problem. The virtue of Eq. (10.38) is that it allows inhomogeneity in direct space. Using the above trace formulas, the derivation of the magnetic susceptibility of dilute alloys to order |2 is straightforward3 . The result given for χ is valid for general Bloch bands to all orders in the impurity potential, and reduces to all known limiting cases discussed by Hebborn and Scannes [6]. In the free-electron-band model the expression for ∆χ per solute atom gives a firm theoretical foundation to the formula used by Henry and Rogers [8], as pointed out by Kohn and Luming [6], which accounts quite well for their experimental results on dilute alloys of Zn, Ga, Ge, and As with Cu. This is given as

∆χ = −

e2 6mc2

d3 q ∆ρ (q) |q|2 + χorb (0)

∆g1 ∆g2 + χspin (0) , g g

(10.45)

3 F. A. Buot, “Magnetic Susceptibility of Dilute Nonmagnetic Alloys”, Phys. Rev., B11, 14261436 (1975).

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Discrete Quantum Mechanics of Bloch Electrons

133

where ∆ρ (q) =

2 3

(2π|) V

d3 p

 

f (Σo ) +

 + f (Σo ) +

1 24 f

2

2

(Σo ) | m∇ VI (q) − f 2

2

(Σo ) | m∇ VI (q) − f

1 24 f

p2 2m p2 2m

f (Σo ) is the Fermi distribution function, and g=

∆g1 =

2 (2π|) V

2

d3 p d3 q

3

(2π|) V

∆g2 = ∆g1 +

d3 p d3 q f

3

2

χspin (0) = −

χorb (0) =



.

(10.46)

,

p2 2m

f (Σo ) − f

d3 p d3 q f (Σo )

24 (2π|)3 V Σo =

p2 2m

p2 3m

 

,

|2 ∇2 VI (q) , m

p2 + VI (q) , 2m

2 3

(2π|) V 2 3

3 (2π|) V

d3 p d3 q f

p2 2m

µ2B ,

d3 p d3 q f

p2 2m

µ2B .

The general result for χ can also be applied to bismuth, where the eﬀect of the impurity potential should be large [6, 9]. Using the Lax k · p model for bismuth, which is equivalent to the Dirac Hamiltonian by a simple velocity scale factor [9] one may make use of the result of Blount [10] and Suttorp [11] for H (p, q) in the form given for Dirac particles. The power of the formalism is applicable to a vast number of quantummechanical problems. Indeed the formalism yields a rigorous quantum-mechanical basis of the distribution-function method in potential screening, the Thomas-Fermi method [12] in the absence of the magnetic field, and the quasiclassical approximation for nonzero magnetic field [13]. By means of the formula N =−

∂ T rF (H) ∂µ

(10.47)

=

d3 q n (q) ,

(10.48)

the local particle density n (q) can thus be obtained as a series expansion in powers in |2 , which can, in principle, be obtained to all orders in |2 . Then the selfconsistent potential due to an impurity charge Ze located at the origin must satisfy

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Nonequilibrium Quantum Transport Physics in Nanosystems

the Poisson equation: ∇2 V (q) = 4πen (q) − 4πeno − 4πZeδ (q) . The result for the potential screening at zero field, low field, and very high magnetic field has been reported4 and agrees with the general result of Horing [13], to order |2 , obtained by the Green’s function method and random-phase approximation using linear-response theory. The present formalism has the advantage that it can be applied to particles of arbitrary dispersion law. For example, an immediate extension can be made to the potential screening by a relativistic quantum plasma using H (p, q) for the Dirac particle given by Blount [10] and Suttorp [11].

4 F. A. Buot, “Formalism of Distribution-Function Method in Impurity Screening”, Phys. Rev. B14, 977-989 (1976).

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Chapter 11

The Eﬀective Hamiltonian

An arbitrary operator Aop in solid-state band theory can be expressed in terms of the crystal momentum operator and lattice-point coordinate operators: Aop = N |3

−1

Aλλ σσ (p, q) ∆λλ σσ (p, q) ,

(11.1)

p,q,λ,λ σ,σ

where Aλλ σσ (p, q) and ∆λλ σσ (p, q) are given by Aλλ σσ (p, q) = v

2i e( | )p·v q − v, λ, σ| Aop q + v, λ , σ ,

∆λλ σσ (p, q) = u

2i e( | )q·u |p − u, λ, σ

p + u, λ , σ ,

(11.2) (11.3)

or by the equivalent expressions, Aλλ σσ (p, q) = u

2i e( | )q·u p + u, λ, σ| Aop p − u, λ , σ ,

∆λλ σσ (p, q) = v

2i e( | )p·v |q + v, λ, σ

q − v, λ , σ .

(11.4) (11.5)

N is the number of lattice points, p is the crystal momentum (limited to the first Brillouin zone), q is the lattice point coordinate, σ, σ label the spin indices, and λ, λ label the band indices. The operator nature of Aop has been transferred to ∆λλ σσ (p, q) in Eq. (11.1). Indeed ∆λλ σσ (p, q) can be rewritten to exhibit its operator nature in terms of P and Q as 2i 2i e( | )(q+v−Q)·u e( | )(p−P )·v Ωλλ σσ

∆λλ σσ (p, q) = v,u

2i e( | ){(q−Q)·u+(p−P )·v} Ωλλ σσ ,

=

(11.6)

v,u

where Ωλλ σσ = q

|q, λ, σ

q, λ , σ = p

135

|p, λ, σ

p, λ , σ .

(11.7)

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Nonequilibrium Quantum Transport Physics in Nanosystems

P and Q are the crystal momentum and lattice-point coordinate operator, respectively, they are the canonical conjugate variables for solid-state problems, [Pl , Qj ] =

| δ lj . i

(11.8)

The eigenvalues and eigenfunctions of Q are the lattice-point coordinates and Wannier functions, respectively. Likewise, the eigenvalues and eigenfunctions of P are the crystal momentum (limited to the first Brillouin zone) and Bloch functions, respectively. Aλλ σσ (p, q) defined by Eq. (11.2) or (11.4) is the lattice Weyl transform of the quantum operator Aop in the Weyl-Wigner formulation of quantum mechanics. Equations (11.1) and (11.2) can be viewed as providing us with the exact mathematical prescription for associating classical c-number quantities with quantum-mechanical operators and vice versa. It should be pointed out that the exact corresponding classical (i.e., energy-band index diagonal) quantities may still contain Planck’s constant |. Indeed the diagonal in energy-band Aλλ σσ (p, q) of Eq. (11.2) can often be achieved only after decoupling the band of interest1 . This decoupling of energy bands is responsible for the g-factor for the spin degree of freedom for each band in the same manner that spin arises from the decoupling of the positive and negative energy states of the relativistic Dirac Hamiltonian. Furthermore, it should be pointed out that if the corresponding classical function (or lattice Weyl transform to be exact) is of the form of a sum of a function of crystal momentum, p, and a function of lattice coordinate, q, then the eﬀective Hamiltonian operator of Eq. (11.1) is obtained by simply replacing the real-valued p and q by the the operator P and Q, respectively. This is the familiar quantization rule of classical particle dynamics.

11.1

Two-Body Eﬀective Hamiltonian

We write down the P -Q representation of a two-body quantum operator as (2)

Aef f (Q, Q , P, P ) =

˜, q, q˜) exp A[λσ][λ˜ ˜ σ] (p, p p,q,λ,λ ,σ,σ ,u,v ˜λ ˜ ,˜ p,˜ ˜ q ,λ, σ ,˜ σ ,˜ u,˜ v

2i (q + v − Q) · u |

2i (p − P ) · v | 2i 2i ˜ ·u q˜ + v˜ − Q ˜ exp p˜ − P˜ · v˜ Ω[λσ][λ˜ × exp ˜σ] , | | (11.9)

× exp

1 The energy-band decoupling scheme is similar to the Foldy-Woutheysen transformation for decoupling the positive and negative energy bands of relativistic Dirac electrons (See for example, F. A. Buot, “Magnetic Susceptibility of Interacting Free and Bloch Electrons”, Phys. Rev., B14, 3310-3328 (1976).)

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The Eﬀective Hamiltonian

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137

˜ σ for λ˜ ˜ σλ ˜σ where [λσ] stands for symbol strings λσλ σ , λ˜ ˜ , and (2)

A[λσ] λ˜ (p, p˜, q, q˜) [ ˜σ] 2i p.v exp exp = | v,˜ v

2i p˜.˜ v |

˜ σ ˜ ,σ × q − v, λ, σ| q˜ − v˜, λ, ˜ A(2) ˜ + v˜, λ ˜ op q Ω[λσ][λ˜ ˜σ] = q ,˜ q

˜ σ |q , λ, σ q˜ , λ, ˜

˜ ,σ q˜ , λ ˜

q + v, λ , σ ,

q ,λ ,σ .

(11.10)

(11.11)

Equivalent expressions corresponding to Eqs. (11.4) and (11.5) can be obtained by using the unitary lattice Fourier transformation connecting the Wannier functions to the Bloch functions.

11.2

Eﬀective Hamiltonian in Second Quantization

The formalism of the second quantization operators in solid-state band theory via the first-stage Weyl correspondence is discussed in Appendix C. From Eqs. (C.23) and (C.24), we can immediately write down the general expression for the eﬀective Hamiltonian in q (lattice coordinate) space as Hef f =

(1)

q1 ,q2 ,λ,λ ,σ,σ

Wλλ σσ (q1 − q2 ; q1 + q2 ) ψ†λσ (q1 ) ψλ σ (q2 ) (2)

+ q1 ,q2 ,λ,λ ,σ,σ ˜λ ˜ ,˜ q˜1 ,˜ q2 ,λ, σ,˜ σ

W[λσ] λ˜ (q − q2 , q˜1 − q˜2 ; q1 + q2 , q˜1 + q˜2 ) [ ˜ σ] 1

× ψ†λσ (q1 ) ψ†λ˜ q1 ) ψλ˜ σ˜ (˜ q2 ) ψλ σ (q2 ) , ˜ σ (˜ (1)

(11.12)

where to begin with, Aop is the sum of electron kinetic energies, relativistic eﬀects (2) plus the periodic potential of the crystal lattice. Aop represents all the electronelectron interaction potential of the relevant electrons under consideration. The presence of the sum and diﬀerence of lattice coordinates in Eq. (11.12) indicates that the corresponding lattice Weyl transform is necessarily a function of both p and q in general. It is important to emphasize here that although the c-number quantities occurring in Eq. (11.12) are simply matrix elements in the Wannier-function representation of one and two-body quantum operators, as expressed by Eqs. (C.25) and (C.26), respectively, there are significant calculational advantages in obtaining these c-number quantities via the first-stage Weyl correspondence, Eqs. (C.21) and (C.22). The Weyl-Wigner formalism of the quantum theory of solids has a

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138

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Nonequilibrium Quantum Transport Physics in Nanosystems

built-in mechanism for the decoupling of interacting bands for obtaining a physically transparent Weyl transform as a power series in the electromagnetic field strength. Moreover, for very-high-field (e.g., high uniform magnetic field) generalization2 of the Bohr-Sommerfield quantization rules may further be applied, to eﬀectively include all powers of the magnetic field strength, to the obtained Weyl transform. Therefore, in this manner physically meaningful calculational progress can easily be made, the Wannier functions and/or Bloch functions become a little more than a formalistic tool rather than a real calculational tool from the beginning, i.e., one does not begin by actually calculating all matrix elements between Wannier functions of the “bare” one and two-body operators to obtain Eq. (11.12) above. Through the first-stage Weyl correspondence, Eqs. (C.21) and (C.22), one is able to start the quantum-dynamical problem from the already calculated zero-field selfconsistent energy band structure. We will restrict ourselves to normal systems, i.e., the average of products of field operators is identically zero if in the given product the number of creation operators is not equal to the number of destruction operators (e.g., ground-state Fermi-sea quantum averaging). Only symmetry breaking terms in the Hamiltonian remove the degeneracy of the statistical equilibrium state and for this case average values of field operators are no longer zero. In obtaining electron-electron correlations, attention must be paid to the number of interchanges of the field operators in Eq. ((11.12) to form combinations of nonzero averages. The idea of obtaining an optimal one-body Hamiltonian, i.e., of the form of the first term of Eq. ((11.12), is to include all average eﬀects of the electron-electron interaction and hence to separate out the true electron-electron correlation eﬀects. This leads us to incorporate in the first term of Eq. ((11.12) the following: 

(1)

δHef f = q1 ,q2 ,λ,λ ,σ,σ

 (2) W[λσ] λ˜ − q , q ˜ − q ˜ ; q + q , q ˜ + q ˜ ) (q 1 2 1 2 1 2 1 2 ˜ [ σ]   ˜λ ˜ ,˜ σ,˜ σ  q˜1 ,˜q2 ,λ,  † q1 ) ψλ˜ σ˜ (˜ q2 ) ψλ˜ ˜ σ (˜

×ψ†λσ (q1 ) ψλ σ (q2 )  −

˜ ,σ,˜ q1 ,˜ q2 ,λ,λ σ

 q˜ ,q ,λ,λ  1 2 ˜ ,˜σ,σ

q2 ) , ×ψ†λσ (q1 ) ψλ˜ σ˜ (˜

 (2) W[λσ] λ˜ (q − q2 , q˜1 − q˜2 ; q1 + q2 , q˜1 + q˜2 ) [ ˜ σ] 1   ψ †λ˜ (˜ q ) ψ (q ) 1 2 λ σ ˜σ (11.13)

to obtain the band-theoretic Hamiltonian. The same expression is subtracted from the second term of Eq. (11.12), the resulting term will then describe the true correlation between electrons. For treating correlation eﬀects, the field-theoretical Green’s function technique has proved to be a very useful tool. We will make some simplifications. Denoting the optimal band-theoretical 2 See, F. A. Buot, Formalism of distribution-function method in impurity screening, Phys. Rev. 14, 977 (1976).

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The Eﬀective Hamiltonian

139

Hamiltonian as Ho and the two-body operator which describe the electron correlation by the sum of Hc and Hex , we have Hef f = Ho + Hc + Hex ,

(11.14)

where after a self-consistent procedure, discussed above for Ho , we have Ho = Hc =

(1)

q1 ,q2 ,λ,σ

Wλ (q1 − q2 ; q1 + q2 ) ψ†λσ (q1 ) ψλσ (q2 ) ,

(11.15)

(2)c

q1 ,q2 ,λ1 ,λ2 ,σ 1 ,σ2

Wλ1 λ2 (0, 0; 2q1 , 2q2 ) n ˆ λ1 σ1 (q1 ) [ˆ nλ2 σ2 (q2 ) − n ˆ λ2 σ2 (q2 ) ] , (11.16)

Hex = −

(2)ex

q1 ,q2 ,λ1 ,λ2 ,σ 1 ,σ2 ,[λ1 σ1 ]=[λ1 σ 1 ]

Wλ1 λ2 (q1 − q2 , q2 − q1 ; q1 + q2 , q1 + q2 )

ˆ λ2 σ2 (q2 ) ×ψ†λ1 σ1 (q1 ) ψλ1 σ2 (q1 ) ψ†λ2 σ2 (q2 ) ψλ2 σ1 (q1 ) − n

.

(11.17)

Equation (11.17) can be written in a more revealing form by summing over the spin indices. The result is Hex = −

(2)

q1 ,q2 ,λ1 ,λ2

Jλ1 λ2 (q1 − q2 , q2 − q1 ; q1 + q2 , q1 + q2 )

1 × Sλ1 (q1 ) · Sλ2 (q2 ) + nλ1 (q1 ) [ˆ nλ2 (q2 ) − n ˆ λ2 (q2 ) ] , (11.18) 4 where the quantities entering in Eq. (11.18) are defined as (2)

Jλ1 λ2 (q1 − q2 , q2 − q1 ; q1 + q2 , q1 + q2 ) (2)ex

= 2Wλ1 λ2 (q1 − q2 , q2 − q1 ; q1 + q2 , q1 + q2 ) , n ˆ λ (q) =

n ˆ λσ (q) ,

(11.19) (11.20)

σ

Sλz (q) =

1 [ˆ nλ⇑ (q) − n ˆ λ⇓ (q)] , 2

(11.21)

Sλx (q) + iSλy (q) = ψ†λ⇑ (q) ψλ⇓ (q) ,

(11.22)

Sλx (q) − iSλy (q) = ψ†λ⇓ (q) ψλ⇑ (q) .

(11.23)

The basis states (Wannier function) used in Eqs. (11.15)-(11.17) are assumed to be the self-consistent Wannier functions of the Hartree-Fock Hamiltonian. From Eqs. (11.4) and (C.21), it is clear that the general dependence on crystal lattice coordinates of the matrix element in Eq. (11.15) reduces to dependence on the diﬀerence

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Nonequilibrium Quantum Transport Physics in Nanosystems

of lattice coordinates only. In the presence of uniform magnetic field the appropriate basis states to use are the set of magnetic Wannier functions and its associated set of magnetic Bloch functions. It has been shown3 how one can start from the self-consistent localized solution in the absence of magnetic field to the nonzero-field self-consistent localized solution, with corresponding field-dependent renormalization of the bands expressed as a power series in the magnetic field strength. Although the renormalization of the bands can only be expected to be asymptotic, it should be pointed out, however, that the existence of magnetic Wannier functions and magnetic Bloch functions can be argued on the basis of the crystal symmetry (in the presence of uniform magnetic field) alone. In general, the renormalized bands can have very complicated dependence in the magnetic field, incorporating the multiband dynamics of Bloch electrons which are very important, for example, in the theory of diamagnetism of solids. The form of the eﬀective Hamiltonian in the presence of a uniform external magnetic field is formally the same as given by Eqs. (11.14)-(11.17), with additional dependence on the magnetic field strength occurring in the matrix elements. However, for Ho this form alone is not explicitly gauge invariant. The explicit dependence of the matrix element on the vector potential or “gauge field” is required due to the nonlocality aspect of product of field operator ψ† (q1 ) ψ (q1 ). This observation leads us to consider the Peierls phase factor” of the magnetic Wannier functions. An expression of the form 

ie W (r1 , r2 ) exp  |c

r1

r2



A (r) · dr ψ† (r1 ) ψ (r2 ) ,

(11.24)

where W (r1 , r2 ) is a c-number, is a gauge-invariant quantity. Now if the matter fields ψ are only defined on a lattice, then Eq. (11.24) would read, using the meanvalue theorem for the integral, as W (q1 , q2 ) exp

ie A |c

q1 + q2 2

· (q1 − q2 ) ψ† (q1 ) ψ (q2 ) ,

(11.25)

ie A (q2 ) · (q1 − q2 ) ψ† (q1 ) ψ (q2 ) , |c

(11.26)

which can also be written as W (q1 , q2 ) exp

where the last line is obtained using Landau gauge A(q) = 12 B×q, B is the magnetic field strength. We apply Eq. (11.26) to the expression for Ho in Eq. (11.15), at the same time incorporating the dependence on the magnetic field strength B due to the fielddependent renormalization of the bands. Using the Landau gauge for the vector 3 See,

G. H. Wannier, Rev. Mod. Phys. 34, 645 (1962).

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The Eﬀective Hamiltonian

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141

potential A(r) = 12 B × r, we obtain = H(B) o

q1 ,q2 ,λ,σ

˜ (1) (q1 − q2 ; B) exp ie A (q2 ) · q1 ψ† (q1 ) ψλσ (q2 ) . W λσ λ |c

(11.27)

The factor multiplying the field operators in Eqs. (25) and (27) is indeed the correct form of the matrix elements of a one-body Hamiltonian, with symmetry of a crystal lattice in a uniform magnetic field, between two magnetic Wannier functions [2] whose functional form is defined by ωλσ (r − q; B) = exp −

ie A (r) · q ω ˜ λσ (r − q; B) , |c

(11.28)

where explicit vector-potential (gauge-field) dependence is given by the well-known Peierls phase factor [2]. Hc and Hex do not need a gauge-field term because the product of field operators occurring there are essentially “local” terms. We may therefore write a universal eﬀective Hamiltonian for metals, semiconductors and insulators, and magnetic and nonmagnetic crystalline solids in a uniform magnetic field as (B) Heff = Ho(B) + Hc(B) + Hex , (B)

(B)

(11.29)

(B)

where Ho is given by Eq. (11.27), and Hc and Hex are of the same form as Eqs. (11.16) and (11.17), respectively, but with magnetic field strength dependence incorporated in the matrix elements. Equation (11.29) is a generalization to nonzero field of a universal Hef f given by Mattis [14, 15] in the absence of magnetic field. 11.3

Eﬀective Non-Hermitian Hamiltonian in a Magnetic Field

The transformation of the one-particle eﬀective Hamiltonian expressed in “bare” dynamical variables into an eﬀective Hamiltonian in terms of crystal-momentum and lattice-position operators can be carried out in rigorous fashion using the lattice Weyl-Wigner formalism of the quantum theory of solids. We are interested here in the eﬀective one-particle Hamiltonian of a many-body system in a uniform magnetic field H = H0 + Σ (z) ,

(11.30)

where Σ (z) is a nonlocal energy-dependent (z is the energy variable) and complexvalued quantity called the self-energy Σ (r, r , z; B) = exp

−ie ˜ (r, r , z; B) , A (r) · r Σ |c

where the periodic function ˜ (r, r , z; B) . ˜ (r − q, r − q, z; B) = Σ Σ

(11.31)

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In terms of ‘bare’ canonical variables H0 is given by H0 =

1 2m

e | ∇r − A (r) i c

2

+ V (r) − gµB Sz Bz ,

where A (r) = 12 B × r (in symmetric gauge), B is the magnetic field in the zdirection and V (r) is the average periodic potential. The total eﬀective potential, V (r) + Σ (z) is non-Hermitian leading us to employ the biorthogonal eigenfunctions of H. Let ω λ (r, q) be any complete set of localized states labeled by a band index λ and a lattice point q and let bλ (r, p) be its lattice Fourier transform. The corresponding discrete quantum mechanics is introduced by Eqs. (10.1)-(10.11). The lattice-position operator Q and a crystal-momentum operator P satisfy the following eigenvalue relations, P |p, λ = p |p, λ ,

(11.32)

Q |q, λ = q |q, λ .

(11.33)

The significance of P and Q can be appreciated by showing that they provide the appropriate canonically conjugate dynamical variables very useful in discussing energy-band dynamics. The Weyl transform of the commutator of P and Q is Pl , Qj and hence [Pl , Qj ] = can write

| i

λλ

(p, q) =

| δ jl δ λλ , i

(11.34)

δ jl δ λλ . Moreover, corresponding to Eqs. (1.4)-(1.8), we

Q |p, λ =

| ∇p |p, λ , i

(11.35)

P |q, λ =

| ∇q |q, λ , i

(11.36)

and for any wavefunction ψ (r), expanded in terms of the complete set |q, λ or |p, λ , we have P ψ (r) = p

[p fλ (p)] |p, λ , | ∇q fλ (q) i

P ψ (r) = p

Q ψ (r) = p

|q, λ ,

[q fλ (q)] |q, λ ,

(11.37)

(11.38)

(11.39)

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The Eﬀective Hamiltonian

143

| − ∇p fλ (p) i

Q ψ (r) = p

|p, λ .

(11.40)

These results enable us to express the quantum dynamics of energy-band electrons entirely in discrete p-q space. The above results depend, of course, on the existence of continuous functions of q, having infinite radius of convergence, which give the right values at the lattice points. The following identities can easily be verified: |q + v, λ = exp

|q − v, λ

q − v, λ =

1 (N |3 )

−

2i P ·v |

exp u

|q − v, λ ,

(11.41)

2i q − v − Q · u Ωλλ , |

(11.42)

where Ωλλ = p

= q

|p, λ

p, λ

(11.43)

|q, λ

q, λ ,

(11.44)

and by virtue of the above identities we may therefore write ∆λλ (p, q) =

1

exp

(N |3 )2

v,u

2i p − P · v exp |

2i q − v − Q · u Ωλλ | (11.45)

and find Aop ψ (r) =

1

Aλλ (p, q) exp

(N|3 )2

p,q,v,u,q ,λ,λ

× exp

2i (q − v − q ) · u |

2i |

| p − ∇q i

fλ (q ) q , λ .

·v (11.46)

Equating the right-hand side of Eq. (A36) to E ψ (r) to obtain an eigenvalue equation, we have in p-q space, 1 (N |3 )2

Aλλ (p, q) exp p,q,v,u,λ

2i (q − v − q ) · u | = E fλ (q ) . × exp

2i | p − ∇q | i

·v

fλ (q ) (11.47)

Now let us take the operator Aop to be the eﬀective one-particle Hamiltonian H defined in Eq. (11.30). It is be shown in Appendix B that the lattice Weyl transform

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of H is of the form where p and q occur only in the combination p − (e/c)A(q), by using magnetic Wannier functions and magnetic Bloch functions. Using this result in (11.47) we have 1 (N |3 )2

p ,q,v,u,λ

2i | = E fλ (q ) . × exp

Hλλ (p − (e/c)A(q); B, z) | p − ∇q i

· v + (q − q ) · u

fλ (q ) (11.48)

Performing a summation over v and u, we obtain, after some simplification the following Hλλ λ

| ∇q − (e/c)A (q) ; B, z i

fλ (q)

= E fλ (q) ,

(11.49)

which we can rewrite as Wλλ (π; B, z) fλ (q) = E fλ (q) ,

(11.50)

λ

where π=

| ∇q − (e/c)A (q) . i

The corresponding equation in p space can easily be deduced from Eqs. (11.37)-Eq. (11.40). This is given by Wλλ λ

| p − (e/c)A(− ∇p ); B, z i

fλ (p) = E fλ (p) .

Indeed in the absence of the magnetic field and for Σ(z) = 0 in Eq. (11.30), hence we have Wλλ (p) = Eλ (p) δ λλ , where Eλ (p) is the energy-band function, with trivial solutions fλ (p) = constant, i

fλ (q) = e | p·q . Because of the complicated many-body character of Σ(z) due to its dependence in the energy variable, the eﬀective Hamiltonian in Eq. (11.50) is in general nonHermitian; the eigenfunctions, though complete, are not orthogonal. Thus one needs to solve the adjoint problem. Except for the extra dependence on the energy variable z, magnetic field, etc., Eqs. (10.1)-(10.11) hold for the biorthogonal Wannier functions and biorthogonal Bloch functions and using these as basis states the

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eﬀective Hamiltonian assumes a “diagonal” form. Thus in the absence of the field we have o Hef f =

1 (N |3 )2

Hλo (p, z) exp p,q,λ,v,u

2i p−P ·v |

2i q − v − Q · u Ωλλ , |

× exp

(11.51)

where Ωλλ = p

= q

|p, λ, z

p, λ , z

(11.52)

|q, λ, z

q, λ , z .

(11.53)

Hλo (p, z) is given by 2i

Hλo (p, z) =

e|

p.v

v

q − v, λ, z| H0 + Σ (z) |q + v, λ, z ,

where |p, λ, z , p, λ , z , |q, λ, z , and q, λ , z are the biorthogonal Bloch and Wannier functions. In the presence of the magnetic field, it can be shown that the lattice Weyl transform of Ho + Σ(z) and the eﬀective Hamiltonian in p-q space can be reduced to an even (diagonal) form. This suggests the existence of biorthogonal magnetic Wannier functions and biorthogonal magnetic Bloch functions satisfying Eqs. (10.1)-(10.11) and Eq. (11.50) leads to an eﬀective Schrödinger equation Wλ (π; B, z) fλ (q) = E fλ (q) , which is free of interband terms.

(11.54)

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Path Integral Formulation

12.1

Evolution Operator and Sum over Trajectories

This section gives the path-integral formulation of the quantum dynamics of Bloch electrons in an external electromagnetic field, from a single-particle point of view. The evolution operator of the eﬀective Schrödinger equation is i U (t, to ) = exp − (t − to ) Hef f , |

(12.1)

where Hef f has the form of Eq. (11.1). The quantity, i exp − (t − to ) Hef f |

2 mn

gives the probability of transition from state n at t = to to state m at t = t and provides the accurate basis of the “golden rule” for quantum transition rates [16]. We are interested here in expressing the transition amplitude as a “lattice path integral.” A well-known procedure to carry this out is to make use of the following identity obeyed by the evolution operator: n+1

U (t, to ) =

U (tj , tj−1 ) ,

(12.2)

j=1

where tn+1 = t. Using the completeness of the Wannier eigenvectors which we will denote by the Dirac ket and bra notation, we have n+1

q∗ | U (t, to ) |qo∗ =

∗ qj∗ U (tj , tj−1 ) qj−1 ,

.... q1∗

∗ qn

(12.3)

j=1

where q ∗ in general represents all the quantum labels of the Wannier function. By making the time intervals infinitely small or by letting n ⇒ ∞, we can take advantage of the linearity, in incremental time, of the small time evolution operators to calculate the matrix elements, after which it can be written again in exponential form but this time as a c-number. In this manner, one can compose the matrix 146

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elements of a large time evolution operator from the matrix element of the small time evolution operators. We have, from Eq. (12.1), for infinitesimal time intervals i (tj − tj−1 ) Hef f + O (tj − tj−1 )2 . (12.4) | To calculate this matrix element of Hef f between two Wannier functions, we need the matrix element of the ∆ operator, involved in Eq. (12.2), as given by Eq. (11.5), U (tj , tj−1 ) ≈ 1 −

qj λj σ j | ∆λλ σσ (p, q) |qj−1 λj−1 σ j−1 = e(i/|)p·(qj −qj−1 ) δ 2q,qj +qj−1 δ λj ,λ δ λ λj−1 δ σj σ δ σj−1 σ , so that we can write, correct to first order in (tj − tj−1 ), qj λj σ j | U (tj , tj−1 ) |qj−1 λj−1 σj−1 i −1 = N |3 exp p · (qj − qj−1 ) | p,q i × exp − (tj − tj−1 ) H[λσ]j [λσ]j−1 (p, q) δ 2q,qj +qj−1 . |

(12.5)

The matrix element of a finite time evolution operator can therefore be written as q ∗ | U (t, to ) |qo∗

n

n+1

N |3

= lim

n⇒∞

i=1 q∗



i × exp − |

i=1

n+1 j=1

−1



(tj − tj−1 ) 

pi

pj ·

(qj −qj−1 ) (tj −tj−1 )

−H[λσ]j [λσ]j−1 pj , (qj +q2j−1 )



 ,

(12.6)

where (qj + qj−1 )/2 are restricted to lattice points only, as a result of carrying out the summation over q in Eq. (12.5). Note that in this paper the corresponding classical Hamiltonian function is rigorously obtained from the quantum Hamiltonian operator by the use of the lattice Weyl transform. A more revealing form of Eq. (12.6) can be obtained if we assume an eﬀective Hamiltonian derived from Eqs. (11.1)-(11.6) to be of the form P2 + V (Q) , (12.7) 2m∗ which is a Hamiltonian, for example, of a shallow impurity in a semiconductor. Then the lattice Weyl transform [Eqs. (11.2) or (11.4)] of Eq. (12.7) is clearly of the form Heff =

Hλλ σσ (p, q) =

p2 + V (q) . 2m∗

(12.8)

Substituting Eq. (12.8) in Eq. (12.6), and completing the square in the exponent so as to make a Gaussian p summation, we are led to the well-known Feynman path integral [17].

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12.2

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Path Integral in Quantum-Field Theory

In order to extend the notion of sum over trajectories for the finite time evolution operator to many-body problems, whose Hamiltonians are expressed in terms of field operators as dynamical variables, we need to extend the notion of eigenfunction and eigenvalues to field operators, i.e., to canonical variables with large or infinite number of components1 . For Bose field operators, that is, canonical field operators which obey the usual commutation relation as P and Q do, there is no diﬃculty in defining the eigenfunctions and eigenvalues and the path-integral formulation is formally identical as for Hamiltonian systems expressed in terms of P and Q. 12.2.1

Bose Systems

ˆ (R) be a field operator defined on a lattice and let π For a Bose system let φ ˆ (R) defined through the Lagrangian be a field operator which is canonically conjugate ˆ (R). Then we define eigenfunctions and eigenvalues as to φ ˆ λ (R) |q = qλ (R) |q , φ

(12.9)

π ˆ (R) |p = pλ (R) |p ,

(12.10)

ˆ λ (R) and where the field configuration qλ (R) is the eigenvalue of the field operator φ pλ (R) is the canonical conjugate variable to qλ (R); λ and R label the components ˆ and π of φ ˆ . The vector dot product now reads p·q =

pλ (R) qλ (R)

(12.11)

R,λ

With this convention for the vector dot product we have the same rules for the transformation function, completeness and orthogonality of the basis states, as for the continuous momentum and coordinate dynamical variables of N λ components, where N is equal to the total number of lattice points and λ is the total number of bands under consideration. Therefore Nλ is equal to the dimension of p and q in Eq. (12.11). Corresponding to Eq. (11.1) we have 1 2π|

Hef f =

Nλ

dNλ p

dNλ q H (p, q) ∆ (p, q) ,

(12.12)

where H (p, q) and ∆ (p, q) are given by H (p, q) =

∆ (p, q) = 1 See

Table 1.1.

dNλ v exp

dNλ u exp

i p·v |

i q·u |

1 1 q− v H q+ v , 2 2 1 1 p− u H p+ u , 2 2

(12.13)

(12.14)

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or by the equivalent expressions H (p, q) =

dNλ u exp

∆ (p, q) =

i q·u | i p·v |

dNλ v exp

1 1 p+ u H p− u , 2 2

(12.15)

1 1 q+ v H q− v . 2 2

(12.16)

Corresponding to Eq. (12.6), we have for the Bose field system q| U (t, to ) |qo = lim

n→∞

× exp

n

 i

|

n+1

dNλ qi

.... i=1

i=1

dNλ pi (2π|)Nλ

n

j=1

(tj − tj−1 ) pj ·

qj − qj−1 qj + qj−1 − H pj , tj − tj−1 2

  

. (12.17)

Thus, there is no diﬃculty in formulating the path integral for many-boson probˆ ∗ where φ ˆ ∗ is the complex conjugate lems. For a many-particle Bose system π ˆ = i|φ ˆ and hence p = i|φ ˆ∗. of φ, 12.2.2

Path Integral for Fermion Systems

In order to directly construct the path-integral formulation of many-body fermion (Bloch electron) problems, we have to extend the concept of eigenvalues and eigenvectors to canonical field variables that obey the anticommutation relations [18], [19]. Since the fermionic dynamical field operators anticommute, this means that we need a c-number base field of eigenvalues which anticommute (eigenvalues are elements of the Grassmann algebra) rather than the field of ordinary complex numbers. In other words, we will need a quantum representation in terms of external algebra rather than in terms of physical algebra or observables. The elements of this external algebra commute with the elements of the physical algebra or ordinary complex numbers, but anticommute with the canonical field operators. Let us denote the fermionic canonical field operators as ψ†m and ψr , where m and r subsume all indices pertaining to spin, band index, and lattice-point position coordinates. We will assume in what follows that the dimension of ψ† and ψ and their eigenvalues is even. Indeed, for the many-body problems considered here, this dimension is even and is equal to 2N λ, where the factor 2 accounts for spin, N is the total number of lattice points which form a finite field, and λ denote the total number of energy bands under consideration. The canonical field operators satisfy the second quantization or canonical field equations: ψα , ψβ = ψ†α , ψ†β

= 0,

(12.18)

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ψα , ψ†β

= δ αβ .

(12.19)

The following general eigenvector construction holds for both boson and fermion dynamical operators [20]: i |q = exp − P · q |

|q = 0 ,

(12.20)

i p | = p = 0| exp − p · Q . |

(12.21)

For fermions, we substitute P = i|ψ † . Thus for fermions, we have |q = exp ψ† · q

|q = 0 ,

p | = p = 0| exp (p · ψ) ,

ψ |q = 0 = 0,

(12.22)

p = 0| ψ† = 0,

(12.23)

and in addition we also have the following construction: q | = q = 0| exp −ψ† · q

,

|p = exp (−p · ψ) |p = 0 , where the “dot product” is defined by ψ† · q =

q = 0| ψ = 0,

(12.24)

ψ† |p = 0 = 0,

(12.25)

ψ†α qα . In Eqs. (12.22)-(12.25), α

we retained the symbol p and q for the eigenvalues wit the understanding that for fermions p and q are elements of a Grassmann algebra, not ordinary c-numbers. It is important to note that since ψ † and ψ are non-Hermitian, the operator group generating the eigenvectors is not unitary. Thus, in contrast with the {P, Q} canonical variables case, for fermions the eigenvector |0 will acquire a distinguished position in the external quantum-mechanical representation. Using the identity eA B e−A = B + [A, B], where the commutator [A, B] is a c-number, one can easily verify that indeed ψ †α |p = pα |p = |p p | ψ†α =

p | pα = pα p | ,

ψα |q = qα |q = |q q | ψα =

pα ,

qα ,

q | qα = qα q | .

(12.26) (12.27) (12.28) (12.29)

Furthermore, we obviously have the following diﬀerential operator realizations: ← − ∂ ∂ p|= p| − , (12.30) p | ψα = ∂pα ∂pα

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Path Integral Formulation

|q =

∂ − ∂qα

ψα |p =

∂ − ∂pα

ψ†α

ψ†α

q|

∂ ∂qα

=

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|q = |q

← − ∂ ∂qα

,

(12.31)

|p = |p

← − ∂ ∂pα

,

(12.32)

← − ∂ − ∂qα

,

(12.33)

|q = |q

where the arrows denote the left and right derivatives. Using the identity eA eB = eB eA e[A,B] for the case where the commutator [A, B] is a c-number, we have the following expression for the transformation functions: p | q = exp (p · q ) =

q | p = exp (−p · q ) =

epα

qα

=

(1 + pα qα ) ,

(12.34)

α

epα

qα

=

(1 + qα pα ) .

(12.35)

α

Using the infinitesimal form of the operator group generating the eigenvectors, Eqs. (12.22)-(12.25), one can also deduce the following relations [20]: q|q

= (q1 − q1 ) (q2 − q2 ) ..... (q2Nλ − q2Nλ ) = α

p|p

(qα − qα ) = δ (q − q ) ,

(12.36)

= (p2Nλ − p2Nλ ) ......... (p1 − p1 ) T

= α

(pα − pα ) = δ (p − p ) ,

(12.37)

where the delta function symbol is retained to denote the products of eigenvalue diﬀerences. The reason for this is that according to the definition of Grassmann integrals T

d [q ] δ (q − q ) =

k

T

d [p ] δ (p − p ) =

k

−

∂ ∂qk

∂ ∂pk

k

(qk − qk ) = 1,

(12.38)

(pk − pk ) = 1.

(12.39)

T

k

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Furthermore, using the resolution of identity given in Appendix D, particular to the external representation, we can derive the following: p|p

q|q where d [p] =

= δ (p − p ) =

= δ (q − q ) =

d [q] e(p −p )·q ,

(12.40)

d [p] e−p·(q −q ) ,

(12.41)

dpα represents the volume element in a multidimensional space. α

Perhaps the above relations are enough to convince the reader that the properties of external representation in terms of eigenvalues defined on a Grassmann algebra are formally identical to that of a quantum representation in terms of boson canonical variables. Other formally identical properties are derived in Appendix D. However, one must bear in mind that integration over anticommuting variables has an entirely diﬀerent significance, it signifies diﬀerentiation with respect to the integration variables. In all mathematical operations, the anticommutation property must be taken into account. The fact that we are dealing with even-dimensional Grassmann variables {p, λ, σ} and {q, λ, σ} will greatly simplify the task of taking anticommutation into account. We have relegated to Appendix D the details of how to directly construct the functional-integral representation of the matrix element of the evolution operator from the underlying external quantum-mechanical representation (c-numbers are elements of the Grassmann algebra). We have also derived the partition function and lattice temperature Green’s function as path integrals. The reader is referred to the derivation in Appendix D for the following results. The transition amplitude between a state Ψ specified at time to and a state Φ specified at time t is given by n+1

n+1

d [pj ]

Φ| Ψ = j=1

  × exp − 

d [qj ] φ (qn+1 ) j=0 n+1 j=1

pj ·

qj − qj−1

1 qj + qj−1 − H pj , | 2

  

ψ (qo ) , (12.42)

where qo = q at time to , qn+1 = q at time t = tn+1 , and = (t − to )/(n + 1). The wave functions are defined as φ (qn+1 ) = Φ| qn+1 and ψ (qo ) = qo | Ψ . H(p, q) is the Weyl transform of the many-body Hamiltonian H ψ† , ψ and is obtained from H ψ† , ψ by simple replacement of the i|ψ† to p and ψ to q. The zero-temperature i Green’s function is obtained from Eq. (12.42) by substituting Φ = Ψo | e( | )Ht ψr , i and Ψ = ψ† e(− | )Hto |Ψo and dividing the result by i Ψo | Ψo , where |Ψo is the s

ground state of the many-body system. A more useful result is obtained for the

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grand partition function ˆ e−βΩ = T r exp −β H − µN

ˆ . = T r exp −β K

We have the expression for the grand partition function given by [21]  n+1  n+1 qj − qj−1 qj + qj−1 + K pj , d [pj ] d [qj ] exp − pj · e−βΩ =  2 j=1

j=1

(12.43)

  

,

(12.44) where whenever qo appears in the action, we take qo = −qn+1 . We also have the equivalent expression [22]   n  n+1 p − p  p + p j j−1 j j−1 · qj−1 − K , qj−1 d [pj ] d [qj ] exp , e−βΩ =   2 j=0

j=1

(12.45) where = β/ (n + 1), and whenever pn+1 occurs in the action of Eq. (12.45) we take pn+1 = −po . Thus the partition function for finite β forces us to define the eigenvalues q and p to be antiperiodic2 with period β. That the corresponding canonical field operator may also be considered as antiperiodic with period β can be seen from the expression for the finite-temperature Green’s function (τ > τ ) [23]. The antiperiodic boundary condition on the eigenvalues p and q is precisely what is needed to preserve the antiperiodicity in each of the time variables of the Green’s function in the path-integral formulation [21]. Indeed, the finite temperature Green’s function can be written as a path integral: Gαγ (τ, τ ) = −eβΩ T r U (β, τ ) ψα U (τ , τ ) ψ†γ U (τ , 0) n+1

= −eβΩ

lim

→0 (n+1) =β

  × exp − 

= Z −1

n+1 j=1

dp

d [pj ] d [qj ] j=1

pj · qj −qj−1 q +q +K pj , j 2j−1

  

qατ pτγ

dq e−S(p,q) qα (τ ) pγ (τ ) ,

where n+1

dp

2 This

d2Nλ pj d2Nλ qj ,

dq = lim

n→∞

j=1

Z −1 = −eβΩ , can also be seen from the geometric phase viewpoint discussed in Sec. 14.3.

(12.46) (12.47)

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and β

S (p, q) = 0

dt p (t) ·

dq (t) + K (p (t) , q (t)) . dt

(12.48)

If we denote the right-hand side of Eq. (12.46) by qατ pτγ , then it is easy to show that qατ = pτγ = 0. Physically this is the consequence of the conservation of the number of particles, as this result can also be interpreted as ψα (τ ) = ψ†γ (τ) = 0. The one-particle Green’s function is defined as δ ψα (τ ) , η→0 δη γ (τ )

(12.49)

Gαγ (τ, τ ) = lim

where η γ (τ ) (also elements of the Grassmann algebra) are components of artificially introduced external sources in the extra symmetry-breaking term added to the Lagrangian of Eq. (12.48) to have a nonvanishing average value of the field operators; the desired extra term to be added to S(p, q) of Eq. (12.48) is β

β

η† · q dt,

p · η dt +

− 0

(12.50)

0

where η and η† are additional anticommuting variables. The right-hand side of Eq. (12.49) is a functional derivative, but reduces to ordinary diﬀerentiation in the discrete time step case defined by Eq. (12.46). Indeed Eq. (12.49) yields the well-known general formula for the finite-temperature Green’s function ˆ

Gαγ (τ , τ ) = eβΩ T r e−β K T ψ†γ (τ ) ψα (τ ) = T ψ†γ (τ ) ψα (τ ) ,

(12.51)

where T stands for the well-known time ordering symbol. The two- and manyparticle Green’s functions are similarly defined, e.g., δ 2 T ψα (τ 1 ) ψ γ (τ 2 ) η→0 δηµ (τ 1 ) δηδ (τ 2 )

Gαγ,δµ (τ 1 , τ 2 ; τ 1 , τ 2 ) = lim

= T ψ†δ (τ 2 ) ψ†µ (τ 1 ) ψα (τ 1 ) ψγ (τ 2 ) .

(12.52)

All of these Green’s functions can be obtained by diﬀerentiating (functional derivative in the continuum time limit) the generating functional G η, η† , G η, η† = eβΩ

dp



dq exp −S (p, q) +

β

0



p · η + η† · q dt ,

(12.53)

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with respect to the external sources η and η† , and taking the limit where η and η† go to zero. To complete the discussion on the path-integral formulation of the manybody system, we note from the preceding sections that for a system of fermions and bosons, requiring both commuting and anticommuting dynamical variables for its description, one can similarly construct the path integral by considering product eigenvectors. The generating functional will be of the form   β G η, η† , J = eβΩ

dq exp −S (p, q; φ∗ , φ) +

dp

p · η + η† · q − φ · J

0

dt ,

(12.54)

β ∗

S (p, q; φ , φ) == 0

dt p (t) ·

dq (t) dφ (t) + φ∗ (t) · − H (p (t) , q (t) ; φ∗ , φ) , dt dt (12.55)

eβΩ =

dp

dq

dφ∗

dφ exp [−S (p, q; φ∗ , φ)] .

(12.56)

ˆ∗, φ ˆ by the replacement i|ψ† → p, H (p (t) , q (t) ; φ∗ , φ) is obtained from H ψ† , ψ; φ

ˆ → φ, with periodic boundary conditions for the boson variables, ˆ ∗ → φ∗ , φ ψ → q, φ and antiperiodic boundary conditions for the fermion variables as before, over the length of t equal to β. We close this section by applying the results obtained here to the many-body Hamiltonian of Eq. (11.29). We write explicitly the eﬀective Hamiltonian of Eq. (11.29) (α and β do not include spin indices σ): Hef f ψ† , ψ =

(1)

(2)c

ψ†α Wαβ ψβ + α,β

−

α,β

(2)c

ψ†α ψα Wαβ ψ†β ψβ − ψ†α ψα Wαβ

ψ†β ψ β

HF

(2)ex † σ 1 σ 2 ψ β ,σ 2 ψ β ,σ 1

ψ†α ,σ1 ψα ,σ2 Wα β α ,β ,σ 1 ,σ2 (2)ex

−ψ†α σ1 ψα σ2 Wα β σ1 σ2 ψ†β σ2 ψβ σ1

HF

,

(12.57)

where ..... HF may be approximated by the Hartree-Fock ground-state average, and from Eqs. (11.27), (11.16), and (11.17), (1)

Wαβ = exp

ie ˜ (1) (q1 − q2 ; B) δ λλ δ σσ , A (q2 ) · q1 W λ |c

(2)c ˜ (2)c (0, 0; 2q1 , 2q2 ; B) , Wαβ = W λ1 λ2

(2)ex σ1 σ2

Wα β

˜ (2)ex (q1 − q2 , q2 − q1 ; q1 + q2 , q1 + q2 ; B) . =W λ1 λ2

(12.58)

(12.59)

(12.60)

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The Weyl transform H(p, q) or Weyl symbol (corresponding c-number function) of Heff is then obtained by replacing i|ψ† by p wherever it occurs and similarly substituting q for ψ. We then have K(p, q) = H(p, q) − µ p · q which can be substituted in all path-integral formulas of this section.

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Chapter 13

Gauge Theory and Geometric Phase in Quantum Systems

For a more complete treatment of the quantum mechanical methods, we will give in this chapter a short introduction to a new area in quantum physics that has been developing for more than five decades now. This has to do with the geometric and topological aspects of quantum systems1 . The virtue of this new development lies in the universality of new ideas connected with geometric aspect of quantum systems. This development was heralded with the work of Pancharatnam on phase shifts of polarized light which is allowed to undergo a cyclical sequence of polarization changes, with each successive polarization state in phase with the previous one (‘nearest successive state phase diﬀerence is zero’) from initial state and back to initial state forming a geodesic triangle on the Poincare sphere (a Poincare sphere, originally conceived by Henri Poincare in about 1892, provides a convenient way of representing polarized light). Pancharatnam discovered that the phase diﬀerence between the initial and final polarization states equals half of the solid angle subtended by the geodesic triangle on the Poincare sphere. The excitement of this new trend was ushered by the rise of the geometric gauge theories of elementary particles, starting with Yang-Mills gauge theory. All these basically set the stage for Berry’s paper on Berry’s phase, whose impact is to point out the universality of geometric gauge field ideas in physics. On the elementary particle side, further developments today has to do with topological aspects of string theory as theory of knots, as well as the convergence of string theory with gauge theory and broken symmetry of elementary particle physics. In condensed matter physics, gauge theoretical ideas has been notably applied to quantum Hall eﬀect and Jahn-Teller eﬀect, while in molecular physics it is applied to the Born-Oppenheimer approximation. To have a fundamental grasp of the topic of this chapter, we need an understanding of a central notion in geometry, the notion of connection. This notion occupies a central importance in modern geometry because it allows a comparison between local geometry of one point to the local geometry of another point on a curve manifold. This general notion of connection allows in turn the general notion of parallel transport and holonomy. The motivation is to be able to build physical theories out of quantities that can be translated to another coordinate without depending on a particular coordinate 1 This

has to do with the complex structure of the background vacuum condensate. 157

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system. Physics should be independent of the coordinate system. This is the interest placed on tensors in what follows. 13.1

Directional (Covariant) Derivative on Curve Spaces

Let us denote our generalized coordinates as xi , which has local basis vectors, ei . Thus there is a vector space tied to the point xi (example is the tangent vector space at a point on the surface of a sphere). Let V = V j xi ej xi (summation convention holds between two like contravariant and covariant indices, j, of vector components), where ej xi are the basis vectors belonging to point xi . Then

We can expand

∂ej ∂xk

d V j ej = dt

∂V j ∂ej ej + V j k k ∂x ∂x

dxk , dt

d V j ej = dt

∂V j ∂ej ej + V j k ∂xk ∂x

dxk . dt

in terms of the basis vectors ∂ej = Γlkj el . ∂xk

Thus, we have dV j ej = dt dV j ej = dt

∂V j ej + V j Γlkj el ∂xk

dxk , dt

∂V j dxk . + V l Γjkl ej k ∂x dt

The last equation suggests modifying the directional derivative of the components of a vector along direction ek in curve spaces by introducing the notion of covari˘ which is independent of the coordinate system. For covariant ant derivative, ∇, derivation along the direction ek , this is given by ∇ek V j ej =

∂V j + V l Γjkl ej . ∂xk

(13.1)

In particular the covariant derivative of basis vector field ej along ek is ∇ek ej = Γlkj el . Indeed, the change of V due to change of coordinate frame alone is given by ∆V |∆ei = ei (xk + ∆xk ) · V − ei (xk ) · V = ei (xk + ∆xk ) − ei (xk ) · V,

∆V |∆ei = ∂ei /∂xk .V ∆xk .

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From ∂ei = Γjki ej , ∂xk we therefore have ∆V |∆ei = (∆ei ) · V = Γjki ej ∆xk .V. Therefore, the change of basis ej along the curve is determine by the geometrical Christoﬀel symbol of the second kind, Γkij , also known as a metric connection or aﬃne connection, i.e., ∆ej = Γkij ek ∆xi . We further note that em ·

∂ej = em · Γkij ek ∂xi = Γm ij ,

where we have use the relation, em · ek = δ m k . 13.2

Parallel Transport in Curvilinear Space

Take the earth surface as an example of a curvilinear space, then locally any vector will have specified direction with respect to our local coordinate axes. To perform parallel transport to nearby local region, the transported vector must have identical direction with respect to our new axes. The mathematical formulation is as follows. In this local space we suppose the existence of a constant Cartesian vector field, V , that is one with constant magnitude and sense. A vector V is parallel transported along a curve xi = φi (t) if dV dt = 0 along this curve. For parallel transport, dV j ej = dxk

∂V j + V l Γjkl ej = 0, ∂xk

i.e., the covariant derivative of V goes to zero ∇ek V j ej =

∂V j + V l Γjkl ej = 0, ∂xk

∂V j + Γjkl V l = 0. ∂xk The last equation is really a balance equation in that the change of the components of V due to change of coordinates (geometrical) must be balanced by the change of the pertinent component of V as a function of xk along the curve (dynamical). This balance equation defines a constant vector along the curve, since dxk ∂V j = −Γjkl ∂t dt

V l,

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i.e. the global change of V is entirely due to change of coordinate frame belonging to xk (t), hence locally, i.e. in each respective frame the vector looks the same and thus has constant magnitude and sense (phase). If we define the matrix A by A = Γjkl dxk , we then say that A defines the connection on a curvilinear surface. dV j = −Ajl V l . Although the elements of A have been indexed as if they are tensors, a connection is not a tensor, that is Γjkl are not the components of a tensor; but the diﬀerence of two connections, in particular, any infinitesimal variation of a connection, is a tensor. The covariant derivative can be described by a “tensor” in a fixed coordinate chart, but it is not a true tensor in the sense that it is not invariant under coordinate changes. The relation Γjkl = Γjlk is independent of the coordinate system and indicates that the connection is torsionless. The definition of the covariant derivative does not use the metric in space. However, a given metric uniquely defines a special covariant derivative called the Levi-Civita connection. It worthwhile mentioning at the outset that in physics, the idea of axiomatically introducing connections from physical point of view is also accepted, as in the use of Pancharatnam connection in his work on phase shifts of polarized light which is allowed to undergo a cyclical sequence of polarization changes. The germ of this idea goes back nine decades ago to Herman Weyl in his first attempt in gauge theory. Physically and mathematically, it is easy to realize that one should not be limited to transporting just tangent vectors and tensors along curves, and that one should also think of axiomatically attaching vectors to points of the manifold and transporting them using connections axiomatically defined in the appropriate way. This liberation of the concept of a connection from its metric origins, turned out to be important in the application of the ideas of diﬀerential geometry to physics.

13.3

Parallel Transport Around Closed Curve

The mathematical nature of parallel transport along a closed curve can then be taken as parallel transport along two vectors, ∆ui and ∆vi . We shall see that, in the limiting case, for a spherical surface the total change of a vector V is determined by the solid angle subtended by the closed curve. The following calculation describes this situation

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∆VTi ot = −Γijk V j ∆uk − Γijk (x + ∆u)V j (x + ∆u)∆vk

+ Γijk (x + ∆v)V j (x + ∆v)∆uk + Γijk V j ∆vk

= Γijk (x + ∆v)V j (x + ∆v) − Γijk V j ∆uk Γijk (x + ∆u)V j (x + ∆u) − Γijk V j

−

∆v k

∂Γijk V j ∂Γijk V j l k ∆v ∆u − ∆ul ∆vk ∂xl ∂xl ∂Γijk V j ∂Γijk V j k l = ∆u ∆v − ∆vk ∆ul ∂xl ∂xl =

=

j ∂Γijk j i ∂ V V + Γ jk ∂xl ∂xl

∆uk ∆vl −

j ∂Γijk j i ∂ V V + Γ jk ∂xl ∂xl

∆vk ∆ul .

Because this is a parallel transport, a condition is met that the covariant derivative of the vector V i is zero and hence by Eq. (13.1): ∂V i = −Γijk V j , ∂xk ∂ Vj = −Γjml V m . ∂xl Then we have, ∆VTi ot = =

j ∂Γijk j i ∂ V V + Γ jk ∂xl ∂xl

∂Γijk V j + Γijk −Γjml V m ∂xl −

=

∆uk ∆vl −

∆uk ∆vl −

∆vk ∆ul

∆uk ∆v l

∂Γijk V j + Γijk −Γjml V m ∂xl

∂Γijk Vj ∂xl

j ∂Γijk j i ∂ V V + Γ jk ∂xl ∂xl

∆vk ∆ul

∂Γijk Vj ∂xl

∆vk ∆ul

− Γijk Γjml V m ∆uk ∆vl + Γijk Γjml V m ∆vk ∆ul . Upon interchanging dummy indices, m ⇔ j, l ⇔ k in the terms containing product of Christoﬀel symbols, we have ∆VTi ot =

∂Γijk Vj ∂xl

∆uk ∆vl −

∂Γijk Vj ∂xl

∆vk ∆ul

− Γijk Γjml V m ∆uk ∆vl + Γijk Γjml V m ∆vk ∆ul

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Fig. 13.1 Example of parallel transport of a vector around a geodesic triangle on the sphere. The length of the transported vector and the angle it makes with each side remain constant locally. The angle by which it twists, α, is proportional to the area inside the loop. Hence, the final vector has been rotated compared to the initial one, and the rotation angle is the solid angle enclosed by the loop. The global rotation without local rotation is often referred to as the holonomy caused by the curvature of the sphere.

∂Γijk Vj ∂xl

=

∆uk ∆vl −

∂Γijl Vj ∂xk

∆vl ∆uk

j j ∆ul ∆vk + Γimk Γm ∆vk ∆ul − Γiml Γm jk V jl V

∂Γijk ∂xl

=

−

∂Γijl i m − Γiml Γm jk + Γmk Γjl ∂xk

V j ∆uk ∆vl .

This gives a final succinct result of:

∆VTi ot =

∂Γijk ∂xl

−

∂Γijl i m − Γiml Γm jk + Γmk Γjl ∂xk

V j ∆uk ∆v l

= Rijkl V j ∆uk ∆vl , where Rijkl is the Riemann Curvature tensor. For spherical surface Rijkl V j reduces to r12 V i . Therefore ∆VTi ot = ΩV i , k

l

where Ωkl = ∆ur2∆v , is the solid angle and r is the radius of the sphere. Note that for simple cases the above calculation may also be written as dV i = Γ dx Vi = Ω.

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13.4

163

Generalization to Quantum Mechanics

The above results can be generalized to higher dimensions (such as 4-dimensional space-time and other abstract spaces), this is in the realm of diﬀerential geometry, especially in the study of diﬀerential forms and connections. What we are interested here in its generalization to quantum physics. First observation is that the covariant derivative, Eg. (13.1), bears close resemblance to the minimal coupling of vector potential in momentum operator. Indeed in gauge theory, the Christoﬀel symbol emerges as a gauge field A (xµ ) or vector potential in the gauge theory of electrodynamics. In generalizing to quantum mechanics, we make the following correspondence to our tensor analysis given above covariant basis vector ei xi |i, R

contravariant dual basis ei (xi ) i, R|

components V i xi ψi (R)

parameters xi R

The |i, R are the eigenstates of the Schrödinger equation, H (R) |i, R = Ei (R) |i, R ,

(13.2)

and the Hilbert vector space spanned by the eigenstates at point R maybe viewed as corresponding to the tangent space in tensor analysis. Corresponding to the Christoﬀel symbol, we have for the geometry of the parameter space described by ∂ |i, R , ∂R ∂ |l, R . = j, R| ∂R

γm R,i = m, R| γ jR,l

Therefore for parallel transport along a curve in parameter space, the change of the wavefunction ψj due to evolution of R (t) is thus given by ∂ψ j = ∂t =

−γ jRl

dR dt

− j, R|

ψl

dR ∂ |l, R ∂R dt

ψl .

For a nondegenerate eigenstate highly removed from other quantum eigenstates, we may neglect the oﬀ-diagonal matrix elements and assume that j, R|

∂ ∂ |l, R ψl = δ jl j, R| |l, R ψl ∂R ∂R ∂ |j, R ψj . = j, R| ∂R

Therefore, we can also write ∂ dR d log ψj = − j, R| |j, R · , dt ∂R dt

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d log ψj = − j, R|

∂ |j, R · dR. ∂R

Since ψj is normalized, the change amounts to the change in the phase i∆φj of the wavefunction ψj . The total phase change around a closed curve C in parameter space is thus given by i∆φj

total

(C) = −

j, R|

∂ |j, R · dR, ∂R

C

∆φj

total

(C) =

j, R| i

∂ ∂R

|j, R · dR,

C

∆φj

total

(C) =

q |c

A (R) · dR. C

The last line looks like the change of phase in Aharonov-Bohm eﬀect. We can also write ∂ |j, R · dR. (13.3) j, R| ∆φj total (C) = − Im ∂R C

The last expression was reported by Berry in his often-cited 1984 paper2 , where parallel transport in parameter space is physically defined to be an adiabatic process where each eigenstate will evolve with H (R (t)), i.e., there will be no crossing of energy levels. Using Stokes theorem, we rewrite Eq. (13.3) as ∆φj

total

(C) = − Im

j, R|

∂ |j, R · dR ∂R

C

= − Im

dS · ∇R × j, R| ∇R |j, R S

= − Im

dS · ∇R × S

= − Im

dS · S

m=j

m

j, R| m m ∇R |j, R

( ∇R j, R| m ) × ( m |∇R j, R ) , (13.4)

where dS is the element of an area bounded by the curve C in parameter manifold. To evaluate the oﬀ-diagonal matrix elements, we make use of the Heisenberg relation, ∂H (R) = P˙ , ∂R 2 M.

V. Berry, Proceedings of the Royal Society of London, A, 392, 45 (1984).

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and i|P˙ = [P, H (R)] ∂ , H (R) . = i| ∂R We have for the oﬀ-diagonal matrix elements, m = j, m, R|

∂H (R) ∂ , H (R) |j, R = m, R| |j, R ∂R ∂R ∂ Ej (R) |j, R = m, R| ∂R ∂ |j, R −Em (R) m, R| ∂R ∂ Ej (R) δ mj = ∂R + (Ej (R) − Em (R)) m, R|

∂ |j, R . ∂R

We evaluate the matrix elements from Eq. (13.2) to yield m, R| ∇R |j, R =

m, R| ∇R H (R) |j, R . Ej − Em

Therefore ∆φj

total

(C) = −

dS · Vj (R) , S

where Vj (R) = Im

m=j

( j, R| ∇R H (R) |m, R ) × m| ∇R H (R) |j, R (Ej − Em )2

= ∇R × (Im j, R| ∇R |j, R ) q (∇R × A (R)) = |c q = B (R) , |c and A (R) =

|c Im j, R| ∇R |j, R . q

Note that B (R) has zero divergence, ∇R · B (R) = 0.

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13.5

Born-Oppenheimer Approximation

The above analogies for the generalization to quantum mechanics has a strong relevance to the physics of systems consisting of two coupled subsystems, where one has a much faster dynamics compared to the other subsystem. This situation arise in molecular physics using the Born-Oppenheimer approximation. The full Hamiltonian for the fast (p, r) and slow (P, R) dynamical variables is H=

P2 p2 + + V (R, r) . 2M 2m

The wavefunction Ψ is separated into nuclear and electronic components Φn (R) ψn (r, R) .

Ψ= n

The Hamiltonian for the fast subsystem, ψn (r, R), depends parametrically on the slow coordinates, namely, H (R) =

p2 + V (R, r) , 2m

which have ‘instantaneous’ eigenfunctions |n, R (with respect to variable R). H (R) |n, R = En (R) |n, R . Thus the full Schrödinger equation is

n

P2 + H (R) Φn (R) ψn (r, R) = Ξ 2M

Φn (R) ψn (r, R) . n

Upon integrating the electronic degrees of freedom, we have drψ∗m (r, R) n

=Ξ

Φn (R)

P2 + H (R) Φn (R) ψn (r, R) 2M drψ∗m (r, R) ψn (r, R) ,

(13.5)

n

 n

  

2

P drψ∗m (r, R) 2M Φn (R) ψn (r, R)

+

drψ∗m (r, R) H (R) Φn (R) ψn (r, R)

drψ∗m (r, R) n

drψ∗m (r, R) n



  = ΞΦm (R) , 

(13.6)

P2 Φn (R) ψn (r, R) + Em (R) δ mn Φn (R) = ΞΦm (R) , 2M (13.7)

P2 Φn (R) ψn (r, R) + Em (R) Φm (R) = ΞΦm (R) . (13.8) 2M

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Now the kinetic energy term function, thus

P2 2M ,

167

operates on both the slow and fast variable wave-

−|2 ∇2R Φn (R) ψ n (r, R) = −i|∇R [((−i|∇R ) Φn (R)) ψn (r, R) + Φn (R) (−i|∇R ) ψn (r, R)]

−|2 ∇2R Φn (R) ψn (r, R) + ((−i|∇R ) Φn (R)) (−i|∇R ) ψn (r, R)

=

+ (−i|∇R ) Φn (R) (−i|∇R ) ψn (r, R) + Φn (R) −|2 ∇2R ψn (r, R)

.

Therefore the Schrödinger wave equation for the slow variable is    −|2 ∇2R Φn (R) drψ∗m (r, R) ψn (r, R)        + ((−i|∇R ) Φn (R)) drψ∗ (r, R) (−i|∇R ) ψ (r, R)   m n   1       2M   + (−i|∇R ) Φn (R) drψ∗m (r, R) (−i|∇R ) ψn (r, R)  n        +Φn (R) drψ∗m (r, R) −|2 ∇2R ψn (r, R) + Em (R) Φm (R) = ΞΦm (R) ,  n

 1    2M

    

−|2 ∇2R Φn (R) δ mn q + (−i|∇R Φn (R)) −| |c Amn (R) q Amn (R) + (−i|∇R ) Φn (R) −| |c +Φn (R)

drψ∗m (r, R) −|2 ∇2R ψn (r, R)

= ΞΦm (R) .



   + Em (R) Φm (R) 

Now Φn (R)

drψ∗m (r, R) −|2 ∇2R ψn (r, R)

= Φn (R) (−i|∇R ) · − Φn (R)

j

drψ∗m (r, R) {−i|∇R } ψn (r, R)

dr ((−i|∇R ) ψ∗m (r, R) |j, R )

· ( j, R| (−i|∇R ) ψ n (r, R)) . We can again write −

dr (−i|∇R ψ∗m (r, R) |j, R ) · ( j, R| − i|∇R ψn (r, R)) = − −i|∇R m, R| |j, R · j, R| |−i|∇R n, R = m, R| |−i|∇R j, R · j, R| |−i|∇R n, R q 2 = |2 Amj (R) · Ajn (R) . |c

(13.9)

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Substituting in Eq. (13.9), we have Φn (R)

drψ∗m (r, R) −|2 ∇2R ψn (r, R) = Φn (R) |2

q |c

2

Amj (R) · Ajn (R)

The Hamiltonian for the slow subsystem becomes    −|2 ∇2R Φn (R) δ mn q     1  + (−i|∇R Φn (R)) · − cq Amn (R)     + (−i|∇R Φn (R)) · − Amn (R)  c  2M   + Em (R) Φm (R) 2 q    n Amj (R) · Ajn (R)  +Φn (R) c j

= ΞΦm (R) ,

 n



  −|2 ∇2R δ mn + − qc Amn (R) · (−i|∇R ) q  1    + − c Amn (R) · (−i|∇R )   Φn (R) + En (R) δ mn  q 2  2M    Amj (R) · Ajn (R) + c j

= ΞΦm (R) ,

n



 1   2M

j

δ mj (−i|∇R ) −

q c

Amj (R) · δjn (−i|∇R ) −

q c

Ajn (R)

+En (R) δ mn

= ΞΦm (R) ,

n





 1  2M

j

= Ξ Φm (R) .







 Φn (R)



q q δ mj P − Amj (R) · δ jn P − Ajn (R) + En (R) δ mn  Φn (R) c c

In Born-Oppenheimer approximation, the oﬀ-diagonal matrix elements Amj ⇒ δ mj Amj , and Ajn (R) ⇒ δ jn Ajn (R), i.e., the mixing of diﬀerent energy levels are ignored. Therefore, for a nondegenerate electronic energy level, Em (R), we have q 1 q P − Am (R) · P − Am (R) + Em (R) Φm (R) 2M c c = Ξ Φm (R) , where Am (R) = Amm (R). We can also write the eﬀective Schrödinger equation for the nuclear motion as, P − qc Am (R) 2M

2

+ Em (R) Φm (R) = Ξ Φm (R) .

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The resulting eﬀective Hamiltonian looks like the Hamiltonian of a charged particle in the presence of background magnetic potential, Am (R). In fact there is an arbitrariness in the specific form of Am (R) depending on the choice of basis ψn (r, R). If one makes a diﬀerent choice, namely, ψ (r, R (t)) = Ω (R (t)) ψ (r, R (t)) where Ω (R (t)) is the transformation matrix appropriate at point R (t) in parameter space, ψ (r, R (t)) represent the transformed basis eigenstates (a column vector) and ψ (r, R (t)) is the column vector of old basis states. Then the A (R) field will transform like q A (R) = RΩ−1 i∇R Ω |R |c = RΩ−1 (i∇R Ω) |R + Ω−1 R| i∇R |R Ω = iΩ−1 ∇R Ω j, R| 1 |j, R + Ω−1 j, R| i∇R |j, R Ω q = iΩ−1 ∇R Ω + Ω−1 A (R) Ω. |c

(13.10)

Equation (13.10) is given for a general case where the connection A (R) is matrix valued and Ω is the matrix of the basis transformation. We have also implicitly taken out of the integral over the electron coordinate those quantities that are only function of the nuclear coordinate R. In the simplest Abelian case , A (R) is a vector function, everything commutes, and Ω belongs to the U (1) symmetry group represented by eiθ . Then Eq. (13.10) reduces to q q A (R) = A (R) − ∇R θ |c |c q q = A (R) − ∇R χ (R) , |c |c which gives us the well-known electromagnetic gauge transformation of the vector potential, A (R) = A (R) − ∇R χ (R) . It does appear that the nuclei in the Born-Oppenheimer approximation behave like charged particles in a magnetic field given by B (R) = ∇ × A (R). When the nuclei go around a closed path, the nuclear wavefunction will accumulate a geometrical phase proportional to the enclosed magnetic flux, ∆φj

total

(C) = −

dS · Bj (R) . S

This phase ∆φj total (C) is nothing but Berry’s phase that a nondegenerate electron wavefunction accumulates when their external parameters R are slowly varied, this has been passed down to the nuclear wavefunction when we integrate out the electronic coordinates in the full Schrödinger equation in Eqs. (13.5)-(13.8). Note that the Berry phase is based on the assumption of a nondegenerate state without mixing of energy eigenstates.

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Chapter 14

Generalizations of Geometric Phase: Fiber Bundles

Berry’s initial discussion in terms of an adiabatic evolution and a single phase described by U (1) symmetry have been generalized extensively through the unifying mathematical language of fiber bundle. This mathematical technique describes Berry phase as a special case of a holonomy on a fiber bundle. Therefore it is worthwhile to give a brief introduction of the mathematics of fiber bundle in the following section. 14.1

The Fiber Bundle Concept

The fiber bundle is a mathematical concept which may be viewed simply as a generalization of the Cartesian plane in x (x belongs to one-dimensional space X) and y (y belongs to a one-dimensional space Y ) coordinates, where the Cartesian plane is now considered as a collection of one-dimensional Y spaces ‘tied’ or labeled by the point x X . In this sense the collection of Y spaces over X space is a fiber bundle, where the Y space is the fiber and the X space is the base space (one can therefore say that the one-dimensional Y spaces are tied together by the one-dimensional base space to form a ‘bundle of fibers’) . However, the Cartesian plane, CP, is a trivial fiber bundle because it is globally a direct product of the Y space and the X space, i.e., CP = X × Y globally. Trivial fiber bundles do not play a role in recent advances in theoretical physics. It is the nontrivial fiber bundles that bears immense impact in the recent development of modern theoretical physics, resulting in great advances in quantum physics while shedding more light on classical physics. Indeed the fiber bundle concept gives a unifying mathematical structure and powerful analytical tool in geometry, astronomy, classical mechanics, hydrodynamics, dissipative kinetics, stochastic processes, and quantum theory. For example, the Hannay angle in classical mechanics can be obtained as a classical limit of the quantum mechanical Berry phase1 . In general, a fiber bundle has a fiber Y space generalized to a manifold as well as the base X space2 , sort of a further generalization of tensor analysis based on 1 See, for example the book by D. Chruscinski and A. Jamiolkowski, Geometric Phases in Classical and Quantum Mechanics, Birkhauser, (2004). 2 A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated. In discussing

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Riemannian manifold. A fiber bundle is a space which locally3 looks like a product of two spaces but may possess a diﬀerent global structure. Examples of trivial bundles are shown in Figs. 14.1(a) and (d) and nontrivial fiber bundles in Figs. 14.1(b), (c), and (e). From these figures, we can formally say the following. A fiber bundle B = {B, X, π, G, Y } is a topological space B which is locally, but not necessarily globally, a product space. Y is the fiber, and π is defined as a map called the projection from the total topological space B to the base X , i.e., π : B ⇒ X and, for x X, the inverse image Yx = π−1 (x) is the fiber over x (note: x X). A fiber bundle also comes with a group action on the fiber. Fiber bundles that have symmetry group acting upon the fiber is of particular interest in physics. This group action represents the diﬀerent ways the fiber can be viewed as equivalent. For instance, the group on a vector bundle is the group of invertible linear maps, which reflects the equivalent descriptions of a vector space using diﬀerent vector space bases. G is the topological group which acts on the fiber. For example, for trivial fiber bundle B = X × Y which is a globally a product of base X and fiber Y , then G consist only of identity element. In nontrivial fiber bundle, the local trivializations are glued together to form a bundle and this is where the group of the bundle G becomes relevant in terms of connection compatible with the action of the group G. In other words, we need rules to relate two diﬀerent coordinate systems on the fibers residing over overlapping small neighborhoods of the base space. The action of G on the fibers is considered a change of coordinates for each fiber at neighboring points in the base manifold. Indeed, the rule for parallel transport is a rule for changing ones fiber coordinates as one moves infinitesimally over the base space X. We mention here some of the fiber bundles, namely, line bundles, vector bundles, and principal bundles. A line bundle expresses the concept of a line that varies from point to point of a manifold. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organizing these. More formally, a line bundle is defined as a vector bundle of rank 1.4 Vector manifolds, the idea of dimension is important. For example, lines are one-dimensional, and planes two-dimensional. In a one-dimensional manifold (or one-manifold), every point has a neighborhood that looks like a segment of a line. Examples of one-manifolds include a line, a circle, and two separate circles. In a two-manifold, every point has a neighborhood that looks like a disk. Examples include a plane, the surface of a sphere, and the surface of a torus. The trivial example of an n-manifold is the n-dimensional Euclidean space. A more abstract and yet very interesting manifold is the Lie group. Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis 3 Locally pertains to an small neighborhood of the base space. 4 There is a diﬀerence between one-dimensional real line bundles and one-dimensional complex line bundles. The first of those is a space homotopy equivalent to a discrete two-point space (positive and negative reals contracted down), while the second has the homotopy type of a circle. A real line bundle is therefore as good as a fiber bundle with a two-point fiber - a double covering. This appears as the orientation double cover on a diﬀerential manifold: indeed that’s a special case in which the line bundle is the determinant bundle (top exterior power) of the tangent bundle. The Möbius strip corresponds to a double cover of the circle (the θ ⇒ 2θ mapping) and

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Fig. 14.1 (a) The base space of the cyclinder is a circle, S1, and the fibers are line segments, L. The total space is the direct product S1 × L, (b) the topology of the total space of the Möbius Strip cannot be covered continuously with a single product of the base space and the typical fiber, unlike the cyclinder. Moreover, Möbius Strip has only one side and one edge! (c) the Möbius Strip corresponds to a double cover of the circle and can be viewed as having fiber two points, (d) a torus is a closed surface defined as the product of two circles: S1 × S1, (e) the Klein bottle is a certain non-orientable surface, i.e., a surface (two-dimensional manifold) with no distinct “inner” and “outer” sides. Whereas a Möbius strip is a two-dimensional surface with a boundary, a Klein bottle has no boundary. Note that a sphere is an orientable surface with no boundary. The Klein bottle can be constructed (in a mathematical sense only, because it cannot be done without allowing the surface to intersect itself) by joining the edges of two Möbius strips together. By adding a fourth dimension to the three dimensional space, the self-intersection can be eliminated. A useful analogy is to consider a self-intersecting curve on the plane; self-intersections can be eliminated by lifting one strand oﬀ the plane along the third dimension.The Klein bottle shown is immersed in three-dimensional space. The base space B is actually the circle S1, and so the Klein bottle is the twisted S1-bundle (circle bundle or fiber bundle with fiber S1) over the circle.

bundles are those where the fibers are vector spaces5 . A bundle B = {B, X, π, G, G} is called a principal bundle if the fiber Y is the same as the group G, and the group of the bundle acts upon the fiber by left-translations6 . A bundle B = {B, X, π, G, Y } can be viewed as we wish as having fibre two points, the unit interval or the real line: - these data are equivalent. In the case of the complex line bundle, we are looking in fact also for circle bundles where the fiber is the circle, or, more precisely, a principal U(1)-bundle. It is homotopically equivalent to a complex line bundle. 5 The collection of tangent vectors forms the tangent bundle, and a vector field is a section of this bundle. 6 Left translation and conjugation are two actions in which G acts on its own members. In the former, g moves y onto g ∗ y. In the latter, g moves y onto g ∗ yg . Using left translation, any finite group can be represented as a permutation group.

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is said to be associated with the principal bundle B if the group G acts eﬀectively upon the topological space Y . Another example of principal bundle pertains to the universal cover of X , it yields a principal bundle over X with structure group π, the projection from the total space, [see Fig. 14.1(c)]. A connection is a mathematical device that defines a notion of parallel transport on the bundle; that is, a way to “connect” or identify fibers over nearby points. It is mathematically identical to the so-called gauge potential7 . In a certain sense, it captures the idea of Christoﬀel symbols on a Riemannian manifold and re-expresses this idea in a more general and abstract way, so that it is applicable on a principle bundle. A principal G-connection on a principal G-bundle B over a smooth manifold X is a particular type of connection which is compatible with the action of the group G. In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can diﬀerentiate sections of that bundle along tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the frame bundle8 of a smooth manifold. Indeed, principal connection is a very important concept in the recent development of a broad area of theoretical physics, in fact it may be said that gravitational forces arises from communication (‘connection’) between points in space-time manifold, and likewise for gauge theories of elementary-particle interactions introduced through the imposition of local symmetries of the Lagrangian.

14.2

Generalizations of Berry’s Geometric Phase in Quantum Physics

We will not go into details of various generalizations of Berry geometric phase, as guided by the conceptual tool and abstract mathematical structure of fiber bundles. We have given a brief introduction to fiber bundles above in order to appreciate generalizations of Berry’s geometric phase in quantum mechanics. We will only summarized here these generalizations, serious readers can follow all the pertinent references9 and new and continuing developments on the subject. Essentially Berry found that, in addition to familiar dynamical phase factor, a geometrical phase factor appears in an adiabatic and cyclic quantum evolution with a nondegenerate eigenstate of a Hamiltonian. After Berry’s work, Simon[24] pointed out that the Berry phase can be regarded as a holonomy on a line bundle over a parameter space. This early realization relating Berry’s findings to the mathematical structure of fiber bundles opens up various generalizations of the geometric phase 7 The

vector potential in electromagnetism. frame bundle is a principal fiber bundle F (E) associated to any vector bundle E. The fiber of F (E) over a point x is the set of all ordered bases, or frames, for Ex . The general linear group acts naturally on F (E) via a change of basis. Thus, the prototypical example of a principal bundle is the frame bundle of a smooth manifold. The frame bundle of a smooth manifold is the one associated to its tangent bundle. For this reason it is sometimes called the tangent frame bundle. 9 See, for example, H. Gato and K. Ichimura, Phys. Rev. A 76, 012120 (2007), and references therein. 8A

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in quantum mechanics. The Berry phase was generalized to the case of a degenerate eigenspace by Wilczek and Zee (WZ) [25] . Considered from the viewpoint of gauge theory, the Berry and WZ phases correspond to the Abelian and non-Abelian gauge field theories, respectively. As it turns out adiabatic quantum evolution used by Berry is a highly restrictive assumption and can be, at best, only an approximation. Therefore Aharonov and Anandan (AA)[26] introduced the nonadiabatic Abelian geometric phase. Moreover, Anandan generalized the AA phase to the non-Abelian one[27] using degenerate group of quantum states. The AA phase has also been generalized to the noncyclic case by Samuel and Bhandari (SB)[28] by applying Pancharatnam’s geometric phase[29] for the interference between two polarized light beams to quantum interference. For open quantum systems, geometric phases for mixed states have been extensively studied, especially for geometric quantum computations. There are mainly three general approaches to the definition of the mixed-state geometric phase which are based on state purification10 , quantum trajectories for the Lindblad superoperator evolution[30] , and quantum interferometry kinematic approach by defining the parallel transport condition that provides a connection form for obtaining the geometric phase for mixed states. The first approach proposed by Uhlmann defines a parallel transport for a purification of a density operator[31] . A connection is defined on a fiber bundle whose fiber is a Hilbert-Schmidt space and base manifold of all density operators. This approach has a problem that the geometric phase depends on the evolution of the ancillary part of the purification method . The second approach proposed by Carollo et al. is based on the application of the SB approach (Pancharatnam’s phase) to each quantum trajectory of the Lindblad superoperator evolution. A problem with this approach is that the geometric phase depends on the type of unraveling nonphysical parameters [32]. The third approach is a natural generalization of the SB approach (Pancharatnam’s phase) to the mixed-state case. This approach was first based on quantum interferometry assuming unitary evolution and nondegenerate density operators[33]. The generalizations to the cases of degenerate density operators and nonunitary evolution have been achieved by kinematic approaches. Sarandy and Lidar (SL) have proposed a natural generalization of the WZ phase to the mixed-state case[34] , which is based on adiabatic approximation in open quantum systems. A natural generalization of the Anandan phase is given by Goto and Ichimura[35], which naturally includes the SL phase.

14.3

Geometric Phase in Many-Body Systems

Geometric phase in many-body systems is closely tied to spin and statistics. In 3-dimensional space, quasiparticles are either fermions or bosons. There are three spin operators in 3-dimensional space, Sx , Sy , and Sz , which satisfy a non-Abelian algebra. The eigenvalues for the spin are in units of 12 |. On the other hand, in 2-dimensional space there exist only one spin operator, a rotation around the z-axis 10 State purification refers to the fact that every mixed state acting on finite dimensional Hilbert spaces can be viewed as the reduced state of some pure state.

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Fig. 14.2 In (a) the loop encircling particle A in two dimensional space cannot be shrunk to the location of particle B without crossing particle A. Note the presence of magnetic flux, φ, also defines the plane geometry in Aharonov-Bohm eﬀect. In other words, the winding of one particle around another in two dimensions is very-well defined. In (b) the loop is continuously shrinkable to the location point of particle B without crossing A in three dimensional space, or the encircling path is arbitrary and not well defined in (b). In top figure of (b) the xy-plane is drawn parallel to the plane of the orbit of B.

by identifying the 2-dimensional space with the xy-plane. Here the rotation around the z-axis is Abelian and hence the eigenvalue is arbitrary. It is a topological consequence that quasiparticles in 2-dimensional space have fractional spin and statistics11 . These are generally called anyons. This is illustrated in Fig. 14.2. Exchanging the two particles twice is topologically equivalent to the one in which particle A is fixed but particle B goes around particle A once counterclockwise. In three or more dimensions, the path going around the fixed point is topologically equivalent to no exchange at all. Hence, if eiθ is the phase factor for one exchange, then in 3-D we must have ei2θ = 1 or eiθ = ±1, which means there are only fermions and bosons in three or more dimensions. In two dimensions the situation is entirely diﬀerent since the path going around the fixed particle cannot be shrunk to a point without crossing particle A12 . Let eiθ = eiαπ for 2-D in Fig. 14.2(a) . Therefore, the generalized anyon exchange relation must be ψ (t, y) ψ (t, x) = eiαπ ψ (t, x) ψ (t, y)

(14.1)

where α is the statistics parameter. 11 In one-dimensional systems, quantum statistics is not well defined because particle interchange is impossible without one particle going through another. In this 1-dimensional space, boson with hard-core repulsion are equivalent to fermions. 12 Technically, an anyon of interest may be represented as a flux-carrying boson (or fermion) where the flux is described by the Chern-Simons gauge theory, a canonical field theoretical technique for topological phases of matter.

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Fig. 14.3 A simple conceptual visualization of quasihole/quasiparticle creation from the condensate is shown in (a), moving in opposite directions later shown in (b). The ‘flat ’ horizontal line represents the vacuum condensate. An instanton eﬀect occurs when there is tunneling between diﬀerent vacuua (or diﬀerent condensates due to symmetry breaking), such as instanton eﬀect due to quasiparticle tunneling in a double-well potential.

14.3.1

Localized Disturbances of the Ground State of 2+1-D Many-Body Systems

Consider the ground state of a system of electrons 2 + 1-D space-time. We assume that the ground state is separated from the excited states by an energy gap13 (i.e, incompressible ground state), as is the situation in fractional quantum Hall states in 2 − D electron systems14 . The lowest energy electrically-charged excitations are known as quasiparticles or quasiholes, depending on the sign of their electric charge15 . These quasiparticles are local disturbances to the ground state (vacuum or condensate) of the electrons corresponding to a quantized amount of total charge (schematically visualized in Fig. 14.3). Let us denote the positions of the quasiparticles be (R1 , ..., Rk ), and assume that these positions are well spaced from each other compared to the microscopic length scales. For example, this could be built into the system’s Hamiltonian by a scalar potential composed of many local “traps”, each suﬃcient to capture exactly 13 Localized disturbance cost energy, hence the presence of an energy gap from the ‘vacuum’ or ground state (condensate). Condensates are often unstable against long-wavelength (global) disturbances. 14 We will not discuss fractional quantum Hall eﬀect as there are many good reviews on this topic already, some using the quantum-field gauge theoretical techniques basically pointing out similarities with superconductivity and superfluidity [superconductivity is a charge condensate which respond to electromagnetic fields, whereas superfluidity is a neutral condensate which respond to hydraulic and gravitational forces, both without incurring dissipation or any resistance). 15 The relativistic Dirac quasiparticles have ‘bare’ electron/positron charge excited from the Fermi-sea condensate. In general, quasiparticles from other condensates can have arbitrary charge, as exemplified by a hierarchy of anyon quasiparticles in fractional quantum-Hall eﬀect. The term “quasiparticle” is also sometimes used in a generic sense to mean both quasiparticle and quasihole. In the standard model, quasiparticle interaction originates from the structure of the pertinent vacuum in terms of the gauge fields.

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one quasiparticle. A state with quasiparticles at these positions can be viewed as an excited state of the Hamiltonian of the system without the trap potential, or as a ground state in the presence of trap potential16 . The quasiparticles’ coordinates (R1 , ..., Rk ) are parameters both in the Hamiltonian and resulting ground state wavefunction (in the presence of quasiparticle traps). This is the ground state wavefunction addressed in considering the geometric phase in what follows. We are concerned here with the eﬀect of taking these quasiparticles around each other (braiding). We imagine making the quasiparticles coordinates R = (R1 , ..., Rk ) adiabatically time-dependent. In particular, we consider a trajectory in which the final configuration of quasiparticles is just a permutation of the initial configuration17 . If the ground state wave function is single-valued with respect to (R1 , .., Rk ), and if there is only one ground state for any given set of Ri ’s, then the final ground state to which the system returns to after the winding is identical to the initial one, up to a phase. Part of this phase is simply the dynamical phase which depends on the energy of the quasiparticle state and the length of time for the process. In the adiabatic limit, it is

dt E(R(t)). The other part is the Berry

phase, which does not depend on how long the process takes. This Berry phase is,

α=i

dR · ψ (R)| ∇R |ψ (R)

(14.2)

where ψ (R) is the state with the quasiparticles at positions R, and where the integral is taken along the trajectory R(t). It is manifestly dependent only on the trajectory taken by the particles and not on how long it takes to move along this trajectory. The phase α has a piece that depends on the geometry of the path traversed18 (typically proportional to the area enclosed by all of the loops), and a piece θ that depends only on the topology of the loops created19 . If θ = 0, then the quasiparticles excitations of the system are anyons. In particular, if we consider the case where only two quasiparticles are interchanged clockwise (without wrapping around any other quasiparticles), θ is the statistical angle of the quasiparticles. There were two key conditions to the occurrence of the Berry phase, namely, the single valuedness of the wave function and the non-degeneracy of the state. In fact, most situation this condition does not hold. We are more interested in systems in which, once the positions (R1 , .., Rk ) of the quasiparticles are fixed, there remain multiple degenerate states (i.e. ground states, ψa (R), in the presence of traps which capture quasiparticles at positions R = (R1 , .., Rk )), which are distinguished by a set of internal quantum numbers, ψa (R), a = 1, 2, ..., n are the n degenerate states). 16 The

motivation for the use of the trap devices is that this arrangement can be implemented experimentally in MESFET devices sub jected to high magnetic fields in the fractional quantum Hall eﬀect regime. 17 At the end, the positions of the quasiparticles are identical to the intial positions, but some quasiparticles may have interchanged positions with others. 18 This has to do with the geometry of the base manifold in the fiber bundle context. 19 This has to do with the braiding of the worldlines of the quasiparticles.

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When the ground state is degenerate, the eﬀect of a closed trajectory of the Ri ’s is not necessarily just a phase factor. The system starts and ends in energy states, but the initial and final ground states may be diﬀerent members of this degenerate space. Adiabaticity in this case is that the adiabatic evolution of the state of the system is confined to the subspace of degenerate states. Thus, it may be expressed as a unitary transformation within this subspace. The inner product in Eq. (14.2) must be generalized to a matrix of such inner products: mab = ψa (R)| ∇R |ψb (R) Since these matrices at diﬀerent points R do not commute, we must path-order the integral in order to compute the transformation rule for the state, ψa ⇒ Mab ψb where

Mab = P exp i =

∞ n=0

dR · m

i

ds1 0

sn−1

s1

2π n

ds2 ........ 0

0

˙ (s1 ) · mαα1 (R (s1 )) .....R ˙ (sn ) · man b (R (sn ))]. × dsn [R

(14.3)

Where R(s), s [0, 2π], is the closed trajectory20 of the particles and P is the path-ordering symbol. Again, the matrix Mab may be the product of topological and non-topological parts. In a system in which quasiparticles obey non-Abelian braiding statistics, the nontopological part will be Abelian, that is, proportional to the unit matrix. Only the topological part will be non-Abelian. The requirements for quasiparticles to follow non-Abelian statistics are then, (i) that the N-quasiparticle ground state is degenerate. In general, the degeneracy will not be exact, but it should vanish exponentially as the quasiparticle separations are increased; (ii), that adiabatic interchange of quasiparticles applies a unitary transformation on the ground state, whose non-Abelian part is determined only by the topology of the braid, while its non-topological part is Abelian. If the particles are not infinitely far apart, and the degeneracy is only approximate, then the adiabatic interchange must be done faster than the inverse of the energy splitting between states in the nearly-degenerate subspace (but, of course, still much slower than the energy gap between this subspace and the excited states);(iii), the only way to make unitary operations21 on the degenerate ground state space, so long as the particles are kept far apart, is by braiding. The simplest (albeit uninteresting) example of degenerate ground states may arise if each of the quasiparticles 20 Here

we set the time variable in terms of s = 2π

t T

where T is the total time it takes around a closed trajectory. 21 This statement is central to topological quantum computing.

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carried a spin 1/2 with a vanishing g—factor. If that were the case, the system would satisfy the first requirement. Spin orbit coupling may conceivably lead to the requirement (ii) being satisfied. Satisfying (iii), however, is central to topological quantum computing which is vigorously pursued reaching a highly interdisciplinary junction of semiconductor nanodevice physics, anyons, quantum Hall eﬀects, superconductivity, superfluidity, braid groups and knot theory, string theory, symmetry and invariance guiding principle, and symmetry breaking, new phases of condensed matter, in the pursuit of a fault-tolerant quantum computer.22 We will not go into details of this rapidly developing field of holonomic quantum computing. The degeneracy of N -quasiparticle ground states is conditioned on the quasiparticles being well separated from one another. When quasiparticles are allowed to approach one another too closely, the degeneracy is lifted. In other words, when nonAbelian anyonic quasiparticles are close together, their diﬀerent fusion channels are split in energy. This dependence is analogous to the way the energy of a system of spins depends on their internal quantum numbers when the spins are close together and their coupling becomes significant. The splitting between diﬀerent fusion channels is a means for a measurement of the internal quantum state, a measurement that is of importance in the context of geometric quantum computation. 14.3.2

Reconstructing Statistical Quantum Fields in Many-Body Physics

Several occasions arise where seemingly intractable problems in many-body physics which is either fermionic or bosonic becomes easier when the statistical character is mapped to the each other. Even when the character is neither fermionic nor bosonic when formulated in terms of fermions/bosons turn out to become easy, even trivial. Particularly, in the diagrammatic many body physics for the spin-boson models, the major diﬃculty of representing spin operators (which are not fermionic nor bosonic) is that spin operators do not obey Wick’s theorem and an expectation 22 For holonomic quantum computing, unitary transformations, which are unitary representations of the braid groups, are implemented by means of braiding the worldlines of non-abelian anyons, i.e., worldliness cross over one another to form braids in a three-dimensional spacetime (2-D + 1). It makes use of non-abelian anyons which support braid groups and knot theory, the so-called Fibonacci anyons. The initialization is by creating particle/anti-particle pairs from the vacuum, computation proceeds by performing a series of braidings, and the output readout is performed by checking if pairs annihilate. Fractional quantum Hall eﬀects in condensed matter can support non-abelian anyons, as well as other topological models such as the Kitaev toric spin-lattice model. This lattice model possesses two kinds of localized excitations called charges and fluxes, which have mutual anyonic statistics although each of them are bosons. Charges reside on the vertices and fluxes reside on the faces of the lattice. In a n × n square lattice one spin are located on each edges or bonds resulting in 2n2 spins by virtue of the toroidal lattice geometry. Spins interact if they share either a vertex or a plaquette, hence four interacting spins are involved. Each kind of excitations are created in pairs, by the so-called string operation (made up of direct product of Pauli matrices acting on each spin along the string). The created pairs are located at the ends of the string on the lattice. Composite string operations are able to create pairs of particles of same type, move one quasiparticle around closed loop until the pair meet and annihilate each other. In this manner braiding of the two types of excitations can be performed on the lattice vacuum.

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value of the product of many spin operators cannot be decomposed into products of two-operator expectation values, even within a free theory. In order to handle spin-boson problem, there is a need to map to fermionic/bosonic particles. A way out of this diﬃculty is to represent spins as bilinears of fermions or as bosons. One of the disadvantages of these approaches is that the Hilbert space of the fermions or bosons needs to be restricted by the application of constraints.23 Another diﬃculty is that once the spins are represented as bilinears, the spin-spin correlation functions are represented by two-particle Green’s functions. An alternative approach is to take advantage of the anticommuting properties of Pauli matrices, writing the spin operator in terms of Majorana fermions24 , i S =− η×η 2 where η = (η 1 , η2 , η3 ) is a triplet of Majorana fermions which satisfy {ηa , ηb } = δ ab . This representation does not require the imposition of a constraint: the fact that S 2 = 34 follows directly from the operator properties of the Majorana fermions. The mapping between spin- 12 system and real fermions follows the following relations σx = 2iη2 η3 ,

σ y = 2iη3 η1 ,

σz = 2iη 1 η2

where η1, η2, η 3 are fermionic majorana creation/annihilation operators, characterized by the following expressions η i = η†i η i , ηj = δ ij Note that from [σi ,σj ] = 2i

ijk σ k ,

then we have

[σx ,σ y ] = −4η 2 η3 η3 η1 − 4η3 η1 η2 η3 = −4η2 η1 = 2iσ z The spin-boson Hamiltonian derived from the mapping is H = Ho + Hint = −i

∆ (η η − η2 η1 ) + HB − λiφη 2 η3 2 1 2

where φ is the boson operator of the heat bath with Hamiltonian HB . 23 See,

for example: D. P. Arovas and A. Auerbach, Phys. Rev. B 38, 316 (1988); C. Jayaprakash, H. R. Krishnamurthy and S. Sarker, Phys. Rev. B 40, 2610 (1989); C. L. Kane, P. A. Lee, T. K. Ng, B. Chakraborty and N. Read, Phys. Rev. B 41, 2653 (1990). 24 For a good reference on Majorana fermions, see A. M. Tsvelik, “Quantum Field Theory in Condensed Matter Physics” (Cambridge, 1995.)

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Fig. 14.4 Mapping between the spin correlation function to a one-particle Majorana Green’s function [redrawn from W. Mao, P. Coleman, C. Hooley and D. Langreth, Phys. Rev. Lett. 91, 207203, (2003)].

14.3.2.1

Bosonization

One-dimensional abelian bosonization is a way for representing 1-D fermion fields ψη (x), where η is a species (e.g. spin) index, in terms of bosonic fields ϕη (x) through a relation of the form ψη (x) ∼ Fη e−iϕη (x) , where Fη is a so-called Klein factor which lowers the number of η-fermions by one. It has become a popular tool for treating certain strongly-correlated electron systems in 1 dimension. The reason for its popularity is that some problems which appear intractable when formulated in terms of fermions turn out to become easy, even trivial, when formulated in terms of boson fields. Some successful applications include Tomonaga-Luttinger liquid theory (dealing with a quantum wire of interacting 1-D electrons), quantum Hall edge states and quantum impurity problems such as the Kondo problem. In contrast with two-state system coupled to bosons, the Kondo problem is a spin- 12 system interacting with fermionic bath. It should be realized that the low-lying excitations of the electron gas in the Kondo Hamiltonian may be approximately described by bosons. Here the bath oscillators correspond to the spin-density excitations. In its simplest form the Kondo problem is given by the Kondo or ‘s − d’ Hamiltonian HK = k,σ

ε (k) c†kσ ckσ + JS · s (0)

where impurity is at site r = 0 and exchange constant J is positive for an antiferromagnetic interaction.

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Chapter 15

Geometric Phase in Quantum Field Theories: Standard Model

A separate geometrization of quantum mechanics has been occurring, well before Berry’s 1984 paper, in the development of quantum field theory of elementary particles. This is the gauge theory of interactions. It turns out that the formal aspects of gauge theory can be mapped to the theoretical framework of diﬀerential geometry and diﬀerential forms, and in this sense gauge theory is occupied with the theory of connection (corresponding to nonquantized gauge field or vector potential in electrodynamics) on principal bundle together with its gauge-invariant properties, principally its curvature (corresponding to field-strength tensor in electrodynamics). Theso-called standard model of particle physics is a gauge theory of the electroweak and strong interactions with the gauge group SU (3) × SU (2) × U(1). In contrast with Berry’s work, symmetry groups and corresponding invariance properties are the guiding principle in construcitng and discovering the physics of the diﬀerent interactions in quantum field theories.

15.1

Classical Gauge Theory

Consider a set of n non-interacting classical scalar fields, with equal masses m. This system is described by an action which is the sum of the (usual) action for each scalar field φi n

S=

dx4 1

1 1 ∂µ φi ∂ µ φi − m2 φ2i . 2 2

The Lagrangian (density) can be written as n

L=

1

1 1 ∂µ φi ∂ µ φi − m2 φ2i 2 2

1 1 = (∂µ Φ)T ∂ µ Φ − m2 ΦT Φ, 2 2 with the vector of fields given by T

Φ = (φ1 , φ2 , φ3 , ...) , 182

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183

where ∂µ is the Einstein notation to describe the partial derivative of Φ in each of the four dimensions. It is now transparent that the Lagrangian is invariant under the transformation Φ=GΦ whenever G is a constant matrix belonging to the n × n orthogonal group O(n). We have 1 1 T (∂µ Φ ) ∂ µ Φ − m2 Φ T Φ , 2 2 1 1 T µ T L = (∂µ (G Φ)) ∂ (G Φ) − m2 (GΦ) (G Φ) , 2 2 1 1 T = (∂µ Φ) GT G∂ µ (Φ) − m2 ΦT GT G Φ, 2 2 1 1 2 T T µ = (∂µ Φ) ∂ Φ − m Φ Φ = L. 2 2 L =

This is the ‘global’ symmetry (i.e., without regards to the geometry of curved spacetime M , a view of physics in a flat tangent space T Mp at point p in a manifold M ) of this particular scalar field Lagrangian, and the ‘global’ symmetry group is often called the gauge group. Incidentally, Noether’s theorem implies that invariance under this group of transformations leads to the conservation of the current Jµα = i∂µ ΦT T α Φ, where the T α matrices are generators of the SO(n) [or O(n) matrices with determinant = 1] group. There is one conserved current for every generator. Upon demanding that this Lagrangian should have ‘local’ O(n)-invariance requires that the G matrices should now be allowed to become local functions of the space-time coordinates x. When G = G(x), we have the new Lagrangian given by 1 1 T (∂µ Φ ) ∂ µ Φ − m2 Φ T Φ 2 2 1 1 T µ = (∂µ (G(x)Φ)) ∂ (G(x)Φ)− m2 (G(x)Φ)T (G(x)Φ) 2 2 1 1 T = [(∂µ G(x))Φ+G(x)(∂µ Φ)] [(∂ µ G(x))Φ+G(x)(∂ µ Φ)]− m2 ΦT GT (x)G(x)Φ 2 2 T T µ 1 1 [(∂µ G(x)) Φ] [(∂ G(x)) Φ] + [G(x) (∂µ Φ)] [G(x) (∂ µ Φ)] − m2 ΦT Φ = T T µ µ 2 + [(∂µ G(x)) Φ] [G(x) (∂ Φ)]+[G(x) (∂µ Φ)] [(∂ G(x)) Φ] 2

L =

=

1 1 (∂µ Φ)T ∂ µ Φ− m2 ΦT Φ 2 2 1 [(∂µ G(x)) Φ]T [(∂ µ G(x)) Φ] + T 2 + [(∂µ G(x)) Φ] [G(x) (∂ µ Φ)]+[G(x) (∂µ Φ)]T [(∂ µ G(x)) Φ]

,

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which include some extra terms given by Lint =

1 2

[(∂µ G(x)) Φ]T [(∂ µ G(x)) Φ] + [(∂µ G(x)) Φ] [G(x) (∂ µ Φ)] + [G(x) (∂µ Φ)]T [(∂ µ G(x)) Φ] T

.

We shall see that this extra terms gives the interaction between the gauge field and scalar fields. However, the form of the Lagrangian is not covariant, since the G matrices do not commute with the derivatives, i.e., (∂µ GΦ)T ∂ µ GΦ = ∂µ ΦT ∂ µ Φ. Interestingly, we can derive the gauge field from the derivatives of G(x), (∂ µ G(x)) = gAµ (x) G(x), ∂µ G(x) = gAµ (x) G(x). Then the extra terms can be written as   T  ΦT (gAµ (x) G(x)) (gAµ (x) G(x)) Φ 1 , Lint = 2 +ΦT (gAµ (x) G(x))T G(x) (∂ µ Φ)+(∂µ Φ)T GT (x)(gAµ (x) G(x)) Φ Lint =

g2 g g (Aµ (x) Φ)T Aµ (x) Φ + (Aµ (x) Φ)T ∂ µ Φ + (∂µ Φ)T Aµ (x) Φ, 2 2 2

where g is the coupling constant - a quantity defining the strength of an interaction. This suggests defining a “derivative” D with the property Dµ G (x) Φ (x) = G (x) Dµ Φ (x) ,

(15.1)

to make the form of the Lagrangian invariant. We use a covariant derivative in the form Dµ = (∂µ + gAµ (x)) . The gauge field A(x) is defined to have the transformation law Aµ (x) = G (x) (Aµ (x)) G−1 (x) 1 − ∂µ G (x) G−1 (x) , g 1 G (x)−1 Aµ (x) G (x) = (Aµ (x)) − G (x)−1 ∂µ G (x) , g 1 −1 −1 G (x) Aµ (x) G (x) + G (x) ∂µ G (x) = Aµ (x) , g 1 G (x)−1 Aµ (x) + ∂µ G (x) = Aµ (x) , g G (x)−1 [gAµ (x) + ∂µ ] G (x) = gAµ (x) , G (x)−1 [∂µ + gAµ (x)] G (x) = gAµ (x) , [∂µ + gAµ (x)] G (x) = G (x) gAµ (x) .

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Therefore, by virtue of Eq. (15.1), we have [∂µ + gAµ (x)] G (x) Φ (x) = G (x) [∂µ + gAµ (x)] Φ (x) , −1

G (x)

[∂µ + gAµ (x)] G (x) Φ (x) = [∂µ + gAµ (x)] Φ (x) .

Using the transformation law for the gauge field, G (x)−1 [∂µ + gAµ (x)] G (x) = gAµ (x) , G (x)−1 [∂µ + gAµ (x)] G (x) Φ (x) + ∂µ Φ (x) = [∂µ + gAµ (x)] Φ (x) . We arrived at the identity [∂µ + gAµ (x)] Φ (x) = [∂µ + gAµ (x)] Φ (x) . The gauge field is an element of the Lie algebra, and can therefore be expanded as α Aα µ (x) T .

Aµ (x) = α

There are therefore as many gauge fields as there are generators of the Lie algebra. Finally, we now have a locally gauge invariant Lagrangian 1 1 (Dµ Φ)T Dµ Φ − m2 ΦT Φ, 2 2 1 1 = (Dµ G (x) Φ (x))T Dµ G (x) Φ (x) − m2 ΦT Φ 2 2 1 1 T T = (Dµ Φ (x)) G (x) G (x) Dµ Φ (x) − m2 ΦT Φ 2 2 1 1 T = (Dµ Φ) Dµ Φ − m2 ΦT Φ = Lloc , 2 2

Lloc = Lloc

which yields 1 1 ((∂µ + gAµ (x)) Φ)T (∂ µ + gAµ (x)) Φ − m2 ΦT Φ 2 2 1 1 = ((∂µ ) Φ)T (∂ µ ) Φ − m2 ΦT Φ 2 2 g g2 g (∂µ Φ)T Aµ (x) Φ + (Aµ (x) Φ)T ∂ µ Φ + (Aµ (x) Φ)T Aµ (x) Φ 2 2 2 = Lglobal + Lint ,

Lloc = Lloc =

where again Lint is Lint =

g2 g g (Aµ (x) Φ)T Aµ (x) Φ + (Aµ (x) Φ)T ∂ µ Φ + (∂µ Φ)T Aµ (x) Φ. 2 2 2

The diﬀerence between this Lagrangian and the original globally gauge-invariant Lagrangian is seen to be the interaction Lagrangian Lint =

g T T µ g g2 Aµ Φ ∂ Φ + (∂µ Φ)T Aµ Φ + (Aµ Φ)T Aµ Φ. 2 2 2

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Fig. 15.1

A Feynman diagram showing scalar bosons interacting via a gauge boson.

Note that the requirement for local gauge invariance introduces interactions between the n scalar fields. In the quantized version of this classical field theory, the quanta of the gauge field A(x) are called gauge bosons. The interpretation of the interaction Lagrangian in quantum field theory is of scalar bosons interacting by the exchange of these gauge bosons, Fig. 15.1. 15.2

The Yang-Mills Lagrangian for the Gauge Field

Indeed, the formalism of gauge theory of interaction started with the Yang-Mills paper in 19541 . The example above is not a self-consistent theory since in the covariant derivatives D, the gauge field A(x) must be given at all space-time points, i.e., externally applied. A proper theory must incorporate the gauge potential as the solution to a field equation. Therefore we also require the Lagrangian to generate the field equation for the gauge potential, and furthermore this Lagrangian which generates this field equation must be locally gauge invariant as well. A possible form for the gauge field Lagrangian is a generalization from classical electrodynamics, written as 1 Lgf = − T r (F µν Fµν ) , 4 with Fµν = [Dµ , Dν ] , and the trace being taken over the vector space of the fields. This is the YangMills action. Other gauge invariant actions also exist in nonlinear electrodynamics, Born-Infeld action, Chern-Simons model, theta term etc.. Note that in this Lagrangian only involves the A field only. Invariance of Lgf in the Lagrangian under gauge transformations is a particular case of a prior classical symmetry. In performing quantization, the choice of gauge is often restricted by performing a gauge transformation, a procedure known as gauge fixing. The ‘self-contained’ Lagrangian for the O(n) gauge theory is now L = Lloc +Lgf = Lglobal +Lint +Lgf .

1 C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96, 191 (1954). This work probably marks the second instance, after Einstein Theory of Relativity, that invariance due to symmetry considerations have ushered significant developments in theoretical physics.

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15.3

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Electrodynamics as a Gauge Theory

Let us consider the case of electrodynamics, with only the electron field. The action which generates the electron field’s Dirac equation is S=

¯ i|cγ ∂µ − mc2 ψd4 x. ψ µ

The global symmetry for this system is ψ ⇒ eiθ ψ. The gauge group here is U (1), just the phase angle of the field, with a constant θ. Demanding that θ have local symmetry implies the replacement of θ by θ(x). An appropriate covariant derivative is then e Dµ = ∂µ − i Aµ , |c where e (≡ − |e|) is the charge of the electron. Then we have e −iθ(x) ¯ i|cγ µ ∂µ − i Aµ − mc2 eiθ(x) ψ(x)d4 x ψ(x)e |c

S =

−iθ(x) ¯ ψ(x)e i|cγ µ ∂µ − mc2 eiθ(x) ψ(x)d4 x

= +

−iθ(x) ¯ γ µ (eAµ ) eiθ(x) ψ(x)d4 x. ψ(x)e

Identifying the “charge” e with the usual electric charge (the use of the term “charge” in gauge theories is borrowed from electrodynamics), and the gauge field A(x) with the four-vector potential of electromagnetic field results in an interaction Lagrangian e¯ (x) γ µ ψ (x) Aµ (x) = J µ (x) Aµ (x) , Lint = ψ | where J µ (x) is the usual four vector electric current density. The gauge theoretical principle is thus seen to introduce the so-called minimal coupling of the electromagnetic field to the electron field in a straightforward fashion. Upon adding a Lagrangian for the gauge field Aµ (x) in terms of the field strength tensor exactly as in electrodynamics, one obtains the Lagrangian which is used as the starting point in quantum electrodynamics ¯ i|cγ µ Dµ − mc2 ψ − LQED = ψ 15.4

1 (F µν Fµν ) . 4µ0

Quantization of Gauge Theories

Quantum electrodynamics (QED) represents the first gauge theory to be quantized. The first methods developed for this involved gauge fixing and then applying canonical quantization. The Gupta-Bleuler method was also developed to handle

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this problem. Non-abelian gauge theories are nowadays handled by a variety of means. The main point to quantization is to be able to compute quantum amplitudes for various processes allowed by the theory. Technically, they reduce to the computations of certain correlation functions in the vacuum state. This involves a renormalization of the theory. When coupling constant of the theory is small enough, all required quantities may be obtained using perturbation theory. These methods have led to the most precise experimental tests of gauge theories. However, in most gauge theories, there are many interesting questions which are non-perturbative. Quantization schemes which are geared to these problems (such as lattice gauge theory) may be called non-perturbative quantization schemes. Nonperturbative computations often require supercomputing, and are therefore less well developed currently than other schemes.

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Chapter 16

String Theory

For completeness in our discussions on quantum methodology, we give a brief discussion of string theory of elementary particles and forces in nature, which is still under vigorous investigation in present-day theoretical physics. Although mathematically consistent and beautiful, it is not quite an established field of conventional theoretical physics for lack of experimental verification of string theory’s new predictions by virtue of its inaccessibility to current experimental techniques. Quantum mechanics is primarily founded on probabilistic information-theoretic ground and superposition principle, supporting localized atomic-size orbitals and mutually-unbiased extended states. On the other hand, general relativity theory is inherently founded on deterministic and geometric concepts of astronomical-size space-time. String theory sought to unify relativity and quantum mechanics into one framework. In string theory, it postulated that particles were not zero-dimensional points, but were instead tiny strings whose vibrations are reflected in the observed properties of the fundamental particles. The two theories must be unified in order to properly describe conditions such as in the cores of black holes and the original singularity at the beginning of the universe. All matter is composed of atoms, which in turn are made from quarks and electrons. All such particles are actually tiny loops or segments of vibrating string, according to String Theory. The basic concept behind all string theories is that each elementary “particle” is actually a vibrating string (of a very small scale, possibly of the order of the Planck length) at resonant frequencies specific to that type of particle. Thus, any elementary particle should be thought of as a tiny vibrating object, rather than as a point. This object can vibrate in diﬀerent modes like a guitar string can produce diﬀerent notes, with every mode appearing as a diﬀerent particle (electron, photon, etc.).

16.1

Feynman Diagrams

Strings can ‘fan out’ and combine, which would appear as particles emitting and absorbing other particles, presumably giving rise to the known interactions between particles. Thus, many first principle processes in quantum field theory, described by Feynman diagrams, Fig. 16.1, are included and get deeper insights in string theory. 189

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Fig. 16.1 (a) In particle physics, zero-distance interactions are allowed, which does not make sense in the quantum theory of gravity. (b) Interactions in string theory do not occur at zero distance, the Feynman diagram in this theory is a two-dimensional smooth surface, and the loop integrals over such a smooth surface lack the zero-distance, infinite momentum problems of the integrals over particle loops.

A world line of a point-like particle’s motion is the unique path of that point particle as it travels through 4-dimensional space-time. By analogy, a similar graph depicting the progress of a string as time passes by can be obtained. The onedimensional string (a small line by itself) will trace out a surface (a two-dimensional manifold), known as the worldsheet ( a cylinder fork in Fig. 16.1). The diﬀerent string modes (representing diﬀerent particles, such as photon or graviton) are surface waves on this manifold. If a string comes into existence briefly and then vanishes, its world sheet is a sphere. Complicated Feynman diagrams are worldsheets consisting of ‘donuts’1 . Because the Feynman diagrams for strong interaction are so intractable, others resorted to the so-called lattice gauge theory, in which the spacetime is divided into finite set of points, called a lattice. The computer computes all fields at each lattice points, and has gotten to a point where particle masses are computed fairly well. However, this numerical methods is not very illuminating. In addition to one-dimensional strings, this theory also allows objects of higher dimensions, such as D-branes. All string theories predict the existence of extra degrees of freedom which are usually described as extra dimensions. String theory is thought to include some 10, 11, or 26 dimensions, depending on the specific theory and on the point of view.

16.2

The Birth of String Theory

In 1968 by Gabriele Veneziano at CERN made the discovery that Euler betafunction perfectly described many properties of the strong nuclear force. Later on, Yoichiro Nambu of the University of Chicago, Holger Nielsen of the Niels Bohr Institute, and Leonard Susskind of Stanford University found that the nuclear interactions were described exactly by the Euler beta-function if the elementary particles are modeled as one-dimensional strings instead of zero-dimensional particles. This was the birth of string theory. However, later experiments in the early 1970s revealed that many of the theory’s predictions did not match experimental data. 1 Perhaps,

the word ‘doknots’ is more appropriate.

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For example, nature is not symmetric under the exchange of left and right (parity violation). But the first string theory was ‘ambidextrous’. As point-particle theory continued to meet significant successes with experiments, only a few devoted physicists keep working on string theory. In 1984, a paper by Michael Green, of Queen Mary College, and John Schwarz of the California Institute of Technology showed that string theory could encompass the four fundamental forces and all the matter in existence. They found a fix that keep the theory internally consistent while allowing parity to be violated. The result was the first superstring revolution with the introduction of extra dimensions. In 1995, Edward Witten ignited the second superstring revolution. He described a formulation for moving past the approximations used during the first superstring revolution and thus into even deeper areas of this vast and complex theory. In 1997 Maldacena conjectured the equivalence between a string theory defined on one space, and a quantum field theory without gravity defined on the conformal boundary of this space, whose dimension is lower by one or more.

16.3

Need for Extra Dimensions in String Theory

Why the need for extra dimensions in string theory? The answer can only come from symmetry arguments.2 This reflects the need for expanding the symmetry group of the standard model. The need arise because in a diﬀerent number of total dimensions than D = 10, string theory is not consistent. The essence is always that there is an inherent symmetry of classical physics, but in calculation of the quantum corrections, the important symmetries can be destroyed. There is a universal contribution to this “anomaly” (quantum eﬀect breaking a symmetry3 ), and each dimension contributes, too. If D = 10, these contributions cancel and the theory become consistent. The relevant symmetry can either be the Lorentz symmetry - in the light cone gauge - or the Weyl symmetry (that implies that no distance scale on the worldsheet is privileged and physics only depends on “angles”, not absolute distances), depending on the formalism one choose. Having those extra dimensions and therefore many ways the string can vibrate in many diﬀerent directions turns out to be the key to being able to describe all the particles in nature. Indeed, even in classical physics the need for extra fifth dimensions was pioneered by the Kaluza—Klein (KK) theory which is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. In KK theory, the fourth spatial dimension is curled up in a circle of very small radius. By virtue of the successes of the standard model, the search for formal theoretical connections between string and gauge theory is gaining grounds and vigorously 2 The symmetry group of string theory includes the symmetry group of standard model as a subgroup. 3 The spontaneous symmetry breaking has been extensively studied in particle physics as well as in condensed matter physics. The physics of the symmetry breaking is related to many interesting phenomena such as the phase transition of ferromagnetism, dynamical mass generation, Higgs mechanism and so on.

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pursued in recent years. This means working with the other six dimensions until the standard model or gauge theory of interaction emerges. 16.4

Nanoelectronics and String Theory

To have a grasp of the complexity of string theory, for the readers in nanoscience and nanotechnology community who are not string theorists, we oﬀer here a simple interpretation of some of the results of string theory by making an analogy with nanoelectronics. It appears that the vibrating tiny string loops corresponds to localized ‘atomic-energy orbitals’ in a ‘crystalline medium’ (thought to be the spacetime vacuum) with ‘atomic size’ and ‘lattice constant’ of the order of the Planck length.4 The corresponding extended ‘Bloch states’ described the quasiparticles motion in space-time, with quasiparticle spread in space-time corresponding to the size of the ‘Wannier function’ (i.e., the state or mode of tiny closed vibrating loops). Clearly diﬀerent ‘atomic energy levels’, or diﬀerent modes of the vibrating string, yield diﬀerent energy bands, diﬀerent masses, and hence diﬀerent quasiparticles just like in solid-state physics.5 It is the size or spread of the ‘Wannier function’ or localized orbital along the quasiparticle path that is responsible for tracing the worldsheet in space-time. This clearly invoke the importance of the discreteness of space-time at the Planck length scale. From this discrete space-time point of view, the importance of the discreteness of space-time in accurately describing the core of the Black hole and the original singularity at the beginning of the universe is somewhat not surprising. Since this development has an exact parallel in nanoscience and nanotechnology, where the atomic discreteness becomes an overpowering consideration in accurately describing nanoscale devices and functioning nanomaterials. Thus, we have, on one hand, the nanoscale materials science for condensed matter, and the other hand Planckscale space-time quantum gravity for cosmological space-time. The allowance for open strings in string theory necessitates the incorporation of a place in space-time, the D-branes 6 , where the string can begin or end. Any ver4 To quote Edward Witten of the Princeton Institute of Advanced Study: “spreading out the particle into a string is a step in the direction of making everything we’re familiar with fuzzy. You enter a completely new world where things aren’t at all what you’re used to. It’s as surprising in its own way as the fuzziness that much of physics acquired in light of quantum mechanics and the Heisenberg uncertainty principle.” 5 Strangely enough, quasiparticles with complex valued mass, the so-called tachyons also appear in the spectrum of permissible string states, in the sense that some states have negative masssquared, and therefore imaginary mass. The real part corresponds to the ordinary mass and the imaginary part correspond to the decay rate, i.e., instability of the field vacuum. Once the tachyonic field reaches the stable vacuum, its quanta are not tachyons any more but rather have a positive mass-squared, such as the Higgs boson. If the tachyon appears as a vibrational mode of an open string, this signals an instability of the underlying D-brane system to which the string is attached. The system will then decay to a state of closed strings and/or stable D-branes. If the tachyon is a closed string vibrational mode, this indicates an instability in spacetime itself, and signal spontaneous symmetry breaking. 6 In 11 dimensions, there are alternate super theories, the so-called M theory (some theorists believe that the final version of M-theory may not even have any fixed dimension), based on membranes as well as point particles. In lower dimensions, there is moreover a whole zoo of super

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sion of string theory allowing open strings must incorporate D-branes, and all open strings must have their endpoints attached to these branes with Dirichlet boundary conditions7 , hence Dirichlet-brane (D-brane). In a way, an analogous perceived role of D-branes corresponds to the important physical role of contact leads/reservoir, serving as particle and/or energy source and sink, in nanoelectronics. The arrangement of D-branes constricts the types of string states which can exist in a system. For example, if we have two parallel D-branes with string attached on both ends, this correspond to a quantum dot in nanoelectronics which also determined the particle states inside the quantum dot. If one consider a group of N separate p-dimensional Dp-branes, arranged in parallel for simplicity, labeled 1, 2, ..., N , this corresponds to a semiconductlor-heterostructure superlattice in nanoelectronics. The notable diﬀerence is that the Dp-branes exhibits far more richer excitations. The strings beginning and ending on some brane i give that brane a Maxwell field and some massless scalar fields on its volume. This Maxwell field corresponds to phonons in superlattices. The strings stretching from brane i to another brane j have more intriguing properties. These strings only interact with strings stretching from brane j to brane k, this corresponds to the nearest-neighbor coupling in semiconductor superlattices. This is how far we are willing to go in this nanoelectronics analogy, perhaps enough to bring the readers intellectual interest to the topics of the rest of this book which deals with nonequilibrium physics of open nanosystems, nanodevices, and nanoelectronics.

theories based on membranes in diﬀerent dimensions. (For example, point particles are D0-branes, strings are D1-branes, membranes are D2-branes, and so on. D0-branes are related to black holes. 7 Neumann boundary condition is also allowed.

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PART 2

Mesoscopic Physics

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Chapter 17

Mesoscopic Physics

17.1

Introduction

Modern electronic materials fabrication techniques utilizing controlled particles and energy beams coupled with etching and masking technology, such as molecular beam epitaxy (MBE) and advanced lithography, have provided means to fabricate small structures with resolution already approaching the atomic scale. The advent of scanning tunneling microscope (STM) and atomic-scale fabrication using the principle of STM have added a new dimension in the fabrication towards atomic sizes. The term nanofabrication is appropriately applied to modern epitaxial growth ˚ and lithographic techniques that are capable of making artificial structure with 100A feature size or less. Quantum confinement of charge carriers in all directions can now be achieved by means of band gap engineering and lithographic techniques combined with confining depletion-barrier layers induced by applied voltages at the gates. The distinction between mesoscopic physics and nanoelectronics reflects the treatment of transport problems, similar to the distinction between solid-state physics and solid-state electronics.1 It was about in the late seventies that ideas of mesoscopic physics and nanoelectronics were borne, encouraged and stimulated by a host of intriguing theories (e.g., theory of localization in lower dimensions, thin films and “wires”) and speculations regarding the novel physical phenomena that can be observed in finite “atomic-scale” structures and thin films which can be fabricated, [36, 37, 38]. Moreover the electronic community sees a promising potential application of nanostructures and nanostructure physics for the continued downscaling of device size, system architecture and interconnects. A rapid growth of mesoscopic physics and nanoelectronics really started around 1985. The electronic device community has been witnessing, for a number of years, a strong and definite trend in the miniaturization of integrated circuits and components towards the atomic-scale dimensions. Clearly, when transport dimension reaches a characteristic dimension, namely, the charge-carrier inelastic coherence length, and the charge-carrier confinement dimension approaches the Fermi wavelength, then classical devices physics based on the motion of particles and ensemble 1 A major portion of Parts 2 and 3 is taken from the author’s article in Physics Reports 234 (2 & 3) 73-174 (1993).

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averaging are expected to be invalid; the wave nature of the electrons, discreteness of energy levels and sample specific properties must now be taken into account The main goal of nanoelectronics is to sustain a continued downscaling of integrated circuits (ICs), resulting in more complex functions per chip. This goal can be accomplished by scaling down each nonlinear device. Assuming three-terminal devices, the conventional circuit interconnects do not scale with the device dimensions and may ultimately dominate the IC delays. Thus interconnects would present a formidable problem at some point in the continued device downscaling in the 3-terminal device format. It is conceivable that the ultimate solution to the foreseeable wiring crisis may lie in the development of “information-based” physics of coupled quantum devices, which may lead to a novel device concept and IC architecture. The nanoelectronics community has expressed optimism that if all these goals succeed, supercomputers could be built on a single chip. This will have a tremendous impact in science, technology, and society as a whole. Conventional IC technologies will definitely reach a limit and cease to function properly for a number of reasons: (1) the device physics in conventional ICs obey the law of large numbers and/or thermodynamic limits; (2) device dimensions are large compared to coherence lengths so that energy quantization, quantum interference and discreteness of charge carriers do not play a significant role in device transport physics. Thus, in order for the downscaling of electron devices to be fully realized in nanoelectronics one must address quantum transport, tunneling and interference eﬀects, and discreteness of electron charge in small semiconductor structures. The investigation of nonequilibrium phenomena in small structures in mesoscopic physics and nanoelectronics oﬀers unprecedented opportunities for both fundamental scientific research and technological breakthroughs.

17.2

Mesoscopic Quantum Transport

Mesoscopic quantum transport physics basically deals with near-equilibrium and steady-state (i.e., nontransient) transport across a small system, typically with dimension comparable to the phase-randomizing or inelastic coherence length, Lφ which can be as large as 1-3 µm in 2-D layer transport channel of MOSFET at T = 4.20 K. Improvements in materials quality have extended the electron elastic mean free path to 100 µm [39]. These studies have already yielded a wealth of interesting and novel transport phenomena due to quantum interference, energy quantization and weak localization that are not found in macroscopic systems. These are due to the wave-like nature of the charge-carrier motion. The dynamical aspects of the phase of the wavefunction begin to dominate transport physics in this regime. Indeed, one can expect, barring some subtle diﬀerences, that electron motion is not very much unlike that of photon motion, as described by the Maxwell electromagnetic wave equation, in waveguide structures, Fabry-Perot interferometer, and in inhomogeneous dielectric medium. One of the major diﬀerences, which plays a major role in mesoscopic transport phenomena, is the Coulomb forces between electrons and the lattice potential, as well as the impurity potential in a solid. Moreover, the lack of thermalization and complete ensemble averaging, the discreteness of

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electronic charge and the inapplicability of the law of large numbers in mesoscopic systems have also yielded intriguing sample-specific and one-electron dominated transport phenomena, with potential for computer and information processing applications. The laws of macroscopic transport physics may not hold in mesoscopic transport physics; the combination of resistances, in series and in parallel, is entirely diﬀerent and highly quantum mechanical in nature. The “two-probe” measured resistances in these systems, unlike macroscopic systems, may be dominated by the nature of the contact reservoir, outside of the system itself. If the distance between probes in multi-probe measurements is less than the inelastic coherence length, the measured resistance maybe nonlocal, i.e., measured resistance between two points may depend on paths excursioning further away from the region between the two points. Therefore contact and spreading resistances are not negligible in these systems. Historically, interest in quantum transport near equilibrium or quantum diﬀusion was initiated by Anderson for disordered atomic system in the tight-binding limit (strong localization) [40]. However, extensive theoretical and experimental investigations have been directed to weak localization phenomena in mesoscopic transport physics by considering the scattering of Schrödinger waves by a disordered array of elastic scattering potentials, defects or impurities, in an eﬀort to understand more fully the scaling theory of localization [41]-[42]. It is worthwhile 2 −7 to note that eh (≈ 0.25 kΩ−1 and |c Weber) are combinations of e (∼ 2.0 × 10 fundamental physical constants, in units of conductance and magnetic flux, respectively, that have occupied the central quantitative entities in the analysis of charge transport in mesoscopic physics. As we shall see in what follows, the prominent role of time-independent S-matrix theory in the formulation of mesoscopic transport has resulted in major advances in mesoscopic physics. Moreover, the basic unitarity property and simple timereversal symmetry of the S-matrix have resulted in remarkably illuminating some of the subtle aspects of the symmetry of the electrical conduction experimentally found in mesoscopic systems. It is worth mentioning at the outset that, as is wellknown in scattering theory, unitarity results from the conservation of particles and implies SS † = 1 or S −1 = S † , where S † is the adjoint matrix, and time-reversal symmetry implies S = S where S is the transpose of S. In the presence of external magnetic field, time reversal symmetry implies S(B) = S(−B) where B is the magnetic field. Indeed, earlier major developments in mesoscopic quantum transport theory are centered around the so-called Landauer-Büttiker formulas, which account for selfconsistency between charge and potential at the scattering center as well as boundary interface eﬀects. These eﬀects are not inherent in the traditional Kubo linear-response theory of conductivity in large sample of traditional solid-state physics.

17.3

Electrical Resistance Due to a Quantum Scattering Event

The understanding of quantum transport near equilibrium in mesoscopic systems has advanced considerably, largely due to the “quantum label” counting arguments

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Fig. 17.1 [Reprinted from Ref. [44 ] with permission.] (a) The elastic scatterer is connected to the randomizing reservoir by ideal 1-D perfect conductors. (b) The chemical potential levels in single channel case: µA for the quasi-Fermi level to the left and µB for the quasi-Fermi level to the right of the sample containing the scatterer, µ1 and µ2 are the Fermi levels of the reservoirs.

initiated by Landauer [43]. This is based on his self-consistent (employing Poisson equation) treatment of charge transport where a scatterer essentially create a dipole or localized voltage drop in the immediate vicinity of the scatterer. More importantly, the eﬀects of the interface between reservoir and conducting leads of a mesoscopic conducting sample is taken into account. He essentially gave the formulation of the macroscopic steady-state conductance due to an elementary quantum scattering event. Initially, Landauer introduces the single-channel (i.e., “one-dimensional” wire)2 version of this counting argument by eﬀectively and implicitly claiming that in quantum transport, the “renormalized” concept of left-of-the-barrier conducting lead L, and right-of-the-barrier conducting lead R, are good quantum labels which serve as a channel for incoming and outgoing quantum states (here the channel quantum label is dropped since only one channel is considered). However, in order to “support” asymptotically free-from-scattering incoming and outgoing quantum states, the conducting leads must be free of scattering centers and hence must be considered perfect conductors. Near equilibrium, a completed scattering event is therefore described by a 2 × 2 S-matrix given by S=

rt tr

,

(17.1)

where r and t are the reflection and transmission amplitude, respectively, of the wave incident on the left side of the barrier, ΨL ; r and t are reflection and trans2 Physically, we assumed the conducting channel to be a tube whose cross-section is narrow enough so that only one (the lowest) of the transverse eigenstates can accept all the longitudinally moving electrons from the left to the right reservoir within some specified energy range. This makes the channel eﬀectively one dimensional.

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mission amplitude, respectively, of wave incident on the right side of the barrier, ΨR . Moreover, if there are three one-dimensional wire leads connected by the scatterer, as in ring geometry, it follows that the S-matrix is 3×3 matrix [45], etc. Note that particle conservation requires S to be unitary. In the presence of time reversal symmetry S = S . Landauer further assumed that quasi-chemical potentials or quasi-Fermi levels µL and µR exist on the ideal conductors on the two sides of the barrier whose diﬀerence measures the voltage drop across the barrier, as is usually employed in electron device physics analyses. At steady-state and assuming the diﬀerence of chemical potential between reservoirs is small, Fig. 17.1 shows that a highly localized potential drop, defined by the diﬀerence between µA and µB , only occurs in the immediate vicinity of the barrier or scattering center. Moreover, there are also contact potentials at the reservoir/conducting-lead interfaces. There are no additional potential drops within the perfectly-conducting leads (regions between the barrier sides and reservoir contacts). Thus, there are no imbalance of mobile charges (charge neutrality condition) in this field-free regions of the conducting leads. All these are reasonable assumptions for small applied voltages, as demonstrated experimentally, for the localized voltage drop across a barrier, and by a numerical simulations which exhibit the interface potential drop between reservoirs and conducting leads. A simple counting argument goes as follows. The quasi-Fermi levels µA and µB on both sides of the sample in Fig. 17.1 are determined such that the number of electrons above µA [or µB ] is equal to the number of empty levels (‘holes’) below µA [or µB ]. Assuming the terminal-reservoir chemical potential to be µ1 on the left and µ2 on the right, and assuming that the diﬀerence between µ1 and µ2 is small (so that the energy dependence of |r|2 and |t|2 can be neglected), the number of electrons above µA [or µB ] is 1 + |r|2 (dn/dE) (µ1 − µA ) [or |t|2 (dn/dE) (µ1 − µB )],

(17.2)

where 2 2 |r| + |t| = 1

normalizes the total event probabilities of the traveling electrons. The maximum possible value of 1 + |r|2 is 2, which counts the filled forward and backward directional states to 2 at each energy in the density of states calculated below. 2 Under steady-state condition, 1 + |r| electrons from the reservoir fills the empty levels (filled forward and backward directional hole states) below µA . Similarly, the 2 transmitting |t| electrons fills the empty levels below µB . Thus, using similar considerations, the number of unoccupied states or ‘holes’ below µA [or µB ] is 2 − 1 + |r|2

(dn/dE) (µA − µ2 ) [or

2 − |t|2 (dn/dE) (µB − µ2 )],

(17.3)

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where the density of states per unit volume (spin not counted) is determined by 1 dpdq, 2π| 1 dp dn = 2π| 2m 1 dE √ = , 2π| 2 2mE 1 m = , (dn/dE) = hmv hv dndq =

(17.4)

the density of states for carrier with velocity v moving in one direction only. Equating the number of electrons and empty levels (‘holes’) in the left side of the barrier which defines the quasi-Fermi level µA , we obtain 1 + |r|2 (µ1 − µA ) = 2 − 1 + |r|2

(µA − µ2 ) ,

|r|2 (µ1 − µ2 ) = 2µA − (µ1 + µ2 ) .

(17.5)

Similarly, equating the number of electrons and ‘holes’ in the right side of the barrier which defines the quasi-Fermi level µB , we obtain 2 2 |t| (µ1 − µB ) = 2 − |t| (µB − µ2 ) ,

|t|2 (µ1 − µ2 ) = 2 (µB − µ2 ) .

(17.6)

Subtracting Eq. (17.6) from Eq. (17.5) we obtain |r|2 − |t|2 (µ1 − µ2 ) = 2µA − (µ1 + µ2 ) − 2 (µB − µ2 ) , |r|2 − |t|2 + 1 (µ1 − µ2 ) = 2 (µA − µB ) , (µ1 − µ2 ) =

(µA − µB )

.

(17.7)

e |t|2 (µ1 − µ2 ) , h

(17.8)

|r|2

The steady-state current can be calculated from −e |t|2 vdn

J= µ2

=

−e |t|2 v dE hv

µ1

=

where −e is the electron charge, and µ1 and µ2 are measured at the leads. Substituting the expression of Eq. (17.7), we have the expression of the current in terms

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of the quasi-Fermi levels, µA and µB , across the barrier as J= =

e 2 (µA − µB ) |t| h |r|2

e2 |t|2 (VA − VB ) . h |r|2

(17.9)

Note the critical role of the reservoir in the counting argument. It allows us not to invoke the Kramer’s conjugate or complex-conjugate (in the absence of spin) quantum channels by virtue of the irreversibility introduced by the reservoir, i.e., no coherence exist between electrons emitted by the reservoirs and most importantly the net current is one directional. At steady state, the most elementary contribution to conductance due to a single quantum scatterer alone is obtained by dividing Eq. (17.9) by the diﬀerence of the quasi-Fermi levels across the barrier, (VA − VB ), to yield 2

2e2 h

G=

|t| |r|2

,

(17.10)

where t and r are the transmissions and reflection amplitudes, and the factor of 2 in Eq. (17.10) comes from spin, for spin-independent interaction. Equation 17.1 accounts only for the contribution of the barrier itself. However if one measures the resistance by connecting the voltage probes to the particle reservoirs or heat baths (i.e., the current source and drain), characterized by chemical potentials at thermal equilibrium, then the expression for the total conductance is Eq. (17.8) divided by the voltage diﬀerence at the leads, G=

2e2 h

2 |t| .

(17.11)

Note that for opaque barrier the resistance is dominated by the barrier contribution, and both equations are approximately identical. However, for transparent barrier, i.e., |t|2 ∼ 1, Eq. (17.11) is dominated by the so-called “contact resistance”, Rc , or the source and drain contact resistances, Rc =

h 2e2

1 |t|2

−

|r|2 |t|2

=

h , 2e2

and hence G in Eq. (17.10) becomes Gc =

2e2 . h

Note that the resistance of the barrier itself plus the contact resistance yields the total resistance of Eq. (17.11). The source and drain contact resistances and the resistance of the barrier itself do obey the “addition rule” of macroscopic physics. Later on, we discuss the role of inelastic scatterings in the addition rule of resistances. The concept of contact resistance has created much controversy for some time. After diﬀerent theoretical calculations were performed for calculating G above,

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it is now understood that contact resistances are due to the geometry of a narrow channel feeding into a large (with width much greater than Lφ , possessing “continuous” degrees of freedom and a large number of screening charges) particle reservoir with the charge carrier thermalizing in the heat bath by inelastic scatterings [46][47]. The result of Eq. (17.10) is often referred to as a 4-probe measurements (implying the existence of weakly-coupled zero-current voltage probes), which measures the resistance across barrier itself, whereas the result of Eq. (17.11) is referred to as the 2-probe measurement which measure the resistance including the contact resistances, i.e., voltages are measured well inside the thermalizing reservoirs. Equation (17.10) was applied to the scaling theory of localization [43], by considering the barrier problem described by Eq. (17.10), in 1-D case, to be any segment of a linear chain of barriers. Thus Eq. (17.10) can be used to calculate the conductance of any 1-D problem as a function of the length of the chain, or more appropriately, as a function of the number n of randomly placed barriers. Landauer further calculated the addition law for two such barriers (for generalization it is convenient to define a barrier from here on as any small regime with a voltage drop i.e., a “quantum resistor”), which lead him to calculate, by induction, the resistance of a linear chain of n randomly placed “quantum resistors”. The exponential increase of the resistance with n for n larger than a characteristic value (corresponding to the localization length in the scaling theory of localization), obtained by Landauer by his counting argument [43], was later formalized by the scaling theory of localization initiated by Anderson, et al. [48]. This result points to the power of the simple quantum-label counting argument of Landauer to non-equilibrium quantum transport problems, particularly to the scaling theory of localization, which was largely unnoticed for sometime. It has also lead the people concerned with mesoscopic transport phenomena to seriously reexamined the simple quantum-label counting argument of Landauer and to generalize it to the multichannel case (higher dimensions) [44], which is also of great interest to the scaling theory of localization in more than one dimension. 17.4

The Multichannel Conductance Formula

In the multichannel and finite temperature generalizations of the Landauer formula [44], the ideal wires have a finite cross section which act as leads connecting a general elastic scattering system (quantum resistor) to the particle reservoir. Additional quantum labels are attached to the diﬀerent conducting channels, characterized at zero temperature by wavevector kn at the Fermi energy EF , due the size quantization in the transverse directions leading to discrete transverse energies, En . The number of conducting channels, N, is determined at zero temperature by the equation En +

|2 kn2 = EF , 2m

n = 1, 2, ...., N.

(17.12)

At finite temperatures, the right-hand side of Eq. (17.12) is allowed to take values around EF with width of about kB T , which will give a total-energy dependence to

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kn . In order for the channel number, n, to be good quantum labels, it is assumed that there is no phase relationships among electrons in diﬀerent channels (which is obtained if source and drain have a well defined chemical potential at thermal equilibrium). Therefore a completed scattering event is now described by a 2N ×2N S-matrix given symbolically by (r) (t ) (t) (r )

S=

,

(17.13)

where each element is a N × N matrix to account for the N conducting channels in each lead. It follows that for scattering in a ring geometry with one extra lead to a particle reservoir, S becomes a 3N × 3N matrix. Again conservation of particles lead to the unitary condition for S, as in Eq. (17.1), which provide relations between primed and unprimed quantities. The multichannel and finite temperature generalization of the two-probe measurement given by Eq. (17.10) becomes [46], [44] G2 =

2e2 h

dE −

∂f ∂E

i,j

|tij (E)|2 ,

for T = 0,

(17.14)

and 2e2 h

G2 =

i,j

|tij (E)|2 ,

at T = 0.

(17.15)

2 Note that in the presence of time-reversal symmetry, i,j |tij (E)| = T r(tt∗ ). The corresponding generalization of the 4-probe measurement given by Eq. (17.11) becomes

G4 =

4e2 ηγ , h ς

T = 0,

(17.16)

where η= γ= ς=

dE −

∂f ∂E

i,j

|tij (E)|2 ,

(17.17)

i

−

∂f −1 (vi (E)) , ∂E

i

−

∂f ∂E

dE dE

1+ −

(17.18) 2

|rij (E)| 2 |t j ij (E)| j

(vi (E))−1 ,

(17.19)

which in the zero temperature limit reduces to [44] G4 =

2e2 h ×

i,j

i

|tij (E)|2 2 1+ −

(vi (EF ))−1 i 2

|rij (EF )| 2 |t j ij (EF )| j

(vi (EF ))

−1

−1

,

(17.20)

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where vi (E) is the longitudinal velocity associated with channel i. For a single channel case, Eqs. (17.14) and (17.16) give the nonzero temperature generalization of Eq. (17.10) and (17.11), respectively. It should be emphasized that Eqs. (17.14) and (17.16) do not at all correspond to the addition of channel resistances in parallel. In the limit of large N, Eq. (17.20) has been shown in Ref. [44] to reduce to Eq. (17.15), in agreement with Ref. [48]. However, we could expect that for transparent barriers the residual or contact resistances in the two-probe measurement, represented by Eq. (17.15), should obey the addition rule of macroscopics, since the contact resistances are due to the thermalization eﬀects at the particle reservoirs. Indeed for this case Tr tt† = N , the total number of channels, and G2 ⇒ N

2e2 , h

(17.21)

showing the validity of the parallel addition rule for contact resistances. Another interesting novel feature of G4 formulas which contain velocity factors, in Eqs. 17.16 and 17.20, has to do with the lack of inelastic or phase-randomizing scattering in these formulas, which lead to the failure of long-standing Onsager-type relationships among transport coeﬃcients. For example, it was found that G4 (B) = G4 (−B) ,

(17.22)

where B is the magnetic field. However, because of the presence of inelastic scattering and associated randomization or entropy production, which plays the central origin of Onsager relationship, G2 still exhibits the Onsager-type relationship, i.e., G2 (B) = G2 (−B) . 17.5

Quantum Interference in Small-Ring Structures

The Landauer-Büttiker formulas are expressed in terms of a minimal number of scattering parameters, the elements of the scattering matrix. Moreover, particle conservation, time-reversal symmetry, and current/voltage probes configuration can significantly reduce the number of independent parameters. Applying the formulas amounts to the calculation of independent scattering probabilities. It is in the calculation of the independent parameters that further sophistication in the calculation of mesoscopic resistance can really arise, in much the same manner as calculating T -matrices in many-body quantum scattering problems. The study of ideal two-terminal ring-geometry small structures (not to be confused with closed-loop ring structure) with a magnetic field through its opening, has also created surprising and enlightening discoveries. In the single-channel calculations, the conductance yields the expected fundamental periodicity due to direct interference between two paths propagating along the two branches of the ring, with period φo = |c e , whose amplitude is typically larger than those of the first harmonic which has a diﬀerent periodicity of φo /2. On the other hand, in the multichannel case the first harmonic contribution with period φo /2 becomes dominant. The φo /2 period contribution originates from the coherent backscattering at zero magnetic

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Fig. 17.2 (a) Coherent backscattering paths (clockwise and counterclockwise with time-reversed scattering sequence in a two-terminal small-ring structures, with a large probability to return to the origin. The constructive interference between two diﬀusion paths, traveling in opposite φ senses around the cylinder yield the 2o -period contribution to the conductance. At high fields φ

the 2o -period contribution coming from longer-lived diamagnetic orbits around the ring becomes dominant. (b) [Reprinted with permission from Ref. [49].] Coherent backscattering in chargecarrier conduction often referred to as the cooperon terms in the conductivity is also responsible for weak localization. (c) Feymnan graph for the two-particle propagation P (x, x ) corresponding to the two time-reversed paths in (b). Magnetic flux threading the loop in (b) destroys the timereversed symmetry of the paths.

field, i.e., constructive interference between two diﬀusion paths, traveling in opposite senses around the ring, Fig. 17.2a. This type of quantum diﬀusion, with large quantum probability to return to point of origin is the same mechanism responsible for weak localization [15]. When the two partial waves, in coherent backscattering, surround an area containing the magnetic flux, φ, then the relative changes of the two phases is 2eφ |c which leads to theφo /2-period contribution to the magnetoconductance of small rings. It was found that for N channel ring, the terms with φo /2 periods have the same dependence with N as the fundamental φo -period contribution, i.e., both proportional to 1/N [44]. Since the flux-independent phases of the two directly propagating waves along the two branches of the ring contain random

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contribution as they emerge from the reservoir, or as they encounter sample-specific impurity configuration in diﬀerent portions of a long series of small rings (similar situation exists in small-radius long cylinder), the contribution to the conductance with period φo can be at any phase of its oscillation at zero magnetic field, with equal probabilities. Thus, when an experiment involves an eﬀective averaging over an ensemble of many rings, such as those preformed in small-radius long cylinder and arrays of small circuits, there are cancellation of the φo -period contributions and the dominance of the φo /2 contributions. The reason that φo /2-period contributions survive ensemble averaging is that this term is defined or constrained (in each ensemble member) to have a maximum resistance or minimum conductance at zero magnetic field (no spin-orbit coupling), providing a definite phase for each ensemble. In coherent backscattering, only the amplitudes of the “scattered” waves interfere. There is no interference between the original wave function and its scattered component in the theory of weak localization. It is expected that at high magnetic field this φo /2 contribution will eventually be completely diminished due to the breakdown of time-reversal symmetry. At high magnetic fields and lowvoltage bias, one should begin to see instead the contribution to the subharmonics of the longer-lived Landau orbits around the ring, with φo /2-period contribution as the leading term. The above discussions pertain to Aharanov-Bohm type phenomena, where the magnetic flux is confined to the opening of a ring or cylinder. Another novel physical phenomena could be expected arising from the coherent backscattering in weak localization theory, through the eﬀect of magnetic field within the material. Indeed the same novel phenomena dominate in the small-ring experiment when a substantial amount of magnetic field strength B goes through the arms of the ring, in the form of aperiodic fluctuations in the conductance. This is the only phenomena observed in singly connected “wires” in a magnetic field and manifest in the form of random-like fluctuations in the conductance. The random-like fluctuation is reproducible for a given system as long as the disorder-scattering configuration is not annealed. The aperiodic nature of the resistance changes in connected systems is due to the sample-specific random configuration of impurities, where the magnetic field modifies the electron interference of the “backscattering loop” and all the phases of all paths leading to a point. In the tight-binding picture, each transfer matrix element Hij can be obtained as sum, over all paths through the sample, of the transmission amplitudes through each path from lattice point i to j. Thus Hij clearly results from an interference process. Basically, each intermediate Hjk is modified by the magnetic field through the Peierls phase factor. The aperiodic oscillations, being sample specific, can be averaged out by ensemble averaging. Aperiodic oscillations also do not strictly require that the whole sample length be equal to the inelastic coherence length, i.e., for L Lφ , one can simply have a combination of L/Lφ fluctuating resistances. More detailed discussions on aperiodic conductance fluctuations are given in Ref. [46, 47].

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Fig. 17.3 [Reprinted with permission from Ref. [22]. Disordered sample of arbitrary shape, containing a hole where magnetic flux can pass through. It is connected via perfect conducting leads to four randomizing reservoirs at chemical potential µ1 , µ2 , µ3 , and µ4 , respectively, in a four-probe measurement.

17.6

Generalized Four-Probe Conductance Formula

The Landauer counting argument has been applied by Büttiker to analyze the conductance of a sample of arbitrary shape (which can be of disconnected geometry, i.e., containing a hole where a magnetic flux, Φ can pass through, Fig. 17.3), with an arbitrary number of probes or terminals, in the limit where the carrier can transverse the whole sample without suﬀering inelastic scattering [50, 51]. By taking the direction of the current at each probe to be the direction away from the respective reservoir, and counting the number of available and mobile carriers at T = 0 from each respective reservoir, one can immediately write down the current at each single-channel conducting lead to be given by   e Ii = (1 − Rii ) µi − (17.23) Tij µj  , h j=i

where by virtue of particle conservation and unitarity, the lowest reference chemical potential, below which the carriers are immobile by virtue of the Pauli exclusion principle, do not appear in the above equation. Tij describes the probabilities for carriers incident in lead j to be transmitted into lead i, and Rii describes probabilities for carrier incident in lead i to be reflected into lead i. The matrix defined by aii = 1 − Rii , and aij = Tij has rows and columns adding to zero by virtue of particle conservation and unitarity of the S-matrix. In a typical four-probe experiment, current is measured between one pair of terminals and the voltage is measured with the other pair of terminals. Specifically let us assume that current is fed into lead m, pass through the sample and exit the sample through lead n, i.e., Im = −In . The chemical potentials measured are µk and µi , under the condition that Ik = Il = 0, where lead k and lead l are the voltage leads. Using the current relations Im + In = 0, Ik + Il = 0, and Ik = Il = 0, one

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can express Im or In in terms of the measured µk − µl = e (Vk − Vl ). The resulting measured resistance is given by Rmn,kl =

h {Tkm Tln − Tlm Tkn } , e2 D/Smn,kl

(17.24)

where Smn,kl = Tmk + Tml + Tnk + Tnl = Tkm + Tlm + Tkn + Tln , D = α11 α22 − α12 α21 ,

α11 α22

  (1 − Rmm ) (1 − Rnn ) (1 − Rkk ) (1 − Rll )       +Tmn Tkl Tnm Tlk = , − (1 − Rmm ) (1 − Rnn ) Tkl Tlk       − (1 − Rkk ) (1 − Rll ) Tnm Tmn

−α12 α21 =

Tml Tkm Tnk Tln + Tmk Tkn Tnl Tln −Tmk Tkm Tnl Tln − Tml Tkn Tnk Tlm

.

(17.25)

Note that the following symmetry relations between the scattering probabilities hold by virtue of the time-reversal symmetry of the S-matrix in the presence of external magnetic flux Φ, Rii (Φ) = Rii (−Φ) , Tij (Φ) = Tji (−Φ) .

(17.26) (17.27)

Therefore the denominator of Eq. (17.24) is symmetric under reversal of the mag4! netic field. Furthermore, D = α11 α22 − α12 α21 is basically independent of the 2!2! electrode configurations. Thus, we have the following reciprocity relation for the four-probe resistance Rmn,kl (Φ) = Rkl,mn (−Φ) ,

(17.28)

which means that the resistance measured in the presence of the flux Φ is equal to the resistance measured in the presence of the flux −Φ, if the reversal of the magnetic field is accompanied by the exchange of the role of the current and voltage leads. Moreover, one can construct a symmetric and asymmetric resistances, RS and RA

1 {Rmn,kl + Rkl,mn } , (17.29) 2 1 (17.30) RA = {Rmn,kl − Rkl,mn } . 2 Indeed, the symmetries of the experimental measurements in the universal conductance fluctuations in wires and Aharanov-Bohm oscillations in loops [52] are consistent with Onsager symmetries for four-probe measurements described above, and therefore were seen to have an entirely quantum origin. To compare with real experiments, it is necessary to extend the above arguments, which hold for single-channel leads, to multichannel leads. The scattering RS =

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probability matrix becomes 4N × 4N for N quantum channels per lead. The idea is to “integrate out” the channel indices (channel quantum labels). Let Rii,mn denotes the probability of a carrier incident in lead i, channel n, to be reflected into lead i, channel m. Also let Tij,mn denote the probability of a carrier incident in lead j, channel n, to be transmitted into lead i, channel m. The current in lead i becomes Ii = N −

µi −

Rii,mn m,n

Tij,mn µj .

(17.31)

j=i,mn

Introducing the “reduced” probabilities Rii =

Rii,mn ,

(17.32)

Tij,mn ,

(17.33)

mn

Tij = mn

then the current formulas are formally the same as before, except that I − Rii is replaced by N − Rii . Consequently, the multichannel four-probe resistance formula is formally exactly the same as before, and therefore has the same symmetry as the single-channel case. Below, we discuss various special cases 17.6.1

Two-Probe Conductance Formula

In a multichannel two-port conductor, we have the following relation for the scattering probabilities, by virtue of particle conservation, N1 − R11 − T12 = 0,

(17.34)

N2 − R22 − T21 = 0,

(17.35)

whereN1 and N2 are the number of channel to the left and right of the sample, respectively. The above relations imply T12 (Φ) = T12 (−Φ), T21 (Φ) = T21 (−Φ). Therefore, T12 = T21 = T , whereT (Φ) = T (−Φ). The current equation, Eq. (17.23), for two ports i = 1, 2, yields for the two-probe conductance formula the familiar expression G2 =

eI = (µ1 − µ2 )

e2 h

T

per spin.

(17.36)

Indeed, all known two-probe experiments obey the symmetry of G2 with regard to flux or magnetic field reversal. It is this two-terminal conductance formula that is the source of the “universality” nature of the conductance fluctuations, i.e., the variation of conductance from sample to sample due to sample-specific configuration of the elastic scattering centers.

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17.6.2

Three-Probe Conductance Formula: Scatterers

Model of Inelastic

The eﬀect of an additional third terminal, where probe 3 is used to measure the chemical potential µ3 (I3 = 0) to the two-terminal conductance is very interesting since it suggests and gives insight on how to model an inelastic scattering event. From either I1 + I2 = 0 or I3 = 0, we obtain the measured µ3 as µ3 =

(T31 µ1 + T32 µ2 ) . (T31 + T32 )

(17.37)

The measured resistance with current flowing from lead 1 into lead 2, with the voltage measured between lead 1 and lead 3 is obtained by expressing I1 or I2 in terms of (µ1 − µ3 ). We have the result R12,13 =

h e2

T32 , D

(17.38)

where D = T21 T31 + T21 T32 + T31 T23 = T12 T13 + T12 T23 + T13 T32 .

(17.39)

Similarly, expressing I1 = I2 in terms of (µ3 − µ2 ), we have R12,32 =

h e2

T31 . D

(17.40)

The ‘diagonal’ conductance R12,12 can be obtained by expressing I1 = I2 in terms of (µ1 − µ2 ), R12,12 =

h e2

(T31 + T32 ) = R12,13 + R12,32 . D

(17.41)

Thus R12,12 expresses the combined resistances, since µ1 − µ2 = (µ1 − µ3 ) + (µ3 − µ2 ) . The interesting result is that the two-terminal conductance in the presence of additional probe, G2 = (R12,12 )−1 can be written as G2 =

e2 h

(Tel + Tin ) ,

(17.42)

where Tel = T21 is the elastic transmission probability describing the transmission of carriers from reservoir 1, characterized by µ1 , into reservoir 2, characterized by µ2 without ever entering reservoir 3, characterized by µ3 . Tin is the inelastic transmission probability, which describes the transmission of carriers from reservoir 1 to reservoir 2, through reservoir 3. Tin is given by T31 T23 T31 + T32 1 , = (Ω31 + Ω23 )

Tin =

(17.43) (17.44)

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where 1 T31 1 = . T23

Ω31 = Ω23

T32 T23

,

(17.45) (17.46)

In the absence of magnetic field, T32 = T23 , and Eq. (17.42) represents the familiar classical addition of conductances of a three-branch and three voltage-node resistor circuit. Furthermore, if Tel = 0, T23 (Φ) = T23 (−Φ), i.e., T23 = T32 also holds, and the result is a classical addition of resistances in the remaining two branches describe by Tin Eq. (17.43), which describes the transport of carriers from reservoir 1 to reservoir 2 via reservoir 3 (the third circuit node), where their energy and phase are randomized. Thus the additional voltage lead (connected to reservoir 3) also acts like an inelastic scatterer. Therefore, the classical addition of series resistances becomes operative only through the intermediary of an inelastic scatterer between two elastic scatterings, with the inelastic scatterer destroying the quantum coherence between elastic collisions. The role of inelastic scattering in quantum coherence will be discussed in more detail in the later section. Here, we should just point out that a one-channel wire connected to a reservoir cannot model complete randomization since the maximum value of Tin in Eq. (17.43) is 12 . In later sections, the inelastic scattering reservoir will be coupled via two channels to obtain complete randomization.

17.6.3

Weakly-Coupled Voltage Probes: Barrier Point Contacts

For voltage leads which are very weakly coupled to the conductor, e.g., through the fabrication of limiting barriers which separate the conductor and voltage-reservoir leads, coupling between reservoirs in lead 3 and lead 4 in a four-probe measurement is of second order and can be neglected. Thus to the lowest order in the voltageprobe coupling parameter ε, describing the transmission probability from voltage leads to the conductor, the existence of one probe does not aﬀect what is measured at the other probe. Therefore, we can use the three-probe formula for obtaining the values of the measured chemical potential at each voltage probe. From Eq. (17.37), we have for µ3 (1)

µ3 =

(1)

T31 µ1 + T32 µ2 (1)

(1)

.

(17.47)

,

(17.48)

T31 + T32

Similarly for the second probe, we have (1)

µ4 =

(1)

T41 µ1 + T42 µ2 (1)

(1)

T41 + T42

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where the superscript indicates the order in the voltage-probe coupling parameter ε. Taking the voltage diﬀerence, we have (1)

µ3 − µ4 =

(1)

(1)

(1)

T31 T42 − T32 T41

(1)

(1)

T31 + T32

(1)

(µ1 − µ2 ) .

(1)

T41 + T42

(17.49)

Substituting in the expression for the currents: I1 = −I2 = he T (µ1 − µ2 ), the expression for (µ1 − µ2 ) in terms of the measured voltage (µ3 − µ4 ), we obtain R12,34 = (0)

h e2

(1)

(1)

(1)

(1)

1 T31 T42 − T32 T41 , (1) T T + T (1) T (1) + T (1) 31 32 41 42

(17.50)

(0)

where T = T12 = T21 has the symmetry, to the lowest order, of the transmission probability of two-terminal conductor indicated by Eqs. (17.34) and (17.35). In general, Eq. (17.50) can be written as (here k is the voltage probe on the m-side of the sample and l is the voltage probe on the n-side of the sample) Rmn,kl =

h e2

1 (0)

(1)

(1)

(1)

(1)

Tkm Tln − Tlm Tkn

(1) (1) Tnm Tkm + Tkn

(1)

(1)

.

(17.51)

Tlm + Tln

(1)

(1)

Since the two voltage probes are uncoupled to first order, the sums Tkm + Tkn and

(1)

(1)

Tlm + Tln

are symmetric with respect to flux or magnetic field reversal. (0)

Moreover, to zeroth order, Tnm has the symmetry of the transmission probability of a two-terminal conductor. Therefore, Rmn,kl has precisely the same symmetry as the generalized four-probe resistance, Eq. (17.24). It is important to note here that the resistance measured depends very much on the nature of the voltage probe contacts. Indeed, there is a distinct possibility of measuring negative resistances in four-probe measurements. 17.6.4

The Landauer Four-Probe Conductance Limit

A four-probe interpretation of the single-channel Landauer conductance formula for a sample containing an elastic scatter has been given by Engquist and Anderson [53], who originally annunciated the counting argument for a definition of a voltage probe. The configuration assumed in Ref. [53] is that the voltage probes are connected to perfect leads and the conductor contains elastic scatterers only between the voltage probes, characterized by T and R, as envisioned by Landauer in deriving the original Landauer four-probe conductance formula. The simplifications and assumptions of Ref. [53] lead to the following relations for the transmission probabilities: Tmn = Tnm = T (1 − ε), Tkm = Tmk = Tln = Tnl = ε (1 + R) − ε2 R, Tkn = Tnk = Tml = Tlm = (1 − ε) T , and Tlk = Tkl = ε (1 − ε) T ε. By substituting these values in Rmn,kl , Eq- (47), we recover the original Landauer conductance formula for a

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disordered sample, Rmn,kl =

h e2

R , T

(17.52)

to zeroth order in ε. For this special case it is clearly no longer necessary to exchange current and voltage leads when reversing the magnetic field. Further, note that for this special Landauer four-probe measurements, Rmn,kl ≥ 0.

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Chapter 18

Model of an Inelastic Scatterer with Complete Randomization

Following the lead given by Engquist and Anderson [53], in using the Landauer counting argument for defining voltage probes to introduce phase randomization, Büttiker calculated the contribution to the measured conductance due to a singlechannel model of a localized inelastic scatterer [54]. Inelastic scattering may flip the electron spin, so in its single-channel formulation it is important to emphasize that a transport channel is understood to refer to only one spin state, i.e., more than one spin is considered multichannel. Hence the density of states is given by dn 1 dE = 2π|υ , where υ is the particle velocity at energy E. Basically an inelastic scatterer at any point in a conductor is modeled by a “coupler” whose function is to connect the conductor to a thermal reservoir via a current lead with two quantum channels, channel 3 and channel 4, Fig. 18.1. The segment of the conductor to the left of the coupler is designated as channel 1 and the segment to the right of the coupler as channel 2. Therefore in the single-channel formulation introduced by Büttiker, an inelastic scatterer has four single-channel leads and is described by a unitary 4 × 4 S-matrix which determines the amplitudes of the outgoing waves in terms of the amplitude of the incoming waves. A reduction of the number of distinct relevant variables for this problem can be accomplished as follows. Let Sb denote the incoherent backward scattering and Sf denote the incoherent forward scattering. For single channel leads, these are the only scattering probabilities. Then we have, Sb = T13 + T14 = |S13 |2 + |S14 |2 ,

(18.1)

Sf = T23 + T24 = |S23 |2 + |S24 |2 .

(18.2)

On the other hand, the coherent processes (i.e., without undergoing inelastic scattering) are denoted by 2

2

(18.3)

2

(18.4)

TC,L = T21 = |S21 | ,

RC,L = R11 = |S11 | ,

TC,R = T12 = |S12 |2 ,

RC,R = R22 = |S22 | .

Current conservation and unitarity imply 216

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Fig. 18.1 [Reprinted with permission from Ref. [54]]. (a) Model of the inelastic “coupler” (scatterer). (b) µL and µR are the chemical potentials of the left and right randomizing reservoirs, respectively; µ is the chemical potential of the randomizing inelastic scatterer. The quasi-Fermi levels µA and µB characterize the polarization charges to the left and right sides of the sample containing the inelastic scatterer, with their diﬀerence determining the voltage diﬀerence across the sample.

Tij + Rii = 1,

(18.5)

j=i

which leads to relations between derived quantities as RC,L + TL + Sb = 1,

(18.6)

TC + RC,R + Sf = 1,

(18.7)

where TC = TC,R = TC,L = T21 = T12 , in the absence of the magnetic field. The inelastic scatterer basically randomizes the phase of the charge-carriers and therefore feeds the conductor, through Sb and Sf , with incoherent carriers up to energy µ, which can be determined by requiring that no net particle current flows into the reservoir. In contrast to this zero particle current flow, it will be shown later that inelastic scattering implies the existence of a net energy flux towards the

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reservoir of chemical potential µ. We have for the current in channels 3 and 4, I3 = (1 − R33 − R34 ) I (µR , µ) − T31 I (µR , µL ) ,

(18.8)

I4 = (1 − R44 − R43 ) I (µR , µ) − T41 I (µR , µL ) ,

(18.9)

where I (µR , β) represents the saturated current values from a reservoir at chemical potential µβ to terminal reservoir at chemical potential µR in the absence of any scatterers. We have for each electron spin I (µR , µ) =

e (µ − µR ) , h

(18.10)

I (µR , µL ) =

e (µL − µR ) . h

(18.11)

By requiring that I3 + I4 = 0, one readily obtains µ=

(Sf µR + Sb µL ) , (Sf + Sb )

(18.12)

which is reminiscent of the expression for the chemical potential measured by voltage probe in three-probe measurements. In order to calculate the voltage across the sample which contains an inelastic scatterer, one makes use of the standard counting argument of balancing the number of electrons and holes above the chemical potentials in the left, as well as in the right, side of the sample. This counting argument basically leads to the diﬀerence between the charges piled up at the scatterer characterized by the chemical potential diﬀerence between two sides of the sample, Fig. 18.1b, given by µA − µB =

1 (1 + R11 − T21 ) (Sb + Sf ) + Sb (Sb − Sf ) (µL − µR ) . 2 Sb + Sf

(18.13)

Substituting the expression for (µL − µR ) in terms µA − µB , in the expression of for I1 = I2 = I given by I = T 21 I (µR , µL ) + Sf I (µR , µ), we obtain I=

e (T21 + χSf ) 2 (µA − µB ) , h 1 + R11 − T21 + χ (Sb − Sf )

(18.14)

where χ=

Sb (Sb + Sf )

Therefore, we obtain the conductance per spin measured across the sample as G=

e2 (T21 + χSf ) . π| 1 + RC,L − TC,L + χ (Sb − Sf )

(18.15)

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In the absence of inelastic scattering, we again recover the original four-probe Landauer conductance formula, per electron spin, as G=

e2 TC,L h RC,L

per spin.

(18.16)

However, in the limit of complete phase randomization, i.e., TC,L = RC,R = RC,L = 0 and Sf = Sb = 1, we have Gin =

e2 h

Sf

per spin,

(18.17)

reminiscent of the two-probe Landauer conductance formula. The energy dissipation in an inelastic process indeed follows from the model. The net heat flow to the reservoir (which represent external large number of degrees of freedom at chemical potential µ), or Joule heating, can be obtained by calculating the net particle current with kinetic energy (µL − µR ) from channel 1 to channels 3 and 4, minus the particle current with kinetic energy (µ − µR ) from channel 3 and 4 to the conductors, channel 1 and 2, where µL > µ > µR . The results are energy flux (1 ⇒ 3, 4) =

1 h

2 Sb (µL − µR ) ,

(18.18)

energy flux (3, 4 ⇒ 1, 2) =

1 h

χSb (µL − µR )2 .

(18.19)

The Joule heating, W , is the diﬀerence of the energy fluxes, W =

1 Sb Sf (µ − µR )2 . h Sb + Sf L

(18.20)

Since the number of particles undergoing inelastic scattering is equal to 1 h Sb (µL − µR ), it follows that the heat dissipation per particle is given by w=

Sf (µ − µR ) Sb + Sf L

per particle per spin.

(18.21)

In order to investigate the eﬀect of a series of combination of elastic and inelastic scatterers, it is necessary to have a specific model of a single inelastic scattering process itself. The S-matrix for this single inelastic scatterer necessitates that RC,L = RC,R = 0, i.e., the absence of coherent reflections. We can allow TC,L = TC,R = TC = 1 − ε where ε measures the strength of randomization. From the sum rule, Eqs. (18.6) and (18.7) we must therefore have Sf = Sb = ε. From the expression for the conductance, Eq. (18.15), the contribution of a single inelastic scatterer becomes Gin =

e2 h

(2 − ε) . ε

(18.22)

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For complete randomization ε = 1, it follows that Gin =

e2 h

,

(18.23)

2

also in agreement with Eq. (17.11) with |t| = 1, reminiscent of the contact resistance per spin. Here the contact resistance is due to the entry and exit of particles through a region with a large number of degrees of freedom, i.e., the reservoir at chemical potential µ. The situation is similar to the Sharvin’s contact resistance [55]. The following two sections are intended as further demonstration of the Smatrix calculational technique in the Landauer-Büttiker counting argument.

18.1

Conductance Formula for a Sample Containing an Inelastic Scatterer between Two Elastic Scatterers

The calculation of conductances due to combinations of elastic and/or inelastic scatterers makes heavy use of scattering-matrix algebra, and indeed shows a marked departure from the standard circuit analysis in electrical engineering. However, it closely parallels the calculation technique for microwave guides. From the requirement of unitarity and time-reversal symmetry in the absence of external magnetic field, we can construct the 4 × 4 S-matrix, which relates the scattered amplitude as functions of incident amplitudes, of an inelastic scattering process Sin as a symmetric and unitary matrix √   √ 1−ε ε 0 √ 0 √   1−ε 0 0 √ ε . (18.24) Sin =    √ε 0 0 − 1 − ε √ √ 0 ε − 1−ε 0

The elastic scattering matrix in a single-channel conductor is constructed as a symmetric and unitary matrix (S −1 = S † ) √ √ 1−δ √ δ rt i √ Sel = = , (18.25) tr δ i 1−δ which means that the transmission probability for this scatterer T = |t |2 = |t|2 = δ. For the combination of scatterers in series, a more useful matrix entity is the transfer matrix, which relates the incident and scattered amplitude in one lead (i.e., to the left of the scatterer) with the incident and scattered amplitude in the other lead (i.e., to the right of the scatterer). The transfer matrices can be determined from the scattering matrices, by rearrangement of the resulting linear equation and the linear equations obtained from the inverse matrix. Thus, for example in Fig.18.2, we have, using the scattering matrix for the elastic scatterer, the following relation, √ √ b1 b1 1−δ √ δ i √ = . (18.26) a1 a1 δ i 1−δ

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221

Fig. 18.2 Phase randomizing scatterer between two elastic scatterers. Outgoing amplitudes are related to the incoming amplitudes through the S-matrix, while amplitudes to the left of the inelastic scatterer are related to the amplitudes at the right side through the transfer matrix.

By rearrangement of the resulting equations, and that of its inverse, the transfer matrix is defined by the following equation √ 1 a1 b1 i 1−δ √1 =√ . (18.27) a1 b1 1 δ −i 1 − δ The above equation also define the “canonical” form of the transfer matrix for a single elastic scatterer to be given by √ 1 1√ i R . (18.28) ST = √ T −i R 1 In the absence of inelastic scattering, the elastic transfer matrices are cascaded to obtain the equivalent scattering matrix for a series of elastic scatterers. Thus in Fig. 18.2, if ε = 0, i.e., no inelastic scatterer between two identical elastic scatterers, we have a2 = a1 , a2 = a1 , and hence b2 b2

=

δ 2−δ

−1

√ 2i 1 − δ √1 −2i 1 − δ 1

b1 b1

.

(18.29)

Therefore Teq =

δ 2−δ

2

,

Req =

4 (1 − δ) (2 − δ)2

.

(18.30)

Applying the four-probe Landauer formula for the combined elastic scatterers, we have therefore G4eq =

e2 h

δ2 4 (1 − δ)

per spin,

(18.31)

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which clearly shows the nonadditivity of a series resistances due to the coherence between the elastic scatterers. In the presence of inelastic scattering, ε = 0, what we need is the combined S-matrix S defined by     b1 b1  b2      = S  b2  . (18.32)  a3   a3  a4 a4

From the matrix elements of S, the equivalent conductance can readily be determined by using Eq. (18.15) for the conductance of a sample with an inelastic scatterer. We have to work with the following matrix equations, Eqs. (18.33)-(18.35), √      √ 1−ε ε 0 a1 a1 √ 0 √   a2   a2   1 − ε 0 0 ε  ,  = √ √ (18.33)  a3   ε − 1 − ε   a3  √0 √0 a4 a4 0 ε − 1−ε 0 a1 a1

1 =√ δ

√ i 1−δ √1 −i 1 − δ 1

b1 b1

,

(18.34)

a2 a2

1 =√ δ

√ 1−δ 1 −i √ i 1−δ 1

b2 b2

.

(18.35)

The idea is to eliminate the “internal” amplitudes a1 , a1 , a2 and a2 from the coupled linear equations obtained from the matrix equation of Eq. (18.33), using the coupled linear equations obtained from the matrix equation, Eqs. (18.34) and (18.35). From the above matrix equations, the equivalent scattering matrix S defined by Eq. (18.32) can be calculated. The result for S is

S=

Sa Sb Sc Sd √ Z

,

(18.36)

where Sa =

√ √ i 1√− δ (2 − ε) √ 1 − εδ 1 − εδ i 1 − δ (2 − ε)

,

Sb =

√ εδ i (1 − √ ε) (1 − δ) εδ εδ i (1 − ε) (1 − δ) εδ

,

Sc =

√ εδ i (1 − √ ε) (1 − δ) εδ εδ i (1 − ε) (1 − δ) εδ

,

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Sd =

√ − (1 i 1 − δ (ε) √ − ε) (2 − δ) i 1 − δ (ε) − (1 − ε) (2 − δ)

223

,

2 and Z = [1 + (1 − ε) (1 − δ)] . From the matrix elements of S we identify the following quantities 2

TC,L = T12 = S12 2

RC,L = T11 = S11 2

Sb = T31 + T41 = S31

= (1 − ε)

δ2 , Z

= (1 − δ) (2 − ε)2

(18.37)

1 , Z

2

+ S41 2

Sf = T32 + T42 = S32

= [1 + (1 − ε) (1 − δ)]

εδ , Z

(18.39)

= [1 + (1 − ε) (1 − δ)]

εδ . Z

(18.40)

2

+ S42

(18.38)

Substituting the above derived quantities in the expression for the conductance, Eq. (18.15), we arrived at G=

e2 h

2δ 2 (1 − ε) + εδ [1 + (1 − ε) (1 − δ)] . Z + (1 − δ) (2 − ε)2 − (1 − ε) δ 2

(18.41)

The resulting expression for the resistance does not give any hint of resemblance to the standard rules of circuit analysis. Indeed, for some values of ε and δ, G exhibits a maximum [54]. However, for complete randomization between elastic scatterers, i.e., ε = 1, one immediately recover the standard rule for the additivity of series resistances. We have for ε ⇒ 1, R (ε = 1) =

h e2

2−δ , δ

(18.42)

which can easily be seen to be R (ε = 1) = Rel1 + Rel2 + Rin ,

(18.43)

where Rel1 = Rel2 =

Rin =

h e2

h e2

ε ⇒ 2−ε

1−δ , δ h e2

(18.44)

.

(18.45)

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Fig. 18.3 Model of complete phase randomizing scatterer. Channels i and i + 1 are uncoupled, creating a transport loop with part of the loop undergoing randomization by the reservoir.

18.2

Quantum Coherence in a Chain of Elastic and Inelastic Scatterers

Let us consider one-dimensional chains of alternating elastic and inelastic scatterers with inelastic scatterer between two elastic scatterers. For complete randomization, ε = 1, the scattering matrix for an inelastic scattering event becomes   0010 0 0 0 1  (18.46) Sin =  1 0 0 0, 0100

which clearly shows that channels 3 and 4, connected to the scattering reservoir, are decoupled. Only channels 1 and 3 are coupled, as well as channels 4 and 2, Fig. 18.3 (refer to the notation of Fig. 18.2).From the current conservation equation one can construct the following tridiagonal matrix equation for the chemical potentials of the scattering reservoirs,    µ1 T1 + T2 −T2 0 ... 0 0  −T2 T2 + T3 −T3 ...   µ2  0 0       µ3  0 −T3 T3 + T4 ... 0 0     ...   ...  ... ... ... ... ... µN−1 0 0 0 ... −TN−1 TN−1 + TN   T1 µL  0     = (18.47)  0 ,  ...  TN µR

indicating that the µ’s depend very much on the transmission probabilities Ti . Here, µL and µR are the chemical potential at the terminal reservoirs. Since the µ’s basically also probed the voltages, in the manner used by Engquist and Anderson [53], of the conductor at the inelastic scattering sites, this means that the voltage will drop strongly across very opaque elastic scatterers and will drop only weakly

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across transparent scatterers, as expected. For N = 3, we have µ1 =

[T1 (T2 + T3 ) µL + T2 T3 µR ] , D

(18.48)

µ2 =

[T1 T2 µL + T3 (T1 + T2 ) µR ] , D

(18.49)

where D is the determinant of a 2x2 tridiagonal matrix D = det

T1 + T2 −T2 −T2 T2 + T3

.

(18.50)

The total resistance for a chain of N elastic and N − 1 inelastic scatterers can readily be deduced from the combination of amplitudes derived above for an inelastic scatterer between two inelastic scatterers for the case of complete randomization, ε = 1. For this case, one can easily deduce that the addition of individual resistances still holds, thus N

RN (ε ⇒ 1) =

(i)

i=1

Rel + (N − 1) Rin ,

(18.51)

where (i)

Rel = Rin =

h e2 h e2

(1 − Ti ) , Ti

(18.52)

.

(18.53)

On the other hand, in the absence of inelastic scatterers between elastic scatterers, ε = 0, the coherence between elastic scatterers will completely invalidate the addition of individual scattering resistances. The calculation of the resistance of N elastic scatterers basically involves the cascaded multiplication of scattering matrices. This is more conveniently done by transforming the transfer matrices to their diagonal forms. The canonical form for the diagonalized transfer matrix is   √ (1+√1−T ) 0 T  √ STD =  (18.54) (1−√1−T ) . 0 T

If one considers the case where each elastic scatterers are identical, i.e., Ti = δ, then the equivalent diagonalized transfer matrix must be equal to the cascaded individual diagonalized transfer matrices   N √   √ (1+√1−δ) (1+√1−T ) 0   0 δ T   = √ (18.55) N . √  (1−√1−T ) (1−√1−δ) 0 0 T δ

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From the above matrix equation, the combined transmission and reflection probabilities can be calculated N 4λN + λ−

T =

2,

N λN + + λ−

2

N λN + − λ−

R=

(18.56)

2,

N λN + + λ−

(18.57)

√ where λ± = 1 ± 1 − δ. Substituting in the four-probe Landauer conductance formula, we obtain G=

e2 h

4δ N

e2 h

T = R

λN +

−

λN −

per spin,

2

(18.58)

and hence (N) Rel

=

N λN + − λ−

h e2

2

per spin.

4δ N

(18.59)

From the last matrix equation, Eq. (18.55), we can also solve for δ in terms of T and N . This “inverse” problem is particularly useful for determining the transmission probabilities of each identical parts, i.e., identical segments of a chain when one knows the transmission probability of the whole linear chain. The result is 1

4η N

δ=

1

2,

(18.60)

ηN + 1 where η is the square of the diagonal matrix element of the combined transfer matrix in Eq. (18.55), η=

1+

√ 1−T √ T

2

.

(18.61)

Clearly, as N ⇒ ∞, δ ⇒ 1 for constant η. A very interesting application of the inverse relation derived above is to investigate the influence of the number of inelastic scatterers within a barrier of known transmission probability T . As a theoretical exercise, we assume that the whole barrier is divided into N equal segments with totally randomizing scatterers between them. Then the combined resistance is given by Eq. (18.51) which can be written as RN (ε ⇒ 1) =

h e2

N −δ , δ

(18.62)

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where δ is given by the inverse relation, Eq. (18.60). We can now easily show that, indeed, the resistance of a barrier exhibits a minimum for some value of N, which satisfies d 1 = ln δ, (18.63) N dN for a given value of T . For N not too large and T small, we have, from Eq. (18.56), T ≈ δ N . Then we can approximate the value of N at the resistance minimum to be given by N = − ln T.

(18.64)

On the other hand for very large number of scatterers within the barrier δ ⇒ 1 and we have RN (ε = 1) ⇒

h e2

N,

which looks like the addition of contact resistances.

(18.65)

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Chapter 19

Other Applications of Landauer-Büttiker Counting Argument

The Landauer-Büttiker counting argument (LBCA) in conductance theory is basically based on the “asymptotic” treatment of a transport process. This approach depends on being able to identify current-carrying quantum channels or good “incoming and outgoing” particle-trajectory quantum labels. The role of the particle reservoir is to bring about a complete lack of correlation of the phases of all the particles emitted by all the particle reservoirs, to act as an electronic blackbody, to introduce irreversibility, i.e., no complex-conjugate (in the absence of spin) quantum channel, and to enforce the “asymptotic” final state of the quantum process which have taken place in the sample. Whether the current-carrying quantum channels which can readily be identified near equilibrium, remain to be good current-carrying quantum channels or good particle-trajectory quantum labels far from equilibrium can not generally be assumed. Thus the LBCA for conductance theory is valid at least near equilibrium situations, it is even expected to be able to treat stationary frequency-dependent conductance behavior by employing the full machinery of many-body time-independent S-matrix theory to calculate the reflection and transmission coeﬃcients, exhibiting reactive behavior and resonance. Other problems dealing with questions relating to the temporal development of a transport process, such as the tunneling times in portions of the scattering region, temporal eﬀect of virtual processes far from equilibrium causing reactive delays and resonance, response to finite signal pulse and step voltage pulse and other highly nonlinear, non-stationary and far-from equilibrium processes are clearly impractical for, if not beyond, the LBCA method. But these are just the kind of problems of great interest in nanodevice physics. It is precisely the present dichotomy of interest in transport problem areas that prompt us to distinguish mesoscopic physics and nanoelectronics. The above-mentioned limitations of the LBCA technique are irrelevant in the domain of near equilibrium mesoscopic physics, where it has enjoyed a widespread acceptance by virtue of its astounding and remarkable success in giving physicallyappealing quantitative and straightforward explanations of a host of complex nearequilibrium mesoscopic transport phenomena. In what follows, we will briefly described applications to quantum Hall eﬀect (integral and fractional), persistent currents in closed loop, sample-dependent conductance fluctuations, transport in Luttinger liquid, thermal noise and excess noise in mesoscopic systems, and only

228

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mention attempts at explaining high-frequency behavior (this will be treated further, using non-equilibrium time-dependent S-matrix theory, in later sections).

19.1

Integral and Fractional Quantum Hall Eﬀect

The quantization of Hall conductance was originally predicted by Ando, et al. [56]. It was experimentally observed with great precision by Klitzing, et al. [57]. Using a gauge-symmetry argument, Laughlin [58] gave an elegant and general explanation of the eﬀect. Implied in the Laughlin’s argument is that the quantized Hall current may be viewed as arising from “edge diamagnetic currents.”The introduction of quasi one-dimensional states or edge states for a finite conducting sample in a strong magnetic field, and the clarification of the role of the edge states in the quantization of the Hall conductance, were first given by Halperin [59]. This work paves the way for the LBCA method to be applied to the explanation of quantum-Hall eﬀect in open multi-probe conductors. For an excellent pre-1987 reviews of quantum-Hall eﬀect, the reader is referred to the book by Prange and Girvin [60]. Streda et al. [61] were the first to make the connection between the LandauerBüttiker type of conductance formulas and the quantum Hall eﬀect, based on the observation that the fundamental unit of conductance appears in both cases. The applicability of the four-probe conductance formula to quantum-Hall eﬀect was first pointed out by van Houten et al. [62] and by Peeters [63]. Büttiker [64], emphasizing the suppression of backscattering in quantum-Hall eﬀect, discussed in detail the role of contacts, elastic and inelastic scattering of the edge states in open multi-probe conductors, and the condition for the breakdown of the quantized quantum Hall eﬀect. Indeed, the edge state picture has provided the current-carrying quantum channels for the LBCA to be applied. Recently, the edge-state picture has been extended to the fractional-quantumHall-eﬀect (FQHE) regime (where the electron-electron interaction play the major role) by Beenaker [65]. MacDonald [66] generalizes the Büttiker multi-channel multi-probe resistance formula to the FQHE regime. In the FQHE regime there is no one-to-one correspondence between edge channels and bulk Landau levels. An infinite hierarchy of energy gaps exist, in principle, corresponding to the infinite number of possible edge channels of which only a small number (corresponding to the largest energy gaps) will be realized in practice; these are the edge-state branches. Indeed, earlier experiments in the resistance of a barrier in the FQHE regime were interpreted in terms of conduction via some form of edge channels. Clearly, the work of MacDonald and that of Beenaker have provided the significant step since it is not obvious that the concept of independent current channels within the same Landau level has any meaning at all for interacting electrons. The parametrization of the relevant many-body states, analyzed in Ref. [66], in terms of single-particle edge-state occupancy has the similarity to the Landau Fermi-liquid phenomenology for strongly interacting metals which allows the understanding of low-energy properties using single-particle concepts. Experiments in quantum Hall eﬀects often employ two-dimensional systems

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obtained in Si − Si O2 MOSFET inversion layers, as well as in heterojunction highmobility two-dimensional electron gas (2-DEG) obtained in GaAs/AlGaAs HEMT devices, where the conduction channel size in the vertical direction is of the order of the Fermi wavelength of electrons. 19.2

Universal Conductance Fluctuations

Another mesoscopic phenomenon is the so-called universal conductance fluctuations in disordered mesoscopic systems. This has also been explained by the generalized multichannel conductance formula viewpoint. This behavior has been observed in the conductance of a narrow-gated channel (wire) of Si − Si O2 MOSFET accumulation layer, as a function of gate voltage [49]. It has also been observed in thin metal wires as a function of the magnetic field. As a function of magnetic field, the phenomena of aperiodic oscillations observed is similar to that discussed in Sec. 17.5. Therefore this sample-specific aperiodic fluctuations were observed as a function of magnetic field and chemical potential (gate voltage) or change of the Fermi wavelength of the mobile electrons, by virtue of the Pauli exclusion principle. For samples sizes comparable to the inelastic coherence length, Lφ , the fluctuation is 2 equal to eh , hence the name “niversal.” The fluctuation results from the quantum mechanical interference of the carrier wavefunctions. This quantum interference is a function of gate voltage by virtue of the change of the wavelength of mobile electrons (the same can be said in the change of current); it depends randomly on the placement of scattering impurities and in magnetic field as discussed in Sec. 17.5. Localization also serves to make some channels inactive in the transport process. 19.3

Persistent Currents in Small Normal-Metal Loop

Discussions of persistent current in a ring of normal metal enclosing the AharonovBohm magnetic flux have been started in the early 1960’s [67, 68]. The paper by Büttiker, Imry and Landauer [69] put the idea in new perspectives by proposing that the persistent current is flux periodic, as a result of the elimination of the reservoirs by closing the conducting specimen, whose size is less than the inelastic coherence length, making it into a closed loop. The coherent wavefunction extending over the whole circumference of the loop lead to a periodic persistent current with period equal to the unit of magnetic flux. This eﬀect is directly related to the nature of the eigenfunctions of an isolated loop, very much in analogy with the solution of Schrödinger equation in a periodic potential with the “lattice constant” corresponding to the perimeter of the loop. Büttiker [45] calculated the eﬀect of inelastic scattering in normal-metal loop and found that the weak inelastic scattering does not destroy the eﬀect. Cheung et al. [70] calculations show that the persistent current decreases exponentially with temperature, decreases as W 2 for weak disorder parameter W , and as exp − Lξ for strong W , where ξ denotes the localization length. They have also found that, aside from the periodicity of I with period φo = |c e , certain averaging procedures

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in ensembles of loops containing diﬀerent number of particles lead to period change from φo to φ2o . It should be emphasized that this period halving is not related to the “backscattering mechanism” mentioned earlier which is responsible for the period halving in two-terminal ring structures discussed in Sec. 17.5. Persistent currents have been detected in metal loops by using 107 loops to build up the signal from the magnetic moment associated with the current in each metal loop [71]. Since persistent currents are orbital diamagnetic eﬀect in small closed-normal-metal loop, it would be quite interesting to see the nature of the persistent current in closed loop of inherently diamagnetic materials such as bismuth and bismuth-antimony alloys. The anomalous large diamagnetism of bismuth and bismuth-antimony alloys is basically of interband origin [9] and one should see quite diﬀerent characteristics than those found in normal-metal loops. This problem is similar to the relativistic Dirac particles [4].

19.4

Transport in One-Channel Luttinger Liquid

The low-energy excitations of an explicitly soluble Luttinger model, when used as a basis for describing general interacting Fermi gas in one dimension, leads to the Luttinger liquid theory by analogy with the Fermi liquid theory, which is a description of low-energy excitations of the explicitly soluble model - the free Fermi gas. A paper by Kane and Fisher [72] calculates the transport of one-channel Luttinger liquid through a weak link. For repulsive electron-electron interactions, the electrons are completely reflected by even the smallest barrier, leading to a truly insulating barrier; this is in striking contrast to the noninteracting electrons. On the other hand, for attractive interactions a Luttinger liquid is perfectly transmitted through even the largest barriers. Kane and Fisher also calculated the twoterminal conductance for Luttinger liquids and show that the Landauer-Büttiker two-terminal conductance per channel is multiplied by a parameter g, where g = 1 for noninteracting electrons, g > 1 for attractive interactions and g < 1 for repulsive interactions. A paper by Loss [73] discusses the particle number parity dependence of the direction of the persistent current in a Luttinger liquid with twisted boundary condition in a closed loop, i.e., whether diamagnetic or paramagnetic. Because of the lack of experimental evidence in this area most of the eﬀorts are quite theoretical.

19.5

Mesoscopic Thermal Noise and Excess Noise

The mesoscopic thermal noise in a two-terminal conductor has been calculated by Landauer [74] and was shown to obey the Johnson-Nyquist formula. Since the Landauer conductance approach assumes only elastic scattering, the thermal noise originates from the thermal agitations in the contacts of the sample. These are the equilibrium noise. Nonequilibrium noise or noise in the presence of transport causes fundamental fluctuations in excess of the equilibrium noise. Büttiker [75]

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calculated the thermal noise and excess noise or quantum shot noise for open multiprobe conductors. These multi-probe results were applied to the conductors in quantized Hall-eﬀect measurements, where it was proposed that the absence of backscattering in quantum-Hall conductance results in no excess noise. Results of Shimizu and Sakaki [76] also addressed a new “coupled noise” resulting from the coupling of excess noise and the fluctuations of the external modulation signal such as that imposed on a gate. 19.6

High-Frequency Behavior

We have alluded previously that the LBCA method should be applicable to the treatment of frequency-dependent behavior of stationary mesoscopic problems by employing the full machinery of many-body time-independent S-matrix theory. The reason for this is that questions about high-frequency behavior could readily lead to examining the energy dependence of many-body problems. There have been several attempts to extend the LBCA to this problem but they are all plagued with ambiguous procedures, faulty and unrealistic assumptions. Landauer outline the diﬃculties associated with the calculation of high-frequency conductance [77]. In later section, we will address the high-frequency behavior of nanodevices operating far from equilibrium, where a number of numerical results have been obtained.

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Chapter 20

“Gated” Schrödinger Waveguide Structures and Ballistic Transport

The domain of ballistic transport (i.e., contact-resistance contriolled current flow) has, enlightening surprises. Indeed, the resistance is independent of the conductor length in the ballistic regime and only depends on the number of subbands occupied, or transport channels, as discussed in Sec. 17.4. The experiments are done in splitgate geometry of a heterojunction HEMT structures [78]-[79]. Indeed, conservation of channel number, by virtue of conservation of transverse momentum, is found for transport from a narrow to a wide geometry, i.e., the net resistance of a chain of ballistic resistors is equal to the value of the highest resistor (i.e., having lowest number of channels) in the chain. Other interesting geometrical eﬀects in ballistic transport occurs in “gated” Schrödinger waveguide structures. For example, negative resistance and resonance due to virtual localized states have been observed in stub [80]-[81], recently reviewed by Hess and Iafrate [81], and in double and multiple constriction, single-bend or multiple-bend geometry [82], [83], [84] Schrödinger waveguide structures. Except for the tight-binding Green’s function method of Ref. [80], the Schrödinger waveguide structures are generally studied through numerical simulation [82], [83], [84] of the Schrödinger wave equation.

20.1

Phenomena Associated with the Quantization of Charge

A diﬀerent sort of fluctuation in mesoscopic systems, which inherently has nothing to do with energy quantization and interference of wave functions, is due to the quantization of the electronic charge. These so-called single-electron eﬀects exist in small-area thin tunnel junctions [85] and small Coulomb islands at low temperature [86]-[87]. The single-electron related phenomena are exciting and intriguing not only from the fundamental point of view, but more importantly for its potential technological impact and potential breakthroughs in accomplishing the goals of nanoelectronics and beyond. The basic physical scale in these phenomena is the charging-energy scale. The charging energy of a system capable of localizing electrons, or separated by a tunnel 2 junctions is U = Q 2C , where Q is the stored charge and C is the capacitance of the system. This energy can be large if C is small. For a small area junction or a small Coulomb island or small quantum well, C is proportional to the cross233

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Nonequilibrium Quantum Transport Physics in Nanosystems

section of the junction or the size of the Coulomb island and can be of the order of 10−18 F in GaAs-based embedding medium. Thus, the energy of charging by one electron can become significant and hence a single-electron behavior begins to dominate the carrier flow across the tunnel junction or Coulomb island at low enough temperatures, T ∼ 10 K. In the purely classical case, a tunnel junction only serves as a tunnel barrier between conducting wires delivering a quasi continuous supply of charge (polarization charge due to arbitrary shift of electrons and ions in the wire). The parabolic dependence of the charging energy with the charge Q and the discrete transfer of electronic charge across the junction leads to an instability for electron transfer across junction when |Q| > 2e , Fig. 20.1a. When |Q| > 2e , an electron transfer becomes favorable, reducing the charge by one electronic charge (i.e., the states, −e e 2 and 2 are degenerate and diﬀer by one electron charge). In the presence of a current source the process of charging and recharging periodically repeats yielding a periodic voltage and charge oscillation across the junction, with average frequency ¯ −e equal to Ie . The range of polarization charge −e 2 < Q < 2 is the Coulomb blockade range, i.e., the transfer of electrons across the junction is energetically impossible. The situation does not change if there is a built-in charge (surface charge) Qo in the tunnel junction at equilibrium and the extra charging energy becomes U=

(Q − Qo )2 , 2C

(20.1)

where Qo = −Cφi , φi is the “built-in” voltage, due for example to the diﬀusion of electrons from a higher band gap tunnel barrier. The other type of single-electron dominated phenomena is much more promising for potential direct applications to nanoelectronics [85], [86], [87], [88]. What is even more interesting is that this phenomena has been observed in small structures fabricated by simple or minor modification of existing novel heterojunction devices (also true for single-electron tunnel junction discussed above). By means of patterning the metal gates of a GaAs HEMT device structure or a Si − Si O2 MOSFET device structure [88], one can implement a single-electron field-eﬀect transistor as schematically shown in Fig. 20.1b, where the region between the two potential barriers defines a Coulomb island (classical) or “quantum dot” (quantum mechanical). The point is that the number of charges in the Coulomb island can be controlled by the gate voltage in a rather continuous manner by means of the polarization by the gate voltage. If we assume that the Coulomb island can have integral number of transferred electrons, i.e., quantized number of transferred charges, the polarization charge induced by the gate allows us to shift the charge in the Coulomb island in continuous manner. Again, the total charging energy can be written as U=

(Q − Qo )2 Q2 = −QVG + + cons tan t, 2C 2C

(20.2)

where Qo = CVG is the induced polarization charge induced by the gate voltage, −QVG is the attractive interaction between opposite charges on the gate electrode 2 and the isolated island, Q 2C is the repulsive interaction between electrons in the iso-

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Fig. 20.1 (a) [From Ref. [85] reprinted with permission.] Charging energy of small junction as a function of the polarization charge Q. The discreteness of the electronic charge e leads to the instability for electron transfer at |Q| > (e/2).(b) [Reprinted with permission from Ref. [86]. Coulomb island and corresponding energy levels. The Coulomb island has an energy gap e2 in its tunneling density of states. (c) Charging energy versus polarization charge Q of width 2C in a Coulomb island. Qo is the charge that minimizes the charging energy by varying the gate voltage VG . Because charge is discrete, only quantized values of the energy will be possible for a e2 for current to flow, this is the given Qo . When Qo = Ne, there will be an activation energy 2C Coulomb blockade. However, when Qo = N + 12 e, the state with Q = Ne and Q = (n + 1)e are degenerate, and the energy gap of the tunneling density of states vanishes. This results in the conductance with periodic sharp peaks as a function of VG , with period e/C. (d) Schematic energy-level diagram showing the eﬀect of the quantization of energy in the Coulomb island. The conductance peak at Qo = N + 12 e also requires that the Fermi level of the emitter be equal to one of the discrete energy levels. This means that the gate voltage diﬀerence between the (N −1)th N , where ∆E and Nth peaks, ∆VN , is ∆VN = (e/C) + ∆E N is the energy level spacing. e

lated island, and Q = eN where N is the number of transferred electrons. Therefore by adjusting the gate voltage so that Qo = N +

1 2

e,

(20.3)

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U becomes degenerate for Q = N e and Q = (N + 1)e, i.e., it does not cost energy to add one more electron, Fig. 20.1c; this indicates that transport across the Coulomb island by one electron is most favorable. On the other hand, when Qo = N e, 2

(20.4)

e to transfer additional electron to or out of the Coulomb island it will cost energy 2C (increase/decrease of charge by one electron). Under this condition, there will be e2 for a current to flow, this is the Coulomb blockade. an activation energy equal to 2C Thus, one expects the conductance to be zero for Qo equal to integral number of electron charge and to become large for odd-half integral number of electron charge. Indeed, experiments on Coulomb island in a 600 nm diameter size in a 2-DEG of GaAs/AlGaAs HEMT-like heterostructures have observed the conductance to go to zero periodically as a function of the gate voltage [88]. The promise of single-electron transistor and the goals of nanoelectronics for ultra-dense IC’s point to a serious consideration on how the inherent classicallybased single-electron eﬀects change with the reduction of the Coulomb island to quantum dot, where there are quantization of energy eigenstates in addition to the quantization of charge. The quantization of energy will surely eﬀect the tunneling. Several research groups have observed Coulomb blockade in quantum dots [89]. Coulomb blockade behavior have also been observed in quantum wells [90], [91], [92]. The eﬀect of the quantization of energy on the Coulomb blockade behavior is schematically explained in Fig. 20.1d.

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Chapter 21

Steady-State Nonlinear Many-Body Quantum Transport

We now examine mesoscopic transport from the nonequilibrium many-body quantum transport formalism point of view. A more accurate and more detailed treatment of the energy dependence of many-body eﬀects and scatterings, as well as the treatment of multidimensionality, becomes a formidable task for present-day computational resources. The accurate treatment of the energy dependence of selfenergies in the transport process is of vital importance in the understanding of the eﬀects of nonuniform dissipative transport processes, spectroscopy-related transport phenomena and a host of thermoelectric eﬀects within the region of the device. In the study of mesoscopic physics, interest is almost always focused on the steady-state quantum transport problems, whereas the time-dependence and highspeed characteristics of nanodevices are often of interest in nanoelectronics. Moreover, the integral form of quantum transport equations are usually treated in mesoscopic physics, this is in contrast with the use of two-time-argument integrodiﬀerential form of quantum transport equations used in nanoelectronics in order to examine the eﬀects of strong nonlinearity, spatio-temporal inhomogeneity, and the possibility of ultrafast autonomous-oscillatory device behavior.

21.1

Correlation Functions

The following correlation functions play principal roles in nonequilibrium manybody quantum physics, therefore it is worth familiarizing with them at the outset1 . The retarded Green’s function, Gr (1, 2), is defined by the anti-commutator of fermion field variables

i|Gr (12) = T r ρH ψH (1) , ψ†H (2)

θ (t1 − t2 )

= i| G> (1, 2) − G< (1, 2) θ (t1 − t2 ) , 1A

(21.1)

complete theoretical formulation of nonequilibrium quantum physics is given in Part 4. 237

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ˆ (q, t) is the annihilation fermion-field variable where {A, B} = AB + BA, with ψ H † ˆ H (q, t) is the creation fermion field variable. We have define2 and ψ

i|G> (1, 2) = T r ρH ψH (1) ψ†H (2)

,

−i|G< (1, 2) = T r ρH ψ†H (2) ψH (1)

,

and

θ (t1 , t2 ) = 1, = 0,

if t1 is later than t2 , if t1 precedes t2 .

The subscript H indicates the use of the Heisenberg representation. The last line gives the same expression for fermions and bosons retarded Green’s functions. Now take the Hermitian conjugate of i|Gr to yield

(i|Gr (12))† = T r ρH ψH (1) , ψ †H (2) †

θ (t1 − t2 )

− G< (1, 2)

†

= −i|

G> (1, 2)

= −i|

−G> (2, 1) − −G< (2, 1)

θ (t1 − t2 ) θ (t1 − t2 ) ,

−i| (Gr (12))† = −i| (Ga (21)) = i| G> (2, 1) − G< (2, 1) θ (t1 − t2 ) , †

where we have used the relation, (G> (1, 2)) = −G> (2, 1)3 . Therefore, from (Gr (12))† = Ga (21), we can define Ga (12) = − G> (1, 2) − G< (1, 2) θ (t2 − t1 ) . 2 In the literature, one may find a diﬀerent definition without the factor | of the Green’s functions, hence our Green’s function has the dimension of |1 when compared to the other definitions which do not have the factor |. 3 For Grassman field variables, A and B, we have the transpose: (AB)T = −B T AT , and complex conjugate: (AB)∗ = −A∗ B ∗ . While these might be unusual, the Hermitian conjugate “†” behaves in a familiar manner: (AB)† = [(AB)T ]∗ = B † A† .

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We define the spectral density function, A (1, 2), as4 A (1, 2) = i| (Gr (12) − Ga (12)) = i| G> (1, 2) − G< (1, 2) . We also have the relation i| (Gr (12) + Ga (12)) = i| G> (1, 2) − G< (1, 2) [θ (t1 − t2 ) − θ (t2 − t1 )] = A (1, 2) [θ (t1 − t2 ) − θ (t2 − t1 )] . The advanced Green’s function, Ga , in terms of field variables for fermions is thus given by −i|Ga (1, 2) = T r ρH ψH (1) , ψ†H (2)

θ (t2 − t1 )

= i| G> (1, 2) − G< (1, 2) θ (t2 − t1 ) , where again the last line gives the same expression for advanced Green’s functions of bosons and fermions. Thus, the spectral function is also given by A (1, 2) = T r ρH ψH (1) , ψ†H (2) = T r ρH ψH (1) , ψ†H (2)

[θ (t1 − t2 ) + θ (t2 − t1 )] .

(21.2)

In the presence of Cooper pairing, as in the case of the transport problems in nonequilibrium superconductivity, the following correlation functions also enter into the theory < T rρH ψH (2) ψH (1) = −i|ghh (1, 2) , > (1, 2) , T rρH ψH (1) ψH (2) = i|ghh < (1, 2) , T rρH ψ†H (2) ψ †H (1) = −i|gee > T rρH ψ†H (1) ψ†H (2) = i|gee (1, 2) , a r a r with ghh , ghh , gee , and gee similarly defined. 4 For

a system at steady-state, we can write A (12) = i| (i2 Im Gr ) = −2| Im Gr

Similarly, we may write Γ (12) = i| (i2 Im Σr ) = −2| Im Σr Other but traditional definition of Green’s function make the above definitions without the factor |.

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Other important quantities which one will encounter in nonequilibrium quantum physics is the chronological (correlation) Green’s function, Gc , and antichronological (correlation) Green’s function, Gac . These are defined as follows, i|Gc (12) = T r ρH ψH (1) ψ†H (2) θ (t1 − t2 ) − T r ρH ψ†H (2) ψH (1) θ (t2 − t1 ) = i|G> (1, 2) θ (t1 − t2 ) + i|G< (1, 2) θ (t2 − t1 ) ,

(21.3)

−i|Gac (12) = T r ρH ψ†H (2) ψH (1) θ (t1 − t2 ) − T r ρH ψH (1) ψ†H (2) θ (t2 − t1 ) = − i|G< (1, 2) θ (t1 − t2 ) + i|G> (1, 2) θ (t2 − t1 ) .

(21.4)

The following quantity is sometimes useful i|F = i| G> + G< = T r ρH ψH (1) , ψ†H (2)

,

(21.5)

where [A, B] = AB − BA is the commutator of A and B. This should be contrasted with Eq. (21.2) for the spectral function which involves the anti-commutator. From the expressions for Gc (12) and Gac (12), we have another expressions for r G (12) and Ga (12) as Gr (12) = Gc (12) − G< (12) = G> (12) − Gac (12) , Ga (12) = Gc (12) − G> (1, 2) = G< (12) − Gac (12) . All of the above correlation functions are related to their respective chronological correlation function, denoted by a generic symbol F c , and the corresponding antichronological correlation function, F ac , through the following typical expressions

21.2

F c = F r + F < ; F ac = F < − F a ,

(21.6)

F c = F a + F > ; F ac = F > − F r ,

(21.7)

Integral Equations of Mesoscopic Physics

In steady-state quantum transport, it is worth pointing out that the absence of time consideration renders the equation of the theory to become a pure finitematrix equations. This is clearly possible since one is basically interested in a finite region of the device with a finite range of energies. The resulting equation is then solved by standard sparse-matrix solver or by using finite iterative and recursive techniques centered on the Dyson equation. Coupling of the device to the rest of the universe can always be formulated by a finite boundary condition. The integral formulation of nonequilibrium Green’s function technique is of central concern for all steady-state many-body quantum transport calculations, rather than the partial diﬀerential and integro-diﬀerential time evolution transport equations.

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In what follows, we will not consider Cooper pairing dynamcs. The integral form of quantum transport equation in mesoscopic physics simplifies to the following expression ˜ ˜ = Go + Go ΣG, G

(21.8)

where

where

˜= G

˜ < (1, 2) ˜ c (1, 2) G G ˜ > (1, 2) G ˜ ac (1, 2) G

Σ=

Σc Σ< Σ> Σac

,

Go =

Gco G< o ac G> o Go

,

,

= −1 for fermions. Thus in matrix form, we have Gc (1, 2) G< (1, 2) G> (1, 2) Gac (1, 2) Gco (1, 2) G< o (1, 2) > ac Go (1, 2) Go (1, 2)

=

+

Gco (1, 2) G< o (1, 2) > ac Go (1, 2) Go (1, 2)

Σc Σ< Σ> Σac

Gc (1, 2) G< (1, 2) G> (1, 2) Gac (1, 2)

. (21.9)

In the nonequilibrium many-body Green’s function technique, the principal quantities of interest are the “reduced” or single-particle correlation functions ≷ ρ (q, q ; t, t ) defined as ρ< (q, q ; t, t ) = −i|G< (q, q ; t, t ) , ρ> (q, q ; t, t ) = i|G> (q, q ; t, t ) . Indeed, the diagonal correlation function ρ< (q, q; t, t) gives the particle density. The oﬀ-diagonal elements of this function contain all the dynamical information about the particle while the diagonal elements contain information about density of states, Pauli exclusion principle, and particle occupation of quantum states. Indeed, as we shall see, the equation for ρ< (q, q ; t, t ) is coupled to the equation for the retarded Green function describing the energetics of the motion of particles. Likewise, ρ> (q, q ; t, t ) pertains to holes or unoccupied states. From Eq. (21.9), we obtain the following integral equations by taking the equations for the matrix components. After simplification we obtain r r < r < a < a a G< = G< o + Go Σ G + Go Σ G + Go Σ G a a r < a r r < = (1) G< o (1 + Σ G ) + Go Σ G + Go Σ G .

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Iterating with respect to G< yields expression which does not contain Gro r r < r r r r < r r r < a G< = G< o + Go Σ Go + Go Σ Go Σ G + Go Σ Go Σ G a a r < a < a a + Gro Σr G< o Σ G + Go Σ G + Go Σ G < r r < < a a r r < a a = Go + Go Σ Go + Go Σ G + Go Σ Go Σ G + (Gro + Gro Σr Gro ) Σ< Ga + Gro Σr Gro Σr G< a a r r r r < a r r r r < = (1 + Gro Σr ) G< o (1 + Σ G ) + (Go + Go Σ Go ) Σ G + Go Σ Go Σ G a a r < a ⇒ (1 + Gr Σr ) G< o (1 + Σ G ) + G Σ G ,

where the arrow indicates that continued iteration with respect to G< converges to the indicated expression. Thus a a r < a G< = (1 + Gr Σr ) G< o (1 + Σ G ) + G Σ G .

One can also show that the first term containing G< o is identically zero for a system initially at thermodynamic equilibrium without the external field and many-body eﬀects. To show this, we note that we can also write −1 a a r < a G< = Gr (Gr ) + Σr G< o (1 + Σ G ) + G Σ G .

The first term reads as −1 a a r r −1 < Go (1 + Σa Ga ) , Gr (Gr ) + Σr G< o (1 + Σ G ) = G (Go )

since (Gr )

−1

+ Σr = (Gro )−1 . But (Gro )−1 G< o is the Schrödinger equation written

in the form, i| ∂t∂1 − Ho ψ†o (2) ψo (1) , which is identically zero. Therefore G< is independent of the initial distribution existing before the system was taken out of equilibrium, and the particular solution is G≶ = Gr Σ≶ Ga ,

(21.10)

where we have also included the solution for G> in the above expression. The so-called Dyson equation for Gr is 2˘ 2 Gr ˘ 22 . Gr (12) = Gro (12) + Gro (1¯ 2) Σr ¯ We can also iterate this expression with respect to Gr , which yields Gr = Gro + (Gro Σr Gro + Gro Σr Gro Σr Gr ) = Gro + Gro Σr Gro + Gro Σr Gro Σr Gr = Gro + Gro Σr Gro + Gro Σr Gro Σr (Gro + Gro Σr Gro Σr Gro Σr Gro + Gro Σr Gro Σr Gro Σr Gro Σr Gr) = Gro + Gro Σr (Gro + Gro Σr Gro ) + (Gro Σr Gro Σr Gro ) Σr Gro + (Gro Σr Gro Σr Gro ) Σr Gro Σr Gr = Gro + Gro Σr Gr + Gr Σr Gro + Gr Σr Gro Σr Gr .

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We can rewrite5 the last term as Σr (12) Gr (23) = Σ> (1, 2) − Σ< (1, 2) θ (t1 − t2 ) G> (2, 3) − G< (2, 3) θ (t2 − t3 ) = −Γr (12) θ (t1 − t2 ) Ar (23) θ (t2 − t3 )

Since the product Γr (12) Ar (23) is real, it is equal to its adjoint. Thus, we may write Σr (12) Gr (23) = G> (2, 3) − G< (2, 3)

†

Σ> (1, 2) − Σ< (1, 2)

†

θ (t1 − t2 ) θ (t2 − t3 )

=

G> (3, 2) − G< (3, 2) θ (t2 − t3 )

Σ> (2, 1) − Σ< (2, 1) θ (t1 − t2 )

=

G> (2, 3) − G< (2, 3) θ (t3 − t2 )

Σ> (1, 2) − Σ< (1, 2) θ (t2 − t1 )

= Ga (2, 3) Σa (1, 2) = Σa (1, 2) Ga (2, 3) 5 Note

that from i|Gr (12) = θ (t1 − t2 ) T r ρH ψH (1) , ψ †H (2) r

>

,

<

G (12) = G (1, 2) − G (1, 2) θ (t1 − t2 ) , (Gr (12))† = −G> (2, 1) + G< (2, 1) θ (t1 − t2 )

= − G> (1, 2) − G< (1, 2) θ (t2 − t1 ) = Ga (12)

we have −i Gr (12) − Gr (12)† = Im Gr (12) = − Im Ga (12) = 2 −i (Gr (12) − Ga (12)) = Im Gr (12) = − Im Ga (12) = 2 Im Σr (12) = − Im Σa (12)

i Ga (12) − Ga (12)† 2 i (Ga (12) − Gr (12)) 2

Im Gr (12) = − Im Ga (12)

We also define spectral function A and parameter Γ to be A (12) = i G> (1, 2) − G< (1, 2) = i (Gr (12) − Ga (12)) Γ (12) = i Σ> (1, 2) − Σ< (1, 2) = i (Σr (12) − Σa (12))

Thus, we also have A (12) = −2 Im Gr (12)

A (12) = 2 Im Ga (12)

Γ (12) = −2 Im Σr (12)

Γ (12) = 2 Im Σa (12)

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Therefore we can also express Gr = Gro + Gro Σa Ga + Gr Σr Gro + Gr Σr Gro Σa Ga = (1 + Gr Σr ) Gro (1 + Σa Ga ) . In fact, the particular solution of ρ≷ can readily determine from the equation satisfied by G≷ and Gr , namely, (Gro )−1 − Σr Gr = 1,

(21.11)

(Gro )−1 − Σr G≷ = Σ≷ Ga .

(21.12)

The particular solution G≷ can immediately be written down by inspection of the above two equations as G≷ = Gr Σ≷ Ga ,

(21.13)

The treatment of the dependence on the initial condition denoted by subscript “o” has not been quite clear in the literature on steady-state quantum transport [93, 94]. In the tight-binding recursive-iterative technique initiated by Caroli, et al. [93]], the subscript “o” pertains to a system of uncoupled regions, with each point at a diﬀerent potential due to an applied bias. The uncoupled regions are, for example, the left metal electrode, insulating barrier, and right metal electrode, and do not represent the thermodynamic equilibrium state of the whole system. In this case, ≷ they have found nonzero contribution of ρh . 21.3

Tight-Binding Recursive Technique

The tight-binding recursive technique for doing steady-state quantum transport has been a popular method for doing analytical work in tunneling phenomena [93], [95, 96], [97, 98]. In this method, a closed coupled “linear” equations for the matrix elements (in the site indices) of the equation for G< and Gr are obtained. This means that the following implied matrix equation a a r < a G< = (1 + Gr Σr ) G< o (1 + Σ G ) + G Σ G ,

(21.14)

Gr = (1 + Gr Σr ) gor ,

(21.15)

is reduced to a small number of coupled linear equations in the unknown critical matrix elements, the number of “c-number” equations being equal to the number of unknowns. The casting of the above equation into a closed coupled linear equations in the unknowns is a trick employed to make the resulting equations tractable and solvable. The resulting coupled equations are then recursively solved for all the unknowns.

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Tight-Binding Expression for the Current

First, let us start with the well-known expression for the electrical dipole moment, d, as a vector. In almost all situations, it is defined as d≡ =

r dc,

any neutral system

ri ci ,

neutral system, discrete charges

with no minus signs in the definition. Note that this is also the “first moment” of the charge distribution, i.e. the first term in the multipole expansion. This definition implies that in simple cases, the dipole moment vector is directed from minus charge and towards positive charge, where ci is the charge at site i. We will thus based our expression for the tight binding current as the polarization current, i.e., the time derivative of the dipole density, D, given by q ρ (q, q, t) ,

D=e q

where q is the coordinate of discrete points, and ρ (q, q, t) is the electron density around site q. Thus, the tight-binding expression for the current may be obtained by taking the time derivative of the dipole density or polarization. j (t) =

j (t) = e

q

d ψ†H (q, t) ψ H (q, t) dt

q

=e

q

dρ< (q, q, t) dt

q

−i|dG< (q, q, t) , dt

q

=e q

dD (t) , dt

where the subscript H indicates the use of Heisenberg representation for the operators. We are thus interested in the equation obeyed by G< which is given by our quantum transport equation. We have for noninteracting system i|

∂ ∂ + ∂t1 ∂t2

G< = Ho , G< .

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By making use of Eq. (31.29) to recast the total time derivative, we have ∂ G< (q1 , q2 , E, t) = Ho , G< (q1 , q2 , E, t) ∂t = q1 | Ho |q G< (q , q2 , E, t)

i|

− G< (q1 , q , E, t) q | Ho |q2 .

Hence i|

∂ G< (q, q, E, t) ∂t = Ho , G< (q, q, E, t) q

j (E, t) = −e = −e

G< (q , q, E, t) − G< (q, q , E, t) q | Ho |q ,

q| Ho |q

=

q q

q,q

i|dG< (q, q, E, t) dt

q q| Ho |q

G< (q , q, E, t) − G< (q, q , E, t) q | Ho |q .

Note that in some definitions ψ†H (q, t) ψH (q, t) = −iG< (q, q, t)other , whereas we used the definition ψ†H (q, t) ψH (q, t) = −i|G< (q, q, t)our , so that our G< (q, q, t)our has the unit of |1 . If we use the other definition, the resulting expression diﬀers only in the presence of | in the denominator, then j (E, t) = =

−e q q| Ho |q | q,q −e |

−e = | −e = |

q,q

q,q

q,q

G< (q , q, E, t) − G< (q, q , E, t) q | Ho |q e |

q q| Ho |q

G< (q , q, E, t) −

q q| Ho |q

e G< (q , q, E, t) − |

(q − q ) q| Ho |q

q,q

q,q

q G< (q, q , E, t) q | Ho |q q G< (q , q, E, t) q| Ho |q

<

G (q , q, E, t) .

For steady state system, this is independent of time, j (E) =

−e |

q,q

(q − q ) q| Ho |q

G< (q , q, E) .

(21.16)

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For a linear system composed of N equal segments of length a, the total current is the same as the one calculated for each discrete point. Substituting q − q = a for each segment, then j (E) =

−e |

a q + a| Ho |q G< (q, q + a, E) − a q| Ho |q + a G< (q + a, q, E) ,

q

where in the summation over q we assume nearest neighbor site coupling only. If we take the site coupling Hamiltonian to be Hermitian, i.e., q + a| Ho |q = q| Ho |q + a = Tq,q+a , then we can define the current for each discrete point, q, −ea Tq,q+a G< (q, q + a, E) − G< (q + a, q, E) | −ea = Tq,q+a G< (q, q + a, E) − G< (q, q, E) | − G< (q + a, q, E) − G< (q, q, E) .

j (q, E) =

In the continuum limit, a ⇒ 0, and for a simple discretization of the Schrödinger equation, we have the site-coupling term given by Lima⇒0 Tq,q+a =

|2 = Tq+a,q . 2ma2

Therefore j (q, E) −ea Tq,q+a = |

G< (q, q+a, E)−G< (q, q, E) − G< (q +a, q, E)−G< (q, q, E)

=

−e |2 Lima⇒0 | 2m

=

−e| Limq 2m

=

e| Limq 2m

⇒q

⇒q

G< (q +a, q, E)−G< (q, q, E) G< (q, q +a, E)−G< (q, q, E) − a a ∂ ∂ − ∂q ∂q ∂ ∂ − ∂q ∂q

G< q , q , E G< q , q , E .

In the presence of magnetic field or vector potential, the space derivative must be replaced by a covariant spatial derivative, namely, ∂ ⇒ ∂q

∂ e −i A ∂q |c

(covariant derivative),

where e (≡ − |e|) is the charge of the electron. This covariant derivative yields the

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familiar expression for the momentum operator, −i|

∂ e ∂ ⇒ −i| − (−i|) i A ∂q ∂q |c ∂ e = −i| − A ∂q c e ˆ = P − A. c

The covariant derivative is a reflection of the gauge invariance in quantum mechanics where a gauge transformation of the electromagnetic field must result in the phase transformation of the wavefunction in a manner so that the form of the Schrödinger equation is written in exactly the same way before the transformation, or that the Lagrangian in terms of the field variables is invariant6 . Thus, in the presence of the magnetic field the explicit dependence of G< (q , q , E) on the vector potential or gauge field is required due to the nonlocality of the product of field operators of the form ψ † (q1 ) ψ (q2 ). An expression of the form   q

 ie G (q , q , E) = exp − |c <

q

 A (r) dr ψ† (q ) ψ (q )

will yield a gauge invariant current, where the only dependence in the vector potential is as shown. If the matter fields were only defined on a lattice, then using the mean value theorem we write the above expression as G< (q , q , t , t ) = exp

q +q 2

ie A |c

· (q − q )

ψ† (q , t ) ψ (q , t ) .

6 The gauge symmetry of quantum fields is characterized by the following transformations of the field and covariant derivative, i

ψ = ψe− | c gχ where χ is a real function, g is the charge, and ∂µ + i

g Aµ |c

= ∂µ + i

g (Aµ + ∂µ χ) |c

Therefore, ∂µ + i

g Aµ |c

ψ

i

g g ∂µ χ + i (Aµ + ∂µ χ) |c |c g ∂µ + i Aµ ψ |c

= e− | c qχ ∂µ − i i

= e− | c qχ

Thus, the covariant derivative of ψ transforms like ψ, and hence ψ

†

∂µ + i

g Aµ |c

ψ = ψ † ∂µ + i

g Aµ |c

which leads to gauge-invariant interaction terms in the Lagrangian.

ψ

ψ

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Therefore, in the presence of magnetic field, the nonlocal G< (q , q , E) acquires a Peierls phase factor to render the current expression gauge invariant, q +q ie A · (q − q ) |c 2 ie A (q) · (q − q ) . ⇒ G< (q , q , E) exp |c

G< (q , q , E) ⇒ G< (q , q , E) exp

Therefore, we may write ∂ e e ∂ −e| Limq ⇒q + i A (q) − − i A (q) 2m ∂q |c ∂q |c ∂ −e| ∂ Limq ⇒q − = G< (q , q , E) 2m ∂q ∂q ie2 A (q) G< (q, q, E) . − mc

j (q, E) =

G< (q , q , E)

(21.17)

Now since (we use here the definition of Green’s function without the |) −iG< (q , q , E) = T r ρH ψ† (q ) ψ (q ) , ∂ ∂ − ∂q ∂q = iT r ρH

G< (q , q , t , t ) ∂ † ψ (q t ) ψ (q t ) − iT r ρH ψ† (q t ) ∂q

∂ ψ (q t ) ∂q

,

we have the familiar expression for the current operator as ˆ =

−ie| Limq⇒q ∇ψ† (q t ) ψ (q t ) − ψ† (q t ) (∇ψ (q t )) 2m e2 A (q) ψ† (q ) ψ (q ) , + mc

which can also be written, interchanging the ordinary derivative before going over to covariant derivative, as e| Limq⇒q ψ† (q t ) (∇ψ (q t )) − ∇ψ† (q t ) ψ (q t ) 2mi e2 A (q) ψ † (q ) ψ (q ) . − mc

ˆ = −

This is the well-known expression for the current operator in the continuum limit7 . We have used above the other definition of the Green’s function to make contact 7 A. Fetter and J. D. Walecka, “Quantum Theory Of Many-Particle Systems” (Mcgraw-Hill, NY, 1971), p. 455.

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with other works. Using our definition which diﬀers by the presence of the Planck’s constant, |, the current expression becomes ˆ = − − 21.3.2

e Limq⇒q 2mi

ψ† (q t ) (∇ψ (q t )) − ∇ψ† (q t ) ψ (q t )

e2 A (q) ψ† (q ) ψ (q ) . m|c

Multidimensional Current Expression

We can generalize the linear system to a cubic lattice of points of lattice constant a. Assuming nearest neighbor hopping term q| Ho |q , we have jx (E, t) =

e |

q

ax qx + ax | Ho |qx G< (qx , qx + ax , E)

− ax qx | Ho |qx + ax G< (qx + ax , qx , E) . Similar consideration holds for the other directions. In the continuum limit, a ⇒ 0, Lima⇒0 Tqx ,qx +ax =

|2 = Tqx +ax ,qx . 2ma2x

Thus, we can write in the continuum limit j (q, E) =

21.3.3

−e| Limq ⇒q (∇q − ∇q ) G< (q , q , E) 2m ie2 A (q) G< (q, q, E) . − mc

Mesoscopic Transport Along a Linear Atomic Chain

For a linear atomic chain, we may consider the particle ‘density’ at site i to be given by

−iG< ii −iG< ij

=

=

c†i ci a c†j ci a

≡

−iG+ ii , a

−iG+ ij ≡ , a

where c†i is the creation operator at site i, ci is the annihilation operator at site i, and a is the lattice constant. With nearest neighbor coupling, the current equation,

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Eq. (21.16), becomes

j (q) = = = = = =

c†q+a cq c†q−a cq −e (a) Tq,q−a −a | −ia −ia −ie † † T cq− a cq+ a2 − cq+ a cq− a2 2 2 | −ie † † T ca cα − cα ca | e T c†a cα − c†α ca i| −e + T G+ αa − Gaα | e + T G+ aα − Gαa , |

(21.18)

(21.19)

where T = Tαa = Taα is the site-coupling matrix. The last line of Eq. (21.18) is the basis of the current calculation in a series of papers by the French authors, who were concerned with the microscopic basis of quantum tunneling calculations. As a bit of history, this series of papers8 marks the earliest calculation of mesoscopic + quantum transport. In what follows, we will not distinguish between G< ij and Gij + with the understanding that G< ij is equivalent to Gij for one-dimensional transport. The treatment of arbitrary voltage across the system generally follows the following arguments. The zeroth-order Hamiltonian is made up of uncoupled regions of say, metallic electrodes and diﬀerent heterostructure materials within the system. The voltage values at each lattice point is included in the zeroth-order Hamiltonian, the voltage at each lattice point Vi acts simply to modify the one electron energies (Stark eﬀect). The only “visible” eﬀect of the bias is the resulting diﬀerence in the electrochemical potentials between the source and drain which enters only in the integration over the energy in the current expression. The nonequilibrium condition, by which current can flow, is established adiabatically by switching on the coupling between the diﬀerent regions of the device. The electron-phonon interaction can not in general be included in the zerothorder Green’s functions since highly-correlated many-body states do not allow Wick’s theorem to work. What has essentially been done in the recursive techniques is to treat the electron-phonon interactions and the coupling between diﬀerent regions as the “perturbation” Hamiltonian, when applying the nonequilibrium Green’s function technique. The coupling self-energies are defined as T

Σrij = T Σaij = Tij ,

(21.20)

T

Σ> ij

(21.21)

=

T

Σ< ij

= 0,

8 The following are the earliest papers on mesoscopic transport calculations: C. Caroli, R. Combescot, D. Lederer, P. Nozieres , and D. Saint-James, J. Phys. C 4, 2598 (1971). C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, J. Phys. C 4, 916 (1971). C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, J. Phys. C. 5, 21 (1972). R. Combescot, J. Phys. C 4, 2611 (1971).

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Fig. 21.1 Schematic “tight-binding” representation of metal-insulator-metal junction. Greek indices describe the atomic sites of the metallic conductors.

where Tij is the matrix elements between Wannier function at sites i and j, and generally refers to the coupling between diﬀerent regions. At steady-state condition, the electron-phonon self-energies is given by the Migdal approximation whose diagram is similar9 to the Hartree-Fock exchange diagram of the electron-electron scatterings, which maybe written as ph

≷ ≶ Σ≷ ij = Gij Dji ,

(21.22)

where D denotes the phonon propagator which is second order in the electronphonon coupling. In calculating the complete Green’s function, one is generally faced with selfconsistency considerations inherent in many-body problems. For very small regions, such as a heterojunction barrier, it is more accurate to first renormalize the zerothorder Green’s function of the barrier by taking into account all possible excursions into the surrounding regions. Thus the renormalized “small-region” Green’s function does depend on the matrix elements T . Moreover, the electron-phonon selfenergies given above does depend on the complete Green’s function, and therefore depend on the matrix elements T . This, of course, assumed that we have already solved the complete Green’s functions. As approximation technique which is often used basically leads to the “electron-phonon” renormalization of the Green’s functions in diﬀerent regions, and hence retain the integrity of the uncoupled diﬀerent regions. It is assumed that the phonon self-energies ph Σij can not cross partitions (i lying on one side and j on the other). This means that for calculating ph Σij the Gij terms include only even number of T vertices. This approximation is equivalent to the assumption that Dij is localized (proportional to Dirac-delta function δ ij ) or very short-ranged. Indeed, in metals ph Σij is short ranged, of atomic size. Physically, ph Σij describes the emission of a virtual phonon together with electron density fluctuations. Since metallic Fermi velocity is much larger than the sound velocity, ph Σij is short ranged, mostly because the phonon propagation has a weak dependence on electron wavevector k. Thus it is not entirely a bad approximation to assume ph Σij to be localized. 9 See Appendix G.1.2. Here the interaction vertex is approximated by a delta function in its arguments.

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The only way for a complete propagator to couple across regions is then via the transfer matrix elements T . The calculation of the complete Green’s function proceeds as follows. First one calculates g by inserting only phonon self-energies on the zeroth-order Green’s functions g o ; then the full Green’s function is obtained by inserting transfer elements T on g. We note that for uniform medium, we can easily write r = gij k

[e (k) − ω − Σr (k, ω)]−1 ,

(21.23)

and if one neglects dispersion of Σr on k, the self-energy acts only to shift the frequency, i.e., r o (ω) = gij (ω + Σr (ω)) , gij

(21.24)

o is the zeroth-order propagator. Since the relevant electron energies are where gij of the order of the Fermi energy and the shift correction is of the order of Debye frequency ω D , the relative magnitude in metals is ωµD , which is very small. For structures which are not translationally invariant, and to the extent that Σ is small in the metallic regions then a lowest order expansion is appropriate r ro = gij + gij

ro r ro giλ Σλµ gµj .

(21.25)

λµ

In the small regions, such a insulating barrier, one has to renormalize the propagator due to phonon self-energies and self-energies connected with excursion to surrounding regions. The zeroth-order g o≷ in the insulating region is zero since the energy range of interest lies in the forbidden band of the insulator. Denoting the surrounding regions on the both sides of the small region by Greek indices, α and α , and using italics indices for the medium inside the thin region, we have, for example, ≷

r ≷ a r a r G≷ aa = Gaa T gαα T Gaa + Gab T gα α T Gba + Gai

ph

≷

Σij Gaja ,

(21.26)

where retarded and advanced Green’s function in the thin region is also renormalized by the presence of the surrounding medium, for example, Grab = r r Dr = 1 − T 2 gαα gaa

r gab , Dr

r r r r gba gαα gαr α , 1 − T 2 gαr α gbb − T 2 T 2 gab

(21.27) (21.28)

where the g’s are propagator in the absence of T and T at the interfaces on both sides, but with phonon self-energies included. We further note that for the insulating regions ph Σij may have an appreciable range, because the electron-phonon interaction is not screened as in metals. In evaluating the expression of the current, various simplification can be made by working to second-order in the electron-phonon coupling. Another simplification in the insulating region, for carrying out the analytic calculations, is to retain the local phonon self-energy approximation.

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The Four-Probe Landauer Current Formula

A small nondissipative system between two conducting leads with transparency less than one has previously been analyzed using the four-probe Landauer formula. The same system can now be analyzed using the steady-state nonequilibrium quantum transport theory discussed here. In recursive technique it is more convenient to recast the expression for G≷ in an iterative form as a a r ≷ a r ≷ . G≷ = G≷ o (1 + Σ G ) + Go Σ G + Σ G

(21.29)

In the absence of dissipation, we only have eﬀective one-body time-independent coupling self-energies between the diﬀerent regions. Then we have to solve the following matrix equation recursively a a r r ≷ G≷ = G≷ o (1 + Σ G ) + Go Σ G ,

(21.30)

where the matrix elements of Σa and Σr are given by Eq. (21.20), and in the absence of dissipation Σ≷ = 0. In the barrier region, G≷ o ≡ 0 since we are interested in the < tunneling energy range. By recursively solving for for G< αa and Gaα we readily obtain ie < T ρ< aα − ραa | e dω a G (ω) Grab (ω) = T 2T 2 | 2π ba

Jel =

< (ω) gα> α (ω) gαα > −gαα (ω) gα< α (ω)

,

(21.31)

where the transparency, T , of the barrier is given by

T (ω) = T 2 T 2 Gaba (ω) Grab (ω) = T 2 T 2 |Grab (ω)|2 .

(21.32)

The expression within the curly bracket in Eq. (21.31) describes the balance of current flow in opposite directions with full account of the Pauli exclusion principle; < gαα (ω) represent the electron occupation of the density of states at energy ω in site α of the left conducting lead, similarly gα< α (ω) stands for the corresponding quantity for the right conducting lead. The hole occupation of the density of states > in the leads is described by gαα (ω) and gα> α (ω). At zero temperature and in the presence of voltage bias V , then we have Jel =

e 2 T T |

µ+eV 2 µ

dω a < G (ω) Grab (ω) gαα (ω) gα> α (ω) . 2π ba

(21.33)

By making linear response and single channel approximations on the above expression and use of Eqs. (21.27) and (21.28), one obtains a formula that resembles the four-probe Landauer formula. 21.3.5

Current Formula in the Presence of Real Phonon Scatterings

The inelastic contribution to the current can readily be calculated of considering the contribution of the Σ< term in the expansion for G< . Recall from Eq. (21.19)

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the current across an interface can be written as e < J = T G< aα − Gαa . |

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(21.34)

In the above form G< αa is a complex quantity which describe the oﬀ-diagonal correlation between barrier and conducting leads. We know that this quantity can be expressed in terms of the transfer matrix elements T and quantities belonging to the separate regions. Furthermore, note that gαa = gaα = 0. We have, using the definition of coupling self-energies, Eq. (21.20), the following “linearized” identities, < a r < G< αa = gαα T Gaa + gαα T Gaa ,

(21.35)

r < < a G< aα = Gaa T gαα + Gaa T gαα .

(21.36)

Using Gr − Ga = G> − G< , we arrived at a more convenient expression e < > < J = T 2 gαα G> aa − gαα Gaa . |

(21.37)

< G> aa and Gaa are the quantities that can account for the dissipation in the barrier region. We have

G> aa =

a Grai Σ> ij Gja ,

(21.38)

a Grai Σ< ij Gja ,

(21.39)

ij

G< aa = ij

where the summation of i and j extends over the whole barrier. Therefore e < > < (21.40) Jin = T 2 Grai Gaja gαα Σ> ij − gαα Σij . | ij ≷

It remains to show that the scattering self-energies, Σ≷ , depends on T and gα α of the right conducting region. This procedure is important to dispel the unsymmetrical appearance of the last equation, in contrast to the symmetric appearance of the elastic contribution, Eq. (21.31). We replace Σ≷ by it explicit form, with each interaction vertex approximated by a delta function in its arguments,10 ≷

≶

Σ≷ = iGij Dji ,

(21.41)

and truncate at second-order in the electron-phonon coupling. Then we have e ao < >o < > Jin = i T 2 Gro (21.42) ai Gja gαα Gij Dji − gαα Gij Dji , | ij where Go denotes the value of G in the absence of phonon scattering, but with transfer matrix elements not set equal to zero (note that in calculating g, T and T are set equal to zero). From Eq. (21.20), we can write the equation for the nonzero 10 See

Appendix G.1.2.

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Fig. 21.2 Schematic diagram for Tij (ω, ω ). It describes the electron from a toi with energy ω, interacts with phonon, and then proceeds to b with energy ω ; the return trip is a “complex conjugate” or a reverse process. ≷o

Gij by taking into account the renormalization caused by the excursions in the electrodes, ≷o

≷

≷ ao ro ao Gij = Gro ia T gαα T Gaj + Gib T gα α T Gbj .

(21.43)

We are only interested in the last term of Eq. (21.43) since the first term of can be shown to give zero contribution in Jin , The proof relies on the symmetry of Gro and Gao matrices and the assumption that the phonon system is in thermodynamics equilibrium. Substituting the second term in Jin , and transforming the time variables, we finally end up with Jin =

ie (2π)2 |

ij

dωdω Tij (ω, ω ) Fji ,

(21.44)

where ro ao ao Tij (ω, ω ) = T 2 T 2 Gro ai (ω) Gib (ω ) Gbj (ω ) Gja (ω) , < < > > Fji = gαα (ω) gα> α (ω ) Dji (ω − ω ) − gαα (ω) gα< α (ω ) Dji (ω − ω ) .

(21.45) (21.46)

Tij (ω, ω ) measures the inelastic transparency of the barrier, and describes a fairly simple situation as indicated in Fig. 21.2, which describes the forward and return trip of the electron across the barrier with phonon emission at i and phonon absorption at j. Note that the electron and phonon propagators are convolved in energy between site i and j. Fji contains all information about energy densities, occupation of states, and Pauli exclusion principle. For phonons at equilibrium, we have for ω > 0 < > Dij (ω) = Dij (−ω) = −iBij (ω) N (ω) ,

(21.47)

> < Dij (ω) = Dij (−ω) = −iBij (ω) [1 + N (ω)] ,

(21.48)

where Bij (ω) is the phonon spectral density correlating sites i and j. The above expressions represent the Kadanoﬀ-Baym ansatz [99] applied to phonon corre< > lation function D≷ , since −i Dij (ω) − Dij (ω) = Bij (ω) sgn ω for all val< = ρα (ω) Aα (ω), and ues of ω. For the metallic electrodes, we have −igαα > igαα = [1 − fα (ω)] Aα (ω), where Aα (ω) is given by Eq. (25.7).

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The total current across the barrier is the sum of Jel + Jin , It should be emphasized that Jel written before also includes those phonon processes which simply renormalized the electron energies, excite virtual phonons without change of energies, and do not involve real phonon excitations. On the other hand, Jin includes bonafide many-body processes in which the electron exchange energy with the phonon as they traverse across the barrier. The resulting total current expression serves to generalize the phenomenological Landauer-Büttiker four-probe conductance formula to include phonon scattering at various sites in the insulating barriers.

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Numerical Matrix-Equation Technique in Steady-State Quantum Transport

The inherently analytical tight-binding recursive technique discussed in Sec. 21, becomes a tedious and impractical calculational method for real multi-dimensional finite nanodevices. Thus, one has to resort to computational methods and numerical techniques in solving the nonequilibrium Green’s function equations. A large-scale computational and numerical task it is convenient to consider only the particular solution of G< , Eq. (21.10). For the present purpose, it will be helpful to write the charge transferred per unit time between two lattice points from q to q [Eq. (2) and Eq. (58) of Ref. [100]] as I

1 (q + q ) , E 2

=

e < G (q, q , E) Tq ,q (E) − G< (q , q, E) Tq,q (E) , (22.1) |

where Tq ,q (E) is the coupling matrix elements between the lattice points q and q . The current density for a cluster of lattice points occupying a volume V is given by J (E) =

−e |

q ,q ,α,δ

(q − q ) G< (q , q , E)αδ T (q , q )δα ,

(22.2)

where α and δ pertain to band and spin indices, written explicitly in the last equation for clarity. If we take the units in which −iG< (q, q, E) = n (q, E), where n (q, E) is the density of electrons at point q, this being a situation obtained in the discretization of a continuum nonequilibrium Green’s function theory (nearest neighbor coupling) as opposed to the lattice nonequilibrium Green’s function theory, then it is possible to define the current density at point q, from Eq. (21.16), as J (q, E) =

e Limq |

→q

η

G< (q , q , E) T (q , q,E) , −G< (q , q , E) T (q , q , E)

(22.3)

where η = q − q , and the sum over q is restricted between neighbors of point q. Moreover, as we have discussed in the discretized continuum theory in Sec. 21.3.1, we have T (q , q ,E) = 258

|2 2m |η|2

.

(22.4)

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Substituting T (q , q,E) above in the expression for J (q, E), we obtained the following familiar expression in the absence of the magnetic field, J (q, E) =

e| lim 2m q ⇒q

∂ ∂ − ∂q ∂q

G< (q , q , E) .

(22.5)

In the presence of magnetic field, the current expression is given by Eq. (21.17). 22.1

Kinetic Equation at Low Temperatures

The particular solution for G< is given by Eq. (21.10). We note that the following sum rule holds, and originates from the fundamental relation i (G> − G< ) = A, where A is the spectral function for the particles, dq dq

Gr (q, q , E) Γ (q , q , E) Ga (q , q , E) = 1. A (q, q , E)

(22.6)

Another observation is that for local phonon self-energy approximation, the collision term in the quantum transport equation can be written as i A (q, q , E) Σ< (q, E) + Σ< (q , E) 2| 1 − G< (q, q , E) [Γ (q, E) + Γ (q , E)] . 2|

Lc (q, q , E) = −

(22.7)

The diagonal element of Lc (q, q , E) which enters in the conservation of particle ∂ equation ∂t ρ = 0 (steady-state) is then simply Lc (q, E) = −

i i A (q, E) Σ< (q, E) − G< (q, E) Γ (q, E) . 2| 2|

(22.8)

Detailed balance condition at thermal equilibrium with zero bias corresponds to the vanishing of the collision term, i.e., Lc (q, E) = 0. Thus at zero bias we have the following relation −iΣ< (q, E) =

n (q, E) Γ (q, E) , A (q, E)

(22.9)

where n (q, E) is the number density of particles. If one writes n (q, E) = f (q, E) A (q, E) ,

(22.10)

then f (q, E) is an arbitrary function multiplying the spectral function A (q, E) chosen to yield the exact value of n (q, E). Note that f (q, E) in Eq. (22.10) does not in general correspond to the diagonal element of the density matrix, as implied in the Kadanoﬀ-Baym Ansatz. Then the detailed balance condition yields −iΣ< (q, E) = f (q, E) Γ (q, E) .

(22.11)

This is a very interesting relation within the local phonon self-energies approximation, since it amounts to a statement that a bonafide many-body nonequilibrium

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quantity Σ< (q, E) can be expressed in terms of single-particle quantities f (q, E) and Γ (q, E). Although the detailed balanced condition strictly holds only at zero bias, in the computational matrix-equation technique, it maybe assumed that the relation can be used in the expression for the particular solution for G< . This point needs some clarification. By using this relation, the resulting G< obtained still allows for transport which eﬀectively accounts for collisions “within” the device region. Indeed, the particular solution for G< is mainly due to to scatterings, as manifested by the presence of Σ< factor. In eﬀect, the use of the detailed balance relation simply weaken the highly nonlinear aspect of integral expression for the particular solution for G< . This particular trick is typical in many-body problems, to render a formulation solvable. As always, extreme caution has to be exercised as this step is only approximately valid within some range of validity, e.g., for small bias range. When the relation given by Eq. (22.11) in substituted in the particular solution for G< given by Eq. (21.13), and taking the diagonal element, then we obtain a recursive equation for the mysterious function f (q, E), f(q, E) =

dq K (q, q , E) f (q , E),

(22.12)

|Gr (q, q , E)|2 Γ (q , E) . A (q, E)

(22.13)

where K (q, q , E) =

From Eq. (22.6), we have the following sum rule for K (q, q , E) dq K (q, q , E) = 1.

(22.14)

Having “extracted” the principal nonlinearity or “non-linear feedback element” in the expression of n(q, E) from the particular solution for G< , we can therefore expect a much less sensitivity of the kernel K (q, q , E) to the solution f(q, E). Indeed, comparing the sensitivity of Γ and Σ< to the solution f (q, E) we have [99] γ 2 (q , ω) [N (q , ω) + f (q , E + |ω)] A (q , E + |ω) . γ 2 (q , ω) [N (q , ω) + 1 − f (q , E − |ω)] A (q , E − |ω) (22.15) What we are interested, for numerical purposes, is the sensitivity of Γ as a function of f(q , E) compared to Σ< , which can be expressed as Γ (q , E) =

dω

γ 2 (q , ω) [N (q , ω) + 1] f (q , E + |ω) A (q , E + |ω) . +γ 2 (q , ω) N (q , ω) f (q , E − |ω) A (q , E − |ω) (22.16) A comparison of the last two equations clearly shows that Γ (q , E) is much less sensitive to the solution f(q , E) for all energies, than Σ< (q , E). For elastic scattering, Γ (q , E) is independent of f(q , E) for all energies. Therefore it is expected that in numerical computation, the equation for f(q, E) will converge much faster Σ< (q , E) = i

dω

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than the equation for the electron density n(q, E) [94]. Once f (q, E) is obtained together with Gr , then G< is reconstructed as G< (q1 , q2 , E) = i

dqGr (q1 , q , E) Gr∗ (q2 , q , E) Γ (q , E) f(q , E),

(22.17)

from which all nonequilibrium quantities can be calculated. The numerical procedure for solving f(q, E) has been detailed in Ref. [94]. The bulk of the numerical calculation lies in solving the Dyson matrix equation for Gr . It has been shown that for low-enough bias at the contacts, described by the Fermi-Dirac distribution, the solution for f (q, E) inside the device will be of the form f(q, E) = e[βE−eµ(q)] + 1

−1

,

(22.18)

where µ (q) is defined as the local electrochemical potential. Thus, at lower bias, knowing µ (q) is equivalent to solving f(q, E). The equation for µ (q) is obtained by Taylor series expanding around µo on both sides of the equation for f (q, E), Eq. (22.12), which yields, µ (q) =

dq K (q, q , µo ) µ (q ) ,

(22.19)

where use is made of the relation given by Eq. (22.14). Once µ (q) and Gr are obtained, then the correlation function G< can be reconstructed using Eq. (22.17), which for low bias can be written explicitly as < G< (q1 , q , E) = G< o (q1 , q , E) + δG (q1 , q , E) ,

(22.20)

G< o (q1 , q , E) = i

dq Ω (q, q , q , E) fo (E) ,

(22.21)

∂fo (E) ∂E

(22.22)

where

δG< (q1 , q , E) = i

dq Ω (q, q , q , E)

−

e [µ (q ) − µo ] ,

Ω (q, q , q , E) = Gr (q, q , E) Gr∗ (q , q , E) Γ (q , E) .

(22.23)

From Eq. (21.17) for the current expression, it follows that we can also write the current at low bias as J (q, E) = Jo (q, E) + δJ (q, E) ,

(22.24)

where Jo (q, E) =

δJ (q, E) = e

dq JΩ (q, q , E) fo (E) ,

dq JΩ (q, q , µo ) [µ (q ) − µo ] ,

(22.25)

(22.26)

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∂ e| ∂ lim − Ω (q, q , q , E) 2m q→q ∂q ∂q ie A (q) Ω (q, q , q, E) . − mc The terminal current is given by JΩ (q, q , E) =

dE |

I=

contact

J (q, E) • dS =

contact

J (q) • dS.

(22.27) (22.28)

(22.29)

By applying the divergence theorem, we can also write I=

dq ∇ • J (q) .

(22.30)

But at steady-state ∇ • J (q) is precisely equal to Lc (q) where Lc (q, E) is the collision term given by Eq. (22.8). Hence we have ∇ • J (q, E) =

dq ∇ • JΩ (q, q , E) f (q , E)

e A (q, E) Γ (q, E) f (q, E) − (−i) G< (q, E) Γ (q, E) | e A (q, E) Γ (q, E) δ (q − q ) = dq 2 − |Gr (q, q , E)| Γ (q , E) Γ (q, E) |

=

× f (q , E) .

22.2

(22.31)

Kinetic Equation at Higher Temperatures and Arbitrary Bias

The preceding discussions, Sec.21.3.4, can be generalized to explicitly account for the coupling of the transport-energy channels. Let us rewrite the particular solution to G< given by Eq. (21.13) as G< (q1 , q , E) =

dq Gr (q, q , E) Σ< (q , q , E) Ga (q , q , E) .

dq

(22.32) In the local self-energy approximation and making use of the relation,1 which holds at steady-state condition, Σ< (1, 2) = G< (1, 2) D> (2, 1)

(22.33)

we have n (q, E) =

dq dE K (q, q , E, E ) n (q , E ) ,

(22.34)

where K (q, q , E, E ) =

1 2 |Gr (q, q , E)| D> (q , E − E ) . 2π|

(22.35)

1 See Sec. G.1.2 for the electron-phonon self-energy in Appendix G. Here we approximated the interaction vertex by a delta function in its arguments.

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Equations (22.34) and (22.35) explicitly show the coupling of all transport-energy channels in determining the particle density n(q, E). The above equation for n(q, E) is a highly nonlinear equation since D> (q , E − E ) also depend on the solution n (q , E ). Equation (22.34), together with the Dyson equation for Gr , can be used [101] to simulate a RTD at large bias. Open-ended boundary condition for Gr was used [94] to simulate perfectly absorbing contact. Kadanoﬀ-Baym ansatz and FermiDirac distribution at electrochemical potential µi is used as boundary condition for n(q, E) at contact i. Note that by integrating both sides of the equation for n(q, E) over the variables q and E, we deduce that dqdE K (q, q , E, E ) is equal to one. Indeed, for Eq. (22.34) to be satisfied, the integral Kernel has to be unitary so that dq dE K (q, q , E, E ) is also of order unity. Thus if one expands the potential µ (q, E) about the equilibrium, to first order in δµ (q, E) valid for small bias, then one readily arrives at µ (q, E) =

dq dE Keq (q, q , E, E ) µ (q , E ) .

(22.36)

This relation is a generalization of Eq. (22.19) and explicitly accounts for the realistic energy coupling or “vertical flow.” 22.3

Relation with Multiple-Probe Büttiker Current Formula

The multiple-probe Büttiker current formula, Eq. (17.23),relating currents Ii at metal electrodes i to the electrochemical potential µj at electrodes j, is rewritten here as   e Ii = (22.37) T i µi − Tij µj  , | j=i

where,

Ti =

Tij = j

Tji ,

(22.38)

j

Tij (B) = Tji (−B) .

(22.39)

Here B is the magnetic field. It would be more helpful to consider the exact tightbinding many-body quantum transport equation in the energy-time space and in the Wannier representation. This is derived in Ref. [100] and is given by i|

∂ < G (q1 , q2 , E, t) = H, G< (q1 , q2 , E, t) ∂t + Σ< , Re Gr (q1 , q2 , E, t) i Σ< , A (q1 , q2 , E, t) + 2 i − Γ, G< (q1 , q2 , E, t) . 2

(22.40)

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For time-independent problems, the above equation reduces to (after multiplying by −1) H, G< (q1 , q2 , E) = − Σ< , Re Gr (q1 , q2 , E) i − Σ< , A (q1 , q2 , E) 2 i + Γ, G< (q1 , q2 , E) . 2

(22.41)

The diagonal elements, q1 = q2 = q, of the left-hand side (LHS) of the last equation yields ∇ • J (q, E), i.e., ie e < Σ , Re Gr (q, q, E) − Σ< , A (q, q, E) | 2| ie + Γ, G< (q, q, E) . 2|

∇ • J (q, E) = −

For inelastic scattering, ∇ • J (q, E) = 0, but across any surface S is given as usual by IS =

dE 2π|

q S

dE | ∇

(22.42)

• J (q, E) = 0. The current

dS • J (q, E) .

(22.43)

From our definition of the current, Eq. (22.3), we can also write in discretized form I=

e |

dE 2π|

q= q

×δ q −q+

+q 2

dS• (q − q )

G< (q , q , E) H (q , q , E) −H (q , q , E) G< (q , q , E)

η η δ q −q− , 2 2

(22.44)

where η = (q − q ) is parallel to a unit vector normal to the surface S, q and q are lattice points on opposite sides of S. In the continuum limit, and with S being the contact interface and the current in Eq. (22.44) directed towards the thermal reservoir, it is easy to see that indeed, ∇ • J (q, E) =

e H, G< (q, q, E) . |

(22.45)

Another way to calculate the current, particularly at the contacts, is simply to integrate the right-hand side (RHS) of Eq. (22.41) over a volume bounded by the contact interface, S, and closed by an extended surface S where the current densities across the extension S are known to be zero or negligible. In the local self-energy approximation, the first term on the RHS of Eq. (22.41) does not contribute. The remaining contribution to the current reads I = −i

e |

dE 2π|

q Ωi

dq A (q, E) Σ< (q, E) − G< (q, E) Γ (q, E) ,

(22.46)

where Ωi is the volume of the ith metallic electrode, and a portion of the bounding surface, S, representing the contact interface. Within the metallic electrode of

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volume Ωi , we can assume detailed balance condition, i.e., the carriers are distributed in energy according to the Fermi-Dirac distribution with local electrochemical potential which may depend on energy, µ (q, E). Therefore we can use the relation given by Eq. (22.9) expressing Σ< (q, E) in terms of Γ (q, E), A (q, E), and G< (q, E). However when this particular relation is substituted in Eq. (22.46), the whole integrand becomes identically equal to zero. This is expected since ∇ • J (q) = 0 at steady state, deep within the electrodes and within the device, dE dq A (q, E) Σ< (q, E) − G< (q, E) Γ (q, E) = 0. 2π|

(22.47)

On the other hand, since the current is not identically equal to zero for nonzero bias, then at steady state the integrand must be nonzero only in a small region near the interface, i.e., at the metal-semiconductor contact. Therefore we can replace the region of integration over Ωi by the integration over a thin layer surrounding the contact interface. For more general self-energies, we can write e dE lim dS · η 2| q →q 2π|  Σ< (q , q , E) A (q , q , E)      +A (q , q , E) Σ< (q , q , E)      Γ (q , q , E) G< (q , q , E) − × +G< (q , q , E) Γ (q , q , E)       Σ< (q , q , E) Re Gr (q , q , E)     −i − Re Gr (q , q , E) Σ< (q , q , E)

I = −i

×δ q −q+

η η δ q −q− , 2 2

                    

(22.48)

where q = q +q , q lies at the contact interface S, η = (q − q ), q and q lies on 2 the opposite sides of S. The incremental volume element dq in Ωi is approximated by dS · η. In the local self-energy approximation the Re Gr terms dot not contribute. The last expression above for the net current is precisely equal to the expression obtained by directly integrating over the contact interface S the current density, Eq. (22.44), but with opposite sign since I in Eq. (22.48) is directed away from the particle reservoir. Therefore we can view the integrand multiplying dS as the current density, Jc (q, E), across the contact interface. In the local energy approximation, we may thus write e Jc (q, E) = −i η A (q, q, E) Σ< (q, q, E) − G< (q, q, E) Γ (q, q, E) , |

(22.49)

where η is of the order of aˆ n, with a the finite discretization length or the lattice constant, and n ˆ is the unit vector normal to the surface at point q pointing towards

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the device. Using the relations, Eqs. (22.9) and (22.35), we have Jc (q, E) =

eη Γ (q, E) n (q, E) − |

dq dE K (q, q , E, E ) n (q , E ) ,

(22.50) where K (q, q , E, E ) is given by Eq. (22.35). The above results emphasizes the presence of the factor η which brings the units of a current density and a direction away from the reservoir, as stipulated in the original derivation of the multiple-probe Büttiker current formula. In order to make contact with Büttiker’s “general” multiple-probe current formula, discussed in Sec. 17.6, we make use of an approximate decoupling of the energy channels at low bias in Eq. (22.50). The procedure can be carried out by applying the detailed-balance relation, Eq. (22.9), valid at low bias, as follows. Firstly, we carry out the integration over E and make use of the following exact relation obtained from Eq. (22.33), −iΣ< (q, q, E) =

dE > D (q , E − E ) n (q , E ) . 2π|

(22.51)

Then we use the detailed-balance relation, Eq. (22.9), which will approximately hold at low bias, and imposed the “exact” relation given by Eq. (22.10). The result is Jc (q, E) =

e T (q, E) f (q, E) − |

dq T (q, q , E) f (q , E) ,

(22.52)

where T (q, E) = A (q, E) Γ (q, E) , 2

T (q, q , E) = |Gr (q, q , E)| Γ (q , E) Γ (q, E) .

(22.53) (22.54)

Note that Eq. (22.52) for the current density at the contact interface diﬀers from Eq. (22.31) for the expression for the gradient of the current density by a factor η, as expected. We can easily see that, by using the exact sum rule, Eq. (22.6), applied to local electron-phonon interaction, for each energy channel (the actual number of one-dimensional channels is determined by Eq. (17.12) at zero temperature) the following relation holds T (q, E) =

dq T (q, q , E) .

(22.55)

Moreover, in the presence of an external magnetic field, the symmetry properties of T (q, q , E) follows from the symmetry of Gr (q, q , E), i.e., T (q, q , E)|B = T (q , q, E)|−B .

(22.56)

In order to transform Eq. (22.52) to a terminal-contact current in terms of chemical potentials, we make a linear-response approximation, integrate over the energy variable and over the region in real space occupied by the contact interfaces. In the

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linear response regime, f (q, E) becomes a Fermi-Dirac probability function with position-dependent chemical potential. Expanding f (q, E) as a Taylor series about equilibrium value µo , we obtain, after integrating Eq. (22.52) over E, Jc (q) =

e η T o (q) µ (q) − |

dq To (q, q ) µ (q ) ,

(22.57)

where T o (q) =

dE 2π|

−

∂f ∂E

T (q, E) ,

(22.58)

To (q, q ) =

dE 2π|

−

∂f ∂E

T (q, q ,E) .

(22.59)

The terms involving µo do not contribute to Eq. (22.57) by virtue of the relation given by Eq. (22.55). The terminal current at the contact interface labeled by index i is therefore given by Ii = qSi

Jc (q) · dSi .

(22.60)

However, the discussions preceding Eqs. (22.47) and (22.48) allow us to replace the integration region in Eq. (22.60) by a region of integration over the volume of the metallic electrode Ωi with the bounding surface S determined so that current densities across portion of the surface other than at the contact interface are negligible,

qSi

Jc (q) · dSi =

e |

q Ωi

dq η T o (q) µ (q) −

dq To (q, q ) µ (q ) . (22.61)

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Alternative Derivation of Büttiker Multiple-Probe Current Formula

For each contact interface labeled by index j, we have µ (q)|qSj = µj . From Eq. (22.61), for the ith contact interface, Si , we integrate Eq. (22.57) over q belonging to Ωi to obtain the terminal current. We obtain   2 e  o Tii µi − Tijo µj − dq To (i, q ) µ (q ) , (23.1) Ii = | q device j=i

where the measured voltage is Vi = Tiio =

Tijo =

2π e

µi e ,

2π e

and

dqT o (q) ,

dq q Ωi

To (i, q ) =

(23.2)

q Ωi

dq To (q, q ) ,

(23.3)

q Ωj

2π e

dqT (q, q ) .

(23.4)

q Ωi

Note that Eq. (22.61) allows for a symmetrical treatment of all the metallic electrodes in Eqs. (23.2)-(23.4). The last term of Eq. (23.1) represents the contribution due to a continuous distribution of inelastic scattering centers within the device, where the sample size is much larger than the inelastic coherence length. However, if the device size is less than the inelastic coherence length, the last term can be neglected and thus regain the Büttiker multiple-probe current formula discussed in Sec. 17.6; for this case it is easy to see that Tiio = j=i Tijo , a relation rooted on particle conservation similar to Eq. (22.38). The last term of Eq. (23.1) clearly suggests that we can treat each inelastic scattering site q , in the present formalism, as connected to a reservoir characterized by a chemical potential µ (q ) in exactly the same manner as stipulated in Sec. 18. Indeed, the last term of Eq. (23.1) may be viewed as a more rigorous justification of the model of an inelastic scatterer given in Sec. 18. Therefore for each inelastic scattering site q within the device, we can apply Eq. (22.57) for the net current from the associated reservoir with the normal vector η indicating the direction 268

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269

away from the reservoir at µ (q ). We then impose the physical constraint that Jc (q )|q device ≡ 0 to solve for the chemical potential µ (q ). The result for the chemical potential at each inelastic scattering site within the device, µ (q ), can be given in symbolic operator notation as ˆ µ= 1−Ω

−1

ˆ +Ω ˆ 2 + .... + Ω ˆ n + .. µo , µo = 1 + Ω

(23.5)

ˆ is defined by where the matrix element of the operator Ω ˆ (q1 , q2 ) = To (q1 , q2 ) , Ω T o (q1 )

(23.6)

and the vector component of µo is defined by µo (q1 ) = j

To (q1 , j) , T o (q1 )

(23.7)

ˆ where the sum over the index j includes all contact interfaces. The result of Ω operator acting on any vector entity µo is defined as another vector entity F given by ˆ o (q) = F (q) = Ωµ

ˆ (q, q ) µo (q ) . Ω q=q

(23.8)

device

The first few terms of Eq. (23.5) can therefore be explicitly written as µ (q ) = j

To (q , j) µj + T o (q )

+

j

q =q1

dq1 j

×

dq1

q =q1 =q2

dq2

device

To (q , q1 ) To (q1 , j) µj T o (q ) T o (q1 )

To (q , q1 ) To (q1 , q2 ) T o (q ) T o (q1 )

To (q2 , j) µj + ..... T o (q2 )

(23.9)

Note that for a finite number of inelastic scattering centers within the device, the series in Eq. (23.9) terminates. Upon substituting the expression for µ (q ) given above in the last term of Eq. (22.11), the final result can be written as   e2  Tii µi − (23.10) Tij µj  . Ii = | j

The T -matrix can be written in symbolic notation as

Tij = Tijo + Toi (1 − τˆ)−1 Toj ,

(23.11)

where Toi (q) = To (i, q) ,

Toj (q) =

To (q, j) , T o (q)

(23.12)

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and the matrix element of the operator τˆ is defined by τˆ (q1 , q2 ) =

To (q1 , q2 ) . T o (q1 )

(23.13)

The first few terms of Tij can thus be written as Tij = Tijo +

dq1 To (i, q) To (q1 , j) q1

device

To (q1 , q2 ) To (q2 , j) T o (q1 ) T o (q2 ) q1 =q2 device To (q1 , q2 ) + dq1 dq2 dq3 To (i, q1 ) T o (q1 ) q1 =q2 =q3 device To (q2 , q3 ) To (q3 , j) + .... × T o (q2 ) T o (q3 ) +

dq1

dq2 To (i, q1 )

(23.14)

From Eq. (22.55), one can easily deduce that Tiio =

Tij ,

(23.15)

j

another confirmation of the particle conservation law. Tij of Eqs. (23.11) and (23.14) can be interpreted as representing the eﬀects, in the transmission probability of the successive inelastic scattering within the device in the particle transmission from contact terminal interface i to contact terminal interface j. In Eq. (23.11), all the various combination of scattering processes within the device is described by ˆ T defined by, the quantity which we shall denote as the “vertex” operator Σ ˆ T = (1 − τˆ)−1 . Σ

(23.16)

Therefore, Eq. (23.10) is a generalization of Büttiker multiple-probe current formula, discussed in Sec. 17.6, valid for sample or device sizes much larger than the inelastic coherence length. Indeed, Eq. (23.10) is given for the extreme case of a continuous distribution of inelastic scatterers within the device. Therefore Eq. (23.1) serves to justify the heuristic principles discussed in Secs. 17.6 and 18, using a counting argument based on time-independent S-matrix theory.

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PART 3

Heterostructure Quantum Devices: Nanoelectronics

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Chapter 24

Nanoelectronics

24.1

Introduction

Nanoelectronics research aims for a continued down-scaling of integrated circuit (IC) components down to the atomic scale and the consequent upscaling of computational complexity and speed performance per chip. Nanoelectronics is highly interdisciplinary involving “information-based” mathematical, physical, and biological sciences. This involves the study of highly nonlinear and nonequilibrium quantum phenomena in the nanometric and ultrafast time scale. Although nanophotonics and nanophononics are quite intriguing and more recently emerging fields in nanoscience and nanotechnology, we will limit the following discussion to nanoelectronics. The most useful device in solid-state electronics which has brought the information and data processing revolution is the transistor. The transistor is basically a device which accomplishes two very important functions, namely, “control” and power “drive.” The “drive” function is accomplished by applying power and a large voltage diﬀerence across two terminals of the device, known as the source (emitter) and drain (collector). The “control” function is implemented by simply controlling the self-consistent potential barriers to current flow between the two terminals. Both functions are important in obtaining “gain” characteristics of the device. Purely electronic control of the self-consistent potential barrier to current flow between the source (emitter) and drain (collector) is essentially accomplished in only two general ways: (1) by directly introducing a biasing (control) charge in the way of the current flow, i.e., inside the transport channel of the device to modulate the self-consistent potential barrier. The biasing charge or current comes from another electrode in contact with the so-called “base” region in the transport channel, (2) by depleting the charge carriers in the channel through the creation of depletion layer (polarization charge) induced by a voltage applied at another electrode called the gate. Conventional transistors employing the first method above are called bipolar transistors and falls into two categories, i.e., the p-n-p transistor and the n-p-n transistor. In both cases, the biasing charge arise from the base-emitter current and has an opposite charge from the current-carrying charge from the emitter to 273

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Nonequilibrium Quantum Transport Physics in Nanosystems

collector. However, in band-gap engineered case, to be discussed later, the biasing charge can be identical to the current-carrying charge between emitter and collector. Conventional transistors employing the second method are called unipolar transistors or field-eﬀect transistors. Thus an electronic device which delivers voltage gain and current drive is basically a three-terminal device. Note that the role of the power supply is essential to obtain gain. Because of the major role of lateral depletion regions in a conventional bipolar devices, these devices are not scalable compared to field-eﬀect devices. On the other hand, bipolar transistors, are inherently high-speed devices compared to field-eﬀect transistors by virtue of their low base-emitter capacitances. The prominence that three-terminal devices have in the electronics area is due to their ability to transmit and amplify (gain >1) information signals over indefinite distances without attenuation, and to provide good isolation between input and output of a logic gate allowing a computational process to proceed in a predetermined fashion. The speed of the signal transmission is proportional to the current drive, which provides the “coupling” between transistors. In the algorithmic dynamics of general-purpose programmable computers the rules of Boolean logic and “clockgating” of various “transistor-logic chain” pathways allow configuration of “bits” in one register (or data in memory) to “drive” or transform another configuration in another register or memory, via gateways and intermediate registers. Periodic “clock pulses” induce the deterministic time evolution of the computational dynamics, in accordance with the stored program and data as the initial configuration. Thus the elementary property of a transistor driving another transistor has been elevated in the computer circuit environment in which one computer configuration is driven into another computer configuration in the evolution of the algorithmic dynamics, with the configuration at the end-result of a convergent computation serving as an “attractor” at which point the computer configuration becomes stable. It should be emphasized that configurations in memories and registers are represented by numbers, and each memory or register can represent the totality of numbers allowed by the precision of the computers1 . The good isolation between input and output, drive, and gain capability allow for an unlimited depth of the algorithmic process, a capability approaching the mathematical model of a Turing machine which ideally has the capability to simulate all computational processes, finite or infinite. It is therefore easy to see why the three-terminal device is hard to replace when designing general purpose computers. The major attraction of resonant tunneling devices (RTDs), and other proposed quantum-based devices, is their ultrasensitive response to voltage bias in going from the high-transmission state to the low-transmission state. If these devices are able to operate under high bias and far-from-equilibrium condition, this essentially means that a very high transistor transconductance and ultra-fast switching are obtainable. Indeed, numerical simulations of RTD, to be discussed in more detail in later sections, and microwave experimental results, indicate the intrinsic speed limit of RTD to be in the tera-Hz range. This high-sensitivity to bias in far-from 1 The precision is measured by the number of bits. Classical computer memories store numbers sequentially, in stark contrast with quantum memories which can store all the numbers simultaneously.

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equilibrium operating condition can inevitably lead to a very high gain. Moreover, there is a strong indication that the gain can further be improved by appropriate design of the source and drain resistance. However in the continued down scaling of three-terminal device sizes and consequent upscaling of complexity per chip, a “wiring crisis” will result. This is essentially characterized as follows: (a) the number and length of connections will scale up, (b) the cross-section of the wire will have to scale down to allow more communication paths per area in the chip. The first will oﬀset the benefits of faster device-switching, since long connections create delays. The second will aggravate the problem of connection delays since the narrower the wire the larger is its resistance. In fact, for nonideal small metal wire, the resistance can go up exponentially as a function of its length. It is not yet clear whether research is novel computer architecture based on three-terminal devices and multi-valued logic will eventually solve this problem. A direct approach to the interconnect problem is to scale down the interconnects as well, from the conventional architecture, by eliminating long interconnects per chip, through computerized circuit-layout optimization, for example, through the use of simulated annealing techniques. Another realistic approach may lie on the advances made in monolithic IC optical communication technologies, such as the use of optoelectronic integrated circuit (OEIC) chipset for chip-to-chip optical communication. Elimination of wiring interconnects, and/or interconnects whose number does not scale up with computational complexity per chip, is contained in various proposals which drastically employs very diﬀerent architectures, such as the use of cellular automaton architecture in which the devices are connected only to their nearest neighbor. What is intriguing about this architecture is that the coupling is not implemented by physical “wires” but accomplished through capacitive coupling between, say, neighboring quantum dots. However, the “forces” and “rule” of the new computational dynamics appropriate to a cellular automation architecture that correspond to the Boolean logic, input-output isolation, “drive” and “gain” in conventional transistor-based general purpose computers are not entirely clear. General purpose computing requires arbitrary stored programs and data as part of the initial configuration of the computer computational dynamics, and an unlimited depth of computation. Cellular automaton (CA) architecture seems to lack the ability to undertake an unlimited depth of processing by virtue of the presence of the inherent feedback mechanisms. Notably, the ability to tailor the deterministic computational dynamics in a general-purpose computer according to an arbitrary program stored in the memory is not even addressed in all the proposals for the CA architecture. There are also serious diﬃculty of integrating external clock pulses to induce the evolution of the computational dynamics in general purpose CA computing. The lack of concrete ideas of what a CA computer is has led to a host of fragmented arguments and proposals on the subject, some are pessimistic while others are highly optimistic. It is conceivable that the charge patterns in cellularautomaton architecture, coupled with the ability to weaken, erase or add capacitive communication paths, may define a “programmable” configuration to perform specialized tasks similar to the present optical and neural computing. Nevertheless,

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Nonequilibrium Quantum Transport Physics in Nanosystems

the ability to represent a bit of information by one electron in quantum cellular automaton circuit is very intriguing, and clearly represent the ultimate eﬃciency in information representation. General-purpose computer design based on coupled CA building blocks, with OEIC chipset optical communication between CA blocks, also need to be explored. In any case, quantum cellular automaton architecture idea should open up strong interests from interdisciplinary community to establish the necessary analytical knowledge base needed for the search for a new general-purpose computational dynamics. Clearly, the merging of nanophotonics, nanophononics, and nanoelectronics will greatly diminish the limitations imposed by the wiring and interconnect problems in future supercomputers.

24.2

Nanodevices

Research eﬀort in nanoelectronics are centered on nanodevices, optoelectronic nanodevices and nanotransistors. Nanotransistors are essentially three-terminal nanodevices. Serious research eﬀorts in nanodevices are centered on the utilization of resonant tunneling switching phenomena and quantum interference (Bragg interference, as in periodic crystals) phenomena in superlattices, for enhanced inelastic coherence lengths and high-eﬀective charge-carrier mobility. Resonant tunneling phenomena have attracted the attention of the device community initially in the form of two-terminal GaAs/AlGaAs/GaAs diodes with significant voltages applied at the source and drain. The resulting quantum transport process is far from equilibrium and highly nonlinear. The really attractive feature is that resonant tunneling at these voltages is observed even at room temperature. In order to make nanotransistors, one simply needs to control the current flow by any one of the two general purely electronic methods, used in conventional transistors, to control the alignment of the resonant energy level with the bottom of the allowed conduction band of the source (emitter quantization). It is clear that obtaining gain would pose no problem, even at room temperature. As a corollary, a novel quantum transport process has to operate in far-from-equilibrium two-terminal diode format in order to have a clear potential for use in a three-terminal transistor with a reasonable gain. A common practice in the IC community for assessing whether a novel transistor design have a clear potential for insertion into a general purpose programmable computer system maybe mathematically characterized as follows. The iterated maps of the input-output transfer characteristic of a logic gate (e.g., a periodic chain of simple inverter logic gate) over the input-output voltage swing must have an unstable midpoint (defined roughly as half of the total voltage swing, where “low” state and “hi” state can no longer be discriminated) and a stable limit cycle (attractor), where complete discrimination of “hi” and “lo” states occur. A smallest number of iteration to reach the limit cycle is desirable in order to have a reliable restoring logic gate. This type of iterated mapsimplemented in a ring oscillator circuit of cascaded Si-CMOS inverter logic gates, is shown in Fig. 24.1. The instability of the middle point is assured if the gain is greater than one in the input-output

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Fig. 24.1 Iterated maps on a voltage-swing interval in a ring oscillator. Diﬀerent transfer characteristics of logic inverter gate in a ring of cascaded inverter gates are shown. Figures (a) and (b) show limit-cycle operation as restoring (amplying) logic gates in an inverter chain. Figure (c) shows a logic operation with gain less than unity, resulting in eventual collapse of the signal along the chain and hence dead signal at succeeding stages along the chain. Figure (d) corresponds to a low power supply voltage of a CMOS inverter (inset) resulting in hysteresis transfer characteristic not appropriate for logic gate operation.

characteristic of each transistor logic gate. The need for a excellent attractor characteristic stems from the variation in the input and output signals in a computer circuit environment. This variation is brought about mainly by the randomness caused by circuit-parasitic voltage drops, non-uniform heating, and by the manufacturing variation of the devices themselves. Indeed, the right combination of high gain and simple switching behavior of a conventional-transistor logic gate, which basically depends on a single triggering point beyond which it assumes either a conducting or nonconducting state, has fostered a high degree of insensitivity of the logic function to the exact value of the controlling input signal, clearly a very important attribute of a logic gate of a reliable computer system. The occurrence of threshold-voltage shift, due to material defects and traps, is indeed a nontrivial reliability problem even in conventional transistor ICs. Although this high degree of tolerance to the variability of input signal has been taken for granted in computer design using conventional transistors, it will become a critical issue for RTDs proposed for a general purpose computer. Unlike

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conventional transistors, the most fundamental problem arises from the fact that high-transmission conducting state occurs only for particular values of the input signal and for particular values of the structural and physical parameters describing the device structure. Tunneling current depends exponentially on the thickness of the barriers (moreover, symmetric and asymmetric barriers may have completely diﬀerent I − V characteristics at very low temperatures, and high-transmission voltage depends on the thickness of the quantum well. All these seems to add up to a challenging set of device specifications and manufacturing-yield uniformity. However, in designing an IC logic gate, as compared to an individual transistor, the designer basically has a larger number of combined geometrical and physical design parameters to optimize and integrate, relating to the coupling and interaction of two or more transistors, in order to produce the desired outputs from the given inputs. It is not quite clear yet if some synergetic interaction which reinforces the “pull-up” and “pull-down” to standardized values within the specified range can be achieved, given a high gain obtainable from RTDs, i.e., one cannot exclude the possibility that the new logic-gate design problem, of providing input signals to lie within a specified range commensurate with expected variation in a computer circuit environment, can be adequately solved. As examples of design-parameter trade-oﬀs for optimization, it is worth mentioning that increasing the tolerance for the “hi” input signal generally leads to a decrease in tolerance of the “lo” input signal as well as to a degradation of the useful voltage swing; however, an increase tolerance for “hi” input signal, without aﬀecting the inherently larger tolerance of the “low” input signal and without degrading the voltage swing, can be achieved by a clever RTD design which incorporates another lower band-gap material within the quantum well. On a related vein, any proposal which advocates the use of multi-valued logic over the binary logic appears to aggravate the signal tolerance and discrimination problem. Clearly, the high-demands placed on manufacturing control are expected to be a challenging manufacturing problem which will continue to confront researchers in quantum tunneling devices and ICs. We estimate that cascaded logic circuits with well-fabricated RTDs, employing binary logic, will have about less than 20% signal tolerance, compared to about 50% larger signal tolerance in conventional CMOS. Variability of the RTDs would call for a much larger signal tolerance, if this is at all possible. Although, there is no question that RTD will find a niche in real circuit applications, it is premature to give a final judgment regarding its fate for general-purpose computer applications. What is needed is an experimental demonstration of a ringoscillator logic-circuit configuration. The IC community almost always makes use of ring oscillator measurements for evaluating logic-gate performance of a competing IC technology for general purpose computer. A ring oscillator consists of a cascaded chain of logic gates (e.g., inverter logic gates), or input/output stages, where the output signal of the preceding stage serves as the input signal of the succeeding stage to physically simulate a very long computation (infinite in case of ring oscillator) in the computer circuit environment. Indeed, the ring oscillator physically implements the iterated maps of the logic-gate input-output transfer characteristic over its voltage-swing interval, as mentioned above. The question usually asked by the conventional IC community in this type of measurements is how fast and

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how energy eﬃcient the information signal is being propagated; this is measured in terms of the power-dissipation/gate-delay product. Clearly, for RTD IC we also need to determine the range of insensitivity or tolerance of the logic function to the variation of the gate signal for the ring oscillator to perform satisfactorily. Simple RTD integrated circuits have been experimentally demonstrated initially in the form of single-stage input/output circuit configurations, such as single-stage analog and digital-logic circuits, as frequency multiplier/divider, as exclusive NOR gate, as parity generator, as multi-state memory, and analog-to-digital converter. Two-stage logic coupling has also been demonstrated, i.e., two-stage XNOR logic gate combination in the form of a fuller adder circuit, and perhaps more complex proprietary circuits have been constructed. It is most critical for the future of RTD applications to general purpose computer, that as the technology for RTD-based IC further develops, a satisfactory experimental demonstration of coupled and closedloop array of input-output logic stages must be realized to physically simulate a very long computation. Indeed, a ring oscillator IC of inverter-logic-gate chain with fan-out and fan-in logic blocks should simulate an infinitely long computational process. Schrödinger wave guide and Aharonov-Bohm eﬀect device concepts have been introduced basically as three-terminal devices from the very start, but with no demonstrated “drive” and gain capability, by virtue of the fact that the source and drain terminals are simply current-biased near equilibrium. The present emphasis placed in this research is on the novel controls, which is similar to electromagnetic wave-guide devices (including “Faby-Perot resonance” type of localization resonance), obtained upon varying the geometrical parameters of a “stub,” “double constriction” or a “bend,” by means of the control of the confining depletion-layer wall in portions of the stub, double constriction or bend through the voltage applied at the gate terminal. These devices also very much depend on the device size to be less or equal to the inelastic coherence length to operate. Thus they are expected to operate only at very low temperatures, for devices that can be fabricated today. To provide gain, these devices must still be able to shut oﬀ at a high drain voltage (with the source at zero voltage). However, a high drain voltage will accelerate the carriers changing their wavelengths and therefore aﬀects quantum interference in a complicated nonlinear manner, and an oﬀ-state at high drain voltage is not easily met by the electrostatic (depletion-layer controlled) Aharonov-Bohm eﬀect devices, thereby seriously limiting the gain. Perhaps with ingenuous design of the waveguiding channels, reasonable gain can be obtained for these devices, particularly for Schrödinger waveguide devices with controlled localization at a stub, double-constriction or bends. Coulomb blockade phenomena, as applied to the concept of a single electron transistor are expected to have attractive drive and gain capabilities. They are expected to be the natural consequence of “lateral” scaling down of the present resonant tunneling transistors for ultra high density ICs. Single electron transistor has been vigorously pursued for coupled quantum-logic gate applications.

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Vertical vs Lateral Transport in Nanotransistor Designs

Rapid developments in advanced lithography, beam epitaxy and deposition processing coupled with advanced fabrication and etching techniques, have resulted in material fabrication control in vertical direction down to one mono-atomic layer (few angstroms), and about two orders of magnitude less fabrication control in the lateral directions. Progress in the application of STM principles is expected to impact on the fabrication control in lateral directions toward atomic sizes. The diﬀerent degree of fabrication control in lateral and vertical directions has resulted in a dichotomy of pioneering-electron-device research eﬀorts. The planarIC community eﬀorts tend to be lateral-quantum transport based and utilize the transport of carriers in the lateral direction, whereas the second group of researchers utilizes the quantum transport of carriers in the vertical direction, across heterojunction interfaces. Indeed, band-gap engineering of materials brought about by molecular beam epitaxy (MBE) and metal-organic chemical vapors deposition (MOCVD) was immediately employed by the second group to improve the performance of conventional bipolar and field-eﬀect transistors, yielding heterojunction transistors. Indeed, the initial development of bipolar transistor based on band-gap engineering makes use of vertical transport in the form of a tunneling hot electron transfer amplifier (THETA) device [102], a resonant transmitting hot electron transistor (RHET) [103], [104], and an abrupt junction heterojunction bipolar transistor (HBT) and a hot electron transistor (HET) [105, 106]. The various designs basically increased current gain with low base resistance and low emitter-base capacitance necessary for high-frequency operations, approaching measured cut-oﬀ frequency as high as 200 GHz. Later development of bipolar transistors also makes use of lateral transport of a high mobility two-dimensional electron goes (2DEG) at the interface between GaAs/AIGaAs heterojunction [102]. On the other hand, the initial development and improvements of field-eﬀect transistors based on band-gap engineering make use of lateral transport in high-electron mobility two-dimensional electron gas (2DEG) of GaAs/AlGaAs heterojunction channel where the high mobility results from the spatial separation of donor impurities at the AlGaAs layer from the mobile (2DEG) carriers at the GaAs/AlGaAs interface [107]. Research eﬀorts in nanotransistors also follow the same dichotomy of eﬀorts. Lateral transport-based nanotransistor research eﬀorts and vertical transport-based nanotransistor research is being pursued in various places around the world. It should be pointed out that the two orders of magnitude larger inelastic coherence length in lateral transport compared to vertical transport makes up for the two orders of magnitude less control in the fabrication of lateral dimensions. Further, it is worth mentioning that most of the mesoscopic research makes use of lateral transport, by virtue of the longer inelastic coherence lengths obtainable. Moreover, confinement in all directions is readily achieved and electrically controlled in lateral transport, by means of lithographically patterned metal gates, than in vertical transport. However abrupt confinement is never achieved in lateral transport-based designs based on depletion barriers. Lateral transport is favored for planar ultradense IC applications. Eﬀorts based on vertical transport are further grouped into

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eﬀorts focused on discrete mesa device configuration, intended mainly for demonstration purposes, and eﬀorts geared toward planar IC integration. The lack of abrupt confinement in lateral transport using confining depletion layers, which resists lateral scaling, urgently calls for some form of lateral heterojunction technology for nanostructures. Indeed, this is an active field of nanofabrication research and this technology is vigorously being pursued. If this technology becomes viable, quantum dots with atomic-scale dimensions and stronger coupling can be realized.

24.4

Nanotransistor Designs

In what follows, we will discuss the earlier and basic approach to nanotransistor designs. Several proposals and experimental nanotransistor devices are based on the diﬀerent ways of controlling resonant tunneling current behavior in multibarrier structures. Because of the high resolution in the process control of vertical dimensions down to 2 Å monolayer uniformity, well-defined characterization and experimental results are usually obtained for nanotransistor designs based on vertical transport. However, the use of high-electron mobility 2 DEG for lateral transportbased nanostructures have also yielded results, with added dimension of control by virtue of the ability to change confinement potentials through the manipulation of gate voltages. The following discussions are only intended to give the underlying principle in the search for a viable nanotransistor based on quantum tunneling phenomena. 24.4.1

Vertical Transport Designs

The obvious approach to controlling the resonant-tunneling current behavior in heterojunction double-barrier structure is to introduce a biasing charge in the quantum well (as in conventional bipolar transistors in which case we call the source as the emitter and the drain as the collector), thereby altering the self-consistent potential of the quantum-well. However, if the introduced electron charge occupies the same quantized energy level as the current-carrying electrons then the base-collector leakage current can become significant so as to short out the base and make transistor control impossible, as can be seen from, Fig. 24.2. As in conventional bipolar transistor, one needs to distinguish the introduced charge from the main current carrying charge between emitter and collector. A way to do this is to somehow assign one energy level (e.g. ground level) in the quantum well for the charge introduced (to change the self-consistent potential) and the next excited energy level as the current-carrying channel between the emitter and collector. It is also desirable to “hide” the ground level from both the emitter and collector Fermi level, to completely eliminate leakage current. These requirements are accomplished by using a quantum-well layer made up of semiconductor material with a narrower band-gap than both the emitter and collector material, as shown in Fig. 24.3. It should be emphasized that this design results in a unipolar transistor, i.e., current is carried only by one carrier, the electrons. Note however that

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Fig. 24.2 [After Ref. [108] with permission.] Unworkable unipolar RTT IC layout design. Section AA shows tunneling behavior, however BB shows parasitic base current making transistor action impossible.

“recombination-type” of behavior (inherent in conventional p − n − p or n − p − n bipolar transistors) involving the two energy levels may occur in the quantum-well. A new set of problems is introduced when designing nanotransistors for planar IC layout configuration compared to the discrete mesa-device configuration. These problems have been discussed by Bate [108]. A naive heterojunction implementation of a quantum-based bipolar, two current-carrier transistor, is to simply have the quantum-well layer p-doped, as shown in Fig. 24.4. However, by inspection of the symmetry in the band-edge diagram of Fig. 24.4, the possibility of large emitter-base hole currents becomes immediately obvious, whenever there is a resonant emitter-collector electron current. The catastrophic parasitic current will degrade the current gain completely. This problem can be overcome by using a semiconductor material in the quantum-well with narrower band-gap than both the emitter and collector layer. In the transport channel, it eﬀectively “hides” the holeenergy level in the quantum well from seeing the available states in both the emitter and collector, eliminating the leakage currents. The above vertical transport design have been implemented [109],[108], this has been fabricated in a bipolar quantum resonant tunneling transistor (BiQuart), Fig. 24.5. A diﬀerent approach to nanotransistor designs is to interchange the base and collector regions of the original-naive approach of Fig. 24.2, discussed earlier. In this new scheme, the biasing charge is introduced on the other side of the double barrier, mainly creating electric field and aﬀecting the self-consistent potential in the quantum well, with the main current flow extracted by contacting the quantum well, i.e., the quantum well becomes the collector region. In this design, the two barriers are generally and highly asymmetrical, with the second barrier many times

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Fig. 24.3 [After Ref. [108 ] with permission.] Workable unipolar RTT IC layout design which eliminates the parasitic base current. Emitter-collector current channel is through the excited quantum-well state. Control charge-carriers are supplied to the ground energy level by the base contact, and prevented from flowing to the collector by the medium-band gap collector region.

Fig. 24.4 [After Ref. [108] with permission.] Unworkable bipolar RTT IC layout design. The emitter-base bias must be greater than the bandgap to achieve resonant tunneling. But contacting the holes in the quantum well requires a region in the base that is p-doped. Unfortunately a p region in contact with n-type emitter region subjected to forward biases greater than the bandgap would result in catastrophic parasitic current, making transistor action impossible.

larger than the first barrier, Fig. 24.6, to eliminate tunneling leakage current from the “base.” Classified as a bipolar transistor this design has a negligible base current and large current-transfer ratio. However, according to our classification since the biasing charge is not directly introduced into the current channel but through the highly capacitive control of the “base” electrode, this design can also be classified as a field-eﬀect nanotransistor. Because the quasistationary states in the well are

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Fig. 24.5 [After Ref. [108 ] with permission.] Workable bipolar RTT IC layout design, which eliminate p -n junction catastrophic parasitic current. A narrow-bandgap quantum well sandwiched between wide-bandgap barriers and medium bandgap emitter and collector regions allows for a reduced forward bias which does not create catastrophic parasitic current across the p − n contact junction.

modulated by an electric field to produce transistor I − V characteristics, Bonnefoi, et al. [110] propose to call this design as a Stark-eﬀect transistor (SET) which was later experimentally demonstrated [111]. One can also add a potential step in the quantum well using a narrow band-gap thin layer to further decrease the base-collector leakage current, to eliminate intervalley scattering and to enhance current drive. Note however that the base or collector contacts to the quantum well are made laterally, i.e. current has to flow laterally along the quantum well towards the electrode contact. Thus, lateral field will also exist along the quantum well which may destroy the coherence of wave functions across the entire doublebarrier structure. Moreover, this quantum-well region is doped to obtain ohmic contact. The device also has less gate control and hence small transconductance. In the Caltech design, this problem is solved by adding a wider well, with closelyspaced energy levels, for the collector contact. This is a triple-barrier geometry, with undoped symmetrical double-barrier emitter structure, Fig. 24.7, which also looks like the result of the interchange of the base and collector regions of the RHET design proposed earlier by Yokayama, et al. [103]. As before, the third barrier is much wider in order to isolate the base input or biasing charge. This new design yields a well-defined negative diﬀerential resistance controlled by the base-emitter voltage. Bonnefoi, et al. [110] proposed to call the modified SET as a negative resistance Stark-eﬀect transistor (NERSET). The NERSET design has been extended by adding another gate to control the lateral flow of current along the wider quantum well, a control gate similar to MESFETs. The resulting design is called the field-eﬀect resonant tunneling transistor (FERTT) [112]-[113], in which

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Fig. 24.6 [From Ref. [110] reprinted with permission.] Workable unipolar field-eﬀect RTT mesa design, with relative position of the base and collector being interchanged. Control is by means of the polarization charge at the base electrode, therefore this design can be classified as a field-eﬀect RTT. Thus, the emitter maybe called the source, the collector can be referred as the drain and the base metallization as the gate.

case the emitter has been referred to as the source, the collector as the drain, the base as the “back gate” and the MESFET-like gate as the “front gate,” Fig. 24.8, which has been experimentally demonstrated. Eﬀorts aimed at developing multistate resonant tunneling transistors [115], in discrete mesa device configurations, has been undertaken [114]. The earlier design is based on the modification of abrupt emitter-base heterojunction (wide bandgap emitter region) n − p − n heterojunction bipolar transistor by incorporating a double barrier or a superlattice structure in the base region [92]. The device has a multivalued transfer characteristic, having as many peaks as the number of resonances in the resonant tunneling structure in the base, Fig. 24.9. These devices have been demonstrated for potential multivalued logic applications [116]. Another multi-state resonant tunneling bipolar transistor (RTBT) being pursued is based on the “cascaded” vertical integration of double-barrier (DB) resonant tunneling diodes embedded in the emitter region [117]. An RTBT with a single DB in the emitter was first reported by Futatsugi, et al. [118]. These DB’s are separated by heavily-doped cladding layers to quantum mechanically decouple the adjacent DB’s from each other. The band diagram is shown in Fig. 24.10. The

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Fig. 24.7 [From Ref. [110] reprinted with permission.] Improved unipolar field-eﬀect RTT mesa design. This design avoids contact to the quantum well, which tend to degrade the coherence of wavefunction across the entire double-barrier structure of Fig. 12.

idea is to obtain current peaks at the same current level with similar peak-to-valley ratios, which naturally lead to designs using analogous resonance of a series of quantum wells. The operation of multistate RTBT can be described as follows. Assume that collector-emitter bias (VCE ) is kept fixed so as all DB’s are resonantly conducting, and let the base-emitter bias (VBE ) be increased. For VBE smaller than the p − n built-in voltage, most of the VBE is dropped across the emitterbase p − n junction. At this point the device behaves as a conventional bipolar transistor, with the emitter-collector current increasing with VBE until the baseemitter junction reaches a flat-band condition. Beyond the flat-band condition, the impedance across p − n becomes negligible, and the additional increase in VBE is dropped across the series of DB’s. Because of the screening eﬀect of the cladding layers, quenching of the resonant tunneling (RT) is initiated across the DB adjacent to the base and anode regions. Once RT is suppressed across a DB its voltage drop across it quickly increases with bias because of increased resistance, and a non-RT current build-up provides the continuity for the RT current through other DB’s operating in RT mode. As VBE increases further, the high-field region widens (i.e., needs more cladding layers to screen the field). Therefore, quenching of RT sequentially propagates toward the cathode end and negative diﬀerential regions are obtained in I − V characteristics corresponding to the quenching of RT through

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Fig. 24.8 [From Ref. [113] reprinted with permission.] A four-terminal field-eﬀect RTT (FERTT) IC layout design. The three terminals at the top surface: source, front gate, and drain, make up a configuration that behave just like a conventional HEMT.

each DB. Thus with n diodes embedded in the emitter region, n peaks are present in the I − V giving saw-tooth features. This was experimentally demonstrated by Capasso, et al. [116]. We note that except for the stark-eﬀect transistors, most of the nanotransistor designs based on resonant tunneling phenomena are essentially modification of conventional heterojunction bipolar transistors. A diﬀerent approach based the modification of field-eﬀect conventional heterojunction transistor (e.g., HEMT structure) has also been proposed [119]. The device consist of an epitaxially grown undoped planar GaAs quantum well and a double AlGaAs barrier sandwiched between two undoped GaAs layers, the whole stack being further sandwiched between two heavily-doped GaAs layers for the source and drain contact. The device area is produced by a V-groove etching which is subsequently overgrown epitaxially with a thin AlGaAs layer, Fig. 24.11. Gate metallization resembles that of the conventional HEMT structure. The fabrication technique and structure was also proposed earlier by Sakaki [120]. The novel feature of the proposed design is that tunneling transport occurs across a quantum “wire,” not across a planar layer as in previous nanotransistors. The operation is as follows. The application of positive gate voltage VG induces 2-D electron gases at the two interfaces on both sides of the double-barrier structures. These are polarized surface charges coming from the heavily-doped contact layers. Because of large zero-point energy and transverse energy-level spacing in the “quantum wire,” there is a range of VG in which electrons are not yet induced in the wire. Applications of drain voltage VD bring about resonant tunneling condition where the transverse energy of electrons in the source matches the unoccupied quantized transverse energy in the wire. The transport process is a tunneling between 2-D electrons to the 1-D density of states in the quantum wire. Assuming conservation of longitudinal momentum kz , and denot-

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Fig. 24.9 [From Ref. [114] reprinted with permission.) Band diagram of multistate resonant tunneling bipolar transistor (RTBT). (a) Design with graded emitter and abrupt double-barrier at the base (at resonance). (b) RTBT with parabolic quantum well in the base, for equally spaced resonances. (c) RTBT with superlattice base.

ing the transport momentum direction as kx , where | (kxo )2 /2m∗ = ∆, kxo being the resonant momentum, the available kz -components contained within the Fermi level in the 2-D source lie in the band of energies Eo + ∆ ≤ E ≤ EF . Here Eo is assumed to be the zero point energy of the 2-D source. Therefore the number of tunneling electrons grows with VD until ∆ = 0, where there is no more matching of the electrons in the source with the quantized energy level in the wire. This situation gives rise to negative diﬀerential resistance, similar to that of “planar” quantum-well nanotransistors discussed before. The present design, however, oﬀers the possibility of transferring control of aligning the quantized energy level of the “wire” with respect to the band of energies in the source, from VD to VG . It has been demonstrated that the gate potential is nearly as eﬀective in lowering the quantized energy level of the quantum wire with respect to the zero-point level in the source as is the drain potential. This implies the very interesting possibility of achieving negative transconductance for a unipolar transistor. This kind of transistor can perform the function of, and hence replace, a p-channel transistor in silicon CMOS logic, resulting in a low-power inverter, in

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Fig. 24.10 [From Ref. [116] reprinted with permission.] Energy-band diagram of multiple state RTBT, operating as a common-emitter transistor with a series of double-barrier (DB) structures in the emitter region, for diﬀerent base-emitter bias conditions. (a) Uniform voltage drop across each DB; in this condition the transistor operates as in conventional bipolar transistor. (b) nonuniform voltage drop, and quenching of RT through DB adjacent to p − n function give rise to negative diﬀerential resistance in collector current Ic vs. base-emitter voltage VEB . The successive quenching of RT for the succeeding DB in the series away from the p − n function produces multipeaks in Ic − VEB characteristics.

which current flows mainly during switching characteristic of CMOS logic. This feature should find important applications in GaAs ICs. 24.4.2

Lateral Transport Designs

The last transistor design discussed above may be considered to be based on the combination of lateral transport and vertical transport. Vertical transport since current flows across heterojunction layers, and lateral transport since current flows along the surface of the V-grove etching. Whereas most vertical-transport-based nanotransistor designs are essentially modifications of conventional heterojunction bipolar transistors, all lateral-transport-based nanotransistors designs are essentially modifications of unipolar or field-eﬀect conventional heterojunction transistors. Specifically, these devices are modifications of high-electron mobility transistor (HEMT), also known as MODFET, TEGFET and SDHT. These are fieldeﬀect-controlled heterojunction semiconductor structures. Lateral-transport-based nanotransistor designs make use of patterned multigate HEMT structure instead of a single gate conventional HEMT structure [107], in a manner similar to the one proposed earlier for MOSFET structures [38]. The longer coherence length and

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Fig. 24.11 Unipolar surface resonant tunneling transistor and energy-band diagram along the gated “surface” conduction channel. The thickness of the two undoped GaAs layers outside of the double-barrier region is suﬃciently large to prevent parasitic resonant tunneling current in the bulk.

mean-free paths in 2-DEG transport channel in HEMT structure enables state-ofthe-art lithographic techniques to fabricate lateral-transport-based nanotransistors. Lateral-transport based designs oﬀer a latitude of design parameters defined by the shape and geometrical pattern of the gates. This feature allows one to design nanotransistors whose function depend on the localization of charge carriers in more than one dimension, such as the use of “quantum wires” and “quantum dots,” where carrier scattering is further reduced due to multi-dimensional size quantization, thus enhancing the coherence length and low-field mobility. It was estimated that quantum wires oﬀer about two orders of magnitude improvement in low-field mobility, a significant higher saturation velocity and longer coherence lengths compared to 2DEG [120]. Thus an obvious approach to further improvement of conventional field-eﬀect high-speed devices based on the HEMT structure is to replace 2-DEG transport channel, controlled by the gate, by an array of quantum wire, Fig. 24.12, say few hundreds of parallel wires to maintain reasonable current drive capability [121]. Indeed, size quantization by itself has immediate applications to conventional (i.e., nonquantum-transport-based) high-speed electronics and optoelectronic devices. Ground-state miniband transport across an array of quantum dots has also been estimated to suppress optical phonon scattering if the three conditions are met: (a) Eg1 EF + kB T , (b) BW < Eop , and (c) Eg1 > Eop , where Eg1 is the width of the first minigap, BW is the width of the first miniband and Eop is the optical phonon energy. Thus, one can expect an even further improvement if the 2-DEG transport channel in HEMT is replaced by an

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Fig. 24.12 [After Ref. [120] with permission.] Schematic cross section of a parallel single-channel quantum wire array FET showing the gate metallization on the top layer.

Fig. 24.13

Schematic cross section of quantum box array FET with top-layer gate.

array of coupled quantum dots, Fig. 24.13. Similar conditions hold for uncoupled quantum dots structures which are considered eﬃcient electro-optic materials. Because size quantization eﬀects have immediate impact in optoelectronics, the optoelectronics community has strong stake on nanostructure research. Lateral device-array designs are of course the preferred choice since light can readily be coupled into lateral structures. The advantage of abrupt confinement across heterojunction has lead to vertical device designs in the form of a mesa device. The use of arrays of quantum wires and quantum dots, providing confinement of injected carriers in the active region of laser diodes has been shown to dramatically improve the lasing characteristics. In the case of quantum-dot array active region, the temperature dependence of the threshold current is virtually eliminated [122, 123]. It was also shown theoretically that the electroabsorption and the associated electrorefraction in array of quantum wires and quantum dots are greatly enhanced over that of “planar” array of quantum wells by virtue of multidimen-

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sional quantum confinement Stark eﬀect (i.e., bunching of quantum states). Such enhanced eﬀects would lead to electroabsorptive and electrorefractive modulators and optical switches with even lower energy requirements than existing quantumwell devices [124]. Another interesting phenomena in optoelectronics is the so-called carrier-induced bleaching of optical absorption or the blue shift of absorption edge, primarily caused by band filling that quenches both exciton and band-to-band absorption, in quantum-wells array. The subsequent changes in refractive index (recall that refraction index in photon transport is akin to electrostatic potential in electron transport) have interesting applications, for example, it can be externally controlled by the injection of carriers or by a voltage applied at the gate in field-eﬀect or HEMT configurations, where the 2-DEG channel is replaced by the multidimensionally confined arrays so as to construct optical modulators and optical bistable switches [124]. Another example in the optoelectronics area is the self-electrooptic eﬀect (SEED) which may have important applications in photonic switching [125]. 24.4.3

GaAs/AlGaAs MODFET-Based Nanotransistors

Based on the modifications of the Schottky-barrier-gate structure of conventional heterojunction HEMT (or MODFET) transistor, a number of lateral-transportbased nanotransistors have been demonstrated. The various structural modifications are: (a) dual gate or split-gate geometry, resulting in the so-called planar resonant-tunneling field-eﬀect transistors (PRESTFET), (b) triple-gate geometry with independently controlled middle gate, this has added features over (a) in that the quantum well depth and barrier heights are controlled independently, (c) “grating-gate” geometry, which lead to induced array of quantum wires under the gate area, where lateral transport is across the wires and Bragg interference in this direction determines the overall transistor characteristics, and (d) “gridgate” geometry, which lead to induced array of quantum dots under the gate area; two-dimensional Bragg interference (e.g., more bunching of states) determines the transistor characteristics. At fixed drain bias, Vds , as the gate bias, Vgs , is increased in quantum arrays, there are essentially two correlated changes that happen, namely, (a) the quantum-well depth increases resulting in “stronger” periodicity and consequent larger negative-mass portion of the miniband, and (b) the electron concentration increases in each “unit cell” resulting in “filling” of the minibands. Thus as Vgs is increased in a continuous manner, the Fermi energy is expected to pass through minibands and minigaps. Whenever the Fermi energy falls on the energy range of the minigap the current should drop to zero at T = 0. For T = 0, due to spread of energies around EF of the mobile carriers, the current should drop to a nonzero minimum at low-enough temperatures. Note that the drain bias, Vds , also create a spread of energies along the transport channel proportional to the electron “quasitemperature” by virtue of electron heating by the applied drain voltage. Also since the minigap widths decreases at higher energies, the current minimum should increase with Vgs . This behavior is indeed what is observed, with much enhance eﬀects for grid-gate geometry [126]. Thus the Bragg interference nanotransistors exhibit gate-controlled negative resistance and negative transconductance.

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A rather very complicated behavior for grid-gate geometry lateral-surface superlattice was found in the measured Ids as a function of Vds , with Vgs fixed to values corresponding to weak conduction. It should be realized that for these values of Vgs , the quantum-well depth is decreased (yielding a decreased negative-mass portion of the miniband), resulting in what we referred to as a “weak periodicity.” The number of electrons in each “unit cell” is also vanishing leading to unoccupied minibands. Thus, at small Vds , no current will flow. The application of higher drain bias is expected to be nonlinear and aﬀects the depletion layers created by the gates (hence aﬀects the occupation of each quantum dot, the depth of each quantum dot potential and the height of the confining potential barriers) nonuniformly, mainly along the bias-field direction. The nonlinear and nonuniform interplay between carrier occupation, self-consistency, and lattice potential changes, particularly at the drain-side of the device, will produce fluctuating quantum energy levels in the superlattice region near the drain. At some values of Vgs and Vds , onset of localization can occur, caused by the energy-band mismatch due to the energy-level fluctuations. Broadening of the energy levels will reduced the “impedance mismatch” and therefore localization can be completely eliminated at higher temperatures. Indeed, it has been found that for higher values of Vgs , the Ids − Vds , behavior do not exhibit localization but exhibit the usual “supply-limited” I − V characteristics. However for lower values of Vgs , the Ids − Vds behavior does exhibit negative diﬀerential resistance presumably due to the onset of localization in the drain-side portion of the device. This behavior goes over into the “supply-limited” current behavior when temperature is raised from 4.2’ K to 77 K. At still lower Vgs only weak localization behavior in the Ids − Vds was also observed, presumably due to the small occupation of the quantum dots. In any case, the result of the Ids −Vds characteristics points to the rather complex and interrelated current control exerted by the gate and drain electrodes in lateraltransport based devices, which rely on a configuration of confining depletion-barrier layers produced by the patterning of the gate electrode.

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Chapter 25

Nanodevice Physics

25.1

Introduction

So far, the theoretical tools employed in analyzing mesoscopic structures, mainly along the lines based on the Landauer formula, are based on near-equilibrium transport physics at steady-state condition using linear response approximation. However, in order for nanodevices to functionally operate in present logic gate-based computational dynamics, it must be subjected to high source-drain voltages and high-frequency bandwidths, in far from equilibrium, highly transient and highly nonlinear regimes. We have encountered at the end of the last section a diﬃculty connected with the understanding concerning the Ids − Vds characteristic of gridgate geometry lateral-surface superlattice subjected to high-drain bias leading to inhomogeneities and nonlinearities within the device. This problem is only the “tip of the iceberg.” It appears that nanodevices and nanoelectronic IC architectures call for a whole new design methodology and new computer-aided design aids to be developed. What is needed is a charge-carrier transport physics well beyond the classical Boltzmann equation. The classical Boltzmann transport equation has been the workhorse of nonlinear, time-dependent and far-from-equilibrium conventional solid-state device physics and computer-aided design of conventional ICs. This transport equation is numerically implemented for device simulation in two general approaches: (1) based on the “truncated” moments of Boltzmann equation, and (2) based on direct numerical solution of the Boltzmann equation through the ensemble particle Monte Carlo techniques. From these two general approaches, equivalent circuit parameters are extracted for use in more complex IC circuit analysis. Depending on the quality of results needed and computational resources available, any of the two approaches can also be directly coupled to the circuit-passive components in more accurate large-scale IC simulation. The classical Boltzmann transport equation, however, inherently fails to account for the nonlocality, correlation and interference quantum eﬀects brought about by the wave nature of the charge-carrier motion. These quantum eﬀects will completely dominate when device localization size approaches the charge-carrier wavelength, and transport channel size approaches the inelastic coherence lengths, as is usually the case with nanodevices based on quantum eﬀects. It is therefore clear that, for characterizing nanodevices, a complete quantum

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transport equation which includes quantum eﬀects, many-body eﬀects and phasebreaking or dissipative eﬀects has to be developed to replace the classical Boltzmann transport equation. For nanostructures with underlying crystalline lattice, it is also desirable to cast the exact quantum transport equation in terms of the canonical dynamical variables (phase space) to bring the formulation to more familiar language aﬀording perfect matching with the classical phase-space distribution at the boundaries and ease in the numerical implementation and in the simulation of highly-transient nanodevice phenomena. The phase-space description allows the utilization of some of the powerful techniques already used in numerically implementing the Boltzmann transport equation. Indeed, the fully time-dependent highly nonlinear numerical quantum transport simulations have provided the resolution of a long-standing controversy over what turns out to be novel highly-nonlinear, and autonomous high-frequency phenomena in symmetric double-barrier nanostructures [127]-[128]. The underlying theoretical basis makes use of the lattice-space Weyl-Wigner formulation of the quantum dynamics of particles in solids [129],[99]-[130], coupled with the nonequilibrium Green’s function or nonequilibrium S-matrix theory.

25.2

Time-Dependent Nonequilibrium Green’s Function

The quantity of major interest in nonequilibrium processes is the “reduced´’ or singleparticle correlation function ρ< (q, q ; t, t ) defined as ˆ (q, t) , ˆ † (q, t) ψ ˆH ψ ρ< (q, q ; t, t ) = T r ρ H H

(25.1)

ˆ † (q, t) is the creˆ (q, t) is the annihilation Fermion-field operator and ψ where ψ H H ation Fermion field operator, the subscript H signifies that the operators are taken in the Heisenberg representation. We can immediately associate the single-particle correlation function to be given by1 ρ< (q, q ; t, t ) = −iG< (q1 , t1 ; q2 , t2 ) .

(25.2)

Therefore the object of central concern in nanodevice nonequilibrium Green’s function technique is the time-evolution equation of G< for electrons (or G> for holes). The resulting equation for G< or G> is coupled to the retarded and advanced Green’s function, Gr and Ga , defined here by Gr (t1 , t2 ) = G> (t1 , t2 ) − G< (t1 , t2 ) θ (t1 − t2 ) , †

Ga = (Gr ) .

(25.3)

(25.4)

1 Here we use the definition of Green’s function without the factor |, following traditional treatments.

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For our purpose, the equation for G≷ (t1 , t2 ) may be simply written as i|

∂ ∂ + ∂t1 ∂t2

G≷ (1, 2)

= Ho , G≷ (1, 2) + Σr G≷ − G≷ Σa (1, 2) + Σ≷ Ga − Gr Σ≷ (1, 2) ˜ G≷ (1, 2) + i Im Σr , G≷ (1, 2) + Σ≷ Ga − Gr Σ≷ (1, 2) , (25.5) = H, ˜ = Ho + Σδ + Re Σr , with Σδ being the ‘singular part’ (delta function in where H time) of the self-energy. Ho = E (p) + V (q), where E (p) is the electron dispersion relation or energy-band function and V (q) is the eﬀective single-particle potential † function. Making use of the relations, (Ga ) = Gr , and Σ≷† (1, 2) = −Σ≷ (2, 1), we can write the last two terms as i Im Σr , G≷ (1, 2) + Σ≷ Ga − Gr Σ≷ (1, 2) = i Im Σr , G≷ (1, 2) + Σ≷ Ga + Σ≷ Ga

†

(1, 2)

= i Im Σr , G≷ (1, 2) + 2 Re Σ≷ Ga (1, 2) ,

(25.6)

which can also be written as i Im Σr , G≷ (1, 2) + Σ≷ Ga − Gr Σ≷ (1, 2) = i Im Σr , G≷ (1, 2) + Σ≷ , Re Gr − i Σ≷ , Im Gr (1, 2) . We will make the following ansatz, which is exact at steady state, iA (1, 2) = −2i Im Gr = − G> (1, 2) − G< (1, 2) , r

>

<

iΓ (1, 2) = −2i Im Σ = − Σ (1, 2) − Σ (1, 2) .

(25.7) (25.8)

Therefore, we can also write the collision terms as i Im Σr , G≷ (1, 2) + Σ≷ Ga − Gr Σ≷ (1, 2) i Γ, G≷ (1, 2) + 2 Re Σ≷ Ga (1, 2) 2 i i Γ, G≷ (1, 2) + Σ≷ , Re Gr + Σ≷ , A (1, 2) . =− 2 2 =−

(25.9)

The second term in Eq. (25.9) is a commutator, like the first term of Eq. (25.5), and serve to complicate the kinetics of particle motion, in addition to the eﬀects of Re Σr , a sort of complex interference phenomena similar to zitterbewegung in Dirac quantum mechanics.2 Thus, the second term in Eq. (25.6) have the character of scattering-in, and additional complex kinetic interference phenomenon, which seems of highly quantum mechanical nature. The first term, which is proportional to G≷ , is the scattering-out term, and the last term which is proportional to the spectral 2 We

shall see that indeed this term contributes to the noise kernel, see Sec. 48.1.1.

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density is the scattering-in term, in analogy with the collision terms in the classical Boltzmann equation. Let us therefore separate the collision terms as −

i i Γ, G≷ + Σ≷ , A 2 2 1 1 Σ> − Σ< , G≷ − Σ≷ , G> − G< =+ 2 2 1 1 = − Σ< , G> + Σ> , G< 2 2 1 1 = − Σ< , G> + Σ> , G< . 2 2

Summarizing, we can write Eq. (25.5) as i|

∂ ∂ + ∂t1 ∂t2

G≷ (1, 2)

˜ G≷ (1, 2) + Σ≷ , Re Gr − 1 Σ< , G> + 1 Σ> , G< , (25.10) = H, 2 2 or in terms of ρ≷ of Eq. (25.2) as i|

∂ ∂ + ∂t1 ∂t2

ρ< (1, 2)

˜ ρ< (1, 2) − i Σ< , Re Gr + 1 Σ< , ρ> + 1 Σ> , ρ< . = H, 2 2

(25.11)

The equation for Gr and Ga = Gr† are simpler and are given as follows: ˆ −1 Gr = 0, G ˆ −1 Ga G

†

= 0.

(25.12) (25.13)

ˆ −1 ρ≷ (1, 2) is prescribed by where the symbolic operation G ˆ −1 Gr (1, 2) = i| G

∂ ∂ + ∂t1 ∂t2

˜ + Σr , Gr (1, 2) . Gr (1, 2) − H

(25.14)

A very important transformation to render the above equation useful for timedependent, highly nonlinear quantum transport problems in nanoelectronics, makes use of Buot lattice Weyl-Wigner formulation of the band dynamics of electrons in a solid3 . By performing this transformation on the above equations, the whole theory can be cast in the language of phase-space distribution, similar to that of the classical Boltzmann equation. In the continuum approximation, let us first transform the double-space and time arguments as follows: τ v t1 = t − , q1 = q − , (25.15) 2 2 τ v q2 = q + . (25.16) t2 = t + , 2 2 3 Refer

to the discussion in Chapter 10 of Part I in this book.

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We also need to define a (3 + 1)-dimensional canonical variables, namely, p = (p, E) ,

q = (q, t) ,

(25.17)

which allows us to write v = (v, τ ). All the variables designated by (1, 2) may now be replaced by (v, τ, q, t). The Weyl transform a(p, q) of any operation Aˆ is defined by the following relation a(p, q) =

ip · v |

dv exp

q−

v v ˆ A q+ 2 2

≡ WA (v, τ, q, t) ,

(25.18)

where the matrix element in the integrand is evaluated between two Wannier functions. At this point, we should emphasize that this transformation strongly impact the numerical simulation of real time-dependent dynamical open systems. The crucial step occurring in Eq. (25.18) is the transformation of nonlocal (double space-time arguments) functions to numerically manageable and physically meaningful local functions in phase space, leading to the exact quantum origin of the Boltzmann transport equation. For our purpose, we transform Eq. (25.5) to the phase-space representation, using the result of Eq. (25.6) for the collision term. Performing the above transformation on the equation for ρ≷ , we have i|

8

2 |

∂ ≷ ρ (p, E, q, t) = ∂t

s ≷ dp dq KH ˜ (p, q; p q ) ρ (p , q )

2 |

8

−

2 |

8

−

dp dq KΣ≷ (p, q; p q ) Ga (p , q )

2i Re i 2

dp dq KΓc (p, q; p q ) ρ≷ (p , q ) ,

(25.19)

˜ = H + Re Σr . Similarly, for Gr,a , Eqs. (25.12)-(25.13), we have where H i|

∂ r G (p, E, q, t) = ∂t

2 |

8 s r dp dq KH+Σ r (p, q; p q ) G (p , q ) ,

(25.20)

where Kas (p, q; p q ) =

Kac (p, q; p q ) =

2i [(q − q ) · u + (p − p ) · v] | × [a (p + u, q − v) − a (p − u, q + v)] , du dv exp

2i [(q − q ) · u + (p − p ) · v] | × [a (p + u, q − v) + a (p − u, q + v)] ,

KΣ≷ (p, q; p q ) =

(25.21)

du dv exp

du dv exp

2i |

[(q − q ) · u + (p − p ) · v]

× Σ≷ (p + u, q − v) .

(25.22)

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Equation (25.19) can also be written in terms of the Poisson bracket diﬀerential operators as i|

≷

≷

∂H ∂ρ ∂H ∂ρ · − · ∂q ∂p ∂p ∂q

∂ ≷ | ρ (p, E, q, t) = 2i sin ∂t 2

≷

− 2i Re exp − i cos

| 2

H (p, q) ρ≷ (p, q) ≷

∂A ∂Σ ∂A | ∂Σ · − · 2i ∂q ∂p ∂p ∂q ≷

≷

∂Γ ∂ρ ∂Γ ∂ρ · − · ∂q ∂p ∂p ∂q

Σ≷ (p, q) Ga (p, q)

Γ (p, q) ρ≷ (p, q) . (25.23)

The electron Wigner distribution function (WDF) is obtained from the correlation function ρ< by performing integration over the energy variable fw (p, q, t) =

dE ρ< (p, E, q, t) .

(25.24)

Note that in the gradient expansion of Eq. (25.23), the leading terms immediately give the form of the classical Boltzmann transport equation, ∂H (p, q) ∂fw (p, q, t) ∂H (p, q) ∂fw (p, q, t) · − · ∂q ∂p ∂p ∂q 1 2 dE Σ≷ (p, q) Ga (p, q) − Γ (p, q) fw (p, q, t) − Re | | ∂fw (p, q, t) −p + collision terms, = ∗ · ∇q fw (p, q, t) − eF · m ∂p

∂ fw (p, q, t) = ∂t

where eF = − ∂V∂q(q) . The equation for the WDF can be obtained by integrating Eq. (25.19) over the energy variable. In the relaxation-time approximation, Eq. (25.19) has been implemented in a fully-dynamical characterization of resonant-tunneling double-barrier structures [99],[131]. By calculating the Wigner trajectories, as constant-value contours of the WDF solution at steady state, the tunneling times and “particletrajectory” characterization of sequential and coherent tunneling have been characterized. The quantum distribution function (QDF) transport equation in the eﬀectivemass approximation can readily be derived from Eq. (25.19). The result is an equation for the Wigner-distribution function which includes the leading collision terms, given by ∂ −p 2π V q − v2 fw (p, q, t) = ∗ · ∇q fw (p, q, t) + 4 dp dv −V q + v2 ∂t m | (p − p ) · v fw (p , q, t) × sin | 1 2 Re (Σ< (p, q, t) Ga (p, q, t)) dE − . −Γ (p, q, t) G< (p, q, t) |

(25.25)

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25.2.1

Electron-Electron Interaction via Exchange of Phonons

In order to bring the calculation of self-energy due to electron-phonon interaction in a familiar electron-electron interaction setting, one writes the eﬀective equation for the electron field operator in a form such that the interaction between Bloch electrons is already renormalized and is described by the exchange of phonons. This is second order in the electron-phonon matrix element, γ αβ , γ αβ (12; ξ) =

Wα∗ (x − 1) ∇rξ

zξ e |x − rξ |

Wβ (x − 2) dx3 ,

where Wβ (x − 2) is the Wannier function, zξ is the eﬀective charge of the nucleus at rξ . Thus, one writes4 i|

∂ ψ (x) = Eα (−i|∇x ) ψ (x) + γ 2 ∂t

dx2 Do (x, x2 ) ψ† (x2 ) ψ (x2 ) ψ (x)

dx2 V (x, x2 ) ψ† (x2 ) ψ (x2 ) ψ (x) ,

+

where γ 2 Do (x, x2 ) represents the interaction between Bloch electrons through exchange of phonons, and V (x, x2 ) represents the electron-electron Coulomb interaction. Thus, the perturbation treatment and self-energy calculation of electron phonon scattering is formally similar to that of the electron-electron interaction. Indeed, a leading contribution to the electron-phonon self-energy is given by the ’one-phonon’(γ 2 Do (x, x2 )) diagram similar to the Hartree-Fock exchange diagram of the electron-electron scattering to first order in V (x, x2 ). Note that the Coulomb interaction, V (x, x2 ) is instantaneous, whereas Do (x, x2 ) is nonlocal in time. The electron-phonon self-energy will be discussed further, where the completely nonlocal forms are given in Appendix G.1.2. By approximating the spectral function for electrons to be given by A(p, E) = δ(E − Ep ), the last term can be reduced to the Boltzmann collision operator [132] −

i |

dE Σ< (p, q, t) A (p, q, t) − iΓ (p, q, t) ρ< (p, q, t) = k

[Wk⇐k fw (k , q, t) − Wk ⇐k fw (k, q, t)] ,

(25.26)

where Wk⇐k = hδ (Ek − Ek + Ωk −k ) γ 2k −k (1 + Nk −k ) [1 − fw (k, q, t)] + hδ (Ek − Ek − Ωk −k ) γ 2k −k Nk −k [1 − fw (k, q, t)] ,

(25.27)

Wk ⇐k = hδ (Ek − Ek − Ωk −k ) γ 2k −k (1 + Nk −k ) [1 − fw (k , q, t)] + hδ (Ek − Ek + Ωk −k ) γ 2k −k Nk −k [1 − fw (k , q, t)] .

(25.28)

4 A good discussion is given in J. C. Inkson, Many-Body Theory of Solids (Plenum Press, New York, 1984).

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Relaxation-Time Approximation

In the relaxation-time approximation, we simply let

p

Wp ⇐p =

1 . τ

(25.29)

From the detailed balance condition at equilibrium we can also write τ −1 fo (p, q) =

Wp⇐p fo (p , q) = Wp⇐p ρo (q) .

(25.30)

p

We can thus obtain the approximate average value of the scattering matrix Wp⇐p to be simply given by Wp⇐p = τ −1

fo (p, q) . ρo (q)

(25.31)

Therefore the collision operator in the relaxation-time approximation can be written as

k

1 Wk⇐k fw (k , q, t) = −Wk ⇐k fw (k, q, t) τ

ρ(q,t) ρo (q) fo (p, q)

−fw (p, q, t)

,

(25.32)

where ρ (q, t) = |13 dp fw (p, q, t) is the particle density, fo (p, q) is the equilibrium WDF, and τ is the averaged relaxation time calculated by considering all scattering processes and using the Matthiessen’s rule [99],[131]. The resulting transport equation must then be solved together with Poisson equation, namely, ∇2 φ (q) =

1 {eρ (q, t) − eno (q)} , ε

(25.33)

where eno (q) represent the background charge density. Therefore V (q) is given by V (q) = φ (q) + ∆Ec (q) ,

(25.34)

where ∆Ec (q) is determined empirically by the band-edge discontinuities at the interfaces. The numerical technique for solving the WDF transport equation has been published [99] and will not be repeated here. In the following section, we discuss the resolution of a controversy concerning the intrinsic bistability of a symmetric double-barrier structure and a novel fully time-dependent highly-nonlinear quantum-transport phenomenon in RTD. 25.3

Intrinsic Bistability of RTD

Perhaps, a significant result of the fully time-dependent, far-from-equilibrium and highly nonlinear quantum transport simulation of a symmetric RTD, using the relaxation-time approximation, is the discovery of the intrinsic high-frequency oscillation at fixed source-drain bias in the negative diﬀerential region associated with

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Fig. 25.1 Current-voltage characteristics of an RTD for increasing and decreasing voltage sweep. Scattering and self-consistency of the potential is taken into account. The dotted curve shown for comparison, do not account for self-consistency, i.e., the voltage is linearly dropped across the undoped double-barrier region.

ideal or nonself-consistent I-V characteristics [127]. This result illustrates the usefulness of the QDF approach for realistic device simulations. For the first time, the time averaged I − V fully reproduces all the features of real experimental I − V curve of symmetric double-barrier structures. Indeed, the computer simulation results based on the above transport equations have fully resolved the long-standing controversy regarding the characteristic experimental features of the I − V characteristic of a symmetric double-barrier structure, thereby clarifying that the “plateau-like” and hysteresis features of the I − V characteristics are due to the intrinsic oscillations [128], by virtue of the highly-correlated charging and discharging of the quantum well. Figure 25.1 shows the time-average I − V characteristics of an RTD at T = 77 K. A prominent “plateau-like” structure and double hysteresis loop in the negative diﬀerential resistance (NDR) region are clearly shown. For a length of time, a strong controversy existed regarding the origin of this. Indeed, there are two schools of thought explaining this result. Goldman, et al., [133] attributed the entire behavior to “charge bistability,” that is charge is dynamically built up and ejected from quantum well, and through a self-consistent feedback mechanism, two current states exist. Sollner [134] disagrees with this, claiming that instead of the hysteresis being due to the charging of the well, it is due to external circuit-induced high-frequency oscillations of the current when operating in the NDR region of the device. The theoretical and numerical works which followed, aimed at resolving this issue, were biased on either Goldman’s hypothesis or Sollner’s explanation, all suggesting aﬃrmative results and adding to the confusion. These earlier theoretical and numerical works did not reproduced the experimental I − V behavior and were biased on explaining either the charge-storage

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Fig. 25.2 Time evolution of the current for various biases for the increasing bias sweep points. At time t = 0 fs. the voltage is increased by 0.01V . The initial WDF is given by the steady state distribution of the previous bias step. For a bias of 0.23V . the current settles to a constant value. For biases up to 0.31V , the current is oscillatory, and at 0.32V , the current initially oscillates with an average value of 5.3 × 105 A/cm2 but decays into a non-oscillatory state with a value of 7.0 × 104 A/cm2 over a time scale on the order of 2000 f s ( the bistable condition).

eﬀects or the externally-induced high-frequency oscillations [135, 136, 137, 138, 139]. The fully time-dependent nonlinear quantum transport simulation demonstrates clearly the presence of intrinsic very high frequency (∼ 2.5 THz) current oscillations, as well as the dynamical charge instability in the quantum well, for fixed bias in the NDR region, Figs. 25.2-25.3. The time average of the intrinsic oscillating currents is what produced the “plateau-like” structure and hysteresis in the I − V curve. The dynamic simulation [127, 128] also shows a bistable charging condition of the quantum well, indeed, the plot of the calculated time-averaged charge stored in the quantum well for the RTD simulated is remarkably similar to that of Fig. 25.1. The dynamic bistability occurs only at 0.25 V and 0.32 V in the NDR region, resulting in the overall hysteresis behavior in the I −V characteristics. Thus, it can be concluded that high-frequency current oscillations lead to plateau-like behavior for positive bias voltage sweep, and the unstable charging of the quantum well at some bias leads to the dynamic current bistability and overall hysteresis. A more accurate quantitative explanation in terms of an equivalent “autonomous” nonlinear circuit model, derived from results of the computer simulation of Eq. (25.25), was initally given by Buot and Jensen [128]. This initial nonlinear circuit model will be summarized in the following section. There it is shown that the plateau-like I − V behavior is due to the sustained limit-cycle operation in the NDR region of the intrinsic current and voltage across the RTD, Fig. 25.4. Al-

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Fig. 25.3 Time evolution of the current for various biases for the decreasing bias sweep points. The current for biases above 0.25V are numerically constant and uniform, but at 0.25V , the current initially is nonoscillatory but slowly increases until it becomes oscillatory over a time scale on the order of 800f s. Only for biases of 0.25V and 0.24V does the current oscillate, for 0.23V the value of the current becomes constant and equal to the current for the increasing bias sweep case. The bistable condition occurs for 0.25V .

though high nonlinearity plays a dominant role, a simplified understanding of the fluctuating current phenomenon may be explained in a manner similar to Coulomb blockade phenomenon. In the NDR region, the charging of the quantum-well helps to maintain the alignment of the quantum level with the allowed high-energy density levels of the source. By monitoring the quantum energy levels in our computation, the ground-state energy level is indeed pushed up with increase of charge in the quantum well until a point is reached where it becomes energetically more favorable to discharge a certain amount of charge ∆Q . This means essentially that degeneracy in the total energy exist between a state with charge ∆2Q and a state with less charge by ∆Q . When ∆Q is released from the well, the quantum energy level is decreased; however, the recharging which sets in opposes the continued decrease of energy level and brings the quantum level to a minimum after which it starts to swing back again. The oscillation of the ground-state energy level with respect to the electron supply levels of the source is clearly seen in the computer simulation. However, as the source-drain bias is further increased towards the upper boundary of the NDR, a point is reached when the decrease of the quantum energy level can no longer be compensated by the consequent weaker recharging of the quantum well, and therefore total charge instability sets in and the quantum-well suddenly becomes depleted. In the reverse source-drain bias sweep, the low-charge quantum-well state is highly stable until the bias is very near the lower boundary of the ideal NDR region. At and beyond a certain point before the peak current, another charge instability set in and the quantum-well charge suddenly increases.

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Fig. 25.4 a) Oscillatory solution for v as a function of time, starting near v = 0 at t = 0; b) trajectory patterns in (i, v) phase-space showing the mechanism for sustained oscillations.

Further slight decrease in bias produced another well-defined oscillating current through a similar mechanism as in the positive source-drain bias sweep. The result is an asymmetrical double-hysteresis and highly-defined plateau-like structure in the I-V characteristics, observed in experiments of symmetric double-barrier structures. These results have also been qualitatively confirmed by an entirely diﬀerent method, through a computer simulation which employed a Monte Carlo particle technique incorporating a space and time-dependent quantum tunneling. In the Monte Carlo computer simulation using particles [140], a simple model of particle quantum tunneling dynamics in the double-barrier region was based on the calculation of phase-time delay obtained from the piece-wise linear-potential Airy function solution to time-independent Schrödinger equation, evaluated at each time step.

25.4

Quantum Inductance and Equivalent Circuit Model for RTD

The use of selfconsistent potential in our simulations reveals the presence of a series resistance when compared with the result of the non-selfconsistent calculations (referred to in Fig. 25.1 as the linear-drop model), which assume that the total voltage bias is linearly dropped across the double-barrier region (undoped region) of the RTD simulated. Note that for the self-consistent simulations, we have found that the linear voltage drop across the double-barrier region still approximately

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holds in the NDR region, however it is no longer equal to the total bias applied. The series resistance can readily be calculated by the following expression, Rs =

∆V 0.23 − 0.12 = = 1.63 × 107 Ωcm2 , Ip 6.75 × 105

(25.35)

where ∆V is the voltage shift of the peak value of the current, and Ip is the peakcurrent value of the self-consistent I − V in Fig. 25.1. A good measure of the intrinsic negative conductance can easily be determined from the linear drop model in Fig. 25.1 and this is taken as the average slope of the linear I − V characteristic between 0.14 and 0.18V in Fig. 25.1, which yields the value of G=

(1.55 − 5.55) × 105 = −1.0 × 107 S/cm2 . 0.18 − 0.14

(25.36)

In what follows, we will also calculate the self-consistent G from the high-frequency oscillation data of Fig. 25.2. The capacitance, C, in the NDR region can be calculated from the self-consistent potential calculations, by first calculating the total displaced charge in the doped region in one end of the device as a function of bias voltage applied. We have calculated the total displaced charges of the doped region of the drain side [128]. We estimate, the average C for the NDR region between 0.23V and 0.32V which is the region we are interested, to be around C=

∆Q ∆V

NDR

= 2.3 × 10−7 F/cm2 .

(25.37)

Note that all the circuit parameters given above are obtained directly from the simulations without any assumption. The most revealing aspect of the simulation is that the current is oscillatory in the NDR region, suggesting that the equivalent circuit model is an autonomous second-order non-linear circuit with at least two energy storage elements. A capacitive-circuit element is obvious by virtue of the presence of barriers and charge storage in the quantum well. Thus, clearly an inductive-circuit element should also exist. Gering et al. [141], proposed an equivalent circuit model for RTD shown as circuit 2 in Fig. 25.5(b). However, results of the present simulation suggest that, due to the correlation of the bottom of allowed energy level of the source and the resonant energy level of the quantum well [140] (brought about by virtue of the self-consistency of the potential with the stored charge in the quantum well), an inductive-delay element should exist in the NDR region in parallel with the capacitance C and in series with the nonlinear RTD intrinsic conductance, shown as circuit 1 in Fig. 25.5(a). Indeed, with the circuit parameters calculated above, it becomes trivial to eliminate circuit 2, Fig. 25.5(b), as a possible equivalent circuit model in the NDR, in favor of circuit 1. To simplify the following discussions, it is helpful to transform the currents and voltages in a manner shown in Fig. 25.5(c), which also show the inherent i(v) characteristics of the nonlinear resistor in the transformed coordinates. In terms of the transformed currents and voltages, the circuit equation for circuit 1 and circuit 2

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Fig. 25.5 a) RTD equivalent circuit model derived here; b) RTD equivalent circuit model originally proposed; c) transformation of current and voltage variables used in the circuit equations.

acquire very physically meaningful form as a damped nonlinear oscillator equation. In the “local” approximation to the nonlinear problem, we can approximate the actual current across the nonlinear resistor in the NDR region as Gv + Io (Fig. 25.5(c)), where v is the voltage in the new coordinate system. Then we have v¨ + ηv˙ + ω2o v = 0,

(25.38)

(RC + LG) , LRCG

(25.39)

(RG + 1) , LRCG

(25.40)

(RC + LG) , LC

(25.41)

(RG + 1) . LC

(25.42)

where for circuit 1, we have η1 = ω201 = and for circuit 2, we have η2 = ω202 =

Equation (25.38) has solutions for v given by v = Aeλ1 t + Beλ2 t ,

(25.43)

where η λ1,2 = − ± 2

η 2

2

− ω2o .

(25.44)

We can immediately see that a first valid condition for the NDR region to exhibit oscillations is for − η2 > 0 and ω2o > 0; this ensures that v = 0 is an unstable

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equilibrium point. Oscillations will of course only happen if ω 2o − η2 > 0. Note that if ω 2o < 0, oscillations are completely impossible. Sustained oscillations will occur if for large variation of v, − η2 eﬀectively changes sign, i.e., − η2 < 0, with ω 2o > 0. In this situation, balance between growth and decay of the amplitude of v will occur. Indeed, from Fig. 25.1 and Eq. (25.39), changes in the sign of η is expected as v goes away from the NDR region. When this particular situation occurs, the oscillation acquires a limiting constant amplitude as depicted in Fig. 25.4(a). With the measures of R and G obtained directly from our numerical simulations, we can now eliminate circuit 2 in favor of circuit 1. For periodic solution to be possible in the NDR, we must therefore have for circuit 1: RC > LG, R |G| > 1.

(25.45) (25.46)

On the other hand, for circuit 2, we must have RC < LG, R |G| < 1.

(25.47) (25.48)

Based on the circuit parameters calculated directly from numerical simulation, we have R |G| = 1.63 > 1.

(25.49)

Hence it becomes impossible for circuit 2 to oscillate in the NDR region with the obtained circuit parameter R and G. As we shall see below, calculations based on the oscillation data further validate this claim. In order to gain better insights into the sustained “limit cycle” for the RTD oscillation, it is helpful to consider the corresponding first-order coupled diﬀerential circuit equations. In the transformed coordinate system shown in Fig. 25.5(c), we have the following time-evolution equation in (i, v, ) phase-space for circuit 1 as d dt

i vc

=

1 L|G| − C1

1 L 1 − RC

i vc

,

(25.50)

where i is the current along the nonlinear resistor and vc , is the voltage drop across the capacitor. A couple of observations can immediately be made on Eq. (25.50). First, it defines trajectories in (i, vc ) phase space with a unique slope at each point ˙ given by v˙ic except at (i, vc ) = (0, 0). Second, the point (i, vc ) = (0, 0) is an equilibrium point. In vector notation, we have, ˙ (t) = A X (t) , X

(25.51)

where X (t) =

i vc

.

(25.52)

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The solution for the trajectories X(t) in (i, vc ) phase space is given by X(t) = C1 eλ1 t Y1 + C2 eλ2 t Y2 ,

(25.53)

where λ1 and λ2 are the two eigenvalues, and Y1 and Y2 are the corresponding eigenvectors of matrix A. We have η λ1,2 = − ± 2

η 2

2

− ω2o ,

(25.54)

where η = a11 + a22 , ω 2o = a11 a22 − a12 a21 ,

(25.55) (25.56)

which give exactly the same results as those given by Eqs. (25.39) and (25.40). Similar analysis in vector notation for circuit 2 yields Eqs. (25.41) and (25.42) for η and ω 2o , respectively. A schematic plot of the trajectories for circuit 1, (R |G| > 1), showing (i, vc ) = (0, 0) to be an unstable equilibrium point, is shown in Fig. 25.4(b). The i(v) characteristics of the nonlinear resistor is also plotted, which allows us to consider the (i, v) phase space as well, where v is the voltage across the nonlinear resistor. Indeed, for large excursion of v beyond the NDR region, G changes sign and hence η also change sign yielding a decaying solution instead of the growing solution inside NDR. Because all trajectories have a unique slope at each point except at (0, 0), no trajectories can intersect, this lead to each trajectory eventually reaching a limiting orbit called the limit cycle. This results in steady oscillation in agreement with our numerical simulations. For η = 0, the oscillating current is basically a superposition of two oscillations which have phase diﬀerences brought about by the two eigenvectors and the two eigenvalues, leading to a nonsinusoidal oscillation. However for η = 0, the resulting oscillation should be pure sinusoidal with frequency exactly equal to ωo in Eq. (25.38). Indeed, Fig. 25.2 shows both sinusoidal and nonsinusoidal oscillations at voltage bias of 0.29V and 0.26V , respectively. We can readily calculate the inductance, L, directly from the sinusoidal data at 0.29V in Fig. 25.2. First we have to solve for the absolute value of the eﬀective inductance |G|0.29V consistent with η = 0 and ω 2 = ω2o . The resulting equation for |G| that satisfy both requirements is RC = |G|

2π Ts

−1

2

C

1−

1 R |G|

,

(25.57)

where Ts , is the period of the sinusoidal oscillation in Fig. 25.2 (Ts = 133.3f s). Thus we have |G|0.29V = RC 2

2π Ts

2

+

1 , R

(25.58)

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and hence L is given by L=

RC . |G|0.29V

(25.59)

Substituting the previously obtained values of R and C, we have |G|0.29V = 2.527 × 107 S/cm2 , L = 1.48 × 10−21 H cm2 .

(25.60) (25.61)

The value of |G|0.29V is greater than the average value determined from the linear drop model. Figures 25.1 and 25.2 indicate that η eﬀectively goes to zero as the bias increase toward the bistability point at 0.32V in Fig. 25.1. Using the obtained value of L, we can estimate the eﬀective |G|0.26V of the non-sinusoidal data at 0.26V in Fig. 25.2. |G|0.26V is calculated from the equation ω2o −

η 2

2

=

2π TNS

2

,

(25.62)

where TNS is the period of the nonsinusoidal oscillation in Fig. 25.2 (TNS = 400f s). Substituting the known value of R, C and L, we obtained |G|0.26V = 0.876 × 107 S/cm2 ,

(25.63)

which is closer to the average |G| determined from the nonlinear drop model of 1.0x107 S/cm2 , In actual experimental set-up, the inductance of the leads might actually be significant. In this case, the resulting circuit equation is 3rd-order diﬀerential equation or three coupled first-order diﬀerential equations. The resulting trajectories in three-dimensional phase-space become more complex (three eigenvalues and three eigenvector solutions) and complex behavior other than periodic oscillations, such as period doubling, fractal structures and chaotic behavior becomes possible. Still a more exact treatment of the wiring leads as transmission lines might call for the equivalent circuit model which will lead to [2(n − 1) + 1] th-order complex differential equations, ([2(n − 1) + 1] eigenvalues, and [2(n − 1) + 1] eigenvectors in general). Therefore a very complex behavior is possible in real laboratory experiments. Chaotic behavior with continuous power spectrum could easily occur in real RTD experiments. The QDF transport simulation clarifies that intrinsic high-frequency current oscillations and self-consistent stored charge in the quantum well are inseparable eﬀects in the NDR region. These eﬀects conspire to form the characteristic “plateaulike” I − V behavior for increasing and decreasing voltage sweeps. The bistable self-consistent charging of the quantum well at some bias voltages leads to dynamic current bistability in the I − V characteristics of the resonant tunneling structures. An autonomous nonlinear equivalent circuit model, which is expected to aid in understanding the performance and in designing RTD-based ICs, is directly derived from first principle quantum transport numerical simulations. We have argued that in some real RTD experiments much more complex highly nonlinear behavior other than periodic oscillations discussed here, including chaotic behavior [142]-[143], can

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occur which are externally induced. Indeed, there are experimental results [144] that clearly support this prediction.

25.4.1

Transient Switching Behavior and Small-Signal Response of RTD from the QDF Approach

The large-signal switching behavior of RTD is of great interest to logic designers. This has been numerically investigated by a number of workers using the Wigner distribution-function approach [99, 130],of switching behavior when a step voltage is applied between the peak-current voltage to the valley-current voltage in RTD, an operating condition with potentially large gain, is a highly nonlinear farfrom-equilibrium and a fully time-dependent problem. Thus the Landaur-Büttiker Counting Argument (LBCA) employed in mesoscopic physics problems is not applicable. Methods which depends on the resolution of “asymptotic” current-carrying quantum channels may only hold at steady state and are only practical for nearequilibrium situations. In the area of small-signal response of RTD of interest to analog-system designers, one seeks to calculate the eﬀect of small a.c. perturbation on a far-from equilibrium steady-state situation. Attempts to extend the Landauer-Büttiker method of identifying current-carrying quantum channels into this far-from equilibrium regime, cast in the spirit of time-dependent scattering theory, have been initiated by Cai and Lax [?]y maintaining current-carrying quantum energy channels outside the double-barrier region. This “spectral method,” although very useful for understanding behavior from independent channels, seems impractical when a self-consistent treatment calls for the simultaneous knowledge of the spatial and temporal behavior of all the current-carrying quantum-energy channels by virtue of the “vertical transport” and other virtual processes. Moreover, for including all many-body effects, it is more appropriate to consider the reduced single-particle density matrix, or the nonequilibrium Green’s function discussed in the preceding sections. It should be realized that the set of problems of great interest in nanodevice physics is entirely diﬀerent from the set of problems where LBCA has been successfully applied. Indeed, for nanodevice performance analysis, the most promising method that can readily address all issues of concern is the non-equilibrium Green’s function technique using the quantum distribution-function approach. The reason for this is that “spectroscopic” details in device transport, although very useful for understanding device physics, are generally “integrated out” in evaluating practically useful device performance or “input-output” device properties, such as switching speed, transconductance, cut-oﬀ frequency, output impedance, gain, current drive, etc. The quantum distribution-function transport equation is indeed the exact analog of the Boltzmann transport equation, obtain in the limit when | goes to zero [132], which has been the workhorse in conventional semiconductor device analysis. The switching behavior of RTDs in terms of its current-response to voltage step between peak-current bias and valley-current bias has been given by Frensley [145] and by Buot and Jensen [99] as a function of position and time. Kluksdahl,

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et al. [146] gave the current response, after the application of the same voltage-step pulse, at the device terminal only. All the three papers diﬀered in their numerical implementation of the Wigner distribution-function transport equation. The work in Ref. [99] also accounts for the scattering in the relaxation-time approximation, Eqs. (25.25) and (25.32). The work in Refs. [99] and [145] found damped high-frequency oscillations due to quantum interference eﬀects, following the application of the step voltage pulse in the absence of scattering and selfconsistency. When scattering was taken into account in Ref. [99], the oscillations became more damped and decayed much faster, yielding a much faster switching speed. Further introduction of self-consistency through the solution of the Poisson equation in Ref. [99] did not significantly change the switching time at 300 K partly due to the small voltage step needed, brought about by the introduction of the source and drain resistance in the self-consistent calculations. In Ref. [146], which solved the WDF transport equation and the Poisson equation, the oscillatory terminal currents were only attributed to plasma eﬀects. However, recent work of Cai and Lax [147]does support the strong oscillations due to quantum interference, after a large “peak-to-valley” signal pulse was applied to RTD, for a single-channel transport. The state of understanding of the a.c. small signal response of RTD is also controversial due to various approximations used by several workers. Some results yield capacitive behavior while others yield inductive behavior for the same frequency range, operating in the NDR region. Below, we give a unifying explanation of the small-signal response of RTD based on the equivalent-circuit model derived in the preceding section. The first work for RTD with operating bias in the middle of the NDR region was given by Frensley [148]. This work did not take into account phonon scattering and self-consistency (i.e., Poisson equation was not included). The linear and nonlinear response (rectification and second harmonic generation) were calculated. The real part of the admittance was found to be negative up to about 5 THz; the reactive imaginary part shows a peak at about 6 THz. Rectification was also found to enhanced at about 4 THz. Frensley did not find any inductive behavior [149]. This is understandable since self-consistency was not included, in the light of the discussions of Sec. 25.3; this will be discussed in more detail in the next section. An entirely diﬀerent approach for calculating the frequency-dependent admittance was used in Ref. [146] by Fourier transformation of the step-voltage pulse and the time-dependent current response obtained from their switching calculations. Since scattering within the device is not included (mainly simulating low temperatures within the device), but self-consistency was taken into account through the use of Poisson equation, hence the simulated device may show hysteresis in the I − V behavior in the NDR region, similar to that found by Jensen and Buot [127]. The “very soft hysteresis” I − V in the NDR calculated in Ref. [146], not borne out by experiments and by simple charge-bistability analysis [135]], reflects the diﬀerence in the numerical implementations and approximations used between Refs. [99] and [146], and suggest that their I − V is practically stable in the NDR region. This will be clarified in the next section. It is however interesting that Kluksdahl, et al. found a highly inductive behavior of the current response.

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Both the inductive and capacitive behaviors are supported by the equivalent circuit model derived in Sec. 25.4 for RTD operating in the NDR region. 25.4.2

High-Frequency Behavior and Small Signal Response of RTD using an Equivalent Circuit Model

A much simpler and straightforward way to analyze the small a.c. signal response of an RTD, biased in the middle of the NDR (a high-gain biasing condition), is through a direct application of the equivalent circuit model. This model was derived in Sec. 25.4 from the WDF time-dependent numerical simulation. In this section, the complex-valued conductance obtained through WDF numerical simulations by Frensley [148, 149] and by Kluksdahl et al. [146], respectively, will be explained simply in terms of the equivalent-circuit model. What is most apparent from the WDF numerical simulations at 300 K of Refs. [148, 149] and [146], where scattering inside the device is not taken into account, is that the series resistance of the RTD sample simulated is much smaller than the sample numerically simulated by Jensen and Buot [127], the sample length simulated is about twice as large in Ref. [127] than in Ref. [146] where selfconsistency, which allows the voltage to be dropped across the whole sample, is also taken into account. Self-consistency is not taken into account in Refs. [148, 149], so that the series resistance is expected to be even much smaller (residual series resistance through the double-barrier region only). Since the quantum well and barrier sizes are about equal in all of the WDF numerical simulations, it is therefore expected that the RTD simulated by Frensley and by Kluksdahl et al. do not obey the criteria for oscillations and limit cycle operation to exist, namely, R |G| > 1 and RC > L |G| . Experimental results support the critical role of R in the bistability and oscillatory behavior in the NDR region [150]. Moreover, it is expected that the quantum inductance in Refs. [148, 149] would be quite small, i.e., mainly caused by the inductive delay to adjust to the new voltage by virtue of the quantum nonlocality and quantum interference eﬀects (this residual quantum inductance in the NDR region is revealed by the damped oscillatory current behavior when applying a voltage step between the current-peak voltage and valley-current voltage [99], without taking self-consistency into account). However, the inductance present in the WDF transport simulation of Ref. [146] is expected to be larger than that of Refs. [148]-[149] since self-consistency is taken into account. Jensen and Buot [127] and Salvino and Buot [140], using entirely diﬀerent methods, have found that the quantum inductance is mainly due to the inductive delay caused by the time-dependent correlation between the self-consistent sharp-resonant-energy level of the quantum well and the bottom of allowed energy level of the emitter, this made highly observable when self-consistency is taken into account at low temperatures. 25.4.2.1

Linear Response

Thus the I − V characteristics in the sample simulated in Refs. [148, 149] and [146], respectively, can be considered to be stable in the NDR region. Indeed, when self-consistency and scattering are properly taken into account at 300 K, Buot and

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Jensen [99] have shown that the I − V characteristics do not show bistability and oscillatory behavior; at this temperature the inductive delay is smeared out by the broadening of the energy levels of the quantum well. It is precisely the stable I −V characteristic in the NDR region that is implied in the concept of steady-state complex-conductance linear response to small a.c. signal. To calculate the complex-valued conductance, let us assume that a small complex-valued signal given by vo eiωt is applied across the two terminals of the RTD. For a stable perturbation around the middle of the NDR region, we can approximate the current in the nonlinear resistor in Fig. 25.5 to be given by i (v) = Io + G v.

(25.64)

Then the equation for the voltage drop, v, (measured from the middle of the NDR region) across the nonlinear resistor is given by the following linear-circuit equation RCLG v¨ + (RC + LG) v˙ + (RG + 1) v = vo eiωt .

(25.65)

For calculating current response, we are only interested in the particular solution of Eq. (25.65). This is given as v=

vo eiωt . {[(RG + 1) − ω 2 RCLG] + iω (RC + LG)}

(25.66)

The terminal-current response, ∆i (t) = ∆ic + ∆iL in the equivalent-circuit model of Fig. 25.5 is obtained as ∆i (t) = CLG¨ v + C v˙ + Gv.

(25.67)

Upon substituting the particular solution for v, given by E. (25.66), we are led to the following expression for the complex conductance, σ (ω) = Re σ (ω) + i Im σ (ω) ,

(25.68)

1 G 1 − ω 2 CL ω 2o − ω 2 + ω 2 Cη , RCLG (ω2o − ω 2 ) + (ωη)2

(25.69)

where Re σ (ω) = Im σ (ω) =

ω C ω2o − ω 2 − G 1 − ω 2 CL η . RCLG (ω2o − ω 2 ) + (ωη)2

(25.70)

The expression for ω 2o is given by Eq. (25.40), and for η by Eq. (25.39), in terms of the circuit parameters R, C, L, and G. We immediately observe that if Re σ (ω) represents a negative conductance at ω = 0, as found by Frensley [149] and by Kluksdahl et al. [146], then RG + 1 > 0 or R |G| < 1. This is consistent with our previous assumption regarding the smallness of R. This also implies that the parameter ω2o < 0. Since R is small, then the parameter η > 0 for similar reasons. In the limit ω becomes very large, the inductive element blocks the eﬀect of the negative conductance, Re σ (ω) becomes positive and is given by 1/R (Re σ (ω) will eventually go to zero as ω becomes very large, if one accounts for a smaller inductance due to the electron inertia in series with the

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two-branch RTD equivalent circuit, as shown numerically in Ref. [151]). Therefore Re σ (ω) must cross the frequency axis at some particular value ω c . The crossing frequency ω c is given by the smallest positive root of the following equation |G| 1 − ω 2 CL

ω 2o + ω 2 + ω2 Cη = 0.

(25.71)

This is a quadratic equation in y = ω 2 . One positive root is guaranteed, this is the crossing frequency, ω c , given by ωc = α ±

α2 + γ 2

1 2

,

(25.72)

where α= γ=

|G| + Cη − |G| ω 2o CL , 2 |G| CL |ω2o | . CL

(25.73) (25.74)

Moreover, provided certain conditions are met, compatible with the smallness of R, the extrema of Re σ (ω) can also exist for at least one positive value of ω, obtained by solving another quadratic equation in y = ω 2 , most likely after the crossing of the frequency axis. These results are indeed what was observed in the WDF numerical simulation of Frensley [149] and Kluksdahl et al. [146]. The imaginary part, Im σ (ω), is proportional to ω at low frequencies. It is C inductive at low frequencies provided L |G| > |G| holds. Otherwise, Im σ (ω) is capacitive. At low frequencies, Kluksdahl, et al. [146] found Im σ (ω) to be inductive, whereas, Frensley found it to be capacitive [149]. This supports our claim that the neglect of self-consistency in Ref. [149] resulted in much smaller quantum inductance compared to that of Ref. [146], where self-consistency was taken into account. In all cases, at higher frequencies, the inductance ceases to enter in the reactive component as expected, and Im σ (ω) is always positive and characterized by the RC branch of the equivalent circuit, i.e., Im σ (ω)|ω⇒∞ =

1 R2 ωC

.

(25.75)

C It is interesting to note that when L |G| > |G| is satisfied, a positive frequency, ω Ic , for crossing from the inductive to capacitive behavior exists and is given by



ωIc = 

L |G|2 − C RCL |G| (|G| CLη + C)

 12

 .

(25.76)

This fundamental result agrees with the numerical simulation of Kluksdahl et al. C [146]. On the other hand, if L |G| < |G| as in the numerical simulation of Frensley [149], then crossing of the frequency axis should be absent unless a smaller inductance (roughly an order of magnitude smaller than the quantum inductance)

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is also taken into account which will result in a crossing from the capacitive to the inductive behavior at high frequency. The equation for finding the extrema of Im σ (ω) is a cubic equation in y = ω 2 . Thus three positive frequencies are possible for observing the maxima and minima of Im σ (ω). One needs to know the exact values of R, C, L, and G to determine the existence and nature of the roots of the extrema equation for Im σ (ω). We should note that Frensley [145] observed three clear extrema. in the capacitive Im σ (ω) whereas Kluksdahl, et al. [146], observed at least two extrema, one before the crossing and one after the crossing of the frequency axis, with the behavior at higher frequencies not fully resolved. 25.4.2.2

Nonlinear Response

A more realistic treatment of RTD is to consider the source and drain resistance to be larger than the absolute magnitude of the inherent negative diﬀerential resistance in the middle of the NDR. This was achieved in the WDF numerical simulation of Jensen and Buot [127] by using a longer simulation-box length. Self-consistency and scattering, through the use of a relaxation-time approximation [99], were taken into account. The criteria for oscillations and limit-cycle operation in the NDR region were achieved at low temperatures as discussed in Sec. 25.4, with results in excellent agreement with the experiments. For NDR region operation which exhibits oscillations and limit-cycle behavior, resulting in hysteresis and charge bistability in the averaged I − V characteristic, the middle of the NDR region, v = 0, is an unstable point. Therefore, it is no longer valid to use the linear approximation to the current across the nonlinear resistor. Any small perturbation will bring the values of v to large excursions. Thus, the problem becomes highly nonlinear. To give the problem a definite form, let us approximate the i(v) characteristic of the nonlinear resistor by adding a cubic term as dictated by symmetry in Fig. 25.5, given by i (v) = Io + Gv + βv3 ,

(25.77)

where β may be approximated as β=

|G| . 3vp

(25.78)

The quantity vp is the voltage, measured from the middle of the NDR region, where the current peak is located. Let us take a real a.c. small signal given by vo cos(wt) to be applied across the terminals of the RTD. Then the equation for the voltage drop across the nonlinear resistor, measured from the middle of the NDR region, is now given by v¨ + η (v) v˙ + ζ (v) v˙ 2 + ω2o (v) v = Ω (v) cos (ωt) ,

(25.79)

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where RC + LG + 3βLv2 , D (v) (12βRCLv) ζ (v) = , D (v) RG + 1 + βv2 , ω2o (v) = D (v) D (v) = RCLG + 3βRCLv2 , vo . Ω (v) = D (v) η (v) =

(25.80) (25.81) (25.82) (25.83) (25.84)

Equation (25.79) is a highly nonlinear equation, and represents a nonlinearly driven nonlinear-damped oscillator. It requires the analytical and geometrical tools of nonlinear mechanics [152]-[153] for its numerical solution. We will not go into this in this section.

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Chapter 26

QDF Approach and Classical Picture of Quantum Tunneling

The quantum distribution-function formalism of the exact many-body quantum transport theory discussed in Sec. 25 goes far beyond the continuum level of quantum dynamics. It is in fact intended to be able to deal with the particle dynamics of real solid-state materials. Indeed, the transformation given by Eq. (25.18) should not be viewed as a transformation of basis states. Equation (25.18) should be viewed as an exact formulation of the “correspondence principle” in quantum mechanics, cast in the canonical conjugate variables appropriate to the given “universe” or crystalline medium. It also embodies the eﬀective-Hamiltonian theory of solid-state physics, which is in the heart of the physics of the quantum dynamics of Bloch electrons in external electromagnetic fields [154] and in charge-carrier transport in solid-state electronics, and has been found to simplify (i.e., eliminate zitter-bewegung) the physics of Dirac particles [155]. The general theory that encompass all these has been formulated with its applications, in a series of papers [154], [156], [157],[129]. This lattice Weyl-Wigner formulation of the multi-band dynamics of electrons in solids is based on the use of the Wannier Functions and Bloch functions as the supporting basis states, in place of the Dirac-delta function and plane waves in one-band continuum quantum mechanics which has been so far the domain of the LBCA in mesoscopic transport. Thus, Sec. 25.2 is really a simplified treatment (when band indices are dropped and when atomic discreteness is not critical) of the lattice Weyl-Wigner formulation of the exact many-body quantum transport theory, a fully discrete version was given by the author in Ref. [100]. There is no doubt that transport physics pertaining to real materials is of paramount importance to the characterization of real nanodevices; exact particle energetics or band-structure eﬀects, both in the particle kinematics and in collision processes, are well-known to dominate device behavior in conventional solid-state electronics.

26.1

Lattice Wigner Function and Band Structure Eﬀects

The lattice Wigner function, based on the use of Wannier function and Bloch function introduced by the author in Ref. [129], has been used by Miller and Neikirk [158], [159], [160] to calculate the band-structure eﬀects in the quantum transport 318

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˚ Fig. 26.1 (a) Wigner distribution function for a symmetric GaAs-AlGaAs-GaAs RTD with 50A barrier and quantum-well widths with zero bias at T = 300K.

simulation of RTDs. Their calculations revealed the importance of the multi-valley electron dynamics even at equilibrium, so as to obtain the correct WDF. Moreover, preliminary results [140] of an entirely diﬀerent and less accurate method intended for multi-dimensional problems of arbitrary geometry, i.e., selfconsistent Monte Carlo particle transport with model quantum tunneling dynamics [141], seems to confirm Miller and Neikirk [159, 160] results for the electron concentration of the Γ and X valleys as function of position in RTD. In any case, more work on discrete phase-space many-body quantum transport [99, 100] of real materials needs to be performed to address unexplored questions concerning time-dependent multi-valley dynamics and collision processes far from equilibrium, eﬀects of interfaces, eﬀects of the additional L valleys in GaAs/AlGaAs RTD and interband tunneling dynamics in novel InAs/GaSb/AlSb-based interband tunnel devices [161, 162]. 26.2

Coherent and Incoherent Particle Tunneling Trajectories

Another virtue of the QDF approach is that it allows for a classical picture of quantum transport processes. This is determined from the solutions to the quantum distribution-function transport equation defined in phase space. At steady-state condition, ∂fw (p,q,t) = 0, the Wigner trajectories (the particle trajectory represen∂t tation of quantum transport) are defined to be the constant value contours of the solution to the WDF transport equation defined in phase-space [99].

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(b) Particle quantum trajectories for the RTD in (a) with zero bias at T = 300K.

Figure 26.2 shows the particle quantum trajectories or Wigner trajectories for an RTD simulated at zero bias and hence at equilibrium. Bound trajectories exist not only inside the quantum well but also inside the barriers and even in the source and drain regions adjacent to the outer barrier edges. These bound trajectories outside of the quantum well are represented by the complementary loops at k < 0.0 and k > 0.0, i.e., portions of the k < 0.0 loop are paired with portions of the k > 0.0 loop. This implies that at these regions in real space the Wigner trajectories instantaneously change directions, as though the particles are bouncing between the impenetrable potential barriers. The tunneling trajectories are decelerated inside the barriers and accelerated within the quantum well, the trajectories with this behavior will be referred to as Sequential Tunneling Trajectories (STT). Figure 26.3 shows the tunneling trajectories when a small bias of 0.02V is applied at the drain of the RTD at 77K, with selfconsistency and scattering within the device switched oﬀ (ballistic trajectories). Indeed, as can be observed in the figure, these tunneling trajectories acquire higher energy (higher momentum) about equal to the bias applied at the drain end, demonstrating a conservation of energy behavior. A remarkable new behavior also emerges compared to the tunneling trajectories of 26.2, here the tunneling trajectories are not accelerated within the quantum well, indicating that the associated particle stays longer in the quantum well. It is as if the associated particle sees the double-barrier structure as a whole. These trajectories will be referred to as Coherent Tunneling Trajectories (CTT). The conservation of energy behavior can also be seen in the WDF plotted at the drain end of the RTD simulated. Figure 26.4 shows WDF at the drain end with and without scattering (in all cases selfconsistency is switched oﬀ unless specified),

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Fig. 26.3 Ballistic resonant tunneling at 77K with small bias applied. Note the new behavior compared to 26.2.

Fig. 26.4 WDF plotted at the drain end of RTD when biased at resonance of 0.116eV , at diﬀerent temperatures. The solid circles are values at 77K without scattering, plotted for comparison. Note the presence of tunneling ridges at smaller energies compared to 26.5.

and at various temperatures for a bias of 0.116V which corresponds to the current resonance peak. Figure 26.5 shows the WDF of a ballistic simulation when a bias of 0.5V is applied. The observed WDF ridges are resonant-tunneling ridges and is due to the ballistic resonant-tunneling particles acquiring energies equal to the voltage diﬀerence between the source and drain.

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Fig. 26.5 WDF plotted at the drain end when the RTD is biased at higher voltage of 0.5eV , with ballistic transport across the device. Note the presence of tunneling ridge at higher energies.

Fig. 26.6 for.

Particle quantum trajectories corresponding to 26.3 when scattering at 77K is accounted

Figure 26.6 shows the Wigner trajectories corresponding to Fig. 26.3 when the scattering at 77 K is switched on. The broken paths are trajectories that failed to tunnel through and are turned back. Notice the slight degradation of the CTT shown by a slight acceleration within the quantum well. Figure 26.7 shows the corresponding trajectory pattern when the temperature is raised to 300 K, representing the incoherent or sequential tunneling trajectory pattern. There is

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Fig. 26.7 Particle quantum trajectories corresponding to Fig. 26.6 when the temperature is raised to 300K.

now a significant tunneling trajectory contributions from the higher energy tail of the WDF at the source end, and all tunneling trajectories exhibit the STT behavior similar to Fig. 21.2, even though a finite bias exist for Fig. 26.7. When the selfconsistent potential is turned on in our simulations, the behavior of all trajectories remains basically the same, diﬀerences occurs mainly due to the diﬀerences brought about by diﬀerent selfconsistent potential profiles. In summary, for ballistic transport across the RTD, a CTT distribution ridge develops in the WDF profile at the drain end and his ridge moves to higher energy or momenta as the bias increases. This behavior is a manifestation of the conservation of energy for these trajectories. These CTT WDF ridges are progressively washed out as temperature increases due to increase in dissipation or scattering. This particle behavior has potential applications to the real particles in Monte Carlo device simulations [140]. Diﬃculties in calculating the Wigner trajectories are encountered at the current peak when the WDF oscillates violently within the device. However, it is known that a Gaussian smoothing of the WDF, e.g., the use of normalized minimal Wigner wave packet , will render a smoother Husimi distribution and thus leads to well-behaved trajectories in phase space. For operating bias that does not lead to violently oscillating WDF, the smoothing procedure does not alter the WDF. The smoothing technique has been used in studying nonlinear quantum phenomena and chaos, to study the structure of trajectories in phase-space and its correspondence with that of nonlinear classical phenomena (a discussion of this paragraph is given in more detail in Ref. [151].

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Chapter 27

RTD as a Two-State Memory Device, a Memdiode or a Memristor

A modification of a conventional quantum-well diode by employing a n− -n+ -n− spacer-layer structure in the source and/or the drain region leads to two stable current-voltage and corresponding two charge-state behaviors all the way down to zero bias. These are the salient features of an experimental observation on quantumwell diodes with n− -n+ -n− spacer layers. A simple theory of self-consistent charge buildup and bistability shows that a limited supply of electrons from the emitter at high bias, brought about by the n− -n+ -n− spacer-layer structure, leads to fractional recharging of the quantum well and hence fractional current values during the decreasing voltage portion of a “closed-loop” voltage sweep. This two charge state phenomenon is the basis for a binary-information storage RTD device at zero bias without dissipation, which we refer to here as a memdiode.

27.1

Binary Information Storage at Zero Bias

Indeed, the storage of binary information at zero bias (without heat dissipation) would find very attractive applications in design of computer memories. We note that this type of memory is interrogated by applying a small bias and rewritten by applying a larger bias in a “closed-loop” voltage sweep.

27.1.1

Intrinsic Behavior of Double-Barrier Structures

A conventional RTD exhibits I − V dynamic bistability at a voltage just beyond the negative diﬀerential resistance (NDR) region. The result is a double hysteresis in the I − V behavior in the NDR region as shown in Fig. 25.1. By symmetry, on interchange of the source and drain regions of the quantum-well diode studied by JB, we can plot the same I − V characteristic in the third quadrant of the I − V plane. It is important to realize that the persistence or stability of the low-current state in the decreasing bias sweep described above is directly related to the diﬃculty in recharging the quantum well during the decreasing bias sweep. By changing the structure of the source and drain regions of the quantum-well diode, so as to limit the supply of electrons from the emitter at high biasing condition, the complete recharging of the quantum well during the decreasing voltage 324

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sweep would become energetically impossible. Then one can expect only “fractional recharging” to occur resulting in fractional current values and hence two current curves in the first I − V quadrant, for voltages below the current-valley voltage. 27.1.2

The Physical Picture

The fundamental structural diﬀerence between the conventional quantum-well diode and RTD memory device lies in the configuration of the spacer layers in the source and drain regions. A simple symmetric RTD memory device incorporates an n− -n+ n− layer instead of a simple n+ layer, thus eﬀectively introducing “self-consistent potential-control element” and transport barriers to current flow from both the source and drain regions. The presence of extra depletion barriers, which separate the large particle reservoirs in the source and/or drain regions, alters the distribution of electrons from the emitter or source at high bias. This frustrates the transition from the lowcurrent state to the original high-current state during the decreasing voltage sweep, by virtue of the failure to fully recharge the quantum well, resulting in the persistence or stability of the low-current state down to zero bias. The collector or drain will also have an altered electron distribution. Continued decrease of the voltage bias to negative values maintains the stability of the low-current (highly depleted quantum well) or high-resistance state until a much larger negative bias is reached whereby it becomes energetically possible to recharge the quantum well, by virtue of inadequate screening by the double-barrier region to maintain the barrier between the particle reservoir and emitter. A bistability of the low-current state and the high-current state then occurs leading to a transition of the low-current state to the high-current state (transition from A to A in Fig. 27.1). As long as only one energy level is operative, a continued decrease of the voltage bias would result in the instability of the high-current state by virtue of the continued tilting of the quantum well leading to eventual discharging of the quantum well and a resumption of the high-resistance state. However, if the voltage is immediately increased toward positive values after the transition from the low-current state to the high-current state in the third quadrant, the high-current state becomes stable and is the state followed across the zero bias point. A continued increase of the bias through zero bias reverts the I − V to the original forward-voltage-sweep characteristic in the first quadrant of the I − V plane described earlier. This process can then be repeated, and the expected result is shown in Fig. 27.1. The figure has inversion symmetry for the high-current state about the zero bias point by virtue of the symmetrical role of the source and drain region of the device. The occurrence of at least two discrete energy levels in the AlAs/GaAs/AlAs quantum well and the lead impedance in the experimental setup will greatly influence the shape of the I−V characteristics of the two distinct current states. This ‘bow-tie’ sort of I − V characteristic also characterizes a memristor 1 . Schematic plots of the expected two distinct and stable charge states at zero 1 A memristor is the term given for a device recently fabricated at the Hewlett-Packard Laboratory showing a ’bow-tie’ I-V characteristics [Nature 453: 42-43 (2008)]. Note that no magnetic interaction is involved in the analysis in our memdiode.

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Fig. 27.1 Schematic plot of the two current state I − V curves, when only fractional recharging of the quantum well occurs during the reverse voltage sweep. In the text, this failure to fully recharge the quantum well is argued to occur if an n− -n+ -n− spacer layer is incorporated in the source and drain region of the quantum-well diode simulated by JB. This would result in a fractional recharging point shown here. The realization of one of the two distinct states at zero bias is thus seen to be history dependent. Note also that the shape of the high-current curve has inversion symmetry about the zero bias point by virtue of the symmetry in the interchange of the role of the source and the drain regions, whereas the shape of the low-current curve has an intrinsic asymmetry about the zero bias point. The result shown in this figure is in agreement with the experimental results observed by GTN, including the presence of a narrow NDR of the low-current state in the first quadrant and proximity of the bias points for the current peaks of the high and low current states (note that the first and the third quadrants correspond to the third and first quadrants, respectively, in Ref. [163]).

bias are shown in Figs. 27.2(a) and 27.2(b). We also display schematically in Fig. 27.2(c) a typical result that is expected from the solution of the Wigner transport and the Poisson equations for conventional RTD, shown to possess only one charge state at zero bias in contrast to the two charge-state states in Figs. 27.2(a) and 27.2(b) of the new spacer-layer design of RTD memory. It is important to point out that the high-resistance charge state in Fig. 27.2(b) is caused by the eﬀective widening and increase in the height of the potential barrier for current flow from the source and drain regions, compared to the low-resistance charge state of Fig. 27.2(a). 27.1.3

Analysis of a RTD Memory or Memdiode

We begin our discussion with Fig. 27.3, which is a schematic diagram of the conduction-band-edge profile of the special diode structure under the influence of a bias. To a first-order approximation, we assume that the potential profile can be described by a superposition of the potential profile at zero bias, which is simply determined by the built-in potential between the n− /n+ interfaces, plus two piecewise

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Fig. 27.2 Schematic diagram of the two distinct and stable charge states at zero bias is shown in (a) and (b). In each diagram, the dotted lines indicate the doping levels, the lower solid curves including the potential barriers indicate the potential profile, and the upper solid curves are the corresponding electron density. The states (a) and (b) correspond to the solutions B and A, respectively, of the Schrodinger and Poisson equations suggested from the calculations by GTN (see Ref. [164]). The high-resistance state at (b) is a result of the failure to recharge the quantum well during the reverse bias sweep. Note the much wider and higher eﬀective potential barrier in (b) compared to (a) as a result of self-consistency. For comparison, (c) shows only one charge state at zero bias for conventional quantum-well diodes; this is a typical result at zero bias obtained from the solution of the Wigner transport and Poisson equations by Buot and Jensen (see Ref. [99]).

Fig. 27.3 Conduction-band-edge profile along the z direction of a quantum-well diode, with an n− -n+ -n− spacer layer, under a voltage bias. The relation between the quantum-well energy level, Fermi level of the electron reservoir and the quasi-Fermi level of the n+ -layer emitter is shown. Note that the n+ -layer collector is expected to become nondegenerate before the n+ -layer emitter becomes nondegenerate at a higher voltage.

linear voltage drops (using more than two piecewise voltage drops do not change the basic physics discussed here). The two piecewise voltage drops are determined by two electric fields, ER and EL , with the electric field discontinuity occurring in the middle of the quantum well. These two important values of the electric fields are related to the quantum-well charge per unit area, Qw , through the Poisson

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equation which yields ER − EL =

Qw

.

(27.1)

To account for the n+ -layer screening, we approximate the potential drop induced by the applied bias to start from within the n+ emitter, i.e., region II in Fig. 27.3 which acts as the emitter. Then φ can be written in terms of Qw , and bias voltage V as eφ =

1+η Qw /C eV + , 2+η 2+η

(27.2)

where c and a may be functions of the bias voltage, V , in c η= , C= . a + b + 12 w a + b + 12 w + c We assume that the quantum-well energy level is sharply peaked at Ew , the quantization energy in the z direction. Then tunneling occurs for values of kz which satisfies the following equation: |2 kz2 = eφ + Ew − eV. 2m∗

(27.3)

From Eqs. (27.1) and (27.2) we obtain the expressions for Qw , in terms of kz2 , V , and Ew . This is given by Qw =

2+η C e

|2 kz2 − Ew + eV 2m∗

.

(27.4)

Equation (27.3) maybe viewed as the self-consistency condition for Qw since it is determined from the electrostatic feedback of Qw on the values of kz2 . The quantum-well charge Qw is determined independently by the quantum transport equation. At steady state, we simply describe the exchange of charge carriers between regions I, II, III, IV, and V in Fig. 27.3 in terms of transition rates. We estimate that the occupancy of region I is hardly disturbed in all useful biasing conditions, since it is a reservoir of large number of particles. Clearly, the electron population in region V , also a large particle reservoir, will be undisturbed well beyond the Debye length from the n− /n+ interface. However, the electron population in the n+ layer of region II (which is acting as the emitter) will be disturbed, being governed by the transport rate equation. Therefore, we denote the quasi-Fermi level in region II as EF diﬀerent from that of the particle reservoirs. The reason for this is that, as depicted in Fig. 27.3, as one increases the applied voltage most of the voltage drop occurs mainly between the emitter and the drainside particle reservoir, while leaving the potential hump between the emitter and the nearby particle reservoir practically undisturbed. This promotes the imbalance between EF and EF as the emitter-collector current increases. This condition persists until the voltage reaches the resonance condition which creates a significant imbalance in the rate of supply of electrons from the particle reservoir and the rate of outgoing electrons from the emitter to the collector. This mechanism leads to

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the condition EF < EF in Fig. 27.3. In fact it will be argued in what follows that the change of the electron population in regions II and IV from “degenerate” electron population (or unlimited supply of electrons) to “nondegenerate” electron population (or limited supply of electrons) as the bias increases is the basic physical mechanism behind the occurrence of the “fractional recharging” point illustrated in Fig. 27.1 during the decreasing bias sweep in the first quadrant of the I − V plot. A simplified analysis of the rate equation for the occupancy of the emitter (region II), quantum well (region III), and collector (region IV ) yields the rate of decay of the stored charge in the quantum well as equal to the rate of supply of electrons from the emitter. In the relaxation-time approximation, this is simply given as Qw (kz ) =

τd ρ (kz ) , τe

(27.5)

where 1 1 1 = + , τd τe τc

(27.6)

1 τc

is the eﬀective rate of decay of Qw into unoccupied collector states, τ1e is the eﬀective rate of decay of Qw into unoccupied emitter states which may be assumed to be equal to the rate of supply of electrons from the emitter to the quantum well, 1 τ e . ρ (kz ) is the total supply of the number of electrons/unit area of the emitter with wave vector kz . Including spin degeneracy, this is given by ρ (kz ) =

m∗ |2 kz2 ln 1 + exp β E − F π|2 β 2m∗

τd Θ kz2 . τe

(27.7)

Therefore, the rate equation yields an independent expression for Qw as a function of kz2 as Qw =

em∗ |2 kz2 ln 1 + exp β EF − 2 π| β 2m∗

τd Θ kz2 . τe

(27.8)

In the above, Θ kz2 (is the unit step function of kz2 ) appears since supply of electrons from the emitter is forbidden for kz2 less than zero. For the purpose of this paper, it is important to point out that Eq. (27.8) for degenerate electron population at the emitter is of the form QD w (kz ) =

e τ d 2 kF2 − kz2 k Θ kz2 Θ kF2 − kz2 . 2π τ e F kF2

(27.9)

On the other hand, for the nondegenerate case Eq. (27.8) is QND w (kz ) =

em∗ |2 kz2 exp −β |EF | + 2 π| β 2m∗

τd Θ kz2 , τe

(27.10)

where the eﬀective quasi-Fermi level EF in the above equation lies below the conduction band edge of the emitter. Indeed, Eqs. (27.9) and (27.10) show completely diﬀerent behavior, linear versus exponential, and have diﬀerent maximum values as

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Fig. 27.4 Schematic plots of Eq. (27.4) for Qw vs kz2 , with V as a parameter (sloping straight lines), and of Eq. (27.8) for degenerate and nondegenerate electron population at the n+ -layer emitter. The open circles are solutions for increasing voltage sweep, and the solid circles are solutions for the corresponding decreasing voltage sweep back to zero in the first quadrant of Fig. 27.1. The transition from A to A corresponds to the transition from the low-current to the high-current state in the third quadrant of Fig. 27.1 (third quadrant corresponds to first quadrant of Ref. [163 ].)

function of kz2 . This has a profound eﬀect on the I − V characteristic for the forward and for the corresponding reverse voltage sweep, when the preceding forward voltage sweep has brought the emitter to a “nondegenerate” state. The self-consistent solution to Qw is determined simultaneously from Eqs. (27.4) and (27.8). The steady-state collector current/unit area can then be approximated by Qτ w . To plot the I − V characteristic, we need to determine Qw as a function c of the voltage bias V . The solutions are illustrated graphically in Fig. 27.4, where Eqs. (27.4) and (27.8) are plotted and the points of intersection are found. From the continuity considerations in the voltage-sweep measurements, the intersection with open circles are the solution for Qw for the increasing voltage sweep, assumed to be the beginning of the measurements (corresponding to the first quadrant of the I − V in Fig. 27.1). The intersections denoted by the solid circles are the solutions for Qw , for the following reverse voltage sweep after the increasing voltage sweep has been extended beyond Vb2 resulting in the onset of the nondegeneracy of the emitter (region II) and collector (region IV ). The presence of two charge states down to zero bias is evident in the figure. The bistability transition from A to A in Fig. 27.4 corresponds to the transition from the low-current to the high-current state of Fig. 27.1 in the third quadrant of the I − V , and also indicates the onset of degeneracy of the emitter, presumably due to the decrease of the eﬀectiveness of the n− -layer potential barrier whereby the particle reservoir takes over initially as the emitter and eventually relaying the function to a degenerate n+ layer.

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27.1.4

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Two-State I-V and Two Charge States

Thus, it is clear that the physical mechanism for the failure to recharge the quantum well during the reverse voltage sweep is due to the limited supply or altered distribution of electrons from the emitter because the n− —layer potential hump creates a barrier to the free supply of electrons from the large particle reservoir. Therefore as a memory device at zero bias, one expects the retention period of the low charge state to be a strong function of the n− —layer hump-potential width. Furthermore, the degree of nondegeneracy or limited supply of electrons in the decreasing voltage sweep is unique for a particular n− -n+ -n− spacer-layer design. The use of a n− -n+ -n− structure in the source and conventional structure in the drain entails a loss of the symmetry depicted in Fig. 27.1, and is expected to only produce two state I − V in the first quadrant, whereas the use of n− -n+ -n− in the drain would only lead to two stable I − V in the third quadrant (drain takes up the role of the emitter). It should be pointed out that for V > Vb2 , the current is generally no longer controlled by the Qw of the first energy level. Two possibilities can occur depending on the width of the quantum well, the height of the double barrier, and the eﬀective mass of the electrons: (1) the current becomes a tunneling current through a single potential barrier to the unoccupied continuum states in the drain contact, or (2) the current begins to be controlled by a diﬀerent quantum-well charge when higher quantum-well energy level becomes operative, in which case similar arguments can be used to discuss the resulting I − V characteristics for these higher voltages and one expects similar I − V behavior at this range of voltages, although a much broader excited energy level may preclude the occurrence of a plateaulike behavior. The essential physics of the theory deals with the altered distribution of the electron supply from the emitter which correspondingly creates an altered condition of the device during the decreasing voltage sweep portion of the closed-loop voltage sweep. Depending of course on the exact nature of the altered distribution of the electron supply in the emitter, coupled with the dynamics of the alignment of the electron distribution having longitudinal momentum at the emitter with the energy levels of the quantum well, as well as the possibility of more than one energy level that may exist in the quantum well, many nonlinear I −V shapes are possible, while retaining the physical picture of a memory device. Aside from the relaxation-time approximation and other simplifications, the simplified analysis given here does not take into account the time-dependent dynamics of quantum transport, particularly as the exact treatment relates to the dynamics (e.g., high-frequency current and charge oscillations leading to the plateaulike behavior of the I − V ) of the bistability in the increasing voltage sweep, i.e., in the dynamical transition from Qmax to zero charge for a bias slightly larger than Vb2 in Fig. 27.4, the dynamical transition from zero charge to the fractional recharging point in the following decreasing voltage sweep, as well as the dynamical transition of quantum-well charge from A to A (occurring in the third quadrant of the I − V in Fig. 27.1 which corresponds to the first quadrant of Ref. [163]) in the reverse (negative) voltage sweep. The finite width of the distribution of energy states in the quantum-well energy level, due to a finite barrier height and high temperature,

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is also not accounted for. These additional considerations will significantly increase the stored charge and, coupled with the higher velocities and larger transmission coeﬃcients, the resulting current when the degenerate Fermi level of the emitter coincide with the peak of the distribution of the quantum-well energy states. This will also lead to the replacement of the unit step function in Eq. (27.7) with an exponentially decreasing function for kz2 < 0, when only the first energy level still controls the current. All these, however, will not significantly change the physics of the RTD memory device discussed here. We expect all the features of the I − V shown in Fig. 27.1 holds at low temperatures and if only one energy level in the quantum well is operative in the device.

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Chapter 28

RTD as a Tera-Herz Source

28.1

Type I RTD High-Frequency Operation

The current peak during forward bias is the result of the passage of the quantum well (QW) discrete energy level into the forbidden energy region of the emitter. This forbidden region does not necessarily correspond to the energy gap between the conduction and valence bands of the emitter. This forbidden region maybe created above the emitter conduction band edge by virtue of the quantization of the supply electronic states by the confining emitter quantum well (EQW). This is illustrated in the conduction energy-band edge (EBE) diagram of Fig. 28.1(a), which shows a triangular EQW. The passage of the QW energy level into the forbidden region of the emitter creates a sudden drop of the current across the device, producing a characteristic sharp current peak. As the electrons are build up in the emitter, the interference of the reflected electrons and the incoming electrons eﬀectively broadens the EQW by virtue of the selfconsistency of charge and potential. The broadening is due to the redistribution of the electrons with some regions becoming positive (deficit of electrons) and some regions negative (excess of electrons) in the emitter. This EQW renormalization broadens the EQW and the lowers the quantized energy levels in the emitter towards the conduction band edge. With this EQW broadening, the alignment of the QW discrete energy level with occupied states in the emitter is consequently restored yielding high transmission coeﬃcients and larger currents. This is depicted in Fig. 28.1(b). This feedback is basically a catalytic process since the quantum well charge, through the selfconsistent potential, helps in restoring the QW energy level alignment with the occupied states in the emitter. The resulting emitter discharge and the selfconsistency of charge and potential restore the emitter potential profile which produces 2-D quantization, i.e., back to the situation depicted by Fig. 28.1(a). The process therefore oscillates between that of Figs. 28.1(a) and 28.1(b), and the average is responsible for the plateau-like behavior above the valley-current minimum. The driving source of this oscillatory condition is the build up and redistribution of charge resulting in significant ripple eﬀect due to the interference of the reflected and incoming electrons. We identify this driving source in terms of the total charge buildup, Q, at the emitter in time duration, RC, where R is the access resistance

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Fig. 28.1 (a) Emitter 2D quantization results in premature alignment of the QW energy level with the forbidden region of the emitter. The electrons accumulate in the emitter with density structure and a broadening of the EQW caused by interference of reflected and incoming electrons. The buildup time takes τ B to achieve the condition in (b). (b) Realignment with occupied states causes rapid depletion of the builtup charge in time τ L , after which condition (a) is restored. (c) Device parameters used in the calculations. (d) Equivalent circuit model, where i(v) [Nw (v)] is the current (charge) of an ideal RTD for a voltage drop v across the negative resistor. The presence of Nw (v) indicates the existence of a capacitance across the negative resistor.

and C is the RTD capacitance. Therefore, Q/RC measures the maximum buildup rate of supply electrons at the emitter in the absence of tunneling to the QW. In the presence of tunneling, one needs to solve the proper coupled rate equations discussed below. We expect the maximum values of Q/RC and oscillation amplitude just after the current peak, in the plateau region as depicted in Figs. 28.1(a)-28.1(b). This is because there is a considerable broadening or renormalization of the EQW in going from Figs. 28.1(a) to 28.1(b), i.e., in bringing the allowed EQW allowed states in line with the QW energy level. On the other hand, well within the plateau region, there is only a further broadening of the EQW and hence the amplitude of oscillation will become smaller as the drain bias is further increased. This is indicated in Figs. 28.2(a)-28.2(b). The presence of Nw (v) indicates the existence of a capacitance across the negative resistor in Fig. 28.1(d)1 . 1 This small capacitance was discussed in detail by E. A. Poltoratsky and G. S. Rychkov, Nanotechnology 12, 556 (2001).

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Fig. 28.2 A smaller amplitude of the oscillation occurs within the plateau region as the QW energy level continues to shift downward with an increase in applied bias. Hence the EQW broadening charge Q in Q/RC also decreases with an increase in applied voltage in the plateau region.

28.2

Type II RTD High-Frequency Operation

Another interesting but entirely diﬀerent physical mechanism inducing selfoscillation occurs in type II RTD. In contrast to that in type I RTD, this oscillation occurs before the resonant current peak. The modulation of the position of the discrete energy level in the quantum well relative to the occupied levels in the emitter is driven by the dynamics of trapped holes in the barrier. The trapped holes serve as control charge (similar to the base charge of a bipolar transistor) for modulating the tunneling electron current from the emitter to the drain. The self-oscillation of the type II RTD is brought about by the oscillatory build-up and decay of the polarization pairing between the conduction-band electrons in the quantum well and the trapped holes in the barrier2 We are therefore interested in Coulomb-correlated pairing dynamics between the conduction electron in the quantum well and trapped hole in the barrier, with twobody correlated motion confined transverse to the heterostructure growth direction. Since the pairing is not between the created electron and hole, we refer to this polarization pair as a ’duon’ to distinguish from excitons and electron-hole Cooper pairs. The bottom-of-conduction and top-of-valence EBE diagrams of type I RTD are depicted in Fig. 28.3(a).The bottom-of-conduction and top-of-valence EBE diagrams of type II RTD are depicted in Fig. 28.3(b). The staggered band-edge alignment shown in the figure can be realized for example by using InAs/AlSb heterojunctions, Fig. 28.3(a), or InAs/AlGaSb heterojunctions, Fig. 28.3(b). Due to its inability to confine holes in the barrier, the structure of Fig. 28.3(a) yields a type-I RTD I −V characteristic as measured experimentally. Stronger hole confinement can be obtained by using InAs/AlGaSb heterojunction, 2 This

was first shown by Buot (1997, 1998) in qualitiative agreement with available experiments.

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Fig. 28.3 (a) EBE alignment of RTD using InAs/AlSb heterojunction. Band-edge oﬀsets are indicated in electron volts. εn and εw are the discrete energy levels in the barrier and quantum well respectively. (b) EBE alignment of RTD using InAs/AlGaSb heterojunction

Fig. 28.3(b), which yields a new I − V bistability before the current peak. Unless otherwise specified in what follows, quantum well refers to the conduction band edge and conduction-band electrons. When the localized valence-band electrons confined in the AlGaSb barrier see the available states in the drain region under bias, these electrons tunnel to the drain leaving behind quantized holes. The Zener transition is initiated when a matching of the discrete level, εn , of the right barrier with the unoccupied conduction-band states in the drain first occurs, at (kzD )2 ≥ (kFD )2 in Fig. 28.4, with the ‘>’ sign holding for indirect band-gap Zener tunneling. The Zener tunneled electrons, deposited at the spacer layer, are acted on by the field of the depletion region and quickly recombine at the drain contact. In eﬀect this process creates a polarization between the barrier and the quantum-well, which establishes a high-field domain in this region. The result is a consequent redistribution of the voltage drop across the device. The time-dependent dynamics of the hole charging and discharging that follows is dictated by the self-consistency of the potential. At higher bias, this is described by Fig. 28.5. As hole charging occurs, Fig. 28.5(1), the polarization between the barrier and the quantum-well induced by the trapped hole charge [Fig. 28.5(2)] creates a high-field domain tending to lower the residual potential drop between the contact and the right barrier. Owing to self-consistency of the potential, further polarization leads to the ‘switching’ of

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Fig. 28.4 Average EBE profiles with applied drain potential eV . EF is the Fermi level. The lower right hand corner indicates the unoccupied transverse and longitudinal momentum states outside the Fermi sphere defined by EF of the drain.

the intravalence-band tunneling of the trapped holes from the barrier towards the quantum well region [Fig. 28.5(3)]. When the situation shown in Fig. 28.5(3) is reached, other possible mechanisms for hole discharging may also occur, namely, thermal activation of the valence electrons in the continuum to recombine with localized holes, or loss of any bound hole states in the barrier. The bound hole leakage can be approximated by tunneling through a triangular potential barrier, which is likely to have a smaller barrier height than that of electron Zener tunneling (viewed as a tunneling through a potential barrier). Any or all of the hole-leakage processes mentioned above will also immediately restore the high field between the barrier edge and the right contact. The situation shown in Fig. 28.5(1) is thus revisited, after which the process repeats. Oscillations of the hole charging of the AlGaSb barrier can occur in the THz range, by virtue of the nanometric dimensional features of the device. The oscillatory process limits the average amount of hole charge that can be trapped in the barrier as a function of drain bias. The direct interband recombination process in the barrier is not considered since it can not compete with the conduction-band electron tunneling process. There is not enough time for direct interband recombination to take place, since the velocity of the conduction electrons at the barrier is quite large, owing to the small probability of being inside the barrier region.

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Fig. 28.5 EBE diagram showing the mechnism of oscillation ot trapped hole charge in the barrier: (1) e − h generation by electron Zener tunneling, (2) duon generation is through an autocatalytic process similar to the stimulated emission of radiation in lasers. (3) mechanisms for hole discharging mentioned in the text.

28.3

Regional Block Renormalization: Type-I RTD

The large-scale simulation results of type-I RTD have provided insights into the most relevant macroscopic dynamical variables for characterizing the autonomous current oscillations and bistability in these devices. From our observations on the physically meaningful trends in the redistribution of electron density as obtained numerically by all QDFTE large-scale simulations for type-I RTD [for example, refer to Fig. 12(d) of Ref. [99, 132], Fig. 10 of Ref. [165], and Figs. 4-5 of Ref. [127]], we are justified in making the following assumptions: (a) emitter and QW electron density are strongly coupled, (b) Barrier regions population density are constant at very small values, (c) Drain region electron density is approximately constant, this region basically functioning as a sink, and (d) Charge neutrality for the whole device is obeyed. By virtue of the these observations, the regional kinetic rate equations (RKRE) for the conduction electron for type-I RTD now reads ∂Nec = J1c − J2c , ∂t

(28.1)

∂NbcL = J2c − J3c = 0, ∂t

(28.2)

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Fig. 28.6 Schematic of the regional division of type-I RTD. The boundary for J1c is arbitrarily aligned with the emitter doping boundary. Strictly speaking, the location for this boundary is governed by the criterion that the incremental time-dependent density maximum interchange its location between the emitter region and quantum well region.

∂Nwc = J2c − J4c , ∂t

(28.3)

∂NbcR = J4c − J5c = 0, ∂t

(28.4)

∂Ndc = J5c − J6c ∂t

(28.5)

0,

where Jic ’s indicate current densities in the conduction band at the heterostructure junctions and proximity of device N + -doping boundaries, Fig. 28.6. 28.3.1

Estimation of J2c and J1c

We make the reasonable assumption that J1c is approximately independent of the emitter and QW densities and depends solely on the applied voltage, we denote this current as the driving current G, J1c = GI .

(28.6)

As our first approximation for J2c , we note that this is the leakage current from the emitter to the quantum well. We thus write J2c =

Nec , τe

(28.7)

where τ1e is the tunneling probability rate from the emitter to the quantum well. In order to estimate τ1e , we invoke the many-body density functional theory [166], [167, 168], namely, the density in the emitter is a unique functional of the emitter potential/electric field, while the density of the quantum well is also a unique functional

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of the quantum well potential/electric field. Since τ1e is determined by the square of the tunneling matrix element between the emitter and quantum well, we make ˜ ec Nwc , where ∆ ˜ is serves as a parameter the simplest assumption that τ1e = ∆N c approximately independent of Ne and Nwc . Thus J2c = 28.3.2

Nec ˜ ec2 Nwc = ∆N τe

(28.8)

Elimination of Fast-Relaxing Variable for Type-I RTD

For the type-I RTD, Ndc is the fast-relaxing variable since this is located in the drain region which basically acts as an electron sink. Thus Ndc can be eliminated by the well-known technique of adiabatic elimination of fast-relaxing variable [169]. This is done as follows. Let Dd be the total density of matching states in the drain commensurate with the states in the quantum well. Hence (Dd − Ndc ) is the remaining available matching states in the drain for the quantum-well electrons to tunnel through. The generation rate for Ndc is proportional to the product of (Dd − Ndc ) and Nwc , so we have Dd = density of matching states in the drain, (Dd − Ndc ) = the remaining available matching states in the drain, ∂Ndc = λ(Dd − Ndc )Nwc − γ d Ndc , ∂t

(28.9)

where λ is the quantum-well leakage-rate parameter, and γ d is the decay rate of Ndc . By virtue of the drain acting as an electron sink, the relaxation of Ndc is the fastest process. By the adiabatic elimination of fast variable technique, we can let ∂Ndc ∂t ⇒ 0. We thus obtain λ(Dd − Ndc )Nwc =

λDd Nwc 1+

c Nw

λ

γd

=

αNwc , 1 + βNwc

(28.10)

where α = λDd ,

(28.11)

λ . γd

(28.12)

β=

This expression is precisely the leakage current, J4c , from the quantum well, since there is no accumulation in the right barrier, J4c =

αNwc . 1 + βNwc

(28.13)

Note for ideal electron sink, γ ⇒ ∞ and β ⇒ 0, which leads to J4c ⇒ λDd Nwc which means there are always available matching states equal to the relevant density of

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states Dd at the drain for the leakage current to occur from the quantum well, with maximum probability-leakage rate given by λDd . Summarizing, we have the Regional Kinetic Rate Equations for type-I RTD given by the following equations: ∂Nec ˜ ec2 Nwc , = GI − ∆N ∂t

(28.14)

c ∂Nwc ˜ c2 N c − αNw . = ∆N e w ∂t 1 + βNwc

(28.15)

The solutions to the above RKRE for type-I RTD have discussed in detail in Ref. [165], which agree with microscopic simulations and experimental observations [165, 127]. 28.4

Regional Block Renormalization: Type-II RTD

The RKRE for the conduction electron for type-II RTD now reads ∂Nec = J1c − J2c , ∂t

(28.16)

∂NbcL = J2c − J3c = 0, ∂t

(28.17)

∂Nwc = J2c − J4c , ∂t

(28.18)

∂NbcR = J4c − J5c = 0, ∂t

(28.19)

∂Ndc = J5c − J6c = 0, ∂t

(28.20)

NbTR − Nwc = NB ,

(28.21)

Nec = NB .

(28.22)

Noting in Fig. 28.7 that we may neglect any accumulation of fast-decaying holes in the emitter region of the filled valence band, in the left barrier, in quantum well region, and in the drain region which simply acts as a sink, the RKRE for the valence-band holes for type-II RTD now reads,. ∂Nev = J1v − J2v ∂t

0,

(28.23)

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Fig. 28.7 Schematic of the regional division of type-II RTD. The same considerations as in Fig. 28.6 concerning the boundary for incoming current J1 also holds here.

∂NbvL = J2v − J3v = 0, ∂t

(28.24)

∂Nwv = J2v − J4v ∂t

(28.25)

0,

∂NbvR = J4v − J5v = 0, due to hole generation and decay process, ∂t

(28.26)

∂Ndv = J5c − J6c = 0, by adiabatic elimination of fast variable. ∂t

(28.27)

We can approximate the electric field within the quantum well to be dominated by the dipolar field due to electrons, Nwc , in the quantum well and an equal number of holes in the right barrier, we indicate this type of pairing between electrons in the quantum well and holes in the right barrier as P , referred to as duons in the literature Ref. [170]. Indicating the total number of holes in the right barrier as NbTR , we have NbTR − Nwc = NB ,

(28.28)

where NB is the number of unpaired holes in the right barrier. In the bias region of interest, i.e., before the resonant current peak for type-II RTD, we can assume that P = Nwc

(28.29)

since the production of Nwc in this bias range is basically induced by the presence of the ’polarization’ P , this entails a catalytic sense to the generation of P . Moreover,

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in this bias range we may for simplicity impose the overall neutrality condition, namely, Nec = NB .

(28.30)

Thus, the new independent variables that determine the time-dependent behavior of type-II RTD in the bias range of interest are P and NB . Again, we can assume that J1c is not a function of the variables P and NB , and designate this as J1c = GII . Therefore, we end up with RKREs with 2-D variables, P and NB , and 2-D ˜ and GII , for type-II RTD similar to type-I RTD, parameters, ∆ ∂Nec ˜ B2 P, = GII − ∆N ∂t

(28.31)

∂P ˜ B2 P − αP . = ∆N ∂t 1 + βP

(28.32)

The analytical solutions to RKREs for type-II, involving limit-cycle solutions, have been discussed in detail in Ref.[170, 171], whose results favorably agree with experiments. The equivalent circuit for type-II RTD oscillator has been suggested in Fig. 10 of Ref. [170]. Thus, we have provided a clearer foundation and justification of NCKRE from first principle QDFTE. In the following section, a better understanding of the ˜ can be obtained by treating the quantized energy level of the emitter parameter ∆ and discrete energy level of the QW as a tunnel-coupled two site-states quantum system subject to dielectric feedback. 28.5

Two Sites Bloch-Equation ‘Instanton’ Approach

In this section we discuss the derivation of NCKRE by considering a two-site tunneling problem using density-matrix techniques. 28.5.1

Type-I RTD

By considering the problem as a tunneling problem between two energy states in the emitter and quantum-well, respectively, we are lead to the following Bloch equations ρ˙ ee = Ge − ρ˙ ww =

1 [ρ w| HI (t) |e − e| HI (t) |w ρwe ] , i| ew

1 [ρ w| HI (t) |e − e| HI (t) |w ρwe ] − γρww , i| ew

(28.33)

(28.34)

ρ˙ ew =

1 −i|γ dRT D ρew + e| HI (t) |w (ρww − ρee ) , i|

(28.35)

ρ˙ we =

1 −i|γ dRT D ρwe + w| HI (t) |e (ρee − ρww ) , i|

(28.36)

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where we have included in the rate equation for the oﬀ-diagonal density-matrix elements the dielectric relaxation-rate parameter, γ dRT D , to account for the selfconsistency of the potential at resonance. In principle, this should also aﬀect the rate of change of the diagonal density-matrix elements. However, without changing the physics, we will assume that this dielectric relaxation eﬀects is absorbed in the second term in the rate equation for ρ˙ ee and in the decay rate γ in the equation for ρ˙ ww . This relaxation essentially describes the selfconsistent response of the potential due to the redistribution of electrons between the emitter and quantum well, which causes misalignment of the energy levels leading to a contribution giving the dissipation of the emitter and quantum-well correlations or dissipation of the oﬀ-diagonal terms of the density matrix, ρew and ρwe . 28.5.1.1

Tunneling Matrix Elements

For the tunneling matrix element w| HI (t) |e , we based our calculation on the time-independent tight-binding interaction between two potential wells given by  b  1R |ω o 1 exp  p idq  , (28.37) w| HI (0) |e 2π | b1L

which makes use of the imaginary coordinate inside the barrier. Note that p = 2mr (E − Vb (q)) is the classical expression calculated inside the barrier in Eq. (28.37). This yields the familiar WKB results for the tight-binding tunneling matrix element   b1R 1 |ω o exp − 2mr (Vb (q) − E) dq  , (28.38) w| HI (0) |e = 2π | b1L

where for RTD, Vb (q) is the barrier-potential function and the the integral is over the left barrier region, mr is the reduced mass, and ωo is the electron frequency of hitting the barrier from the emitter side. Indeed, several tunneling calculations has been done by extending the classical particle trajectory inside the barrier and using complex time-coordinate in this region [172, 173]. In order to bring the physics of autonomous current oscillation owing to dielectric-relaxation feedback of a selfconsistent potential, we write the conduction electron Hamiltonian in the eﬀective mass approximation by introducing a time-dependent perturbation of the electric field as H=

p2 + V (q) − eF (t) · q, 2m

(28.39)

where eF (t) · q describes the induced or autonomous time-dependent perturbation due to dielectric relaxation feedback leading to autonomous current oscillations. If one make the transformation in phase space, i.e., (p, q) ⇐⇒ (p , q ), as follows p = po +

eF (t ) dt ,

(28.40)

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q = qo + ξ (t ) ,

(28.41)

where e is the unit charge, po = p, qo = q, and ξ˙ (t ) =

t

eF (t ) dt , it was

shown by Buot and Rajagopal [151] that, in accurate description of quantum dynamics in phase space (domain of classical mechanics), the resulting Wigner QDFTE for type-I RTD has exactly the same form as the QDFTE in the absence of the induced time-dependent perturbation, except for the presence of the driving-force term, ξ (t ), in the argument of the potential, i.e., V (q) ⇐⇒ V (q − ξ (t )). Assuming a harmonically-induced perturbation, F (t ) = eF cos (Ωt ), which is a good approximation in the emitter and left barrier regions, then t

F (t ) dt = to

eF (sin Ωt − sin Ωto ) . Ω

(28.42)

This yields the transformed phase space as eF (sin Ωt − sin Ωto ) , Ω

(28.43)

eF eF sin Ωto (t − to ) − (cos Ωt − cos Ωto ) . mΩ mΩ

(28.44)

p = po +

q = qo + po −

Equation (28.37), using imaginary coordinate-time inside the barrier [172, 174, ] 173 , can be generalized in the presence of the oscillating barrier. For our purpose, it is convenient to use the above transformation of coordinate and momentum and write the transformed tunneling matrix elements as [in what follows, let to = 0 at q = b1L ]  b  1R | ωo 1 exp  w| HI (t) |e = p idq  . 2π |

(28.45)

b1L

Upon substituting the expressions in Eqs. (28.43)-(28.44), we have

w| HI (t) |e =



 | ωo exp   2π



b1R 1 |

po + b1L

× idqo + po idt −

eF Ω

(sin Ωt)

eF mΩ id (cos Ωt −

1)

 . 

(28.46)

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Rearranging the terms, we have  b  1R | ωo 1 exp  po idqo  exp w| HI (t) |e = 2π | 

× exp 

b1L

i |

× exp − × exp

b1R

b1L

1 eF d i sin Ωt | Ω



i eF cos Ωt tpo dpo + p2o dt exp − po | mΩ

i eF po 2 cos Ωt | Ω

i eF eF | Ω 2m

t−

b1R b1L

b1R b1L

sin Ωt cos Ωt 2Ω

b1R

,

(28.47)

b1L

which can be written as  b  1R | ωo i w| HI (t) |e = exp  po dqo  exp 2π | b1L

=

i eF d sin Ωt | Ω

exp

d | ωo exp − 2mr (Vb (b1L , 0) − E) 2π | i i eF d sin Ωt exp ∆S , × exp | Ω |

i ∆S |

(28.48)

where b1R

p2o dt − po

∆S = b1L

+

eF eF Ω 2m

t−

eF cos Ωt mΩ

b1R b1L

sin Ωt cos Ωt 2Ω

− po

eF cos Ωt Ω2

b1R b1L

b1R

,

(28.49)

b1L

and d = (b1R − b1L ) is the width of the left barrier of the RTD. Our simple calculation is based on the accurate representation of quantum dynamics in phase space as described by the Wigner QDFTE. Our result for the first oscillating factor in Eq. (28.48) for the tunneling matrix element diﬀers from the result obtained by integrating the Lagrangian along classical trajectory of an electron in complex time-coordinate space in the barrier region [173] by a constant phase of π2 , i.e., sin Ωt − π2 = − cos Ωt is used in Ref. [173] instead of our sin Ωt, and the expression for the remaining terms represented by ∆S also diﬀers. It is reasonable to neglect the contribution coming from ∆S in considering the time dependence of the tunneling matrix elements discussed here.

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Thus, we will take the tunneling matrix elements as d | ωo exp − 2mr (Vb (b1L , 0) − E) exp 2π | i eF d sin Ωt . = Ao exp | Ω

w| HI (t) |e =

i eF d sin Ωt | Ω (28.50)

The electron barrier potential is due to the discontinuity of the conduction band, Vb (b1L , 0) = ∆Ec . We can expand this expression for w| HI (t) |e as a Fourier series, indicating all subharmonics, which can be derived using the Jacobi-Anger identity, which yields ∞

eiz sin φ =

Jn (z) einφ ,

(28.51)

n=−∞

where the sum is over the integers n, Jn (z) being the Bessel function of the first kind, J−n (z) = (−1)n Jn (z). Therefore, we have w| HI (t) |e = Ao exp i = Ao

∞

eF d sin Ωt |Ω

Jn

n=−∞

28.5.1.2

eF d |Ω

einΩt .

(28.52)

Elimination of Oﬀ-Diagonal Elements of the Density-Matrix

Considering the Bloch equations, Eqs. (28.33)-(28.36), we can eliminate the oﬀdiagonal density-matrix elements by solving the rate equation for ρ˙ ew and ρ˙ we . Because of the induced oscillation of the tunneling matrix elements, which in principle has been shown in Eq. (28.52) to consist of several subharmonics, we expect that ρwe and ρew also oscillate as a function of time. To simplify we only consider the leading time-dependent solution for ρwe in the form ρwe = ρowe eiΩt .

(28.53)

Then we have ρ˙ we = iΩρwe =

1 −i|γ dRT D ρwe + w| HI (t) |e (ρee − ρww ) , i|

iΩ + γ dRT D ρwe =

ρwe =

i |

1 [ w| HI (t) |e (ρee − ρww )] , i|

w| HI (t) |e (ρww − ρee ) . iΩ + γ dRT D

(28.54)

(28.55)

(28.56)

By virtue of the fact that the quantum well generally goes from full to nearly empty during the oscillation we may write, starting with the QW empty at the

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initial, t = 0, which is the moment the resonance is reached, the expression, (ρww − ρee )

−ρww (t) = −

[An cos nΩ t + Bn sin (nΩ t + φ)] ,

(28.57)

n

which means that at initial time when resonance sets in, the quantum-well is initially empty, (ρww − ρee ) −ρoww , i.e., roughly the negative of the amplitude of the oscillating diﬀerence (ρww − ρee ). Since the relation holds for leading harmonics, we may now write ρwe

−

ρew

i |

i |

w| HI (t) |e . ρ iΩ + γ dRT D ww

(28.58)

e| HI (t) |w . ρ −iΩ + γ dRT D ww

(28.59)

Similarly, we have

Thus, we are able to express the oﬀ-diagonal density-matrix elements in terms of the diagonal ρww . Note that in Eqs. (28.58) and (28.59), ρww serves as an promoter of density-correlation functions, or as a catalyst for the interaction between the emitter and the QW subsystems. We believe the Bloch equation calculation presented here has the essence of a quantum theory of catalysis in chemical physics. Substituting the expressions in Eqs. (28.58) and (28.59) in Eq. (28.33) yields   e|HI (t)|w i ρww w| HI (t) |e | (−iΩ+γ d  1  RT D ) , ρ˙ ee = Ge −  (28.60)  i|  w|HI (t)|e i − e| HI (t) |w − | iΩ+γ d ρww ( RT D )

which reduces to

ρ˙ ee = Ge −

 

2γ dRT D

 Ω2 + γ d RT D

2

 2 | w| HI (t) |e |  ρ .  ww |2

(28.61)

We also have for the QW density equation, Eq. (28.34), the following expression   2  d 2γ RT D | w| HI (t) |e | (28.62) ρ˙ ww = ρ − γρww . 2  Ω2 + γ d  ww |2 RT D

The last term, γρww , which describes the current from the quantum well to the drain, can be calculated in the same manner as before, discussed in Sec. 28.3.2. Now consider the coupling term given by w| HI (t) |e =

d | ωo exp − 2π |

2mr (Vb (b1L , 0) − E) exp

i eF d sin Ωt , | Ω (28.63)

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where ω o is the electron frequency of hitting the barrier from the emitter side. The frequency of electron-barrier hits in the emitter side, ωo , can be estimated from the product of the emitter density/unit area and longitudinal velocity, vzF , i.e., ωo = nee vzF ∂ ρee EF (k) , |∂kz

(28.64)

where p = |k is the momentum and EF (k) is the Fermi energy level of the emitter measured from the bottom of the emitter band located near the right-barrier edge. Thus, we may write w|HI (t)|e =

ρee ∂k∂ z EF (k) d exp − 2π |

i eF d sin Ωt . | Ω (28.65)

2mr (Vb (b1L , 0) − E) exp

2 The expression for | w| HI (t) |e | is thus given by   2 ∂ E (k) F d 2 exp −2 2mr (Vb (b1L , 0) − E)  ρ2ee . | w| HI (t) |e | =  ∂kz 2π |

(28.66)

Substituting in the Bloch equations, we have  2  ∂ d   ∂kz EF (k) exp −2 | 2mr (Vb (b1L , 0) − E) d ρ˙ ee = Ge − 2γ RT D 2   (2π|)2 Ω2 + γ dRT D  ρ˙ ww = 2γ dRT D

Writing

   

∂ ∂kz EF

  

˜ ≡ 2γ dRT D ∆

      

2

(k) exp −2 |d (2π|)

∂ ∂kz EF

2

Ω2

2

(k)

2mr (Vb (b1L , 0) − E) +

−2 |d

exp 2

(2π|)

      

2 γ dRT D

2mr (Vb (b1L , 0) − E)

Ω2 + γ dRT D

2

we are lead to the following expressions

      

ρ2ee ρww , (28.67)

ρ2ee ρww −γρww . (28.68)       

,

(28.69)

˜ 2ee ρww , ρ˙ ee = Ge − ∆ρ

(28.70)

˜ 2ee ρww − γρww , ρ˙ ww = ∆ρ

(28.71)

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αρww where γρww = 1+βρ is obtained using the the technique of elimination of the fast ww relaxing electron densities at the drain region, as was done before [Refer to Sec. 28.3.2]. Upon going over to the symbols for the emitter and QW densities used in the renormalization approach, we are lead to basically identical expressions as in the previous rate equations. The virtue of the two site-states Bloch equation ap˜ which was emproach is that it yields an explicit expression for the parameter ∆ pirically employed in the current-continuity renormalization approach. It is essentially the dielectric feedback, through the Poisson equation, characterized here ˜ that brings about the NCKRE for the emitter and by the relaxation-parameter ∆ quantum-well electron densities.

28.5.2

Type-II RTD

The extension of the Bloch equation approach to type-II RTD for deriving the tunneling kinetic-rate equations now becomes trivial when one make the important observation, as was done before, that the resonance condition is readily reached before the conventional type-I RTD current peak owing to the induced dipole potential due to trapped holes in the second barrier, this occurs at relatively smaller drain bias. Again, we can approximate the electric field within the quantum well to be dominated by the dipolar field due to electrons, ρcww , in the quantum well and an equal number of holes in the right barrier, we indicate this type of pairing between electrons in the quantum well and holes in the right barrier as P , which was referred to as “duons" in the literature Ref. [170]. Indicating the total number of holes in the right barrier as NbTR , we have NbTR − ρcww = NB ,

(28.72)

where NB is the number of unpaired holes in the right barrier. In the bias region of interest, i.e., before the resonant current peak for type-II RTD, we can assume that P = ρcww

(28.73)

since the production of ρcww in this bias range is basically induced by the presence of the ’polarization’ P , this entails a catalytic sense to the generation of P . Moreover, in this bias range we may for simplicity impose the overall neutrality condition, namely, ρcee = NB .

(28.74)

Thus, the new independent variables that determine the time-dependent behavior of type-II RTD in the bias range of interest are P and NB . Again, we can assume that generation of ρcee is not a function of the variables P and NB , and designate this as GII . Therefore, we end up with RKREs with 2-D variables, P and NB , and ˜ and GII , for type-II RTD similar to type-I RTD, 2-D parameters, ∆ ∂Nec ˜ B2 P, = GII − ∆N ∂t

(28.75)

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∂P ˜ B2 P − αP . = ∆N ∂t 1 + βP

(28.76)

as obtained before employing the renormalization approach but now with a better ˜ understanding of the parameter ∆. 28.6

Stability Analysis

For completeness, we outline a unified stability analysis of type-I and type-II RTD THz sources. It is convenient for this purpose to unify the fundamental rate equations of type-I and type-II RTDs and write them in dimensionless form as, ∂ Π Π = ∆Q2 Π − , ∂t 1+Π

(28.77)

∂ Q = G − ∆Q2 Π, ∂t

(28.78)

where the transformation of variables which expresses the ‘duality’ of type-I and type-II RTDs is indicated as follows: Dimensionless variable Π Q ∆

Type-I RTD βNw βNe

Type-II RTD βP βNB

h β2 ∆ α Q/RC (α/β)

h β2 ∆ α G (α/β)

αt

αt

G τ

.

(28.79)

Note that ∆, α, and β are somewhat diﬀerent-valued parameters between type-I and type-II RTDs by virtue of their diﬀerent physical mechanisms. The stationary solution to the coupled rate equations, Eqs. (28.77) and (28.78) is given by, G = ∆Q2 Π =

Π . 1+Π

(28.80)

In terms of these dimensionless variables, the stationary values of Π and Q are given by, Π0 = 0

Q =

G , 1−G

1−G ∆

(28.81) 1 2

.

(28.82)

The physical situation corresponds to, 0 < G < 1.0, whereas for G > 1.0, the dynamical system under study cannot be sustained or becomes unbounded in the presence of a catalytic process . The question whether there is a nonstationary solution to our fundamental rate equations can first be answered by examining the stability of the stationary point

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in space. This is done by examining the neighborhood of the stationary point. Let us denote the coordinates of this neighborhood by, Π = Π0 + p, Q = Q0 + q.

(28.83) (28.84)

The solution for the trajectories in space about the equilibrium point is given by p q

= A1 eλ1 τ

V1p V1q

+ A2 eλ2 τ

V2p V2q

,

(28.85)

where λ1 and λ2 are the eigenvalues of the matrix defined by substituting Eqs. (28.83)-(28.84) in Eqs. (28.77)-(28.78) and retaining only linear terms in p and q [refer to Eq. (28.87) below, without the nonlinear terms]. The corresponding eigenvectors are V1 and V2 , respectively. The eigenvalues are given by, λ1,2 =

T r (M) 1 + 2 2

T r (M )2 − 4 det (M).

(28.86)

The character of the stationary point can thus be determined with the help of the invariants of the matrix (M ), namely, T r (M ) , det (M) , and D (M ) = [T r (M )]2 − 4 det (M) . The stationary point can not be a saddle point for physical reason 3 √ since det (M ) == 2G (1 − G) 2 ∆ > 0. The physical processes [165] also suggest that the stationary point can only be any one of the following cases: stable focus (T r (M ) < 0), center (T r (M ) = 0), or unstable focus (T r (M ) > 0 ). On physical grounds, we expect the limit cycle solution for uniqueness and structural stability. For the unstable focus we have to demonstrate that a limit cycle exists. The region in parameter space where the structurally stable limit cycle is possible lies in the area under the bifurcation curve (locus of T r (M ) = 0 ) in this space. 1 The trace of (M ) is given by T r (M ) = G (1 − G) − {∆/ (1 − G)} 2 . Thus,

T r (M ) > 0 implies (1 − G)3 > 4∆ . In the next section, we will employ a nonlinear perturbation technique using the method of multiple time scales with values of the parameter around T r (M) = 0. As we shall show in the following nonlinear analysis the limit cycle indeed occurs at T r (M) > 0. The amplitude and frequency of oscillation is expected to depend on the actual values of the two parameters G and ∆ in this region. For the unstable focus we have to demonstrate that a limit cycle exists. The region in parameter space where the structurally stable limit cycle solution is possible is the area under the bifurcation curve of Fig. 28.8, [locus of T r(M ) = 0]. 28.7

Numerical Results

We present the following numerical results for a point in parameter space chosen below the bifurcation curve of Fig. 28.8 to illustrate the nonlinear physics of the simple kinetic rate equations3 . 3 Numerical rsults courtesy of Thouraya M. Hajjem, Computational and Data Sciences Division, George Mason University, Fairfax, Virginia.

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Fig. 28.8

353

Plot of the bifurcation curve, T r (M) = 0, which is the curve for ∆ = ∆c .

Fig. 28.9 Phase space (Π, Q) portrait of the limit cycle, Π oscillation, Q oscillation, and Π + Q oscillation corresponding to the parameters, G = 0.31 and ∆ = 0.0505.

28.8

Perturbation Theory and Limit Cycle Solutions

Retaining nonlinear terms for p and q measured from the stationary point, the rate equations from Eqs. (28.77) and (28.78), become a matrix equation,

∂ ∂τ

p q

1

=

G (1 − G) 2G [∆/ (1 − G)] 2 1 − (1 − G) −2G [∆/ (1 − G)] 2

p q

+

Np Nq

,

(28.87)

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Fig. 28.10 (a) Surface plot of the amplitude of Π, (b) Surface plot of the frequency of Π, (c) Surface plot of the amplitude-frequency product. (All axes in arbitrary units). Similar results hold for the plots of Q, but with phase diﬀerences.

where the nonlinear components of the last term are given by, 1

N p = (1 − G)3 p2 + 2 [∆ (1 − G)] 2 pq + +

∞ n=3

∆G 2 q + ∆pq 2 (1 − G)

(−1)n (1 − G)n+1 pn , 1

N q = −2 [∆ (1 − G)] 2 pq −

(28.88)

∆G 2 q − ∆pq 2 . (1 − G)

(28.89)

The perturbation technique employed in what follows essentially transforms the above nonlinear equation into a hierarchy of solvable and simpler equations, obtained by equating coeﬃcients of powers of the smallness parameter, ε. Near T r (M ) = 0, we use as our smallness parameter the departure of ∆ from ∆c , where 4∆c = (1 − G)3 . Thus, let the smallness parameter be ε = {∆ − ∆c } /D, where D is determined from the expansion of ∆ in powers of ε. D ≈ ∆2 in the analysis that follows. G is assumed constant at fixed bias, i.e., a function only of the external bias. We make the following expansion, ∆=

∞

εj ∆j , where ∆0 = ∆c .

(28.90)

j=0

We also expand the matrix in powers of ε through direct Taylor expansion in powers of {∆ − ∆c } as, (M ) = (Mc ) + ε∆1 +∆21

∂ (M) ∂∆

2

∂ (M ) ∂∆2

∆=∆c

1 + ε2 ∆2 2

+ O ε2 . ∆=∆c

∂ (M ) ∂∆

∆=∆c

(28.91)

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Using 4∆c = (1 − G)3 , we obtain the following expressions, 1 1 −G −1 −1

(Mc ) = G (1 − G) ∂ (M ) ∂∆

(M1 ) =

(28.92)

= 2G (1 − G)−2

∆=∆c

1 ∂ 2 (M ) 2 ∂∆2

(M2 ) =

, 01 0 −1

= −2G (1 − G)−5

∆=∆c

, 01 0 −1

(28.93) .

(28.94)

We let the solution depends on time τ in a combination τ 0 = τ and τ 1 = (∆ − ∆c ) τ . Thus, instead of determining the solution in terms of τ we seek the solution as a function of τ 0 , τ 1 , and ε. This method of doing the nonlinear perturbation analysis is well-known and is often referred to as the method of multiple time scales [175]. This has the virtue that it separates the dependence of the solution into the fast and slow time scales. For limit cycle behavior, for example, we expect that the amplitude of the oscillation is only a function of the slow time scale. The left side of the rate equation can now be written as, ∂ ∂τ

p (τ 0 , τ 1 , ε) q (τ 0 , τ 1 , ε)

=⇒

∂ ∂ + (∆ − ∆c ) τ0 τ1

p (τ 0 , τ 1 , ε) q (τ 0 , τ 1 , ε)

.

(28.95)

Since the last term in Eq. (28.87) represents the nonlinear term for the solution, we adopt the following expansion, p (τ 0 , τ 1 , ε) q (τ 0 , τ 1 , ε)

=

∞

εj+1

j=0

pj (τ 0 , τ 1 ) qj (τ 0 , τ 1 )

.

(28.96)

Therefore, any finite solution will indicate that the limit cycle occurs for values of the parameter away from the critical point, T r (M) = 0, i.e., away from the bifurcation point. With Eq. (28.96), the nonlinear term in Eq. (28.87) acquires the following expansion in terms of the smallness parameter, Np Nq

= ε2

N2p N2q

+ ε3

N3p N3q

+ O ε4 ,

(28.97)

where N2p N2q

N3p N3q

p0 q0 + (G/4) q02 + (1 − G) p20 2 , = (1 − G) − p0 q0 + (G/4) q02    p0 q1 + p1 q0 + (G/2) q1 q0 + (1−G) p0 q02 4 1    2    ∆1 G ∆2 2 + (1−G) 2p q 3 q0 + 3  0 0   (1−G)   2 3   =   +2 (1 − G) p0 p1 − (1 − G) p0 . (1−G)  2  p0 q1 + p1 q0 + (G/2) q1 q0 + 4 p0 q0   1 −  2 ∆1 G ∆2 2 + (1−G) 2p q 3 q0 + 3 0 0 (1−G)

(28.98)

(28.99)

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We did not show nonlinear terms with fractional powers of ε in Eq. (28.97) associated with ∆1 in Eqs. (28.88) and (28.89), since the left hand side of the rate equation does not contain fractional powers of ε. To eliminate the occurrence of these fractional powers of ε, we have to make ∆1 = 0 in the expansion of ∆, Eq. (28.90), and also in Eqs. (28.91) and (28.99). Upon substituting all the expanded quantities in the nonlinear rate equation, Eq. (28.87), we obtain a hierarchy of simpler equations. Those arising from the first up to the third powers of ε are given below,

L0

p2 q2

L0

p0 q0

= 0,

L0

p1 q1

=

N2p (p0 , q0 ) N2q (p0 , q0 )

+ ∆2 L1

p0 q0

=

N3p (p0 , q0 , p1 , q1 ) N3q (p0 , q0 , p1 , q1 )

(28.100) ,

(28.101) ,

(28.102)

where L0 = L1 =

∂ − (Mc ) , ∂τ 0 ∂ − (M1 ) . ∂τ 1

(28.103) (28.104)

The first equation in the hierarchy is a simple eigenvalue problem, analogous to our linear-stability analysis before. The only diﬀerence is that the present eigenvalue problem has to be solved with values of the parameter at the critical point, where T r (M ) = 0 , using the matrix (Mc ). The solutions to Eqs. (28.100), (28.101), and (28.102) are given in detail in the Appendix of Ref. [165] to second order in ε, where it is shown there that limit cycle exists for ∆ < ∆c . Thus, to second order in the smallness parameter, the limit cycle solution is given as, Π Q

Π = 0 + Q0 +O

1 2

∆ − ∆c ∆2 ∆ − ∆c ∆2

p0 q0

+

∆ − ∆c ∆2

p1 q1

3 2

,

(28.105)

where we have p0 q0

|Θ (∞)| 2 cos Ωτ 1 1 1 −2G 2 cos Ωτ − (1 − G) 2 2 sin Ωτ G2 2 |Θ (∞)| cos Ωτ = , 1 − sin(Ωτ + Φ) G2 =

(28.106)

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357

and Φ = tan−1

G (1 − G)

1 2

,

(28.107)

3

1

Ω = G 2 (1 − G) 2 + Im η (∆ − ∆c ) +

Im σ 2 |Θ (∞)| (∆ − ∆c ) . ∆2

(28.108)

We also have, p1 q1

= |Θ (∞)| +

2

(1 − G) G 

|Θ (∞)|2   3G (1 − G) 

2G − 12

4 (1 − G)3 cos 2Ωτ −ω (1 + 2G) sin 2Ωτ 15 9 2 (1 − G) cos 2Ωτ 2 G− 2 − 3G +ω 8 − 3G−2G −1 sin 2Ωτ



 . 

(28.109)

We note that Eq. (28.109) also contains a time-independent term, indicating a higher-order shift of the center of the limit cycle from the stationary point. Therefore, the average value of is given by, Π Q

∆ − ∆c ∆2

(1 − G) G

=

Π0 Q0

+

=

Π0 Q0

+ higher-order corrections,

average

|Θ (∞)|2

2G − 12

+ O ε3 (28.110)

where the leading higher-order corrections come from the time-independent terms. Thus, we have demonstrated that a unique limit cycle exists away, (∆ < ∆c ), from the bifurcation point, (∆ = ∆c ), and the average value is determined by the time-independent terms. In examining the dependence of various quantities on the driving source,G, we make the assumption that (∆ − ∆c ) is approximately a constant for type-I RTD. The physical reason for this is that ∆ is a measure of the inductive delay in electricalcircuit terminology. ∆ is large for small inductive delay and small for large inductance. For larger driving source, G, which happens immediately after the currentpeak bias, ∆c is smaller and we also expect larger inductive delay at this point, and hence ∆ is also smaller. For larger values of bias in the plateau, meaning weaker driving source G, ∆ is larger and we also expect the inductive delay to be smaller, meaning ∆ is also larger. Therefore, |∆ − ∆c | is approximately constant. It is taken small enough such that the second term in Eq. (28.110) is only a very small correction to the first term, otherwise one has to include other higher order terms. Similar relation also holds for the type-II RTD.

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PART 4

General Theory of Nonequilibrium Quantum Physics

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Chapter 29

General Theory of Nonequilibrium Quantum Physics in Real Time

29.1

Introduction

The theoretical foundation of computational nanoelectronics is based on nonequilibrium quantum physics and lattice Weyl transform techniques [176]. The rationale is that functional nanodevices are essentially characterized by space-time inhomogeneity, nonlinear quantum eﬀects, and openness to input and output boundaries. Research in nanoelectronics and nanodevices, including nano-optics and nano-magnetics, may thus be viewed as nonequilibrium nanoscience and nanotechnology. Nowadays, the study of nonequilibrium nanosystems is pursued in almost all fields of technology, medicine, biology, material science, etc., and will no doubt greatly influence the scientific and technological developments of the 21st century. The emerging and intriguing fields of quantum control and quantum computing are also being vigorously pursued by virtue of its strong relevance to the study of nanosystems and ultrafast physical phenomena. Actual nanodevice implementations of physical concepts from these two-closely related disciplines will also make use of nonequilibrium-physics simulation techniques for their performance and optimization analysis. In the past, the invention of transistors and laser have ushered rapid computational and analytical advances in the study of nonequilibrium systems, mainly using semi-classical physics. However, the continued miniaturization of semiconductor transistors and lasers to nanoscale regime and the growing number of novel device concepts that are continually being proposed are beginning to usher rapid computational and theoretical developments in nonequilibrium quantum physics. The study of nonequilibrium quantum physics seems to have been initiated by Schwinger [177] in 1961. Schwinger basically introduced the concept of time contour or double time axis to diﬀerentiate forward and backward time development, in order to incorporate irreversibility, scattering, nonlinear eﬀects, and interaction with environment. In 1964, Keldysh [178] extended this idea, using the Feynman diagram technique, to statistical system very far from thermodynamic equilibrium under the action of external fields. Earlier, at about the same time as Schwinger, Kadanoﬀ and Baym [179] discussed quantum statistical mechanics using the properties of the so-called nonequilibrium Green’s functions which are the major players in the time contour technique of Schwinger and Keldysh.

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The major complexity of the time contour technique is that products of different correlation functions are determined based on the evaluation of time integral along the time contour [180]. Moreover, quantum transport equations for the distribution functions in phase-space were only given using the slowly-varying approximation [179, 181]. With the emergence of computational and theoretical nanoscience and nanoelectronics, and the study of ultrafast phenomena and devices, it became highly desirable to have a straightforward algebraic manipulation for calculating quantum transport equations and many-body corrections to diﬀerent correlation functions,1 including the nonequilibrium Green’s functions, without determining the pertinent ranges of time integral along the time contour. It was also desirable to avoid the slowly-varying approximation in deriving quantum transport equations for the distribution functions suitable for simulating the performance of high-speed heterojunction devices. These two general requirements are desirable for straightforward ease in calculations of the needed many-body corrections and for analyzing nonlinear-time-dependent performance of heterojunction nanodevices [127]. These are achieved by using (a) Liouville space quantum dynamics to bring about a completely algebraic manipulation similar to the zero-temperature manybody techniques, and (b) lattice Weyl transformation formalism [183] to avoid the slowly-varying approximation. These two developments will be treated here in more detail. First we give a fundamental theoretical foundation of nonequilibrium quantum many-body physics. An alternative and straightforward algebraic approach to nonequilibrium quantum physics can be formulated in terms of quantum Liouville space (L-space) dynamics of superoperators and super-statevector or quantum superfield theoretical techniques [184, 185, 186].2 The hallmark of the quantum superfield theory (QSFT) of nonequilibrium physics, as compared to the time-contour formalism, is its algebraic and straightforward procedure for deriving the equations for the super-correlation functions that enter in the theory, a procedure exactly paralleling that of the many-body quantum field theory at zero temperature. QSFT is a real time approach and the relevant quantum transport equations comes out to be simply the consequent equations for the components of the super-correlation functions. This is in contrast with the time-contour formalism where one has to trace the forward and backward time development along the time contour, in a rather awkward manner, to derive the relevant quantum transport equations [177, 178, 179, 180]. We demonstrate in Appendix G the power of the QSFT approach by calculating the electron self-energy due to Coulomb interaction, including pairing dynamics, to second order in the electron-electron interaction potential. Eﬀectively, what happens between the time-contour and real-time formalism is that instead of the doubling of time axis together with ordinary Hilbert (H)-space quantum-field operators in the time-contour formalism, these are replaced in the quantum superfield theoretical approach by a single real-time axis and a doubling 1 Indeed, the eﬀects of higher-order many-body correlations between quasiparticles have been experimentally shown to dominate the nonlinear optical response of semiconductors [182]. 2 Portions of Part 4 also appeared in the author’s articles in Handbook of Theoretical and Computational Nanotechnology, Vol. 1, edited by M. Rieth and W. Schommers (American Scientific Publishers, Stevenson Ranch, CA, 2006), and J. Comp. Theor. Nanoscience 4, 1037 (2007).

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of quantum field operators [184, 185, 186]. In QSFT, the ordinary quantum field operators become superoperators. It is important to realize that the doubling of the time axis in the time-contour approach or the doubling of the quantum field operators in the quantum superfield theoretical approach serves the same purpose, namely, this doubling diﬀerentiates the quantum dynamics of forward and backward time development, i.e., both formalisms incorporate scattering and interaction with applied driving fields and environment. Indeed, the doubling of the quantum-field operators is quite natural in the QSFT approach. In the QSFT approach the von Neumann density-matrix evolution equation of quantum statistical dynamics in Hilbert space (H-space) becomes a superstatevector quantum dynamical equation in quantum Liouville space (L-space).

29.2

Quantum Dynamics in Liouville Space

The von Neumann density-matrix evolution equation of quantum statistical dynamics in H-space becomes a super-statevector quantum dynamical equation in L-space. Thus, the familiar von Neumann density-matrix operator equation in H-space given by i|

∂ ρ (t) = [H, ρ] ∂t

(29.1)

becomes a super-Schrödinger equation for the super-statevector in L-space expressed as i|

∂ |ρ (t) ∂t

= L |ρ (t) .

(29.2)

In Eq. (29.1), ρ (t) is the density-matrix operator for the whole many-body system in H-space, whereas in Eq. (29.2) |ρ (t) is the corresponding super-statevector in L-space. The superoperator L corresponds to the commutator [H, ρ] of Eq. (29.1), and is referred to in this paper as the Liouvillian. Thus, we may write the Liouvillian L as L = H − H,

(29.3)

which defines H and H. These are superoperators with the property that H |ρ (t) = |Hρ (t) and H |ρ (t) = ρ (t) H† . These relations are valid for fermions and bosons. For number-conserving fermion operator H, H = H† . From Eq. (29.3), we can make the assertion that any H-space operator basically becomes the “hat” and “tilde” superoperators in L-space. This is often referred to as the doubling in the number of the degrees of freedom; this doubling arises due to the need to diﬀerentiate the quantum dynamics for ‘forward’ and ‘backward’ time ˜ which is an arbitrary product of development. In general, any superoperator A, quantum superfield operators, will be denoted as A˜ to take care of the commutation properties of the quantum superfield operators. This will become clear in Sec. 29.2.

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The nature of these superoperators can be analyzed using a set of many-body basis states [185]. If a set {|n } is an orthonormal basis in H-space in the number representation of many-body basis states, then the super-statevector basis are constructed from the direct product space and is given by the set { ||m n| }, which represents the orthonormal super-statevector basis in L-space |n

m || ||m n|

= δ mm δ nn .

This leads to the corresponding super-statevector expansion in L-space |A

= m,n

[ |n m|| |A ] ||m n| ,

(29.4)

where |n m|| |A = m| A |n is the corresponding expansion coeﬃcient, which is identical to the expansion coeﬃcient of the operator A in ordinary H-space A= m,n

m| A |n |m n| .

In general, the super-scalar product of two super-statevectors |A by, A| |B

= T rA† B,

and |B

is given (29.5)

where the right-hand side (RHS) corresponds to taking the trace in H-space. The norm of any super-statevector |A is thus defined as |A| =

A| |A

1 2

= T rA† A

1 2

.

(29.6)

A unit super-statevector is here defined as |1

= m

||m m| .

(29.7)

The trace of any operator A in H-space can thus be written as a super-dot product of the unit super-statevector with the super-statevector corresponding to A, Tr A =

1| |A .

(29.8)

In particular, T r ρA = T r Aρ = 1| |ρA = 1| A˜† |ρ = 1| |Aρ = 1| Aˆ |ρ . Since any H-space operator consists of arbitrary number of annihilation and creation quantum-field operators, we need to determine the property of the corresponding super-annihilation and super-creation operators to be able construct a quantum superfield theory of nonequilibrium systems of elementary excitations. This is discussed in the next section. Doubling of H-Space Operators in L-space The Liouvillian superoperator defined above also induces the “doubling” of the H-space many-body quantum field operators,ψ and ψ† , to become the “hat” and

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†

“tilde” , ψ (ψ) and ψ ψ , superoperators in L-space. They are easily defined for the “hat” superoperators as follows, †

ψ ||m n|

= ψ† |m n|

ψ ||m n|

= ψ |m n|

,

(29.9)

.

(29.10)

Therefore, the “hat” quantum field superoperators obey the usual equal-time commutation (anticommutation) relations for bosons (fermions). The nature of the tilde quantum field superoperators can be determined by imposing the H-space operational constraint implied by the Liouvillian superoperator in Eq. (29.2), and the requirement that the “hat” and “tilde” superoperators commute (anticommute) for bosons (fermions). Then, the time evolution superoperator of Eq. (29.2) factorizes into a product of the “hat” time evolution superoperator and the “tilde” time evolution superoperator, as expected from the formal solution of Eq. (29.1) in H-space. Let us, for the moment, write ψ ||m n| †

ψ ||m n|

= C (m, n) |m n| ψ†

,

(29.11)

= C † (m, n) ||m n| ψ .

(29.12)

Combining Eqs. (29.11) and (29.12), making use of the commutation (anticommutation) relation for the H-space quantum field operators, we must also have C † (m, n) C (m, n + 1) = C (m, n) C † (m, n − 1) = 1.

(29.13)

This is satisfied by writing C † (m, n) = m−n C † (0, 0) and C (m, n) = m−n C (0, 0), where = −1 for fermions and = 1 for bosons. Therefore, we must have C † (0, 0) C (0, 0) = . Thus for fermions, we have two arbitrary choices for the self-consistent values of C † (0, 0) and C (0, 0), namely, (a) C † (0, 0) = 1, C (0, 0) = , and (b) C † (0, 0) = , C (0, 0) = 1 . For bosons, all the C’s can be taken equal to unity. In this paper, we prefer to use the convention (b) for fermions for convenience in what follows, whereas Schmutz [185] use the convention (a). Thus, we have in addition to Eqs. (29.9) and (29.10) the following relations for the ‘tilde’ quantum field superoperators in L-space, ψ ||m n| †

ψ ||m n| †

where ψ 1 , ψ2

†

= ψ1 , ψ2

|m n| ψ†

=

m−n

=

m−n+1

,

(29.14)

||m n| ψ ,

(29.15)

= δ 1,2 , and all other commutation or anti-commutation of ‘hat’ and ‘tilde’ superoperators are zero. Thus, any superoperator A˜ consisting of η(η−1) product of unpaired quantum superfield operators will be defined as A˜ = e 2 A˜

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† η(η−1) and A˜† = e 2 A˜† , where η is the number of unpaired ψ or ψ in A˜ . A˜ and A˜† obey the property given by Eqs. (29.14)-(29.15). Note for number-conserving ˜ =H ˜=H ˜ †. operator, η = 0, hence H To construct QSFT we basically need two more things, (a) the Schwinger equation, i.e., the equation of motion of the average superfield, and (b) a generating super-functional for all correlations. These will allow us to go over from superoperator theory to the statistical mechanics of c-number fields, where the c-number fields are elements of the Grassmann algebra for fermions [129]. With the properties of the superfield operators known, the Schwinger equation can easily be obtained. In the next two sections, we derive the generating super-functional.

Super S-matrix Theory in L-Space From Eq. (29.2), we can formally write the solution for the super-statevector |ρ (t) as |ρ (t)

= T exp

−i |

t to

L (t ) dt

|ρ (to ) ,

(29.16)

where T is the usual real-time ordering operator. We take to as the time when the “perturbation” Liouvillian, L(1) = L − Lo , is turned on. Then we can also write |ρ (to )

to

−i |

= T exp

ξo

Lo (t ) dt

|ρ (ξ o ) ,

(29.17)

where the system, during the time duration from ξ o to to , is acted on only by the “unperturbed” Liouvillian Lo . Therefore, we can also write |ρ (t) where

−i |

= T exp

t

0

L (t ) dt S (t, to ) |ρo (0) ,

(29.18)

 S (t, to ) = Uo (0, t) U (t, to ) Uo (to , 0) ,     t Uo (0, t) = T exp |i 0 Lo (t ) dt ,    t  U (t, to ) = T exp −i , | to L (t ) dt |ρo (0)

= Uo (0, ξ o ) |ρ (ξ o ) .

If we let to ⇒ −∞, and write |ρo (0) |ρ (t)

= T exp

−i |

(29.19)

= ρeq

(29.20)

, then we have

t 0

Lo (t ) dt S (t, −∞) ρeq

.

(29.21)

It follows that |ρ (0)

= S (0, −∞) ρeq

= |ρL

= |ρI (0) ,

(29.22)

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which |ρ (t) define O, we

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defines the super-Heisenberg representation |ρL of the super-statevector . Note that |ρL is independent of time. Equation (29.19) also leads us to the super-interaction representation in L-space. Thus for any superoperator define OI (t) = Uo (0, t) O (t) Uo (t, 0)

(29.23)

as the super-interaction representation of O in L-space, and OL (t) = U (0, t) O (t) U (t, 0)

(29.24)

as its super-Heisenberg representation. Therefore, we have for any superoperator O, OL (t) = S (0, t) OI (t) S (t, 0) ,

(29.25)

which give the relation between the super-Heisenberg representation and the superinteraction representation. Since L, H , and H commute then we have for the “hat” and “tilde” quantum field superoperators the following relations i |

ψHe (t) = T exp

i |

= T exp

t 0

H (t ) dt ψ T exp

−i |

L (t ) dt ψ T exp

−i |

t 0

t

H (t ) dt

0 t 0

L (t ) dt

= ψL (t) , ψHh (t) = T exp

= T exp

i | i |

(29.26) t

H (t ) dt ψ T exp

0 t 0

L (t ) dt ψ T exp

= ψL (t) .

t

−i | −i |

0 t 0

H (t ) dt L (t ) dt (29.27)

Similar relations exist for the canonically conjugate quantum field operators: † † † † ψHe (t) = ψL (t), and ψHh (t) = ψL (t) . Construction of Generating Super-Functional The “super-Schrödinger equation” that we have developed in L-space and its corresponding formal solution given above are not complete. For example, it is not clear what would be the canonically conjugate counterpart of the super-statevector |ρ (t) for defining generating functional, analogous to the existing canonically conjugate pair of H-space states, ψ+ , t and ψ− , t , which naturally occur in zero temperature variational quantum action principle forming the basis for deriving the time-dependent Schrödinger equation, as first given by Dirac [187].

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Time-Dependent Variational Principle in Liouville Space To complete the “super-Schrödinger quantum mechanics”, we need to construct the corresponding variational principle in L-space. We follow the general construction of a variational principle [188, 189] as an optimization of the estimate for the expectation value of any operator O at the time t1 from the knowledge of the initial condition for the density matrix super-statevector |ρ (to ) , subject to the constraint that |ρ (t) obeys the L-space super-Schrödinger equation. Therefore, we have to optimize the following super-functional [189], t1

Φ ( |ρ (t) , |Λ (t) ) =

O (t)| |ρ (t1 ) −

Λ (t)| i| to

∂ − L |ρ (t) , ∂t

(29.28)

where we have introduced the dual space super-statevector Λ (t)| as a ‘Lagrange multiplier’. Upon integration by parts, Φ ( |ρ (t) , |Λ (t) ) is also equal to the following super-functional Φ ( |ρ (t) , |Λ (t) ) =

O (t)| |ρ (t1 ) − i| Λ (t1 )| |ρ (t1 ) +i| Λ (to )| |ρ (to ) t1 ∂Λ (t) − LΛ (t) |ρ (t) . i| + ∂t to

(29.29)

Thus, the optimum condition for Φ ( |ρ (t) , |Λ (t) ) occurs for i| Λ (t1 )| = O (t)| , and when the following equations of motion are obeyed for to < t < t1 , i| ∂ |ρ(t) ∂t i| ∂ |Λ(t) ∂t

= L |ρ (t) ,

(29.30)

= L |Λ (t) .

If we let t1 ⇒ ∞ and to ⇒ −∞, apply the boundary condition: i| Λ (t1 )| = O (t)| , and |ρ (−∞) = ρeq , then the solutions can be written as  t |ρ (t) = T exp −i , | 0 Lo (t ) dt S (t, −∞) ρeq (29.31) 0  |Λ (t) = T ac exp −i | t Lo (t ) dt S (t, ∞) |O ,

where T ac denotes anti-chronological time ordering. Therefore, we can write a “transition probability” in L-space as Λ (t)| |ρ (t)

=

1 i|

O| S (∞, −∞) ρeq

=

1 i|

OL | |ρL ,

(29.32)

where |ρL is the super-Heisenberg representation of the super-statevector |ρ (t) defined by Eq. (29.22), and similarly OL | = O| S (∞, 0) Note that in terms of evaluating the expectation value of the superoperator at t = t1 = ∞ , we can explicitly write this as a time-ordered expectation value by rewriting Eq. (29.32) as Λ (t)| |ρ (t)

=

1 i|

O| S (∞, −∞) ρeq

=

1 i|

1| OS (∞, −∞) ρeq

. (29.33)

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We are particularly interested in O = i| , where is the identity operator corresponding to an asymptotic state of maximum entropy at t = ∞ (corresponding to ψ+ , t indicating asymptotic freedom of the zero temperature variational quantumaction principle). Then the canonically conjugate pair |Λ (t) and |ρ (t) is defined by the “transition probability” in L-space given by the following relation Λ (t)| |ρ (t) where

L|

=

1| S (∞, −∞) ρeq

L|

=

|ρL ,

(29.34)

is given by L|

=

1| S (∞, 0) .

(29.35)

From the time-ordered expectation value deduced by Eq. (29.33), we see that indeed 1| S (∞, −∞) ρeq is the analog to the transition amplitude occurring in zero temperature time-dependent quantum mechanics [187]. The Eﬀective Action and Generating Super-Functional In order to gain further insights into the S-matrix formalism in L-space, we formulate the constrained stationary value [190] of the “super-action” given by dt

Λ (t)| i|

∂ − L |ρ (t) , ∂t

Λ (t)| |ρ (t) = 1 , Λ (t)| Ψ |ρ (t) = Ψ

(29.36)

where Ψ stands for an arbitrary quantum field superoperator (the extension to several species of quantum field superoperators is straightforward). Thus, we consider the stationary variation of the super-functional Ω ( |ρ (t) , |Λ (t) ) =

dt

Λ (t)| i|

∂ − L |ρ (t) ∂t

−

dx

Λ (t)| η (x) · Ψ |ρ (t)

−

dt w (t) Λ (t)| |ρ (t) ,

(29.37)

where η (x) and w (t) are introduced as c-number Lagrange multipliers. Note that η (x) is also playing the role of the Schwinger source field. Carrying out the variation with respect to |Λ (t) and |ρ (t) , and enforcing the stationarity of Ω ( |ρ (t) , |Λ (t) ), we obtain the following equations i| ∂ |ρ(t) − L |ρ (t) − ∂t

i| ∂ |Λ(t) ∂t

− L |Λ (t) −

dx η (x) · Ψ |ρ (t) − w (t) |ρ (t)

= 0,

∗

dx η (x) · Ψ |Λ (t) − w (t) |Λ (t)

(29.38)

= 0.

We write the solutions to these equations as |ρ (t)

η,w

= exp

−i |

|Λ (t)

η,w

= exp

i |

t −∞

dt w (t ) |ρ (t)

η

w∗ (t ) |Λ (t)

η

t dt −∞

 , ,

(29.39)

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where |ρ (t)

η

and |Λ (t)

∂ |ρ(t) ∂t ∂ |Λ(t) i| ∂t

i|

η

η

η

obey the equations

− L |ρ (t)

η

−

dx η (x) · Ψ |ρ (t)

− L |Λ (t)

η

−

dx η (x) · Ψ |Λ (t)

η η

 = 0, 

(29.40)

= 0. 

Therefore, we can calculate the “transition probability” in L-space from the solutions of Eq. (29.38), which is given by Eq. (29.39). We obtain the following relation Λ (t)| |ρ (t)

η,w

= 1 = exp

∞

−i |

dt w (t )

Λ (t)| |ρ (t)

−∞

η

,

(29.41)

which leads to the “transition probability” in the presence of the Schwinger source term given by Λ (t)| |ρ (t)

η

= exp

i |

∞

dt w (t ) = exp

−∞

i W . |

(29.42)

This is the same “transition probability” as that obtained in Eq. (29.34). Therefore, we obtain the equality 1| S (∞, −∞) ρeq

= exp

i W . |

(29.43)

Analogous to the zero temperature S-matrix formalism, we identify W as the generating super-functional for connected n-point super-Green’s functions. This is supported by the time-ordered way of taking the average value, shown by Eq. (29.33). To have a deeper appreciation of this analogy, we take the scalar product with Λ (t)| η,w of both sides of the first line of Eq. (29.38) and integrate over time. We obtain W =

dt w (t )

=

dt

Λ (t)|

i|

∂ |ρ (t) ∂t

−L−

dx η (x) · Ψ

|ρ (t)

η,w

(29.44)

from which we deduced the variational derivative δ W = − Λ (t)| Ψ |ρ (t) δη

η,w

= Ψ

(at the optimum condition) .

(29.45)

Therefore, we have the eﬀective super-action given by Aef f =

∞ −∞

dt

Λ (t)| i|

∂ − L |ρ (t) ∂t

η,w

=W+

dx η (x) · Ψ

(29.46)

from which we also deduced the variational derivative δ Aef f = η. δ Ψ

(29.47)

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Hence, the eﬀective super-action is stationary with respect to the variation of Ψ when η (x) = 0, which is the physical situation, i.e., W = Aeff . Thus for η (x) = 0, we have W = −i| ln 1| S (∞, −∞) ρeq

∞

=

dt

Λ (t)| i|

−∞

∂ − L |ρ (t) . ∂t

(29.48)

The relation derived here for nonequilibrium quantum-field theory between super S-matrix and the eﬀective super-action provides a rigorous basis supporting the functional theory of time-dependent many-body quantum mechanics discussed by Rajagopal and Buot in a series of papers [167, 191, 168]. We will refer to S (∞, −∞) as the complete S-matrix superoperator for quantumfield theoretical methods of nonequilibrium systems. Equation (29.19) allows us to write the evolution equation for the S-matrix operator as i|

∂ (1) S (t, to ) = LI (t) S (t, to ) , ∂t

(29.49)

where (1)

LI (t) = Uo (0, t) (L − Lo ) Uo (t, 0) ,

(29.50)

and (L − Lo ) = [v (1) + u (1)] Ψ (1) + [v (1, 2) + u (1, 2)] Ψ (1) Ψ (2) .

(29.51)

The Schwinger external sources [192] indicated by u(1) and u(1, 2) contain all the time-dependent part of v(1) and v(1, 2) respectively. These Schwinger source terms are to be set equal to zero at the end of all calculations. The solution to Eq. (29.49) immediately yields S (t, to ) = 1 −

i |

t to

(1)

LI (t ) S (t , to ) dt ,

(29.52)

which upon iteration yields S (t, to ) = T exp Thus, we can explicitly exhibit 1| S (∞, −∞) ρeq

−i |

t to

(1)

LI (t ) dt .

1| S (∞, −∞) ρeq =

1| T exp

= exp

i W , |

−i |

(29.53)

as ∞

−∞

(1)

LI (t ) dt

ρeq (29.54)

where W is the eﬀective action. Equation (29.54) shows the relation of Eq. (29.43) (1) to LI (t) similar to zero-temperature many-body quantum-field theory.

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Chapter 30

Super-Green’s Functions

In the use of quantum field theoretical techniques, we need to construct the Green’s function formalism in L-space which incorporates all the nonequilibrium (“hat” and “tilde”) quantum field superoperators, and suﬃciently general enough to be able to account for quantum transport in systems with pairing interaction. This can be conveniently done by introducing a 4-component second quantized quantum field superoperators given by1 

ψ (1)



  †  ψ (1)  Ψ (1) =  †  ≡ {Ψα (1) , α = 1, 2, 3, 4} .  ψ (1) 

(30.1)

ψ (1)

This multicomponent notation for the quantum field superoperator is a further generalization of the multicomponent quantum field operator introduced by De Dominicis and Martin [193] in treating non-normal systems at thermal equilibrium. Their treatment was an extension of that first introduced by Nambu [194]. Since the exact pairing mechanism is left unspecified in what follows, the equations derived here are also applicable to other exotic pairing proposed in high-Tc superconductors. The corresponding quantum transport equations represent more general quantum transport equations including particle-pair dynamics. From the commutation (anticommutation) relations of the ‘hat’ and ‘tilde’ superoperators, we obtain the following equal-time commutation (anticommutation) relations, [Ψ (1) , Ψ (2)] ⇒ [Ψα (r) , Ψβ (r )] = δ (r − r ) (τ 4 )αβ , 1 Sometimes

(30.2)

we use the following designation of the components of the field superoperator, {Ψα (1) , α = i, ii, iii, iv}

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where 

0  0 τ4 =  − 0

Similarly, we have

 01 0 00−  . 00 0  10 0

⇒ Ψα (r) , Ψ†β (r )

Ψ (1) , Ψ† (2)

(30.3)

= δ (r − r ) (τ 4 )αβ σx ,

(30.4)

where σx =

0δ δ0

,

δ=

10 01

,

(30.5)

and Ψ† (1) , Ψ† (2)

⇒ Ψ†α (r) , Ψ†β (r )

= δ (r − r ) (τ 4 )−1 αβ .

(30.6)

The important point to note is that the ‘ ’-commutation relations between the 4-component quantum field superoperators are c-numbers (constant matrices). This is important for the validity of the Wicks’ theorem applied to the averages of products of superoperators in the diagrammatic treatment. We further note that a general Liouvillian superoperator L can be expressed in either of the three diﬀerent schemes, namely, (a) in terms of Ψ alone, (b) in terms of Ψ and Ψ† , analogous to the equilibrium quantum field theoretical treatment, or (c) in terms of the Ψ† alone. Since there seems to be no distinct advantages for our purpose, in formulating a generalized quantum field theory in L-space using either scheme (a), (b), or (c), we are free to choose either of these three schemes. For convenience, we chose scheme (a) since the resulting L-space formalism bears a close resemblance to the theory of classical statistical systems, especially when all potentials are -symmetrized 2 . 2 Consider

∂ the terms occuring in i| ∂t ΨL ξ

=

ΨL ξ

, L . As an example, consider

the term v (1234) [ΨL (ξ ) , ΨL (1) ΨL (2) ΨL (3) ΨL (4)]. Applying the equal-time commutation relations between the fields, we have the following results, v (1234) ΨL ξ = (τ e4 )α

, ΨL (1) ΨL (2) ΨL (3) ΨL (4)

ξ ,α1

δ rξ − r1 ΨL (2) ΨL (3) ΨL (4) × v (1234)

+ (τ e4 )α ,α2 ξ

δ rξ − r2 ΨL (1) ΨL (3) ΨL (4) × v (1234)

+ (τ e4 )α ,α3 ξ

δ rξ − r3 ΨL (1) ΨL (2) ΨL (4) × v (1234)

+ (τ e4 )α ,α3 ξ

δ rξ − r4 ΨL (1) ΨL (2) ΨL (3) × v (1234)

(30.7)

It is desirable to fix the arrangement of indices in the fields. Since the 1234 indices are dummy variables and repeated indices are summed over, we can rearrange the prefactors, involving the fields, of the potential in Eq. (30.7) such that they are brought to the same form as the first prefactor. This can be done if we accompany the rearrangement of the fields by a displacement of the dummy variable 1 in v(1234) by an appropriate number of places to the right and taking into account the appropriate factors due to the commutation relationship indicated in Eq. (30.7).

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In this scheme (a), we define the one-particle super Green’s function as L|

G (1, 2) =

T ΨL (1) ΨL (2) |ρL . L | |ρL

¯ From previous developments, the averages can be written in terms of the S-matrix, e.g., δ2 1| S(∞,−∞) |ρeq δu(1)δu(2)

2

G (1, 2) = (i|)

.

1| S (∞, −∞) ρeq

(30.8)

where in the functional derivative we have assumed that a Schwinger source term, u (i), which couples to the quantum field, ΨL (i), has been added to the interac¯ tion potential occurring in the S-matrix. Clearly, multi time-dependent averages or moments are defined only for time-ordered quantum field superoperators. The Green’s functions of the three diﬀerent schemes,(a), (b), and (c) mentioned above, are related to each other through the constant matrix σ x . We obtained the following, F (1, 2) G (1, 2) GT (1, 2) † F (1, 2)

G (1, 2) = i|

,

(30.9)

where the superscript T indicates the taking of the transpose, and,

†

F (1, 2) =

< c (1, 2) ghh (1, 2) ghh > ac (1, 2) ghh (1, 2) ghh

F (1, 2) =

c < (1, 2) gee (1, 2) gee > ac gee (1, 2) gee (1, 2)

,

(30.10)

,

(30.11)

Therefore, we can also write, v (1234) ΨL ξ

, ΨL (1) ΨL (2) ΨL (3) ΨL (4) 3

= (τ e4 )α

ξ

,α1

n

δ rξ − r1

Dn (1) v (1234)

n=0

where Dn (1) transpose the index 1 n places to the right and the time-ordered average, the following,

n

ΨL (2) ΨL (3) ΨL (4)

= (−1)n . We have, upon taking

3

(τ e4 )α

ξ ,α1

=

δ rξ − r1

(τ e4 )α ,α1 ξ

n n=0

δ rξ − r1

Dn (1) v (1234)

 1  3! 

P (1234)

P

ΨL (2) ΨL (3) ΨL (4)

  v (1234) G (234) 

This observation can be utilized as a trick for recasting nonequilibrium quantum field theory as a pure functional theory, similar to the classical-statistical field theory. To implement this trick, one simply rewrite the external potentials in e-symmetrized form from the very beginning, as employed by De Dominicis and Martin [193].

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G (1, 2) =

GT (1, 2) =

Gc (1, 2) G< (1, 2) G> (1, 2) Gac (1, 2)

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,

GcT (1, 2) G>T (1, 2) G
(30.12)

.

(30.13)

The matrix elements are time-ordered averages. By virtue of the number-conserving Liouvillian and the property of quantum superfield operators, Eqs. (29.14)-(29.15), these are given, with the time-ordered averages in the super-Heisenberg representation translated into their H-space counterparts, with the aid of Eqs. (29.14)-(29.15), by the following expressions: c ˆ (1) ψ ˆ (2) = θ (12) i|g > (1, 2) + θ (21) i|g < (1, 2) = i|ghh T ψ (1, 2) , (30.14) L L hh hh

ˆ (1) ψ ˜ † (2) = −T rρ ψ (2) ψ (1) = i|g < (1, 2) , T ψ L L H H H hh

(30.15)

˜ † (1) ψ ˆ (2) = T rρ ψ (1) ψ (2) = i|g> (1, 2) , T ψ L L H H H hh

(30.16)

˜ † (1) ψ ˜ † (2) = θ (12) i|g < (1, 2) + θ (21) i|g > (1, 2) = i|g ac (1, 2) , (30.17) T ψ hh L L hh hh ˆ (1) ψ ˆ † (2) = θ (12) i|G> (1, 2) + θ (21) i|G< (1, 2) = i|Gc (12) , T ψ L L

(30.18)

ˆ (1) ψ ˜ (2) = T rρ ψ† (2) ψ (1) = −i|G< (1, 2) , T ψ L L H H H

(30.19)

˜ † (1) ψ ˆ † (2) = T rρ ψ (1) ψ† (2) = i|G> (1, 2) , T ψ H H L L H

(30.20)

˜ † (1) ψ ˜ (2) = − θ (12) i|G< (1, 2) + θ (21) i|G> (1, 2) T ψ L L = −i|Gac (12) ,

(30.21)

ˆ † (1) ψ ˆ (2) = − θ (12) i|G< (2, 1) + θ (21) i|G> (2, 1) T ψ L L = −i|GcT (12) ,

(30.22)

˜ (1) ψ ˆ (2) = −T rρ ψ† (1) ψ (2) = i|G
(30.23)

ˆ † (1) ψ ˜ † (2) = −T rρ ψ (2) ψ† (1) = −i|G>T (1, 2) , T ψ L H H L H

(30.24)

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˜ (1) ψ ˜ † (2) = θ (12) i|G> (2, 1) + θ (21) i|G< (2, 1) T ψ L L = = i|GacT (12) ,

(30.25)

ˆ † (2) = θ (12) i|g > (1, 2) + θ (21) i|g< (1, 2) ˆ † (1) ψ T ψ ee ee L L c (1, 2) , = i|gee

(30.26)

< ˜ (2) = T rρ ψ† (2) ψ† (1) = −i|gee ˆ †L (1) ψ (1, 2) , T ψ L H H H

(30.27)

˜ (1) ψ ˆ † (2) = −T rρ ψ† (1) ψ† (2) = −i|g> (1, 2) , T ψ ee L H H L H

(30.28)

< > ˜ (2) = θ (12) i|gee ˜ L (1) ψ (1, 2) + θ (21) i|gee (1, 2) T ψ L ac (1, 2) . = i|gee

(30.29)

The ‘retarded’ and ‘advanced’ correlations are also defined using the above components, for example, we have c < > ac r ghh − ghh = ghh − ghh = ghh ,

(30.30)

c > < ac a − ghh = ghh − ghh = ghh . ghh

(30.31)

Note that G (1, 2) corresponds to the Keldysh nonequilibrium Green’s function for normal systems [178, 99]. In what follows we often refer to the super-Green’s function simply as average or moment quantum distribution function. Later we define quantum super-correlation functions or quantum cumulants, analogous to the classical statistical theory of random and dissipative phenomena. This distinction is very important in treating the quantum transport of particle system which exhibit superfluidity and superconductivity. We will also refer to both classes of distribution functions as generalized quantum distribution functions (GQDF). GQDF and Self-Consistent Equations of Motion In terms of the 4-component field of Eq. (30.1), let us consider a nonequilibrium system whose Liouvillian can in general be expressed as M

L=

N=1

υ (1, 2, 3, ..., N) Ψ (1) Ψ (2) Ψ (3) .....Ψ (N ) + LSchw ext ,

(30.32)

where the Schwinger source term is given by = Hext − H LSchw ext

,ext

= u (1) Ψ (1) + u (1, 2) Ψ (1) Ψ (2) .

(30.33)

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The indices 1, 2, 3...N , each stands for all pertinent quantum labels, and repeated indices are “summed” over. For example, for a system of interacting bosons or fermions, interacting via two-body forces, we have the following expression for the potentials occurring in Eq. (30.32),  υ (1) = 0,     −| 2 2 υ (1, 2) = [δ α1 ,3 δ α2 ,1 − δ α1 ,2 δ α2 ,4 ] 2m ∇1 + v {1} δ (r1 − r2 ) ,   (30.34) υ (1, 2, 3) = 0,    υ (1, 2, 3, 4) = [δα1 ,3 δ α2 ,3 δ α3 ,1 δ α4 ,1 − δ α1 ,4 δ α2 ,4 δ α3 ,2 δ α4 ,2 ]    × {v (r1 − r2 )} δ (r2 − r3 ) δ (r1 − r4 ) , where the subscripts of the delta-function pertain to the components of Ψ in Eq. (30.1). In the super-Heisenberg representation, Ψ obeys the equation of motion as i|

∂ ΨL ξ ∂t

= ΨL ξ

,L .

(30.35)

Note that in the equation of motion for ΨL ξ , all times occurring in L are equal to tξ and therefore we can use the equal-time relations to evaluate the RHS of Eq. (30.35). The results can be written as i|

∂ ΨL ξ ∂tξ

= (τ 4 )αξ

,α1

M

×

δ rξ − r1

N−1 n

Dn (1) υ 1, tξ ; 2, tξ ; ......N, tξ

n=0

N=1

×ΨL 1, tξ

ΨL 2, tξ

......ΨL N, tξ

,

(30.36)

where Dn (1) is a ‘transposition operator’ acting only on υ 1, tξ ; 2, tξ ; ......N, tξ . The eﬀect of Dn (1) is to transpose the index 1, n places to the right in υ 1, tξ ; 2, tξ ; ......N, tξ , while the functional form of υ relative to the order of its indices remains unaﬀected. The average of ΨL ξ , which we denote by ≡ G ξ , obeys the following equation of motion, ΨL ξ i|

∂ G ξ ∂tξ

= (τ 4 )αξ M

×

N=1

,α1

δ rξ − r1

N

i=1

δ tξ − ti

v (1, 2, ......N ) (N − 1)!

G (2, 3, 4, ...N ) , (30.37)

where G (2, 3, 4, ...N ) = ΨL (2) ΨL (3) ΨL (4) ......ΨL (N) , P

v (1, 2, 3, ......N ) = P

υ (1, 2, 3, ......N ) .

(30.38) (30.39)

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Here v (1, 2, 3, ......N) is the -symmetrized potential obtained by summing all the -permutation of the indices in υ (1, 2, 3, ......N ). With Eq. (30.37), the hierarchy of equations for the GQDF, G (1, 2, 3, 4, ...N ), for any N can be generated by using the variational derivative indicated in Eq. (30.8). The variational derivative allows for an exact “closed form” equations to be written down for all the distribution functions. The major task in the calculations lies in expressing the variational derivative operations self-consistently in terms of the pertinent lower-order distribution functions and eﬀective multi-point eﬀective interaction potentials or vertex functions, in such a way as to decouple the hierarchy and obtain a mathematically closed set of coupled equations, the number of equations being equal to the number of unknown functions. Analogous to classical case we also refer to 1| S (∞, −∞) ρeq = exp |i W as the characteristic functional, it is a functional of the Schwinger source fields. This can be evaluated in terms of 1-point and 2-point GQDFs, from which a fully renormalized equation of motions can be obtained. It serves as the generating functional for the quantum-field moment distributions in the presence of the Schwinger source terms. We may thus write i δ δ δ i W G (1, 2, .....N) = i| i| ....i| exp W , | δu (N ) δu (N − 1) δu (1) | (30.40) where to account for anti-commuting Grassman variables u (i), in the case of fermions, we take the convention that the variational derivatives are right-hand derivatives, i.e., variational derivatives involving higher-indexed u’s operates from the right of the variational derivatives involving lower-indexed u’s, in a ‘horizontal stack’ fashion. The anticommutation relationship must be taken into account if any rearrangement diﬀerent from this conventional arrangement is made. This convention must be adhered to make the theory consistent. Equation (30.40) for the quantum-field moment distribution G yields an iterative relation between G of diﬀerent orders, exp

G (1, 2, .....n) =

i|

δ δW − δu (n) δu (n)

G (1, 2, .....n − 1) ,

(30.41)

where −

δ ln 1| S (∞, −∞) ρeq δW = i| δu (n) δu (n)

= G (n) = K (n) .

(30.42)

In Eq. (30.41), the operator within the curly bracket is understood to be righthanded operator in the same way as the repeated variational derivatives operators were defined. Upon expanding the exponential in Eq. (29.54), we are also able to express the characteristic functional in terms of the generalized quantum-field moment distributions, G’s, as ∞

i 1 exp W = u (1) u (2) ....u (n) G (1, 2, .....n) | n! n=0

(30.43)

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since the G’s are the coeﬃcient in the Fourier series expansion of Eq. (29.54) in the limit the u(i)’s go to zero. We note that in the perturbative diagrammatic application of Wick’s theorem (for superfluid Bose system it is assumed that Lo is expanded about the condensate for Wick’s theorem to be applicable) to evaluate the G’s, each term in the summation of Eq. (30.43) can be represented by a sum of diagrams or graphs in a one-to-one correspondence basis. 30.1

Connected Diagrams: Correlation Function K

The set of topologically distinct diagrams corresponding to each G is made up of ‘connected’ and ‘disconnected’ subsets of graphs. An important lemma in graph theory [195] asserts that the sum of the RHS of Eq. (30.43) satisfy the following identity, ∞

1 u (1) u (2) ....u (n) G (1, 2, .....n) n! n=0 = exp

∞

1 u (1) u (2) ....u (n) K (1, 2, .....n) , n! n=0

(30.44)

where the quantum-field cumulant distribution functions or quantum correlation functions, K (1, 2, .....n), represent the topologically distinct ‘connected’ and irreducible subset of graphs occurring in the LHS of Eq. (30.44). Corresponding to the classical theory [196], these quantum correlation functions are generated by variational derivatives of ln 1| S (∞, −∞) ρeq = |i W with respect to the Schwinger external sources. Thus, we have K (1, 2, .....n) = (i|)n

δ n ln 1| S (∞, −∞) ρeq , δu (n) δu (n − 1) ....δu (1)

(30.45)

which yields a simple iterative relation between correlation functions of diﬀerent orders, namely, K (1, 2, .....n) = i|

δK (1, 2, .....n − 1) . δu (n)

(30.46)

Similar functional derivative relations can be obtained between GQDF with even number of indices by using the variation with respect to the external Schwinger source term u(1, 2). Since u(1, 2) is an ordinary c-number, the order of the u(i, j)’s is not critical in taking the functional derivatives. We use the more elementary variation with respect to the u(i)’s, so as to completely resemble classical distribution function theory and to include systems which exhibit superfluidity and superconductivity. The above iterative relations are important in setting up the exact “closed” form of the equations for a GQDF of arbitrary order. For a system of bosons, one expects that the 1st, 2nd, 3rd, and 4th order GQDF essentially determines its characteristic functional, whereas, for unconstrained fermion systems only even-ordered GQDF, 2nd and 4th order, determine its characteristic functional.

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The following relations hold between the correlation quantum distribution functions (CQDFs) and the moment quantum distribution functions (MQDFs), K (1) = G (1) ,

K (1, 2) = G (1, 2) −

1 2!

(30.47)

P P

G (1) G (2) .

(30.48)

To obtain higher-order CQDFs, we have to subtract all possible lower order CQDFs from the MQDFs diﬀerence, K (1, 2, .....n) = G (1, 2, .....n) −

1 n!

P P

G (1) G (2) ....G (n) .

This can be verified by a straightforward application of the variational derivatives defined above.

30.2

Self-Consistent Equations for GQDF

We multiply both sides of Eq. (30.37) by the inverse of the matrix τ 4 and limit ourselves to M = 4 in Eq. (30.32), which will cover almost all of many-body physics, and make use of Eq. (30.41) to rewrite Eq. (30.37) as Go−1 ξ , 2 G (2) = V ξ

+

1 V ξ 23 (3 − 1)!

i|

δ + G (3) G (2) δu (3)

1 δ V ξ 234 i| + G (4) (4 − 1)! δu (4) δ + G (3) G (2) , × i| δu (3)

+

(30.49)

where Go−1 ξ , 2 = (τ 4 )−1 i|δ ξ , 2 V ξ

∂ −V ∂t2

,

(30.50)

= δ rξ − r1 δ tξ − t1 {v (1) + u (1)} ,

(30.51)

1 V ξ 2...n = δ rξ − r1 (n − 1)!

n

i=1

δ tξ − ti

ξ ,2

1 v (12...n) . (n − 1)!

(30.52)

We recast Eq. (30.49) as Go−1 (1, 2) G (2) = V (1) +Ξ (1) ,

(30.53)

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where Ξ (1) =

δ 1 V (123) i| + G (3) G (2) 2! δu (3) δ δ 1 + G (4) i| + G (3) G (2) . (30.54) + V (1234) i| 3! δu (4) δu (3)

Equation (30.53) for the average field is sometimes referred to as the Schwinger equation. In Eq. (30.53), Ξ (1) is the eﬀective (‘internal’) source function contributed by the system. We can utilize Eq. (30.53) to obtain the equation for the propagator or two-point correlation function K (23). We obtained ˘ (12) K (23) = δ (13) , Go−1 (1, 2) − Σ i|

˘ (12) = δΞ (1) , Σ δG (2)

(30.55)

˘ (12) is the particle super self-energy. We will refer to K (23) /i| as the where Σ super-propagator kernel or simply as the propagator. Note that the inverse of the ˘ (12) /i|, whereas the inverse of the quantum correlation function is Go−1 (1, 2) − Σ ˘ (12). propagator is Go−1 (1, 2) − Σ 30.2.1

Schwinger Equation as Generalization of the Kohn-Sham and Gross-Pitaevskii Equations

From Eq. (30.53) and second equation of Eq. (30.55), we may write the Schwinger equation as Go−1 (1, 2) G (2) = V (1) +Ξ (1) ˘ (12) G (2) , = V (1) + Σ

(30.56)

which, in the limit that the Schwinger source term V (1) goes to zero, yields the eﬀective Schrödinger equation with nonlocal eﬀective potential given by the self-energy ˘ (12), namely Go−1 (1, 2) − Σ ˘ (12) G (2) = 0. A very well-known approximaΣ tion scheme for approximating this self-consistent many-body eﬀective Schrödinger equation is given by Kohn and Sham [166] in the form of a density-functional formalism. Equation (30.56) can also be deduced from the equation of motion obeyed by G (1, 2). Corresponding to the Kohn-Sham equation for fermions, a well-known equation for dilute many-body boson systems is the so-called Gross-Pitaevskii equation, or the nonlinear Schrödinger equation, for bosons. In the high-density limit (highly nonlinear regimes) it has been shown that the Gross-Pitaevskii equation can be mapped to a Kohn-Sham scheme of density functional theory and vice versa. 30.2.2

Closure Problem and Renormalization Procedure

˘ (12) are functionals of Equations (30.53) and (30.55) are not closed since Ξ (1) and Σ higher-order GQDF. In what follows we will show that the 3-point CQDF occurring ˘ (12) can be expressed in terms of lower-order CQDFs in combination in Ξ (1) and Σ

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with 3-point vertex function. This 3-point vertex functions in turn obeys an iterative ˘ (12) with equation which can be solved in terms of the functional derivative of Σ ˘ (12) can respect to G (i) and K (ij). Similarly, the 4-point CQDF occurring in Σ ˘ be expressed in analogous fashion. We will see that Σ (12) involves 3-point and 4-point CQDF, which can be expressed in terms of 3-point and 4-point vertices in combination with 1-point and 2-point CQDF. These vertex function can in turn ˘ (12) with respect to the 1-point be expressed in terms of functional derivative of Σ and 2-point CQDF, thus closing the number of unknown functions and eﬀectively decoupling the hierarchy. In the end, we basically end up with eight equations in eight unknowns, which can then be solved self-consistently. ˘ (12) in terms of CQDF. We make use of Eqs. First let us express Ξ (1) and Σ (30.41), (30.48), and the expansion of higher-order cumulants as function of the averages to obtain the following expressions, Ξ (1) =

V (123) {K (23) + G (2) G (3)} 2! V (1234) + {K (234) + 3G (2) K (34) + G (2) G (3) G (4)} , (30.57) 3!

V (1234) K (23l) K−1 (lw) ˘ (1w) = V (123) Σ + + G (3) δ (2w) + G (2) δ (3w) 2! 3!   −1 K (234l) K (lw)          + K (24) δ (3w) + K (34) δ (2w) + K (23) δ (4w)  −1 −1 × + G (3) K (24l) K (lw) + G (2) K (34l) K (lw) . (30.58)    +G (4) K (23l) K−1 (lw) + G (2) G (3) δ (4w)        + G (2) G (4) δ (3w) + G (3) G (4) δ (2w)

We will now show that Eqs. (30.57) and (30.58) are functionals of only one-point and two-point CQDFs. In order to do this, we define the n-point ‘eﬀective interaction potentials’ or vertex functions as ˘ (123...n) = − Γ

δ δG (n)

......

δ δG (3)

K−1 (12)

=

δ ˘ Γ (12.....n − 1) δG (n)

=

δ n F (1) , i|δG (n) δG (n − 1) ....δG (1)

(30.59)

where F (1) is the generating functional for the n-point super vertex function [193], obtained from the eﬀective super-action W by Legendre transformation, F (1) = W + V (1) G (1) +

1 V (12) G (12) . 2!

We can relate the vertex functions to the CQDF, obtaining ˘ (123) = K−1 (1j) K (jil) K−1 (i2) K−1 (l3) , Γ

(30.60)

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which allows us to express the 3-point CQDF in terms of the 3-point vertex function in combination with 2-point CQDF, ˘ (123) K (3¯ K (¯1¯2¯3) = K (¯ 11) Γ 3) K (2¯ 2) .

(30.61)

Similarly, we write ˘ (1234) = i| Γ

δ ˘ δu (m) Γ (123) . δu (m) i|δG (4)

(30.62)

Using Eqs. (30.60) and (30.61), we end up with the following expression, ˘ (1234) = K−1 (1j) K (jilm) K−1 (i2) K−1 (l3) K−1 (m4) Γ ˘ (1j4) K (jy) Γ ˘ (y23) − Γ ˘ (1j3) K (jy) Γ ˘ (y24) −Γ ˘ (12j) K (jy) Γ ˘ (y34) . −Γ

(30.63)

The last expression enables us to express the 4-point CQDF in terms of 4-point vertex and 3-point vertex in combination with only 2-point CQDFs. We have, corresponding to Eq.(30.61), the expression K (1234) = K (1¯ 1) D (¯ 1¯ 2¯ 3¯ 4) K (¯ 22) K (¯ 33) K (¯ 44) ,

(30.64)

˘ (1234) + Γ ˘ (1j4) K (jy) Γ ˘ (y23) D (1234) = Γ ˘ (1j3) K (jy) Γ ˘ (y24) + Γ ˘ (12j) K (jy) Γ ˘ (y34) . +Γ

(30.65) (30.66)

where

Substituting the expressions given by Eqs. (30.61) and (30.64) in Eqs. (30.57) and (30.58), we obtain Ξ (1) =

V (123) V (1234) {K (23) + G (2) G (3)} + 2! 3! ˘ (¯ K (2¯2) Γ 2¯ 3¯ 4) K (¯ 33) K (¯ 44) + K (24) G (3) × +G (2) K (34) + G (4) K (23) + G (2) G (3) G (4)

,

(30.67)

˘ (¯ ˘ (1w) = V (123) K (2¯2) Γ 2¯ 3w) K (¯ 33) + G (3) δ (2w) + G (2) δ (3w) Σ 2!   K (2¯ 2) D (¯ 2¯ 3¯ 4w) K (¯ 33) K (¯ 44)       ˘ (¯   + G (3) K (2¯ 2) Γ 2¯ 4w) K (¯ 44)         ˘ ¯ ¯ ¯ ¯  +G (2) K (3 3) Γ ( 3 4w) K ( 44)  V (1234) ˘ ¯ ¯ ¯ ¯ + . (30.68) +G (4) K (2 2) Γ ( 2 3w) K ( 33)   3!     + K (24) δ (3w) + K (34) δ (2w)       +K (23) δ (4w) + G (2) G (3) δ (4w)       + G (2) G (4) δ (3w) + G (3) G (4) δ (2w)

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30.2.3

Iterative Equations for the Vertex Functions

The structure of the dependence of the vertex functions on and K (ij)can be exhib˘ (ij) with ited by relating these vertex functions to the functional derivatives of Σ ˘ (123). This can be expressed as, respect to G (i) and K (ij). Let us first consider Γ ˘ (123) = Γ

˘ (12) δΣ i|δG (3)

+ K(ij)

˘ (12) δΣ i|δK (45)

G(i)

K (46) −

δK−1 (67) δG (3)

K (75) . (30.69)

˘ (123) as Therefore, we have the iterative equation for Γ ˘ (123) = Γ

˘ (12) δΣ i|δG (3)

+ K(ij)

˘ (12) δΣ i|δK (45)

G(i)

˘ (673) K (75) . K (46) Γ

(30.70)

˘ (123) as functional of only What this means is that it is indeed possible to express Γ ˘ (12) is also a G (i) and K (ij). This is also consistent with Eq. (30.68), in that Σ functional of only G (i) and K (ij) since we can also prove an analogous iterative ˘ (123) in terms of relation for D (1234). Thus, Eq.(30.70) allows us to express Γ ˘ (12) with respect to G (i) and the partial functional derivatives of the self-energy Σ ˘ (ijk) on G (i) K (ij), respectively, and exhibits the structure of the dependence of Γ ˘ (ij) on these same quantities. The formal and K (ij) relative to the dependence of Σ solution of Eq. (30.70) is ˘ (123) = Λ (12, ¯ Γ 1¯ 2)

˘ (¯ δΣ 1¯ 2) i|δG (3)

,

(30.71)

K(ij)

where 

Λ (12, ¯1¯2) = δ (1¯1) δ (2¯2) −

˘ (12) δΣ i|δK (45)

G(i)

−1

K (4¯ 1) K (¯ 25)

.

(30.72)

Now consider the iterative equation for D (1234) in terms of the functional derivative of the self-energy. From Eq. (30.64), we have D (1234) = K−1 (1j) i|

δK (ji) −1 K (i2) K−1 (l3) K−1 (m4) . δu (lm)

(30.73)

This can be re-expressed in terms of functional derivative of the self-energy with respect to 2-point CQDF. We obtain D (1234) =

˘ (12) δΣ i|δK (43)

+ G(i)

˘ (12) δΣ i|δK (56)

G(i)

K (5¯ 5) D (¯ 5¯ 634) K (¯ 66) . (30.74)

The formal solution for is given by D (1234) = Λ (12, ¯ 3¯ 4)

˘ (¯ δΣ 3¯ 4) i|δK (43)

. G(i)

(30.75)

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Substituting Eqs. (30.71) and (30.75) in the expression for Ξ (1), Eq. (30.67) and ˘ (1w), Eq. (30.68), we finally obtain the following expressions, Σ Ξ (1) =

V (1234) V (123) {K (23) + G (2) G (3)} + 2! 3!   ˘ δ Σ(56) ¯ ¯ ¯  K (22) Λ (2356) i|δG(4) K (¯ 33) K (¯ 44)    K(ij) × , + K (24) G (3)     +G (2) K (34) + G (4) K (23) + G (2) G (3) G (4)

(30.76)

˘ δΣ(56) 2) Λ (¯ 2¯ 356) i|δG(w) K (¯ 33) V (1234) V (123) K (2¯ ˘ K(ij) + Σ (1w) = 2! 3! + G (3) δ (2w) + G (2) δ (3w)   ˘ δΣ(56)  K (¯ 33) K (¯ 44)  K (2¯2) Λ (¯ 2¯ 3, 56) i|δK(w   ¯ 4) G(i)       ˘   δΣ(56)   ¯ ¯ ¯ ¯   K ( 44) + G (3) K (2 2) Λ ( 2 4, 56)   i|δG(w) K(ij)       ˘   δ Σ(56)  ¯ ¯ ¯ ¯ K (44)  +G (2) K (33) Λ (34, 56) i|δG(w) K(ij) × . (30.77) ˘ δΣ(56)   ¯ ¯ ¯ ¯   K ( 33) +G (4) K (2 2) Λ ( 2 356)   i|δG(w)   K(ij)         + K (24) δ (3w) + K (34) δ (2w)         +K (23) δ (4w) + G (2) G (3) δ (4w)     + G (2) G (4) δ (3w) + G (3) G (4) δ (2w)

We now summarized the results of this section. We have Eqs. (30.53) and (30.55) ˘ (ij), respectively. Equation (30.57) for for G (i) and K (ij), containing Ξ (i) and Σ ˘ Ξ (1) and Eq. (30.58) for Σ (12) show that Ξ (1) is a functional of 3-point CQDF, ˘ (12) is a functional of 3-point CQDFs and 4-point CQDF. This is a reflection and Σ of the hierarchal nature of the equation for the GQDFs. However, Eq. (30.61) shows that a 3-point CQDF can be expressed in terms of the lower order 2-point CQDFs in combination with 3-point vertex function. Similarly, Eq. (30.64) shows that a 4-point CQDF can be expressed in terms of 2-point CQDFs in combination with 3-point and 4-point vertex functions. Equations (30.71) and (30.75) show that these vertex functions can be expressed in terms of 2-point CQDFs in combination with partial functional derivatives of the self-energy with respect to G (i) and K (ij), thus closing the number of unknown functions. Thus we have a closed set of coupled ˘ (ij), K (ijk), K (ijkl), equations in eight unknown functions, G (i), K (ij), Ξ (i), Σ ˘ ˘ Γ (ijk), and Γ (ijkl), which give self-consistent solutions. In a fully self-consistent and explicitly renormalized formulation, it is desirable to have a universal functional of the GQDFs, whereby the equations of motion describing the system are obtained as a stationary principle for this universal functional subject to constraint on pertinent conservation laws. Since W is the eﬀective action, we expect this procedure to amount to a stationary action principle of W with respect to the GQDFs. We can construct this functional of GQDF by method of Legendre transformation introduced by De Dominicis and Martin [193] for interacting particle systems at thermodynamic equilibrium. This tedious renormalization procedure will not be given here. Moreover, questions about 1-particle irreducibility,

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i.e., existence theorem of well-defined approximate self-energies, vertex functions, etc. will not be addressed here. These are assumed to be not a problem here, at least for nonrelativistic Fermi systems. In general, these are fundamental problems that plague the physics of many-degrees of freedom, namely, quantum field theory and statistical many-body physics. This appears in the guise of the so-called “closure” problem. In non-equilibrium quantum physics, this problem has to do with the proper reduction of the quantum mechanics of the whole system, say A + Bath, which is governed by a unitary evolution operator, to the quantum mechanics of the subsystem A alone which is governed by a non-unitary evolution operator to express irreversibility. The ‘correct reduction’ from the whole system unitary picture to the subsystem non-unitary picture has been the subject of intense fundamental research, and similar question like ‘renormalizability’ in quantum field theory is expected to be a long-standing problem in physics of many-degrees of freedom. With respect to the evolution of reduced density matrix, there is a consensus that the ‘Lindblad criterion’ [197] must be satisfied3 , which does hold in transport equations derived in the paper since the trace of the right hand side of the transport equations reduce to zero, to express total probability equal to 1. This paper only reveals the formal mechanism of the reduction to 1-particle, 2-particle, etc. nonequilibrium descriptions. The reduction procedure is a common theme in any treatment using quantum-field theoretical techniques. There is an even more ‘fundamental’ question whether a non-unitary evolution should be the starting point of quantum mechanics. This relates to the ‘big bang’ as the beginning of the universe and the irreversibility of time. This philosophical question is beyond the scope of this chapter. In strongly correlated systems where higher-order correlations are significant, for example in systems containing electrons, holes, excitons, bi-excitons, trions, etc. the quantum field operators representing the composite particles may have to be employed to make the problem tractable, and the formalism given here can thus be repeated for a composite-particle QSFT.

3 This

is covered in Part 7 on phenomenological superoperator in the Kraus representation.

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Chapter 31

Quantum Transport Equations of Particle Systems

We derived here the general quantum transport equations valid for normal systems, Bose superfluids [198], superconductors [199], and dense electron-hole liquids. This involves the formulation of the total time evolution equations of various GQDF. From Eq. (30.55), we have the equation for the 2-point correlation function given by K−1 (1, 2) K (2, 3) = δ (1, 3) .

(31.1)

Note that for fermions, K (1, 2) = G (1, 2) , however the expression for K (1, 2) in Eq. (30.48), which may involved the condensate wavefunction, must be used for bosons. We write here the explicit expressions of the quantities occurring in Eq. (31.1). We have ˜ (1, 2) (1, 2) G T † ˜ (1, 2) G (1, 2)

K (1, 2) = i|

(31.2)

with matrix elements consisting of 2 × 2 matrices given by  (1, 2) = F (1, 2) − ℘ (1, 2) ,   † (1, 2) =† F (1, 2) − ℘∗ (1, 2) ,  ˜ (1, 2) = G (1, 2) − ℵ (1, 2) ,  G   T ˜ G (1, 2) = GT (1, 2) − ℵT (1, 2) ,

(31.3)

where

℘ (1, 2) =

1 i|2!

ℵ (1, 2) =

1 i|2!

P

gh (1) gh (2)

11 11

P

P ∗ gh (2) gh∗ (1)

11 11

P

  , 

  .

(31.4)

In Eq. (31.4), i|ℵ (1, 2)is the one-particle reduced density matrix of the boson condensate, this is identically zero for fermions. For lack of appropriate terminology, we will simply refer to i|℘ (1, 2) as the “anomalous” one-particle reduced density matrix of the condensate for bosons, this is zero for fermions. The condensate wave ˆ (i) = ψ ˜ † (i) . function is given by gh (i) = ψ 387

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The inverse of the 2-point correlation function is determined from Eqs. (30.50) and (30.55), where V (12) indicates all one-body operators, such as the kinetic energy operator and the external potential. From Eqs. (30.34), (30.39), and (30.50), we have the matrix form of V (12), taking into account the permutation of indices, given by   0 0 υ ¯ (2, 1) 0  0 0 0 −¯ υ (1, 2)   V (1, 2) =   υ ¯ (1, 2) 0 0 0 0 − υ ¯ (2, 1) 0 0 ≡

ϑT (1, 2) 0

0 ϑ (1, 2)

,

(31.5)

2

where υ ¯ (1, 2) = δ (1, 2) i=1

δ (t1 − ti ) υ (1, 2). The first term of Eq. (30.50) has the

matrix form given by (τ 4 )

−1

i|δ (1, 2)

∂ ∂t2

 0 0 − i|δ (1, 2) ∂t∂2 0  0 0 0 i|δ (1, 2) ∂t∂2   =   i|δ (1, 2) ∂ 0 0 0 ∂t2 ∂ 0 0 0 − i|δ (1, 2) ∂t2 

≡

0 (Got )−1T (Got )−1 0

.

(31.6)

Equations (31.5) and (31.6) also define the 2 × 2 matrix elements, ϑ (1, 2) and (Got )−1 . Note that (Got )−1T does not have the exact meaning of transpose since the ˘ (1, 2) is a term in time derivative is still acting to the right. Since the self-energy Σ −1 the expression for the inverse correlation function K (1, 2), we write this in the form   c cT ∆ee ∆< Σ>T ee Σ † ac ∆ (1, 2) ΣT (1, 2) ΣacT  ee ∆ee Σ  (1, 2) . ˘ (1, 2) = (31.7) = Σ c < c   Σ Σ (1, 2) ∆ (1, 2) Σ ∆hh ∆< hh > > ac ac Σ Σ ∆hh ∆hh

Here it is assumed that the ‘singular’ part (delta function in time) of the self-energy has been absorbed in the V (1, 2), otherwise it is understood to be the singular part of the diagonals of ∆ and G, in ∆cee , Σc , etc. We write the resulting equations for the 2 × 2 matrix elements of Eq. (31.1) as ˜ ˜ G−1 o G = δ + ΣG + ∆

†

,

˜ T = δ + ΣT G ˜T + †∆ Go−1T G

(31.8) ,

(31.9)

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Go−1T

†

G−1 o

= ΣT

=Σ

†

+

+

†

389

˜ ∆ G,

(31.10)

˜T . ∆G

(31.11)

We determined from Eqs. (31.1)-(31.7) and Eqs. (30.10)-(30.13) the following relations   ¯ (1, 2) 0 i|δ (1, 2) ∂t∂2 − υ  , G−1 (31.12) o (1, 2) = 0 − i|δ (1, 2) ∂t∂2 − υ ¯ (2, 1) 

Go−1T (1, 2) = 

−

¯ (2, 1) i|δ (1, 2) ∂t∂2 + υ 0

˜ (1, 2) = G

=

(1, 2) =

=

†

(1, 2) =

=

0 ¯ (1, 2) i|δ (1, 2) ∂t∂2 + υ

Gc (1, 2) G< (1, 2)

˜ c (1, 2) G ˜ < (1, 2) G

,

˜ ac (1, 2) ˜ > (1, 2) G G

> ac (1, 2) ghh (1, 2) ghh < c g˜hh (1, 2) g˜hh (1, 2) > ac (1, 2) g˜hh (1, 2) g˜hh

c < (1, 2) gee (1, 2) gee > ac (1, 2) gee (1, 2) gee c < g˜ee (1, 2) g˜ee (1, 2) > ac (1, 2) g˜ee (1, 2) g˜ee

,

(31.13)

− ℵ (1, 2)

G> (1, 2) Gac (1, 2)

< c ghh (1, 2) ghh (1, 2)



(31.14)

− ℘ (1, 2) ,

(31.15)

− ℘ (1, 2) .

(31.16)

Equations (31.8) and (31.11) were also given by Aronov, et al. [199] for superconductors, these equations formally resembles the well-known Gorkov equations [200] for superconductors at thermal equilibrium, except that now each quantity entering in Eqs. (31.8-31.11) are 2 × 2 matrices by virtue of the doubling of the quantum-field operators.

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31.1

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General Quantum Transport Equations

We give here the quantum transport equations for fermions only. We make use of the following relations in deriving the quantum transport equations in what follows,  Gc (12)† = −Gac (21),    G> (12)† = −G> (21),     G< (12)† = −G< (21), (31.17) c ac gee (12)† = −ghh (21),    > > gee (12)† = −ghh (21),     < < † gee (12) = −ghh (21).

Introducing the operators X X , whose eﬀect when operating on a matrix is to interchange the diagonal and oﬀ-diagonal elements, and X OX , whose eﬀect is to interchange of the oﬀ-diagonal elements only, we have the total time derivative of † given by the 2 × 2-matrix equation, i|

∂ ∂ + ∂t1 ∂t2 −1 0 = 0

†

(12)

[(Got )−1 ]T (12)† (22) 10 0

+ X OX

(Got )−1 (22) (21

Similarly, we obtained the total time derivative of i|

∂ ∂ + ∂t1 ∂t2 10 = 0− + XX

†

10 0 −1

.

(31.18)

as follows,

(12) (Got )−1 (12) (22) −1 0 0 1

2) (Got )−1T (1¯

†

(22

† T

0 01

.

(31.19)

˜ is given by The total time derivative of G i{

∂ ∂ + ∂t1 ∂t2

˜= G

10 0− + XX

˜ 22 (Got )−1 12 G 0 01

˜ (Got )−1 G

†

10 0 −1

.

(31.20a)

To obtain the desired quantum transport equations, we take the equations for the matrix elements of the above matrix equations, and express the chronological and anti-chronological quantities in terms of the advanced and retarded ones, using the relations, F c = F r + F < ; F ac = F < − F a , F c = F a + F > ; F ac = F > − F r , where F r and F a are the retarded and advanced function, respectively.

(31.21) (31.22)

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391

Summarizing, we have the following general quantum transport expressions for fermions [{A, B}means anticommutator of A and B]: i|

∂ ∂ + ∂t1 ∂t2

G< = vG< − G< vT + Σr G< − G< Σa + Σ< Ga − Gr Σ< < < − ghh ∆aee + ∆rhh gee

a r < + ∆< hh gee − ghh ∆ee ,

i|

∂ ∂ + ∂t1 ∂t2

(31.23)

G> = vT G> − G> v + Σr G> − G> Σa + Σ> Ga − Gr Σ> > > − ghh ∆aee + ∆rhh gee

a r > + ∆> hh gee − ghh ∆ee ,

i|

∂ ∂ + ∂t1 ∂t2

(31.24)

< < < < = v, ghh + ghh (Σa )T ghh + Σr ghh a r Σ> + Σ< ghh + ghh T

− ∆rhh G>

T

< Σ> − Σ< + ghh

T

+ G< ∆ahh

a T r < − ∆< hh (G ) + G ∆hh > < − ∆< hh G − G

i|

∂ ∂ + ∂t1 ∂t2

T

,

(31.25) T

> > > > = v T , ghh + ghh (Σa ) ghh + Σr ghh a r + ghh Σ< + Σ> ghh T

− ∆rhh G<

T

> + ghh Σ> − Σ<

T

+ G> ∆ahh

T

a r > − ∆> hh (G ) + G ∆hh > < − ∆> hh G − G

i|

∂ ∂ + ∂t1 ∂t2

T

,

(31.26)

< < < < a − (Σr )T gee = − vT , gee + gee Σ gee

−

Σ>

T

a r gee + gee Σ< + Σ> − Σ<

+ ∆ree G< + G>

T

T

< gee

∆aee

a rT < + ∆< ee G + G ∆ee

− G> − G<

T

∆< ee ,

(31.27)

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Nonequilibrium Quantum Transport Physics in Nanosystems

i|

∂ ∂ + ∂t1 ∂t2

> > > > = − v, gee + gee (Σa ) gee − ΣrT gee

−

Σ<

T

a r gee + gee Σ> + Σ> − Σ<

+

G<

T

∆aee + ∆ree G>

T

> gee

a r T > + ∆> ee G + (G ) ∆ee

− G> − G<

T

∆> ee .

(31.28)

Note that the equations for the retarded and advanced quantities can be derived, by using relations of the form of Eqs. (30.30)-(30.31) or Eqs. (31.21)-(31.22). The procedure makes use of the equations for the chronological Green’s functions, obtained from Eqs. (31.18)-(31.20a), coupled with the quantum transport equations given above. We will give physical meaning to the terms occurring in the above expressions by applying these general expressions in Sec. 32 to a simple two-level system which has a rich physical content at the core of quantum optics. First we give a discussion on how to perform the lattice Weyl transformation on the above quantum transport equations.

31.2

Transport Equations and Lattice Weyl Transformation

In the time-dependent analyses of open and active nanosystem and nanodevices, the quantum distribution function approach in phase space has so far been the most successful technique, as evidenced by works on resonant tunneling heterostructures [201]. The quantum distribution-function transport equations in (p, q, E, t)-space are obtained by applying the “lattice” Weyl transformation (although continuum approximation is interchangeably employed in this paper, this is not essential and we use the word “lattice” [183] when referring to solid-state problems) of the propagator and wavefunction equations for various excitations by using the following set of identities. The first set is the lattice Weyl transform of the following diﬀerential operators:  ∂ F (12) ⇔ i| ∂t Fw (p, q, E, t) ,     2 2 ∂ ∂ 2i ∂ − ∂t2 F (12) ⇔ − | E ∂t Fw (p, q, E, t) ,  ∂t21 2    p · ∇ F (p, q) , ∇21 − ∇22 F (12) ⇔ 2i q w |

i|

∂ ∂t1

+

∂ ∂t2

1

(31.29)

1 2 exp |i {p · q − Et} where the Bloch states has the representation, q| p = N| 3 3 3 [N | ⇒ h in the continuum limit], representing a traveling wave in lattice space, λ {p} with group velocity in a band λ given by dEdp . The above identities readily follow from the definition of the lattice Weyl transform, which in the continuum

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Quantum Transport Equations of Particle Systems

approximation can simply be written as i|

∂ ∂ + ∂t1 ∂t2

F (12) = i| ⇔ i|

∂2 ∂2 − ∂t21 ∂t22

∂ Fw (p, q, E, t) , ∂t

F (12)

∂ ∂ + ∂t1 ∂t2

=

i τ v τ v dv dτ e | (p·v−Eτ ) F q − , t − ; q + , t + 2 2 2 2

∂ ∂t

∂ ∂ − ∂t1 ∂t2

i

=

dv dτ e | (p·v−Eτ )

=

dv dτ

−2

∂ ∂τ

∂ ∂t

F (12) −2

i

e | (p·v−Eτ ) ∂ ∂τ

i

e | (p·v−Eτ )

−

dv dτ

−2

=−

dv dτ

i i 2 E e | (p·v−Eτ ) |

i ∂ = −2 E | ∂t ⇔−

τ v τ v F q − ,t − ;q + ,t + 2 2 2 2

∂ ∂τ

τ v τ v F q − ,t − ;q + ,t + 2 2 2 2

∂ ∂t

∂ ∂t ∂ ∂t

τ v τ v F q − ,t− ;q + ,t + 2 2 2 2 τ v τ v F q − ,t − ;q + ,t+ 2 2 2 2

τ v τ v i dv dτ e | (p·v−Eτ ) F q − , t − ; q + , t + 2 2 2 2

2i ∂ E Fw (p, q, E, t) . | ∂t

Similarly, we have ∇21 − ∇22 F (12) = (∇1 + ∇2 ) · (∇1 − ∇2 ) F (12) =

i τ v τ v dv dτ e | (p·v−Eτ ) (∇q ) · (−2∇v ) F q − , t − ; q + , t + 2 2 2 2

=

i τ v τ v dτdv (−2∇v ) · e | (p·v−Eτ ) (∇q ) F q − , t − ; q + , t + 2 2 2 2

+

dτdv

i = 2 p · ∇q | ⇔

τ v τ i v i 2 p· e | (p·v−Eτ ) (∇q ) F q − , t − ; q + , t + | 2 2 2 2 τ v τ v i dτ dv e | (p·v−Eτ ) F q − , t − ; q + , t + 2 2 2 2

2i p · ∇q Fw (p, q) . |

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Nonequilibrium Quantum Transport Physics in Nanosystems

The second set is the lattice Weyl transform of product of arbitrary operators [we used here the notation: p ≡ (p, −E) and q = (q, t)] in terms of “Poisson bracket operator”, AB (p, q) = exp

∂ (a) ∂ (b) ∂ (a) ∂ (b) · − · ∂p ∂q ∂q ∂p

| i

a (p, q) b (p, q) ,

(31.30)

or in terms of integral operator, AB (p, q) = =

1 (2π|)8 1

+ dp dq KA (p, q; p , q ) b (p , q ) − dp dq a (p , q ) KB (p, q; p , q ) ,

(2π|)8

(31.31)

where integral kernels are defined by v u ,q ∓ . 2 2 (31.32) Thus, we may write the lattice Weyl transform of a commutator [A, B] and an anticommutator {A, B} as KY± (p, q; p , q ) =

where Λ =

| 2

du dv exp

i [(p − p ) · v + (q − q ) · u] |

y p±

[A, B] (p, q) = cos Λ [a (p, q) b (p, q) − b (p, q) a (p, q)] −i sin Λ {a (p, q) b (p, q) + b (p, q) a (p, q)} ,

(31.33)

{A, B} (p, q) = cos Λ {a (p, q) b (p, q) + b (p, q) a (p, q)} −i sin Λ [a (p, q) b (p, q) − b (p, q) a (p, q)] ,

(31.34)

∂ (a) ∂p

[A, B] (p, q) =

·

∂ (b) ∂q

−

1 8

(2π|)

∂ (a) ∂q

·

∂ (b) ∂p

. In terms of integral operators, we have,

+ − dp dq KA (p, q; p , q ) b (p , q ) − b (p , q ) KA (p, q; p , q ) ,

(31.35) {A, B} (p, q) =

1 8 (2π|)

+ − dp dq KA (p, q; p , q ) b (p , q ) + b (p , q ) KA (p, q; p , q ) .

(31.36) The above expressions simplify considerably when the lattice Weyl transforms are scalar functions [99, 183]. For this case, we have the ‘lattice’ Weyl transform of a commutator and anti-commutator of two operators,[A, B] and {A, B} , respectively, given by the following expressions [A, B] (p, q) =

{A, B} (p, q) =

1 (2π|)8 1 (2π|)8

s dp dq KA (p, q; p , q ) b (p , q ) ,

c dp dq KA (p, q; p , q ) b (p , q ) ,

(31.37)

(31.38)

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395

where, KYs,c (p, q; p , q ) =

i [(p − p ) · v + (q − q ) · u] | v u v u ∓y p − ,q + . × y p+ ,q − 2 2 2 2 du dv exp

(31.39)

The lattice Weyl transform of Eqs. (31.23)-(31.24), in the absence of pairing between electrons and with the energy variable integrated out, has been numerically simulated and applied to high-speed resonant tunneling diodes by the author [99, 127], and others [165]. The collision terms were simulated using the relaxationtime approximation. For steady-state quantum transport, these equations, in the absence of electron pairing, have been shown to serve as the foundation of the Landauer-Büttiker formula in mesoscopic physics [176], [202], [203].

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Chapter 32

Generalized Bloch Equations

For multi-band quantum transport dynamics in solids, it is more convenient to consider electrons and holes. This is often the case with semiconductor physics. Thus, it is desirable to recast the general quantum transport equations, Eqs. (31.23)(31.28) in the electron-hole picture [204]. In this sense, generalized semiconductor Bloch equations (GSBE) refers to Eqs. (31.23)-(31.28) expressed in the so-called ‘defect’ representation1 (the transformation table is given in Appendix F). In GSBE, the particle kinematics within the respective bands is accounted for, as well as all relevant scattering mechanisms. Moreover, pairing interaction between electrons, between electron and hole, and between holes are also included, as well as creation of electron-hole pairs by optical excitation and impact ionization (other excitation processes is implicitly incorporated in the self-energy, as well as Auger recombination, etc.), and Zener tunneling. The first-order Hartree Coulomb pairing between an electron and a hole which leads to the formation of excitons will not be explicitly treated here as a vast literature already exist on this subject. With the Bloch band as discrete indices, Eqs. (31.23)-(31.24) becomes i|

∂ ∂ + ∂t1 ∂t2

≶

Gαβ (12) ≶

¯ T 22) − G≶ = vα δ αγ δ 1¯2 Gγβ (¯ 22 αγ (12) vγ δ γβ δ ¯ ≶ ¯ ¯ a ¯ + Σrαγ (1¯2) G≶ γβ (22) − Gαγ (12) Σγβ (22) ≶

r ¯ a ¯ ¯ ¯ + Σ≶ αγ (12) Gγβ (22) − Gαγ (12) Σγβ (22) ≶

≶

+ ∆rhh,αγ (1¯ 2) gee,γβ (¯ 22) − ghh,αγ (1¯ 2) ∆aee,γβ (¯ 22) ≶

≶

a r + ∆hh,αγ (1¯ 2) gee,γβ (¯ 22) − ghh,αγ (1¯ 2) ∆ee,γβ (¯ 22) ,

(32.1)

and similarly for Eqs. (31.25)-(31.28). The most general semiconductor Bloch equations follow from these multi-band quantum transport equations by transforming 1 See also, “Foundation of Computational Nanoelectronics” in Handbook of Theoretical and Computational Nanotechnology, edited by M. Rieth and W. Schommers (American Scientific Publishers, Stevenson Ranch, CA, 2006).

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Generalized Bloch Equations Table 32.1

397

Density Matrix and Generalized Notation

Density Matrix

Generalized Notation in Defect Representation

ρvc = cv (1) c†c (2)

e−h,> ⇒ φ†v (1) ψ †c (2) = i|gee,vc (12)

ρcv = cc (1) c†v (2) ρvv = cv (1) c†v (2) ρcc = cc (1) c†c (2)

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e−h,> ⇒ ψ c (1) φv (2) = i|ghh,cv (12)

⇒ φ†v (1) φv (2) = −i|Ge−h, (12) cc

to the semiconductor defect representation using the dictionary given in Appendix F. There the complete derivation is straightforward. It is more instructive to give the results for the multi-band quantum transport equations for α, β ⇒ v and c only in the defect representation since the resulting equations serve as the generalization of the density-matrix treatment of Mandel and Wolf [205] in quantum optics, as well as the coherent-wave treatment of Stahl and Balslev [206], which also provide a closed set of equations.

32.1

Generalized Bloch Equations in Quantum Optics

We identify the density matrix elements used by Mandel and Wolf [205] with the notations used here in Table 32.1, where the averaging account for mixed states. The generalized Bloch equations in quantum optics are thus given by the following equations, i|

∂ ∂ + (12) −Ge−h, ¯ e−h,r e−h,> ¯ + ∆ee,vc (1¯2) ghh,cv (22) − gee,vc (1¯ 2) ∆e−h,a hh,cv (22)

+

e−h,rT ¯ −Σe−h,
e−h,a ¯ e−h,> e−h,r ¯ (1¯2) ghh,cv (22) − gee,vc (1¯ 2) ∆> + ∆ee,vc hh,cv (22) e−h,> ¯ e−h,r e−h,> ¯ (1¯2) ghh,vv (22) − gee,vv (1¯ 2) ∆e−h,a + ∆ee,vv hh,vv (22)

+

e−h,
e−h,a ¯ e−h,> ¯ e−h,> e−h,r (1¯2) ghh,vv (22) − gee,vv (1¯ 2) ∆hh,vv (22) + ∆ee,vv

+

e−h,rT ¯ ¯ −Gcv (22) −Σ
,

(32.2)

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i|

∂ ∂ + ∂t1 ∂t2

e−h,> (12) gee,vc

e−h,> ¯ e−h,> T ¯ (22) − gee,vc (1¯ 2) vcc (22) = vvv (1¯2) gee,vc

+

e−h,> ¯ (1¯ 2) gee,vc (22) −Σe−h,aT vv e−h,a ¯ ¯ (1 2) ∆ − −Ge−h,
e−h,> ¯ e−h,> (¯ 22) − gee,vc (1¯ 2) Σe−h,a (¯ 22) + ∆e−h,r ee,vc (12) Gcc cc

+

e−h,a ¯ (1¯ 2) gee,vc (22) −Σe−h, ¯ ¯ (12) ∆ee,vc (22) − −Gvv

e−h,r ¯ e−h,a (¯ 22) − gee,vc (1¯ 2) Σe−h,> (¯ 22) + ∆e−h,> ee,vc (12) Gcc cc e−h,r e−h,> e−h,a ¯ + ∆hh,vv (1¯2) Ge−h,> (¯ 22) − gee,vv (1¯ 2) Σvc (22) vc

+

e−h,aT e−h,> ¯ (1¯ 2) gee,cc (22) −Σvc e−h,
e−h,r e−h,> + ∆ee,vv (1¯2) Ge−h,a (¯ 22) − ghh,vv (1¯ 2) Ge−h,> (¯ 22) vc vc

+

i|

e−h,a ¯ (1¯ 2) gee,cc (22) −Σe−h, ¯ (1¯ 2) ∆ee,cc (22) − −Gvc

∂ ∂ + ∂t1 ∂t2

(32.3)

e−h,> (12) ghh,cv

e−h,> e−h,> T − ghh,cv vvv = vcc ghh,cv e−h,> e−h, + Σe−h,r ghh,cv − Ge−h,> ∆ahh,cv cc cc e−h,r −Ge−h,rT − ghh,cv −Σe−h, vv vv hh,cv e−h,a e−h,r e−h,> + Σe−h,> ghh,cv − Gcc ∆hh,cv cc e−h,> e−h,a + Σe−h,r ghh,vv − Ge−h,> cv hh,cv ∆hh,vv e−h,> + Σe−h,> ghh,vv − Gcv ∆hh,vv cv e−h,r cv − ghh,cc −Σcv hh,cc

,

(32.4)

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399

∂ ∂ e−h,> + (12) Gcc ∂t1 ∂t2 e−h,> T ¯ Ge−h,> = vcc (12) (¯ 22) − Gcc (1¯ 2) vcc cc

e−h,r e−h,> e−h,> ¯ ¯ (1¯2) gee,vc (22) − ghh,cv (1¯ 2) ∆e−h,a + ∆hh,cv ee,vc (22)

e−h,r e−h,a ¯ (1¯2) Ge−h,> (¯ 22) − Ge−h,> (1¯ 2) Σcc (22) + Σcc cc cc

e−h,> e−h,r e−h,a ¯ ¯ (1¯ 2) gee,vc (22) − ghh,cv (1¯ 2) ∆e−h,> + ∆hh,cv ee,vc (22)

e−h,> e−h,r e−h,> ¯ (1¯2) Ge−h,a (¯ 22) − Gcc (1¯ 2) Σcc (22) + Σcc cc e−h,r e−h,> ¯ e−h,> e−h,a ¯ ¯ ¯ (12) Gvc (22) − Gcv (12) Σvc (22) + Σcv e−h,r e−h,> e−h,> ¯ ¯ (1¯2) gee,cc (22) − ghh,cc (1¯ 2) ∆e−h,a + ∆hh,cc ee,cc (22)

e−h,> e−h,r e−h,> ¯ (1¯2) Ge−h,a (¯ 22) − Gcv (1¯ 2) Σvc (22) + Σcv vc

e−h,> e−h,r e−h,a ¯ ¯ (1¯ 2) gee,cc (22) − ghh,cc (1¯ 2) ∆e−h,> + ∆hh,cc ee,cc (22) .

(32.5)

In applying these equations to quantum optics [205], we will neglect the particleparticle pairing terms2 which are contained in the last four brackets of every equations, we will also neglect terms arising from the particle self-energies Σ as well as terms involving ∆e−h,≶ , and all dissipative terms. These result in a closed set of equations. i|

∂ ∂ + ∂t1 ∂t2

e−h,
e−h,> ¯ e−h,> ¯ ¯ e−h,a ¯ = ∆e−h,r ee,vc (12) ghh,cv (22) − gee,vc (12) ∆hh,cv (22) ,

i|

∂ ∂ + ∂t1 ∂t2

e−h,> (12) gee,vc

e−h,> ¯ e−h,> T ¯ = vvv (1¯2) gee,vc (22) − gee,vc (1¯ 2) vcc (22) e−h, ¯ ¯ ∆ee,vc (¯ + Gvv (12) 22) + ∆ee,vc (1¯ 2) Gcc (22) ,

i|

∂ ∂ + ∂t1 ∂t2

∂ ∂ + ∂t1 ∂t2

(32.8)

(12) Ge−h,> cc

e−h,> ¯ e−h,> ¯ ¯ e−h,a ¯ = ∆e−h,r hh,cv (12) gee,vc (22) − ghh,cv (12) ∆ee,vc (22) . 2 We

(32.7)

e−h,> e−h,> e−h,> T (12) = vcc ghh,cv − ghh,cv vvv ghh,cv e−h, ∆ahh,cv , − ∆e−h,r cc hh,cv Gvv

i|

(32.6)

will only retain the electron-hole pairing terms.

(32.9)

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The equivalent optical Bloch equations corresponding to Eqs. (32.6)-(32.9) are obtained by noting the locality of the spatiotemporal argument and using the follow> < ing identities in the limit of equal time argument: Fαβ = Fαβ for the electron-hole pairing correlations. These are the following equivalent expressions, ∂ ∂ + ∂t1 ∂t2

e−h,< (21) −i|Gvv

e−h,< ¯ e−h,r e−h,< ¯ = ∆ee,vc (1¯2) ghh,cv (22) − gee,vc (1¯ 2) ∆e−h,a hh,cv (22) ,

i|

∂ ∂ e−h,< + (12) gee,vc ∂t1 ∂t2 e−h,< ¯ e−h,< T ¯ = vvv (1¯2) gee,vc (22) − gee,vc (1¯ 2) vcc (22)

e−h,a ¯ e−h,r + Ge−h,< (¯21) ∆ee,vc (22) + ∆ee,vc (1¯ 2) Ge−h,> (¯ 22) , vv cc

i|

∂ ∂ + ∂t1 ∂t2

(32.11)

e−h,< e−h,< e−h,< T (12) = vcc ghh,cv − ghh,cv vvv ghh,cv e−h,> − Ge−h,< ∆ahh,cv , − ∆e−h,r cc hh,cv Gvv

i|

(32.10)

∂ ∂ + ∂t1 ∂t2

(32.12)

e−h,< (12) Gcc

e−h,r e−h,< e−h,< ¯ ¯ = ∆hh,cv (1¯2) gee,vc (22) − ghh,cv (1¯ 2) ∆e−h,a ee,vc (22) .

(32.13)

Taking the zero of energy at the middle of the energy gap, we let vcc = −vvv = |ω2 o . Upon substituting the corresponding quantities in Table 32.1, and identifying3 e−h,a ¯ ¯ ∆e−h,r 2 , and ∆ee,vc (21) ⇒ v| HI (t) |c δ 1¯ 2 , with the relahh,cv (12) ⇒ c| HI (t) |v δ 1¯ e−h,r e−h,a e−h,a ¯ e−h,r ¯ tion ∆ee,vc (21) = ∆ee,vc (21), ∆hh,cv (1¯ 2) = ∆hh,cv (1¯ 2) by virtue of the locality of time-dependence, we end up with ∂ 1 ρ = [ v| HI (t) |c ρcv − ρvc c| HI (t) |v ] , ∂t vv i|

(32.14)

1 ∂ ρ = [−|ω o ρvc + v| HI (t) |c (ρcc − ρvv )] , ∂t vc i|

(32.15)

1 ∂ ρcv = [|ωo ρcv + c| HI (t) |v (ρvv − ρcc )] , ∂t i|

(32.16)

1 ∂ ρcc = − [ v| HI (t) |c ρcv − ρvc c| HI (t) |v ] . ∂t i|

(32.17)

3 This definition is arbitrarily chosen to reproduce the form of the general quantum transport equations, Eqs. (31.23)-(31.28), which account for pairing dynamics. Here, in the defect representation of the optical Bloch equations, we have only electron-hole pairing, representing the oﬀ-diagonal elements.

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Generalized Bloch Equations

401

These equations are exactly the Bloch equations of a two-level atom. These equations are indeed very rich in physics. In what follows, we follow the discussions given by Mandel and Wolf4 in revealing some important physics of the optical Bloch equations as this relates to other two-state systems.

32.2

The Bloch Vector Representation

In what follows we will cast the equations into a more physically meaningful form, which will give a clearer picture of the issuing quantum dynamics of a two-level system. To do this, we first express the evolution equations in terms of the Bloch vector representation. In this representation, we make the following substitutions †

> > Sx = i|gee (v, c) + (i|gee (v, c)) > > = i|gee (v, c) + i|ghh (c, v) > = 2 Re i|gee (v, c) ,

> > i|gee (v, c) − (i|gee (v, c)) > > = |gee (v, c) − |ghh (c, v) > = 2 Im i|gee (v, c) ,     Sz = i| (G> (c, c) + G< (v, v))    > <  ro = i| (G (c, c) − G (v, v)) = T rρ = 1,      1 >  i|G (c, c) = 2 (1 + Sz ) ,     1 <  −i|G (v, v) = 2 (1 − Sz ) ,    1 > i|gee (v, c) = 2 (Sx + iSy ) .

Sy =

It follows that

†

1 i

                      

(32.18)

2

2

> (v, c) + i| G> (c, c) + G< (v, v) Sx2 + Sy2 + Sz2 = 4 i|gee

.

(32.19)

2

Since [i| (G> (c, c) − G< (v, v))] = 1, then we have i|G< (v, v)

2

+ i|G> (c, c)

2

= 1 + 2i|G< (v, v) i|G> (c, c) .

Therefore, we have (by virtue of spatiotemporal locality) > Sx2 + Sy2 + Sz2 = 1 − 4|2 G< (v, v) G> (c, c) + gee (v, c)

2

≤1

> (v, c) = 1 − 4 −i|G< (v, v) i|G> (c, c) − i|gee

= 1 − 4 |φv |2 |ψc |2 − φ†v ψ†c = 1 for pure state.

2

(32.20)

2

(32.21) (32.22)

4 For more discussions, see the book by L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics” (Cambridge University Press, New York, 1995).

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where by virtue of Schwartz inequality |φv |2 |ψc |2 − φ†v ψ†c

2

≥0

with equality only if either φv or ψc is zero, meaning when the ensemble degenerates to a single realization, hence a pure state. We can write Sx = r sin θ cos φ, Sy = r sin θ sin φ, Sz = r cos θ, where r = > (v, c)|2 . Then, we have 1 − 4|2 G< (v, v) G> (c, c) + |gee

 i|G> (c, c) = 12 (1 + Sz ) = 12 (1 + r cos θ) ,  −i|G< (v, v) = 12 (1 − Sz ) = 12 (1 − r cos θ) ,  > i|gee (v, c) = 12 (Sx + iSy ) = 12 r sin θ eiφ ,

which gives cos θ = (i|G> (c, c) + i|G< (v, v)) /r, and tan φ = state, r = 1, we have  i|G> (c, c) = 12 (1 + Sz ) = 12 (1 + cos θ) ,  −i|G< (v, v) = 12 (1 − Sz ) = 12 (1 − cos θ) ,  > i|gee (v, c) = 12 (Sx + iSy ) = 12 sin θ eiφ .

(32.23) Sy Sx .

In the pure

(32.24)

Hence, we arrive at the following expressions, |ψc | = |φv | = > i|gee (v, c) =

1 2

(1 + cos θ) = cos θ2 ,

1 θ 2 (1 − cos θ) = sin 2 , 1 1 2 (Sx + iSy ) = 2 sin θ

eiφ ,

which lead us to

      

(32.25)

arg φv − arg ψc = φ since cos

θ 1 θ sin = sin θ. 2 2 2

Thus, we can represent up to a phase factor an arbitrary pure atomic state field ˆ (in second quantization), expanded in terms of |v and |c , as operator ψ ˆ = sin θ ei φ2 |v + cos θ e−i φ2 |c , ψ 2 2

(32.26)

which in turn leads us to the concept of phase operator θ. However, the exact definition of the proper dynamical variable corresponding to the phase of a quantum field is not without a long-standing controversy since the early days of quantum mechanics. We will not go into this issue here. Equation (32.26) is usually given in ˆ , and hence the phases become terms of wavefunction, which we denote here as ψ c-numbers.

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32.3

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Bloch Vector Equations

We obtain the following equations for the three components of S  ∂ ∂ > ∂ > c) + i| ∂t ghh (c, v)  ∂t Sx = i| ∂t gee (v,  r a  ∆ (v,c)−∆hh (c,v)   Sz = −ω o Sy + ee  i|   Im ∆ree (v,c)   = −ωo Sy + 2 S , z | ∂ ∂ > ∂ > S = | g (v, c) − | g (c, v) y ee ∂t ∂t ∂t hh  r  (v,c)  = ω o Sx − 2 Re ∆ee Sz ,  |   ∂ ∂ ∂ > <  S = i| G (c, c) + i| G (v, v)   ∂t z ∂t ∂t  2 2 r r = | Re ∆hh (c, v) Sy + | Im ∆hh (c, v) Sx .

(32.27)

We define the following

Re ∆ree (v, c) , | Im ∆ree (v, c) By = 2 , | Bz = ω o .

Bx = 2

Then we have

(32.28) 

∂  ∂t Sx = −Bz Sy + By Sz ,   ∂ S = B S − B S , y z x x z ∂t    ∂ ∂t Sz = Bx Sy − By Sx ,

(32.29)

which can be expressed in vector notation as

∂ S = B × S. ∂t

(32.30)

which is a precision of S about B, similar to the precision of angular momentum S in a magnetic field B. Note that S rotates about the z-axis with frequency Bz = ω o . 32.4

Atomic Energy and Dipole Moment

The vector S is related to expectation values of certain physical variables. This is expected since the time evolution equation for S corresponds to the Heisenberg equation of motion for the corresponding operator.5 The atomic energy is related to Sz , i.e., 1 HA = Eo + |ωo Sz . 2

(32.31)

The dipole moment µvc is a real quantity when the transition between the lower ‘s’ and upper ‘p’ states has ∆m = 0 or complex when ∆m = 1, such as might be induced by circularly polarized light. When µvc is real then µ = µvc Sx , when µvc 5 The

corresponding operator are directly related to the Pauli spin matrices.

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2 is complex then µ = Re µvc Sx +Im µvc Sy . For example, for real µvc = 128 243 eao z, 128 for complex µvc = − 243 eao (x + iy), where (x, y, z) are unit vectors in the Cartesian directions. We will thus take

µ = Re µvc Sx + Im µvc Sy

(32.32)

as the general expression for µ . The atomic dipole interaction with classical field is of the form HI = −µ · E(t),

(32.33)

where the operator µ = Re µvc (|c v| + |v c|) + Im µvc

(|c v| − |v c|) i

= µvc |v c| + µ∗vc |c v| .

(32.34)

On the other hand, in a real atom the important interaction is that between the atomic electrons and the field, where the field enters in the combination p − ec A . 2 2 A For weak or moderately intense light beams the e2m term can be neglected. For high intensity light beams where multiphoton interactions dominate, this term becomes important. Hence, we have the simplified interaction term HI = − We identify obtain

e m

p (t) ⇒ i|

∂ ∂t µ.

e p (t) · A(ro , t). m

(32.35)

We make use of the Heisenberg equation of motion to

∂ µ = [µ, HA ] = −iω o (µvc |v c| − µ∗vc |c v|) . ∂t

(32.36)

HI = iω o (µvc |v c| − µ∗vc |c v|) · A(ro , t).

(32.37)

Therefore, we have

e Note that the interaction Hamiltonians −µ · E(t) and − m p (t) · A(ro , t) + diﬀer in the dipole approximation by a gauge transformation [207]. We make use of HI = −µ · E(t) in what follows. We identify

v| HI (t) |c ≡ ∆aee (v, c) = −µvc · E(t), c| HI (t) |v ≡ ∆rhh (c, v) = −µ∗vc · E(t).

e2 A2 2m

(32.38)

We assume that the oscillations of the electric field are centered in some frequency ω 1 close to ωo . We write E(t) = ε |E(t)| eiφ(t)

e−iω1 t + c.c.

= 2 |E(t)| {Re(ε) cos [ω 1 t − φ (t)] + Im(ε) sin [ω 1 t − φ(t)]} ,

(32.39)

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where |E(t)| eiφ(t) is a slowly varying complex amplitude, and ε is a unit polarization vector. Then ∆aee (v, c) = −µvc · ε |E(t)| e−i[ω1 t−φ(t)] + c.c. = ∆ree (v, c) , ∆rhh (v, c) = −µ∗vc · ε |E(t)| e−i[ω1 t−φ(t)] + c.c. = ∆ahh (v, c) .

(32.40)

These should then be substituted in the Bloch equations, Eq. (32.29). The resulting equations simplify in certain cases, for example, for ∆m = ±1 transition we may y) set µvc = |µvc | (ex+ie (x and y are unit vectors), with the external field circularly 2 y) polarized and propagating in the z-direction, ε = (ex+ie . Then µvc · ε = 0 and µvc · 2 ∗ ε = |µvc | , and we obtain Re ∆ree (v, c) | = −Ω cos [ω1 t − φ(t)] ,

Bx = 2

Im ∆ree (v, c) | = −Ω sin [ω 1 t − φ(t)] ,

(32.41)

By = 2

Bz = ω o .

(32.42) (32.43)

The Rabi frequency Ω is defined as Ω=

2 µ · ε∗ |E(t)| . | vc

(32.44)

If we are dealing with ∆m = 0 transition, when µvc may be taken to be a real vector (By = 0), and if the incident light is linearly polarized so that ε is real then (note that the c.c. contributes here resulting in the doubling of Bx ) ∂ Sx = −Bz Sy + By Sz (≡ 0) ∂t = −Bz Sy , ∂ Sy = Bz Sx − 2Bx Sz , ∂t 2 2 ∂ Sz = 2 Re ∆rhh (c, v) Sy + Im ∆rhh (c, v) Sx (≡ 0) ∂t | | = 2Bx Sy ,

(32.45)

and the eﬀective B becomes  Bx = −2Ω cos [ω1 t − φ(t)] ,  By = 0,  Bz = ω o = Bx .

(32.46)

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Rotating Wave Approximation

Observe that the two set of equations for ∆m = ±1 and ∆m = 0 actually diﬀer only by the contributions of certain anti-resonant terms. We may consider the B vector as a sum of two vectors given by B = B+ [−Ω cos (ω 1 t − φ(t)) , Ω sin (ω1 t − φ(t)) , 0] ,

(32.47)

where the second vector, often referred to as auxiliary B -vector, represents a vec∂ tor rotating about the z-axis at frequency ∂t (ω1 t − φ(t)) whereas the Bloch vector rotates in the opposite direction around the z-axis at frequency ω o . Thus relative to ∂ φ(t). the Boch vector S, the auxiliary B -vector rotates at frequency (ω 1 + ω o ) − ∂t Thus when we integrate the equations of motion over any measurable time interval, its eﬀect is expected to be very small. We are therefore justified in neglecting this auxiliary B -vector to a good approximation. This procedure is often referred to as the rotating wave approximation, it allows us to use the same set of equations for ∆m = ±1 and ∆m = 0 , where it is exact for ∆m = ±1 transition. 32.6

Transformation to Rotating Frame

The Bloch vector S described by the equation of motion, Eq. (32.29) still rotates at optical frequency about the z-axis at frequency ωo . It is desirable to describe the motion in a rotating frame of reference in which S rotates more slowly. We can not use the frame rotating at atomic frequency ω o because this could vary from one atom to another in a real medium. It is customary to let the frame rotate at the frequency ω 1 of the applied field. Then the Bloch vector S in the rotating frame is related to the Bloch vector S in the stationary frame by the transformation S = Θ S,

(32.48)

 cos ω 1 t sin ω1 t 0 Θ =  − sin ω 1 t cos ω 1 t 0  . 0 0 1

(32.49)

Sx + Sy = (Sx + Sy ) e−iω1 t .

(32.50)

where 

a transformation which leaves Sz unchanged. We therefore have

In order to transform the Bloch equations [Eq. (32.27), using Eqs. (32.41)-(32.43)], ∂ ∂t Sx ∂ ∂t Sy ∂ ∂t Sz

= −ω o Sy − Ω sin [ω 1 t − φ(t)] Sz , = ω o Sx + Ω cos [ω 1 t − φ(t)] Sz ,

= −Ω cos [ω1 t − φ(t)] Sy + Ω sin [ω1 t − φ(t)] Sx ,

      

(32.51)

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we write this in the form ∂ S =RS, ∂t

(32.52)

where 

 0 −ω o By −Bx  R =  ωo 0 −By Bx 0   − sin [ω1 t − φ(t)] 0 − ωΩo 0 cos [ω1 t − φ(t)]  . = Ω  ωΩo sin [ω 1 t − φ(t)] − cos [ω 1 t − φ(t)] 0

Thus, we have

∂S ∂Θ ∂ S = S+Θ ∂t ∂t ∂t ∂Θ −1 Θ ΘS + ΘRΘ−1 ΘS = ∂t ∂Θ −1 Θ + ΘRΘ−1 S . = ∂t

(32.53)

Upon substituting the expressions for Θ and R, we obtain the Bloch equations in the rotating frame  = (ω1 − ω o ) Sy + Ω sin φ(t)Sz ,    ∂ S = − (ω − ω ) S + Ω cos φ(t)S , 1 o x z ∂t y    ∂ S = −Ω cos φ(t)S − Ω sin φ(t)S , y x ∂t z ∂ ∂t Sx

(32.54)

which can also be written as

∂ S = B ×S , ∂t

(32.55)

where  Bx = −Ω cos φ(t),    By = Ω sin φ(t),    Bz = (ω o − ω 1 ) .

(32.56)

The Bloch vector S precesses about the vector B at the rate determined by orientation and the magnitude of B , namely, B = Ω2 + (ω 1 − ω o )2

1 2

.

(32.57)

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State Preparation and Adiabatic Following Phenomenon

∂ Ω (Rabi frequency) and |ω o − ω 1 | If the detuning |ωo − ω1 | ∂t φ(t), then B points approximately down or up, and its motion is simpler in the rotating frame. If the Bloch vector S initially points in the direction that is approximately parallel to B , then S precesses about B in a cone of small angle, and this cone tends to follow slow variations of B . The Bloch vector S is then said to follow the B vector adiabatically. This adiabatic following phenomenon has been exploited to prepare the atom in a certain quantum state. Thus, if ω1 − ω o goes from a large positive value to a large negative value, then the B -vector rotates almost through 180o , as does the cone of precession. The atom starting in the ground state can thus be brought near to the excited state. Since the actual magnitude |ω o − ω 1 | of the detuning is not very important, the technique can therefore be applied to an inhomogeneously-broadened medium like a gas, in which diﬀerent atoms moving with diﬀerent velocities have diﬀerent natural frequencies in the laboratory frame because of the associated Doppler shifts.

32.7 32.7.1

Analytical Solutions of the Bloch Equations The Rabi Problem

In the special case |E(t)| eiφ(t) = constant, and the phase φ ⇒ 0 by proper choice of the origin of time, the solution to the resulting equations was given by Rabi in connection with the behavior of spin 12 in a magnetic field. If the atom starts in the lower state |v at time t = 0, so that Sz (0) = −1 and Sx (0) = 0 = Sy (0), the Bloch equations in the rotating frame, Eq. (32.54) which reduces to ∂ ∂t Sx ∂ ∂t Sy

= (ω 1 − ωo ) Sy ,

= − (ω1 − ω o ) Sx + Ω Sz , ∂ ∂t Sz

= −Ω Sy ,

   

(32.58)

  

has a solution, satisfying the initial conditions, that takes the following form [by 2 noting that the rate of precession of S is Ω2 + (ω 1 − ω o )

Sx =

(ωo −ω1 )Ω Ω2 +(ω1 −ωo )2

Sy =

−Ω

2

2

1 − cos Ω + (ω1 − ω o ) 1 )2 2

[Ω2 +(ω1 −ωo ]

sin Ω2 + (ω1 − ω o )2 1

Sz = −

(ω1 −ωo )2 +Ω2 cos[Ω2 +(ω1 −ωo )2 ] 2 t . Ω2 +(ω1 −ωo )2

1 2

1 2

1 2

],   t ,       t,        

(32.59)

The most interesting aspect of the solution is the behavior of the atomic excitation as 2 measured by Sz . This oscillates with frequency Ω2 + (ω 1 − ω o )

1 2

and amplitude

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409

about the average value Sz

time

=−

(ω 1 − ω o )2

Ω2 + (ω 1 − ωo )2

.

The physical phenomenon is known as the Rabi oscillation or optical nutation. If Ω is large, i.e., the exciting field is suﬃciently strong or the detuning |ω 1 − ω o | is suﬃciently small, then Sz oscillates almost between the values ±1, and almost at frequency Ω, the Rabi frequency. Remarkably, the applied field has the eﬀect of repeatedly exciting and de-exciting the atom. In a fully quantum mechanical treatment of this problem using quantized electromagnetic field, the eﬀects of spontaneous emission cause Sz to damp out gradually in time, even though the Rabi oscillations continue. This phenomenon has been observed in relatively recent years, in late 1970’s. The Rabi oscillation has the eﬀect of modulating the amplitude of the atomic flourescence in time. This amplitude modulation is reflected in the appearance of sidebands in the spectrum of atomic flourescence centered at frequencies ω 1 ± 1 2

Ω2 + (ω 1 − ωo )2 . To see this, we use Eq. (32.59) to calculate the expectation value of the dipole moment µ = Re µvc Sx + Im µvc Sy, with the help of   cos ω 1 t sin ω1 t 0 Θ =  − sin ω 1 t cos ω 1 t 0  . 0 0 1

If we take µvc to be real, we obtain

µ = µvc Sx cos ω1 t − Sy sin ω 1 t 

(ω o − ω 1 ) cos ω 1 t

1  1 2 2 2 + + (ω − ω ) − (ω o − ω 1 ) Ω 1 o 2   1   × cos ω 1 − Ω2 + (ω1 − ω o )2 2 t µvc Ω  = 1 Ω2 + (ω 1 − ω o )2   1 2 2 2 − + (ω o − ω 1 )  2 Ω + (ω 1 − ωo )  1  2 2 t × cos ω 1 + Ω2 + (ω1 − ω o )

The dipole moment oscillates at three frequencies ω 1,



      .      

2 ω 1 ± Ω2 + (ω1 − ω o )

1 2

.

Indeed, a three-peaked spectrum has been observed in resonance flourescence experiments in the 1970’s. The Bloch equations are sometimes augmented by adding phenomenological damping terms. The exact terms are embodied in Eqs. (32.2)-(32.5). The damped oscillatory solutions of the phenomenologically modified equations were first obtained by Torrey [208] in 1949.

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Response to Light Pulse

Actually the Rabi solution, Eq. (32.59) can be used when the atom is subjected to a rectangular exciting pulse for the duration of the pulse. The general solution for an arbitrary exciting field is much more complicated. We can simplify the light pulse problem by supposing that the φ(t) ⇒ 0 and ω 1 ⇒ ω o . Then the Bloch equations in the rotating frame given by Eq. (32.54) simplify to ∂ ∂t Sx ∂ ∂t Sy ∂ ∂t Sz

= 0,

= Ω Sz ,

= −Ω Sy .

      

(32.60)

The Bloch vector S moves entirely in the y , z -plane and Sx is a constant. Suppose that the initial quantum state is a pure state, S = 1, and rotates in the y , z plane. We put 2

1 2

Sy = 1 − Sx

sin [Θ(t) + C] 2

1 2

= 1 − Sx

{sin Θ(t) cos C + cos Θ(t) sin C} , 2

1 2

Sz = 1 − Sx

cos [Θ(t) + C] 2

= 1 − Sx 2

2

1 2

{cos Θ(t) cos C − sin Θ(t) sin C} , 2

so that Sx + Sy + Sz = 1 at all times. Here Θ(t) is a new variable and C is a constant. Substituting in Eq. (32.60) above leads to the equation for the new variable Θ(t), ∂ Θ(t) = Ω (t), ∂t t

Θ(t) =

Ω (t )dt , −∞

Θ(−∞) = 0.

Expressing C in terms of initial values at t = −∞, when we suppose the exciting field has not yet been turned on, we have the solution Sx (t) = Sx (−∞), Sy (t) = Sy (−∞) cos Θ(t) + Sz (−∞) sin Θ(t), Sz (t) = −Sy (−∞) sin Θ(t) + Sz (−∞) cos Θ(t).

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This solution represents a rotation of the Bloch vector in the y , z -plane about the x-axis through an angle Θ(t), known as the tipping angle associated with the time-dependent optical field. At the end of the pulse whose amplitude is non-zero for a limited time only, then the Bloch vector has been rotated through an angle A ≡ Θ(∞) =

∞

Ω (t )dt ,

−∞

where A is sometimes referred to as an area of the pulse. For example, if the atom starts at |v hence Sx (t) = 0, Sy (t) = 0, Sz (t) = − cos Θ(t) = −1 + 2 sin2

1 Θ(t) , 2

and A = π, then it ends up in the excited state Sz (∞) = −1, i.e., |c at the end of the pulse. Such light pulse is known as the π- pulse. If A = 2π, the Bloch vector makes a complete revolution in the y , z -plane and the atom ends up back in the state |v . Note that the rotation is determined entirely by the area of the pulse and it is independent of the pulse shape. 32.7.3

Self-Induced Transparency

Self-induced transparency is intriguing since this phenomenon is similar to the propagation of a signal in a computer medium, as typified by a ring oscillator consisting of restoring (amplifying) logic gates in an inverter chain. If detuning is non-zero, i.e., ω 1 = ωo , the motion of the Bloch vector no longer depends on only one variable Θ(t). The state in which the atom is left at the end of the pulse depends both on the pulse shape and on the detuning (ω o − ω 1 ) in general. We can still find a special form of 2π-pulse that has the property of leaving a lower state |v in lower state |v irrespective of detuning (ωo − ω 1 ). Thus, instead of the equation for on resonance, Sz (t) = −1 + 2 sin2

1 Θ(t) , 2

which starts at Sz (−∞) = −1, increases to +1 and returns to −1 after a 2π pulse, the plausibility argument is made that an atom that starts and ends in the lower state undergoes a somewhat smaller amplitude cycle with the ansatz Sz (t)(t) = −1 + 2B sin2

1 Θ(t) , 2

which introduced another variable B where B ≤ 1 and depends on detuning such B ⇒ 1 as ω 1 ⇒ ω o . The last two equations of the Bloch equations, Eq. (32.54)

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with φ(t) = 0, Eq. (32.58), give ∂ S (t) = −Ω Sy , ∂t z which in turn yields ∂

Sy = − ∂t

∂ Sz (t) Sz (t) = − ∂t∂Θ(t) Ω ∂t

= −B sin Θ(t). The second equation of Eq. (32.58) is ∂ S = − (ω 1 − ω o ) Sx + Ω Sz , ∂t y which can be solved for the expression of Sx , using the expression for Sy and Sz above, −

∂Θ(t) = − (ω1 − ω o ) Sx , ∂t

−B cos Θ(t) −1 + 2B sin2 12 Θ(t)

which yields Sx =

{1 − B} ∂Θ(t) . (ωo − ω 1 ) ∂t

We can obtained equation for Θ(t) by substituting the values of Sx , Sy , Sz in the first equation of the Bloch equations, Eq. (32.58) to obtain B (ω o − ω 1 )2 ∂2 sin Θ(t) Θ(t) = ∂t2 1−B 1 = 2 sin Θ(t), T 1 2 B ∂ Θ(t) = 2 (ω o − ω 1 ) sin ∂t 1−B = Ω (t),

1 Θ(t) 2

which determines the shape Ω (t) of the 2π-pulse. Note that for any detuning T12 is still arbitrary by virtue of the presence of the parameter B. Hence for any value of T we can calculate B as B=

1 1 + (ω o − ω 1 )2 T 2

,

which satisfies the required solution on B for any T. The equation ∂ Θ(t) = 2 ∂t

B 1−B

1 2

(ω o − ω 1 ) sin

1 Θ(t) 2

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describes a simple pendulum and is directly integrable to yield Θ π

1 dΘ = T 2 sin 12 Θ

t

dt, to

where t = to when Θ = π and is otherwise arbitrary. Hence t − to 1 , ln tan Θ = 4 T t − to , T 2 t − to ∂ Ω = Θ(t) = sec h ∂t T T

Θ(t) = 4 arctan exp

(32.61) ,

and therefore the electric field |E(t)| of the applied pulse has the shape of a hyperbolic secant. Using Eq. (32.61), we can now easily get the expression for sin 12 Θ and sin Θ. These are t − to , T t − to tanh sin Θ = 2 sec h T

1 sin Θ = sec h 2

t − to T

.

Therefore the solutions of the Bloch equations in the rotating frame are {1 − B} ∂Θ(t) (ω 1 − ω o ) ∂t 2 (ω o − ω 1 ) T t − to = 2 2 sec h T 1 + (ω o − ω 1 ) T

Sx =

Sy = −B sin Θ(t) =−

2

2

T2

sec h

t − to T

1 + (ω o − ω1 ) 1 Sz (t) = −1 + 2B sin2 Θ(t) 2 1 = −1 + 2 sec h2 1 + (ω o − ω 1 )2 T 2

,

t − to T

tanh

t − to T

,

,

a solution first given by McCall and Hahn [209]. This result can be applied to inhomogeneously broadened material medium or group of atoms, in which diﬀerent atoms have diﬀerent resonant frequencies ω o . If these atoms are subjected to the hyperbolic secant 2π-pulse at frequency ω 1 , each atom undergoes a diﬀerent cycle of excitation , but in such a way that all of them are back in the lower level at the end of the pulse. Therefore, in eﬀect no energy is absorbed out of the exciting pulse, even though the frequency of the light is close to the atomic resonance, where the medium is expected to be highly absorbing. This is the basis of the phenomenon of self-induced transparency discovered by McCall and Hahn [209] (1967, 1969).

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Fig. 32.1 Bloch sphere traces of the motion of Bloch vector in response to hyperbolic secant 2π pulse for various detunings ∆ω. All the atoms start and end in the lower state |1 . Here the z-axis is drawn with negative values up. (Reproduced from McCall and Hahn, 1969.)

There are other interesting physics of two-level systems that can be obtain by proper quantum control, i.e., in designing pulses and sequence of pulses, such as in photon echo experiments where a sequence of π2 -pulse followed after a certain time interval by a stronger π-pulse will result in photon echo in atomic systems, or spin echo in spin systems. These are beyond the scope of this chapter.

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Chapter 33

Generalized Coherent-Wave Theory

Stahl and Balslev (S-B) [206] introduced the coherent wave theory in discussing the electrodynamics of semiconductor band edge. The correlation functions they
i|

∂ ∂ + ∂t1 ∂t2

−Ge−h,
(¯ 22) − −Ge−h,
T ¯ vvv (22)

e−h,
e−h,> ¯ e−h,r e−h,> ¯ + ∆ee,vc (1¯2) ghh,cv (22) − gee,vc (1¯ 2) ∆e−h,a hh,cv (22)

+

e−h,
e−h,a ¯ e−h,> e−h,r ¯ (1¯2) ghh,cv (22) − gee,vc (1¯ 2) ∆> + ∆ee,vc hh,cv (22) e−h,> ¯ e−h,r e−h,> ¯ (1¯2) ghh,vv (22) − gee,vv (1¯ 2) ∆e−h,a + ∆ee,vv hh,vv (22)

+

e−h,aT (1¯ 2) −Ge−h,
e−h,a ¯ e−h,> e−h,r ¯ + ∆ee,vv (1¯2) ghh,vv (22) − gee,vv (1¯ 2) ∆e−h,> hh,vv (22)

+

¯ −Ge−h,rT (¯ 22) −Σ
,

(33.1)

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Nonequilibrium Quantum Transport Physics in Nanosystems Table 33.1

S-B Tight Binding

Generalized Notation

s†λ,j = c†j d†λ,j

Yij† = c†j d†i

e−h,< ⇒ ψ†c (2) φ†v (1) = −i|gee,vc (12)

sλ,j = dλ,j cj

Yij = di cj

nj = c†j cj

Cij = c†i cj

pλν,j = d†λ,j d†ν,j

Dij = di dj

i|

∂ ∂ + ∂t1 ∂t2

e−h,> ⇒ φv (1) ψc (2) = i|ghh,vc (12)

⇒ ψ†c (1) ψc (2) = −i|Ge−h,
e−h,< (12) gee,vc

e−h,< ¯ e−h,aT e−h,< (22) (1ξ) gee,vc (ξ2) vvv (1¯2) gee,vc −Σvv + e−h,< T ¯ e−h,>T e−h,a ¯ (1ξ) ∆ee,vc (ξ2) −gee,vc (12) vc (22) +Gvv

=

−i|

Coherent-Wave Correlation Functions and Generalized Notations

S-B Atomic

+

e−h,r e−h,< e−h,a (1ξ) Gcc (ξ2) (1ξ) gee,vc (ξ2) −Σe−h,>T ∆ee,vc vv + e−h,< e−h,a e−h,aT e−h,< (ξ2) (1ξ) ∆ee,vc (ξ2) −gee,vc (1ξ) Σcc +Gvv

+

e−h,a e−h,r (ξ2) (1ξ) Ge−h,< (ξ2) ∆ee,vv ∆e−h,< ee,vc (1ξ) Gcc vc + e−h,r e−h,< e−h,< e−h,a (ξ2) (ξ2) −gee,vc (1ξ) Σcc −gee,vv (1ξ) Σvc

+

e−h,aT e−h,< e−h,< e−h,a (1ξ) gee,cc (ξ2) (1ξ) Gvc (ξ2) ∆ee,vv −Σvc + e−h,a e−h,r e−h,< (1ξ) ∆ (ξ2) (1ξ) Σ (ξ2) +Ge−h,>T −g vc ee,cc ee,vv vc

+

e−h,>T e−h,a (1ξ) gee,cc (ξ2) −Σvc , e−h,aT e−h,< (1ξ) ∆ee,cc (ξ2) +Gvc

∂ ∂ + ∂t1 ∂t2

(33.2)

e−h,> (12) ghh,vc

e−h,> e−h,> T ghh,vc (ξ2) vvv (1ξ) − vc (ξ2) ghh,vc (1ξ)

=

e−h,> (ξ2) + − ghh,vc

+

e−h,r e−h,> Σvv (1ξ) − ∆e−h,r (1ξ) hh,cv (2ξ) Gvv

e−h,a e−h,> (ξ2) ∆hh,cv (ξ1) − Σe−h,r (2ξ) ghh,vc (1ξ) −Ge−h,
e−h,r e−h,> (2ξ) Σvv (1ξ) + ∆e−h,> + ghh,cv hh,vc (ξ2)

e−h,rT (ξ1) Gvv

e−h,a e−h,r e−h,> + Gcc (2ξ) ∆vc (1ξ) + Σe−h,
+

+

(2ξ) ∆e−h,a −Ge−h,< cv hh,vv (ξ1)

e−h,r +Σcv

(2ξ)

e−h,< ghh,vv (ξ1)

e−h,< −Ge−h,r hh,cv (2ξ) ∆hh,vv (ξ1)

+ Σe−h,< (2ξ) cv

e−h,a ghh,vv (ξ1)

+

+

e−h,< (2ξ) ghh,cc e−h,r −∆hh,cc

(ξ1) Σe−h,rT ee,cv

e−h,>T (2ξ) Gcv (ξ1)

e−h,r e−h,>T (2ξ) Σcv (ξ1) ghh,cc e−h,< (ξ1) − ∆ee,cc (2ξ) Ge−h,rT cv

,

(33.3)

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−i|

∂ ∂ + ∂t1 ∂t2

417

e−h,
e−h,
+

e−h,r > ¯ ¯ e−h,> ¯ (¯22) ∆e−h,a ghh,vc ee,vc (21) − ∆hh,cv (22) gee,cv (12)

+

e−h,
e−h,r e−h,> ¯ e−h,a ¯ ¯ + ghh,cv (2¯2) ∆e−h,> ee,cv (12) − ∆hh,vc (22) gee,vc (21) e−h,
+

e−h,< e−h,a ¯ e−h,r e−h,< ¯ −Gcv (2¯2) Σvc (21) − Σcv (2¯ 2) −Gvc (21)

+

e−h,< e−h,r e−h,< ¯ ¯ ¯ −ghh,cc (2¯2) ∆e−h,a ee,cc (21) − ∆hh,cc (22) −gee,cc (21)

e−h,< ¯ e−h,a ¯ + Ge−h,r (2¯2) −Σvc (21) − −Σe−h,< (2¯ 2) Gvc (21) cv cv e−h,r e−h,< ¯ e−h,a ¯ ¯ (2¯2) −∆ee,cc (21) − −∆e−h,< + ghh,cc hh,cc (22) gee,cc (21) .

(33.4)

In making contact with the work of Stahl and Balslev [206], we dropped the particle pairing terms [the terms contained in the last four brackets of each equation], and also ignore dissipative terms involving the particle self-energies Σ and ∆≶ . Then the Eqs. (33.1)-(33.4) reduce to the following semiconductor Bloch equations corresponding to those given by Stahl and Balslev [206]. i|

∂ ∂ + ∂t1 ∂t2 T vvv (1¯2)

=

(12) −Ge−h,
e−h,> ¯ e−h,r e−h,> ¯ (1¯2) ghh,cv (22) − gee,vc (1¯ 2) ∆e−h,a + ∆ee,vc hh,cv (22) ,

i|

∂ ∂ + ∂t1 ∂t2

(33.5)

e−h,< e−h,< ¯ e−h,< (12) = (vvv (1¯ 2)) gee,vc (22) − gee,vc (1¯ 2) vcT (¯ 22) gee,vc e−h,>T e−h,a (1ξ) ∆ee,vc (ξ2) + Gvv e−h,r e−h,< + ∆ee,vc (1ξ) Gcc (ξ2) ,

−i|

∂ ∂ + ∂t1 ∂t2

e−h,> (12) = ghh,vc

(33.6)

e−h,> e−h,> T ghh,vc (ξ2) vvv (1ξ) − vc (ξ2) ghh,vc (1ξ) e−h,r e−h,> (2ξ) Gvv (1ξ) − ∆hh,cv e−h,
(33.7)

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∂ ∂ + ∂t1 ∂t2

−i|

(12) −Ge−h,
e−h, e−h,a ¯ ¯ e−h,> ¯ (¯22) ∆ee,vc (21) − ∆e−h,r ghh,vc hh,cv (22) gee,cv (12)

+

33.1

.

(33.8)

The Tight-Binding Limit

In the tight-binding limit, the energy band energies in the coordinate representation correspond to the hopping integrals between diﬀerent sites. Therefore, we may write T v c the one-body terms such as vvv (1¯ 2) = −Eg δ 1¯2 − T¯21 , vcc (¯ 22) = T¯22 . The externally e−h,r (1¯ 2) ⇒ applied light field produces interband transition which we denote by ∆hh,cv e−h,a ¯ c| HI (t) |v δ 1¯2 = −Mo · F (1) δ 1¯2 , and ∆ee,vc (21) ⇒ v| HI (t) |c δ 1¯2 = −Mo · F (1) δ 1¯2 , where F is the electric field and Mo is the dipole matrix element. We also e−h,r e−h,a e−h,a ¯ ¯ ¯ ¯ (21) = ∆e−h,r have the relation ∆ee,vc ee,vc (21), ∆hh,cv (12) = ∆hh,cv (12) by virtue of the locality of time-dependence. Upon substituting the symbols in Table 33.1, we obtain ∂ ∂ + ∂t1 ∂t2 =

D12

1 v v [(−Eg δ 1¯2 − T¯21 ) D¯22 − (D1¯2 ) (−Eg δ ¯22 − T¯22 )] i| +

1 e−h,> ¯ e−h,> (−Mo · F δ 1¯2 ) i|ghh,cv (22) − i|gee,vc (1¯ 2) (−Mo · F δ ¯22 ) , (33.9) i| ∂ ∂ + ∂t1 ∂t2

−

Yij† (12) = −

∂ ∂ + ∂t1 ∂t2

Y12 =

1 † v (−Eg δ 1¯2 − T¯21 ) Y¯22 − Y1†¯2 T2c¯2 −i|

−

1 (1ξ) (−Mo · F δ ξ2 ) i|Ge−h,>T vv i|

−

1 e−h,< (ξ2) , (−Mo · F δ 1ξ ) i|Gcc i|

(33.10)

1 v c Y1ξ − T2ξ (Yξ2 ) −Eg δ 1ξ − T1ξ i| −

1 (−Mo · F δ 2ξ ) i|Ge−h,> (1ξ) vv i|

+

1 [Cξ2 (−Mo · F δ ξ1 )] , i|

(33.11)

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Generalized Coherent-Wave Theory

−

∂ ∂ + ∂t1 ∂t2 =

419

C12

1 c c [T¯ vcc C¯22 − C1¯2 T¯22 vcc ] i| 21 +

1 e−h,> (1¯ 2) Y¯ (−Mo · F δ ¯21 ) − (−Mo · F δ 2¯2 ) i|gee,cv i| 22

.

(33.12)

Making use of the following relations [from equal-time anti-commutation relation] i|G>T (12) = δ 12 + i|G e−h,> (12) = ψc (1) ψ†v (2) = − ψ†v (2) ψc (1) = −i|ghh,vc (21) , i|ghh,cv e−h,> e−h,< (12) = ψv (1) ψ†c (2) = − ψ†c (2) ψv (1) = i|gee,vc (12) , i|gee,vc e−h,> e−h,< (12) = ψ†c (1) ψv (2) = −i|gee,vc (21) , i|gee,cv

and completing substitution, noting that t1 = t2 , these equations become i iMo · ∂ † v v D12 + [D1¯2 T¯22 F (1) Y21 − Y12 − T¯21 D¯22 ] = − F (2) , ∂t | | i ∂ † † † v Y12 − Eg Y12 + T¯21 Y¯22 + Y1†¯2 T2c¯2 ∂t | iMo · [δ 12 F (2) − C21 F (1) − D12 F (2)] , =− |

(33.14)

i ∂ v c Y12 + Eg Y12 + T1ξ Yξ2 + T2ξ Y1ξ ∂t | iMo · [(δ 12 F (2) − F (1) C12 − F (2) D21 )] , = | i iMo · ∂ † c c C12 + [C1¯2 T¯22 F (1) Y12 − F (2) Y21 − T¯21 C¯22 ] = − ∂t | |

(33.13)

(33.15)

,

(33.16)

which is exactly the coherent wave equations derived by Stahl and Balslev [206]. Note that the hopping matrix Tij is assumed to be Hermitian in Ref. [206]. 33.1.1

Flat Band Case

For flat band case, we simply use the symbols under S-B Atomic in Table 33.1, putting i = j and hence taking the hopping integrals to be zero. Here we assume that the valence band consist of multiplet, and we use the indices λ and ν to distinguish two degenerate levels of the multiplet . Note that although HI (t) consist

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Nonequilibrium Quantum Transport Physics in Nanosystems

of dot-product sum over all directions, the dipole matrix element picks up only certain directions. Then we obtain iMo ∂ pλν,j = − Fλ ,j sν,j − s†λ,j Fν,j , (33.17) ∂t | ∂ † i iMo sλ,j − Eg s†λ,j = − [δ λν Fν,j − nj Fλ,j − pλν,j Fν,j ] , ∂t | |

(33.18)

i iMo ∂ sλ,j + Eg sλ,j = [(δ λν Fν,j − nj Fλ,j − pλν,j Fν,j )] , ∂t | |

(33.19)

iMo ∂ nj = − sλ,j − s†λ,j Fλ ,j , ∂t |

(33.20)

which are exactly the coherent wave equations given by in Ref. [206] for flat-band case. When expressed in terms of density, we simply replace the Dirac delta δ λν , which arise from the anti-commutation relation, by δ λν N where N = V1 and replace the index j by a representative coordinate r. We can also cast the above equations in vector-tensor notation as → i ∂← p (r) = − s (r) ⊗E (r) − E (r) ⊗s† (r) , (33.21) ∂t | i ∂n (r) → = − s (r) ·E (r) − E (r) ·s† (r) = T r← p (r) , (33.22) ∂t | i ∂s† (r) ← → → − iωo s† (r) = − E (r) · 1 (N − n (r) ) − ← p (r) · E (r) , (33.23) ∂t | ∂s (r) i ← → → + iω o s (r) = (N − n (r) ) 1 · E (r) − E (r) ·← p (r) , (33.24) ∂t | which describe the dynamics of a coupled flat conduction ‘s’-band and flat valence ‘p’-band with arbitrary state of polarization. Here we assigned the notation Eξ (r) Moξ (r) ⇒ E as used in [206]. It should be noted that the coherent-wave theoretical formalism of Stahl and Balslev [206] does not incorporate scattering as well as the dynamics of real pairing (second-order Coulomb pairing, phonon and other “excitation-induced” pairing) between electrons, between holes, and between electron and hole, which maybe important in strongly-correlated systems such as in high-Tc superconductive materials.

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Chapter 34

Impact Ionization and Zener Eﬀect

Impact ionization and its inverse, Auger recombination, are nonconserving fermion processes. It involves three valence band and one conduction band indices and vice versa [210]. These processes can be described by the second-order Coulomb selfenergies within the two-point correlation function (Green’s function) formalism [211, 212]. On the other hand, Zener eﬀect is a particle-conserving process and involves only one valence and one conduction band indices. For simplicity and with an eye to applying in type II RTD, we will drop all particle-pairing terms. We will also ignore the intraband particle self-energies, assuming that these contribute only to the renormalization of the valence and conduction bands. We will also approximate intraband scattering by a relaxation time approximation [99]. The inverse process to e−h,> e−h,< electron-hole pair creation −i|gee,vc (12) is a recombination process, i|ghh,vc (12) , †

e−h,> e−h,< due to the relation: −i|gee,vc (12) = i|ghh,vc (12). In discussing impact ionization, Auger recombination, and Zener eﬀect, it is more convenient to use the following correlation function equations in the defect representation,

−i|

∂ ∂ + ∂t1 ∂t2

e−h,< (12) Gvv

e−h,< ¯ e−h,< = Gvv (22) vT (1¯ 2) − vT (¯ 22) Gvv (1¯ 2)

+ relaxation-time approximation for intraband scattering e−h,> e−h,a e−h,> + ∆e−h,r ee,vc (2ξ) ghh,cv (ξ1) − gee,vc (2ξ) ∆hh,cv (ξ1) ,

i|

∂ ∂ + ∂t1 ∂t2

(34.1)

e−h,< (12) gee,vc

e−h,< ¯ e−h,< = vvv (1¯ 2) gee,vc (22) − gee,vc (1¯ 2) vcT (¯ 22)

(1ξ) ∆e−h,a + Ge−h,>T vv ee,vc (ξ2) e−h,< (ξ2) , + ∆e−h,r ee,vc (1ξ) Gcc

421

(34.2)

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−i| =

∂ ∂ + ∂t1 ∂t2

e−h,> (12) ghh,vc

e−h,> e−h,> T ghh,vc (ξ2) vvv (1ξ) − vc (ξ2) ghh,vc (1ξ) e−h,> (1ξ) − ∆e−h,r hh,cv (2ξ) Gvv e−h,a − Ge−h,
i|

(34.3)

∂ ∂ + G< cc (12) ∂t1 ∂t2 < ¯ ¯ T ¯ = v (1¯2) G< cc (22) − Gcc (12) v (22) + relaxation-time approximation for intraband scattering e−h,< e−h,< e−h,a + ∆e−h,r hh,cv (1ξ) gee,vc (ξ2) − ghh,cv (1ξ) ∆ee,vc (ξ2) .

(34.4)

e−h,a In the equations above, ∆e−h,r hh,cv , and ∆ee,vc contains the sum of the contribution from the second-order Coulomb pairing self-energies and Zener tunneling interband ∆. To first order Hartree Coulomb potential, these contributions are well-known and manifest in excitonic electron-hole pair. However, the nonequilibrium Coulomb pair potential to second order is seldom investigated. Indeed, the self-energy to second order in the Coulomb potential (Appendix F) readily yields Coulomb pair potential ∆ and † ∆ where †

∆=

∆=

∆cee −∆< ee ac −∆> ee ∆ee ∆chh ∆< hh ac ∆> hh ∆hh

,

.

This means that the e-e and h-h Cooper pairing terms are generally present when many-body Coulomb interaction is taken into account. These contribution may be significant in highly-correlated systems such as in high-Tc superconductivity materials. As we shall see below, these second-order terms in † ∆ and ∆ yield the impact ionization and Auger recombination processes.

34.1

Coulomb Pair Potential ∆ for Impact Ionization and Auger Recombination

The general expressions of the fermion-nonconserving Coulomb self-energy to second order is represented by the following Feynmann diagrams, where the terms corresponding to the Cooper pairing between particles are omitted [see Appendix F for the full second-order diagrams].

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Impact Ionization and Zener Eﬀect

Fig. 34.1

423

e−h,< Direct and exchange diagrams for ∆e−h,< ee,vc (1, 4 ) and ∆hh,cv (1, 4 ) .

The direct and exchange diagrams of Figs. 34.1 and 34.2 correspond to the following expressions   e−h,≶ e−h,≷ T e−h,≶ T   14 G 23 G 3 2 +g   ee,vc vv vv         e−h,≶ e−h,≶ e−h,≷  +gee,vc 14 Gcc  23 G 3 2 cc e−h,≶ −∆ee,vc (14 ) = − , e−h,≷ T e−h,≶ T e−h,≶  13 Gvv 3 2 gee,vc 24     −Gvv        e−h,≶ e−h,≷ e−h,≶  −gee,vc  13 Gcc 3 2 Gcc 24

e−h,≶

−∆hh,cv (14 ) = −

 e−h,≶ e−h,≷ T  23 +ghh,cv 14 Gvv      e−h,≶  +g e−h,≶ 14 Gcc 23 hh,cv

e−h,≶ T

Gvv

e−h,≷

Gcc

 32       32 

e−h,≶ e−h,≶ T e−h,≷ T  3 2 Gvv 24 −ghh,cv 13 Gvv      e−h,≶ e−h,≷ e−h,≶  13 Gcc 3 2 ghh,cv 24 −Gcc

      

.

We can also obtain the corresponding diagrams for the interband retarded and advanced self-energies by making the following replacement in the above graphs, e−h,< e−h,r e−h,> e−h,a e−h,< e−h,r e−h,> e−h,a ⇒ gee , gee ⇒ gee , ghh ⇒ ghh , ghh ⇒ ghh , Ge−h,
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Fig. 34.2

e−h,> Direct and exchange diagrams for ∆e−h,> ee,vc (1, 4 ) and ∆hh,cv (1, 4 ).

Ge−h,rT , Ge−h,>T ⇒ Ge−h,aT , Ge−h,< ⇒ Ge−h,r , Ge−h,> ⇒ Ge−h,a and similarly, e−h,r e−h,a ∆e−h,< (1, 4 ) ⇒ ∆hh (1, 4 ), ∆e−h,> (1, 4 ) ⇒ ∆hh (1, 4 ), ∆e−h,> (1, 4 ) ⇒ ee hh hh e−h,a e−h,< e−h,r ∆ee (1, 4 ), ∆ee (1, 4 ) ⇒ ∆ee (1, 4 ). 34.2

Pair Potential ∆ due to Zener Eﬀect

Zener tunneling is a motion of particles not confined to a single band dynamics, it arise from the interband matrix element of the position operator [213, 214]. In the correlation function and quantum superfield theoretical approach, it arises from the electron-hole pair potential ∆Z i,j (i = j, ij = c, v) due to interband component of the position operator. In the lattice Weyl-transform representation, the position operator is represented by roperator ⇔ q + Xλλ (p) .

(34.5)

In the case where the energy bands are well separated, the interband part Xλλ (p) may be ignored which leads to lattice Weyl transform of the position operator given by roperator ⇔ q, where q is the lattice point coordinate usually approximated as forming a continuum. In this case, the intraband dynamics in the presence of

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425

uniform electric field leads to Stark ladder eﬀects, viewed as a relative shift of an energy level at each lattice point of an array of identical atoms by an amount equal to eF a along the direction of the field. On the other hand, for atomic system only the polarizing part of the matrix element of the coordinate operator exists and form 2 o the so-called oscillator strength, fλλ = 2m | |Xλλ | (Eλ − Eλ ). Hence, for general energy bands in semiconductors, both the intraband band and interband portions of the matrix elements of the coordinate operator should be considered. Moreover, for real nanodevices the lattice Weyl transform of Xλλ will generally be a function of crystal momentum p and lattice coordinate q, i.e., Xλλ (p, q). For type II RTD, we may write the lattice Weyl transform of the self-consistent potential φ as consisting of intraband and interband terms [214], φ (roperator ) ⇔ φ (q) δ λλ + ∇q φ (q) · Xλλ (p, q) ,

(34.6)

where we have expanded to first order in the interband component. For flat band case, Xλλ (q) goes to zero unless Eg (q) also goes to zero, i.e., for time-independent fields interband transitions does not occur for flat bands. Hence, Xλλ (q) is often attributed to inertial eﬀect due fast passage in avoided crossing region in E (K), where K =k − |e F t. Thus, in the GSBE, the electron-hole pair potential ∆ responsible for Zener tunneling is given by (Z)

∆λλ = eF (q) · Xλλ (q) ,

λ = λ , λ, λ = c, v.

(34.7)

Typical result for the Zener tunneling probability for a narrow-gap semiconductor given by Kane and Blount [215] follows the formula√first given by the quantummr (E ) 3 mechanical calculation of Zener and goes like exp −π 2||eFg| 2 , where mr is the reduced eﬀective mass, Eg is the energy gap, e is the electron charge, and F is the uniformly applied electric field. The quantum mechanical calculation of Fredkin and √ mr (Eg ) 32 for Zener tunneling across semiconducWannier [216] goes like exp −5 3||eF | tor p-n junction, about the same as that first given by Zener. Viewed as a one-body barrier tunneling problem, by distorting the triangular potential barrier for Zener tunneling [217], one can obtained exact agreement with the quantum mechanical calculations of Zener. Zener tunneling within the correlation function formalism is due to the interband component of the position operator as discussed by Blount [213], Kane and Blount [215], and Adams [214].

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Chapter 35

Quantum Transport Equations in Phase Space

The quantum distribution-function transport equations are obtained by performing the lattice Weyl transform of Eqs. (34.1)-(34.4). After integrating over the energy variable, we have for the valence-band quantum transport equation, ∂ ∂t

fwh (p, q, t)

=−

p · ∇q fwh (p, q, t) mv

+

2π (2π|4 )

+

∂ ∂t

fwh (p, q, t)

∂ ∂t

fwh (p, q, t)

+

i [(p − p ) · v ] |

dp dv sin

v 2 − v2

Φv q + −Φv q

fwh (p , q, t)

intraband col lision

,

(35.1)

interband transition

For the electron-hole generation quantum transport equation, we have, ∂ ee f (p, q, t) ∂t w p − i|2 ∇q Eg + 2mr

i = |

2π 4 (2π|)

+ 

fwee (p, q, t) −

dp dv sin

2πi dp (2π|)4 e−h,a × ∆ee,vc

  +  − 2πi  (2π|)4

2

dp

i [(p − p ) · v ] |

dv exp q+

v 2

i |

[(p − p ) · v ]

v 2

i | fwc

dp K (p − p , q ) fwee (p , q, t)

(p , q, t) 426

v 2 − v2

eφ q + −eφ q 

  , [(p − p ) · v ]  

1 − fwh (−p , q, t)

dv exp

e−h,r × ∆ee,vc q−

i |

fwee (p , q, t)

(35.2)

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427

For the recombination process, we have, ∂ hh f (p, q, t) ∂t w =− −

i |

p + i|2 ∇q 2mr

Eg +



2π (2π|)4

2

fwee (p, q, t) +

dp dv sin

i |

i [(p − p ) · v ] |

2πi dp dv exp |i [(p − p ) · v ] (2π|)4  v × ∆e−h,r 1 − fwh (p , q, t)  ee,vc q − 2 − 2πi dv exp |i [(p − p ) · v ]  − (2π|)4 dp e−h,a × ∆ee,vc q + v2 fwc (−p , q, t)

dp K (p − p , q ) fwee (p , q, t) eφ q + v2 −eφ q − v2 

fwee (p , q, t)

  , 

(35.3)

and for the conduction band, the following expression, ∂ ∂t

fwc (p, q, t)

=− + + +

p · ∇q fwc (p, q, t) mc 2π (2π|4 ) ∂ ∂t ∂ ∂t

dp dv sin

i [(p − p ) · v ] |

Φc q + v2 −Φc q − v2

fwc (p, q, t)

fwc (p, q, t) intraband col lision

fwc

(p, q, t)

,

(35.4)

interband transition

where φ (q) is the self-consistent potential, and Φv (q) = −eφ (q) + ∆Ev (q) , Φc (q) = −eφ (q) + ∆Ec (q) . The reduced eﬀective mass is mr , and 1 2π| 1 fwee (p, q, t) = 2π| 1 hh fw (p, q, t) = 2π| 1 fwc (p, q, t) = 2π|

fwh (−p, q, t) =

K (p − p , q ) =

1 (2π|)3

e−h, dE i|ghh,vc (p, E, q, t) , e−h,< dE −i|Gcc (p, E, q, t) ,

dv exp

i [(p − p ) · v ] |

∆Ev q − v2 −∆Ec q + v2

.

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For our purpose, the complicated intraband collision term will be approximated by the relaxation time approximation, successfully used in simulating AlGaAs/GaAs RTD [99]. Thus, we have in Eqs. (35.1) and (35.4) ∂ ∂t

fwh (p, q, t) intraband col lision h

= ∂ ∂t =

1 ρ (q, t) h f (p, q, 0) − fwh (p, q, t) , τ h ρho (q) w fwc (p, q, t) intraband col lision

1 ρc (q, t) c f (p, q, 0) − fwc (p, q, t) , τ c ρco (q) w

where fwh (p, q, 0) is the Wigner distribution function of holes at zero bias, ρh (q, t) = fwh (p, q, t) is the hole density, and τ h is a constant hole relaxation time calcup

lated by considering all scattering mechanisms. Similarly, fwc (p, q, 0) is the Wigner fwc (p, q, t) distribution function of conduction electrons at zero bias, ρc (q, t) = p

is the electron density, and τ c is a constant electron relaxation time calculated by considering all scattering mechanisms [99]. The interband transition terms due to Zener tunneling is the basis of the intriguing autonomous current oscillation of type II RTD. The interband transition terms are given by ∂ ∂t

fwh (p, q, t) interband transition



2π (2π|)4

dp dv exp

i |

[(p − p ) · v ]



(35.5)



(35.6)

     v e−h,> (p , q, t) q − g ∆e−h,r = , ee;vc 2  ×  hh;cv  e−h,> e−h,a −∆ee;vc q + v2 ghh;cv (p , q, t) ∂ ∂t

fwc (p, q, t) interband transition



2π (2π|)4

dp dv exp

i |

[(p − p ) · v ]

     v e−h,< (p , q, t) ∆e−h,r q − g = , ee;vc 2  ×  hh;cv  e−h,< e−h,a −∆ee;vc q + v2 ghh;cv (p , q, t) e−h,r(Z)

e−h,a(Z)

(q ) and ∆e−h,a (q ) given by Eq. with ∆e−h,r ee;vc (q ) = ∆ee;vc hh;cv (q ) = ∆hh;cv (34.7) for the Zener electron-hole pair potential. We have used the integral in-

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429 e−h,r(Z)

stead of a gradient expansion since the Zener interband pair potential ∆hh;cv

(q )

e−h,a(Z) ∆ee;vc (q )

and can have a strong variation as a function of q near the drain-side of the barrier edge in type II RTD, Fig. 28.5. The Zener electron-hole pair correlae−h,< tion function gee (or pair wavefunction) can be approximated by neglecting the intraband self-energies and terms involving ∆≶ . Equation (35.2) is a Schrödinger equation of the pair wavefunction in the mixed p-¯ q representation. The time-dependent numerical simulation of the quantum transport of electrons and holes in type-II RTD, using Eqs. (35.1)-(35.3), present a real challenge. This simulation is needed for cost-eﬀective optimization of device structure and material properties in the design of type-II RTD devices.

35.1

Conservation of Particle in Zener Tunneling

We observe from Eq. (35.5) and Eq. (35.6) that the following equality holds, ∂ ∂t

fwh (p, q, t)

= interband transition

∂ ∂t

fwc (p, q, t)

, interband transition

e−h,< e−h,> e−h,< e−h,> since gee;vc (p , q, t) = gee;vc (p , q, t) and ghh;cv (p , q, t) = ghh;cv (p , q, t). This simply states the rate of hole generation by Zener tunneling of electrons from the valence band to the conduction band is equal to the generation rate for electrons in the conduction band.

35.2

Nanosystem Applications

In what follows we will give a brief discussion of the application of the generalized transport equations to the numerical simulation and analysis of semiconductor heterojunction nanodevices. Some of these results are discussed in Part III of this book. Readers interested in the details are referred to Refs. [99, 165, 218, 170, 127]. 35.2.1

Resonant Tunneling Diode (RTD)

Equation (35.4) has been applied to a time-dependent numerical simulation, under diﬀerent applied bias voltages, of GaAs/AlGaAs RTD, referred to here as a typeI RTD. Typical results for the quantum distribution function in phase space or the Wigner distribution function, with their corresponding conduction band-edge potential profiles, are shown in Table 35.1 Aside from the design of RTD device as a switching element, the clever design of the emitter regions of RTD in order to function as a dissipationless memory device, and as a tera-Hertz source nanodevice are discussed in Part 3 of this book.

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Nonequilibrium Quantum Transport Physics in Nanosystems Table 35.1 Potential Profile

RTD Potential and Wigner Distribution Function Wigner Distribution Function

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Chapter 36

QSFT of Second-Quantized Classical Fields: Phonons

Here we give the QSFT formalism for the quantized classical vibrational field in solid. Similar techniques can be followed for QSFT of other classical real fields, such as quantized electromagnetic fields and quantized plasma oscillations. In terms of the nonequilibrium superfield theoretical approach to a nonequilibrium electronphonon system, the Liouvillian can in general be expressed as e−Schw L = υ (1, 2) Ψ (1) Ψ (2) + υ (1, 2, 3, 4) Ψ (1) Ψ (2) Ψ (3) Ψ (4) + Lext o(2)

+ Mio1 i2 (ξ 1 ξ 2 ) U˙ i1 (ξ 1 ) U˙ i2 (ξ 2 ) + Θi1 i2 (ξ 1 ξ 2 ) Ui1 (ξ 1 ) Ui2 (ξ 2 ) ph−Schw − gri Ui (ξ) + Lext

− Xio (12; ξ) Ψ (1) Ψ (2) Ui (ξ) ,

(36.1)

where the electron superfield Schwinger source term is given by Eq. (30.33) and the phonon superfield Schwinger source term is ph−Schw Lext = −εi (ξ) Ui (ξ) ,

(36.2)

where the real phonon superfield is Ui (ξ, t) =

i

ε (ξ) =

√ mk u ˆi (ξ, t) √ mk u ˜i (ξ, t) φiSchw −φiSchw

.

,

(36.3)

(36.4)

The u ˆ (ξ, t) and u ˜ (ξ, t) are the ‘hat’ and ‘tilde’ displacement operator, respectively, of atoms in site ξ. The indices 1, 2, 3, 4 each stands for all pertinent quantum labels, and repeated indices are “summed” over. The index ξ includes the index k which labels the kth atom in the unit cell. Thus mk refers to the mass of the kth atom. Note that the boson field source terms are purposely written with negative sign while the fermion field source term takes a positive sign, following the convention analogous to the minimal coupling interaction of charge particle and electromagnetic vector potential. This leads to the equation for the average boson 431

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Nonequilibrium Quantum Transport Physics in Nanosystems

field with a positive source term. This will result in a diﬀerent sign of the factor 1 i| in the definition of both super-propagators for phonons, plasmons, and photons compared to those of fermion super-propagators, Eq. (30.55) In the Liouvillian for the real phonon superfield, Eq. (36.1,) we have (2)

φi1 i2 (ξ 1 ξ 2 ) o(2) Θi1 i2 (ξ 1 ξ 2 ) = [δ α1 ,1 δ α2 ,1 − δ α1 ,2 δ α2 ,2 ] √ 2 mk1 mk2 (2)

φi1 i2 (ξ 1 ξ 2 ) =

∂ 2 V (ξ 1 ξ 2 ....ξ N ..) ∂ui1 (ξ 1 ) ∂ui2 (ξ 2 )

j=ξ2 j=ξ1

δ (tj − t) ,

(i1 , i2 = x, y, z) ,

(36.5)

(36.6)

where V (ξ 1 ξ 2 ....ξ N ..) is the total eﬀective potential energy function for the nuclear motion, and 1 Mio1 i2 (ξ 1 ξ 2 ) = [δ α1 ,1 δ α2 ,1 − δ α1 ,2 δ α2 ,2 ] δ i1 i2 δ (ξ 1 − ξ 2 ) 2

j=ξ2 j=ξ1

δ (tj − t) ,

(36.7)

where the subscripts of the delta-function pertain to the components of the phonon superfield Ui (ξ, t). The last term in the Liouvillian L represents the electron-phonon interaction, where j=ξ

Xio (12; ξ) =

P P (12)

δ α1 ,3 δ α2 ,1 δ αξ ,1 − δ α1 ,2 δ α2 ,2 δ αξ ,2

χi (12; ξ) δ (tj − t) , √ mkξ j=1 (36.8)

χin,m (12; ξ) = r1 , n| z ∗ e∇irξ

1 |r2 , m , |x − rξ |

(36.9)

where z ∗ is the eﬀective charge of the nucleus located at ξ. The energy-band index n, m included in indices 1, 2 are explicitly written in the definition of the matrix element for χi (12; ξ) ≡ χin,m (12; ξ). In bulk materials, χi (12; ξ) is often determined using knowledge of the deformation potential [219], diagonal in band index, n = m. Various approximation schemes also treat r1 = rξ = r2 . In time-dependent perturbation theory leading to the Fermi golden rule, χi (12; ξ) enters the Fermi golden rule for the transition probability Pif =

2π | i| He−ph |f |2 δ (Ef − Ei ) , |

(36.10)

where i| He−ph |f = i| χi (12; ξ) ui (ξ, t) |f .

(36.11)

Expressing the ion displacement ui (ξ, t) at lattice point ξ in terms of the normal modes of the dynamical matrix expressed in terms phonon annihilation and creation operators and polarization vectors in bulk materials, then one finds that only longitudinal phonons interact with electrons if umklapp processes are neglected. In semiconductors with the diamond or zinc-blende structure, the two nonequivalent

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atoms per unit cell can allow electron interaction with optical vibrational modes corresponding to movement of the two atoms relative to each other. The optical electron-phonon interaction is also determined using knowledge of optical deformation potential. Electrons interact with polar modes, which is important in ionic crystals. 36.1

Liouvillian Space Phonon Dynamics

The free phonon quantum superfield equation in the super-Heisenberg or Liouvillian representation can be written as ∂2 Ui (ξ, t) = − ∂t2

ξ ,j

Dij ξ, ξ Uj ξ , t ,

(36.12)

where summation is explicitly indicated for the repeated indices and the superdynamical matrix Dij ξ, ξ is defined as (2)

Dij ξ, ξ

Θ ξξ = √i1 i2 . mk mk

(36.13)

We write the solution of Eq.(36.12) as Ui (ξ, t) = ui (k) exp we obtain1 E |

2

ui (k) = ξ ,j

Dij ξ, ξ exp

i |

(p · Xlo − Et) . Then

i p · (Xlo − Xlo ) uj (k ) . |

(36.14)

This resulting eigenvalue problem can be solved for the eigenvalues and correspond2 ing eigenvector. Denote the eigenvalues as ω oλ (p) and the corresponding eigenk,j vector as eλ (p), where λ is the index for the vibrational branch and j labels the independent spatial directions. Then we have ωoλ (p)2 ek,i λ (p) exp

i p · Xlo |

= ξ ,j

Dij ξ, ξ exp

i p · Xlo ekλ ,j (p) . |

(36.15)

Instead of the eigenvector ek,j λ (p), it is more convenient to use as basis function the following expression Bλi (p; ξ) ≡ Bλi (p; l.k) = ek,i λ (p) exp

i p · Xlo . |

(36.16)

The following orthonormality and completeness relation hold for ek,i λ (p) and Bλi (p; l.k) ∗ k,j ek,j λ (p) eλ (p) = δ λλ ,

(36.17)

k,j 1 The use of i as dummy label, either subscript or superscript, should not be confused with its √ use as multiplicative factor as i = −1.

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k ,j ek,j (p)∗ = δ kk δ jj , λ (p) eλ

(36.18)

k,j

and ∗

Bλj (p; l.k) Bλj (p; l.k) = δ λλ ,

(36.19)

k,j ∗

Bλj (p; l.k) Bλj (p; l.k ) = δ kk δ jj .

(36.20)

k,j

We also have ∗ k,j ek,j λ (p) = eλ (−p) ,

(36.21)

Bλj (p; l.k)∗ = Bλj (−p; l.k) .

(36.22)

We now expand the ion super-displacement operator in terms of the eigenfunction of the super-dynamical matrix Dij ξ, ξ , 1 Ui (ξ, t) = √ N

Qλ (p, t) Bλi (p; ξ) ,

(36.23)

λ,p

where N is the number of crystal lattice points, Qλ (p, t) is the normal mode coordinate superoperator in the super-Heisenberg representation. Since the displacement superoperator is real , then Qλ (p, t)† = Qλ (−p, t). From the canonical commutation relation, U˙ i (ξ, t) , Uj ξ , t

=

| δ ij δ ξξ i

10 01

,

(36.24)

we also have | Q˙ λ (p, t) , Qλ (p , t) = δ λλ δ p,−p i

10 01

.

(36.25)

Moreover, from the super-Heisenberg equation of motion for the displacement operator, we have ∂2 Qλ (p, t) = −ωoλ (p)2 Qλ (p, t) , ∂t2

(36.26)

which demonstrates the independent harmonic oscillator character of the normal modes. The phonon super-annihilation and super-creation operators αλ (p) and αλ (p)† , respectively, are defined in the usual fashion in terms of the independent oscillator coordinate by the following relations Qλ (p, t) =

| αλ (p, t) + α†λ (−p, t) , 2ωoλ (p)

(36.27)

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|ω oλ (p) αλ (p, t) − α†λ (−p, t) , 2

Q˙ λ (p, t) =

(36.28)

which satisfy the usual commutation relations [αλ (p, t) , αλ (p , t)] = α†λ (p, t) , α†λ (p , t) = 0, α†λ (p, t) , αλ (p , t) = δ λλ δ p,p

10 01

(36.29)

.

(36.30)

In the super-annihilation and super-creation superoperator representation, the Liouvillian for free phonon system becomes a sum of independent harmonic oscillators, L=

α†λ (p) αλ (p) + λ,p

1 2

| Ωλ (p) ,

(36.31)

where Ωλ (p) =

0 ωoλ (p) 0 −ω oλ (p)

,

(36.32)

and similarly, we have αλ (p) =

a ˆλ (p) a ˜λ (p)

α†λ (p) =

,

a ˆ†λ (p) a ˜†λ (p)

.

(36.33)

Corresponding to Eq. (36.27), we define the corresponding Hilbert space operators

36.2

qλ (p, t) =

| aλ (p, t) + a†λ (−p, t) , 2ωoλ (p)

(36.34)

qλ† (p, t) =

| αλ (−p, t) + α†λ (p, t) . 2ω oλ (p)

(36.35)

The Phonon Super-Green’s Function

The phonon super-correlation function Sij ξξ ; tt is defined by the functional derivative of the average super-displacement operator with respect to the Schwinger source for phonon displacement field Sij ξξ ; tt

=

=

δ δεj ξ

Ui (ξ, t)

c ξξ ; tt Sij > ξξ ; tt Sij

< −Sij ξξ ; tt

ac −Sij ξξ ; tt

.

(36.36)

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< The ‘reduced density matrix’ for phonons is defined by Sij ξξ ; tt by < ξξ ; tt Sij

=

i T u ˆi (ξ, t) u ˜j ξ , t |

=

1 N

=

1 N

which is given

Bλi (p, ξ) Bλj p , ξ

i T qˆλ (p, t) q˜λ (p , t ) |

Bλi (p, ξ) Bλj† p , ξ

i T r ρ qλ† (p , t ) qλ (p, t) , |

{λ,p}

{λ,p}

(36.37) where the last line is written in Hilbert space representation. Substituting the relation between qλ (p, t) and qλ† (p, t) in terms of Hilbert space annihilation, aλ (p, t), and creation operators, α†λ (p, t), using Eqs. (36.34)-(36.35), we obtain < ξξ ; tt Sij

=

1 N

Bλi (p, ξ) Bλj† p , ξ {λ,p}

× T rρ

i 2 ω oλ (p) ω oλ (p )

  aλ (−p , t ) aλ (p, t)           +aλ (−p , t ) a† (−p, t)   λ  +a†λ (p , t ) aλ (p, t)      +a†λ (p , t ) a†λ (−p, t)

     

.

(36.38)

For free phonons at thermal equilibrium, only the phonon conserving combinations, with λ = λ and p = p , survived giving the result < −iSij ξξ ; tt

=

1 N

× where

† k ,j ek,i λ (p) eλ (p) exp λ,p

1

i p · (Xlo − Xlo ) |

   (1 + n (ω o (p))) eiωoλ (p)(t−t )  λ

2ωoλ (p)  +n (ω o (p)) e−iωoλ (p)(t−t )  λ

n (ω oλ (p)) =

1 o eβ|ωλ (p) − 1

,

(36.39)

(36.40)

is the equilibrium distribution function for phonons. By virtue of translational invariance with respect to lattice coordinates and time, the Weyl transformation to phase space can immediately be written as < (kk p; ω) = −iSij

† k ,j ek,i λ (p) eλ (p) λ

2π 2ω oλ (p)

(1 + n (ω oλ (p))) δ (ω + ω oλ (p))

. +n (ω oλ (p)) δ (ω − ωoλ (p)) (36.41)

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The terms in the curly bracket can be combined by noting that (1 + n (−ω oλ (p))) = −n (ωoλ (p)), and representing the Dirac-delta functions as limit of their respective analytical expressions. The result is †

k ,j ek,i λ (p) eλ (p)

< (kk p; ω) = 2π sgn ω −iSij

λ

× LimΓ⇒0

Γ ω2

−

ωoλ

2

(p)

2

+

Γ 2 2

n (ωoλ (p))

= Aij (kk p; ω) n (ω oλ (p)) ,

(36.42)

where the spectral function Aij (kk p; ω) for thermal phonons appears expanded in terms of the eigenvectors (polarization vector) of the dynamical matrix † k ,j ek,i λ (p) eλ (p)

Aij (kk p; ω) = λ

2π sgn ω δ ω 2 − ωoλ (p)2

.

(36.43)

Using similar procedure, we also have > −iSij (kk p; ω) = Aij (kk p; ω) (1 + n (ω oλ (p))) .

(36.44)

< > The expressions for −iSij (kk p; ω) and −iSij (kk p; ω) leads to the relation > < (kk p; ω) − Sij (kk p; ω) = Aij (kk p; ω) . −i Sij

(36.45)

The sum rule obeyed by the spectral function Aij (kk p; ω) reads ∞

1 2π

ω Aij (kk p; ω) dω = δ kk δ ij ,

(36.46)

−∞

which also follows from the canonical commutation relation. The phonon density of states in the vibration branch index λ is defined by σ (ω) = p

= =

ω N 2π sgn ω δ ω 2 − ω oλ (p)2 π

N 3

(2π)

ω ω δ ω2 − ω oλ (p)2 dp3 2π π |ω|

N 8π3 ωo λ (p)=ω

dSp , |∇p ω oλ (p)|

(36.47)

where the momentum integral is over the constant energy surface, Sp , at ω = ωoλ (p) . < > (kk p; ω) and −iSij (kk p; ω) has so far only given for The expressions for −iSij a phonon system at thermal equilibrium. When the system is not in equilibrium,

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the reduced density matrices not only depend on the relative coordinates but also depend on the center coordinates. For nonequilibrium system, in analogy with Eq. (36.42) for thermal equilibrium, the following expression, < (k, k ; p, lc ; ω, tc ) = Aij (k, k ; p, lc ; ω, tc ) n (p, lc ; ω, tc ) , −iSij

(36.48)

is obtained after lattice Weyl transformation and expressing the result as expanded in terms of the eigenvectors of the dynamical matrix. By similar token, we can also write the phonon “hole” distribution as > (k, k ; p, lc ; ω, tc ) = Aij (k, k ; p, lc ; ω, tc ) [1 + n (p, lc ; ω, tc )] . −iSij

36.3

(36.49)

Transport Equation for the Phonon Super-Correlation Function

The phonon super-correlation function or super-propagator obeys o Sij ξ ¯ξ

−1

Sjm ¯ξξ −i|

= δ im ξ − ξ + Πij ξ ¯ ξ

Sjm ¯ξξ , −i|

(36.50)

where o Sij

ξ¯ξ

−1

=

o 2Mij

o(2) Θij ξ¯ξ ∂2 ¯ + 2√ . ξξ ∂t2 mkξ mk¯ξ

(36.51)

Taking into account Eqs. (36.7)-(36.5), we may write Eq. (36.50) as ¯ o(2) ξξ Θ ∂2 iy Sij ξξ ; tt + √ 2 ∂t mk mk

Syj ξ ξ ; tt − Πiy ξξ ; tt

Syj ξ ξ ; t t

= δ ij δ ξξ δ (t − t ) ,

(36.52)

¯ o(2) ξξ where diagonal matrix in the ‘hat’ and ‘tilde’ indices, Θ iy

¯ o(2) ξξ Θ iy



=

(2)

φiy ξξ

¯iy ξξ =D o(2)

√ / mk mk 0

, defined by

0 (2) φiy

ξξ

√ / mk mk

 

(36.53)

in contrast to the use of Θiy ξξ of Eq. (36.51), which is defined by Eq. (36.5). The phonon transport equation is determined from Eq. (36.52) and its adjoint, and

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is given by ∂2 ∂2 − 2 2 ∂t ∂t

Sij ξξ ; tt

¯ iy ξξ = −D + XX

Syj ξ ξ ; tt −1 0 0 1

+ Πiy ξξ ; tt − XX

¯ jy ξξ D

Syj ξ ξ ; tt

†

−1 0 0 1

Syj ξ ξ ; t t

−1 0 0 1

jy

ξξ

Syj ξ ξ ; tt

†

−1 0 0 1

,

(36.54)

where X X indicates the operation of interchanging the diagonal as well as the diagonal elements of the resulting matrix. With the “super quantities” as matrices in the “hat” and “tilde” indices, we can write Eq. (36.54) as coupled equations of the matrix elements. The consistency of the resulting coupled transport equations are verified by checking for the equation of the diﬀerent expressions for retarded propagator, as well as for the equation of the advanced propagator.

36.4

Phonon Transport Equations in Phase Space

As usual in deriving the quantum transport equations, we are interested in the equation for which defines the statistics of phonons and phonon “holes”, as well as the equation of which defines the energetics of their particle behavior. From the coupled equations for the matrix elements, we obtain the following expressions, ∂2 ∂2 − 2 2 ∂t ∂t

S >,< = Re Πr − d(2) , S >,< + Π>,< , Re S r + i Im Πr , S >,< − i Π>,< , Im S r = Re Πr − d(2) , S >,< + Π>,< , Re S r ±

∂2 ∂2 − 2 2 ∂t ∂t

1 1 Π>,< , S <,> ∓ Π<,> , S >,< , 2 2

S r,a = − d(2) , S r,a + [Πr,a , S r,a ] ,

(36.55)

(36.56)

where we have used the definition, (2)

(2)

dij

ξξ

φij ξξ , = √ mk mk

(36.57)

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and the following identities i Im S r =

S> − S< , 2

i Im Πr =

Π> − Π< . 2

(36.58)

What remains to be done is to take the lattice Weyl transform (LWT) of Eq. (36.55) and (36.56) to obtain the transport equations in p, Xl , ω, t ≡ (p, q, E, t) phase-space. First we note that from Eqs. (36.48) and (36.49), by virtue of the orthonormality and completeness of the eigenvectors of the dynamical matrix, the lattice Weyl transforms are matrices in the indices, λ, k, i where k refers to the kth nucleus in a unit cell, λ refers to the vibration branch index, and i refers to (2) the Cartesian directions. The lattice Weyl transform of dij ξ, ξ is diagonal in the vibration branch index with the diagonal elements as matrices in the remaining indices, i.e., (2)

† k ,j [ωoλ (p)]2 ek,i λ (p) eλ (p) ,

dij (kk ; p, q, E, t) =

(36.59)

λ

which is diagonal in the vibration index and independent of (q, E, t) set of variables. In view of the trivial dependence on indices of the resulting matrix of all the lattice Weyl transforms we make further simplification by simply considering the vibration branch index only. Making use of the relations given in Eqs. (31.33)-(31.34), we obtain the following phonon transport equation in phase space, ≷

2 ω oλ (p) fλλ (p, q) −2i ∂ ≷ E fλλ (p, q) + cos Λ ≷ 2 | ∂t −fλλ (p, q) ω oλ (p)

− i sin Λ

≷ ω oλ (p)2 fλλ (p, q) ≷

2 +fλλ (p, q) ω oλ (p)

= cos Λ Re Πr (p, q) , f ≷ (p, q) r

λλ

≷

− i sin Λ Re Π (p, q) , f (p, q)

λλ

+ cos Λ Re Π≷ (p, q) , Re f r (p, q)

λλ

− i sin Λ Re Π≷ (p, q) , Re f r (p, q) + i cos Λ Im Πr (p, q) , f ≷ (p, q) + sin Λ Im Πr (p, q) , f ≷ (p, q) ≷

λλ

λλ

λλ

r

− i cos Λ Π (p, q) , Im f (p, q) − sin Λ Π≷ (p, q) , Im f r (p, q)

λλ

λλ

,

(36.60)

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where the Λ’s are the appropriate diﬀerential “Poisson bracket” operators, and ≷

≷

fλλ (p, q) = −iSλλ (p, q, −E, t) . We can also express the transport equation as integro-diﬀerential equations by using Eqs. (31.35) and (31.36), as ≷

∂|ω oλ (p) ∂fλλ (p, q, E, t) ∂ ≷ f (p, q) = − · ∂t ∂p ∂q 1 1 2i (h4 )2  

−

dp dq

du dv exp

i [(p − p ) · v + (q − q ) · u] |  

v Re Πr (p+ u 2 ,q− 2 ) ≷ f (p , q ) ω     r   Re Π (p− u ,q+ v2 ) 2 ≷  −f (p , q )   ω    ≷   Π (p+ u ,q− v2 ) r 2   f (p , q ) ω  +     v Π≷ (p− u   2 ,q+ 2 ) r   −f (p , q ) ω  × , ≷ u v Π p+ ,q− ( 2    2) ≶  f (p , q )   ω ±  ≷ u v     Π p− ,q+ ( ) 2 2 ≶   (p , q ) + f ω   ≶    Π (p+ u2 ,q− v2 ) ≷   f (p , q )   ω ∓  v   Π≶ (p− u 2 ,q+ 2 ) ≷ + f (p , q ) ω

(36.61)

where the quantities without subscripts are matrices in branch index λ and do not necessarily commute. Equation (36.61) will serve as a basis for simulating the transport of phonon across interfaces and boundaries of intervening diﬀerent materials. The LWT of the retarded correlation function equation is given by the following diﬀerential form, 2

r,a (p, q) ω oλ (p) fλλ −2i ∂ r,a E fλλ (p, q) + cos Λ 2 r,a | ∂t −fλλ (p, q) ω oλ (p)

− i sin Λ

r,a (p, q) ω oλ (p)2 fλλ r,a (p, q) ω oλ (p)2 +fλλ

= cos Λ [Πr,a (p, q) , f r,a (p, q)]λλ − i sin Λ {Πr,a (p, q) , f r,a (p, q)}λλ ,

(36.62)

or by its corresponding integral form through the use of Eqs. (31.35) and (31.36).

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36.5

The Phonon Boltzmann Equation

More revealing equations can be seen by neglecting oﬀ-diagonal or “interbranch” terms, and expanding the equations in terms of the gradients. Using the diﬀerential “Poisson bracket” operator in Eqs. (31.33)-(31.34) to implement the gradient expansion to the leading order, we obtain −E ∂ ≷ f (p, q) − | ∂t λλ

| ≷ ∇p Ω2λλ (p, q) · ∇q fλλ (p, q) 2

∂ ≷ | ∂ Re Πrλλ (p, q) fλλ (p, q) − 2 ∂E ∂t

=

| ∂ ∂ ≷ Re Πrλλ (p, q) f (p, q) 2 ∂t ∂E λλ

−

| ≷ r ∇p Πλλ (p, q) · ∇q Re fλλ (p, q) − 2

| ∂ ≷ ∂ r Π (p, q) Re fλλ (p, q) 2 ∂E λλ ∂t

+

| ≷ r ∇q Πλλ (p, q) · ∇p Re fλλ (p, q) − 2

| ∂ ≷ ∂ r Πλλ (p, q) Re fλλ (p, q) 2 ∂t ∂E

i < i < > + Π> λλ (p, q) fλλ (p, q) − Πλλ (p, q) fλλ (p, q) , 2 2

(36.63)

where we have defined the renormalized kinematic frequency by Ω2λλ (p, q) = ω oλ (p)2 − Re Re Πrλλ (p, q). The leading terms in the right-hand side of Eq. (36.63) containing the Planck’s constant | are bonafide quantum corrections to the phonon Botlzmann equation. The Boltzmann equation for the distribution function of phonons, nλλ (p, q), from vibration branch λ readily follows by neglecting these leading quantum corrections. The result upon substituting E | = ω is given by the following expression, ∂ Ωλλ (p, q) nλλ (p, q) + ∇p Ωλλ (p, q) · ∇q nλλ (p, q) ∂t ω =−

i 2

Π> λλ (p, q) ω

nλλ (p, q) +

i 2

Π< λλ (p, q) ω

[1 + nλλ (p, q)] ,

(36.64)

with the spectral function satisfying the equation ∂ Ωλλ (p, q) Aλλ (p, q) = ∇p |Ωλλ (p, q) · ∇q Aλλ (p, q) . ∂t ω

(36.65)

We thus obtain the familiar interpretation [180] relating as the scattering-out rate and as the scattering-in rate. The simplest solution to the second line of Eq. (36.65) is to assume that the spectral function, Aλλ (p, q), to be independent of crystal lattice coordinates and time. Further by taking the self-consistent and renormalized frequency to be given by the solution of the equality Ωλλ (p, q) = ω, with this

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solution denoted by ω λ (p, t), we finally obtain ∂ nλ (p, q, ωλ , t) = −∇p |ω λ (p) · ∇q nλ (p, q, ωλ , t) + ∂t ∂ nλ (p, q, ωλ , t) ∂t

=i c

Π< λ (p, q, ω λ , t) [1 + nλ (p, q, ω λ , t)] 2ω λ (p)

−i

Π> λ (p, q, ω λ , t) nλ (p, q, ω λ , t) 2ωλ (p)

∂ nλ (p, q, ω λ , t) ∂t

, c

‘scattering in’ ‘scattering out’. (36.66)

The calculation of Πij ξ, ¯ξ and corresponding Feynman diagrams are given in the Appendix F. Equation (36.66) is similar to the frequently used form of the Boltzmann equation for phonons characterized by the use of self-consistently renormalized phonon frequency and phonon group velocity [220] arising from the real part of the phonon self-energy. A study of the collision term of Eq. (36.66) for anharmonic phonons, with phonon-phonon interaction arising from the cubic terms in the ionic displacements, can be found in the work of the author [221] and in the study of phonon hydrodynamics and second sound [222, 223].

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PART 5

Operator Space Methods and Quantum Tomography

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Chapter 37

Operator Hilbert-Space Methodology in Quantum Physics

The study of quantum transport of highly nonlinear and nonequilibrium high-speed quantum devices centers on the quantum distribution transport equations in phase space, given in Part 4. Indeed, the Wigner distribution functions (WDFs) in phase space are the coeﬃcient of the expansion of the density operator in terms of the complete set of traceless basis operators and an identity operator. Unitary operators considered in themselves also form a vector space, with the unitary scalar or dot product of two operators, A and B in operator space defined by the trace of the product of operators in a manner as follows A| B = T rA† B = B † A† .

(37.1)

This chapter is a preparation for the chapter on discrete phase space over finite fields 37.1

The Density Operator in Operator Vector Space

The density-matrix operator ˆρ can be expanded in terms of the phase-space pointˆ (p, q). The coeﬃcient of expansion is the Wigner function, projection operators ∆ W (p, q). Hence, we have1 ρ ˆ = (2π|)−3

ˆ (p, q) . dp dq W (p, q) ∆

(37.2)

ˆ (p, q) forms the complete operator basis, and obeys the completeness relation Here ∆ 1 (2π|)3

ˆ (p, q) = 1. dp dq ∆

(37.3)

ˆ (p, q) also forms an orthogonal basis operators in the following sense The ∆ ˆ (p, q) ∆ ˆ (p , q ) = (2π|)3 δ (p − p ) δ (q − q ) , Tr ∆

(37.4)

1 For continuum phase space, we follow the tradition of referring to a point in phase space by the ordered pair (p, q). In our discussion of discrete phase space in later chapters, it is more convenient to refer to a discrete point in phase space by the ordered pair (q, p) analogous to the use of the pair (x, y) in the cartesian coordinate frame.

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ˆ (p, q). Therefore, there where the right hand side is also the Weyl transform of ∆ 2 are N − 1 independent basis operators for an N -dimensional Hilbert space. By ˆ (p, q)’s, the coeﬃcient of the expansion of ˆ virtue of the orthogonality of the ∆ ρ in Eq. (37.2), which is the Wigner function W (p, q), or the Weyl transform of ˆ ρ, is thus given by ˆ (p, q) , W (p, q) = T r ρ ˆ∆

(37.5)

and for any operator A, the Weyl transform of A is also given as ˆ (p, q) a (p, q) = T r A∆ = =

1 1 i due | q·u p + u A p − u 2 2 1 1 i dve | p·v q − v A q + v . 2 2

Note that ∗ a (p, q) =

i du exp − u · q |

1 1 ˆ† p+ u . p− u A 2 2

If Aˆ is Hermitian then 1 1 i p − u Aˆ p + u du exp − u · q | 2 2 1 1 i u·q p + u Aˆ p − u = du exp | 2 2 = a (p, q) ,

∗ a (p, q) =

so that the Weyl transform a (p, q) is real. The Wigner function is a quasiprobability function which normalizes to unity, 1 3

(2π|)

1

ˆ (p, q) dp dqT r ρ ˆ∆ (2π|)3 1 ˆ (p, q) T rˆ ρ dp dq ∆ = (2π|)3 = T r ρ = 1,

dp dqW (p, q) =

where the last line makes use of Eq. (37.3). ˆ (p, q), the first two equivalent exThere are three equivalent expressions for ∆ pressions are: ˆ (p, q) = ∆

i 1 dve( | )p.v q + v 2

1 q− v , 2

(37.6)

ˆ (p, q) = ∆

i 1 due( | )q.u p − u 2

1 p+ u . 2

(37.7)

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For an N-dimensional Hilbert space, Eqs. (37.6) - (37.7) clearly reflect the oneto-one correspondence between the N 2 − 1 independent basis operators |q q | ˆ (p, q), or |p p |, and the N 2 − 1 independent phase-space point projectors, ∆ through the discrete Fourier transformation. Note that, by virtue of completeness, |q q| = |p p| = 1. Thus, it generally requires N 2 − 1 independent q

p

parameters (components) to characterize any operator such as the density operator, ρ ˆ. ˆ (p, q) is given by The third equivalent and symmetric expression for ∆ ˆ (p, q) = (2π|)−3 ∆ = (2π|)−3

i du dv e( | )[(q−Q).u

i du dv e( | )(q.u

+ (p−P ).v]

+ p.v)

i e(− | )(Q.u

+ P.v)

,

(37.8)

where Q and P are the position and momentum operators, respectively. Equation i (37.8) is the basis of quantum tomography since e(− | )(Q.u + P.v) is related to the − phase-space line projectors. The generalized Pauli-matrix operator, Yuv , to be discussed below, diﬀers from the phase-space line projectors in the sign of u, namely, i − Yuv = e(− | )( P.v−Q.u ) ˆ (p, q), Eq. (37.8), invites us to consider the The symmetric expression for ∆ operator Q.u + P.v, akin to coherent-states annihilation operator2 . We denote the eigenvalue, c, of this operator as indicating a ‘line’ in phase space given by q.u + p.v = c

(37.9)

Equation (37.9) can also be written in terms of the angle, φ, that the shortest distance, r = √u2c+v2 , from the line to the origin makes with the q-axis, i.e., q. √

v c u + p. √ 2 = √ 2 , u2 + v2 u + v2 u + v2 q cos φ + p sin φ = r.

(37.10)

Thus for parallel lines, φ is fixed while the shortest distance, r, from the origin ˆ (p, q) in Eq. (37.2) for the changes. Substituting the symmetric expression for ∆ 2 The combination of the operators, P and Q, in the coherent-state formalism is due to the form of the harmonic oscillator Hamiltonian, with coherent-state wavefunction   2  mω 2| ψ α (q) = C exp − q− α  2|  mω

ˆ (p, q) is where C is the normalization constant. The natural expansion of any operator in terms ∆ an integral of line projectors, with line wavefunction given by ψ u,v,c (q) =

where

√ 1 2π| |v|

1 2π| |v|

is the normalization constant.

exp −i

u c q− 2|v u

2

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expansion of the density operator ρ ˆ, we obtain −6

ρ ˆ = (2π|)

i dp dq W (p, q) e( | )(q.u

du dv

−3

ρ ˆ = (2π|)

i e(− | )(Q.u

+ p.v)

˜ (u, v) e(− |i )(Q.u du dv W

+ P.v)

+ P.v)

,

where ˜ (u, v) = (2π|)−3 W

i dp dq W (p, q) e( | )(q.u

+ p.v)

,

(37.11)

is the Fourier transform of the WDF. The inversion thus give the WDF W (p, q) =

˜ (u, v) e(− |i )(q.u du dv W

+ p.v)

.

(37.12)

ˆ (p, q), which is very useful in invesFor completeness, we give an expression for ∆ tigating the transformation properties of Radon transform and Wigner distribution function. This follows from Eq. (37.8), which can be rewritten as i du dv e(− | )(Q.u

ˆ (p, q) = (2π|)−3 ∆

∂ = (2π|)−3 e(−Q. ∂q ∂ = e(−Q. ∂q

∂ − P. ∂p )

∂ − P. ∂p )

i e( | )(q.u

+ p.v)

i du dv e( | )(q.u

+ p.v)

+ P.v)

δ (q) δ (p) ⇒ δ (q − Q) δ (p − P ) .

(37.13)

Thus the expression in Eq. (37.2) becomes ˆρ = (2π|)−3

37.2

dp dq W (p, q)

∂ e(−Q. ∂q

∂ − P. ∂p )

δ (q) δ (p) .

Formulation in Terms of Translation Operators

As we shall see below, the Pauli spin matrices for spin 12 particles can be derived from the generalized Pauli-matrix (GPM) operators, Yuv , given by i ± Yuv = exp − (v · P ± u · Q) . | The phase-space translation operator, T (q, p)sym , diﬀers only by a phase factor i u·v −i (v · P ± u · Q) exp | 2 | ∓i −i (v · P ) exp (u · Q) , | |

±sym Tuv = exp ±

= exp

(37.14)

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where exp −i | (v · P ) is a translation operator for positive displacement, v, in the coordinate axis, and exp ±i | (u · Q) represent translation operator for displacement, ±u, in the momentum axis. For convenience in what follows, we will take the exp |i (u · Q) for positive displacement, u, in the momentum axis. The formulation in terms of phase-space translation operators is useful in extending the continuous WDF theory to discrete WDF over finite fields. Since the generalized − ’s, diﬀer from the translation operators only by phase Pauli-matrix operators, Yuv − factors it is more convenient to examine the formulation in terms of Yuv rather than in terms of the translation operators. Any operator can be expanded in terms of phase-space point projector basis ˘ (p, q), operators, ∆ 1 Aˆ = 3 h

˘ (p, q) , dpdq a (p, q) ∆

where ˘ (p, q) = ∆

∞

−∞

1 q− v 2

i dv exp − p · v |

1 q+ v 2

ˆ (p, q) , =∆ which may also be written as (compare with Eq. (37.8)) ˘ (p, q) = 1 ∆ h3

dudv exp

i ˆ i ˆ ·u {p · v − q · u} exp − P ·v−Q | |

,

(37.15)

− which is an expansion in terms of the GPM operator, Yˆuv . In what follows, we will − ˆ drop the ‘minus’ superscript in Yuv . Any operator Aˆ can thus be expressed as

1 Aˆ = 3 h

dudv χ (u, v) exp −

i ˆ ˆ·u P ·v−Q |

,

(37.16)

where χ (u, v) =

a (p, q) =

1 h3

dpdq a (p, q) exp

dudv χ (u, v) exp

i {p · v − q · u} , |

−i {p · v − q · u} . |

(37.17)

(37.18)

We shall see that the Pauli-spin matrix, σ ˆx, σ ˆ y , and σ ˆ z can easily be derived from the GPM operator, Yˆp q , given by the expression i ˆ ˆ·p P ·q −Q Yˆp q = exp − |

.

(37.19)

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Except for a phase factor, namely exp − |i p 2·q , this operator is identical to the translation operator for positive displacements in phase space given by i p ·q sym Tˆ (p , q ) = exp − | 2

37.2.1

i ˆ − q · Pˆ p ·Q |

exp

.

Weyl Transform of GPM Operator

Note that the Weyl transform of Yp q is a phase function, Yp q (p, q) =

du exp

= exp

i u·q |

i p ·q | 2

1 i p + u exp {Q · p − P · q } 2 |

1 p− u 2

i u·q |

du exp

1 i × p + u exp − q · P 2 | = exp

i i p · q exp − q · p | |

= exp

i (q · p − p · q ) |

i p ·Q |

exp

= exp

1 p− u 2

i (rp,q × rp ,q ) , |

(37.20)

where rp,q ×rp ,q is the cross product of two vectors, rp,q and rp ,q , with components T T qp and q p , respectively. The last line indicates that the Weyl transform q of Yp q equals unity when the two nonzero vectors in phase space, namely, and p q are collinear, i.e., has vanishing symplectic product, q · p −p· q = 0, otherwise p it is an oscillatory phase function, with the oscillation frequency proportional to triangular area form by the vectors rp,q and rp ,q . This is in contrast to the Weyl ˘ (p, q) (p , q ), which has a value transform of the phase-space point projector, ∆ unity when the points (p, q) and (p , q ) exactly coincide, otherwise its value is zero. Moreover, the Weyl transform of the translation operator, T (p , q )sym , is thus given by sym

T (p , q )

(p, q) = exp −

i p ·q | 2

exp

i (q · p − p · q ) . |

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The coeﬃcient of operator expansion, χ (u, v), in Eq. (37.16) may also be expressed as χ (u, v) = T r (AYuv ) =

=

1 3

dpdq a(p, q)Yuv (p, q)

3

dpdq a(p, q) exp

(2π|) 1 (2π|)

=

exp

i (q · u − p · v) |

i (q · u − p · v) | .

average

We may refer to χ (u, v) as the characteristic function of the phase-space quasiprobability distribution a (p, q). The trace inner products of the GPM operators can easily be evaluated using their respective Weyl transforms as T r Yp† q Yp

q

=

=

1 3

dpdqYp† q (p, q) Yp

3

dpdq exp

(2π|) 1 (2π|)

q

(p, q)

i (q · p − p · q ) exp |

−i (q · p − p · q ) |

= (2π|)3 δ (p − p ) δ (q − q ) , showing that the set of GPM operators are orthogonal. Therefore, set of operators Yαβ form an orthonormal set. The GPM operators are traceless since (2π|)3/2 T r (Yp q ) =

=

1 3

dpdqYp q (p, q)

3

dpdq exp

(2π|) 1 (2π|)

i (q · p − p · q ) |

= (2π|)3 δ (p ) δ (q ) , which shows that the operators Yp q are traceless for nonzero p or q , except for Y00 .3 This property is characteristic of Pauli spin matrices. This is in contrast to ˘ (p, q), whose trace is unity. However, we can the phase-space point projectors, ∆ 3 For discrete phase space of prime number N, there will be N 2 − 1 basis operators of zero traces.

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show that GPM operators form a complete set since 1 3

(2π|)

dp dq Yp q =

=

1 (2π|)

3

dp dq exp

−i {Q · p − P · q } |

3

dp dq exp

i p q · P− | 2

1 (2π|)

p 2

−i p ·Q |

p exp 2

=

dp

=

dp dq

p 2

p |q 2

=

dp dq

p 2

1

=

dq

∞  

3/2

(2π|)

dp

−∞

=

dq |q

−i p ·Q |

exp

1 (2π|)3/2

q | exp

−i p ·q |

exp

i p ·q 2|

exp

−i p ·q 2|

−i p ·q |

q | exp

 p  q| 2 

q | = 1,

as expected since any operator can be expanded in terms of the GPM basis operators.

37.2.2

Weyl Transform of the GPM Eigenstate Projector

As we shall see in Sec. 38.2 below, the eigenstate,ψ p ,q ,c , of the GPM operator in the position representation is given by

q| ψ p ,q ,c = ψp ,q ,c (q) =

1 p exp i 2|q 2π| |q |

q−

c p

2

,

where c is a real number. We can thus calculate the Weyl transform of the eigenstate projector, Pp ,q ,c given by Pp ,q ,c = ψp ,q ,c

ψp ,q ,c

(37.21)

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ˆ (p, q) in Eq. (37.6) as by using the following expression of ∆ ˆ (p, q) Pp ,q ,c (p, q) = T r Pp ,q ,c ∆ =

=

1 ψ p ,q ,c q + v 2

1 dve(i/|)p.v q − v ψp ,q ,c 2 1 (2π| |q |) × exp −i

c 1 q+ v− 2 p

p 2|q

c 1 q− v− 2 p

p 2|q

dve(i/|)p.v exp i

2

i (pq − qp + c) .v |q

=

1 (2π| |q |)

=

1 2π| |q | δ [c − (qp − pq )] (2π| |q |)

dv exp

2

= δ [c − (qp − pq )] . We see that the Weyl transform of the eigenstate projector of the GPM operator is a projector on a line in phase space described by the line equation qp − pq = c. Using the trace formula for products of operators in terms of their Weyl transforms, we obtain T r (APp ,q ,c ) = T r A ψp ,q ,c =

1

ψ p ,q ,c

= ψp ,q ,c A ψp ,q ,c

dpdq a (p, q) δ [c − (q · p − p · q )] ,

(2π|)d

(37.22)

where d is the dimensionality. The expression of the right hand side is of the form of Radon transform which will be discussed in Sec. 38.1. sym , and Therefore the trace of the product of the translation operator, T (p, q) the line projector, Pp ,q ,c , is given by the integral of the product of their respective Weyl transform, sym

T r T (p , q )

sym

Pp ,q ,c = ψp ,q ,c T (p , q ) =

1 d

(2π|) ×

=

exp −

dpdq exp 1 d

(2π|)

exp −

ψp ,q ,c

i p ·q | 2 i (q · p − p · q ) δ [c − (q · p − p · q )] | i p ·q | 2

exp

i c , |

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where the right hand side is the eigenvalue of the translation operator, T (p , q ) with ψp ,q ,c as the eigenfunction.

37.3

sym

,

Point Projector in Terms of Line Projectors

Using the knowledge of the Weyl transform of the point projector and GPM operator, we are now able to determine the expression of the point projector in terms ˘ (p, q) in terms of Yu,v , we have of the line projectors. Upon expanding ∆ ˘ (p, q) = ∆ =

˘ (p, q) Yu,v dvdu T r Yu,v ∆ ˘ (p, q) e− ic| Pu,v,c , dvdudc T r Yu,v ∆

(37.23)

where we have, from Eqs. (37.20) and Eq. (37.4) ˘ (p, q) T r Yu,v ∆ =

=

1 (2π|)3 1 (2π|)

3

˘ (p, q) dp dq Yu,v (p , q ) ∆ p ,q dp dq exp

i (v · p − u · q ) (2π|)3 δ (q − q ) δ (p − p ) |

i = exp − (q · u − p · v) . | Upon substituting in Eq. (37.23), we have ˘ (p, q) = ∆

dvdudc exp

i (p · v − q · u − c) Pu,v,c . |

(37.24)

Thus, the expansion has significant contribution from line projectors whose line in phase space passes through the fixed point (p, q), which is the ‘addressed’ point ˘ (p, q). It would be interesting to see what would be the of the point projector, ∆ expression that can be obtained if indeed we restrict the integral to exactly only to those lines. Obviously, we will over-estimate the integral in Eq.(37.24). 37.3.1

˘ (p, q) in Terms of Intersecting Lines at Point (p, q) ∆

˘ (p, q) in terms of the sum of all lines The following derivation which expresses ∆ intersecting at point (p, q) is also true in the formulation of discrete phase space over finite field, and in fact the same idea is crucial to the construction of the discrete Wigner function in Part VI of the book.

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Therefore, let us try to evaluate the integral, ˘ (p, q) = ∆

dvdudc δ (p · v + q · u − c) Pu,v,c ,

where the prime restricts the integral only to all those lines passing through the ˘ (p, q) |q point (p, q). A way to determine this is to take the matrix element q | ∆ so that we can use the expression for Pu,v,c in the q-represention. Therefore, we obtain ˘ (p, q) |q q |∆ = q|

dvdudc δ (p · v + q · u − c) Pu,v,c |q

=

dvdudc q | ψu,v,c

=

du dv

× =

ψu,v,c |q

c 1 u q − exp −i 2|v u 2π| |v|

dc

c 1 u q − exp i 2|v u 2π| |v| dv δ

δ (p · v + q · u − c) 2

2

δ (p.v + q.u − c)

p (q − q ) 1 (q − q ) [(q + q ) − 2q] exp i 2 |

.

Writing4 δ

1 (q − q ) [(q + q ) − 2q] 2

= δ (q − q ) + δ

(q + q ) −q , 2

we have q|

dvdudc δ (p · v − q · u − c) Pu,v,c |q dv {δ (q − q )} exp i

= + 4 We

dv

δ

(q + q ) −q 2

p (q − q ) | exp i

p (q − q ) |

.

(37.25)

have made use the relationship δ(f (x)) = a

δ(x − a)/|f (a)|,

where a runs through all the zeros of f and the denominator is the magnitude of the value at a of the derivative of f .

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ˆ (p, q) as The first term can be identified with the expression involving the ∆ dv {δ (q − q )} exp i

p (q − q ) |

=

i 1 dve(− | )p.(q −q ) q q + v 2

=

i 1 dve(− | )p.v q q + v 2

1 q− v q 2

1 q− v q 2

with the variable v set equal to zero, hence this yield δ q q for fixed values of q and q . The second term can be satisfied by restricting the variable of integration v such that q = q + 12 v and q = q − 12 v in the integrand, then we have by virtue of the orthogonality of the basis states we can write dv =

δ

(q + q ) −q 2

exp i

i 1 dve( | )p.v q q + v 2

p (q − q ) |

1 q− v q 2

˘ (p, q) |q ≡ q |∆

(37.26)

for fixed values of q and q . Thus we have shown that q|

dvdudc δ (p · v − q · u − c) Pu,v,c |q

˘ (p, q) q ≡ q ∆

+ δ q q . (37.27)

We have arrived at a very important result, Eq. (37.27) above, whose extension to discrete phase space over finite fields is at the core of the discrete WDF construction, ˘ (p, q) = ∆ ˘ (p, q) − I, i.e., and conclude that ∆ ˘ (p, q) = ∆ ˆ (p, q) = ∆

dvdudc δ (p · v + q · u − c) Pu,v,c − I.

(37.28)

The WDF is now expressed as ˆ (p, q) W (p, q) = T r ρ ˆ∆ =

ρ dv du dc δ (p · v + q · u − c) T r (ˆ ρPu,v,c ) − T rˆ

=

ρ ψu,v,c − T rˆ dv du dc δ (p · v + q · u − c) ψu,v,c ˆ ρ

.

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We shall see later that this result, Eq. (37.28), is readily extended to the discrete phase space in the form   1 ˆ (p, q)  ∆ ψ u,v,c ψu,v,c − I  (37.29) = N f inite f ield uvc|(q,p)

where the summation is restricted to ‘discrete’ lines intersecting at point (q, p) [in Eq. (41.2)], and we have used the expression of the line projector in Eq. (37.21).

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The Wigner Function Construction

The WDF can be reconstructed from measurements of the Radon transform, to be discussed below.

38.1

The Quasi-Probability Distribution and Radon Transform

In one dimension, we can bring in the geometrical-line concept in phase space, described by Eq. (37.9) to Eq. (37.11) by writing this equation in the form ˜ (u, v) = (2π|)−1 W

bc

ei | dc (2π|)−1

dp dq W (p, q) δ q.

v u + p. − c b b

,

dp dq W (p, q) δ q.

v u − p. − c b b

.

and similarly writing Eq. (37.17) as χ ˘ (u, v) = (2π|)−1

bc

e−i | dc (2π|)−1

The quantity enclosed by the curly bracket is of the form of the Radon transform of the WDF, which we will designate as W ub , vb ; c and χ ub , vb ; c , respectively, W

u v , ; c = (2π|)−1 b b

dp dq W (p, q) δ q.

χ

u v , ; c = (2π|)−1 b b

dp dq W (p, q) δ q.

v u + p. − c , b b

v u − p. − c , b b

and hence −1

W (u, v; c) = (2π|)

−1

χ (u, v; c) = (2π|)

dp dq W (p, q) δ (q.u + p.v − c) ,

dp dq W (p, q) δ (q.u − p.v − c) . 460

(38.1)

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˜ (u, v) and the Radon transform W (u, v; c) of the Thus the Fourier transform W Wigner distribution function are related by the expression ei | dc W

bc

= (2π|)−2

dp dq

= (2π|)−2

dpdq e | (q.u

−1

χ ˘ (u, v) = (2π|)

38.1.1

u v , ;c b b

bc

˜ (u, v) = (2π|)−1 W

ei | dc W (p, q) δ q. i

bc

e−i | dc χ

+ p.v)

v u + p. − c b b

W (p, q) ,

(38.2)

u v , ;c b b

bc

= (2π|)−1

e−i | dc (2π|)−1

= (2π|)−2

dpdq e− | (q.u−p.v) W (p, q) .

dp dq W (p, q) δ q.

v u − p. − c b b

i

(38.3)

The Radon Transform

In what follows we will restrict our discussion on the conventional Radon transform, ˜ (u, v) and the Radon transform W (u, v; c) W (u, v; c). The relation between W is often referred to as the Fourier slice theorem. The importance of the Radon transform W (u, v; c) is that practical measurements of the the ‘image’ of W (p, q) is often obtained as projections described by W (u, v; c). Thus the recovery or reconstruction of W (p, q) from the measurements consists of two mathematical steps, namely, (a) transition from Radon transform W ub , vb ; c to Fourier transform ˜ (u, v), Eq. (38.2), W ˜ (u, v) = (2π|)−1 W = (2π|)−2 = (2π|)−2

bc

ei | dc W bc

ei | dc

u v , ;c b b dp dq W (p, q) δ q.

v u + p. − c b b

i

dp dqe | (q.u+p.v) W (p, q) ,

˜ (u, v) to yield W (p, q) in Eq. (37.12), and (b) the Fourier transform inversion of W W (p, q) =

˜ (u, v) e(− |i )(q.u du dv W −1

= (2π|)

= (2π|)−1

+ p.v)

i u v , ;c du dv dc e(− | )(q.u+p.v−bc) W b b i du dv dc e(− | )(q.u+p.v−c ) W (u, v; c ) ,

(38.4)

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which yields the identity W (p, q) = (2π|)

−2

dp dq e | (q .u+p .v) W (p , q ) i

du dv

= (2π|)−2

i e(− | )(q.u+p.v)

dp dq W (p , q ) (2π|)2 δ (q − q) δ (p − p) .

Note that the integration over u and v in Eq. (37.12) amounts to integration over all angles that the normal vector from the origin to the line makes with the ‘x’ or ‘q’ axis, i.e., over all line-image projections.

38.2

Line Eigenstates and Line Projection Operators

The position representation of the eigenfunction is denoted as ψu,v,c (q). Noting that P ψu,v,c (q) = −i|∇q ψu,v,c (q), we obtained the following first-order diﬀerential equation for the wavefunction ψu,v,c (q), in one-dimensional systems as uq − iv|

∂ ψu,v,c (q) = cψu,v,c (q) . ∂q

(38.5)

The general solution may be written as q| ψ u,v,c = ψu,v,c (q) =

c 1 u q− exp −i 2|v u 2π| |v|

2

.

(38.6)

The normalization of q| ψ u,v,c was chosen so that, as a projection operator ψu,v,c ψu,v,c must yield identity when integrated over the eigenvalues c, i.e., all parallel lines, ψu,v,c

ψu,v,c dc = =

dq

q| ψu,v,c

ψu,v,c q dc

dq δ (q − q )

= 1, c2

ψu,v,c

and therefore

ψu,v,c dc, with c2 > c1, is a projection operator in these

c1

range of c-eigenvalues. We have ψu,v,c

ψu,v,c dc =

dqδ (q − q )

c u 1 q− exp −i 2|v u 2π| |v|

c 1 u q − exp i 2|v u 2π| |v| u (0) = 1. = exp −i 2|v ×

2

dc

2

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In other words, the normalization constant ment that ψ u,v,c

1 2π||v|

463

is determined from the require-

ψu,v,c dc = 1.

(38.7)

We can write the operator i du dv e( | )(q.u

ˆ (p, q) = (2π|)−3 ∆

+ p.v)

i du dv dc e( | )(q.u

= (2π|)−3

i e(− | )(Q.u

+ p.v−c)

+ P.v)

ψu,v,c

ψu,v,c

ψu,v,c dc

ψu,v,c .

(38.8)

From Eqs. (37.3) and (38.7), considering one-dimensional system, we have 1 (2π|)

ˆ (p, q) = dp dq ∆

ψu,v,c

ψu,v,c dc = 1

This means that the projection along parallel lines in phase space integrated over all the parallel lines is equal to the projection on each point in phase space integrated over all points. In view of this, we may thus write the following equality in terms of the projection on a single line in phase space as a projection on the eigenstates ψu,v,c , namely, 1 (2π|)

ˆ (p, q) = ψ u,v,c δ (c − qu − pv) dp dq ∆

ψu,v,c .

(38.9)

Equation (38.9) for the operators corresponds to Eq. (38.1) for the quasi-probability distribution. For emphasis, we summaryize these corresponding relations as follows, W (u, v; c) = (2π|)−1 ψu,v,c

ψu,v,c =

1 (2π|)

dp dq W (p, q) δ (q.u + p.v − c) , ˆ (p, q) δ (q.u + p.v − c) , dp dq ∆

From Eq. (37.8), written for one-dimensional system as, ˆ (p, q) = (2π|)−1 ∆ = (2π|)−1

i du dv e( | )[(q−Q).u+(p−P ).v]

i du dv e( | )[(q.u+p.v)−(Q.u+P.v)]

(38.10)

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we must have the identity du dv =

i ψ ub , vb ,c e( | )[(q.u+p.v)]

bc

e−i | dc ψ ub , vb ,c

i du dve( | )[(q.u+p.v)−(Q.u+P.v)]

bc

e−i | dc ψ ub , vb ,c

i ψ ub , vb ,c e( | )[(q.u+p.v)]

i = e( | )[(q.u+p.v)−(Q.u+P.v)] bc

e−i | dc ψ ub , vb ,c

ψ ub , vb ,c

i = e(− | )(Q.u+P.v) .

(38.11)

Note that for u = 0 in Eq. (38.10), using Eq. (38.11), representing lines parallel to the q-axis for various values of c, we can take q ψ0, vb ,c = t 1 |v| exp {icbq/|v}, 2π|

b

Eq. (38.6), so that bc

i ψ0, vb ,c e( | )

p.v

dv

e−i | dc ψ0, vb ,c

=

dv

dq dqδ (q − q )

e−i | dc q ψ0, vb ,c

=

dv

dq dqδ (q − q )

dc

bc

1 2π| |v| b

i exp {ic [b (q − q + v)] /|v} e( | )p.v

i e( | )p.(x−x ) 2π|

= 2π|

dx dxδ (x − x )

= 2π|

dx dx δ (x − x ) x| p p| x

= 2π|

dx x| p p| x

= 2π| |p p| =

i ψ0, vb ,c q e( | )p.v

i dve( | )(p−P ).v .

A more revealing results is obtained when v = 0, representing lines parallel to the p-axis for various values of c. The eigenfunction of Eq. (38.5), can be taken as

q ψ ub ,0,c =

1 u b

δ q−

bc u

.

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We have i ψ ub ,0,c e( | )q.u

bc

e−i | dc ψ ub ,0,c

du

i ψ ub ,0,c q e( | )q.u

bc

=

du

dq

e−i | dc q ψ ub ,0,c

=

du

dq

bc 1 bc e−i | dc |u| δ q − u

δ q −

b

=

du

= 2π|

dq

ei

u(q−c |

)

bc u

i e( | )q.u

dc δ (q − c ) δ (q − c ) ,

dxδ (x − q) = 2π|

dx q |x x |q

i ˆ du e( | )(q−Q).u .

= 2π| |q q| =

ˆ (p, q) is obtain using The matrix element of the phase space point projector ∆ the phase space line-states projector as ˆ (p, q) |q q| ∆ = (2π|)−1

= (2π|)

du dv

−1

bc

ei | dc q | ψ ub , vb ,c

du dv

bc

ei | dc

1 2π| |v| b

× =δ =

1

exp i

2π| |v| b (q + q ) −q 2

u bc q − 2|v u

2

ψ ub , vb ,c |q exp −i

i e(− | )(q.u+p.v)

u bc q − 2|v u

2

i e(− | )(q.u+p.v)

i e( | )p.(q −q )

i 1 dve( | )p.v q q + v 2

1 q− v q 2

,

(38.12)

where the last line agrees with that obtained using Eq. (37.6). 38.2.1

Density Operator in Terms of Line Projectors

In Eq. (37.2), the density operator is expanded in terms of the phase space point projectors. In one dimensional system, the corresponding expansion in terms of the

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phase space line projectors is ρ= ˆ

dc ψu,v,c

dc

ˆ ψu,v,c ψu,v,c ρ

ψu,v,c

= (2π|)−3

dp dqW (p, q)

dc

dc ψu,v,c

= (2π|)−6

dp dqW (p, q)

dc

dc

i × ψ u,v,c e(− | )(Q.u+P.v) ψu,v,c

= (2π|)−6

dp dq

ψu,v,c

ˆ (p, q) ψu,v,c ∆

i du dv e( | )(q.u+p.v) ψu,v,c

ψu,v,c

du dv W (p, q) e( | )(q.u+p.v−c) ψu,v,c i

dc

ψu,v,c

ψu,v,c .

We can check this expression by calculating the Weyl transform of ˆ ρ which is W (p, q). Taking the Weyl transform of the last expression amounts to taking the Weyl transform of the line projector, ψu,v,c ψu,v,c , equal to δ [c − (q u + p v)]. Therefore, we have W (p , q ) 1 1 ρ q + v dv e(i/|)p .v q − v ˆ 2 2

=

= (2π|)−6 ×

dp dq

dc

i du dv W (p, q) e( | )(q.u+p.v−c)

1 dv e(i/|)p .v q − v 2

ψu,v,c

1 ψu,v,c q + v 2

i du dv W (p, q) e( | )(q.u+p.v−c) δ [c − (q u + p v)]

= (2π|)−6

dp dq

= (2π|)−6

dp dq W (p, q) (2π|)3 δ (q − q ) (2π|)3 δ (p − p )

dc

= W (p , q ) . Note that, as before, Tr ρ ˆ ψu ,v ,c =

ψu ,v ,c

1 (2π|)

d

= ψu ,v ,c ρ ˆ ψu ,v ,c

dpdq W (p, q) δ [c − (q · u + p · v )]

= W (u , v ; c ) .

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467

Translational Covariance of the Wigner Function

From Eq. (37.14), we have the symmetric positive displacement operator in phase space given by T (q, p)sym = exp −

i p·q | 2

−i (q · P − p · Q) . |

exp

The transformed density operator is ρ ˆ = exp

i i (y · Q − x · P ) ρ ˆ exp − (y · Q − x · P ) . | |

Then we have i ˆ (p, q) exp W (p, q) = T r ρ ˆ exp − (y · Q − x · P ) ∆ |

i (y · Q − x · P ) |

.

We have i ˆ (p, q) exp exp − (y · Q − x · P ) ∆ |

i (y · Q − x · P ) |

i = (2π|)−3 exp − (y · Q − x · P ) | ×

i du dv e( | )[(q.u+p.v)−(Q.u+P.v)] exp

i (y · Q − x · P ) . |

ˆ Pˆ = i|, By the use of the Campbell-Baker-Hausdorﬀ operator identity, using Q, the transformed phase-space point projector is given by i ˆ (p, q) exp exp − (y · Q − x · P ) ∆ | i = (2π|)−3 exp − (y · Q − x · P ) | × exp

i (y · Q − x · P ) |

= (2π|)−3

du dv exp

i (y · Q − x · P ) | i du dv e( | )[(q.u+p.v)−(Q.u+P.v)]

i |

i [(q.u + p.v)] exp − [(yv + ux)] |

i × exp − (Q.u + P.v) | = (2π|)−3

du dv exp

i |

i [((q − x) .u + (p − y) .v)] exp − (Q.u + P.v) |

= (2π|)−3

du dv exp

i |

[([(q − x) − Q] .u + [(p − y) − P ] .v)] ,

(38.13)

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ˆ (p, q) becomes a which shows that the transformed phase-space point projector ∆ phase-space point projector on displaced phase-space point [(q − x) , (p − y)]. This can also be written as i ˆ (p, q) exp exp − (y · Q − x · P ) ∆ | = (2π|)−3

[([(p − y) − P ] .v)]

1 (q − x) − v − Q .u 2

i |

× exp =

i |

du dv exp

i (y · Q − x · P ) |

i [([(p − y)] .v)] |

dv exp

1 (q − x) + v 2

1 (q − x) − v , 2

or written as i ˆ (p, q) exp exp − (y · Q − x · P ) ∆ | = (2π|)−3

× exp =

du dv exp i |

i |

i (y · Q − x · P ) | 1 (p − y) − u − P .v 2

[([(q − x) − Q] .u)]

1 du (p − y) − u 2

1 (p − y) + u exp 2

i |

[((q − x) .u)] . (38.14)

The transformed Wigner distribution function can be obtained from the expression ˆ (p, q) W (p, q) = T r ˆρ ∆

=

dv exp

i |

1 [([(p − y)] .v)] T r ˆ ρ (q − x) + v 2

=

dv exp

i |

[([(p − y)] .v)]

= W ((p − y) , (q − x)) ,

1 (q − x) − v 2

1 1 ρ (q − x) + v (q − x) − v ˆ 2 2 (38.15)

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or we can also obtain the same expression for W (p, q) by using the equivalent ˆ (p, q) expression for ∆ ˆ (p, q) W (p, q) = T r ˆρ ∆ =

du exp

i |

[((q − x) .u )] T r

=

du exp

i |

[((q − x) .u )]

1 ρ ˆ (p − y) − u 2

1 (p − y) + u 2

1 1 ρ (p − y) − u (p − y) + u ˆ 2 2

= W ((p − y) , (q − x)) .

(38.16)

Thus the transformation of the density operator ˆρ = exp

i i (y · Q − x · P ) ˆ ρ exp − (y · Q − x · P ) | |

expanded in terms of phase-space point projector basis as ρ ˆ = (2π|)−3

ˆ (p, q) dp dq W (p, q) ∆

leads to the coeﬃcient of expansion W (p, q) given by ˆ (p, q) W (p, q) = T r ρ ˆ∆ resulting in a covariant transformation of the Wigner distribution function, namely, W (p, q) = W (p − y, q − x). This means that if the density matrix is translated by a positive displacement, the shape of the Wigner distribution function follows rigidly and hence changes phase-space origin. More specifically, the Wigner distribution function is covariant since their structure will not change (rigid) as you move them along the phase-space manifold.

38.4

Transformation Properties of the Radon Transform

A convenient tool for investigating the transformation properties of the Wigner disˆ (p, q) tribution function and its Radon transform is aﬀorded by the expression of ∆ as given by Eq. (37.13). We have ˆρ = (2π|)−3 = (2π|)−3

ˆ (p, q) dp dq W (p, q) ∆ dp dq W (p, q)

∂ ∂ e(−Q. ∂q −P. ∂p ) δ (q) δ (p) ,

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ˆ (p, q) W (p, q) = T r ρ ˆ∆ = Tr ρ ˆ

∂ ∂ e(−Q. ∂q −P. ∂p )

δ (q) δ (p) .

Suppose we have a density operator ρ ˆo corresponding to the Wigner distribution function Wo (p, q). As was done above, let us make a displacement of the state described by ˆρo by using the displacement operator T (q)sym . Then we have ρ ˆ = exp

i i (y · Q − x · P ) ρ ˆo exp − (y · Q − x · P ) . | |

Then W (p, q) is given by W (p, q) ∂ ∂ = T r ˆρ e(−Q. ∂q −P. ∂p )

= T r exp

δ (q) δ (p)

i i (y · Q − x · P ) ρ ˆo exp − (y · Q − x · P ) | |

∂ ∂ × e(−Q. ∂q −P. ∂p )

δ (q) δ (p)

i = Tr ρ ˆo exp − (y · Q − x · P ) | × exp

i (y · Q − x · P ) |

∂ ∂ e(−Q. ∂q −P. ∂p )

δ (q) δ (p) .

By the use of the Campbell-Baker-Hausdorﬀ operator identity, we thus obtain W (p, q) 

= Tr 

ˆ − x · Pˆ ρ ˆo exp − |i y · Q × exp

i |

ˆ ∂ ˆ ∂ e(−Q. ∂q −P . ∂p )

ˆ − x · Pˆ y·Q

∂ ∂ ˆ ˆ = Tr ρ ˆo e(−(Q+x). ∂q −(P +y). ∂p )



 δ (q) δ (p)

δ (q) δ (p)

= Wo (p − y, q − x) , which simply shows a rigid displacement of the old WDF, Wo (p, q). The Radon transform of the displaced WDF is then connected by a displacement of the variable c, i.e., W (u, v; c) = Wo (u, v; c − ux − vy) ,

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whereas the Fourier transforms are connected by multiplication with a phase factor, exp − |i (ux + vy) , ˜ o (u, v) . ˜ (u, v) = exp − i (ux + vy) W W | 38.5

Intersection of Line Projectors: Mutually Unbiased Basis (1)

(2)

Let Pl be a phase-space line projector and Pl be another phase-space line pro(1) jector that project on a line not parallel to that of Pl . Then the intersection, (1) (2) Pl , of these two projectors is defined by Pl (1)

(2)

Pl

Pl

(1)

(2)

= T r Pl Pl

is a mutual probability measure. For example, let us consider the special case described by the following wavefunctions, (1)

Pl

c 1 = q ψu,0,c = √ δ q − u u ψu,0,c ,

= ψu,0,c and (2)

Pl

= q ψ0,v,c = ψ0,v,c

=

1 exp {ic q/|v} 2π| |v|

ψ0,v,c .

Then (1)

(2)

T r Pl Pl

= T r ψu,0,c =

ψu,0,c ψ0,v,c

ψu,0,c ψ0,v,c

=

dq

=

1 2π| |vu|

=

1 . 2π| |vu|

ψ0,v,c

= ψu,0,c ψ0,v,c

2

dq q ψu,0,c dq δ q −

ψu,0,c q c exp ic u

q ψ0,v,c c − q /|v u

ψ0,v,c q

ψ0,v,c ψu,0,c

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Since u and v are fixed for these intersecting lines, we have for this special case, u = cos (0) = 1 and v = sin π2 = 1. Therefore, we have (1)

(2)

=

T r Pl Pl

1 . 2π|

Note that the probability of observing the eigenvalue q in the state |p is given 2

1 1 by | q| p |2 = √2π| exp |i pq = 2π| . This means that the operator Q yields an equally probable or uniform distribution of eigenvalues when measured in the eigenstate of P , and vice versa. Moreover, the above result is independent of c and (1) c , i.e., result holds for all pair of non-parallel lines one taken from Pl and the (2) other from Pl . We can generalize the above result to arbitrary pair of nonparallel phase-space line projectors. Let these pair of nonparallel projectors be (1)

Pl

(2)

Pl

⇒ q ψu,v,c = = ψu,v,c

ψu,v,c

⇒ q ψu ,v ,c

=

= ψ u ,v ,c

1 exp 2π| |v|

−i u | 2v

1 exp 2π| |v |

−i |

u 2v

q−

c u

q−

2

,

c u

2

ψu ,v ,c .

Then (1)

(2)

T r Pl Pl

= T r ψu,v,c

ψu,v,c ψu ,v ,c

=

dq

dq

ψu,v,c q

=

dq

dq

c 1 u q − exp i 2|v u 2π| |v|

× =

q ψ u ,v ,c

2

× exp i

1 |v | |v|

dq

dq exp

1 (q − q) (q + q) 2|

ψu,v,c ψu ,v ,c

2

ψu ,v ,c q q ψu,v,c

2

1 u c exp i q− 2| |v | u 2π| |v | 1 2π|

=

ψu ,v ,c

i |

u u − v v

2

1 u exp −i 2|v 2π| |v |

c 1 u q− exp −i 2|v u 2π| |v| c c − v v .

(q − q)

q − 2

c u

2

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Make a change of variables, let 1 q = x + y, 2 1 q = x − y. 2 Then (1)

(2)

T r Pl Pl =

1 2π|

2

1 |v | |v|

1 × exp i yx |

dy

dx exp

i |

c c − v v

y

u u − v v

x

u u − v v c c − v v

=

1 2π|

1 |v | |v|

=

1 2π|

1 |v | |v|

=

1 , provided (u v − uv ) = 0, i.e., nonparallel lines, 2π|

dx δ |v | |v|

dx δ x −

+

(c v − cv ) (u v − uv )

for if (u v − uv ) = 0, Eq. (37.10) indicates that the two lines are parallel since the equality arccos

1 1+

u2 v2

= arccos

1 1+

u2 v2

leads to φ = φ in Eq. (37.10). The above result is remarkably the same as for the special case before and also (1) independent of c and c , i.e., result holds for all pair of lines one taken from Pl (2) and the other from Pl . This means that the operator uQ + vP yields an equally probable or uniform distribution of eigenvalues when measured in the eigenstate of u Q + v P , and vice versa, whenever (u v − uv ) = 0, i.e., eigenstate of nonparallel lines in phase space are mutually unbiased.

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PART 6

Discrete Phase Space on Finite Fields

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Chapter 39

Discrete Phase Space on Finite Fields

The original formulation of lattice-space discrete Weyl transform and discrete WDF1 is based on crystalline solid with inversion symmetry, and hence based on an odd number of discrete lattice points, (q, p), obeying the Born-von Karman boundary condition. Thus, this formulation is generally based on a finite field represented by a finite prime number, N , of lattice points obeying modular arithmetic, closed under addition and multiplication2 . The presence of multiplicative inverses in the formulation assumes that the finite number of lattice points is a prime number, since primality is required for the nonzero elements to have multiplicative inverses. Indeed, in the limit that the number of lattice points is very large, the lattice point coordinates obeying the Born-von Karman boundary condition, assumes the field of prime integers3 .

39.1

Discrete Wigner Function on Finite Fields

One can also formulate the discrete WDF more generally based on the algebraic concept of finite fields, which are extension of prime fields, where q and p are field elements (mod irreducible polynomial ). Discrete phase space based on finite fields is particularly useful when the quantum numbers involved, specifying the quantum states, are discrete configurations other than the particle position and momentum quantum numbers. A simplest example is that of spin- 12 particle, which will be discussed below. We give a brief introduction on the algebraic concept of finite field in Appendix I.

1 The original formulation is entitled, “Method for Calculating T r Hn in Solid State Theory”, Phys. Rev., B10, 3700-3705 (1974). 2 There are infinitely many prime numbers. Examples of prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 73, 79, 83, 89, 97, 101, 103, 107, 179, 181, 191, 193, 197, 199, 211, 223, 227, and 229, etc. When written in base 10, all prime numbers except 2 and 5 end in 1, 3, 7 or 9. 3 A review article by Kasperkovitz and Peev, Ann. Phys. 230, 21 (1994) makes a number of misleading statements about the discrete phase-space formulation of the quantum theory of solids by failing to recognize the finite-field aspects of the theory.

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39.1.1

Line in Discrete Phase Space: Pure Quantum State

For each line λ = {u, v, c} in phase space, let Q (λ) be a rank-1 projection operator representing a pure quantum line state. The following equality in terms of the projection on a single line in phase space as a projection on the eigenstates ψ u,v,c follows from Eq. (38.9), namely, Q (λ) = ψu,v,c 1 = (2π|)

ψu,v,c ˆ (p, q) . δ (c − qu − pv) dp dq ∆

We need the commutation properties of Q (λ) and the translation operator, T (q, p)sym .4 With an eye on discretization, let us recall the displacement operator in continuous phase space as given by T (q, p)sym = exp −

i p·q | 2

exp

i (p · Q − q · P ) , |

where the operators acting on the position wavefunctions are defined by P ψ (q) ⇒ −i|∇q ψ (q) and Qψ (q) ⇒ qψ (q) . In terms of momentum eigenstates we also have P ψ (p) ⇒ pψ (p), Qψ (p) ⇒ i|∇p ψ (p). We call the operator Xq = exp − |i q · P as the ‘horizontal’ translation operator Xq and the operator Zp = exp |i p · Q as the ‘vertical’ displacement operator Zp . We have Xq |q = |q + q , Zp |q = exp

i p · q |q , |

and Zp |p = |p + p , i Xq |p = exp − q · p |p . | sym

Thus, T (q, p) function yields,

acting on the momentum representation of the Schrödinger wavei p·q | 2

i (p · Q − q · P ) ψ (p ) | i i (p · Q) exp − (q · P ) ψ (p ) = Zp Xq ψ (p ) = exp | | i = exp − p · q ψ (p + p) . |

T (q, p)sym ψ (p ) = exp −

exp

4 Note that a point in discrete phase space is referred to here as (q, p) instead of (p, q) used in the continuum phase space.

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The transformed phase-space point projector is given by Eq. (38.13) i ˆ (p, q) exp i (y · Q − x · P ) exp − (y · Q − x · P ) ∆ | | i = (2π|)−3 du dv exp [([(q − x) − Q] .u + [(p − y) − P ] .v)] , (39.1) | ˆ (p, q) becomes a which shows that the transformed phase-space point projector ∆ phase-space point projector on displaced phase-space point [(q − x) , (p − y)]. Equation (39.1) can also be written as i ˆ (p, q) exp i (y · Q − x · P ) exp − (y · Q − x · P ) ∆ | | 1 1 i (q − x) − v . = dv exp [([(p − y)] .v)] (q − x) + v | 2 2 39.1.2

sym

Commutation Relation Between Q (λ) and T (q, p)

We can show that Q (λ) commutes with the translation operator, as long as the displacements in phase space are colinear with the line described by λ = {u, v, c}. We have i ˆ (p, q) δ (c − qu − pv) dp dq exp − (y · Q − x · P ) ∆ | i (y · Q − x · P ) × exp | i i (y · Q − x · P ) . = exp − (y · Q − x · P ) ψu,v,c ψu,v,c exp | |

1 (2π|)

We make use of the result of Eq. (38.13) 1 (2π|)4

δ (c − qu − pv) dp dq

× exp =

1 (2π|)4 ×

=

[([(q − x) − Q] .¯ u + [(p − y) − P ] .¯ v)]

δ (c − (q + x) u − (p + y) v) dp dq

d¯ u d¯ v exp 1

(2π|)4 ×

i |

d¯ u d¯ v

i |

u + [p − P ] .¯ v)] [([q − Q] .¯

δ (c − q u − p v − (xu + yv)) dp dq

d¯ u d¯ v exp

i |

u + [p − P ] .¯ v)] . [([q − Q] .¯

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If the origin is displaced such that xu + yv = 0 then we have for all values of c, i.e., for all lines of its striations i ˆ (p, q) δ (c − qu − pv) dp dq exp − (y · Q − x · P ) ∆ | i (y · Q − x · P ) × exp | 1 = δ (c − q u − p v) dp dq (2π|)

1 (2π|)

×

1 3

(2π|)

= ψu,v,c

i |

d¯ u d¯ v exp

u + [p − P ] .¯ v)] [([q − Q] .¯

ψu,v,c ,

where the last line makes use of Eqs. (37.8) and (38.9). Therefore i = exp − (y · Q − x · P ) | = ψu,v,c ψu,v,c ,

ψu,v,c

ψu,v,c exp

i (y · Q − x · P ) | (39.2)

rendering the line—state projection operator covariant under the displacement such that xu + yv = 0, i.e., the displacement is collinear with the line described by the projector ψ u,v,c ψu,v,c . Moreover, under this condition, ψu,v,c ψu,v,c commutes with the translation operator, ψu,v,c

ψu,v,c exp

= exp

i (y · Q − x · P ) |

i (y · Q − x · P ) |

ψu,v,c

ψu,v,c .

(39.3)

Therefore the projection ψu,v,c ψu,v,c that is associated with the line in phase space commutes with the translation operator for displacements along the line.

39.2

Generalized Pauli Matrices

Let us recall the shift operator, Xq , and phase operator, Zp , which obey the following equations i Xq |q = exp − q · P |

Zp |q = exp

|q = |q + q ,

i p · Q |q = exp |

i p · q |q . |

(39.4)

(39.5)

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The product of Xq and Zp is thus given by i Xq Zp = exp − q · P |

exp

i p ·Q |

sym

= T (q , p )

− |i q · P, |i p · Q 2

i = exp − (q · P − p · Q) exp | ip | ip = exp − | = exp −

·q 2 ·q 2

i exp − (q · P − p · Q) | (39.6)

Yq p

where Yq p is the generalized spin-matrix operator5 . Clearly we have, from Eq. (37.19), Yˆq† p = exp = exp

39.2.1

i ˆ† q · Pˆ † − p · Q | i ˆ q · Pˆ − p · Q = Yˆ−q −p . |

Commutation Relations and Products of Yq

(39.7)

p

The use of Campbell-Baker-Hausdorﬀ operator identity allows us to evaluate the following products of Yq p . Yq p Yq

p

i i = exp − (q · P − p · Q) exp − (q · P − p · Q) . | |

Therefore, for Q and P in the same direction, as in one-dimensional system, then Yq p Yq

p

= Yq +q

p +p

exp

i [p q − q p ] . 2|

(39.8)

Hence, we have the commutation relation, [Yq p , Yq

p

] = Yq +q

Note that Yq p and Yq

exp p +p

− exp

i 2|

[p q − q p ]

i 2|

[p q − q p ]

.

(39.9)

commute if and only if the symplectic product q q and are collinear. Similarly, [p q − q p ] = 0, i.e., the two vectors p p p

5 Since X , Z , and Y q p q p are exponential function of the canonical operators, their eigenvalues for a qubit system is the set {1, −1} instead of the set {0, 1} for the canonical operators themselves.

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we have Yq p Yq

p

Yq

p

= Yq +q

p +p

= Yq +q

+q

Yq

i [p q − q p ] 2| i [p q − q p ] exp 2|

exp

p

p +p +p

i [[p + p ] q − [q + q ] p ] 2| i [p [q + q ] − q [p + p ]] = Yq +q +q p +p +p exp 2| i [p q − q p ] . (39.10) × exp 2| × exp

This can be generalized as

Yq1 p1 Yq2 p2 Yq3 p3 .....Yqn pn = Y

n

n

qj j=1

pl l=1

   n n  i  p1 exp qj − q1 pj   2|  j=2 j=2

   n n  i  p2 qj − q2 pj  × exp  2|  j=2+1 j=2+1

× ............. i [pn−1 qn − qn−1 pn ] . exp 2|

If all the qi ’s are equal to q and all pl ’s are equal to p, then we have (Yqp )n = Ynq,np exp

i n (n − 1) [pq − qp] . 4|

We can use Eq. (39.10) to evaluate the following expression Yq p (Yqp )n Yq† p = Yq p (Yqp )n Y−q −p = Yq p Ynq,np Y−q −p exp = Yq +nq−q ,p +np−p exp

i n (n − 1) [pq − qp] 4| i [p (nq − q ) − q (np − p )] 2|

i [np (q ) − nq (p )] 2| in (p q − q p) . = Ynq,np exp | × exp −

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39.2.2

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Expansion of Operators: Hamiltonian in Terms of Generalized Pauli Matrices

Equation (37.16), is seen as expansion of any operator in terms of generalized Pauli operators, Eq. (37.19), or matrices (on basis of a finite Hilbert space). For example, when applied on a finite field GF(2), then we have 1

π

ˆ

ˆ

χ (u, v) e−( | )(P ·v−Q·u) i

Aˆ = v=0 u=0 1

π

=

i ˆ i ˆ i χ (u, v) e( | )Q·u e−( | )P ·v e 2 u·v

(39.11)

v=0 u=0

The above expression reduces to 1

π i

Aˆ =

e 2 u·v χ (u, v)

ˆv Zˆu X

v=0 u=0 1

π

=

i ˆ ˆ χ (u, v) e( | )(Q·u−P ·v)

v=0 u=0 1

π

χ (u, v) Yˆu,v

= v=0 u=0

i ˆ ˆ Pˆ ·v) ˆ v = e−( |i )Pˆ ·v , Yˆu,v = e( |i )(Q·u− and Zˆu = e( | )Q·u are where the symbols X the Pauli operators. Performing the sum explicitly, we have i ˆ i ˆ ˜0,1 e−( | )P ·1 Aˆ = a ˜0,0 e( | )Q·0 + a i ˆ i ˆ i ˆ i +a ˜π,0 e( | )Q·π + e 2 π·1 a ˜π,1 e( | )Q·π e−( | )P ·1

ˆ1 + a =a ˜0,0 Iˆ + a ˜0,1 X ˜π,0 Zˆ1 + a ˜π,1 Yˆ1,1 ,

(39.12)

ˆ 1 represents the Pauli matrix σ where Iˆ is the identity operator, X ˆ x , Zˆ1 represents ˆ ˆ y . Thus, for example, any Hamiltonian operator for spin- 12 σ ˆ z , and Y1,1 represents σ system can, in general, be expanded in the form, ˆ + cYˆ + dZ, ˆ H = aIˆ + bX

(39.13)

where a, b, c, d, are real coeﬃcients. The case of non-Hermitian set of operators, a and a† , where a and a† are boson operators in various chosen order has been discussed under section on ‘boson annihilation and creation operators, ordering’ in Part 1. It is worthwhile to point out that with Eq. (39.13), unitary evolution operator,

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ˆ (t), is now given by U ˆ (t) = exp U = exp = exp

i ˆ + cYˆ + dZˆ t aIˆ + bX | i i ˆ + cYˆ + dZˆ t aIˆ exp t bX | | iE0 iE ˆ + ny Yˆ + nz Zˆ t n0 Iˆ exp t nx X | |

as the exponential map of the Hamiltonian operator. Ignoring the overall global phase factor, and writing H=E n ˆ·σ ˆ, this produces a unitary transformation exp

i Ht |

i = exp t E n ˆ·σ ˆ | Et ˆ = cos I + i sin |

Et |

n ˆ · σ,

(39.14)

where n ˆ = (nx , ny , nz ), with n2x + n2y + n2z = 1. In the Bloch sphere picture, this corresponds to a rotation around the n ˆ -axis of the Bloch sphere at the rate 2E | . As we have seen before, Eq. (39.14) represents the most general unitary transformation for spin- 12 or two-level system. 39.2.3

Pauli Matrices

We shall see below that these operators, X, Z, and Y , lead to the generalized Pauli matrices. Note that these results hold for discrete eigenvalues q and p belonging to finite fields, i.e., whose number of elements are power of a prime number. Now let us take the smallest field of prime number consisting of two elements, obeying modular arithmetic. Then the eigenvalues q, may be designated by the number 0 and 1, and the eigenvalues p designated by 2π|n N , {n = 0, 1 and N = 2}. Observe , where n and m are integers, the number N that the product of the form pq = 2π|nm N must be a prime number such as the number 2. In this discrete quantum mechanics, we may still designate the “Wannier function” basis states by the Dirac notation |q , in this case by |0 and |1 for N = 2. In the orthonormal “Wannier function” basis states the matrix elements of X, Z, and Y , are calculated as follows q | Xq |q = δ q q | Zp |q = exp

,q+q

,

i p · q δq |

,q ,

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ip | ip | ip |

q | Yq p |q = exp = exp = exp

·q 2 ·q 2 ·q 2

485

q | Xq Zp |q i p · q q | Xq |q | i p · q δ q ,q+q . |

exp exp

(39.15)

Thus, we may express the generalized spin matrix operator expanded in terms of the ‘Wannier function’ basis states as Yq p =

exp q ,q

= exp

i p ·q | 2

i p ·q | 2

exp

i p · q δq |

exp

i p · q |q + q |

q

,q+q

|q q| .

Substituting the possible values of q and p, namely, {0, 1} and tively, and using modular arithmetic, we have X0 = Z0 = Y0,0 = 01 10

X1 =

Z1 =

1 0 0 −1

10 01

q|

2π|0 2π|1 2 , 2

(39.16) , respec-

= I,

= σx ,

= σz ,

i p · q δ q ,q+q | 1 0 = Z1 = σ z , 0 −1

Y01 = exp =

Y10 = δ q

,q+q

=

01 10

= X1 = σ x ,

i p ·q i p · q δq exp | 2 | i p · q δ q ,q+q = i exp | 0 −i = = σy , i 0

Y1,1 = exp

,q+q

where σ x , σ y , and σz are the well-known Pauli matrices, the transformation matrices for particles with spin 12 .

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Likewise, in the orthonormal “Bloch function” basis states the matrix elements of X, Z, and Y , can be calculated from the matrix expression, ip p | Y˜p q |p = exp − | ip = exp − | ip = exp − |

·q 2 ·q 2 ·q 2

˜ p Z˜q |p p |X i ˜ p |p exp − q · p p | X | i exp − q · p δ p ,p+p . |

with only Y˜1,1 diﬀering by a phase, Y˜1,1 =

39.3

0 i −i 0

= −σ y

Discrete Fourier Transform and Generalized Hadamard Matrix

Moreover, the transformation from the “Wannier function”, |q , to the corresponding “Bloch function”, |p , for the two-state system we are considering, yields the Hadamard matrix which is well-known in the quantum computing community. Consider the identity |p =

q

q| p |q ,

where the q| p is the transformation function. For discrete quantum mechanics, this is given by the discrete Fourier transform function, i 1 q| p = √ exp − p · q . | N 2π|1 Upon substituting the possible values of q and p, namely, {0, 1} and 2π|0 , 2 , 2 respectively, for our two-state system, we obtained the matrix for q| p

1 q| p = √ 2

1 1 1 −1

≡ H,

which is the Hadamard matrix, H, which is really a “two-state discrete Fourier transform”. Noting that the “Bloch function” basis states can be obtained from the “Wannier function” basis states through the Hadamard transformation function matrix, we may thus write p= p=

2π| 2 0 2π| 2 1

1 =√ 2

1 1 1 −1

|q = 0 |q = 1

,

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which give the “Bloch function” as a linear combination of the “Wannier functions” p=

2π| 0 2

1 = √ (|0 + |1 ) , 2

(39.17)

p=

2π| 1 2

1 = √ (|0 − |1 ) . 2

(39.18)

It follows that the operator in the “Bloch function” representation are related to the operator in the “Wannier function” representation through the Hadamard matrix. Indeed, we have the following relations 1 √ 2

1 1 1 −1

0 −i i 0

1 √ 2

1 1 1 −1

=

0 i −i 0

= Y˜1,1 = −σy = σTy ,

1 √ 2

1 1 1 −1

=

1 0 0 −1

= Z˜1 = σz ,

(39.19)

01 10

˜ 1 = σx , =X

(39.20)

1 √ 2

1 1 1 −1

01 10

1 √ 2

1 1 1 −1

1 0 0 −1

1 √ 2

1 1 1 −1

=

Equations (39.19) and (39.20) reflect the exchange of the roles of the shift and phase operator when viewed from the “Wannier function” basis space to the “Bloch function” basis states. 39.3.1

Eigenfunctions and Eigenvalues of X1 , Z1 , and Y1,1

From Eqs. (39.4) and (39.5), it is clear that the eigenfunctions of σz or exp |i p · Q are the “Wannier functions” |0 and |1 , with eigenvalues 1 and −1, respectively. Similarly the eigenfunctions of σ x or exp − |i q · P are the “Bloch functions” 2π| √1 √1 p = 2π| 2 0 = 2 (|0 + |1 ) and p = 2 1 = 2 (|0 − |1 ), with eigenvalues 1 and −1, respectively. However, the eigenfunctions of exp − |i (q · P − p · Q) , which is related to σ y ⇒ exp − |i (1 · P − 1 · Q) = iX1 Z1 in the “Wannier function” representation, must be a linear combination of the “Wannier function” basis states, namely, |0 and |1 . To find this combination it is easier to directly diagonalize the σy Pauli matrix by a transformation matrix P , i.e., P −1 σy P = D,

(39.21)

where D is a diagonal matrix. The columns of P gives the eigenvectors of σy . Indeed, one can readily verify the the matrix P given by 1 P =√ 2

1 1 i −i

diagonalizes σy according to Eq. (39.21), with eigenfunctions |T1 = √12 (|0 + i |1 ) and |T−1 = √12 (|0 − i |1 ), with eigenvalues 1 and −1, respectively.

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In terms of matrices and eigenvectors, we have the following operators and eigenvectors, in the ‘Wannier function’ or ‘position’ basis states representation, 1 |0 |0 0 |0 0 ⇒ and ⇒ , [here we explicitly write the projector 0 |1 0 1 |1 1 characterizing the matrix element for clarity], i exp − P |

⇒ σx = =

0 |0 0| 1 |0 1| 1 |1 0| 0 |1 1|

0 |0 1| |1 0| 0

⇒

01 10

1 ±1

.

with eigenstates 1 ψ±1 = √ 2

|0 ± |1

1 ⇒√ 2

We also have i Y11 ≡ exp − (P − Q) | σy = =

⇒ σy ,

0 |0 0| −i |0 1| i |1 0| 0 |1 1| 0 −i |0 1| i |1 0| 0

0 −i i 0

⇒

,

with eigenstates 1 ψ±1 = √ 2

|0 ±i |1

1 ⇒√ 2

1 ±i

.

Similarly, we have exp

i Q |

⇒ σz = =

1 |0 0| 0 |0 1| 0 |1 0| −1 |1 1|

1 |0 0| 0 0 −1 |1 1|

⇒

1 0 0 −1

,

with eigenstates as the ‘coordinate eigenfunctions’ also referred to here as the ‘Wannier functions’ ψ+1 = ψ−1 =

1 |0 0 |1 0 |0 1 |1

⇒

1 0

,

⇒

0 1

.

Moreover, an operator of the form ˆ = √1 (σ x + σy + σz ) O 3

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can readily be written as ˆ = √1 O 3 1 = √ 3

01 10

+

1 1−i 1 + i −1

0 −i i 0

+

1 0 0 −1

,

which has the eigenvalues ±1, with eigenvectors given by ψ±1 = 39.3.2

1 √ 3∓ 3

1/2

1 |0

√ ±( 3∓1) (1−i)

⇒

|1

1 √ 3∓ 3

1

1/2

√ ±( 3∓1) (1−i)

General Quantum State of a Two-Level System: Sphere

.

Bloch

The normalized quantum state of a discrete system is given by |Ψ =

q

ψ (q) |q ,

(39.22)

where q

|ψ (q)|2 = 1,

in the “Wannier function” basis states, or given by |Ψ =

p

ψ (p) |p ,

where 2

p

|ψ (p)| = 1,

in the “Bloch function” basis states. Thus for the two-level system, we have from Eq. (39.22) |Ψ = α |0 + β |1 ,

(39.23)

where |α|2 + |β|2 = 1. Since α and β are complex numbers, we may thus write |Ψ = rα eiφα |0 + rβ eiφβ |1 with four real parameters rα , rβ , φα , and φβ . We can choose the coeﬃcient of |0 to be real and positive by factoring out eiφα . Since the global phase has no observable

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consequences, we may now write Eq. (39.23) as a three-parameter expression |Ψ = rα |0 + rβ ei(φβ −φα ) |1 = rα |0 + rβ eiφ |1 . By writing rβ eiφ = x + iy, the normalization constraint becomes rα2 + |(x + iy)|2 = 1 = rα2 + x2 + y2 which is the equation of a unit sphere in real 3-dimensional space with Cartesian coordinates (x, y, rα ). In terms of spherical polar coordinates for the parameter space these Cartesian coordinates a re given by x = r sin θ cos φ, y = r sin θ sin φ, rα ⇒ z = r cos θ, where r ≡ 1 by virtue of the normalization constraint on |Ψ . Therefore, we have |Ψ = z |0 + sin θ (cos φ + i sin φ) |1 = cos θ |0 + eiφ sin θ |1 . Now we have just two parameters, θ and φ, describing points on the surface of a unit sphere. Note that for θ = π4 , we obtain the Hadamard transformation from the “Wannier function” basis states to the “Bloch function” basis states for the choices of φ = 0 and φ = π. Moreover, we also obtain the eigenfunctions of σy for the choice of φ = π2 and φ = − π2 , for the eigenvalues 1 and −1, respectively. Note that all these eigenfunctions are orthonormal and are defined by opposite points on the horizontal plane cutting the hemisphere in the parameter space at θ = π4 . Moreover, quantum states defined by points on the hemisphere for θ = 0 [|0 “Wannier function” basis state] and those defined by θ = π2 [eiφ |1 ≡ |1 “Wannier function” basis state] are orthonormal. In general two quantum states defined by (θ, φ) and π2 − θ, φ + π , respectively, i.e., points in the Euclidean hemisphere diﬀering in parameter θ by 90◦ are orthogonal. We have |Ψ = cos θ |0 + eiφ sin θ |1 , |χ = cos

π π − θ |0 + ei(φ+π) sin − θ |1 . 2 2

The dot product of their corresponding vectors is π π − θ − sin θ sin −θ 2 2 π π = 0, = cos θ + − θ = cos 2 2

χ |Ψ = cos θ cos

(39.24)

demonstrating the orthogonality of the quantum states defined by these points on the Euclidean hemisphere.

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39.3.2.1

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Range of the Parameters

We notice that |Ψ = |0 for θ = 0 and |Ψ = eiφ |1 (equivalent to |1 ) for θ = π2 . This suggests that the range 0 ≤ θ ≤ π2 generates all the quantum states of the two-level system as far as this parameter is concerned. Since the surface of the Euclidean sphere is defined in the range of θ, 0 ≤ θ ≤ π, this means that it is only necessary to consider the upper hemisphere in the parameter space. We expect that corresponding points in the lower hemisphere are equivalent to the points in the upper hemisphere. Indeed, the point corresponding to parameter coordinates (θ, φ) lies in the opposite point of the sphere defined by the coordinates (π − θ, φ + π). For this opposite point on the sphere, we have |Ψ = cos (π − θ) |0 + ei(φ+π) sin (π − θ) |1 = − cos θ |0 + eiφ sin θ |1 = − |Ψ , indicating that the opposite points in the sphere only diﬀer by a global phase factor of eiπ , which has no observable consequences, and therefore are equivalent. 39.3.2.2

Bloch Sphere

The Bloch sphere is obtained by scaling the parameter θ by a factor of 2, i.e., θB = 2θ. This maps the points on the Euclidean hemisphere onto points on a Bloch sphere. We have |Ψ = cos

θB θB |0 + eiφ sin |1 , 2 2

(39.25)

where 0 ≤ θB ≤ π, 0 ≤ φ ≤ 2π are the range of the coordinates of the points in the Bloch sphere. The scaling θB = 2θ is a one-to-one mapping of the points in the Euclidean hemisphere to the Bloch sphere except for the points on the equator of the Euclidean sphere which is mapped to a single point θB = π, the ‘south pole’ on the Bloch sphere. But as we have remarked above, all the points on the equator of the Euclidean sphere, eiφ |1 , are equivalent, and indeed the phase, eiφ , becomes meaningless on the ‘south pole’ of the Bloch sphere. It is worthwhile mentioning at this point that shift operator X1 or its matrix representation σ x interchanges the ‘south’ and ‘north’ pole of the Bloch sphere and may be viewed as a 180◦ degree rotation of the Bloch sphere about the x-axis. To see this, note that rotation by angle π of the Bloch sphere about the x-axis would send any vector from the first quadrant to the fourth quadrant, or from the second to third quadrant, with the accompanying change of parameters φ ⇒ −φ, θB ⇒ π − θB .

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Hence Eq. (39.25) becomes |Ψ

π − θB π − θB |0 + e−iφ sin 2 2 θB − π θB − π |0 − e−iφ sin = cos 2 2 θB θB |0 + e−iφ cos |1 = sin 2 2 θB θB |0 + cos |1 , ⇒ eiφ sin 2 2 = cos

|1 |1

where the last line is obtained by multiplying by a global factor eiφ . Since X1 is Hermitian, this correspond to the ‘contravariant’ transformation of the component vector or wavefunction cos θ2B

01 10

eiφ sin

θB 2

eiφ sin θ2B

=

.

cos θ2B

The phase operator Z1 or its matrix representation σ z rotates the Bloch sphere about the z-axis by angle π, as can be seen from 1 0 0 −1

cos θ2B eiφ sin

θB 2

=

cos θ2B ei(φ+π) sin θ2B

,

since π−rotation about the z-axis only changes the angle φ by π. The Y1,1 operator or its matrix representation given by σ y rotates the Bloch sphere about the y-axis by an angle π, which transform the vector from the first quadrant to the second quadrant, or from the third to the fourth quadrant, with the accompanying change of parameters φ ⇒ π−φ θB ⇒ π − θB Thus the new quantum state is |Ψ

π − θB π − θB |0 + ei(π−φ) sin 2 2 θB θB |0 − e−iφ cos |1 = sin 2 2 θB θB |0 − cos |1 . ⇒ eiφ sin 2 2 = cos

|1

Since Y1,1 is Hermitian, this correspond to the ‘contravariant’ transformation of the component vector or wavefunction 0 −i i 0

cos θ2B eiφ sin θ2B

=i

−eiφ sin θ2B cos θ2B

⇒

eiφ sin (θB2+π) cos (θB2+π)

.

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Moreover, successive application of any one of the three operators, X1 , Z1 , Y1,1 for even number of times yields identity, whereas when applied for odd number of times yields the same operator. 39.3.2.3

Quantum States on Opposite Points of Bloch Sphere

We note that θ = π4 is mapped to θB = π2 . Thus the two orthonormal “Bloch function” basis states are now defined on opposite points of the Bloch sphere, as well as the two orthonormal eigenfunctions of the σ y Pauli matrix. Indeed the relation given by Eq. (39.24) now reads, using θB instead of θ, π θB π θB θB θB cos − sin − − sin 2 2 2 2 2 2 π − θB π − θB θB θB cos sin − sin = cos 2 2 2 2 π = 0, = cos 2

χ |Ψ = cos

(39.26)

which shows that quantum states defined by opposite points on the Bloch sphere are orthogonal.

39.3.3

Exponential Map

Recall our basis for defining the shift and phase operators in discrete phase space, Eqs. (39.4) - (39.7), which lead to the Pauli matrices σ x , σy , and σz . In Sec. 39.3.2.2, we have shown that the operators X1 , Z1 , and Y1,1 correspond to rotation of the Bloch sphere by a specific angle π about the axes x, z, and y, respectively. In this section, we are interested in rotation of the Bloch sphere about the axes x, z, and y, by an arbitrary angle. Furthermore, we are also interested in the rotation of the Bloch sphere by arbitrary angle about an arbitrary axis of rotation. To do this end we employ the exponential map to parametrize by an arbitrary angle the X1 , Z1 , and Y1,1 rotation operators about their respective axis of rotation. We can proceed in either of the two ways, namely, obtained the exponential map of the operators themselves, or obtained exponential map through the matrix representation of these operators. We should mention that exponential map is a fundamental construction in the theory of Lie groups. Case 39.1

Exponential map using the operators: We note the following i X1 = exp − P |

,

(X1 )2n = I, i (X1 )2n+1 = exp − P |

= X1

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Z1 = exp

i Q , |

(Z1 )2n = I, (Z1 )2n+1 = exp

i Q |

= Z1

i Y1,1 = exp − (P − Q) , | (Y1,1 )2n = I, i (Y1,1 )2n+1 = exp − (P − Q) |

= Y1,1

where we have used modular arithmetic for the prime field of two elements. Now the exponential of an operator A parametrized by θ is given by exp {iθA} = I + iθA −

(θA)3 (θA)4 (θA)5 (θA)2 −i + +i + ...... 2! 3! 4! 5!

For the special case that A2n = I, and A2n+1 = A, we have θ3 A θ4 I θ5A θ2 I −i + +i + ...... 2! 3! 4! 5! θ3 θ5 θ2 θ4 + ..... I + i θ − + ..... A = 1− 2! 4! 3! 5! = cos (θ) I + i sin (θ) A

exp {iθA} = I + iθA −

Therefore, we have for arbitrary angle θ B the following rotation operator on the Bloch sphere θB X1 2

= cos

θB 2

I − i sin

θB 2

X1

(39.27)

θB Y1,1 2

= cos

θB 2

I − i sin

θB 2

Y1,1

(39.28)

θB Z1 2

= cos

θB 2

I − i sin

θB 2

Z1

(39.29)

Rx (θB ) = exp −i

Ry (θ B ) = exp −i

Rz (θ B ) = exp −i

From Eqs. (39.27) - (39.29), the matrix representation can readily be obtained by using σ x , σ y , and σz . Case 39.2 It is also interesting to see the exponential map in terms of the operators Q and P . Then the exponential map looks like an exponential of an exponential.

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We have θB X1 2

Rx (θ) = exp −i = exp −i

i θB exp − P 2 |

= exp −i

θB 2

= exp −i

θB cos 2

1 1 cos P − i sin P | | 1 P |

exp i

θB i sin 2

1 P |

(39.30)

We make use of the Jacobi-Anger identity to evaluate the first factor. We have

exp −i

1 P |

θB cos 2

=

∞

θB 2

in Jn

n=−∞

1

ein | P

We can rewrite the right hand side as ∞

in Jn

n=−∞

θB 2

∞

1

ein | P =

i2n J2n

n=−∞ ∞

+

θB 2

1

ei2n | P

i2n+1 J2n+1

n=−∞,n=0

θB 2

1

1

ei(2n+1) | P

1

1

Using modular arithmetic, ei2n | P = I for all n, and ei(2n+1) | P = ei | P for all n. Therefore, we have ∞

i2n J2n

n=−∞

θB 2

∞

1

ei2n | P =

θB 2

(−1)n J2n

n=−∞

=

J0

θB 2

+2

∞

I

(−1)n J2n

n=1

θB 2

I

The expression within the curly bracket is the well-known expression for cos in terms of the Bessel function of the first kind, namely,

cos

θB 2

= J0

θB 2

− 2J2

θB 2

+ J4

θB 2

− J6

θB 2

+ ....

θB 2

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θB 2

i2n+1 J2n+1

n=−∞,n=0 ∞

1

ei(2n+1) | P θB 2

i2n+1 J2n+1

=

n=−∞,n=0

=

−∞ n=−1

= =

−∞

1

ei | P

θB 2

i−(2n+1) J−(2n+1)

θB 2

i (−1)n J(2n+1)

n=1

=i 2

∞

θB 2

n=1

∞

1

ei | P +

∞

1

ei | P +

θB 2

1

ei | P θB 2

i2n+1 J2n+1

n=1

i (−1)n J2n+1

n=1

θB 2

(−1)n J(2n+1)

i2n+1 J2n+1

n=1

i−(2n+1) (−1)(2n+1) J(2n+1)

n=−1 ∞

∞

1

ei | P +

θB 2

1

ei | P

1

ei | P

1

ei | P

the expression within the curly bracket of the last line is the well-known expression of sin θ2B in terms of the Bessel functions as θB 2

sin

θB 2

= 2J1

− 2J3

θB 2

θB 2

+ 2J5

− .........

Thus the first factor of Eq. (39.30) yields exp −i

1 P |

θB cos 2

θB 2 θB 2

= cos = cos

I − i sin I − i sin

θB 2 θB 2

1

ei | P X1

We are left with second factor of Eq. (39.30) exp i = =

∞ n=−∞ ∞

Jn i

θB 2

J2n

θB i 2

n=−∞

= J0 i

=

1 P |

θB i sin 2

θB 2

J0 i

θB 2

I +2

n

ei | P i 2n | P

e

+

J2n+1 i

n=−∞,n=0 ∞ n=1

+2

∞

∞ n=1

J2n i

θB 2

J2n i

θB 2

θB 2

∞

I+

2n+1 P |

J2n+1 i

n=−∞,n=0

I+

ei

∞ n=−∞,n=0

θB 2

J2n+1 i

θB 2

ei

2n+1 P |

ei

2n+1 P |

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The expression in the curly bracket is a well-known sum formula for the Bessel functions [Abramowitz and Stegun, 1972, p.363] which is equal to identity, i.e., J0 i

θB 2

∞

+2

J2n i

n=1

θB 2

=1

We are left with the last term which evaluates to zero as ∞

J2n+1 i

n=−∞,n=0 ∞

=

ei

2n+1 P |

(−1)2n+1 J(2n+1) i

= =

θB 2

n=1 ∞ n=1 ∞ n=1

(−1) J(2n+1) J(2n+1) i

θB i 2

θB 2

θB 2

ei

−1 | P

+

∞

J(2n+1) i

n=1

i −1 | P

e

∞

+

J(2n+1) i

n=1 i −1 | P

1

ei | P − e

θB 2

θB 2

1

ei | P 1

ei | P

=0

We could have readily concludedin Eq. (39.30) that exp i −1 1 ei | P −ei | P

θB 2 i

sin

1 |P

=

≡ 0 by virtue of modular arithmetic for the I, since sin |1 P = 2 coeﬃcient of the operator P , where −1 = 1 (mod 2). therefore we end with the same result as before, Rx (θB ) = cos

θB 2

I − i sin

θB 2

X1

Similar procedure can be followed for the rotation about the y and z axes, yielding Ry (θ B ) = cos

θB 2

I − i sin

θB 2

Y1,1

Rz (θB ) = cos

θB 2

I − i sin

θB 2

Z1

For θB = π, Rx (π) = −iX1 , Ry (π) = −iY1,1 , and Rz (π) = −iZ1 . Although the presence of −i occurs here as a non-significant global factor, the unitary operator that exactly yield X1 , Z1 , and Y1,1 is obtain for the exponential mapping of the whole quantum operators, X, Z, and Y , as we will show below. But first let us examine rotation about an arbitrary axis. 39.3.3.1

Rotation about an Arbitrary Axis in Real 3-D Space

The parametrization of the rotation the Bloch sphere about an axis in real space by an arbitrary angle θ = θ2B can likewise be carried out by the exponential map of the projection on a unit vector in real three dimensions, n ˆ = (nx , ny , nz ), multiplied by

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θB 2 ,

of either X1 , Z1 , and Y1,1 or σ x , σy , and σz . The expression for the rotation operator about the n ˆ -axis is given by Rnˆ (θB ) = exp i

θB n ˆ·σ , 2

(39.31)

where σ denotes the three component vector (X1 , Y1,1 , Z1 ) of Pauli operators. One can readily show that (ˆ n · σ)2 = (nx X1 + ny Y1,1 + nz Z1 )2

= n2x (X1 )2 + n2y (Y1,1 )2 + n2z (Z1 )2

= n2x + n2y + n2z I =I since the cross products anticommute, by virtue of Eq. (39.8). The expression for Rnˆ (θ B ) is therefore similar to the expressions given by Eqs. (39.27) - (39.29), giving Rnˆ (θB ) = cos

θB 2

I + i sin

θB 2

n ˆ · σ.

Note that for θB = π, we obtain the Hadamard transformation from the “Wannier function” basis states to the “Bloch function” basis states for the choices of φ = 0 and φ = π. 39.3.3.2

Arbitrary Unitary Operator for a Qubit: Quantum Control

So far, we have only given an exponential map of the shift and phase operators and a portion of their products, X1 , Z1 , and Y1,1 out of the whole quantum operators X, Z, and Y . This is the treatment following traditional discussions the literature. However, from Eqs. (39.4) - (39.7), it is clear that a complete unitary operator should be an exponential map of (I + X1 ), (I + Z1 ), and (I + Y1,1 ), and for arbitrary axis of rotation, (n0 I + n ˆ · σ), where I is the identity matrix. Denoting the arbitrary unitary operator for a two-level system, also known as a qubit, as Un˜ (θB ), then Eq. (39.31) must now read as Un˜ (θB ) = exp i

θB [n0 I + n ˆ · σ] , 2

where n ˜ = (n0 , nx , ny , nz ). Since I commutes with X1 , Z1 , and Y1,1 , we can write θB θB n0 I exp i n ˆ·σ 2 2 θB = exp i n0 Rnˆ (θB ) . 2

Un˜ (θ B ) = exp i

Similarly, we have Ux (θB ) = exp i

θB n0 Rx (θB ) , 2

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Uy (θB ) = exp i

θB n0 Ry (θ B ) , 2

Uz (θB ) = exp i

θB n0 Rz (θB ) . 2

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By letting n0 = 1 and θ B = π, we indeed have exactly Ux (π) = X1 , Uy (π) = Y1,1 , Uz (π) = Z1 . It is also through the unitary operator Un˜ (θB ) that one can exactly recover the Hadamard transformation matrix, which is a two-state discrete Fourier transformation between the “Wannier function” basis states and “Bloch function” basis states, or contravariant transformation between components (wavefunction) based on “Wannier function” basis states and based on “Bloch function” basis states . Thus for n0 = 1, θ B = π, and n ˆ = √12 , 0, √12 , we have Un˜ (π) = exp i

Un˜ (π) = exp i

π 2

= exp i

π 2

1 = √ 2

cos

π Rnˆ (π) 2

π π I − i sin 2 2

1 −i √ 2

1 1 1 −1

n ˆ = √12 ,0, √12

,

1 √ (X1 + Z1 ) 2

1 1 1 −1

.

The general unitary transformation Un˜ (θB ) constitutes the Lie group SU (2). The Lie group SU (2) is a continuos map and represents a universal covering of the three-dimensional rotation group SO (3). This can be seen from the following relations Un˜ (0) = I, Un˜ (2π) = exp i = exp i

2π n0 Rnˆ (2π) 2 2π n0 (−I) 2

= (−1)

n0

= (−1)

n0 +1

(−I) I,

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Un˜ (4π) = exp i

4π n0 Rnˆ (4π) 2

= exp {i2πn0 } Rnˆ (4π) = exp {i2πn0 } I = I. If n0 = 0, Un˜ (2π) = −I. Since, in general, we need 4π rotation to identically restore the state of a two-level system, SU (2) is a double covering of SO (3) in group theoretical terminology. 39.3.4

Density Operator for a Two-Level System: Disordered and Pure States

From Eq. (39.12), we can write the density operator for single qubit as ˆρ = n0 I + nx X1 + ny Y1,1 + nz Z1 , and hence the density matrix will be of the form of a Hermitian operator as ρ=

n0 + nz nx + iny nx − iny n0 − nz

.

From the requirement that T r ρ = 1, we must have n0 = 12 . Thus, we can rewrite ρ as ρ=

1 2

1 + αz αx + iαy αx − iαy 1 − αz

which we cast in terms of the Pauli matrices as 1 ρ = (I + α · σ) . 2

,

(39.32)

The eigenvalues of ρ are then (1±|α|) , so for physical reasons, the range of α is 2 0 ≤ α ≤ 1. Since the vector α corresponds to direction in the Bloch sphere, we have here a representation of the density matrix in which the center of the sphere, α = 0, corresponds to completely disordered density matrix, ρ = 12 I and the surface of the Bloch sphere with |α| = 1 corresponds to the pure states with diﬀerent spin orientations (in the direction of the vector α).

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Chapter 40

Discrete Quantum Mechanics on Finite Fields

We are interested in extending the phase space (p, q) treatment of quantum mechanics based on the simplest prime field, GF(2), i.e., a two-level system or single spin- 12 system to a larger system composed of spin- 12 subsystems1 . Recall that in the treatment of the quantum mechanics over GF(2) above, the ubiquitous dot product p·q 2π | , where p = n| 2 and q = m, with n and m being elements of GF(2) obeying modular arithmetic, governs the theoretical development for spin- 12 particle. The theoretical treatment to be discussed in what follows also holds for other extension of any prime field, e.g., extension of GF(3), GF(5), etc.. 40.1

Tensor Product of Operators

Suppose |v and |w are unentangled states on m-dimensional and n-dimensional space, C m and C n , respectively. The state of the combined system is |v ⊗ |w on mn-dimensional space, C mn . If the unitary operator A is applied to the first subsystem, and B to the second subsystem, the combined state becomes A |v ⊗ B |w . In general, the two subsystems will be entangled with each other, so the combined state is not a tensor product state. We can still apply A to the first subsystem and B to the second subsystem. This gives the operator A ⊗ B on the combined system, defined on entangled states by linearly extending its action on unentangled states. Example 40.1 (A ⊗ B) (|0 ⊗ |0 ) = A |0 ⊗ B |0 (A ⊗ B) (|1 ⊗ |1 ) = A |1 ⊗ B |1 1 1 (A ⊗ B) √ (|0 |0 + |1 |1 ) = √ [(A ⊗ B) |0 |0 + (A ⊗ B) |1 |1 ] 2 2 1 = √ (A |0 ⊗ B |0 + A |1 ⊗ B |1 ) 2 1 A good reading on discrete quantum mechanics on finite fields is given by K. S. Gibbons, M. J. Hoﬀman, and W. K. Wootters, Phys. Rev. A 70, 062101 (2004).

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Let {e1 , e2 , e3 , ....em } be the basis of the first subsystem and let {f1 , f2 , f3 , ....fn } be the basis of the second subsystem. Thus we write A= i,j

B= k,l

aij |ei ej | ,

bkl |fk fl | .

The basis of the combined system is {e1 , e2 , e3 , ....em } ⊗ {f1 , f2 , f3 , ....fn }. The operator A ⊗ B is A⊗B =

i,j

aij |ei ej | ⊗

= i,j,k,l

= i,j,k,l

k,l

bkl |fk fl |

aij bkl |ei ej | ⊗ |fk

fl |

aij bkl (|ei ⊗ |fk ) ( ej | ⊗ fl |) ,

which yields a mn by mn matrix. In terms of the larger basis system, {e1 , e2 , e3 , ....em } ⊗ {f1 , f2 , f3 , ....fn }, we can also write the operators A and B as A= i,j,k,l

= i,j

aij δ kl (|ei ⊗ |fk ) ( ej | ⊗ fl |)

aij |ei ej | ⊗

k

|fk fk |

= A ⊗ IB , where IB = k

|fk

fk |. Similarly, we have B= i,j,k,l

= i

δ ij bkl (|ei ⊗ |fk ) ( ej | ⊗ fl |)

|ei ei | ⊗

= IA ⊗ B,

k,l

bkl |fk fl |

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where IA =

|ei ei |. Therefore, we can recast

i

A ⊗ B = (A ⊗ IB ) ⊗ (IA ⊗ B) = 

i,j

=

aij |ei

i,j,k

But

503

ej | ⊗

k

|fk

fk |



⊗

|ei

i





aij bkl (|ei ⊗ |fk ) ( ej | ⊗ fk |) ⊗ 

ei | ⊗

i ,k,l

k,l

bkl |fk



fl |



(|ei ⊗ |fk ) ( ei | ⊗ fl |) .

fk | fk = δ k,k , ej | ei = δ j,i , ( ej | ⊗ fk |) ⊗ (|ei ⊗ |fk ) = δ k,k δ j,i . Upon summing over i and k , we end up with A⊗B =

i,j,k,l

= i,j

aij bkl (|ei ⊗ |fk ) ( ej | ⊗ fl |)

aij |ei ej | ⊗

k,l

bkl |fk fl |

as obtained before. Thus, the Hilbert spaces of composite systems are represented by the tensor products of the Hilbert spaces of the composing subsystems. If the Hamiltonian of the composite system is a sum of terms which act only on the individual subsystems, namely, H = H(1) ⊗ I2 + I1 ⊗ H(2) , then the unitary evolution operator, U (t), is given by i U (t) = exp − H t | i H(1) ⊗ I2 + I1 ⊗ H(2) t = exp − | i i H(1) ⊗ I2 t exp − I1 ⊗ H(2) = exp − | |

t .

The terms of the right hand side in the expansion of the exponential as power series would involve products of the form  x  y 

(1)

i,j

Hij |ei ej | ⊗

k

|fk

fk | 

i

|ei

ei | ⊗

(2)

k,l

Hkl |fk

fl | ,

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where x and y are integers. First assume x = y. Then  

(1)

i,j

Hij |ei ej | ⊗ 

=  

=  

=  

= 

x 

fk | 

|fk

k

(1)

i,j

Hij |ei ej | ⊗

i,j

i,j

(1) Hij |ei

⊗

i



k



(1)

i,j

|fk  

k

|fk   

Hij |ei ej | ⊗ 

i



fk | 

|fk 

(1)

Hij |ei ⊗

k

|ei

i

i

ej |ei

k,l

Hkl |fk fl | (2)

ei | ⊗

ei | ⊗

δ i ,j ei | ⊗

k,l

x

fl |

Hkl |fk

x

(2)

fk |fk fl |

Hkl k,l

(2) Hkl δ k ,k k,l

x

(2)

k,l

(2)

ei | ⊗

|ei

x

x

fl |

Hkl |fk fl | .

If x = y, then assume that x < y. Then we have  

(1)

i,j

Hij |ei ej | ⊗ 

× 

= 

i

i,j

× =  

=

i

Hij |ei

|ei

(1)

i,j

i,j

k

ei | ⊗

(1)





|ei

|fk



fk | 



k,l



ej | ⊗  x



(1) Hij |ei ej | ⊗ 

(2)

k,l

Hkl |fk

x

fl |

x

Hkl |fk fl | y−x

Hkl |fk fl | 

ei | ⊗

y−x

(2)

(2)

k,l

i

|ei

(2) Hkl |fk fl |

ej | ⊗ 

ei | ⊗

Hij |ei

k,l

x 

k,l

k,l

x 

(2) Hkl |fk fl | 

y

(2) Hkl |fk fl | .

(2)

k,l

y−x

Hkl |fk fl |

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Similar results holds for x > y. Therefore, U (t) = exp −

i H(1) ⊗ I2 |

t exp −

i I1 ⊗ H(2) |

t

i i = exp − H(1) t ⊗ exp − H(2) t | | = U (1) (t) ⊗ U (2) (t) , which shows that the unitary operators of a composite system is a tensor product of each subsystem’s unitary operators. If U (1) (t) ⊗ U (2) (t) acts on a product state |v ⊗ |w , the results will remain a product state, A |v ⊗ B |w . When the Hamiltonian includes interaction terms which act on both subsystems, this is no longer true. 40.1.1

Entanglement Due to Interactions

If the Hamiltonian includes terms which act on both subsystem, namely, H = H(1) ⊗ I2 + I1 ⊗ H(2) + Hint , where Hint represents the terms which act on both subsystems, or the interaction terms, then the unitary evolution operator, U (t) = exp − |i H t , will no longer be a tensor product in general. In this case, when U (t) acts on a product state, the state becomes entangled in general. Example 40.2

For spin- 12 particle, Hint might take the form as Hint = Eint Z (1) ⊗ Z (2)

where Z (1) and Z (2) are Pauli matrices pertaining to particle 1 and particle 2, respectively. The corresponding unitary transformation is of the form U (θ) = cos (θ) I − i sin (θ)

Z (1) ⊗ Z (2)

This transformation acting on initial product state given by |ψ ⊗ |φ = α1 β 1 |↑↑ + α1 β 2 |↑↓ + α2 β 1 |↓↑ + α2 β 2 |↓↓ will transform this into U (θ) (|ψ ⊗ |φ )

= e−iθ α1 β 1 |↑↑ + eiθ α1 β 2 |↑↓ + eiθ α2 β 1 |↓↑ + e−iθ α2 β 2 |↓↓

= e−iθ (α1 β 1 |↑↑ + α2 β 2 |↓↓ ) + eiθ (α1 β 2 |↑↓ + α2 β 1 |↓↑ )

which is no longer a product state for θ = m π2 . For θ = m π2 , m = even, the result is the same as as for θ = 0 product state. For θ = m π2 , m = odd, the result is corresponds to the product of (α1 |↑ − α2 |↓ ) (β 1 |↑ − β 2 |↓ ) which is still a product state. However, for θ = m π4 , the result becomes a superposition of singlet, St = 0, and triplet states, St = 1, and no longer a product state.

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The No-Cloning Theorem

The no cloning theorem results from the unitary transformation process in quantum mechanics. This theorem forbids making identical copies of an arbitrary unknown quantum state. The term cloning refers to a unitary transformation process whose end result is a separable state with identical factors. Since copying is supposed to be a unitary process, U, the initial state should also be a product state. We will now show that no such unitary U exists. Proof. Suppose the state in system A that we wish to copy is |ψ A . In order to make a copy, we take a system B with the same state space and initial state |b . The initial, or blank, state must be independent of |ψ A , of which we have no prior knowledge. The composite system is then described by the tensor product, and its state is |ψ

⊗ |b

A

B

Note that we cannot perform any observation, since this irreversibly collapses the system into some eigenstate of the observable, corrupting the information originally contained in |ψ A . What we could do is control the Hamiltonian of the system, and thus the time evolution operator U up to some fixed time interval, which is an unitary operator. Then U acts as a copier provided that U |ψ

A

⊗ |b

B

= |ψ

A

⊗ |ψ

(40.1)

B

holds for any given state, such as |ψ A or |φ A , in system A. By definition of unitary operator, U preserves the inner product. Therefore we ought to have b|B ⊗ φ|A U † U |ψ b|B ⊗ φ|A |ψ

A A

⊗ |b ⊗ |b φ| ψ

B B A

= φ|B ⊗ φ|A |ψ A ⊗ |ψ = φ|A |ψ A φ|B |ψ B = φ| ψ A φ| ψ B

B

This is clearly not true in general except, for example, φ| ψ = 0, in which case |φ and |ψ are eigenfunctions in system A. Therefore no such U exists for copying an unknown quantum state. This proves the no cloning theorem. Alternatively, one can argue using just linearity of U . If cloning succeeds, as in Eq. (40.1), then one must have for |φ = |ψ U (α |ψ

A

+ β |φ

A) ⊗

|b

B

= α |ψ A ⊗ |ψ B + β |φ A ⊗ |φ B = (α |ψ A + β |φ A ) ⊗ (α |ψ B + β |φ

which is clearly a contradiction. Moreover, if β = 0, we have U α |ψ αU |ψ

⊗ |b A ⊗ |b A

B B

= α |ψ = α |ψ

⊗ α |ψ B A ⊗ |ψ B A

Therefore α |ψ

A

⊗ |ψ

B

= α2 |ψ

A

⊗ |ψ

B

This is again a contradiction, therefore no-cloning holds.

B)

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Generality: In the statement of the theorem, two assumptions were made: the state to be copied is a pure state and the proposed copier acts via unitary time evolution. There is no loss of generality. If the state to be copied is a mixed state, it can be purified. Similarly, an arbitrary quantum operation can be implemented via introducing an ancilla (the definition of Ancilla is someone who helps through a process. Syn: A broker) and perform a suitable unitary evolution. Thus the no cloning theorem holds in full generality. 40.1.2.1

Consequences

Some of the consequences of the no-cloning theorem is that ‘fan-out’ in logic gates are impossible. In contrast, the no cloning theorem is a desirable virtue in quantum cryptography, as it forbids eavesdroppers from creating copies of a transmitted quantum cryptographic key. The no-cloning theorem is closely related to the uncertainty principle in quantum mechanics. If cloning an unknown state is possible, then one could make arbitrarily many copies, and measure each dynamical variable with arbitrary precision, thereby bypassing the uncertainty principle. 40.2

Quantum Control

For many experimental systems, the quantum evolution can be controlled by turning precisely-tuned lasers on and oﬀ for precise lengths of time. For spin- 12 system, this implies that we can perform arbitrary combinations of unitary transformations, Eq. (39.14), of the form U (θ B ) = cos

θB 2

I + i sin

θB 2

n ˆ · σ.

By turning one Hamiltonian on for a particular length of time, τ 1 , and another Hamiltonian for another length of time, τ 2 , then we can perform the unitary evolution U (τ 1 + τ 2 ) = U1 (τ 1 ) U2 (τ 2 ) . The essential feature is that any rotation around any axis n ˆ can be decomposed as a sequence of three rotations around two axes. For example any rotation can be achieved by the sequence of rotations, e.g., Rnˆ (θ) = Rz (φ)Ry (η)Rz (ξ), for some values of φ, η, and ξ, where rotations are applied from right to left. Example 40.3

A rotation about the x-axis is given by the unitary transformation Rx (θ) =

cos (θ) i sin (θ) i sin (θ) cos (θ)

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This can be done by performing three rotations in a row around the z and y axes, namely, Rx (θ) = Ry (η)Rz (−θ)Ry (−η) where cos (θ) − i sin (θ) 0 0 cos (θ) + i sin (θ)

Rz (−θ) =

e−iθ 0 0 eiθ

=

Ry (η) =

cos (η) sin (η) − sin (η) cos (η) cos (η) − sin (η) sin (η) cos (η)

Ry (−η) = Substituting η = π4 , then we have Rx (θ) =

1 2

1 1 −1 1

e−iθ 0 0 eiθ

=

1 2

1 1 −1 1

e−iθ −e−iθ eiθ eiθ

=

1 −1 1 1

cos (θ) i sin (θ) i sin (θ) cos (θ)

We can also arrive at the same result by performing the following diﬀerent combination Rx (θ) = Rz (η)Ry (−θ)Rz (−η) eiη 0 0 e−iη

= Substituting η =

3π 4 ,

e−iη 0 0 eiη

then we have

Rx (θ) =

as before.

cos (θ) − sin (θ) sin (θ) cos (θ)

eiη 0 0 e−iη

e−iη cos (θ) −eiη sin (θ) e−iη sin (θ) eiη cos (θ)

=

cos (θ) −ei2η sin (θ) cos (θ) e−i2η sin (θ)

=

cos (θ) i sin (θ) i sin (θ) cos (θ)

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40.2.1

509

Pauli Operators over Power-of-Prime Finite Fields

We have seen before that the operators, X, Z, and Y , lead to the Pauli matrices over the prime field, GF(2), enabling us to formulate the quantum mechanics of spin- 12 systems. We will refer to X, Z, and Y as the Pauli operators. We are here interested in the formulation of quantum mechanics over finite fields which are extension of the prime fields. Let us rewrite the various Pauli operators, namely, the shift operator, Xq , and phase operator, Zp , which obey the equations i Xq |q = exp − q · P |

|q = |q + q ,

i p · Q |q = exp |

Zp |q = exp

(40.2)

i p · q |q , |

(40.3)

and the operator Yq p i Yq p ≡ exp − (q · P − p · Q) . |

(40.4)

The product of Xq and Zp is thus given by i Xq Zp = exp − q · P |

exp

i p ·Q |

i = exp − (q · P − p · Q) exp | = exp −

i p ·q | 2

− |i q · P, |i p · Q 2

i exp − (q · P − p · Q) . |

(40.5)

Therefore Yq p = exp

i p ·q | 2

Xq Zp .

(40.6)

The variables, q and p, we are considering now belongs to the elements of the finite fields, GF(rn ), where r is a prime number and n an integer. In the following discussion, we will focus our treatment on finite fields extension given by GF(2n ), i.e., r = 2. For GF(rn ), the translation operator T (q)sym = exp −

i p·q | 2

exp

i (p · Q − q · P ) |

= Xq Zp ,

which we now label here as T(qp) = exp −

i p·q | 2

exp

i (p · Q − q · P ) , |

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acquires diﬀerent meaning from those of prime fields, such as GF(r)) or GF(2) (discussed above), when applied to GF(rn ). For GF(rn ), the translation operator becomes a product of operators acting on each respective composite ‘prime’ system, namely, (1)

(2)

(3)

(n)

n = T(qp) ⊗ T(qp) ⊗ T(qp) ⊗ .... ⊗ T(qp) T(qp)

= exp −

i |

n

n

pi qi 2

i=1

⊗ exp

l

i (p · Q − q · P ) |

(l)

,

(40.7)

where pi = | 2π r wi , with wi , qi , q, and p as field elements of GF(r), The symbol n

used, l

⊗ indicates direct product. In terms of the field elements of GF(rn ), we

n as can rewrite T(qp)

T(˜qp) ˜ = exp −

i p˜ · q˜ exp | 2

i ˜ − q˜ · P˜ p˜ · Q |

,

(40.8)

where p˜, and q˜ are now considered elements of the extension field, GF(rn ). It is ˜ and P˜ are the ‘position’ and ‘momentum’ operators over also understood that Q n GF(r ). In terms of the field bases of GF(rn ), q˜ and p˜ have the following expansions n

q˜ =

qi ei , i=1

n

p˜ =

pj fj , j=1

where the set E = {e1 , e2 , e3 , ...en } and F = {f1 , , f2 , f3 , ....fn } are field bases of GF(rn ), and the coeﬃcients qi and pi are basically field elements of GF(r), i.e., of base prime-field. Therefore, the simple product pi qi obeys the modular arithmetic of the base prime-field. Since 1 ≤ i ≤ n and 0 ≤ qi ≤ r − 1, indeed, there are rn diﬀerent q˜ elements and rn diﬀerent p˜ elements. Since in the power-of-prime field algebra, the ‘dot product’ is carried out as a trace operation, we have   n  n  p˜ · q˜ = T r (˜ q p˜) = T r qi ei pj fj   i=1

j=1

n

qi pj T r (ei fj ) .

=

i,j=1

Clearly, we can reproduce the first factor in Eq. (40.7) if we choose the field element basis F = {f1 , , f2 , f3 , ....fn } to be the dual basis of E = {e1 , e2 , e3 , ...en }. If we

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denote this dual basis as F = {e1 , e2 , e3 , ...en }, then T r ei ej = δ ij . Thus, we obtained n

q˜ · p˜ =

qi pj T r ei ej i,j=1 n

=

n

qi pj δ ij = i,j=1

pi qi , i=1

in agreement with the terms in Eq. (40.7). First let us consider the simplest system, a single qubit. 40.2.1.1

Phase Space for a Spin- 12 System or Single Qubit

A spin 12 particle can be considered a single qubit. We take the z-component of spin to be associated with the diﬀerent quantum labels of the “Wannier function”. Likewise we take the x-component of spin to be associated with the diﬀerent quantum labels of the “Bloch function”. Thus the phase space consists of exactly four points. However, instead of labeling by spin 12 or − 12 , it is more convenient to label the “Wannier function” by q = 0 or 1, respectively, where now 0 and 1 are interpreted as elements of the binary finite field GF(2). Correspondingly, the momentum labels will be p = 2πq 2 = πq, where addition and multiplication of q and p will be mod 2, by virtue of the q’s being elements of GF(2). Moreover, for convenience in the calculations that follow, it is not necessary to carry the constant multiplier π for labeling the momentum coordinate. This simplifies the phase space to have identical labels for the horizontal and vertical axis. The translation operator X is associated with transforming the “Wannier functions” (translation of coordinates) and is characterized by the Pauli matrix, σx . The translation operator Z is associated with transforming the “Bloch functions” (translation of momentum) and is characterized by the Pauli matrix, σz . Note that the Pauli matrix, σz , is a phase operator when operating on the “Wannier functions” but become a translation operator when acting on the “Bloch functions”. Furthermore, note that the Pauli operator, σ x , also becomes a phase operator when acting on the “Bloch functions”. This reciprocal relationship leads us to associate the vertical or momentum axis with the x-component of spin (since it renders the vertical striation invariant). Moreover, there is also a one-to-one mapping of the momentum coordinate (‘Bloch function eigenvalue) with the x-component of spin. Thus, the phase space of a single qubit consists of exactly four discrete points. Mathematically of course, the (horizontal) coordinate axis, q, is labeled by 0 and 1, and the (vertical) momentum axis, p, is labeled by π0 and π1. Aside from the factor π in the momentum coordinate, the addition and multiplication of q and p will be mod (2). In the following counting argument, we will ignore the presence of π in the momentum coordinate. As in the continuum case, a line in this discrete phase space satisfies a linear equation qu + pv = c,

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where u, v and c also takes values from the prime field GF(2). Therefore, 0 ≤ u ≤ 1, 0 ≤ v ≤ 1 and 0 ≤ c ≤ 1. We have the following three set of parallel lines (i.e., the same slope) (1) Vertical lines q = 0 consisting of points (0, 0) and (0, 1) , q = 1 consisting of points (1, 0) and (1, 1) .

(40.9)

(2) Horizontal lines p = 0 consisting of points (0, 0) and (1, 0) , p = 1 consisting of points (0, 1) and (1, 1) .

(40.10)

(3) Diagonal lines q + p = 0 consisting of points (0, 0) and (1, 1) , q + p = 1 consisting of points (0, 1) and (1, 0) .

(40.11)

Observed how the lines are labeled: the vertical lines are labeled by the q’s, the horizontal lines by the p’s, whereas the diagonal lines are labeled by the diﬀerent values of c. Following the terminology in the literature, we will refer to each set of parallel lines in discrete phase space as ‘striation’ corresponding to a set of basis states. We thus have three striations in the phase space of a single qubit. 40.3

Striations and Mutually Unbiased Bases

As in the continuum case, we wish to establish that the diagonal striation forms the third mutually unbiased basis, in addition to the “Wannier function” and “Bloch function” conjugate bases. To do this, we first recall how the conjugate basis for the horizontal and vertical striations were deduced. For the horizontal striations, Eq. (40.10), we note that translating the points in phase space in the horizontal directions leave the horizontal striation invariant, by virtue of the cyclical character of the elements of the field along the line. This transformation is given by σx =

01 10

i ⇒ exp − q · P |

.

It is the eigenfunction of σx which define the basis states represented by the horizontal striation. Indeed, the two horizontal parallel lines in phase space defines the “Bloch function” basis states, |p , which are the eigenfunctions of X or σ x operator discussed in Sec. 39.3.1 1 |p = 0 = √ 2 1 |p = 1 = √ 2

1 1 1 −1

,

.

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Similarly, the vertical striation, Eq. (40.9), is invariant with respect to the translation in the vertical directions, as represented by the operator Z = exp |i p · Q ⇒ σz . The eigenfunction of σz which define the vertical striation are the “Wannier function” basis states, |q , which are given in Sec. 39.3.1 |q = 0 =

1 0

,

|q = 1 =

0 1

.

The completeness relation is |q = 0 q = 0| + |q = 1 q = 1| =

1 0

10 +

=

10 01

= I.

0 1

01 (40.12)

Likewise |p = 0 p = 0| + |p = 1 p = 1| 1 =√ 2 =

1 2

1 1 √ 11 +√ 2 2

1 1 11 11

+

1 2

1 −1 −1 1

=

1 −1

1 √ 1 −1 2

10 01

= I.

It is important to note that the other lines or basis states labeling the vertical striations are generated by the translation operator, σx ⇒ exp − |i q · P , as can be seen from Eq. (40.9), for example, |q = 1 =

0 1

=

01 10

1 0

.

The other basis states of the horizontal striations are generated by the translation operator, σz ⇒ exp |i p · Q , as can be seen from Eq. (40.10), for example 1 |p = 1 = √ 2

1 −1

=

1 0 0 −1

1 √ 2

1 1

.

Thus there will be no ambiguity in the assignment of the basis state to each line in a striation once the ‘vacuum line’ (or first line, which may be chosen to be the one belonging to the line aq + bp = 0) is assigned a definite basis state, since the other basis states for each striation are generated by an appropriate translation operator. Moreover, note that the transformation-matrix operator from the q-basis to the

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conjugate (mutually unbiased) p-basis, and vice versa, follows from the completeness of each basis. For example, |p = 0 |p = 1

=I

|p = 0 |p = 1

= (|q = 0 q = 0| + |q = 1 q = 1|)

|p = 0 |p = 1

(|q = 0 q = 0| |p = 0 + |q = 1 q = 1| |p = 0 ) (|q = 0 q = 0| |p = 1 + |q = 1 q = 1| |p = 1 )   q| |p = 0 |q   q , =   q| |p = 1 |q

=

q

so that |p = 0 |p = 1

=

q = 0| |p = 0 q = 0| |p = 1

q = 1| |p = 0 q = 1| |p = 1

|q = 0 |q = 1

,

1 1 is a discrete Fourier transform 1 −1 matrix. Moreover, the square of the transformation-matrix element is equal to 12 , i.e., the probability of finding a momentum eigenstate is ‘unbiasedly’ distributed among the position eigenstates (equal superposition), and vice versa. Similarly, the requirement of translational covariance for the quantum state projector represented by the diagonal striation, Eq. (40.11), will also lead us to the third conjugate basis states. We immediately see that the diagonal striation is invariant under a combined vertical and horizontal translation, this combined translation is represented by ZX or σz σx (XZ or σx σz will work as well). Thus the diagonal striation yields the third basis states given by the eigenfunctions of σz σx = iσ y . From Sec. 39.3.1, except for the global phase factor, these are by virtue of Eq. (40.12), and the matrix

√1 2

1 |T1 = √ (|0 + i |1 ) , for eigenvalue 1, 2 1 |T−1 = √ (|0 − i |1 ) , for eigenvalue − 1, 2 which in vector notations are given by 1 |T1 = √ 2 1 |T−1 = √ 2

1 i 1 −i

,

.

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We also have |T1 T1 | + |T−1 T−1 | =

1 2

1 i

=

1 2

1 −i i 1

=

1 −i +

10 01

+

1 2

1 2

1 −i

1i

1 i −i 1

= I.

We can also write |T1 |T−1

|T1 |T−1

=I

= (|q = 0 q = 0| + |q = 1 q = 1|) = 

(|q = 0 q = 0| |T1 + |q = 1 q = 1| |T1 ) (|q = 0 q = 0| |T−1 + |q = 1 q = 1| |T−1 )

  = 

so that |T1 |T−1

|T1 |T−1

=

q| |T1 |q

q

q| |T−1

q

  , |q 

q = 0| |T1 q = 0| |T−1

1 =√ 2

1 i 1 −i



q = 1| |T1 q = 1| |T−1 |q = 0 |q = 1

|q = 0 |q = 1

.

In this case, we clearly observed that since the construction based on the translation operators only tell us which basis states is associated with the striation, there is ambiguity in assigning each basis states to each line in a striation. However, it is the cyclical character of the basis states which allows us to arbitrarily assign one of the basis states to the first line (‘vacuum line’ ) of the striation. The other basis states of this striation follows via the translation operator for the striation, which can easily be determined from Eq. (40.11) to be σz , i.e., involving translation of the momentum coordinate, p, only. Remarkably, this is the same generator for the basis states of the horizontal striations shown above. For example, if we assign |T1 to the ‘vacuum’ line passing through points (0, 0) and (1, 1), then by performing σz translation on |T1 yields |T−1 for the second line passing through points (0, 1)

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and (1, 0), 1 √ 2

1 0 0 −1

|T−1 =

1 = √ 2

1 −i

1 i

.

Likewise, upon transforming |T−1 , we recover |T1 |T1 =

1 0 0 −1

1 = √ 2

1 i

1 √ 2

1 −i

.

Note that the probability of finding the Bloch states, p = in any one of the diagonal states is given by T±1 p =

2π| 0 2

2

1 = √ ( 0| ± i 1|) 2 =

2π| 2 0

1 √ (|0 + |1 ) 2

, Eq. (39.17), 2

1 |(1 ± i)|2 4

1 . 2 Moreover, as shown above, the probability of finding any of the Wannier states among any of the Bloch states or among any of the diagonal states is N1 , where N = 2 for the prime field GF(2). Hence there are three mutually unbiased bases (MUB) for the two-state system. =

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Chapter 41

Discrete Wigner Distribution Function Construction

A straightforward general algorithm for calculating the discrete Wigner distribution function is guided by our previous construction in the continuous case which has notable contribution from all the lines (Radon transform) intersecting the point (q, p) in phase space. The exact procedure is a straightforward extension of the result in the continuous phase-space as embodied in Eq. (37.29), in terms of summing all projectors of the lines intersecting the point (q, p) in phase space. This is a discrete WDF construction in terms of line projectors in discrete phase space. In the present discrete case, this result can also be independently verified as follows. To see this, let us first sum the probability function, T r ρ ˆ ψu,v,c ψu,v,c , from all the lines intersecting the point (q, p). We must have for the N + 1 lines defining the mutually unbiased bases, T r ˆρ ψu,v,c

ψu,v,c

= (N + 1) W (q, p) +

uvc|(q,p)

W (q , p ) (q ,p )=(q,p)

= N W (q, p) +

W (q , p ) (q ,p )no

restriction

= N W (q, p) + 1. Therefore, we end up with  1  W (q, p) = N

Tr ˆ ρ ψu,v,c

ψu,v,c

uvc|(q,p)

ˆ (q, p) = Tr ˆ ρ∆ f inite field ,



− 1

(41.1)

which yields ˆ (q, p) ∆

f inite f ield

=



1  N

ψu,v,c uvc|(q,p)



ψu,v,c − I  .

(41.2)

This corresponds to Eq. (37.28) in the continuum version. The factor N1 is a normalization factor. In what follows we will drop the subscript finite field for the discrete case. 517

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We can easily check the above results since the expression ˆ (q, p) W (q, p) = T r ρ ˆ∆    1 ψu,v,c ψu,v,c − I  = T r ˆρ  N uvc|(q,p)   1  = ρI) Tr ˆ ρ ψu,v,c ψu,v,c − T r (ˆ N uvc|(q,p)   1  Tr ˆ ρ ψu,v,c ψu,v,c − T r (ˆ = ρI) , N uvc|(q,p)

yields the identity,  1  W (q, p) = N

Tr ρ ˆ ψu,v,c

ψu,v,c

uvc|(q,p)



− 1 .

Indeed, for the discrete case, we can also independently check the following properties. (a) Point projectors form an orthogonal operator-basis set, ˆ (q, p) ∆ ˆ (q , p ) = N δ qq δ pp . Tr ∆ The proof follows the one given by Gibbons, Hoﬀman, and Wooters1 Proof. The proof involves some simple counting or bookeeping. Starting with the expression of Eq. (41.2), we have   ˆ (q, p) ∆ ˆ (q , p ) =  Tr ∆

=

ψu,v,c

uvc|(q,p)



 ×

ψu,v,c − I 

ψu ,v ,c u v c |(q

,p

)

T r ψu,v,c



 ψu ,v ,c − I 

ψu,v,c ψu ,v ,c

‘ uvc|(q,p) uv c

−2 1 Phys.

Rev. A70, 062101 (2004).

|(q ,p ) T r ψu,v,c uvc|(q,p)

ψu,v,c

+ Tr I

ψu ,v ,c

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ψu,v,c ψu ,v ,c

=

519

‘ uvc|(q,p) uv c

−2

|(q ,p ) ψu,v,c ψu,v,c + T r I

(41.3)

uvc|(q,p)

Suppose (q, p) = (q , p ). For this case, only one line from each striation contains the point (q, p). Then the summand in the double sum consists of unity for u, v, c = u , v , c and N1 for u, v, c = u , v , c . The sum of the former gives the number of striations, N + 1, whereas the sum of the later gives N(N+1) . The last two terms N of Eq. (41.3) always give −2

uvc|(q,p)

ψu,v,c ψu,v,c + T r I = −2 (N + 1) + N = − (N + 2)

Thus for (q, p) = (q , p ), we have right hand side of Eq. (41.3) given by ˆ (q, p) ∆ ˆ (q , p ) Tr ∆

(q,p)=(q ,p )

= N +1+

N (N + 1) − (N + 2) N

=N Now suppose, (q, p) = (q , p ). Since only one line can contain the two points, the condition stipulated before, namely, u, v, c = u , v , c can occur only once which give unity. The remaining contribution from the double sum comes from u, v, c = u , v , c . Therefore, we have ˆ (q, p) ∆ ˆ (q , p ) Tr ∆

(q,p)=(q ,p )

= 1+

N (N + 1) − (N + 2) N

=0

(b) 

ˆ (q, p) = 1 = T r  Tr ∆

ψu,v,c uvc|(q,p)



ψu,v,c − I  = N + 1 − N = 1.

(c)

q,p



ˆ (q, p) =  ∆

q,p

 

ψu,v,c uvc|(q,p)



ψu,v,c − I  = [(N + 1) − N ] = 1.

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(d) We have corresponding to Eq. (38.9),

q,p

ˆ (q, p) δ (qu + pv − c ) = ∆

q,p



δ (qu + pv − c ) 

ψu,v,c uvc|(q,p)

ψ u ,v ,c .

= N ψu ,v ,c



ψu,v,c − I 

Thus, we have prove that for the discrete phase space, the set of point projectors, ˆ ∆ (q, p) , constitute a complete orthogonal operator basis, just like the continuum case.

41.1

Discrete Wigner Function for a Single Qubit

For a single qubit, let us assume that the density-matrix operator is given, in the coordinate representation, for pure state as ρ ˆ = |0 0| .

(41.4)

We associate the plus spin state with the coordinate eigenstate ψ↑ =

|0 0

.

Thus, ρ ψ↑ = ψ↑ , ˆ

Tr ˆ ρ ψ↑

ψ↑

= 1.

To calculate the Wigner function at the origin in phase space, defined by the point (q = 0, p = 0), we have  1  W (0, 0) = N

Tr ρ ˆ ψu,v,c

ψu,v,c

uvc|(0,0)



− 1 .

There are three lines passing through the origin with projectors (for +1 eigenvalues):

|q = 0 q = 0| ,

|p = 0 p = 0| = |T1 =

√1 2

√1 2

(|0 + |1 )

(|0 + i |1 )

|T1 =

√1 2 √1 2

(|0 + |1 ) ,

(|0 + i |1 ) .

(41.5)

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From the expression above, we obtain

W (0, 0) =

=

=



1   2



T r (ˆ ρ |q = 0 q = 0|) √1 2

+T r ρ ˆ √1 2

Tr ρ ˆ



√1 2

(|0 + |1 )

(|0 + i |1 )

√1 2

(|0 + |1 ) 

q = 0| ρ ˆ |q = 0

1   2

√1 2

+ √1 2



ˆ (|0 + |1 ) ρ

ρ (|0 − i |1 ) ˆ

1

√1 2



√1 2

−1

(|0 − i |1 )

(|0 + |1 )

(|0 + i |1 ) − 1

   

   

 1  +1  = 1. 2   2 2 1 +2 − 1

Also consider the density-matrix operator given by the ‘+’ eigenstate of σ y ρ ˆ = |T1 T1 | =

1 (|0 + i |1 ) ( 0| − i 1|) 2

1 1 1 1 |0 0| + |1 1| − i |0 1| + i |1 0| , 2 2 2 2

=

(41.6)

W (0, 0)

=

1 2



  +  

√1 2

=

√1 2

1 2

+ 14 +

1 4

1 4

+

+

1 4

1 2

|0 0| +

|0 0| +

1 2

|1 1| − i 12 |0 1| + i 12 |1 0| |0

1 2

|1 1| − i 12 |0 1| + i 12 |1 0|

|1 1| − i 12 |0 1| + i 12 |1 0|



1 2

+ 14 +

|0 0| + 1 2

(|0 + |1 )

(|0 − i |1 )



1  2

1 2

0|

1 4

−1

√1 2



    (|0 + i |1 ) − 1 √1 2

(|0 + |1 )

 1 = .  2

Now consider the point in phase space, (q = 1, p = 0). Again there are three lines passing through this point, the corresponding three line projectors are: |q = 1 q = 1| ,

|p = 0 p = 0| = |T−1 =

√1 2

√1 2

√1 (|0 2 |T−1 = √12 (|0

(|0 + |1 )

(|0 − i |1 )

+ |1 ) , − i |1 ) .

(41.7)

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Thus we have 

1   2

W (1, 0) =

√1 2

+T r ρ ˆ √1 2

Tr ρ ˆ



√1 2

(|0 + |1 )

(|0 − i |1 )

√1 2

(|0 + |1 )

√1 2

+ √1 2



ˆ (|0 + |1 ) ρ

ρ (|0 − i |1 ) ˆ

√1 2



0

√1 2

−1

(|0 − i |1 ) 

q = 1| ρ ˆ |q = 1

1   2

=



T r (ˆ ρ |q = 1 q = 1|)

(|0 + |1 )

(|0 − i |1 ) − 1

   

   

 1  + 1  = 0. 2   2 1 2 −1

=

Also consider the density-matrix operator given by the ‘+’ eigenstate of σ y ρ ˆ = |T1 T1 | =

1 (|0 + i |1 ) ( 0| − i 1|) 2

1 1 1 1 |0 0| + |1 1| − i |0 1| + i |1 0| , 2 2 2 2

=

W (1, 0) 

√1 2

+ √1 2



1 =  2

1 2

q = 1|

 1 =  2 

1 2

+ 14 + +

1 4

−

1 4

1 4

−

1 2

|0 0| +

|0 0| +

1 2

1 4

−1



|1 1| − i 12 |0 1| + i 12 |1 0| |q = 1 1 2

|1 1| − i 12 |0 1| + i 12 |1 0|

|1 1| − i 12 |0 1| + i 12 |1 0|

√1 2

√1 2

(|0 + |1 )

(|0 − i |1 ) − 1



1 2

1 4

1 2

(|0 + |1 )

(|0 + i |1 )

|0 0| +

    

  = 0. 

Similarly, we have for the point (q = 0, p = 1), the following three intersecting lines:

|q = 0 q = 0| ,

|p = 1 p = 1| = |T−1 =

√1 2

√1 2

√1 (|0 2 |T−1 = √12 (|0

(|0 − |1 )

(|0 − i |1 )

− |1 ) , − i |1 ) .

(41.8)

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Therefore, we have 

1   2

W (0, 1) =

√1 2

+T r ρ ˆ √1 2

Tr ρ ˆ



√1 2

(|0 − |1 ) √1 2

(|0 − i |1 )

(|0 − |1 )

(|0 − i |1 )

q = 0| ρ ˆ |q = 0

1  =  2

√1 2

+ √1 2



ˆ (|0 − |1 ) ρ

ρ (|0 − i |1 ) ˆ 

√1 2

√1 2

(|0 − |1 )

(|0 − i |1 ) − 1

1 1 1  1 + = .  2 1 2 2 2 −1

=



T r (ˆ ρ |q = 0 q = 0|)



−1

   

   

Also consider the density-matrix operator given by the ‘+’ eigenstate of σ y 1 (|0 + i |1 ) ( 0| − i 1|) 2 1 1 1 1 |0 0| + |1 1| − i |0 1| + i |1 0| , 2 2 2 2

ρ ˆ = |T1 T1 | = = W (0, 1) 

√1 2

+ √1 2



1 =  2

1 2

q = 0|

1  =  2

1 2

+ 14 + +

1 4

−

1 4

1 4

−

1 2

|0 0| +

|0 0| +

1 2

1 4

−1



|1 1| − i 12 |0 1| + i 12 |1 0| |q = 0

1 2

|1 1| − i 12 |0 1| + i 12 |1 0|

|1 1| − i 12 |0 1| + i 12 |1 0|

√1 2

√1 2

(|0 − |1 )

(|0 − i |1 ) − 1



1 2 1 4

1 2

(|0 − |1 )

(|0 + i |1 )

|0 0| +

   

  = 0.

For the diagonal point (q = 1, p = 1), the three line projectors are: |q = 1 q = 1| ,

|p = 1 p = 1| = |T1 =

√1 2

√1 2

(|0 − |1 )

(|0 + i |1 )

|T1 =

√1 2 √1 2

(|0 − |1 ) ,

(41.9)

(|0 + i |1 ) .

This immediately gives W (1, 1) = W (0, 1) = 0. Also consider the density-matrix operator given by the +-eigenstate of σ y 1 (|0 + i |1 ) ( 0| − i 1|) 2 1 1 1 1 |0 0| + |1 1| − i |0 1| + i |1 0| , 2 2 2 2

ρ ˆ = |T1 T1 | = =

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Nonequilibrium Quantum Transport Physics in Nanosystems

W (1, 1)  =

1  + 2 

1  2 1 4

1 2

q = 1|

√1 2

=

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√1 2

(|0 − |1 )

(|0 − i |1 )

+

1 2 1 2

1 4

+

1 4

1 4

+

1 4

−1

1 2

|0 0| +

|0 0| +



1 2

+ 14 +

|0 0| + 1 2

|1 1| − i 12 |0 1| + i 12 |1 0| |q = 1

1 2

|1 1| − i 12 |0 1| + i 12 |1 0|

|1 1| − i 12 |0 1| + i 12 |1 0|



   √1 (|0 + i |1 ) −1 2 √1 2

(|0 − |1 )

 1 = . 2

Thus, the resulting discrete Wigner distribution function for the density operator of Eq. (41.4) has nonzero elements only in the first column of the single-qubit discrete phase space, whereas the discrete Wigner distribution function of the density operator of Eq. (41.6) has only nonzero oﬀ-diagonal elements. If the density operator is for pure state given by ρ ˆ = |1 1|

(41.10)

then we can readily deduce that the discrete Wigner function has nonzero matrix elements in the second column of the single-qubit discrete phase space given by W (0, 0) = 0, 1 W (1, 0) = , 2 W (0, 1) = 0, 1 W (1, 1) = . 2 On the other hand, if the density operator is given by a more complex form, such as the −1 eigenstate of the following operator, √ ˆ = (σ x + σ y + σz ) / 3 = √1 O 3

1 1−i 1 + i −1

,

(41.11)

with normalized eigenfunctions given by ψ±1 =

1 √ 3∓ 3

1 ⇒ √ 3∓ 3

1/2

1/2

then we can write for pure state, ρ ˆ = ψ−1

ψ−1

1 √ = 3+ 3

√ 3+1 |1 1 |0 − (1 − i)



1 |0



√ ±( 3∓1) (1−i)



√ ±( 3∓1) (1−i)



1

|1 

 

,

√ 3+1 1 |0 − |1 (1 − i)

.

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We have W (0, 0)  1  =  2 =



q = 0| ρ ˆ |q = 0 √1 2

+

+ 1

2 3+ 

     +  ×  +   

√1 2

√1 2

(|0 + |1 ) ρ ˆ

√1 2

(|0 − i |1 ) ˆ ρ

(|0 + |1 )

(|0 + i |1 ) − 1

   

√ 3 q = 0| √1 2 √1 2

1 |0 −

√

(

(|0 + |1 )

1 |0 −

(|0 − i |1 )

1 |0 −

3+1) (1−i)

|1

√ ( 3+1) (1−i)

(

√ 3+1) (1−i)

1 |0 −

|1

1 |0 −

|1

1 |0 −

+1 3 + 

1

=

 √   2 3+ 3 1

− 3+

√ 3 +1 + + √1



1  =  √ 2 3+ 3 4

4 − 3+

(

(

√

(

√

√1 2 √1 2

√ 3+1) (1−i)

1−

√ + (1 − i) + i 3+1

1 √ 2 3+ 3 4

4+4+ 1+

=

1 √ 2 3+ 3 4

8−2 1+

=

1 8 − 2 (7.464) √ 2 3+ 3 4

√ 3

√ 3

|1

3+1) (1+i) 3+1) (1+i)

          1−

√ i( 3+1) (1+i)

(

√

|1

√1 2

|1

√1 2

3+1) (1+i)

√1 2

√1 2

(1 + i) −

√ (1 + i) − i 3+1

√ √ −4 3 − 4 + 2 3 + 2



|q = 0



√ √ 3 + 1 + (1 − i) − 3+1

=

=

1+

2

√1 1− 2 √ i( 3+1) (1−i)

√ 3+1) (1+i)

√ 3

 √ √  ( 3+1)  + √1 1 − ( 3+1) 1 − (1+i)  (1−i) 2 1  = √ √ √  i( 3+1) i( 3+1) 2 3+ 3   + √1 1 + (1−i) 1 − (1+i) 2   √ −1 3 + 3 

(

    (|0 + |1 )     (|0 + i |1 )    

   

√ 3+1

  

√ 3+1

1 8 − 14.93 √ 3 4

2 3+ =

1 −6.93 = −0.183, this is a negative quasi-probability. 37.856

Now consider the point in phase space, (q = 1, p = 0). Again there are three lines passing through this point, enumerated in Eq. (41.7).

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We have W (1, 0)  =

=

=

= =

1  2

2

2

2 2



q = 1| ρ ˆ |q = 1 1 √ ˆ √12 (|0 + |1 ) + (|0 + |1 ) ρ 2

  

√1 2

ˆ √12 (|0 − i |1 ) − 1 (|0 − i |1 ) ρ   √ √ ( 3+1) ( 3+1) 2 (1−i) (1+i)     √ √  ( 3+1)   + 1 − ( 3+1)  1 1 − (1−i) (1+i)   √   √ √ 3+ 3 2  ( 3+1)   + 1 − i ( 3+1)  1 + i (1−i) (1+i)   √ −1 3 + 3 2 √  2 2 3+1 √ √  + (1 − i) − 3 + 1 1 (1 + i) − √ 3+1  √ √ 3 + 3 4  + (1 − i) − i 3 + 1 √(1 + i) + i 3 + 1 − 3+ 3 4   √ 2 3+1 1  √  √ √ +1 3+ 3 − 3 3+1 1 3.732 √ (8.464 − 4.732) = √ = 0.394. 3+ 3 2 3+ 3

   

Considering the point (q = 0, p = 1), again, we have the three intersecting lines are enumerated in Eq. (41.8). Therefore, we have W (1, 0) =



1  2

+ √1 2

√1 2

q = 0| ˆ ρ |q = 0 ˆ (|0 − |1 ) ρ

ˆ (|0 − i |1 ) ρ 

√1 2

√1 2

(|0 − |1 )

(|0 − i |1 ) − 1

4  + ((1 − i)) ((1 + i))  √ √  + 3+1 + + 3 + 1  √  + ((1 − i)) + 3 + 1  √  1  + + 3 + 1 ((1 + i)) √ =  i)) 2 3+ 3 4 √((1 − i)) ((1 +√   −i 3 + 1 +i 3+1  √  ((1 − i)) +i 3 + 1  √   −i 3 + 1 ((1 + i)) √ − 3+ 3 4

   

                  

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Fig. 41.1

527

√ Wigner distribution function for −1 eigenstate of (σ x + σy + σ z ) / 3.

 4   +4 1   √ √ =   +6 2 3+ 3 4 √ √3 + 1 − 3 3+1 2 14.928 = = 0.394. 37.856 

For the diagonal point (q = 1, p = 1), the three line projectors are enumerated in Eq. (41.9). We have W (1, 1)

=



1  2

+

ˆ (|0 − |1 ) ρ

√1 2

(|0 − |1 )

   

ˆ √12 (|0 + i |1 ) − 1 (|0 + i |1 ) ρ   √ 2 3+1 2     + ((1 − i)) ((1 + i))   √  + + √3 + 1 + 3+1    √     + ((1 − i)) + 3 + 1   √   1  + + 3 + 1 ((1 + i))  √ =   2 3+ 3 4 i))  √((1 − i)) ((1 +√    +i 3 + 1 −i 3 + 1    √   ((1 − i)) −i 3 + 1   √   +i 3 + 1 ((1 + i))   √ − 3+ 3 4 =

+

√1 2

√1 2

q = 1| ρ ˆ |q = 1

1 √ 2 3+ 3

√ (3.732) = 0.394. 3+1 +1 = 2 (4.732)

Collecting the results for the operator of Eq. (41.11) yields Fig. 41.1 for the discrete

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Wigner distribution fucntion. Note that the sum of quasi-probabilities equals unity, W (p, q) = 3 (0.394) − 0.183 = 1.183 − 0.183 = 1.00. 41.2

Discrete Phase Space Structure for Two Qubits

The state space for two qubits has dimension equals to N = 22 = 4, and hence the discrete phase space consists of a 4 × 4 array of ‘(p, q)’ points. Recall that the number of independent point projectors is N 2 −1, and the corresponding number of points along a line in the phase space is N − 1. Thus, the number of line projectors equals N + 1, which is the number of mutually unbiased bases (MUB’s). For two qubits, there are five line projectors or five MUB’s. The discrete phase space of two qubits illustrate the use of finite field arithmetic (mod irreducible polynomial) instead of the modular arithmetic (mod prime real number). This means that instead of ordered pair ‘(p, q)’ of real numbers, in the single qubit case, where p and q belong to the real number finite field, the ‘(p, q)’ points for the two qubit case form a vector space over the finite-field extension of the prime number. The new variables are reminiscent of the creation of complex numbers, and p and q are now extension field elements. As a bit of historical fact, as early as the 16th century, mathematicians were devising new “numbers” as a way of solving polynomial equations; they were thinking in terms of algebraic formulas rather than graph or line representation or pictures. Since 4 = 22 , the prime finite field is GF(2) whose elements we will represent by 0, and 1, and where addition and multiplication are done modulo 2. We seek an extension of degree 2 over the prime field, so our first task is to find a monic2 irreducible polynomial of degree 2 in GF (2)[x], where the variable x belongs to GF(2). Recall that a polynomial is irreducible if the substitution of any element belonging to the prime field GF(2) does not yield zero, that is if any element of GF (2) is not a root of the polynomial, hence the need to extend the prime number field, analogous to the creating complex number from the reals. The candidate monic quadratics are: x2 + 1 and x2 + x + 1. However, x2 + 1 has root in GF(2) since 12 + 1 = 0(mod 2). Therefore the only such irreducible monic polynomial is x2 + x + 1 since there is no solution in GF(2) for the equation x2 + x + 1 = 0 Letting ω be a root of x2 + x + 1 = 0, or ω 2 = ω + 1, so the powers of ω are: ω1 = ω, ω2 = (−1) ω + (−1) = ω + 1, ω3 = ω (ω + 1) = ω 2 + ω = 0 + 1 = 1, 4

ω = ωω 3 = ω (1) = ω, 2A

polynomial in which the coeﬃcient of the highest order term is 1.

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which also shows that all the elements of GF 22 can be expanded in terms of the basis: ω 0 , and ω 1 . Observe here that the multiplicative group is cyclic, and for any nonzero element ξ belonging to the set ω 1 , ω2 = ω + 1 of GF(22 ), ξ 4−1 = 1. Thus for GF(22 ), ω 4 = ω 1 = ω, and ω 3 = 1 = ω 0 . It is also true that (x + y)2 = x2 + y 2 over GF(22 ). This can easily be seen using the binomial theorem with the coeﬃcients evaluated modulo 2. With the created ‘new’ numbers, the momentum and position, p and q, variables will now be labeled by the elements of GF 22 , which consists of field elements GF 22 ⇒ 0, 1, ω, ω 2 = {0, 1, ω, ω ˘} with arithmetic perform modulus irreducible polynomial, mod x2 + x + 1 . 41.2.1

Striations Construction

˘ . Moreover, in conFor convenience in what follows, we denote ω 2 = ω + 1 by ω structing the diﬀerent striations in discrete phase space, it is simpler to tentatively label the momentum coordinate by an identical label as the position coordinate, namely, by the set of discrete elements {0, 1, ω, ω ˘ }. Later on we will transform the momentum coordinate label by the order of the binary string (cardinal integers) in terms of the dual-field basis, to be able to consistently map the product translation operators in phase space. By using the binary-string notation, we free ourselves from the specific dual basis taken for the ‘momentum’ axis, as demonstrated in what follows. In the 4×4 phase space grid, the five striations are defined by the linear equation aq + bp = c, where a, q, b, p, and c belong to GF 22 . Therefore, we have the following five sets of parallel lines (i.e., having the same slope) (1) Vertical striation q q q q

= 0 consisting of points (0, 0) , (0, 1) , (0, ω) ,and (0, ω ˘) , = 1 consisting of points (1, 0) , (1, 1) , (1, ω) ,and (1, ω ˘) , = ω consisting of points (ω, 0) , (ω, 1) , (ω, ω) ,and (ω, ω ˘) , =ω ˘ consisting of points (˘ ω, 0) , (˘ ω , 1) , (˘ ω, ω) ,and (˘ ω, ω ˘) .

(41.12)

(2) Horizontal striation p = 0 consisting of points p = 1 consisting of points p = ω consisting of points p=ω ˘ consisting of points

(0, 0) , (1, 0) , (ω, 0) and (˘ ω , 0) , (0, 1) , (1, 1) , (ω, 1) and (˘ ω , 1) , (0, ω) , (1, ω) , (ω, ω) and (˘ ω, ω) , (0, ω ˘ ) , (1, ω ˘ ) , (ω, ω ˘ ) and (˘ ω, ω ˘) .

(41.13)

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(3) Diagonal striation q + p = 0 consisting of points q + p = 1 consisting of points q + p = ω consisting of points q+p = ω ˘ consisting of points

(0, 0) , (1, 1) , (ω, ω) , and (˘ ω, ω ˘) , (0, 1) , (1, 0) , (ω, ω ˘ ) , and (˘ ω , ω) , (0, ω) , (1, ω ˘ ) , (ω, 0) , and (˘ ω, 1) , (0, ω ˘ ) , (1, ω) , (ω, 1) , and (˘ ω , 0) . (41.14)

(4) ‘Low-slope’ striation (also referred to as ‘Belle’ striation) ωq + p = 0 consisting of points ωq + p = 1 consisting of points ωq + p = ω consisting of points ωq + p = ω ˘ consisting of points

(0, 0) , (1, ω) , (ω, ω ˘ ) , and (0, 1) , (1, ω ˘ ) , (ω, ω) , and (0, ω) , (1, 0) , (ω, 1) , and (0, ω ˘ ) , (1, 1) , (ω, 0) , and

(˘ ω , 1) , (˘ ω , 0) , (˘ ω, ω ˘) , (˘ ω , ω) . (41.15)

(5) ‘High-slope’ striation (also referred to as ‘Beau’ striation) q + ωp = 0 consisting of points q + ωp = 1 consisting of points q + ωp = ω consisting of points q + ωp = ω ˘ consisting of points

(0, 0) , (ω, 1) , (˘ ω, ω) , and (1, 0) , (˘ ω , 1) , (ω, ω) , and (ω, 0) , (0, 1) , (1, ω) , and (˘ ω , 0) , (1, 1) , (0, ω) , and

(1, ω ˘) , (0, ω ˘) , (˘ ω, ω ˘) , (ω, ω ˘ ) . (41.16)

Observe that the coordinates of all points on a line intersecting the origin (0, 0) (referred to here as the ‘vacuum’ state line) of phase space are proportional to each other, just as in the continuum case. Furthermore, all points in a striation are diﬀerent since all lines are parallel, and hence do not meet. Moreover, two lines each from diﬀerent striations meet at only one point. The following lines are not separate lines since these have identical slopes as some of the previous lines already given, since multiplying these linear equations by ω yields the indicated line equations already defined above, q+ω ˘ p = 0 consisting of points (0, 0) , (˘ ω, 1) , (1, ω) , and (ω, ω ˘) ⇒ ω (q + ω ˘ p) = 0 = ωq + p, q+ω ˘ p = 1 consisting of points (0, ω) , (1, 0) , (ω, 1) , and (˘ ω, ω ˘) ⇒ ω (q + ω ˘ p) = ω = ωq + p, q+ω ˘ p = ω consisting of points (0, ω ˘ ) , (1, 1) , (ω, 0) „ and (˘ ω , ω) ⇒ ωq + p = ω ˘, q+ω ˘p = ω ˘ consisting of points (0, 1) , (1, ω ˘ ) , (ω, ω) , and (˘ ω, 0) ⇒ ωq + p = 1. (41.17)

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Similarly, we have, ω ˘ q + p = 0 consisting of points (0, 0) , (1, ω ˘ ) , (ω, 1) , and (˘ ω, ω) ⇒ ω (˘ ω q + p) = 0 = (q + ωp) , ω ˘ q + p = 1 consisting of points (ω, 0) , (0, 1) , (1, ω) , and (˘ ω, ω ˘) ⇒ (q + ωp) = ω, ω ˘ q + p = ω consisting of points (˘ ω, 0) , (1, 1) , (0, ω) , and (ω, ω ˘) ⇒ (q + ωp) = ω ˘, ω ˘q + p = ω ˘ consisting of points (1, 0) , (˘ ω , 1) , (ω, ω) , and (0, ω ˘) ⇒ (q + ωp) = 1.

(41.18)

These results are consistent with the fact that all the elements of GF 22 can be expanded in terms of the finite field basis: ω0 , and ω1 (or 1 and ω). 41.2.2

Binary String Encoding of Points in Discrete Phase Space

The translation ‘lattice’ step size for discrete finite field diﬀers from that of primenumber field, using modular arithmetic, such as that in discrete phase space of crystalline solid-state lattice or spin 12 particle. Whereas, for the crystalline lattice or spin 12 particle, the step size is uniform, which is the lattice constant or unit step, in the extension finite field GF(2n ) the ‘lattice’ size are elements of GF(2n ). However, we can make the proper connection, or mapping, between a translation vector (q, p) where q and p are now labeled by elements in GF(2n ) and individual two-state particles by expanding q and p in the finite-field basis, allowing for diﬀerent field basis for q and p. The number of finite-field basis in GF(2n ) is n (the number of bits), whereas the coeﬃcient of the expansion are elements of GF(2) which is the binary set {0, 1}. Hence, the expansion route will give exactly the number 2n elements of GF(2n ). Hence the binary strings of coeﬃcients in the expansion is a one-to-one mapping of the elements of GF(2n ), which clearly allows us to use the coeﬃcient vector to make independent translation on each qubit. Let us therefore expand q and p as   qe1  qe2  n    q= qei ei ⇒   ..  ,  ..  i=1 qen 

 pf1  pf2  n    p= pfi fi ⇒   ..  ,  i=1 ..  ffn

where the set E = {e1 , e2 .......en } and F = {f1 , f2 .......fn } are finite field bases.

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The coeﬃcient of the expansions, qei and pfi , are elements of GF(2) belonging to the binary set {0, 1}. The idea is to associate the binary coeﬃcients qei and pfi with the ith particle, where i belongs to the set {i = 1, .., n} for n particles. Each member of the basis sets {ei } and {fi } are therefore also associated with the ith particle. This interesting scheme allows us to uniquely characterize a translation by any vector (q, p) in GF(2n ) by using their vector components (represented by binary strings), to enable digital unitary operation on each bit in an array of n qubits, thus preserving the desire to maintain the physically-meaningful product Hilbert space of n qubits. From the expression of translation operator for a single bit, where q and p are elements in GF(2), given by i p ·q | 2 i = exp − q · P | = Xq Zp ,

T (q , p ) = exp −

i (p · Q − q · P ) | i p ·Q |

exp exp

we have the resulting expression for the translation operator of the extension finite field, GF(2n ), n

T ⊗ (q, p) =

Xqei Zpfi ,

(41.19)

i=1

where q and p of T ⊗ (q, p) in Eq. (41.19) are now elements of GF(2n ), qei and pfi are the coeﬃcients of their respective field-basis expansion. Although the finite-field basis can be chosen arbitrarily, once the field basis for q is chosen, then the field basis for p must be proportional to the dual field basis, n

as a consistency requirement. Clearly, if we define p · q = p · q ⇒ p, q = T r (pq)  n

= Tr 



n

qei ei

i=1

j=1

n

qei pfi , then i=1

pfj fj 

qei pfj T r (ei fj )

= i,j=1 n

n

qei pfj δ ij =

= i,j=1

qei pfi , i=1

hence the field basis fj for the element p in GF(2n ) is related to the dual basis, e˜j . The construction of the dual basis {˜ ej } is as follows.

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41.2.3

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Construction of Dual Field Basis for Two Qubits

We will illustrate here the use of general construction of dual field basis for two qubits, as an example. From the discussion of Sec. I.2, for the field basis of the ‘position coordinate’, i.e., a set E = (e1 , e2 ...en ) ⇒ (e1 , e2 ), we may consider primal bases of the form {1, α1 , α2 ......αn−1 } where α does not lie in GF (2), hence for n = 2, {1, α1 , α2 ......αn−1 } ⇒ (e1 , e2 ) = (1, ω). Here (1, ω) associated with (first particle, second particle). We should point out that in binary string notation, the order of the bits is reversed, i.e., (01) is associated with (second particle ‘0’ first particle ‘1’). Hence the expansion of (01) ⇒ 1 · 1 + ω · 0. To construct the dual field basis, we first build the matrix A defined as A = (T r (ei ej )) = (aij ) ,

i, j = 1, n

and let B = A−1 = (bk,j ) ,

k, j = 1, n .

en } is given by Then the dual basis F = {˜ e1 , e˜2 ...˜ m−1

e˜j =

ek bk,j ,

(41.20)

k=0

since n

T r (ei e˜j ) = T r ei

ek bk,j k=1

= AA−1

ij

= δ ij .

Consider GF 22 with the basis {1, ω} where ω is a primitive root which satisfies ω 2 = ω + 1. We have T r (ω) = ω + ω 2 = ω + (ω + 1) = 1, and therefore T r (1) = 1 + 1 = 0, T r (ω) = 1, T r ω 2 = T r (ω + 1) = 1. 2 2

Thus if q ∈GF 2

qi ei , then T r (q) = q2 . The matrix A is given by

and q = i=1

A = Tr =

01 11

e1 e1 e1 e2 e2 e1 e2 e2 .

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The matrix B = A−1 11 = 10

,

where AA−1 = I (mod 2). Therefore the dual basis, from Eq. (41.20), is given by (1, ω)

11 10

= (˘ ω, 1) ,

yielding e˜1 = 1 + ω = ω ˘, e˜2 = 1, ω, 1} is associated with (first particle, second where the dual basis F = {˜ e1 , e˜2 } = {˘ particle). 41.2.3.1

Commutation Relation

The basic commutation relation for Xq Zp follows from Eq. (39.9) which yields, exp −

i p ·q | 2

Yq p , exp −

i p ·q | 2

Yq

p

i ((p + p ) · (q + q )) Yq +q p +p 2| i i [p · q ] − exp [p · q ] . × exp | |

= exp −

(41.21)

Therefore, two translation operators, Xq Zp and Xq Zp commute if the following relation hold p ·q =p ·q . By extending this result to product translation operator T ⊗ (q, p), we obtain T ⊗ (q , p ) , T ⊗ (q , p ) = 0, where q and p of T ⊗ (q, p) are now elements of GF(2n ), if the following equality holds n

n

pfi fi , i=1



Tr 

n

pfi fi i=1

n

=

qej ej j=1

n

p ·q = p ·q ,

j=1



n

pfi fi , i=1



qej ej  = T r 

qej ej j=1

n

n

pfi fi i=1

j=1

, 

qej ej  .

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Note that the ‘dot’ product, p · q is performed as a trace operation. We obtain n

n

pfi qej T r (fi ej ) = i,j=1

pfi qej T r (fi ej ) , i,j=1

n

n

pfi qei = i=1

pfj qej ,

(41.22)

j=1

where we take the basis for the momentum equal to the dual basis, fi = e˜i , such that ei ej ) = δ ij . T r (fi ej ) = T r (˜ As in the continuum case, this relation, Eq. (41.22) holds for all translations along a line in discrete phase space. Moreover, for the special case of a ray line, i.e., the line intersecting the origin, where all (q, p) points on the line can be generated by displacement (q, p) from the origin (0, 0), i.e., T ⊗ (q, p) for each striation, then (q , p ) = (sq , sp ) and Eq. (41.22) is obviously satisfied, where the scaling factor s ranges over the elements of GF(2n ). For other lines not intersecting the origin, Eqs. (41.12) - (41.16), one can easily verify that successive generation of neighboring points along a given line given by T ⊗ (∆q, ∆p) also satisfy ∆p · ∆q = ∆p · ∆q yielding commuting translation operators on a given line not intersecting the origin. For this reason, it is more convenient to to investigate the eigenvectors of commuting translation operators along a ray for each striation of the discrete phase space structure.

41.3

Line Projectors for Two Qubit Systems

To find the line projectors associated with the five striations, we need to find the simultaneous eigenvectors of the translation operators which renders the striation covariant. To do this we need to extend the procedure used for the single bit system, i.e., we need to find the simultaneous eigenvectors of commuting translation operators along a reference line of each striation in discrete phase-space over finite fields. 41.3.1

Product Hilbert Space for a Two Qubit System

For two qubits, we associate the product σ z -spin states with the coordinate, ‘q’, eigenstates

|q = 0 = ψ↑↑



  |0 2 |0 1    0  , |q = 1 = ψ↑↓ =  |0 =    0 0

 0  2 |1 1  ,  0 0

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

  0    0  = ˘ = ψ↓↓ =   |1 |0  , |q = ω  2 1 |1 0

|q = ω = ψ↓↑

 0  0  .  0 |1 2 1

(41.23)

These ‘q’ eigenstates corresponds to the elements of the finite field GF(22 ), namely, ψ↑e2 ↑e1

⇒ 0, q = 0 · e1 + 0 · e2 ⇒ (00) ,

ψ↑e

↓e1

⇒ 1, q = 1 · e1 + 0 · e2 ⇒ (01) ,

ψ↓e2 ↑e1

= ω, q = 0 · e1 + 1 · e2 ⇒ (10) ,

ψ↓e2 ↓e1

= ω ˘ , q = 1 · e1 + 1 · e2 ⇒ (11) ,

2

(41.24)

where we also show the expansion of the coordinate q in terms of the field basis, with the corresponding binary notation (in ‘memory tape register’) of the particle σz -spin configuration also given. Observe the arrangement of the memory register and the corresponding product eigenstates, i.e., the first particle is the rightmost bit and the second-particle bit stands on the left of the first-particle bit, the third to the left of the second, ad infinitum, corresponding to the physically meaningful configuration of the system. Note that the binary-string cardinal order of the ‘position’ (horizontal) coordinate follows the order of the field elements of GF 22 , i.e., {0, 1, ω, ω ˘ }. √1 (|0 + |1 ) and p = 2π| 1 = √1 (|0 − |1 ), Similarly, with p = 2π| 2 0 = 2 2 2 with eigenvalues 1 and −1, respectively, we have the product p states given in the q-representation by |p = 0 2 |p = 0

1

= ψ→e˜2 →e˜1 |p = 0 2 |p = 1 = ψ→e˜

2

←e˜

1

|p = 1 2 |p = 0 = ψ←e˜

2

1

→e˜

1

1



|0  1  |0 =  2  |1 |1

2 |0 1



  , p = 0 · e˜1 + 0 · e˜2 = 0 ⇒ (00) , 2 |0 1  2 |1 1 2 |1 1 



 |0 2 |0 1  1  − |0 2 |1 1  =  ˘ ⇒ (01) ,  , p = 1 · e˜1 + 0 · e˜2 = ω 2  |1 2 |0 1  − |1 2 |1 1 

 |0 2 |0 1  1  |0 2 |1 1  =   , p = 0 · e˜1 + 1 · e˜2 = 1 ⇒ (10) , 2  − |1 2 |0 1  − |1 2 |1 1

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|p = 1 2 |p = 1 = ψ←e˜

2

1

←e˜

1

537



 |0 2 |0 1  1  − |0 2 |1 1  =   , p = 1 · e˜1 + 1 · e˜2 = ω ⇒ (11) , (41.25) 2  − |1 2 |0 1  |1 2 |1 1

where we also show the expansion of the coordinate p in terms of the dual field basis, with the corresponding binary notation (in ‘memory register’) of the particle σx -spin configuration. Clearly, the binary-string cardinal order of the ‘momentum’ (vertical) coordinate is not the same order as the horizontal coordinate, namely the set {0, 1, ω, ω ˘ }. Therefore, in analyzing the momentum (vertical) translation operators, the assumed order {0, 1, ω, ω ˘ } in deriving the diﬀerent striations must be mapped to the correct cardinal order of the vertical axis in terms of the order of the corresponding binary string, p = |p = 0 2 |p = 0

1

= (00) = ψ→e˜2 →e˜1

⇒ 0,

p = |p = 0 2 |p = 1

1

= (01) = ψ→e˜2 ←e˜1

⇒ω ˘,

p = |p = 1 2 |p = 0

1

= (10) = ψ←e˜2 →e˜1

⇒ 1,

p = |p = 1 2 |p = 1

1

= (11) = ψ←e˜2 ←e˜1

⇒ ω.

(41.26)

The p-product states in Eq. (41.25) are given in the q-representation, we have, for example,

ψ→e˜2 ←e˜1

= |0 p2 |1

p 1

1 =√ 2





1 |0 1 |1

2 2

1 ⊗√ 2

1 |0 1 −1 |1 1

|0 2 |0 1  1  − |0 2 |1 1   =   , p = 1 · e˜1 + 0 · e˜2 . 2  |1 2 |0 1  − |1 2 |1 1 The translation operators can now be defined in terms of product translation operators, whose matrix-product form is given by the matrix-element projector template, where the matrix-element projectors are entries of a multiplication table given by

|0 |0 |1 |1

⊗ 2 |0 2 |1 2 |0 2 |1

1 1 1 1

|0 |0 |1 |1

0|1 |0 2 1 2 |1 1 2 |0 1 2 |1 1

0|2 0|1 0|1 0|1 0|1

0|2 0|2 0|2 0|2

|0 |0 |1 |1

1|1 |0 2 1 2 |1 1 2 |0 1 2 |1 1

0|2 1|1 1|1 1|1 1|1

0|2 0|2 0|2 0|2

|0 |0 |1 |1

0|1 |0 2 1 2 |1 1 2 |0 1 2 |1 1

1|2 0|1 0|1 0|1 0|1

1|2 1|2 1|2 1|2

|0 |0 |1 |1

1|1 |0 2 1 2 |1 1 2 |0 1 2 |1 1

1|2 1|1 1|1 1|1 1|1

1|2 1|2 1|2 1|2

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Thus, with the above multiplication table entries, the matrix product reads |0 0| |0 1| |0 0| |0 1| ⊗ |1 0| |1 1| 1 |1 0| |1 1| 2  |0 0| |0 1| |0 0|2 |0 2  |0 2 |1 0| |1 1| |1 1 =  |0 0| |0 1| |0 |1 2 0| |1 |1 0| |1 1| 1 2 2 |1  |0 2 |0 1 0|1 0|2 |0 2 |0 1 1|1 0|2 |0  |0 |1 0| 0| |0 |1 1| 0| |0  = 2 1 1 2 2 1 1 2  |1 2 |0 1 0|1 0|2 |1 2 |0 1 1|1 0|2 |1 |1 2 |1 1 0|1 0|2 |1 2 |1 1 1|1 0|2 |1

 1|2    1|2

0| |0 0| |1 0| |0 0| |1

1| 1| 1| 1|

1

2 |0 1

0|1 0|1 0|1 0|1

1|2 1|2 1|2 1|2

2 |1 1 2 |0 2 |1

1 1

1

|0 |0 |1 |1

2 |0 1 2 |1 1 2 |0 1 2 |1 1

1|1 1|1 1|1 1|1

 1|2 1|2   . (41.27) 1|2  1|2

If the operator only acts on the first particle, then matrix-product is given by the following template a |0 0| b |0 1| ⊗ c |1 0| d |1 1| 1  a |0 2 |0 1 0|1 0|2  c |0 |1 0| 0|  2 1 1 2 =  0 |1 2 |0 1 0|1 0|2 0 |1 2 |1 1 0|1 0|2  a |0 2 |0 1 0|1 0|2  c |0 |1 0| 0|  2 1 1 2 =  0 0

1 |0 0| 0 |0 1| 0 |1 0| 1 |1 1| b |0 d |0 0 |1 0 |1

|0 2 |1 2 |0 2 |1

2

b |0 2 |0 d |0 2 |1

1 1 1 1 1 1

1|1 1|1 1|1 1|1

0|2 0|2 0|2 0|2

2

0 |0 0 |0 a |1 c |1

2 |0 1 2 |1 1

2 |0 1

2 |1

1|1 0|2 1|1 0|2

1|2 1|2 1|2 1|2

0 |0 0 |0 b |1 d |1

2 |0 1 2 |1 1

2 |0 1

2 |1 1

0 0 a |1 2 |0 c |1 2 |1

0 0

1

0|1 0|1 0|1 0|1

1 1

0 0 0|1 1|2 b |1 2 |0 0|1 1|2 d |1 2 |1

1 1

1|1 1|1 1|1 1|1

 1|2 1|2    1|2  1|2 

  . 1|1 1|2  1|1 1|2

Similarly, if acting only on the second particle we obtain 1 |0 0| 0 |0 0 |1 0| 1 |1  e |0 2 |0 1  0 |0 |1  2 1 =  g |1 2 |0 1 0 |1 2 |1 1  e |0 2 |0 1  0  =  g |1 2 |0 1 0

1| 1| 0|1 0|1 0|1 0|1

1

⊗

0|2 0|2 0|2 0|2

e |0 0| f |0 1| g |1 0| h |1 1| 0 |0 e |0 0 |1 g |1

2 |0 1

2 |1 1

2 |0 1 2 |1

0|1 0|2

1

1|1 1|1 1|1 1|1

0|2 0|2 0|2 0|2

0

2

f |0 0 |0 h |1 0 |1

2 |0 1

|1 2 |0 2 |1 2

f |0 2 |0

1

0|1 0|1 0|1 0|1

1

0|1 1|2

1 1

1|2 1|2 1|2 1|2

0 |0 f |0 0 |1 h |1

2 |0 1

2 |1 1

2 |0 1

2 |1 1

0

1|1 1|1 1|1 1|1

 1|2 1|2    1|2  1|2 

1|1 1|2   .  0|1 0|2 0 h |1 2 |0 1 0|1 1|2 0 0 h |1 2 |1 1 1|1 1|2 g |1 2 |1 1 1|1 0|2 e |0 2 |1

1

1|1 0|2

0

f |0 2 |1

1

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Thus when acting only on the first particle of an initial product-state vector in the q-representation, we have  a |0 2 |0 1 0|1 0|2 b |0 2 |0 1 1|1 0|2 0 0   c |0 |1 0| 0| d |0 |1 1| 0| 0 0   2 1 1 2 2 1 1 2 =   0 0 a |1 2 |0 1 0|1 1|2 b |1 2 |0 1 1|1 1|2  0 0 c |1 2 |1 1 0|1 1|2 d |1 2 |1 1 1|1 1|2 



|0  |0  ×  |1 |1



2 |0 1



  2 |0 1  2 |1 1 2 |1 1 

(a + b) |0  (c + d) |0  =  (a + b) |1 (c + d) |1

2 |0 1



 . 2 |0 1  2 |1 1 2 |1 1 

(41.28)

Note that the same result is obtained if we isolate the first particle and operate on it, i.e., a |0 0| b |0 1| c |1 0| d |1 1| =

1

(a + b) |0 (c + d) |1

1

|0 |1

1 1

,

1

and then form the product state. We obtained after forming the resulting product state vector

|0 |1

2 2

⊗

(a + b) |0 (c + d) |1

1 1



(a + b) |0  (c + d) |0  =  (a + b) |1 (c + d) |1

2 |0 1

 , 2 |0 1  2 |1 1 2 |1 1 

which is the same result as that of the product matrix 1 |0 0| 0 |0 1| 0 |1 0| 1 |1 1|



a |0 0| b |0 1| c |1 0| d |1 1|

acting on the initial product-state vector, Eq. (41.28). 2

1

⊗

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When acting only on the second particle, we have   e |0 2 |0 1 0|1 0|2 0 f |0 2 |0 1 0|1 1|2 0  0 f |0 2 |1 1 1|1 1|2  0 e |0 2 |1 1 1|1 0|2      g |1 2 |0 1 0|1 0|2  0 h |1 2 |0 1 0|1 1|2 0 0 h |1 2 |1 1 1|1 1|2 0 g |1 2 |1 1 1|1 0|2   |0 2 |0 1  |0 |1    × 2 1  |1 2 |0 1  |1 2 |1 1   (e + f ) |0 2 |0 1  (e + f ) |0 |1   2 1 (41.29) = .  (g + h) |1 2 |0 1  (g + h) |1 2 |1 1

Thus



0 |0  1 |0  σx ⊗ I =   

0 1  = 0 0 

1 0 0 0

0 0 0 1

0 |0

2

  I ⊗ σx =   1 |1 

|0 2 |1 2

0 0  = 1 0

0 0 0 1

2

1 0 0 0

1 1

0 0 

0|1 0|2 1 |0 0|1 0|2 0 |0

|0 2 |1 2

1 1

1|1 0|2 1|1 0|2

0 0

0 0

0 |1 1 |1

0

1 |0

|0 2 |1 2

1 1

0 0 0|1 1|2 1 |1 0|1 1|2 0 |1

|0 2 |1 2

1 1

0 0 , 1 0 |0 |0

1

0|1 0|2

0 1

0 

0|1 0|2

|1

0 |0

2

1 |1

2 |1

1

1|1 0|2

0 1

1|1 0|2

0 |1

2

2

|0 |0

1

0 1

0

0 1 . 0 0

Upon operating on

|ψ→←



0|1 1|2

 |0 1 |0 2  1  − |0 2 |1 1  =  , 2  |1 2 |0 1  − |1 2 |1 1

0|1

0

1|1 1|1



   1|2  1|2



1|1 1|2     1|2 0 0 |1 2 |1 1 1|1 1|2 1 |0

2

|1

1

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we obtained the eigenvalue +1,   0 |0 2 |0 1 0|1 0|2 0 1 |0 2 |0 1 0|1 1|2 0  0 1 |0 2 |1 1 1|1 1|2  0 0 |0 2 |1 1 1|1 0|2      1 |1 2 |0 1 0|1 0|2  0 0 |1 2 |0 1 0|1 1|2 0 0 0 |1 2 |1 1 1|1 1|2 0 1 |1 2 |1 1 1|1 0|2   |0 2 |0 1  1  − |0 2 |1 1  ×   2  |1 2 |0 1  − |1 2 |1 1   1 |0 2 |0 1  1  − |0 2 |1 1  =  , 2  |1 2 |0 1  − |1 2 |1 1 which is the same eigenvalue if we operate on |ψ→→ .

41.3.2

Eigenvectors of Commuting Translation Operators

We need 2n −1 commuting T ⊗ (q, p)’s that leave a reference or ‘vacuum’ line in each striation invariant, since the simultaneous eigenvectors of these 2n − 1 commuting T ⊗ (q, p)’s determine the line projectors of each striation. We need N −1 translation operators associated with one of the line in a striation containing N points to generate the remaining N − 1 points from a given fixed point on the line, where N = 2n for GF(N = 2n ). These N − 1 translation operators leave each line in a striation invariant. Moreover, these N − 1 translation operators, along a line, commute with each other, analogous to the continuous case, Eq. (39.9). Since the striations determine the mutually unbiased bases, the situation is similar to commuting translation operators operating on the Wannier functions in solid-state physics. Thus for GF(N = 22 ), we need three commuting translation operators determined by each striation (given slope), whose simultaneous eigenvectors constitute the basis states to be associated with the given striation. We take the line passing through the origin,often called a ray in each striation, as the ‘vacuum’ or reference line for each striation. For the five striations in Eqs. (41.12) - (41.16), we consider the rays in each of the five striations. As discussed before, we use the field basis E = {e1 , e2 } = {1, ω} ω , 1} for the momentum for the coordinate q and dual field basis F = {˜ e1 , e˜2 } = {˘ p, where the subscripts 1 and 2 pertains to particle-1 and particle-2, respectively. 41.3.3

Vertical Striation Ray and ‘Position’ Basis

The vertical striation ray is given by q = 0 consisting of points (0, 0) , (0, 1) , (0, ω) ,and (0, ω ˘) .

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We have the following three commuting unitary operators associated with the translations (0, 1),(0, ω), and (0, ω ˘ ) from the origin, determined from Eq. (41.19)   10 0 0 0 1 0 0   ω + 1) = I ⊗ Z =  T ⊗ (0, 1) = T ⊗ (0 + 0ω, 0˘  0 0 −1 0  , 0 0 0 −1   1 0 0 0  0 −1 0 0   T ⊗ (0, ω) = T ⊗ (0 + 0ω, ω ˘ + 1) = Z ⊗ Z =   0 0 −1 0  , 0 0 0 1   1 0 0 0  0 −1 0 0   T ⊗ (0, ω ˘ ) = T ⊗ (0 + 0ω, ω ˘ + 0) = Z ⊗ I =  0 0 1 0 , 0 0 0 −1 which have eigenstate, Eqs. (41.23)-(41.24)

|q = 0 = ψ↑e2 ↑e1 . Note that from Eq. (41.12), the q = 1 line can be obtained from the q = 0 line by translating all the q-coordinate of the q = 0 line by unity. Likewise the q = ω line can be obtained from the q = 0 line by translating all the q-coordinate of the q = 0 line by ω. Moreover, the q = ω ˘ line can be obtained from the q = ω line by translating all the q-coordinate of the q = ω line by unity. Thus, the corresponding translation operator operating on the eigenstate will generate the eigenstate of the translated line in a striation. In what follows, we will denote the translation operator operating on the eigenstate of a line in a striation by U ⊗ (q, p). This method of generating the eigenstates of parallel lines in a striation eliminates the arbitrariness in the assignment of the eigensates of all the lines in a striation once the eigenstate of the reference or ‘vacuum’ line is fixed. q q q q

= 0 consisting of points (0, 0) , (0, 1) , (0, ω) ,and (0, ω ˘) , = 1 consisting of points (1, 0) , (1, 1) , (1, ω) ,and (1, ω ˘) , = ω consisting of points (ω, 0) , (ω, 1) , (ω, ω) ,and (ω, ω ˘) , =ω ˘ consisting of points (˘ ω, 0) , (˘ ω , 1) , (˘ ω, ω) ,and (˘ ω, ω ˘) .

(41.30)

We give here the eigenstate translation operators. Referring to Eq. (41.12), we can generate the eigenstate for the q = 1 line from the eigenstate of q = 0 line by using the following translation operator U ⊗ (q (0 → 1) , ∆p = 0) = U ⊗ (1 · 1 + 0 · ω, 0 · ω ˘ + 0 · 1)   0100 1 0 0 0  = X ⊗I = 0 0 0 1. 0010

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Therefore, if the |q = 0 = ψ↑e2 ↑e1

|q = 0 = ψ ↑e2 ↑e1

then the |q = 1 = ψ↑e2 ↓e1 |q = 1 = ψ↑e2 ↓e1

543

line eigenstate is given as 

1 · |0  0 · |0  =  0 · |1 0 · |1

  1  0 |1   2 1  =  ,  0 |0 2 1 0 |1 2 1 2 |0 1



line eigenstate can be generated = U ⊗ (1 · 1 + 0 · ω, 0 · ω ˘ + 0 · 1) |q = 0 .

We obtain   1 · |0 2 |0 1 0100  1 0 0 0   0 · |0 |1    2 1 |q = 1 =   ,  0 0 0 1   0 · |1 2 |0 1  0010 0 · |1 2 |1 1     0 0 · |0 2 |0 1  1 · |0 |1   1     2 1 ψ↑e2 ↓e1 =   =  .  0 · |1 2 |0 1   0  0 0 · |1 2 |1 1 

Similarly, we have

U ⊗ (q (0 → ω) , ∆p = 0) = U ⊗ (0 · 1 + 1 · ω, 0˘ ω + 0)   0010 0 0 0 1   = I ⊗X = , 1 0 0 0 0100

and hence it follows that

 1 · |0 0010  0 0 0 1   0 · |0   |q = ω = ψ↓e2 ↑e1 =    1 0 0 0   0 · |1 0100 0 · |1     0 0 · |0 2 |0 1  0 · |0 |1   0     2 1 =  =  .  1 · |1 2 |0 1   1  0 0 · |1 2 |1 1 

|0 2 |1 2 |0 2 |1 2

1



  1 1 1

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hence

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U ⊗ (q (ω → ω ˘ ) , ∆p = 0) = U ⊗ (1 · 1 + 0 · ω, 0˘ ω + 0)   0100 1 0 0 0   = X ⊗I = , 0 0 0 1 0010  0 · |0 0100  1 0 0 0   0 · |0   |q = ω ˘ = ψ↓e2 ↓e1 =    0 0 0 1   1 · |1 0010 0 · |1     0 0 · |0 2 |0 1  0 · |0 |1   0     2 1 =  =  .  0 · |1 2 |0 1   0  1 1 · |1 2 |1 1 

|0 2 |1 2 |0 2 |1 2

1



  1 1 1

Horizontal Striation Ray and ‘Momentum’ Basis

The horizontal striation ray is given by p = 0 consisting of points (0, 0) , (1, 0) , (ω, 0) and (˘ ω , 0) . We have the following three commuting unitary operators associated with the translations (1, 0),(ω, 0), and (˘ ω , 0) from the origin   0100 1 0 0 0   T ⊗ (1, 0) = T ⊗ (1 + 0ω, 0˘ ω + 0) = X ⊗ I =  , 0 0 0 1 0010   0010 0 0 0 1   T ⊗ (ω, 0) = T ⊗ (0 + 1ω, 0˘ ω + 0) = I ⊗ X =  , 1 0 0 0 0100   0001 0 0 1 0   T ⊗ (˘ ω, 0) = T ⊗ (1 + 1ω, 0˘ ω + 0) = X ⊗ X =  , 0 1 0 0 1000

which have eigenstate, Eqs. (41.25)-(41.26)

|p = 0 = ψ→e˜2 →e˜1 .

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Consider the horizontal striation which we rewrite as p = 0 consisting of points (0, 0) , (1, 0) , (ω, 0) and (˘ ω , 0) , p = 1 consisting of points (0, 1) , (1, 1) , (ω, 1) and (˘ ω , 1) , p = ω consisting of points (0, ω) , (1, ω) , (ω, ω) and (˘ ω, ω) , p=ω ˘ consisting of points (0, ω ˘ ) , (1, ω ˘ ) , (ω, ω ˘ ) and (˘ ω, ω ˘) . p p 2 |1 1 )

(41.31)

line can be obtained from the Note that from Eq. (41.13), the p = (|0 p = (|0 p2 |0 p1 ) line by translating the p-coordinate of the first particle by unity, i.e., (|0 p2 |0 p1 ) ⇒ (|0 p2 |1 p1 ) . Likewise the p = (|1 p2 |0 p1 ) line can be obtained from the p = (|0 p2 |0 p1 ) line by translating the p-coordinate of the second particle by 1. Moreover, the p = (|1 p2 |1 p1 ) line can be obtained from the p = (|1 p2 |0 p1 ) line by translating the p-coordinate of the first particle by unity. Thus, we can generate the vertical striation eigenstates of Eq. (41.25) by using the following translation operators, U ⊗ ((|0 p2 |0 p1 ) ⇒ (|0

p 2

Thus if |p = 0 2 |p = 0

1

|1 p1 )) = U ⊗ (0 · 1 + 0 · ω, 1 · ω ˘ + 0 · 1)   1 0 0 0  0 −1 0 0    = Z ⊗I = . 0 0 1 0  0 0 0 −1 = |0 p2 |0

p 1

= ψ→e˜2 →e˜1

then |p = 0 2 |p = 1

1

= |0 p2 |1

|0  |0 1 =  2  |1 |1

2 |0 1



 , 2 |0 1  2 |1 1 2 |1 1 

p 1

= ψ→e˜2 ←e˜1 



  |0 1 0 0 0  0 −1 0 0  1  |0    =    0 0 1 0  2  |1 0 0 0 −1 |1 

 |0 2 |0 1  1  − |0 2 |1 1  =  . 2  |1 2 |0 1  − |1 2 |1 1

2 |0 1



  2 |0 1  2 |1 1 2 |1 1 

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Similarly, we have the following generator U ⊗ ((|0 p2 |0 p1 ) ⇒ (|1

p 2

so that we have, |p = 1 2 |p = 0

1

|0 p1 )) = U ⊗ (0 · 1 + 0 · ω, 0 · ω ˘ + 1 · 1)   10 0 0 0 1 0 0   = I ⊗Z =  0 0 −1 0  , 0 0 0 −1

= |1 p2 |0

p 1

= ψ←e˜2 →e˜1 

|0 2 |0 1  1 |0 2 |1 1  . =    − |1 |0 2 2 1 − |1 2 |1 1

We also have

U ⊗ ((|1 p2 |0 p1 ) ⇒ (|1

so that we obtain |p = 1 2 |p = 1

  |0 10 0 0  0 1 0 0  1  |0   =  0 0 −1 0  2  |1 0 0 0 −1 |1  

1

= (|1

p 2



  |0 2 1 |1 2 1 2 |1 1 

|1 p1 )) = U ⊗ (0 · 1 + 0 · ω, 1 · ω ˘ + 0 · 1)   1 0 0 0  0 −1 0 0   = Z ⊗I = 0 0 1 0 , 0 0 0 −1

p p 2 |1 1 )

= ψ ←e˜2 ←e˜1 

2 |0 1

   |0 2 |0 1 1 0 0 0  0 −1 0 0  1  |0 |1  2 1    =  0 0 1 0  2  − |1 |0  2 1 0 0 0 −1 − |1 2 |1 1  

|0 2 |0 1 1 − |0 2 |1 1   . =  2 − |1 2 |0 1  |1 2 |1 1

Note that the order of the momentum coordinate, as elements of GF 22 when expanded in terms of the dual basis, is p = 0 = [00] (|0 p2 |0 p1 ), p = ω ˘ = [01] (|0 p2 |1 p1 ), p = 1 = [10] (|1 p2 |0 p1 ), p = ω = [11] (|1 p2 |1 p1 ), in accordance with Eq. (41.25). Therefore, the order of the momentum coordinate in what follows should be decomposed when expanded in terms of field basis according to Eq. (41.25). For example, in what follows, if the line pass through a point (1, 1) then this indices should expand as (1 · 1 + 0 · ω, 1 · ω ˘ + 0 · 1) since in binary strings this

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corresponds to (01, 01), i.e., the first field basis state in (1, ω) and its dual (˘ ω, 1) belongs to the first particle. The reason for this is that for ease in constructing the diﬀerent striations, which determine the mutually unbiased bases of the discrete phase space, the order of the momentum coordinate is simply assumed as (0, 1, ω, ω ˘ ). However, by virtue of our dual field basis chosen, the actual binarystring order as given in Eq. (41.25) is (0, ω ˘ , 1, ω), Eq. (41.26). This corresponds to the cardinal-number bit arrangement of (00, 01, 10, 11), in terms of the dual field basis, (˘ ω , 1), where the rightmost bit belongs to the first particle. Thus in what follows, the mapping (0, 1, ω, ω ˘ ) ⇒ (00, 01, 10, 11) in terms of our dual basis chosen will be applied, For example, a maximum change in momentum coordinate by δp = ω ˘ between points in a line or between points in each of two parallel lines will be mapped to [11] ⇒ δp = 1 · ω ˘ + 1 · 1 = ω in terms of our dual field basis. 41.3.5

Diagonal Striation Ray and ‘Y Y ’ Basis

The diagonal striation ray is determined by the line q + p = 0 consisting of points (0, 0) , (1, 1) , (ω, ω) , and (˘ ω, ω ˘) with the following commuting unitary operators associated with the translations (1, 1),(ω, ω), and (˘ ω, ω ˘ ) from the origin, 

 0 −1 0 0 1 0 0 0    T ⊗ (1, 1) = T ⊗ (1 · 1 + 0 · ω, 1 · ω ˘ + 0 · 1) = XZ ⊗ I =  ,  0 0 0 −1  0 0 1 0   0 0 −1 0  0 0 0 −1    T ⊗ (ω, ω) = T ⊗ (0 + 1 · ω, 0 · ω ˘ + 1 · 1) = I ⊗ XZ =  , 1 0 0 0  01 0 0   0 0 0 1  0 0 −1 0    T ⊗ (˘ ω, ω ˘ ) = T ⊗ (1 · 1 + 1 · ω, 1 · ω ˘ + 1 · 1) = XZ ⊗ XZ =  .  0 −1 0 0  1 0 0 0 The following is a simultaneous eigenvector to the translations along (q + p = 0) line     1 |0 2 |0 1 1    1  −i  1  −i |0 2 |1 1   ψp+q=0 =   ⇒  . 2  i |1 2 |0 1  2 i  1 1 |1 2 |1 1

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We have the diagonal striation as q + p = 0 consisting of points (0, 0) , (1, 1) , (ω, ω) , and (˘ ω, ω ˘) , q + p = 1 consisting of points (0, 1) , (1, 0) , (ω, ω ˘ ) , and (˘ ω, ω) , q + p = ω consisting of points (0, ω) , (ω, 0) , (1, ω ˘ ) , and (˘ ω , 1) , q+p = ω ˘ consisting of points (0, ω ˘ ) , (˘ ω , 0) , (1, ω) , and (ω, 1) .

(41.32)

We have the transformation of the eigenstates between diﬀerent parallel lines in the striation given as follows ˘ + 1 · 1) U ⊗ ((q + p = 0) ⇒ (q + p = 1)) = U ⊗ (0 · 1 + 0 · ω, 0 · ω   10 0 0 0 1 0 0    = I ⊗Z = ,  0 0 −1 0  0 0 0 −1 corresponding to no translation in the horizontal axis and translation of vertical axis in terms of the binary strings indicated as (00) ⇒ (10) , (10) ⇒ (00) , (11) ⇒ (01) , (01) ⇒ (11) , which clearly shows the translation of the ‘momentum’ of the second particle by unity in order to translate the (q + p = 0) line to the (q + p = 1) line of the same striation. It is worthwhile to point out that in general, in analogy to the continuum case, identical translation is obtained if one keep one of the coordinates, namely q or p, constant between two points, each one belonging to each of the two parallel lines. Therefore, in determining the transformation operator U ⊗ ((q + p = 1) ⇒ (q + p = ω)), we need to consider the following pair of points, each belonging to (q + p = 1) line and (q + p = ω) line, arranged in terms of the order of the q-coordinates, q + p = 1 consisting of points (0, 1) , (1, 0) , (ω, ω ˘ ) , and (˘ ω , ω) , q + p = ω consisting of points (0, ω) , (1, ω ˘ ) , (ω, 0) , , and (˘ ω , 1) , which clearly show that the two parallel lines diﬀer by a displacement ω ˘ of the momentum (vertical) axis. In terms of the binary strings, based on the dual field basis, this corresponds to a displacement of momentum axis by (01), a σz operation

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on the first particle. Hence, we have ˘ + 0 · 1) U ⊗ ((q + p = 1) ⇒ (q + p = ω)) = U ⊗ (0 · 1 + 0 · ω, 1 · ω   1 0 0 0  0 −1 0 0   = Z ⊗I = 0 0 1 0 . 0 0 0 −1

Similarly, we have the following pair-arrangement for the last two parallel lines q + p = ω consisting of points (0, ω) , (1, ω ˘ ) , (ω, 0) , and (˘ ω, 1) , q+p = ω ˘ consisting of points (0, ω ˘ ) , (1, ω) , (ω, 1) , and (˘ ω, 0) , (41.33) which correspond to a displacement of the momentum coordinate by unity. In terms of the binary strings, this corresponds to a displacement of momentum axis by (10), a σz operation of the second particle. Hence, we have ˘ )) = U ⊗ (0 · 1 + 0 · ω, 0 · ω ˘ + 1 · 1) U ⊗ ((q + p = ω) ⇒ (q + p = ω   10 0 0 0 1 0 0   =I⊗Z =  0 0 −1 0  . 0 0 0 −1

Now we can assign the projectors of all the lines in the striation by fixing the projector of the reference line, which we have referred to as the ‘vacuum’ (q + p = 0) line. If we assign the following eigenvector to the reference ‘vacuum’ (q + p = 0) line     1 |0 2 |0 1 1  −i   1 1 −i |0 |1 2 1 ⇒  , ψ p+q=0 =  2  i |1 2 |0 1  2 i  1 1 |1 2 |1 1 then we can obtain the eigenvector of the rest of parallel lines in the striation by using the translations between parallel lines. We have ψp+q=1 = I ⊗ Z ψp+q=0     1 10 0 0  0 1 0 0  1  −i      =   ,  0 0 −1 0  2  i  1 0 0 0 −1   1  1  −i   ψp+q=1 =  , 2  −i  −1

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ψp+q=ω = Z ⊗ I ψp+q=1     1 1 0 0 0  0 −1 0 0  1  −i      = ,    0 0 1 0  2  −i  −1 0 0 0 −1   1  1  i  ψp+q=ω =   , 2  −i  1 ψp+q=˘ω = I ⊗ Z ψp+q=ω     1 10 0 0 0 1 0 0  1  i      =     0 0 −1 0  2  −i  1 0 0 0 −1   1  1  i  =  . 2 i  −1 41.3.6

Low-Slope-Striation Ray and ‘Belle’ Basis

The smaller slope striation ray is the line given by ωq + p = 0 consisting of points (0, 0) , (1, ω) , (ω, ω ˘ ) , and (˘ ω , 1) with the following three commuting unitary operators associated with the translation by vectors (1, ω),(ω, ω ˘ ), and (˘ ω, 1) from the origin (0, 0),   01 0 0 1 0 0 0   T ⊗ (1, ω) = T ⊗ (1 + 0 · ω, 0 · ω ˘ + 1) = X ⊗ Z =   0 0 0 −1  , 0 0 −1 0   0 0 −1 0 0 0 0 1  T ⊗ (ω, ω ˘ ) = T ⊗ (0 + ω, ω ˘ + 1) = Z ⊗ XZ =  1 0 0 0, 0 −1 0 0   0 0 0 −1 0 0 1 0   T ⊗ (˘ ω , 1) = T ⊗ (1 + ω, 1˘ ω + 0) = XZ ⊗ X =   0 −1 0 0  . 1 0 0 0

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The simultaneous eigenvector is

ψωq+p=0



   1 |0 2 |0 1 1   1 1  1 |0 2 |1 1   1  =   ⇒  . 2  i |1 2 |0 1  2 i  −i −i |1 2 |1 1

Rewriting the ‘low-slope’ striation,

ωq + p = 0 consisting of points (0, 0) , (1, ω) , (ω, ω ˘ ) , and (˘ ω , 1) , ωq + p = 1 consisting of points (0, 1) , (1, ω ˘ ) , (ω, ω) , and (˘ ω , 0) , ωq + p = ω consisting of points (0, ω) , (1, 0) , (ω, 1) , and (˘ ω, ω ˘) , ωq + p = ω ˘ consisting of points (0, ω ˘ ) , (1, 1) , (ω, 0) , and (˘ ω , ω) , (41.34) then the eigenstates translation operator between diﬀerent parallel lines in the striation given as follows. A unity translation, δp = [01] (mod 2) exists between the ωq + p = 0 line and ωq + p = 1 line. Thus, we have U ⊗ ((ωq + p = 0) ⇒ (ωq + p = 1)) = U ⊗ (0 · 1 + 0 · ω, 1 · ω ˘ + 0 · 1)   1 0 0 0  0 −1 0 0    = Z ⊗I = . 0 0 1 0  0 0 0 −1

Using similar considerations, we have the following transformation operators between parallel lines in the striation as δp = [11] (mod 2) : U ⊗ ((ωq + p = 1) ⇒ (ωq + p = ω)) ˘ + 1 · 1) = U ⊗ (0 · 1 + 0 · ω, 1 · ω   1 0 0 0  0 −1 0 0    = Z ⊗Z = ,  0 0 −1 0  0 0 0 1

δp = [01] (mod 2) : U ⊗ ((ωq + p = ω) ⇒ (ωq + p = ω ˘ )) ˘ + 0 · 1) = U ⊗ (0 · 1 + 0 · ω, 1 · ω   1 0 0 0  0 −1 0 0    = Z ⊗I = . 0 0 1 0  0 0 0 −1

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Now, if we assign the eigenvector to the vacuum line,

ψωq+p=0



   1 |0 2 |0 1 1   1 1  1 |0 2 |1 1   1  =   ⇒  , 2  i |1 2 |0 1  2 i  −i −i |1 2 |1 1

then we have the eigenvectors belonging to the rest of the parallel lines in the striation as

ψωq+p=1 = Z ⊗ I ψωq+p=0

     1 1 1 0 0 0   0 −1 0 0  1  1  1  −1      = ,   ⇒  0 0 1 0  2  i  2 i  −i i 0 0 0 −1 

ψωq+p=ω = Z ⊗ Z ψωq+p=1

ψωq+p=ω



 1  1  1  =  , 2  −i  i

ψωq+p=˘ω = Z ⊗ I ψωq+p=ω 

 1  1  −1  =  . 2  −i  −i 41.3.7

   1 1 0 0 0  0 −1 0 0  1  −1      = ,    0 0 −1 0  2  i  i 0 0 0 1 

   1 1 0 0 0  0 −1 0 0  1  1      =     0 0 1 0  2  −i  i 0 0 0 −1 

High-Slope-Striation Ray and ‘Beau’ Basis

The larger slope striation ray is described by the line q + ωp = 0 consisting of points (0, 0) , (ω, 1) , (˘ ω, ω) , and (1, ω ˘)

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553

with the following three commuting unitary operators associated with the translations (ω, 1),(˘ ω , ω), and (1, ω ˘ ) from the origin (0, 0),  0 0 1 0  0 0 0 −1   ˘ + 0 · 1) = Z ⊗ X =  T ⊗ (ω, 1) = T ⊗ (0 + ω, 1 · ω 1 0 0 0 , 0 −1 0 0   0 0 0 −1  0 0 −1 0   T ⊗ (˘ ω , ω) = T ⊗ (1 + ω, 0 · ω ˘ + 1) = X ⊗ XZ =  0 1 0 0 , 10 0 0   0 −1 0 0 1 0 0 0  T ⊗ (1, ω ˘ ) = T ⊗ (1 + 0ω, 1 · ω ˘ + 1 · 1) = XZ ⊗ Z =  0 0 0 1. 0 0 −1 0 

The following is the simultaneous eigenvector to the ‘vacuum’ line, q + ωp = 0,

ψ q+ωp=0

   1 |0 2 |0 1 1   −i  1 1 −i |0 |1 2 1 ⇒  . =  2  1 |1 2 |0 1  2 1  i i |1 2 |1 1 

For convenience, we rewrite the ‘high-slope’ striation q + ωp = 0 consisting of points q + ωp = 1 consisting of points q + ωp = ω consisting of points q + ωp = ω ˘ consisting of points

(0, 0) , (ω, 1) , (˘ ω, ω) , and (1, 0) , (˘ ω , 1) , (ω, ω) , and (ω, 0) , (0, 1) , (1, ω) , and (˘ ω , 0) , (1, 1) , (0, ω) , and

(1, ω ˘) , (0, ω ˘) , (˘ ω, ω ˘) , (ω, ω ˘ ) . (41.35)

A translation of the momentum coordinate by ω ˘ (mod 2) exists between the q+ωp = 0 line and q + ωp = 1 line. Thus, we have U ⊗ ((q + ωp = 0) ⇒ (q + ωp = 1)) = U ⊗ (0 · 1 + 0 · ω, 1 · ω ˘ + 1 · 1)   1 0 0 0  0 −1 0 0   = Z ⊗Z =  0 0 −1 0  . 0 0 0 1 Using similar considerations, we have the following transformation operators between parallel lines in the striation, q + ωp = 1 consisting of points (0, ω ˘ ) , (1, 0) , (ω, ω) , and (˘ ω , 1) , q + ωp = ω consisting of points (0, 1) , (1, ω) , (ω, 0) , and (˘ ω, ω ˘) ,

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as ˘ + 1 · 1) U ⊗ ((q + ωp = 1) ⇒ (q + ωp = ω)) = U ⊗ (0 · 1 + 0 · ω, 0 · ω   10 0 0 0 1 0 0   = I ⊗Z =  0 0 −1 0  . 0 0 0 −1 Similarly, for the lines, q + ωp = ω consisting of points (0, 1) , (1, ω) , (ω, 0) , and (˘ ω, ω ˘) , q + ωp = ω ˘ consisting of points (0, ω) , (1, 1) , (ω, ω ˘ ) , and (˘ ω , 0) , (41.36) we have U ⊗ ((q + ωp = ω) ⇒ (q + ωp = ω ˘ )) = U ⊗ (0 · 1 + 0 · ω, 1 · ω ˘ + 1 · 1)   1 0 0 0  0 −1 0 0   = Z ⊗Z =  0 0 −1 0  . 0 0 0 1 If we assign the following eigenvector to the ‘vacuum’ line, q + ωp = 0,

ψq+ωp=0

   1 |0 2 |0 1 1  −i   1 1 −i |0 |1 2 1 ⇒  , =  2  1 |1 2 |0 1  2 1  i i |1 2 |1 1 

then it follows that

ψq+ωp=1 = Z ⊗ Z ψq+ωp=0

     1 1 1 0 0 0  0 −1 0 0  1  −i  1  i       =  0 0 −1 0  2  1  = 2  −1  , i i 0 0 0 1 

ψq+ωp=ω = I ⊗ Z ψq+ωp=1  1 1 i  , =  2 1  −i 

   1 10 0 0 0 1 0 0  1  i     =  0 0 −1 0  2  −1  i 0 0 0 −1 

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Discrete Wigner Distribution Function Construction

ψq+ωp=˘ω = Z ⊗ Z ψq+ωp=ω  1 1  −i  . =  2  −1  −i 

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   1 0 0 0 1  0 −1 0 0  1  i     =  0 0 −1 0  2  1  0 0 0 1 −i 

Now, let P be the matrix with these eigenvectors of the striation as its columns   1 1 1 1 1  −i i i −i   P =  2  1 −1 1 −1  i i −i −i

and hence

P −1

 1 i 1 −i 1  1 −i −1 −i  . =  2  1 −i 1 i  1 i −1 i 

One can readily verify that P diagonalizes A, as a simple computation confirms P −1 AP = D, where the matrix A is any of the three commuting translation operators, T ⊗ (ω, 1), ω , ω), or T ⊗ (1, ω ˘ ) given above. For example, T ⊗ (˘     1 i 1 −i 1 1 1 1  1  −i i i −i  1   1 −i −1 −i  T ⊗ (ω, 1)  (Z⊗X)   2 1 −i 1 i 2  1 −1 1 −1  1 i −1 i i i −i −i     1 1 1 1 0 0 1 0 1 i 1 −i    1  1 −i −1 −i    0 0 0 −1   −i i i −i  =       1 −1 1 −1  1 0 0 0 4 1 −i 1 i i i −i −i 0 −1 0 0 1 i −1 i    1 −1 1 −1 1 i 1 −i   −i −i i i  1 1 −i −1 −i   =  4  1 −i 1 i   1 1 1 1  i −i −i i 1 i −1 i     1 0 0 0 4 0 0 0   1  0 −4 0 0   =  0 −1 0 0  . =     0 0 1 0  4 0 0 4 0 0 0 0 −1 0 0 0 −4

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

   1 i 1 −i 1 1 1 1   1 1  1 −i −1 −i  ⊗  −i i i −i  ω, ω)    T(X⊗XZ) (˘  2  1 −i 1 i  2  1 −1 1 −1  1 i −1 i i i −i −i      0 0 0 −1 1 i 1 −i 1 1 1 1     1  1 −i −1 −i   0 0 −1 0  1  −i i i −i  =      2  1 −i 1 i   0 1 0 0  2  1 −1 1 −1  10 0 0 1 i −1 i i i −i −i    −i −i i i 1 i 1 −i    1  1 −i −1 −i   −1 1 −1 1   =    4  1 −i 1 i   −i i i −i  1 1 1 1 1 i −1 i     −1 0 0 0 −4i 0 0 0  0 −1 0 0   1    0 −4i 0 0  =  .  = i  0 0 1 0 4  0 0 4i 0  0 0 01 0 0 0 4i These results reflect the fact that the three commuting unitary translation operators along a line in a striation have simultaneous eigenvectors (have identical Hilbert space), although their respective eigenvalues are generally diﬀerent.

41.4

Discrete Wigner Function for Two Qubits

To calculate the Wigner function at various points in phase space, we have to evaluate  1  W (q, p) = N

Tr ρ ˆ ψu,v,c

ψu,v,c

uvc|(q,p)



− 1 ,

where the sum is over all lines passing through the point (q, p) with each line projector, ψu,v,c ψu,v,c , determined by the eigenvector assigned to each of the pertinent lines. First let us enumerate the various projectors passing each point in phase space graph.

41.4.1

The Origin in Phase Space, q = 0, p = 0

There are five lines intersecting the origin in phase space, with line projectors as follows:

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|q = 0

2

|q = 0

1

|p = 0

2

|p = 0

1

ψ p+q=0 ψ ωq+p=0 ψ q+ωp=0

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q = 0|1 q = 0|2 ,

|0 2 |0 1 + |0 2 |1 1 + |1 2 |0 1 + |1 2 |1 1 × 0|1 0|2 + 1|1 0|2 + 0|1 1|2 + 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 + i |1 2 |0 1 + |1 2 |1 1 , ψ p+q=0 = 14 × 0|1 0|2 + i 1|1 0|2 − i 0|1 1|2 + 1|1 1|2 |0 2 |0 1 + |0 2 |1 1 + i |1 2 |0 1 + −i |1 2 |1 1 ψωq+p=0 = 14 , × 0|1 0|2 + 1|1 0|2 − i 0|1 1|2 + i 1|1 1|2 1 |0 2 |0 1 − i |0 2 |1 1 + |1 2 |0 1 + i |1 2 |1 1 ψq+ωp=0 = 14 . × 0|1 0|2 + i 1|1 0|2 + 0|1 1|2 − i 1|1 1|2 p = 0|1 p = 0|2 =

1 4

,

The Point (1, 0) in Phase Space Structure

41.4.2

The five intersecting lines are: q = 1, p = 0, q + p = 1, ωq + p = ω, and q + ωp = 1 in phase space picture. We have the following line projectors: |q = 0

2

|q = 1

1

|p = 0

2

|p = 0

1

ψ p+q=1 ψ ωq+p=ω ψ q+ωp=1

q = 1|1 q = 0|2 ,

|0 2 |0 1 + |0 2 |1 1 + |1 2 |0 1 + |1 2 |1 1 × 0|1 0|2 + 1|1 0|2 + 0|1 1|2 + 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 − i |1 2 |0 1 − |1 2 |1 1 ψ p+q=1 = 14 , × 0|1 0|2 + i 1|1 0|2 + i 0|1 1|2 − 1|1 1|2 |0 2 |0 1 + |0 2 |1 1 − i |1 2 |0 1 + i |1 2 |1 1 , ψ ωq+p=ω = 14 × 0|1 0|2 + 1|1 0|2 + i 0|1 1|2 − i 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 − |1 2 |0 1 + i |1 2 |1 1 . ψ q+ωp=1 = 14 × 0|1 0|2 − i 1|1 0|2 − 0|1 1|2 − i 1|1 1|2 p = 0|1 p = 0|2 =

1 4

,

The Point (ω, 0)

41.4.3

The point (ω, 0) has intersecting lines, q = ω, p = 0, q + p = ω, ωq + p = ω ˘ , and q + ωp = ω, with the following line projectors: |q = 1

2

|q = 0

1

|p = 0

2

|p = 0

1

ψ p+q=ω ψ ωq+p=ω˘ ψ q+ωp=ω

41.4.4

q = 0|1 q = 1|2 ,

|0 2 |0 1 + |0 2 |1 1 + |1 2 |0 1 + |1 2 |1 1 × 0|1 0|2 + 1|1 0|2 + 0|1 1|2 + 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 − i |1 2 |0 1 + |1 2 |1 1 ψ p+q=ω = 14 , × 0|1 0|2 − i 1|1 0|2 + i 0|1 1|2 + 1|1 1|2 |0 2 |0 1 − |0 2 |1 1 − i |1 2 |0 1 − i |1 2 |1 1 ψ ωq+p=ω˘ = 14 , × 0|1 0|2 − 1|1 0|2 + i 0|1 1|2 + i 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 + |1 2 |0 1 − i |1 2 |1 1 ψ q+ωp=ω = 14 . × 0|1 0|2 − i 1|1 0|2 + 0|1 1|2 + i 1|1 1|2 p = 0|1 p = 0|2 =

1 4

,

The Point (˘ ω , 0)

The point (˘ ω , 0) has intersecting lines, q = ω ˘ , p = 0, q + p = ω ˘ , ωq + p = 1, and q + ωp = ω ˘ , with the following line projectors:

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|q = 1

2

|q = 1

1

|p = 0

2

|p = 0

1

ψ p+q=ω˘ ψ ωq+p=1 ψ q+ωp=ω˘

q = 1|1 q = 1|2 ,

|0 2 |0 1 + |0 2 |1 1 + |1 2 |0 1 + |1 2 |1 1 × 0|1 0|2 + 1|1 0|2 + 0|1 1|2 + 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 + i |1 2 |0 1 − |1 2 |1 1 , ψ p+q=˘ω = 14 × 0|1 0|2 − i 1|1 0|2 − i 0|1 1|2 − 1|1 1|2 |0 2 |0 1 − |0 2 |1 1 + i |1 2 |0 1 + i |1 2 |1 1 ψωq+p=1 = 14 , × 0|1 0|2 − 1|1 0|2 − i 0|1 1|2 − i 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 − |1 2 |0 1 − i |1 2 |1 1 ψ q+ωp=ω˘ = 14 . × 0|1 0|2 + i 1|1 0|2 − 0|1 1|2 + i 1|1 1|2 p = 0|1 p = 0|2 =

1 4

,

The Point (0, 1)

41.4.5

The point (0, 1) has intersecting lines, q = 0, p = 1, q + p = 1, ωq + p = 1, and q + ωp = ω, with the following line projectors: |q = 0

2

|q = 0

1

|p = 0

2

|p = 1

1

ψ q+p=1 ψ ωq+p=1 ψ q+ωp=ω

q = 0|1 q = 0|2 ,

|0 2 |0 1 − |0 2 |1 1 + |1 2 |0 1 − |1 2 |1 1 × 0|1 0|2 − 1|1 0|2 + 0|1 1|2 − 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 − i |1 2 |0 1 − |1 2 |1 1 ψ q+p=1 = 14 , × 0|1 0|2 + i 1|1 0|2 + i 0|1 1|2 − 1|1 1|2 |0 2 |0 1 − |0 2 |1 1 + i |1 2 |0 1 + i |1 2 |1 1 ψωq+p=1 = 14 , × 0|1 0|2 − 1|1 0|2 − i 0|1 1|2 − i 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 + |1 2 |0 1 − i |1 2 |1 1 ψ q+ωp=ω = 14 . × 0|1 0|2 − i 1|1 0|2 + 0|1 1|2 + i 1|1 1|2 p = 1|1 p = 0|2 =

1 4

,

The Point (0, ω)

41.4.6

The point (0, ω) has intersecting lines, q = 0, p = ω, q + p = ω, ωq + p = ω, and q + ωp = ω ˘ , with the following line projectors: |q = 0

2

|q = 0

1

|p = 1

2

|p = 0

1

ψ q+p=ω ψ ωq+p=ω ψ q+ωp=ω˘

41.4.7

q = 0|1 q = 0|2 ,

|0 2 |0 1 + |0 2 |1 1 − |1 2 |0 1 − |1 2 |1 1 × 0|1 0|2 + 1|1 0|2 − 0|1 1|2 − 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 − i |1 2 |0 1 + |1 2 |1 1 ψ q+p=ω = 14 , × 0|1 0|2 − i 1|1 0|2 + i 0|1 1|2 + 1|1 1|2 |0 2 |0 1 + |0 2 |1 1 − i |1 2 |0 1 + i |1 2 |1 1 ψ ωq+p=ω = 14 , × 0|1 0|2 + 1|1 0|2 + i 0|1 1|2 − i 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 − |1 2 |0 1 − i |1 2 |1 1 ψ q+ωp=ω˘ = 14 . × 0|1 0|2 + i 1|1 0|2 − 0|1 1|2 + i 1|1 1|2 p = 0|1 p = 1|2 =

1 4

,

The Point (0, ω ˘)

The point (0, ω ˘ ) has intersecting lines, q = 0, p = ω ˘, q + p = ω ˘ , ωq + p = ω ˘ , and q + ωp = 1, with the following line projectors:

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|q = 0

2

|q = 0

1

|p = 1

2

|p = 1

1

ψ q+p=ω˘ ψ ωq+p=ω˘ ψ q+ωp=1

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q = 0|1 q = 0|2 ,

|0 2 |0 1 − |0 2 |1 1 − |1 2 |0 1 + |1 2 |1 1 × 0|1 0|2 − 1|1 0|2 − 0|1 1|2 + 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 + i |1 2 |0 1 − |1 2 |1 1 , ψ q+p=˘ω = 14 × 0|1 0|2 − i 1|1 0|2 − i 0|1 1|2 − 1|1 1|2 |0 2 |0 1 − |0 2 |1 1 − i |1 2 |0 1 − i |1 2 |1 1 ψ ωq+p=ω˘ = 14 , × 0|1 0|2 − 1|1 0|2 + i 0|1 1|2 + 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 − |1 2 |0 1 + i |1 2 |1 1 ψq+ωp=1 = 14 . × 0|1 0|2 − i 1|1 0|2 − 0|1 1|2 − i 1|1 1|2 p = 1|1 p = 1|2 =

1 4

,

The Point (1, 1)

41.4.8

The point (1, 1) has intersecting lines, q = 1, p = 1, q + p = 0, ωq + p = ω ˘ , and q + ωp = ω ˘ , with the following line projectors: |q = 0

2

|q = 1

1

|p = 0

2

|p = 1

1

ψ q+p=0 ψ ωq+p=ω˘ ψ q+ωp=ω˘

q = 1|1 q = 0|2 ,

|0 2 |0 1 − |0 2 |1 1 + |1 2 |0 1 − |1 2 |1 1 × 0|1 0|2 − 1|1 0|2 + 0|1 1|2 − 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 + i |1 2 |0 1 + |1 2 |1 1 ψ q+p=0 = 14 , × 0|1 0|2 + i 1|1 0|2 − i 0|1 1|2 + 1|1 1|2 |0 2 |0 1 − |0 2 |1 1 − i |1 2 |0 1 − i |1 2 |1 1 ψ ωq+p=ω˘ = 14 , × 0|1 0|2 − 1|1 0|2 + i 0|1 1|2 + i 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 − |1 2 |0 1 − i |1 2 |1 1 ψ q+ωp=ω˘ = 14 . × 0|1 0|2 + i 1|1 0|2 − 0|1 1|2 + i 1|1 1|2 p = 1|1 p = 0|2 =

1 4

,

The Point (ω, ω)

41.4.9

The point (ω, ω) has intersecting lines, q = ω, p = ω, q + p = 0, ωq + p = 1, and q + ωp = 1, with the following line projectors: |q = 1

2

|q = 0

1

|p = 1

2

|p = 0

1

ψ q+p=0 ψ ωq+p=1 ψ q+ωp=1

41.4.10

q = 0|1 q = 1|2 ,

|0 2 |0 1 + |0 2 |1 1 − |1 2 |0 1 − |1 2 |1 1 × 0|1 0|2 + 1|1 0|2 − 0|1 1|2 − 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 + i |1 2 |0 1 + |1 2 |1 1 ψ q+p=0 = 14 , × 0|1 0|2 + i 1|1 0|2 − i 0|1 1|2 + 1|1 1|2 |0 2 |0 1 − |0 2 |1 1 + i |1 2 |0 1 + i |1 2 |1 1 ψωq+p=1 = 14 , × 0|1 0|2 − 1|1 0|2 − i 0|1 1|2 − i 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 − |1 2 |0 1 + i |1 2 |1 ψq+ωp=1 = 14 . × 0|1 0|2 − i 1|1 0|2 − 0|1 1|2 − i 1|1 1|2 p = 0|1 p = 1|2 =

1 4

,

The Point (˘ ω, ω ˘)

The point (˘ ω, ω ˘ ) has intersecting lines, q = ω ˘, p = ω ˘ , q + p = 0, ωq + p = ω, and q + ωp = ω, with the following line projectors:

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|q = 1

2

|q = 1

1

|p = 1

2

|p = 1

1

ψ q+p=0 ψ ωq+p=ω ψ q+ωp=ω

q = 1|1 q = 1|2 ,

|0 2 |0 1 − |0 2 |1 1 − |1 2 |0 1 + |1 2 |1 1 × 0|1 0|2 − 1|1 0|2 − 0|1 1|2 + 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 + i |1 2 |0 1 + |1 2 |1 1 , ψ q+p=0 = 14 × 0|1 0|2 + i 1|1 0|2 − i 0|1 1|2 + 1|1 1|2 |0 2 |0 1 + |0 2 |1 1 − i |1 2 |0 1 + i |1 2 |1 1 ψ ωq+p=ω = 14 , × 0|1 0|2 + 1|1 0|2 + i 0|1 1|2 − i 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 + |1 2 |0 1 − i |1 2 |1 1 ψ q+ωp=ω = 14 . × 0|1 0|2 − i 1|1 0|2 + 0|1 1|2 + i 1|1 1|2 p = 1|1 p = 1|2 =

1 4

,

The Point (ω, 1)

41.4.11

The point (ω, 1) has intersecting lines, q = ω, p = 1, q + p = ω ˘ , ωq + p = ω, and q + ωp = 0, with the following line projectors: |q = 1

2

|q = 0

1

|p = 0

2

|p = 1

1

ψ q+p=ω˘ ψ ωq+p=ω ψ q+ωp=0

q = 0|1 q = 1|2 ,

|0 2 |0 1 − |0 2 |1 1 + |1 2 |0 1 − |1 2 |1 1 × 0|1 0|2 − 1|1 0|2 + 0|1 1|2 − 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 + i |1 2 |0 1 − |1 2 |1 1 ψ q+p=˘ω = 14 , × 0|1 0|2 − i 1|1 0|2 − i 0|1 1|2 − 1|1 1|2 |0 2 |0 1 + |0 2 |1 1 − i |1 2 |0 1 + i |1 2 |1 1 ψ ωq+p=ω = 14 , × 0|1 0|2 + 1|1 0|2 + i 0|1 1|2 − i 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 + |1 2 |0 1 + i |1 2 |1 1 ψq+ωp=0 = 14 . × 0|1 0|2 + i 1|1 0|2 + 0|1 1|2 − i 1|1 1|2 p = 1|1 p = 0|2 =

1 4

,

The Point (˘ ω , 1)

41.4.12

The point (˘ ω , 1) has intersecting lines, q = ω ˘ , p = 1, q + p = ω, ωq + p = 0, and q + ωp = 1, with the following line projectors: |q = 1

2

|q = 1

1

q = 1|1 q = 1|2 ,

|p = 0

2

|p = 1

1

p = 1|1 p = 0|2 =

ψ q+p=ω

ψ q+p=ω =

1 4

ψ ωq+p=0

ψωq+p=0 =

1 4

ψ q+ωp=1

ψq+ωp=1 =

1 4

41.4.13

|0 2 |0 1 − |0 2 |1 1 + |1 2 |0 1 − |1 2 |1 1 × 0|1 0|2 − 1|1 0|2 + 0|1 1|2 − 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 − i |1 2 |0 1 + |1 2 |1 1 , × 0|1 0|2 − i 1|1 0|2 + i 0|1 1|2 + 1|1 1|2 |0 2 |0 1 + |0 2 |1 1 + i |1 2 |0 1 − i |1 2 |1 1 , × 0|1 0|2 + 1|1 0|2 − i 0|1 1|2 + i 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 − |1 2 |0 1 + i |1 2 |1 1 . × 0|1 0|2 − i 1|1 0|2 − 0|1 1|2 − i 1|1 1|2 1 4

,

The Point (˘ ω , ω)

The point (˘ ω , ω) has intersecting lines, q = ω ˘ , p = ω, q + p = 1, ωq + p = ω ˘ , and q + ωp = 0, with the following line projectors:

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|q = 1

2

|q = 1

1

|p = 1

2

|p = 0

1

ψ q+p=1 ψ ωq+p=ω˘ ψ q+ωp=0

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q = 1|1 q = 1|2 ,

|0 2 |0 1 + |0 2 |1 1 − |1 2 |0 1 − |1 2 |1 1 × 0|1 0|2 + 1|1 0|2 − 0|1 1|2 − 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 − i |1 2 |0 1 − |1 2 |1 1 , ψ q+p=1 = 14 × 0|1 0|2 + i 1|1 0|2 + i 0|1 1|2 − 1|1 1|2 |0 2 |0 1 − |0 2 |1 1 − i |1 2 |0 1 − i |1 2 |1 1 ψ ωq+p=ω˘ = 14 , × 0|1 0|2 − 1|1 0|2 + i 0|1 1|2 + i 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 + |1 2 |0 1 + i |1 2 |1 1 ψq+ωp=0 = 14 . × 0|1 0|2 + i 1|1 0|2 + 0|1 1|2 − i 1|1 1|2 p = 0|1 p = 1|2 =

1 4

,

The Point (1, ω)

41.4.14

The point (1, ω) has intersecting lines, q = 1, p = ω, q + p = ω ˘ , ωq + p = 0, and q + ωp = ω, with the following line projectors: |q = 0

2

|q = 1

1

|p = 1

2

|p = 0

1

ψ q+p=ω˘ ψ ωq+p=0 ψ q+ωp=ω

q = 1|1 q = 0|2 ,

|0 2 |0 1 + |0 2 |1 1 − |1 2 |0 1 − |1 2 |1 1 × 0|1 0|2 + 1|1 0|2 − 0|1 1|2 − 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 + i |1 2 |0 1 − |1 2 |1 1 ψ q+p=˘ω = 14 , × 0|1 0|2 − i 1|1 0|2 − i 0|1 1|2 − 1|1 1|2 |0 2 |0 1 + |0 2 |1 1 + i |1 2 |0 1 − i |1 2 |1 1 ψωq+p=0 = 14 , × 0|1 0|2 + 1|1 0|2 − i 0|1 1|2 + i 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 + |1 2 |0 1 − i |1 2 |1 1 ψ q+ωp=ω = 14 . × 0|1 0|2 − i 1|1 0|2 + 0|1 1|2 + i 1|1 1|2 p = 0|1 p = 1|2 =

1 4

,

The Point (ω, ω ˘)

41.4.15

The point (ω, ω ˘ ) has intersecting lines, q = ω, p = ω ˘ , q + p = 1, ωq + p = 0, and q + ωp = ω ˘ , with the following line projectors: |q = 1

2

|q = 0

1

|p = 1

2

|p = 1

1

ψ q+p=1 ψ ωq+p=0 ψ q+ωp=ω˘

41.4.16

q = 0|1 q = 1|2 ,

|0 2 |0 1 − |0 2 |1 1 − |1 2 |0 1 + |1 2 |1 1 × 0|1 0|2 − 1|1 0|2 − 0|1 1|2 + 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 − i |1 2 |0 1 − |1 2 |1 1 ψ q+p=1 = 14 , × 0|1 0|2 + i 1|1 0|2 + i 0|1 1|2 − 1|1 1|2 |0 2 |0 1 + |0 2 |1 1 + i |1 2 |0 1 − i |1 2 |1 1 ψωq+p=0 = 14 , × 0|1 0|2 + 1|1 0|2 − i 0|1 1|2 + i 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 − |1 2 |0 1 − i |1 2 |1 1 ψ q+ωp=ω˘ = 14 . × 0|1 0|2 + i 1|1 0|2 − 0|1 1|2 + i 1|1 1|2 p = 1|1 p = 1|2 =

1 4

,

The Point (1, ω ˘)

The point (1, ω ˘ ) has intersecting lines, q = 1, p = ω ˘ , q + p = ω, ωq + p = 1, and q + ωp = 0, with the following line projectors:

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|q = 0

2

|q = 1

1

q = 1|1 q = 0|2 ,

|p = 1

2

|p = 1

1

p = 1|1 p = 1|2 =

ψ q+p=ω

ψ q+p=ω =

1 4

ψ ωq+p=1

ψωq+p=1 =

1 4

ψ q+ωp=0

ψq+ωp=0 =

1 4

41.5 41.5.1

|0 2 |0 1 − |0 2 |1 1 − |1 2 |0 1 + |1 2 |1 1 × 0|1 0|2 − 1|1 0|2 − 0|1 1|2 + 1|1 1|2 |0 2 |0 1 + i |0 2 |1 1 − i |1 2 |0 1 + |1 2 |1 1 , × 0|1 0|2 − i 1|1 0|2 + i 0|1 1|2 + 1|1 1|2 |0 2 |0 1 − |0 2 |1 1 + i |1 2 |0 1 + i |1 2 |1 1 , × 0|1 0|2 − 1|1 0|2 − i 0|1 1|2 − i 1|1 1|2 |0 2 |0 1 − i |0 2 |1 1 + |1 2 |0 1 + i |1 2 |1 1 . × 0|1 0|2 + i 1|1 0|2 + 0|1 1|2 − i 1|1 1|2 1 4

,

Examples of Two-Qubit Discrete Wigner Function Example 1

Take a density matrix operator to be given by ψ↑↑ = |↑ 2 |↑

ρ = ψ↑↑

= |q = 0 2 |q = 0

1

1

↑|1 ↑|2

q = 0|1 q = 0|2

then from  1  W (q, p) = N 

1  = N

Tr ˆ ρ ψu,v,c

ψu,v,c

uvc|(q,p)

ˆ ψu,v,c ψu,v,c ρ uvc|(q,p)



− 1 

− 1

and examination of the intersecting line projectors at each point in phase space listed in the preceding tables, we readily obtain the discrete WDF shown in Fig. 41.2. 41.5.2

Example 2

A density-matrix operator given by ψ−→↑ = |↑ 2 |−→

ρ = ψ↑−→

= |q = 0 2 |p = 0 = |q = 0

2

1

1

−→|1 ↑|2

p = 0|1 q = 0|2

1 √ (|0 + |1 ) 2

1

1 √ (|0 + |1 ) 2

q = 0|2 1

1 (|0 2 |0 1 + |0 2 |1 1 ) ( 0|1 0|2 + 1|1 0|2 ) 2 1 = {|0 2 |0 1 0|1 0|2 + |0 2 |1 1 1|1 0|2 + |0 2 |0 2 =

1

1|1 0|2 + |0 2 |1

1

0|1 0|2 }

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Fig. 41.2

Wigner distribution function for two qubit in state, ψ ↑↑ .

Fig. 41.3

Wigner distribution function of two qubits in state, ψ ↑−→ .

Upon examination of the intersecting lines at each point in phase space, we obtain Fig. 41.3. 41.5.3

Example 3

A density-matrix operator given by 1 ψ↑↓ − ψ↓↑ ψ ↑↓ − ψ↓↑ 2 1 ρ = (|0 2 |1 1 − |1 2 |0 1 ) ( 1|1 0|2 − 0|1 1|2 ) 2 1 = {|0 2 |1 1 1|1 0|2 + |1 2 |0 1 0|1 1|2 − |0 2 |1 2 ρ=

1

0|1 1|2 − |1 2 |0

1

1|1 0|2 }

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Fig. 41.4 Discrete Wigner function for 1 |0 2 |1 1 − |1 2 |0 1 1|1 0|2 − 0|1 1|2 . 2

a

density-matrix

operator

given

by

ρ

=

Note that we can also express the pure state given above, including irrelevant global phase factor, as, 1 ±√ 2

ψ↑↓ − ψ↓↑

1 = √ (|0 2 ∓ |1 2 ) (|0 1 ± |1 1 ) ± |0 2 |0 1 ∓ |1 2 |1 2 √ = 2 |p = 1, 0 2 |p = 0, 1 1 ± |0 2 |0 1 ∓ |1 2 |1 1

1

Upon examination of the intersecting lines at each point in phase space, we obtain Fig. 41.4. 41.6

Quantum Nets: Arbitrary Assignment to a ‘Vacuum’ Line

Since only one line from each striation intersect at a chosen point (q, p) in discrete phase space, the contribution of each striation to the discrete Wigner function, Eq. (41.1), will depend on the particular assignment of line eigenstate to the ‘vacuum’ line or reference line. Since there are N + 1 striations (bits) and there N line eigenstates (digits) for each striation, there will be N N+1 diﬀerent assignments of eigenstates in all the striations. However, we can generally fixed the eigenstate assignment of the horizontal and vertical striations from physical grounds. Thus assuming the assignment of the eigenstates of the horizontal and vertical striations are fixed, then we are left with N + 1 − 2 = N − 1 striations, and hence N N−1 diﬀerent configuration of line eigenstates. Each configuration is often referred to in the literature as a quantum net. These diﬀerent quantum nets will therefore lead in general to diﬀerent discrete Wigner distribution functions. However, although in general diﬀerent, the sum of the Wigner function, W (q, p), over all points on any line in phase space is invariant and represent the probability of the measurement outcome associated with that line. Fixing the quantum net amounts to a specific

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assignment of the vacuum state in each of the striations to obtain a specific Wigner function representation. The freedom to specify the field basis for each of the horizontal and vertical axes is not crucial if one use the binary-string representation, in cardinal order, for each of the two axes. Fixing the field basis for the horizontal axis, using the power of primitive roots of the irreducible polynomial, also fixes the dual field basis for the vertical axis. In constructing the striations in phase space, it greatly simplifies the task by using the same levels for the points of the vertical axis as that of the horizontal axis (using the elements of GF(2n )). Only when calculating the commuting translation operators between points along a line, and transformation operators between lines in a striation does one have to use the binary-string representation of the vertical axis to reflect the use of its proper dual field basis. 41.7

Potential Applications

The virtue of discrete phase space over finite fields is that it reveals some of the fundamental aspects of quantum mechanics. Unitary operation of each qubit can be systematically carried out in this formalism, and discrete Wigner function provides a new tool for picturing n-qubit computations, as a change in the ‘Wigner register ’. Moreover, promising research is underway in the phase space representation, as a natural representation, of error-correction code in quantum computing. Its potential use in analyzing quantum information processes, such as in Grover’s search algorithm and teleportation seems promising. It may be that the inherent freedom to choose the Wigner function representation in the formalism becomes a virtue when appropriately fixing to a preferred choice in order to bring ease in computation, similar to the benefits of gauge fixing in the quantum-field theory of electromagnetic fields.

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PART 7

Phenomenological Superoperator of Open Quantum Systems: Generalized Measurements

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Chapter 42

Interference and Measurement

We give here a simple demonstration that repeated Fourier transform of single qubit enhances one state and cancels or destructively superpose the other, an interference eﬀect. This repeated quantum Fourier transform, for example can be implemented for photons with two polarization states using a series of two beam splitters (‘spectral analyzer’). In the language of Wannier states (q-states) and Bloch states (p-states), assume that the original state of the photon is given by ˆ (Z) ˆ or of the Pauli matrix |p = 0 ( |q = 0 ), an eigenfunction of the operator X σx (σ z ). Then after passing through the first beam splitter the state becomes a superposition of ‘spectral’ q states, 1

2πi 1 1 |p = 0 = √ e | p.q |q = √ (|q = 0 + |q = 1 ) , 2 q=0 2

where the two states, |q = 0 and |q = 1 , comes out as separate beams from the beam splitter. In matrix form this is given by 1 =√ 2

1 0

1 1 1 −1

1 1

,

or by taking the inverse, to write the output state in terms of the input state 1 √ 2

1 1

out

1 =√ 2

1 1 1 −1

1 0

. in

In the second alternative case, a superposition of p states, 1

−2πi 1 1 e | p.q |p = √ (|p = 0 + |p = 1 ) , |q = 0 = √ 2 q=0 2

which in matrix form is given by 1 0

1 =√ 2

1 1 1 −1 569

1 1

,

(42.1)

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or by taking the inverse to write the output state in terms of the input state 1 √ 2

1 1

out

1 =√ 2

1 1 1 −1

1 0

.

(42.2)

in

Observe the identical matrix formulations of the two cases, where the 2×2 matrix is the well-known Hadamard matrix. This is the reason that in the literature these two cases are often not diﬀerentiated. However, without any caveat, this is very confusing to the uninitiated since the q-basis states form a complete set, as well as the p-basis states. Therefore any q-basis state can only be ‘spectrally’ decompose into an equal superposition of the p-basis states and vice versa, as implemented by the beam splitter. Moreover, it is worth pointing out that the initial prepared states for the two cases are eigenfunction of two diﬀerent operators, namely, the Zˆ and ˆ operators, which do not commute. Also in both cases the states after the first X Fourier transform, implemented by the beam splitter serving the function of the Hadamard matrix, is an equal superposition of pertinent states from corresponding mutually-unbiased bases, 1 0

1 ⇒√ 2

1 1

,

1 |p = 0 ⇒ √ (|q = 0 + |q = 1 ) , 2 or 1 |q = 0 ⇒ √ (|p = 0 + |p = 1 ) . 2 After the second beam splitter, we have the splitting of states 1 1 √ (|q = 0 + |q = 1 ) ⇒ {(|p = 0 + |p = 1 ) + (|p = 0 − |p = 1 )} 2 2 = |p = 0 . In the second alternative case, 1 1 √ (|p = 0 + |p = 1 ) ⇒ {(|q = 0 + |q = 1 ) + (|q = 0 − |q = 1 )} 2 2 = |q = 0 . What these all means is that there will always be a count in detector for state |0 and never in detector for state |1 because the relative phases of the state |1 that arrives in detector ‘1’ destructively interfere. In terms of matrix equation, both cases is covered by applying the Hadamard matrix to both output of Eqs. (42.1) and (42.2) 1 0

out

1 =√ 2

1 1 1 −1

1 √ 2

1 1

. in

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Note that what comes out of the two beam splitters is exactly the same as the initial 1 prepared state , this is of course predicted since the whole process is described 0 by the following matrix equation 1 0

1 √ 2

= out

1 √ 2

1 1 1 −1

1 1 1 −1

1 0

, in

where the Hadamard matrix is its own inverse, and hence the cascaded two Hadamard transformation of the two beam splitters yields identity. Thus, for example, the two separate beams, |q = 0 and |q = 1 , from the first beam splitters enhances the |p = 0 state after both beams pass through the second beam splitter, i.e., the |p = 0 beams interfere constructively at detector ‘0’, while the |p = 1 beams interfere destructively at detector ‘1’.

42.1

Projective Measurements

The familiar projective or orthogonal measurement of von Neumann is described by ˆ a Hermitian operator on the Hilbert space of a closed an observable operator M, system being observed. In particular, we have ˆ = M m

|m m m| =

mPˆm = m

† ˆ m Pˆm Pm , m

ˆ with eigenvalue m, where Pˆm = |m m| is the projector onto the eigenspace of M ˆ |m = m |m . M Upon measuring the state |ψ , we have ˆ |ψ = ψ| M

2

m

m | ψ| |m | .

Thus the probability of getting a result m is given by † ˆ p (m) = | ψ| |m |2 = ψ| Pˆm |ψ = ψ| Pˆm Pm |ψ .

Given that the measurement result is m, the quantum system immediately after measurement is |Φ ⇒

Pˆm |ψ

,

ψ| Pˆm |ψ

so that |Φ is a normalized state, Φ |Φ =

† ˆ ψ| Pˆm |ψ Pm |ψ ψ| Pˆm = = 1, † ˆ ˆ ψ| Pˆm |ψ ψ| Pm Pm |ψ

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and |Φ Φ| = =

† Pˆm |ψ ψ| Pˆm ψ| Pˆm |ψ

† ˆ † ˆ Pm |ψ ψ| Pˆm Pm Pˆm † ˆ ψ| Pˆm Pm |ψ † ˆ † ˆ Pˆm Pm |ψ ψ| Pˆm Pm

=

† ˆ † ˆ Pm |ψ ψ| Pˆm Pm Pˆm

Tr

Pˆm =

Observe that for orthogonal projectors Pˆm ,

.

† ˆ † ˆ Pˆm Pm = 1, and Pˆm Pm =

m

m

† ˆ Pm is clearly a positive operator. We can therefore regard δ mm Pˆm . Moreover, Pˆm the whole measurement process as that of replacing/mapping,

ρ ˆS → =

pm m f ρ ˆef S ,

† Pˆm ρ ˆS Pˆm = pm

† Pˆm ρ ˆS Pˆm ,

(42.3)

m

(42.4)

† = i Pˆm = ˆ 1S . where by construction m Pˆm Pˆm † ˆ We shall see that, except for the orthogonality condition, Pˆm Pm = δ mm Pˆm , which can be considered only as special case, all these properties can be generalized to measurement operators of subsystem alone where the focus is on the nonnegative † ˆ † ˆ ˆm operator of the form, Pˆm Pm (orthogonal, von Neumann) to ⇒ M Mm (nonorthogonal for the subsystem alone). This generalization is the so-called positive valued measurement (POVM), discussed in more detail below. We shall see that the generalization is described by the following equations

ˆ† M ˆ p (m) = ψ| M m m |ψ , |Φ ⇒

ˆ m |ψ M

,

† ˆ ˆm ψ| M Mm |ψ † ˆ ˆm M Mm = I,

m

p (m) = 1 = m

m

† ˆ ˆm ψ| M Mm |ψ .

The eﬀect of the whole process on the reduced density matrix of the system is to take † ˆ mρ ˆm M ˆS M † ˆ mˆ ˆm ˆρS → pm = M ρS M . (42.5) pm m m

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Interference and Measurement

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The process of generalized measurement is therefore equivalent to a quantum operˆ m } are called the Kraus ation in which the generalized measurement operators {M operators. Note this is more general than the case of a von Neumann measurement, ˆ m need not be projectors. because the operators M 42.1.1

Eﬀects of Measurements

Now what would happen if a projective measurement is performed between the two beam splitters, i.e., we measured which path the photon took after the first beam splitter? Then we would have determined the output of the first beam to be either |0 or |1 with equal probability, in either case no longer a superposition. After the second beam splitter, we either have 1 |0 ⇒ √ (|0 + |1 ) 2 or 1 |1 ⇒ √ (|0 − |1 ) , 2 where it is to be understood that the basis states in both sides of the arrow belongs to diﬀerent and mutually-unbiased bases. In any case, there will be a photon count in detector for state |0 and in detector for state |1 . So the state measurement performed between the two beam splitters drastically changes the outcome after the second beam splitter, hence the measurement destroy the enhancing interference of state |0 . Since most quantum information processing rely on constructive interference eﬀects to work, measurement before complete processing can destroy their eﬀectiveness. Indeed, interaction with environment also eﬀectively ‘measure’ the quantum system, an eﬀect called decoherence. 42.1.2

Eﬀects of Measurements on Entanglement

For composite system, we can ask the question on the eﬀect of measurement on entanglement. Consider the maximally-entangled Bell basis state for two qubits 1 |Ψ+ = √ (|0 ⊗ |0 + |1 ⊗ |1 ) 2 1 = √ (|q = 0 ⊗ |q = 0 + |q = 1 ⊗ |q = 1 ) . 2 If we measure the first qubit in the Zˆ eigenstates or q-basis, |0 or |1 , we get either state with equal probability; after the measurement the two-qubit system is left in the product state, either |0 |0 or |1 |1 , i.e., no longer entangled. The same result is the same if we measure the second qubit in the Zˆ eigenstates. We can also ˆ eigenstates by first expanding |Ψ+ in terms measure each qubit in terms of the X

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ˆ We have of the complete eigenstates of X. 1 |Ψ+ = √ {(|0 + |1 ) ⊗ (|0 + |1 ) + (|0 − |1 ) ⊗ (|0 − |1 )} 2 2 1 = {|p = 0 ⊗ |p = 0 + |p = 1 ⊗ |p = 1 } . 2 Clearly, whatever the outcome, the final state will also be a product state. What we have done is referred to as complete local measurement (a complete measurement of one of the subsystems). Thus, a complete local measurement of entangled system will result in the system being left unentangled, therefore in addition to destroying interference, measurement can destroy entanglement. However, projective measurement of composite system such as a two-qubit system can also create entanglement. The fundamental reason for this is that the states of composite system can be described by the set of entangled basis states which also form a complete set, such as the Bell basis states for two-qubit system. Projective measurement using the entangled basis states is often called joint measurements, as opposed to complete local measurements (nondegenerate projective measurements) described above. Thus given any product state of two-qubit system, after a joint projective measurement on the Bell basis the system will be left in one of the four Bell basis states, namely, 1 |Φ+ = √ (|0 2 1 |Φ− = √ (|0 2 1 |Ψ+ = √ (|0 2 1 |Ψ− = √ (|0 2 42.1.3

|0 + |1 |1 ) , |0 − |1 |1 ) , |1 + |1 |0 ) , |1 − |1 |0 ) .

Measurements in Quantum Teleportation

We can see both these eﬀects, complete local measurement and entanglement measurement in quantum teleportation, where Alice and Bob initially shared an entangled Bell states, one qubit belongs to Alice and the other qubit belongs to Bob. In addition, Alice also have an unknown state to send to Bob. Then Alice makes a projective measurement on Bell basis, thus entangling her unknown qubit with the qubit that she shared as an entangled system with Bob’s qubit. However, with respect to Alice and Bob original system, Alice is making a complete local measurement thus unentangling her qubit with Bob’s qubit, but entangling her qubit with her unknown qubit. This is the reason why Alice still has to send, through classical communication channel, the result of her measurement so that Bob can act accordingly in order to receive the unknown qubit state. Quantum teleportation will be discussed further in Part 8.

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Chapter 43

Quantum Operations on Density Operators

Here, we present a phenomenological treatment of some aspects of the general formulation of quantum physics of open systems given in Part 4. In what follows, we refer to quantum operation as a broad class of transformations that a quantum mechanical system can undergo. This formalism describes not only time evolution or symmetry transformations of isolated systems, but also transient interactions with an environment for purposes of measurement. This description is formulated in terms of the density operator description of a quantum mechanical system. A ˆ from density operators to other density quantum operation is defined as a map, M, operators satisfying the following conditions. (i) Preserve the normalization of the state: ˆ (ˆρS ) = Tr TrM

† ˆ mρ ˆm M ˆS M = 1 if Trˆ ρS = 1.

(43.1)

m

(ii) Linearity:

ˆ M

pi ρ ˆi

=

i

i

ˆ (ˆρi ) pi M ˆ m( M

= m

(43.2)

† ˆm pi ˆ ρi )M = i

† ˆ mρ ˆm M ˆi M .

pi i

(43.3)

m

(iii) Complete positivity: for any possible environment E and any possible joint density matrix ρ ˆ of the system and environment, the result of the composite operaˆ (ˆρ) is another positive operator or density operator. (This requirement tion (I ⊗ M) ˆ ρS ) be positive for any includes, but is more general than, the requirement that M(ˆ system density matrix ˆρS .) f ψ|S ρef S |ψ

S

= µ

ˆ µ† |ψ ˆ µ ρS M ψ|S M 575

S

≥ 0.

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The Kraus Representation Theorem

Any quantum operation satisfying the conditions given above can be expressed in the form ˆ ρ) = M(ˆ

ˆk ρ ˆ†, E ˆE k

(43.4)

k

with Ek† Ek = ˆ 1.

(43.5)

k

The formula (43.4) is known as the Kraus representation or operator-sum represenˆk } known as the Kraus operators. tation of the quantum operation; the operators {E 43.2 43.2.1

Examples of Quantum Operations Unitary Evolution

Unitary evolution of the system by itself trivially has the form of a quantum operation: ˆS ρ ˆ†, ˆS U ρ ˆS → U S

(43.6)

ˆ†U ˆ ˆ U S S = 1S .

(43.7)

with

43.2.2

Probabilistic Unitary Evolution

Suppose our system remains isolated, but its Hamiltonian is uncertain because of some (classical) random process. The result is that diﬀerent Hamiltonians may be applied with probabilities pi ; the resulting evolution is ρ ˆ→

ˆSi ˆ ˆ† , pi U ρS U Si

(43.8)

i

ˆSi is the unitary evolution associated with Hamiltonian by virtue of linearity, where U √ ˆ i. This has the form of a quantum operation with Kraus operators pi U Si . 43.2.3

Von Neumann Measurements

Suppose we make a projective (von Neumann) measurement on our system. If ˆ = ˆ the operator we measure is O m om |m m| ≡ m om Pm , then according to the standard von Neumann measurement postulate, result om is measured with ρS |m = TrS [Pˆm ρ ˆS ]. In this event the state of the system is probability pm = m|ˆ † ˆ ˆ replaced by Pm ρ ˆS Pm /pm .

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We can therefore regard the whole measurement process as that of replacing ρS →

pm m

† Pˆm ρ ˆS Pˆm = pm

† Pˆm ρ ˆS Pˆm ,

(43.9)

m

† = i Pˆm = ˆ 1S . The von Neumann measurement where by construction m Pˆm Pˆm is therefore a special case of a quantum operation in which the Kraus operators are the projection operators Pˆm . This leads us to defined a more general measurement, the so-called positive operator valued measure (POVM)

43.2.4

POVMs

As an introduction to generalized measurements, we give a brief discussion of the generalization of the von Neumann measurement. Allow the system and environˆ which simultaneously applies ment to interact by applying a unitary operator U ˆ the operators Mm to the system, and takes the environment from the fixed starting state |e0 to some one particular environment state, say |em . Then ˆ |ψ |e0 = U

m

ˆ m |ψ |em . M

(43.10)

Thus, the system and the environment are correlated. Normalization of this new state requires ˆ †U ˆ |ψ |e0 = e0 | ψ|U

m

† ˆ ˆm ψ|M Mm |ψ = 1.

(43.11)

If this is true for any ψ, we conclude that † ˆ ˆm M Mm = ˆ 1S . m † ˆm ˆ mM are often said to form a positive operator-valued meaThe operators M sure. With the system and the environment are correlated in this way, we now measure the state of the environment (rather than of the system), using the operator

ˆ = IˆS ⊗ O

m

om |em em | ≡

om Pˆm . m

The probability of outcome m is ˆ † (IˆS ⊗ |em em |)U ˆ |ψ |e0 pm = e0 | ψ|U ˆ † (IˆS ⊗ |em em |)M ˆ m |ψ |em = em | ψ|M m mm

= m

† ˆ ˆm ψ|M Mm |ψ ,

(43.12)

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and in this event the state of the whole system is ˆ m |ψ |em ˆ |ψ |e0 M Pˆm U = . √ √ pm pm

(43.13)

The eﬀect of the whole process on the reduced density matrix of the system is to take † ˆ mρ ˆm M ˆS M † ˆ mˆ ˆm ˆρS → M pm = ρS M . (43.14) p m m m The process of generalized measurement is therefore equivalent to a quantum operation in which the Kraus operators are the generalized measurement operators ˆ m }. Note this is more general than the case of a von Neumann measurement, {M ˆ m need not be projectors. because the operators M

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Chapter 44

Generalized Measurements

There is a need to generalize the orthogonal measurement considered by von Neumann which is most often treated in most introductory books in quantum mechanics. The advantage of generalized concept of measurement is that many important practical measurements are not projective measurements. Clearly the eﬀects of the measurement devices on the observed system renders the system to be no longer closed and thus not necessarily evolving according to a unitary evolution. Let us now generalize the measurement concept beyond orthogonal measurements considered by von Neumann. The generalized measurement concept arise from the need to make measurements on the subspace of a larger Hilbert space. Then, an orthogonal measurement in the Hilbert space of a closed system described by H, cannot in general be described as an orthogonal measurement in subsystem A alone. Suppose our system of interest is described by the Hamiltonian HA , which is part of a larger system which is a tensor product, H = HA ⊗ HB . For the product ˆ acting on the Hilbert space of H, the partial trace of Aˆ ⊗ B ˆ over operator, Aˆ ⊗ B, space of HB is ˆ = T rH B ˆ Aˆ T rB Aˆ ⊗ B B = operator in HA . Similarly ˆ = T rH Aˆ B ˆ T rA Aˆ ⊗ B A = operator in HB . In terms of the bases {|i } on HA and {|j } on HB , we may represent general ˆ on H = HA ⊗ HB as operators O ˆ= O ii ,jj

O(ij)(i j ) |i i | ⊗ |j j | . 579

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Then ˆ = T rB O ii ,jj l

=

O(ij)(i j ) |i i | ⊗ l |j j | l

O(ij)(i j) |i i | ,

ii ,j

and similarly, ˆ = T rA O kii ,jj l

=

O(ij)(i j ) k |i i | k ⊗ |j j |

O(ij)(ij ) |j j | .

i,jj

Now the joint state of H = HA ⊗ HB can be expressed as |Ψ = |φ

A

= i,j

⊗ |θ

B

αi |i ⊗ β j |j .

ˆ in H is The expectation value of O

ˆ |Ψ Ψ| O = Ψ |Ψ

 

k,l



α∗k k| ⊗ β ∗l l| 

O(ij)(i =

j )

ii ,jj



k,l



k,l



O(ij)(i ii ,jj

α∗k k| ⊗

β ∗l

O(ij)(i j ii ,jj

)

j |

|i i | ⊗ |j



l| 

k ,l



k ,l

α∗i αi

⊗

β ∗j β j



αk |k ⊗ β l |l  

αk |k ⊗ β l |l  

α∗k k| |i ⊗ β ∗l l| |j  ⊗  k,l

=

j )



k ,l

(α∗k αk ⊗ β ∗l β l )



αk i | |k ⊗ β l j | |l 

.

Therefore, upon taking the trace in the second indices belonging to HB , we obtained ˆ |Ψ = T rB Ψ| O

ii ,j

= ii ,j

= ii

O(ij)(i j) α∗i αi ⊗ β ∗j β j O(ij)(i j) α∗i αi ⊗ β j  

βj j

2



2

O(ij)(i j)  (α∗i αi ) .

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ˆ The last line is the same as taking the expectation value in HA of an operator E defined by   

ˆ= E

ii

where

2

βj

j

O(ij)(i j)  |i i | ,

βj

Eii =

2

O(ij)(i j)

j

and ˆ |φ ˆ |Ψ = φ| E T rB Ψ| O A

A.

Suppose we have measured HB so that the summation over the index j collapses to only one value1 , βj

Eii = j

2

O(ij)(i j) ⇒ β j

2

O(ij)(i j) ,

and therefore ˆj = E

βj

2

ii

O(ij)(i j) |i i | .

(44.1)

ˆ is a positive operator, we may thus write, after summing over i Assuming that O and i ˆ⇒E ˆj = M ˆ †M ˆ j |φ E j

A

φ|A .

Then, after measurement in HB , we have ˆ |Ψ = φ| M ˆ †M ˆ j |φ T rB Ψ| O j A ˆj |φ . = φ| E A

A

A

ˆ is a local joint measurement, namely, If we assume that the operator O ˆ = I ⊗ |j j| , O ˆj |φ then the probabilities p (j) = φ|A E and also j

φ|A Eˆj |φ

A

A

(44.2)

are indeed nonnegative for every |φ

=

A

p (j) = 1, j

1 We could have assumed from the beginning that the system B is a pure state, which is not a real approximation since one can always introduce an additional ‘far environment’ (‘far’ from the system A), F , with an orthonormal set of at least the same number of states states {|fi }. Forming an entangled state with system B yields one of the basis states of the combined B + F system, and hence a pure state. We will come to this in more detail in a later section.

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since the total probabilities must sum up to one. Therefore, we also have the resolution of identity, Eˆj = I. j

ˆj But note that in contrast with projective measurements, |j j|, the operators E need not be projectors since it is now operating in the Hilbert space of the subsystem A. Since Eq. (44.2) can be considered an orthogonal projector operating on the joint system, we have seen that viewed from the subsystem A this appears to be the ˆj . This generalization is called Positive Operator Valued Measurement operator E (POVM), which includes projective measurements as a special case. What we have done is a simple demonstration of the existence of generalized measurement for a subsystem. The trace out system B is often referred to as the ancilla. In quantum information and quantum computing, most actual measurement are done indirectly through an ancilla. Oftentimes, we have the following situation: (1) An extra system B (an ancilla) in a known initial state is prepared, (2) Perform a unitary transformation, such as a controlled unitary transformation, which leave the joint system A and B entangled, (3) Measure the ancilla B and discard it. The joint state of the system A plus B is |Ψ = |ψ

A |0 B

.

ˆ , is performed on |Ψ Then a joint unitary transformation, U ˆ |Ψ = U ˆ |ψ U

A |0 B

.

The ancilla subsystem B is then measured. The measurement is given by some set of orthogonal projectors Pˆj = Pˆj† = Iˆ ⊗ φj B φj B , Pˆj Pˆk = δ jk Pˆj , Pˆj = Iˆ j

where j labels the possible results. We have assume a complete local measurement ˆ j = φj where Q φj B are all nondegenerate or one-dimensional, where the set B φj B are the orthonormal basis on the ancilla subsystem B. Therefore after the measurement the state of A plus B is left unentangled. The probability of the outcome labeled by j denoted as pj is given by ˆ † Pˆj U ˆ |ψ pj = ψ|A 0|B U

A |0 B

,

(44.3)

ˆ † Pˆj U ˆ is still a set of orthogonal projectors by virtue of the where the set U ˆ as a joint measurement of the system ˆ . Thus, we may view U ˆ † Pˆj U unitarity of U A and ancilla B. We now demonstrate that this can also be viewed as a generalize measurement on subsystem A alone. This means that we can find a set of positive ˆj on the space of the system alone, i.e., operators E ˆj |ψ pj = ψ|A E

ˆj = I. ˆ E

A, j

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ˆ=U ˆ † Pˆj U ˆ from Eq. (44.3). Again, the We can easily find Eˆj from Eq. (44.1) with O ˆ operators Ej must be positive because pj is non-negative for every |ψ A , moreover, ˆj = I; ˆ however E ˆj E ˆk = δ jk E ˆj . pj = 1 ⇒ E j

44.1

j

Distinguishing Quantum States

The POVM concept is very useful in quantum computation and information. Since POVM is more general than projective measurements, often this concept gives us a guide for our intuition to do some tricks in quantum information and quantum computation. One of the problems which finds POVM to be of great help is in the problem of distinguishing nonorthogonal quantum states, |ψ and |φ , where ψ |φ = 0. ˆ 0 = |ψ ψ| and With projective measurement, the best we can do is to use M ˆ ˆ ˆ 0. ˆ M1 = I − M0 , i.e., M1 is a positive square root of non-negative operator I − M ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ From these we get E0 = M0 and E1 = I − M0 , where E0 + E1 = I. Since ˆ1 |ψ = 0, we can conclude that if the result of measurement is p1 then the ψ| E state cannot be in state |ψ . Let us decompose |φ = a |ψ o + b |ψ ⊥ , i.e., into component parallel to |ψ and a component perpendicular to |ψ . Now if the result ˆ0 |ψ = ψ| |ψ ψ| |ψ = 1 if the state |ψ is is p0 , this can arise either as ψ| E prepared before the measurement, or as φ| |ψ ψ| |φ = |a|2 < 1 if the state |φ is prepared, i.e., there is a nonzero probability of getting the result p0 when |φ is prepared. Therefore there will be an error in identifying which state, |ψ or |φ , was prepared before the measurement if the outcome is p0 . A more rigorous proof that non-orthogonal states cannot be reliably distinguished by projective measurements can be demonstrated by contradiction. Deˆφ = |φ φ|, and E ˆψ = |ψ ψ|. If the measurement is possible, then we fine E ˆα = I, ˆψ |ψ = 1. Since ˆ it follows ˆ E must have φ| Eφ |φ = 1, and ψ| E α

that α

ˆα |ψ = 1. Since ψ| E ˆψ |ψ = 1, then φ| E ˆψ |φ = 0, which means ψ| E

ˆψ |φ = 0. Writing |φ = a ψ E o and |φ are non-orthogonal, then

+ b |ψ ⊥ , with |a|2 + |b|2 = 1, |b|2 < 1 since |ψ

ˆφ |φ = b E ˆφ |ψ E

⊥

ˆφ |φ = |b|2 ψ⊥ | Eˆφ |ψ⊥ ⇒ φ| E ˆα |ψ = |b|2 < 1. ≤ |b|2 ψ| E α

The last inequality contradicts the assumption that a measurement is possible, ˆφ |φ = 1, to be able to distinguish between two non-orthogonal states, since φ| E ˆφ |φ < 1. This is where the concept of the last inequality yields contradiction, φ| E POVM becomes indispensable.

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Utility of POVM

With the use of POVM, it is possible to perform measurement which distinguishes between non-orthogonal states some of the time, but never makes an error of misidentification. Consider the following example. Let |ψ = |0 and |φ = √12 (|0 + |1 ). Take ˆ2 , E ˆ3 , where the POVM containing the following three elements, P OV M = Eˆ1 , E √ 2 √ |1 1| , 1+ 2 √ 2 (|0 − |1 ) ( 0| − 1|) ˆ √ E2 = , 2 1+ 2

ˆ1 = E

ˆ3 = Iˆ − E ˆ1 − E ˆ2 , E ˆ3 is a positive operator. Now where the coeﬃcient is arbitrarily chosen such that E ˆ1 and E ˆ2 are cleverly suppose the state is prepared in state |ψ = |0 . Note that E ˆ ˆ chosen such that ψ| E1 |ψ = 0 and φ| E2 |φ = 0. We have ˆ2 |φ = φ| E

√ (|0 − |1 ) ( 0| − 1|) 2 √ φ| |φ 2 1+ 2   φ| |0 2 0| |φ       + φ| |1 1| |φ

√ 2 2 √ = |0 1| 1+ 2   − φ| 2 |φ      − φ| |1 2 0| |φ

ˆ2 |φ = φ| E

√ 1+

              

,

  1 ( 0|) |0 0| (|0 )   4          1   + 4 ( 1|) |1 1| (|1 ) 

2 √ 2  − 14 ( 0|) |0 1| (|1      1 − 4 ( 1|) |1 0| (|0

 )      )

= 0.

Hence, if the measurement outcome is p1 then it is certain that the prepared state is |φ . Similarly if the measurement outcome is p2 then it is certain that the prepared state is |ψ . However, if the outcome of the measurement is p3 then one can not distinguish the state that was prepared. The important observation is that one never make a mistake identifying the prepared state, but this comes at the cost of sometimes having no information at all about the identity of the state. For the non-orthogonal quantum state |ψ and |φ , we can also form the following

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simpler elements of POVM, ˆ1 = a Iˆ − |ψ ψ| , E ˆ2 = a Iˆ − |φ φ| , E ˆ1 − E ˆ2 . ˆ3 = Iˆ − E E

ˆ1 |ψ = 0 and where 12 ≤ a ≤ 1, and we still have cleverly chosen such that ψ| E ˆ2 |φ = 0. Then the same conclusion as above will result. φ| E

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Chapter 45

Phenomenological Density Matrix Evolution

We have seen that an orthogonal measurement of an isolated system consisting of two subsystems, A and B, may be viewed as a (generally nonorthogonal) POVM on A alone. We now consider the unitary evolution of an isolated system consisting of two subsystems, A and B, if we described this evolution on A alone. Let the bipartite system be in a product state of the form ρA ⊗ |0

B

0|B ,

indicating that the system A has density matrix ρA and the system B is assumed to be in pure state, simply designated as |0 B . This is not technically a real approximation, by virtue of the following ‘purification’ argument. Suppose we had an environmental density operator corresponding to the mixed state ρ ˆB = i

pi |ψ i ψi |,

(45.1)

where the N states {ψi } are not necessarily orthogonal but are normalized, and i pi = 1. Then we can always introduce an additional ‘far environment’ (‘far’ from the system), F , with an orthonormal set of at least N states {|fi }. The following entangled state is one of the basis states of the combined B + F system, and hence a pure state , |Ψ =

i

√ pi |ψi |fi ,

(45.2)

It has the property that its reduced density matrix in the original environment B is √ TrF [|Ψ Ψ|] = pi pj |ψi ψj |TrF [|fi fj |] = pi |ψi ψi | = ρ ˆB , (45.3) ij

i

and it is therefore indistinguishable (as far as any measurement within B only is concerned) from the original density matrix ˆ ρB . We can consider this as a ‘purification’ of the environment ρ ˆB . Thus, we will suppose this has been done, and the original environment B replaced by a new, bigger, environment (which we will still, however label as B) in a pure state. The system evolves in time according to the 586

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587

ˆAB unitary time evolution operator U ˆAB ρ ⊗ |0 U A

ˆ† . 0|B U AB

B

Now we are interested in describing the evolution of the subsystem A alone. To achieve this, we have to perform the partial trace over HB to find the eﬀective density matrix of A, which we designate as ρeff A f ˆ ρef A = T rB UAB ρA ⊗ |0

B

µ

ˆAB ρA ⊗ |0 µ|B U

µ

ˆAB |0 µ|B U

= =

B

ˆ† 0|B U AB B

ˆ † |µ 0|B U AB

ˆ † |µ ρA 0|B U AB

B

B

ˆ µ ρA M ˆ µ† , M

=

(45.4)

µ

ˆ µ = µ| U ˆ where the set {|µ B } forms the orthonormal basis for HB and M B AB |0 B ˆAB that the operator is an operator acting on HA . It follows from the unitarity of U ˆ µ satisfy the relation M ˆ µM ˆ µ† = M µ

B

ˆAB |µ 0|B U

B

µ

ˆAB |0 µ|B U

B

ˆAB |0 µ|B U

B

µ

ˆAB |µ 0|B U

=

ˆ µ† M ˆ µ = IˆA . M

= µ

f Equation (45.4) is exactly the Kraus representation of the map of ρA to ρef A . A quantum operation, also called superoperator, can be regarded as a linear map that f takes density operators to density operators. The ρef A has the following properties: f is Hermitian: (1) ρef A f† ρef = A

ˆ µ ρ† M ˆ† M A µ = µ

ˆ µ ρA M ˆ µ† = ρef f , M A µ

f (2) ρef has a unit trace: A f T rρef A =

ˆ µ† M ˆ µ = T r ρA T r ρA M µ

ˆ µ† M ˆµ M

= T rρA = 1,

µ

f is positive: (3) ρef A f ψ|A ρef A |ψ

A

= µ

ˆ µ† |ψ ˆ µ ρA M ψ|A M

A

≥ 0.

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f Thus, ρef is a density matrix operator if ρA is a density matrix operator. It A Kraus representation, Eq. (45.4) is not unique since it depends on the basis chosen

for HB . If one change the basis to f ρef A = ν

ˆAB |0 ν|B U µ

= ν

µ

Tνµ µ|B

B

ˆAB |ν ρA 0|B U



ˆ µ ρA  Tνµ M

B

B

ˆAB ρA 0|B U

µ



ˆν ρA N ˆν† . N

=

then

µ

ˆAB |0 Tνµ µ|B U

= ν

ν|B =

µ

|µ

B

∗ Tµν

ˆ † Tµ∗ ν  M µ

ν

Superoperators provides a formalism for discussing the general theory of decoherence, i.e., the evolution of pure states to mixed states. When there is only one term in the resulting operator sum, Eq. (45.4), then ρA undergoes a unitary evolution which is a special case. If there are two or more terms in the operator sum, then there are pure initial states of HA that become entangled with HB under the ˆAB , evolution governed by U |ϕ

A

f ϕ|A ⇒ ρef (mixed final state). A

It should be mentioned that unitary evolution operators form a group, but superoperators define a dynamical semigroup1 . 45.1

Quantum Channels

We may refer to the general evolution, Eq. (45.4), of the density matrix of subsystem A as a process describing the fate of a qubit (or information) that is transmitted, in 1 A mathematical ob ject defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an identity element and its elements need not have inverses within the semigroup [taken from WolframMathworld]. Semigroups describing the time evolution of open quantum systems in finite dimensional spaces have generators of a special form, known as Lindblad generators: Every generator of a semigroup of completely positive trace preserving maps Tt : ρ(s) → ρ(s + t) for t ≥ 0 on the set of finite dimensional density matrices ρ, can be written in the form

ρ˙ = D(ρ) = −i[H, ρ] +

Dhα (ρ) α

where, Dh (ρ) = hρh† −

1 (h†hρ + ρh†h) 2

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communication theory, with some loss of fidelity from the transmitter to the receiver, referred to as quantum channel. Quantum channels are completely positive, trace preserving (CPTP) maps between spaces of operators. 45.1.1

Time Evolution

For a purely quantum system, the time evolution, up to certain time t, is given by ˆ ρU † , ρ⇒U i

ˆ = e− | Ht , where H is the Hamiltonian at time t. Clearly this give a CPTP where U map in the Schrödinger picture and is therefore a channel. The dual map in the Heisenberg picture is ˆ ˆ † AU. Aˆ ⇒ U 45.1.2

Partial Trace

Consider a composite quantum system with state space, HA ⊗HB , with the density matrix operator ρ ∈ HA ⊗HB . The reduced density matrix on system A is obtained by taking the partial trace of ρ with respect to B system as ρA = T rB ρ. The partial trace operation is a CPTP map, therefore a quantum channel in the Schrödinger picture. In the Heisenberg picture, the dual map of this channel is ˆ AB ⇒ O ˆA ⊗ IB , O ˆA is an observable of system A. where O 45.2

Depolarizing Channel

The depolarizing channel is an instructive model of decohering qubit. If the set {|0 , |1 } is an orthonormal basis for a qubit, the three types of errors are characterized as follows, 01 , 10 1 0 , (2) Phase flip error: |ψ ⇒ σ z |ψ , where σz = 0 −1 0 −i (3) Both errors2 occur: |ψ ⇒ σ y |ψ , where σ y = i 0 (1) Bit flip error: |ψ ⇒ σx |ψ , where σx =

.

2 In discrete quantum mechanics, the generalized Pauli matrix is related to the translation and phase operators: i Xq |q = exp − q · P |q = q + q |

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If an error occurs, where the three types of errors can occur with equal likelihood, then |ψ evolves to an ensemble of the three states σ x |ψ , σ y |ψ , and σz |ψ with equal probability. 45.2.1

Unitary Representation of the Channel

ˆAE , acting on The depolarizing channel can be represented by a unitary operator, U HA ⊗HE , where HE , the environment space has dimension 4 to be able to record ˆAE should acts on HA ⊗HE as the four possible evolution. The unitary operator, U ˆAE : U |ψ A ⊗ |0 E ⇒

1 − p |ψ +

A

p [(σx |ψ 3

⊗ |0 A

E

⊗ |x

E)

+ (σ y |ψ

A

⊗ |y

E ) + (σ z

|ψ

A

⊗ |z

E )] ,

where if we could only measure the environment basis {|µ E , µ = 0, x, y, z}, then we would know what kind of error had occurred. We would then be able to intervene and correct the error in qubit A. 45.2.2

Kraus Representation of the Channel

We have, after the partial trace of the environment, HE , in the {|µ ˆ ˆ µ = µ| U M E AE |0

E

,

so we have ˆ0 = M ˆx = M ˆy = M ˆz = M Zp |q = exp

1 − pIˆA , p σx, 3 p σy , 3 p σz , 3

i p · Q |q = exp |

i p ·q |

The product of Xq and Zp is thus given by i i Xq Zp = exp − q · P exp p ·Q | | i p ·q = exp − | 2 i q ·P −p ·Q × exp − | where the right hand side is related to σ y for two-state systems.

|q

E}

basis,

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which satisfy [note: σ 2µ = IˆA ] the relation

µ

ˆ µ† M ˆ µ = 1 − p + 3 p IˆA M 3 = IˆA .

Thus, the density matrix ρA of the qubit evolves as f ρA ⇒ ρef A = (1 − p) ρA +

p [σx ρA σx + σ y ρA σy + σz ρA σ z ] . 3

(45.5)

The right hand side is a sum over the four (in principle distinguishable) ways that the environment could evolve. The depolarizing channel reduces the radius of the Bloch sphere by a factor 1 − p, while preserving its shape. 45.2.3

Relative-State Representation

ˆA The relative-state method is a method to completely characterize an operator M ˆ A ⊗ IˆB acts on a single pure maximally entangled acting on HA by describing how M state in HA ⊗HB where dim HB ≥ dim HA . This method maybe viewed as realizing an ensemble of pure states in HA by performing measurements in HB on an entangled state in HA ⊗ HB . This means that we can also characterize the channel by describing how a maximally-entangled state of two qubits evolves when the channel acts only on the first qubit. This is what we mean by relative-state representation of the channel. There are four mutually orthogonal entangled states, namely, the Bell complete basis states, φ+

AB

φ−

AB

ψ+

AB

ψ−

AB

1 (|00 2 1 = (|00 2 1 = (|01 2 1 = (|01 2 =

+ |11 ) , − |11 ) , + |10 ) , − |10 ) .

If we assume the initial state is φ+ AB , then the depolarizing channel acts on the first qubit. The initial entangled state thus evolves as φ+

AB

φ+

AB

p φ− AB φ− AB ⇒ (1 − p) φ+ AB φ+ AB + 3 p ψ+ AB ψ+ AB + ψ− AB ψ− AB . + 3

(45.6)

The worst case scenario for the quantum channel has the value of probability p = 34 ,

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which leads to φ+

AB

φ+

AB

⇒

1 + φ 4 +

=

1 4

AB

φ−

φ+

AB

φ−

AB

AB

+ ψ+

ψ+

AB

AB

+ ψ−

AB

ψ−

AB

1ˆ IAB . 4

the identity operator in the last line follows from the completeness of the Bell basis states. Thus, the entangled state becomes the totally random matrix on HA ⊗ HB . This amounts to the channel evolving the initial quantum state to a final state which is a completely random mixture. Therefore, by the relative state method, we find that after measurement in HB , T rB

φ+

AB

φ+

AB

= T rB

1ˆ IAB 4

= T rB

1ˆ IA ⊗ IˆB 4

=

1 2ˆ IA = IˆA , 4 2

hence, irrespective of the initial value of the state in HA , we can conclude that a pure state, |ϕ A ϕ|A of qubit A evolves as |ϕ

A

1 ϕ|A ⇒ IˆA , 2

that is a random matrix in HA irrespective of the initial state |ϕ A . There is another way to express the evolution of maximally-entangled state, Eq. (45.6) by writing φ+

AB

φ+

AB

4 ⇒ 1− p 3

φ+

AB

φ+

AB

4 + p 3

1ˆ IAB , 4

with p ≤ 34 . A useful measure of how well the channel preserves the original quantum information or qubit state, is the entanglement fidelity, Fe . The value of Fe is a measure of how close the final density matrix is to the original maximally entangled state φ+ AB : Fe = φ+

AB

f φ+ ρef A

AB

.

From Eq. (45.6), Fe = 1 − p, an thus entanglement fidelity, Fe is the probability that no error occurred.

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Bloch Sphere Picture

Here we discuss how depolarizing channel reduces the radius of the Bloch sphere by a factor 1 − p, while preserving its shape. We have in the Bloch representation ρ=

1 ˆ I +P ·σ , 2

where P is the polarization of the qubit. Let P = Pz ez which can always be prepared by rotating the axes of the Bloch sphere. Then, from Eq. (45.5), we have f ρA ⇒ ρef A

(45.7)

1 ˆ I + Pz σ z (45.8) 2 p + σx Iˆ + Pz σz σx + σ y Iˆ + Pz σ z σ y + σ z Iˆ + Pz σ z σ z (45.9) 3 2p 1 ˆ p 1 ˆ I + Pz σz + I − Pz σz (45.10) 1−p+ 3 2 3 2 2p 1 ˆ 2p 1 ˆ I + Pz σ z + I − Pz σ z (45.11) 1− 3 2 3 2 4p 1 1 ˆ I + Pz σz − (Pz σz ) (45.12) 2 3 2 1 ˆ 4p (45.13) I + 1− Pz σ z , 2 3

= (1 − p)

= = = =

which shows that contraction of the Bloch sphere in the z-direction. From rotational symmetry, upon considering the full density operator, ρ = 12 Iˆ + P · σ , we see that the Bloch sphere contracts uniformly under the action of the channel. The spin polarization P ⇒ 1 − 43 p P , hence the name depolarizing channel. This corresponds to the polarization that is totally randomized, with no net polarization, with probability, 43 p. 45.2.5

Semigroup Property

The contraction of the Bloch sphere corresponds to positive map of the density operator, resulting in density operator with positive eigenvalues (dissipative map). Thus, decoherence can shrink the Bloch sphere but no physical process can inflate it again, i.e., the superoperator have a semigroup property, which means that the mapping is not invertible. This is a statement of irreversibility due to dissipation. This semigroup property is the characteristics of superoperators.

45.3

Phase Damping Channel

A phase damping channel is a mapping where the on-diagonal terms of ρ remains invariant while the oﬀ-diagonal terms decay.

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Unitary Representation for the Whole System

The whole channel is represented by a unitary transformation √ |0 A |0 E ⇒ 1 − p |0 A |0 E + p |0 A |1 E , √ |1 A |0 E ⇒ 1 − p |1 A |0 E + p |1 A |2 E . Here, unlike the depolarizing channel, the qubit A does not make any transition. It is the environment which scatters oﬀ the qubit, with probability p of being scattered into state |1 E if A is in state |0 A and into state |2 E if A is in state |1 A . Note that the channel picks a stable basis for qubit A, namely, |0 A and |1 A . 45.3.2

Kraus Operators

Upon evaluating the partial trace with respect to Hilbert space of HE in the {|0 E , |1 E , |2 E } basis set, we have the Kraus operators M0 =

1 − pIˆ =

M1 =

√ p

10 00

,

M2 =

√ p

00 01

,

10 01

1−p

,

ˆ Therefore, an initial density matrix operator ρ evolves where M02 + M12 + M22 = I. to ρ ⇒ M0 ρM0† + M1 ρM1† + M2 ρM2† ρ00 0 = (1 − p) ρ + p 0 ρ11 =

(1 − p) ρ01 ρ00 (1 − p) ρ10 ρ11

,

(45.14)

showing that the on-diagonal terms of ρ remains invariant while the oﬀ-diagonal terms decay. Observe that in this case, the mapping by the three Kraus operators above can also be carried out by just two Kraus operator to show phase damping, N0 = N1 =

1√ 0 0 1−p 0 0 √ 0 p

,

,

where N0 N0† + N1 N1† = 1.

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We thus obtained ρ ⇒ N0 ρN0† + N1 ρN1† √ ρ00 1 − pρ01 = √ + 1 − pρ10 (1 − p) ρ11 √ ρ 1 − pρ01 , = √ 00 1 − pρ10 ρ11

0 0 0 pρ11

which gives equivalent phase damping mechanism of the oﬀ-diagonal elements. In Eq. (45.14), let us assume that the probability, p, of a scattering event in time ∆t is p = Γ∆t, where Γ is the scattering rate. Then for t = n∆t is governed by successive mapping consisting of n steps. Therefore the oﬀ-diagonal elements t are decayed by a factor (1 − p)n = (1 − Γ∆t) ∆t . In the limit ∆t ⇒ 0 t

Lim∆t⇒0 (1 − Γ∆t) ∆t ⇒ e−Γt ,

(45.15)

which means that after a long time the density matrix becomes diagonal or classical. 45.4

Amplitude-Damping Channel

Consider an operation which produces a ‘downward’ decay only, from excited state |1 A to the ground state |0 A , with probability p. The environment is, for example, the electromagnetic field assumed initially to be in its vacuum state |0 E (This would be a suitable model for spontaneous emission from an atom, or for a T1 process in spin resonance at very low temperature.) With the unitary evolution that acts on the system A and the environment, E, according to |0

|1

A |0 E A |0 E

⇒ |0 ⇒

A |0 E

,

1 − p |1

A |0 E

+

√ p |0

A |1 E

,

we see that the coherent superposition of system A ground and excited states evolves as √ (a |0 A + b |1 A ) |0 E ⇒ a |0 A + b 1 − p |1 A |0 E + p |0 A |1 E . (45.16) This describes the probability p that the excited state of A decays to the ground state and a photon has been emitted, resulting in the environment transition from |0 E to |1 E . Thus one of the Kraus operators should be M1 = From the requirement that to complete the set would be

k

√ p

01 00

.

(45.17)

Mk Mk† = 1, we see that a suitable Kraus operator

M0 =

1√ 0 0 1−p

.

(45.18)

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The operator M1 induces a quantum jump, i.e., the decay from |1 A to |0 A , and M0 describes how the state evolves if no jump occurs. We have the mapping, ρ ⇒ M0 ρM0† + M1 ρM1† =

√ 1 − pρ01 ρ00 √ 1 − pρ10 (1 − p) ρ11

+

=

√ ρ00 + pρ11 1 − pρ01 √ 1 − pρ10 (1 − p) ρ11

.

pρ11 0 0 0 (45.19)

The eﬀect on the Bloch sphere is to ‘squash’ it towards the North pole into an ellipsoid, so that in the z-direction its height √ is reduced by a factor 1 − p, while its radius in the xy-plane is reduced by a factor 1 − p. Again by applying the channel successively n times, then the probability that t the excited state is still populated for time t is (1 − Γ∆t) ∆t ⇒ e−Γt . Therefore as t ⇒ ∞, (1 − p) ⇒ 0, and ρ⇒

ρ00 + ρ11 0 0 0

.

The system A always ends up in its ground state after a long time. Thus the quantum operation takes the mixed initial state to a pure final state, e.g., ρ00 0 0 ρ11

⇒

ρ00 + ρ11 0 0 0

.

The consideration of ‘upward’ as well as ‘downward’ decay processes generalizes the channel so that it becomes appropriate for an environment at finite temperature. 45.4.1

POVM and Unchanging Environment

In Eq. (45.16), representing the unitary evolution of amplitude-damping channel, let us assume that we monitor the environment with a photon detector. POVM is based on the entanglement between the system and environment. Therefore, if we detect a photon, we are certain that the initial state of the system A was an excited state, |1 A , since the ground state could not have decayed. On the other hand, if no photon is detected, we have projected out the state |0 A from the initial state, a |0

A

+ b 1 − p |1

A

,

which has evolved by virtue of the measurement of system E to become more likely in the ground state |0 A . in other words, the information becomes more biased towards the ground state |0 A and hence the information entropy has decreased. The unitary transformation which entangles A and E, Eq. (43.10), followed by an orthogonal measurement of E describes quantum operation in which the Kraus ˆ m }, and Eq. (43.12) gives operators are the generalized measurement operators {M

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the probability of the outcome of measurement in E that projects onto the {|m basis set,

E}

p (m) =

A m

† ˆ ˆm Mm |ϑ ϑ|M

A

= T r (Fm ρA ) , † ˆ ˆm where Fm = M Mm . Thus in the amplitude-damping channel, we have

F0 = F1 =

1 0 0 1−p 00 0p

,

,

where p (1) is the probability of a photon detection, and p (0) is the complementary probability that no photon is detected. Therefore, for t Γ−1 , so that p ⇒ 1, then M0 =

10 00

,

(45.20)

M1 =

01 00

,

(45.21)

F0 =

10 00

,

F1 =

00 01

,

hence T r (M0 M1 ) = 0, and

and therefore the POVM approaches an orthogonal measurement in the {|0 A , |1 A } basis set. Here we can project project the state |0 A via POVM F0 by not detecting a photon.

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Chapter 46

Master Equation for the Density Operator

For a closed system of coupled subsystem of interest, A, and reservoir, R, the unitary evolution of the density matrix, ρAR (t), is given by the nondissipative Liouville-von Neumann equation i|ρ˙ AR (t) = [HAR , ρAR (t)] = LρAR (t) , where L is the Liouvillian superoperator. The evolution of the reduced density matrix on system A is obtained by taking the partial trace with respect to the reservoir, R, as f i|ρ˙ ef A (t) = T rR [HAR , ρAR (t)] .

In general, the resulting equation of motion will be of the form t f i|ρ˙ ef A

(t) =

ef f f HA , ρef A

f dτLD (t, τ ) ρef A (τ ) ,

(t) + 0

which is a non-Markovian evolution, i.e., the time evolution of the density operator depends upon the past, the system develops a memory. However, under the Markovian approximation, the memory eﬀects can be neglected, and the equation of motion turn into the form f ef f ef f eff i|ρ˙ ef A (t) = HA , ρA (t) + LD ρA (t) .

To find an explicit expression for the last term, we use the axiomatic and phenomenological Lindblad semigroup approach, semigroup because the inverse of the dynamical map does not exist, t ≥ 0, reflecting an irreversible evolution. To start with, let’s define three timescales. (i) τ S , the characteristic timescale on which the system itself evolves; (ii) τ E , the characteristic timescale on which the environment evolves and hence ‘forgets’ information about its initial state; (iii) τ R , the rate at which the relaxation of the system as a result of its interaction with the environment occurs. 598

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The Lindblad Master Equation

The theory of quantum operations essentially gives the end result of quantum operations to the system’s density matrix, not the diﬀerential time evolution dynamics. We make use of the fact that a general superoperator has a Kraus representation f ρef A (t) =

ˆ µ (t) ρA (0) M ˆ µ† (t) . M µ

ˆ µ (t) are This is a general form of completely positive dynamical map, where M eff † ˆ ˆ ˆ Mµ (t) Mµ (t) = I. We can easily manage bounded operators on HA satisfying µ

to examine the evolution by looking at the dynamics on a small timescale δt (i.e., quantum operation in timescale δt) that has to satisfy two condiitons: δt τS and δt τ E . This means the system density matrix only evolves by a change proportional to δt in this time interval, and however, long enough compared with the time over which the environment ‘forgets’ its information about the system. Since δt τ E , the evolution of the system will depend only on the present system density matrix, and not on anything that has happened in the past (no memory). The idea is to look for a suitable quantum operation such that ρ ˆS should be altered only to order δt. If the elapsed time is infinitesimal interval δt, then f 1 ρef A (t) should only be altered to order δt . Thus, ρeff A (δt) =

ˆ µ ρA (0) M ˆ µ† M µ

= ρA (0) + O (δt) . ˆ 0 , must be M ˆ 0 = IˆA + O (δt) and It follows that one of the Kraus operators, say M √ ˆ µ , µ > 0 describe the transitions the others must be O δt . The operators, M that the subsystem A must undergo due to interactions with the reservoir R. Under the Kraus normalization condition requirement, we may write ˆ − i H δt, ˆ 0 = IˆA + K M | √ ˆ µ = δtL ˆµ, M ˆ are both Hermitian operators, but otherwise arbitrary at this where HA and K ˆ µ are also arbitrary and are known as Lindblad operators, stage; the operators L ˆ by using the these need be neither unitary nor Hermitian. We can determine K 1 Strictly speaking, δt should be long enough compared with the time over which the reservoir losses memory about the subsystem, A (i.e., δt τ E ), but short enough compared to the characteristic timescale of the subsystem, A (i.e., δt τ A ).

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Kraus normalization condition ˆ µ† M ˆ µ = IˆA M µ

=

ˆ + i H δt IˆA + K | † ˆµL ˆ µ δt + O δt2 + L

ˆ − i H δt IˆA + K |

µ

ˆ + = IˆA + 2Kδt

ˆ †µ L ˆ µ δt + O δt2 . L µ

Therefore, we have ˆ = −1 K 2

ˆ †µ L ˆµ. L µ

Upon substituting in Eq. (45.4), we have f ρef A (δt) =

ˆ − i H δt ρA (0) IˆA + K ˆ + i H δt IˆA + K | | † ˆ µ ρA (0) L ˆ µ δt + L µ

ˆ + i H δt = IˆA ρA (0) IˆA + IˆA ρA (0) K | ˆ − i H δtρA (0) IˆA + ˆ µ ρA (0) L ˆ †µ δt + K L | µ = ρA (0) −

i ˆ δt + [H, ρA (0)] δt + ρA (0) , K |

= ρA (0) −

i 1 [H, ρA (0)] δt − | 2

ˆ µ ρA (0) L ˆ †µ δt L µ

ˆ †µ L ˆµ L

ρA (0) , µ

ˆ µ ρA (0) L ˆ †µ δt, L

δt + µ

ˆ B} ˆ represents the anti-commutator AˆB ˆ +B ˆ A. ˆ Taking the limit δt → 0 where {A, we obtain the master equation, d 1 ˆ ˆρ = [H, ˆρA ] + dt A ı|

µ

ˆ †µ − 1 {ˆ ˆ †µ L ˆ µ} . ˆµˆ ρ (0), L ρA (0)L L 2 A

(46.1)

ˆ µ , this formula would reduce to Note that if there were no Lindblad operators, L equation describing unitary quantum Liouville evolution. We would then identify ˆ as the Hamiltonian of the (closed) system. Moreover, if we choose a basis which H ˆ then the unitary part of the evolution does not contribute. In general diagonalizes H ˆ appearing in equation(46.1) is not the ‘bare’ Hamiltonian of the the operator H isolated system, but that there are corrections to it that come from the interaction

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ˆ is not even unique; the equation of motion with the environment, R. Indeed, H remains invariant under the changes ˆµ → L ˆ µ + lµ ˆ1A , L

ˆ →H ˆ+ 1 H 2ı

µ

ˆ µ − lµ L ˆ †µ ) + bˆ (lµ∗ L 1A ,

(46.2)

where {lµ } and b are arbitrary scalars. The equation of motion also remains invariant under an arbitrary unitary transformation of the Lindblad operators: ˆµ → L

ˆj . uµj L

(46.3)

j

Equation (46.1) is the well-known Lindblad master equation. The first term describes a unitary transformation, whereas the summation term describe nonunitary evolution. It is a positive Markovian evolution of the density matrix. It follows from the Kraus representation that Lindblad master equation preserves density matrif ces, ρ˙ ef is Hermitian and T rρ˙ eff = 0. The right-hand side of equation (46.1) A A is a linear functional of ρ ˆA , hence it defines the Lindbladian superoperator Llindb . Symbolically, we can write the Lindblad master equation as i|

dˆ ρ = Llindb ρ ˆ dt = [H, ρ ˆ] +

i 2| µ>0

Vµ ρ, Vµ† + Vµ , ρVµ†

,

(46.4)

ˆ µ , describe the interaction between the system and environment. where Vµ = |L Following various analyses on inherent decoherence, we may also add a small time-reversal breaking decoherence (phase-destroying) term in the form of a double commutator involving the Hamiltonian, i|

i dˆ ρ = [H, ρ ˆ] + dt 2 µ>0

Vµ ρ, Vµ† + Vµ , ρVµ†

−

i τPl [H, [H, ρ ˆ]] , 2 |

(46.5)

where τ P l is the smallest time scale (Planck time 5.39121×10−44 s). This constant τ P l , smallest measurable unit of time2 , is thus a measure of the deviation from strictly deterministic time evolution. Remark 46.1 Note that on the other hand, we can have a controlled and deterministic evolution using measurement, this is embodied in the so-called ‘quantum Zeno eﬀect’. It is a quantum mechanical phenomenon in which an unstable particle, if observed continuously, will never decay. This occurs because every measurement causes the wavefunction to “collapse” to a pure eigenstate of the measurement basis. Given a system in a state A, which is the eigenstate of some measurement operator. Assume the system under free time evolution will decay with a certain probability into state B. If measurements are made periodically, with some finite interval between each one, at each measurement, the wave function collapses to an eigenstate 2 The smallest measurable unit of space is the Planck length 1.6×10−35 meters. By definition, the ratio of the Planck length to the Planck time is the speed of light in a vacuum.

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of the measurement operator. Between the measurements, the system evolves away from this eigenstate into a superposition state of the states A and B. When the superposition state is measured, it will again collapse, either back into state A as in the first measurement, or away into state B. However, its probability of collapsing into state B, after a very short amount of time t, is proportional to t2 , since probabilities are proportional to squared amplitudes, and amplitudes behave linearly. Thus, in the limit of a large number of short intervals, with a measurement at the end of every interval, the probability of making the transition to B goes to zero. Furthermore, this double-commutator decoherence term is a specific case of Lindblad evolution since [H, [H, ρ ˆ]] = − ([Hˆ ρ, H] + [H, Hˆ ρ])

(46.6)

and also has a Kraus representation of the corresponding map. Equation (46.6) is obtained by allowing some internal parameters of the system to evolve in a random fashion, proposed to be a physical mechanism for the transition from quantum to classical mechanics. This last term in Eq. (46.5) basically dissipate the oﬀ-diagonal elements of the density matrix, a phase-destroying term. In what follows, we will simply ignore the terms in Eq. (46.6). The formal solution to Eq. (46.1) or (46.4) can be written in the form of a time-evolution superoperator: ρ ˆA (t) ≡ Tˆ← exp[

t 0

Llindb (t )dt ]ˆ ρA (0).

(46.7)

Here Tˆ← is the familiar Tˆ, the time-ordering operator that puts earliest times to the right and latest times to the left. A way to solve this systematically is by the S-matrix techniques of Schwinger. Provided the Lindbladian is time-independent, this can be simplified to ρ ˆA (t) = exp(Lt)ˆ ρA (0).

(46.8)

If the dimension of the system’s Hilbert space is N, a matrix representation for L would contain N 2 × N 2 elements; directly exponentiating it would therefore require O(N 12 ) operations.3 The term involving the Lindblad operators on the RHS of equation (46.1) is known as the dissipator, written D[ˆ ρ]; thus we have in Schrödinger representation L[ˆ ρA ] =

1 ˆ [H, ρ ˆA ] + D[ˆ ρA ]. ı|

(46.9)

An alternative way of representing the information is to transfer the timedependence to the operators: we then require that the expectation value of any 3 Numerical methods for computing matrix exponentials have been studied by many authors. One popular approach is to use the Pade approximation together with a scaling and squaring strategy. Another method is based on the Schur decomposition and evaluating the exponential of a triangular matrix. The Schur approach is signicantly simplified when the matrix is real symmetric or Hermitian. For an n × n matrix, A, the required number of operations to evaluate eA is generally of order O n3 .

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ˆ be the same in either picture, (system) operator O ˆ ρA (t)] = T rA [O(Vˆ ˆ ρA (0))] T rA [Oˆ ˆ ρA (0)] = T rA [(V † (t)O)ˆ ˆ ≡ T rA [OH (t)ˆ ρA (0)], t where V † (t) ≡ Tˆ→ exp[ 0 L† (t )dt ], and the operator Tˆ→ orders in the opposite ˆH obeys the equation sense to normal (i.e. earliest times to the left). Note that O of motion,

ˆH dO ˆ = V † (t)L† (t)O dt ˆ H (t). = V † (t)O

(46.10) (46.11)

In the case of a time-independent Lindbladian things simplify again, ˆH (t) = exp[L† t]O, ˆ O

46.2 46.2.1

ˆH dO ˆ H (t). = L† (t)O dt

(46.12)

Examples Spontaneous Emission

Consider a two-level atom represented using the Pauli matrices, and take a Hamiltonian as, ˆ = − |ω 0 σz , H 2

(46.13)

(where ω0 is the energy diﬀerence between the ground and excited states, and the minus sign gives us the usual convention that | ↑ = |0 is the ground state and | ↓ = |1 the excited state). The single Lindblad operator for spontaneous emission is, ˆ= L

√ Γ

01 00

.

(46.14)

Therefore, from Eq. (46.1), we have ∂ ∂t

ρ00 ρ01 ρ10 ρ11

= ıω 0

0 ρ01 −ρ10 0

+Γ

ρ11 − 12 ρ01 − 12 ρ10 −ρ11

.

(46.15)

The resulting solutions are ρ00 (t) ρ01 (t) ρ10 (t) ρ11 (t)

= ρ00 (0) + ρ11 (0)[1 − exp(−Γt)], = ρ01 (0) exp[(ıω 0 − Γ/2)t], = ρ10 (0) exp[(−ıω0 − Γ/2)t], = ρ11 (0) exp(−Γt).

(46.16)

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From Eq. (46.16), the population in the excited state |1 decays exponentially with a time constant T1 = 1/Γ, whereas the oﬀ-diagonal elements of the density matrix (‘coherences’) decay with a longer time constant T2 = 2/Γ. Therefore T1 = 12 T2 corresponding to the fact that the xy-plane of the Bloch sphere is ‘squashed’ more slowly than the shrinking of the z-axis in the amplitude damping channel. 46.2.2

Bloch Equations in Magnetic Resonance for Spin 1/2

A similar Hamiltonian given by Eq. (46.13) describes a spin precessing in a static magnetic field (chosen to be in the z-direction), ˆ = − |ω 0 σz , H 2

(46.17)

where ω 0 = µB is the Larmor frequency. We may want to three types of process to describe with Lindblad operators: ˆ1 = L

Γ1 σ + =

Γ1

01 00

;

ˆ2 = L

Γ2 σ − =

Γ2

00 10

;

ˆ3 = L

Γ3 σ z =

Γ3

1 0 0 −1

.

ˆ 1 describes relaxation from |1 to |0 with the emission of energy (see section where L ˆ 2 describes the reverse relaxation from |0 to |1 on amplitude damping channel); L ˆ with the absorption of energy; L3 describes pure dephasing processes which do not transfer energy between the spin and its environment (the phase-flip error channel). The Lindblad equation becomes ∂ ∂t

ρ00 ρ01 ρ10 ρ11

= ıω 0 + Γ2

0 ρ01 −ρ10 0

ρ11 − 12 ρ01 − 12 ρ10 −ρ11

+ Γ1

−ρ00 − 12 ρ01 − 12 ρ10 ρ00

+ Γ3

0 −2ρ01 −2ρ10 0

.

(46.18)

The solutions are now eqm ρ00 (t) = ρeqm 00 + (ρ00 (0) − ρ00 ) exp(−t/T1 ), ρ01 (t) = ρ01 (0) exp[(ıω0 − T2−1 )t], ρ10 (t) = ρ10 (0) exp[(−ıω 0 − T2−1 )t], eqm ρ11 (t) = ρeqm 11 + (ρ11 (0) − ρ11 ) exp(−t/T1 ), eqm where the equilibrium populations ρeqm 00 and ρ11 satisfy eqm Γ1 ρeqm 11 = Γ2 ρ00 ,

(46.19)

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and the relaxation times T1 and T2 are now given by T1−1 = Γ1 + Γ2 , (Γ1 + Γ2 ) . T2−1 = 2Γ3 + 2

(46.20) (46.21)

If Eq. (46.19) correspond to the thermal equilibrium populations of the spin in an applied field, we must have the detailed balance condition Γ2 = exp(−β|ω 0 ). Γ1

(46.22)

ˆ 1 process involves the spontaneous or stimulated emission of phonons, and The L ˆ 2 process describes the absorption of phonons. Hence we may take the L Γ1 = γ[1 + n(ω 0 )];

Γ2 = γn(ω 0 ),

(46.23)

where n(ω) is the Bose occupation number n(ω) =

1 . 1 − exp(−β|ω)

(46.24)

From this we deduce that T1−1 = Γ1 + Γ2 = γ[2n(ω 0 ) + 1] = γeβ|ω0 coth

β|ω0 2

,

T2−1 = 2Γ3 + T1−1 /2. Observe that if Γ3 is large or the pure dephasing process is fast, as is frequently the case, then T2 may be much shorter than T1 . A more familiar way of writing the equation of motion (46.18) is in terms of the components of the Bloch vector α is as follows, (αz − αeqm ) dαz z =− ; dt T1 dαx = −ω0 αy − αx /T2 ; dt dαy = ω 0 αx − αy /T2 , dt

(46.25)

= tanh(β|ω 0 /2). This makes it explicit where the mean magnetization is αeqm z that the motion is a combination of free precession about the z-axis and relaxation towards the equilibrium magnetization (0, 0, αeqm ). z 46.3

The Pauli Master Equation

The spontaneous emission in Sec. 46.2.1 and magnetic resonance in Sec. 46.2.2 share the property that the equation of motion for the populations (diagonal elements of ρ) involve only other diagonal elements, in the form of the rate equation. This is generally true if the basis states chosen are eigenstates of the Hamiltonian, e.g.,

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ˆ in Eqs. (46.13) and (46.17), and make a random-phase approxieigenstates of H mation at other initial times so as to make ρ diagonal in the chosen basis states at time t. It follows that the unitary part of the evolution does not cause a change of the populations, only the dissipator of the Lindblad operator connects each basis state to at most one other basis state, having a non-zero entry in each row and column. In that case the Lindblad operators contribute the following terms to the equation of motion for the diagonal element ρnn : (46.26)

µ

[(Lµ )nmµ ρmµ mµ (L†µ )mµ n − |Lµ,nmµ |2 ρnn ]

(46.27)

µ

|Lµ,nmµ |2 (ρmµ mµ − ρnn ),

∂t ρnn (t) = =

where it is assumed that Lindblad operator Lµ couples state n only to state mµ . The resulting set of equations for the populations P (n, t) = ρnn (t) can be written in the more familiar ‘scattering-in’ and scattering-out’ terms, ∂t P (n, t) = m

[W (n ← m)P (m, t) − W (m ← n)P (n, t)],

(46.28)

where the rates are given by W (n ← m) =

µ

|Lµ,nmµ |2 δ m,mµ .

(46.29)

Equation (46.28) is the well-known Pauli master equation; it constitutes a set of purely classical kinetic equations describing the evolution of the populations of the system’s quantum states. 46.4

Lindblad Equation for a Damped Harmonic Oscillator

Consider a Harmonic oscillator interacting with the electromagnetic field. The interaction Hamiltonian is Hint =

λi ab†i + a† bi , i

where a is a system operator, and bi is an environment operator. Suppose the excitation level of the oscillator can cascade down by successive emission of photons, but no absorption of photons will occur. This situation is the case with the reservoir at zero temperature. √ Hence there is only one transition operator (Lindblad jump ˆµ = L ˆ 1 = Γa, where Γ is the rate for the oscillator to decay from the operator ), L first excited state, n = 1, to the ground state, n = 0. We expect that this situation described a damped harmonic oscillator, with ‘scattering out’ as the only physical process, the decay from level |n to |n − 1 is nΓ because anyone of the n excitations/particles can decay, i.e., the nth level considered as a state of n noninteracting particles.

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The corresponding Lindblad master equation is i 1 1 d ρ = − [Ho , ρ] + Γ aρa† − a† aρ − ρa† a , dt | 2 2 where Ho = |ω o a† a is the Hamiltonian of the system. We may include phenomenologically the radiative renormalization of the frequency ωo of the oscillator, the so-called Lamb shift and replaces Ho ⇒ Ho with |ωo ⇒ |ω. It is convenient to place all the time dependence on the operators to study the eﬀect of the Lindblad jump operators. Therefore we adopt the interaction picture, where i

i

i

i

ρ = e− | Ho t ρI e | Ho t , a = e− | Ho t aI e | Ho t , yielding 1 1 d ρ = Γ aI ρI a†I − a†I aI ρI − ρI a†I aI , dt I 2 2 which reduces to 1 1 d ρI = Γ aρI a† − a† aρI − ρI a† a dt 2 2

(46.30)

since aI = ae−iωt , a†I = a† eiωt . Therefore, we have Tr a

d ρ dt I

= Tr

1 1 Γ a2 ρI a† − aa† aρI − aρI a† a 2 2 1 1 aρ a† a − aa† aρI 2 I 2

= ΓT r =

Γ Tr 2

a† , a aρI ,

d Γ Γ T r (aρI ) = − T r (aρI ) = − a ˜ , dt 2 2 Γ d a ˜ =− a ˜ , dt 2

(46.31)

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where a ˜ = T r (aρI ) i

i

= Tr

ae | Ho t ρe− | Ho t

= Tr

e− | Ho t ae | Ho t ρ

i

i

= T r (˜ aρ) iωt = e T r (aρ) . Integrating Eq. (46.31), we obtain Γ

˜ (0) . a ˜ = e− 2 t a

(46.32)

This equation is indicative of decoherence taking place (see Sec. 46.6 below), in the absence of damping, a ˜ is a constant. In the limit t ⇒ ∞, a ⇒ 0, and there will be no coherent state. The occupation number operator of the oscillator, n = a† a, evolves as d n = Tr dt

d † a aρI . dt

Upon substituting the expression in Eq. (46.30), we have Tr

d † a aρI dt

= Γ Tr = Γ Tr

1 1 a† aaρI a† − a† aa† aρI − a† aρI a† a 2 2 1 1 a† a† aaρI − a† aa† aρI − a† aa† aρI 2 2

= Γ T r a† a† , a aρI , d T r a† aρI = −Γ T r a† aρI , dt which can also be written as d n = −Γ n , dt which readily integrate to n = e−Γt n (0) .

(46.33)

As expected, Γ is the damping rate of the oscillator, with exponential damping law. 46.5

Lindblad Equation for Phase Damped Harmonic Oscillator

For phase-damped harmonic oscillator, the coupling of the oscillator to the reservoir is Hint =

λi b†i bi a† a. i

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ˆµ = L ˆ1 = Hence there is only one transition operator (Lindblad jump operator ), L √ † Γa a. The master equation in the interaction picture is 1 † d ρ = Γ a† aρI a† a − a a dt I 2

2

1 ρI − ρI a† a 2

2

.

(46.34)

The parameter Γ can be interpreted as the rate at which the reservoir photons are scattered when the oscillator is singly occupied. If the occupation number is n then the scattering rate is Γn2 as reflected in the last two terms of Eq. (46.34). This is because the scattering amplitude, proportional to n, due to each of the n particles’ add coherently, hence the rate (probability) is proportional to n2 . Using the occupation number representation, we have ρI = n,m

ρnm |n m| .

Equation (46.34) becomes 

n,m

 n,m d ρ |n m| = Γ   dt nm =

ρnm a† a |n m| a† a − − 12

n,m

1 2

ρnm a† a n,m

ρnm |n m| a† a

1 1 Γ nm − n2 − m2 ρnm |n m| 2 2 n,m

= n,m

−

2

2

|n m|

   

Γ (n − m)2 ρnm |n m| . 2

The equation for the oﬀ-diagonal elements, is therefore Γ d ρ = − (n − m)2 ρnm , dt nm 2

(46.35)

which readily integrates to Γ

2

ρnm = ρnm (0) e− 2 (n−m) t , which is a similar sort of behavior as for the damping of single qubit, Eq. (45.15). Indeed, if we prepare the state as |ψ = 12 (|0 + |1 ), i.e., m = 0, n = 1, then the expression for the decay of the oﬀ-diagonal element is identical with that of the Γ 2 single qubit. If |ψ = 12 (|0 + |n ), then ρn0 = ρn0 (0) e− 2 n t , showing the rate for reservoir photons to scatter oﬀ the excited oscillator in state |n . Decoherence occurs in the number-eigenstate basis because it is the occupation number operator that appears in the coupling Hamiltonian, Hint , of the oscillator to the reservoir.

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46.6

Coherent State and Decoherence

In the amplitude-damping model, Hint =

λi ab†i + a† bi , so we can expect that i

decoherence will occur in the a-eigenstate basis. Now the normalized coherent state, with eigenvalue α, in the number representation is |α = e−

|α|2 2 2

− |α| 2

=e

†

eαa |0 ∞

αn √ |n . n! n=0

These states, although complete, are not orthogonal, 2

2 | α1 |α2 | = e−(|α1 |

+|α2 |2 +2 Re α∗ 1 α2 ) 2

= e−|α1 −α2 | ,

so that the overlap is small if |α1 − α2 | is large. We can write Eq. (46.30) in terms of coherent state basis, ρI = αα

ραα |α α | ,

as d dt

αα

ραα |α α |

=Γ αα

1 1 ραα a |α α | a† − a† a ραα |α α | − 2 2 αα

αα

ραα |α α | a† a .

Therefore,

αα

d ρ β| |α α | |γ dt αα = αα

1 1 Γ αα ∗ − αβ ∗ − α ∗ γ ραα β| |α α | |γ . 2 2

Since the states β| and |γ are arbitrary, we can let β = α, and γ = α in both sides of the equation, and since ρ∗α α = ραα , we have d dt

αα

ραα α| |α α | |α

=Γ αα

1 1 1 ρ αα ∗ + ραα α α∗ − 2 αα 2 2

αα

ραα αβ ∗ −

1 2

ραα α ∗ γ αα

α| |α α | |α ,

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d ρ =Γ dt αα

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1 1 1 1 2 αα ∗ + α∗ α − |α|2 − |α | ραα 2 2 2 2

Γ 2 |α − α | ραα . 2 Thus in the coherent state representation, the oﬀ-diagonal element already has a 2 factor e−|α1 −α2 | , and hence the oﬀ-diagonal terms in ρ decay like =−

2 1 ρα1 ,α2 = e−{ 2 Γ(|α1 −α2 | )}t ρα1 ,α2 (0) ,

2 which shows the decoherence rate Γdecoher = 12 |α1 − α2 | Γdamping , i.e., decoherence rate is much faster, for |α1 − α2 | 1, than the damping rate of Eq. (46.33). The loss of energy corresponds to the decay of the diagonal elements of ρ, and the simultaneous loss of phase coherence corresponds to the decay of the oﬀ-diagonal elements of ρ to zero. In the case of superposition of minimum uncertainty wavepackets centered at diﬀerent points, say x and −x, a ‘cat’ state

1 |cat = √ (|α1 + |α2 ) . 2

(46.36)

In the coherent state, the expectation value of the occupation number operator is α| a† a |α = |α|2 . For the superposition of Eq. (46.36), if α1 and α2 have comparable moduli but significantly diﬀerent phases, then Γdecoher = 2 2 1 1 − ei(θ2 −θ2 ) Γdamping . This decoherence rate is of the order of the rate 2 |α| for a single photon emission, since this emission enables approximate location of the wavepackets and hence destruction of the coherence of superposition. This rate 1 decoher is large compared to rate of energy loss, i.e., ΓΓdamping 2 n oscillator . Moreover, for an oscillator coupled to a reservoir at finite temperature, the rate depends on the rate for single photon to be emitted or absorbed. This rate is expected to be much faster than at zero temperature, since we now have to includes photon modes with frequency comparable to the oscillator frequency ω. For temperature, T |ω, the number of photon modes that gets ‘scattered’, 1 kT nγ . Thus |ω |ω e kT −1

Γdecoher Γdamping

1 n oscillator nγ 2 mω2 x2 kT E kT ∼ ∼ |ω |ω |ω |ω x2 mkT x2 ∼ ∼ 2, |2 λT √

(46.37)

2π | where x here is the oscillator amplitude and λT = √mkT is the thermal de Broglie wavelength; hence the decoherence is naturally fast.

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Chapter 47

Microscopic Considerations of a Two-Level System Revisited

We write the Hamiltonian of two level system in the form 1 |ωo (|2 2| − |1 1|) 2 1 = |ωo σ z . 2

Ho =

(47.1)

More generally, the density matrix is describe by an ensemble (superposition) as ˆρ = ij

ρij |i j| , i, j = 1, 2,

where ρij is the ensemble average, ρij = ci c∗j , i, j = 1, 2, and ρ11 + ρ22 = 1 2

2

since |c1 | + |c2 | = 1, i.e., any pure state of two-level system can always be written in the form: |ψ = c1 |1 + c2 |2 . The time-evolution equation in the Schrödinger picture is ih

dˆ ρ (t) = [Ho + HI , ρ ˆ (t)] , dt

where HI has entirely oﬀ-diagonal matrix elements, r · E (R, t) , HI (t) = −eˆ where E (R, t) is the classical electric field operator, R is a parameter describing the position of the atom, and eˆ r is the dipole operator for the two-level system. We 612

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therefore have i|

∂ ∂t

ρ11 ρ12 ρ21 ρ22

− 12 |ωo HI12 HI21 21 |ω o

= −

ρ11 ρ12 ρ21 ρ22 − 12 |ωo HI12 1 H21 I 2 |ω o

ρ11 ρ12 ρ21 ρ22

− 12 |ω o ρ11 + HI12 ρ21 − 12 |ωo ρ12 + HI12 ρ22 1 HI21 ρ11 + 12 |ω o ρ21 H21 I ρ12 + 2 |ω o ρ22

=

− 

−ρ11 21 |ω o + ρ12 HI21 ρ11 HI12 + ρ12 21 |ωo −ρ21 21 |ω o + ρ22 HI21 ρ21 HI12 + ρ22 21 |ωo

− 12 |ωo ρ11 + HI12 ρ21 − 12 |ω o ρ12 + HI12 ρ22  +ρ 1 |ω − ρ H21 −ρ H12 − ρ 1 |ω o o  11 2 12 I 11 I 12 2 =  HI21 ρ11 + 12 |ω o ρ21 HI21 ρ12 + 12 |ωo ρ22 +ρ21 21 |ω o − ρ22 HI21 −ρ21 HI12 − ρ22 21 |ω o   −|ωo ρ12 12 21 H ρ − ρ H 12 I  I 21 +H112 (ρ22 − ρ11 )  . =   |ω o ρ21 21 12 HI ρ12 − ρ21 H1 21 +HI (ρ11 − ρ22 )

     (47.2)

We thus readily obtain Bloch equations,

47.1

∂ 1 ρ = [ 1| HI (t) |2 ρ21 − ρ12 2| HI (t) |1 ] , ∂t 11 i|

(47.3)

1 ∂ ρ12 = [−|ωo ρ12 + 1| HI (t) |2 (ρ22 − ρ11 )] , ∂t i|

(47.4)

1 ∂ ρ21 = [|ωo ρ21 + 2| HI (t) |1 (ρ11 − ρ22 )] , ∂t i|

(47.5)

1 ∂ ρ = − [ 1| HI (t) |2 ρ21 − ρ12 2| HI (t) |1 ] . ∂t 22 i|

(47.6)

Quantized Radiation Field

Consider the same problem but quantize the radiation field. For convenience in what follows, first let us consider the properties of Pauli spin operators σx =

01 10

, σy =

0 −i i 0

, σz =

1 0 0 −1

.

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We have the following relations σ 2i = 1,

i = x, y, z,

[σ i , σ j ] = 2iεijk σ k , ˆ {σ i , σ j } = 2δ ij I, σi , |σ|2 = 0, σi σj = δ ij Iˆ + iεijk σk , ˆ the comwhere εijk is the Levi-Civita symbol. Including the identity operator, I, pleteness relation for the Pauli spin operators is ˆ σi σ†i = 4I,

Iˆ Iˆ† + i=x,y,z

ˆ

so that we can expand any two dimensional operators in terms of σ2i and I2 . It is also customary to express the two-dimensional eigenkets in terms of the eigenkets if σz , which also form a complete basis states ˆ |+ +| + |− −| = I,

ˆ |0 0| + |1 1| = I,

1 0

10 +

0 1

01 =

10 01

,

where in our notation, σz |0 = +1 |0 , σ z |1 = −1 |1 , i.e., |0 =

1 0

,

|1 =

0 1

.

In the treatment of quantized radiation fields, it is convenient to define two nonHermitian operators, σ± , by σ+ =

1 (σ x + iσy ) = 2

01 00

= |0 1| ,

σ− =

1 (σ x − iσy ) = 2

00 10

= |1 0| ,

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where one can easily verify the ladder operation properties of these operator for a two-level system σ + |1 σ + |0 σ − |1 σ − |0

= |0 , = 0, = 0, = |1 ,

which means that σ+ creates an excited level |0 when acting on the ground state |1 . On the other hand, σ − annihilate an excited level |0 back to the ground state |1 . Now let us express the interaction Hamiltonian HI (t) operator in terms of σ± and annihilation/creation operators of the quantized radiation field. We have ˆ R, ˆ t , r·E HI (t) = −eˆ ˆ R, ˆ t is the electric field operator, R ˆ is the position of the atom, and eˆ where E r is the dipole operator. Thus 1| HI (t) |2 = 0| HI (t) |1

ˆ R, ˆ t |1 = 0| − eˆ r·E

ˆ R, ˆ t = − 0| eˆ r |1 · E ˆ R, ˆ t . = −P·E Similarly, 2| HI (t) |1 = 1| HI (t) |0

ˆ R, ˆ t . = −P ∗ ·E

Hence we can write HI (t) = 0| HI (t) |1 |0 1| + 1| HI (t) |0 |1 0| , which can also be written in terms of σ ± as ˆ R, ˆ t HI (t) = − P |0 1| + P ∗ |1 0| ·E ˆ R, ˆ t . = − Pσ + + P ∗ σ− ·E Now the radiation field operator correspond to the time dependent classical field variable, hence it is usually given as a time-dependent quantum operator in the Heisenberg representation, namely, ˆ R, ˆ t =i E q

|Ωq uq (R) a ˆq (t) − a ˆ†q (t) , 2εo Vq

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where uq (R) is the mode function, εo is the electric permittivity and Vq (= V8 in three dimensional box) is the eﬀective mode volume, and a ˆq (t) and a ˆ†q (t) are the timedependent annihilation and creation operators. Here we neglect the quantization ˆ ⇒ R, just a parameter. For simplicity, we will of atomic motion (phonons), i.e., R consider only a single mode q and write in the Schrödinger representation (SR) ˆ R, ˆ t = iΞ0 u (R) a E ˆq (t) − a ˆ†q (t) , where Ξ0 = HI (t) is

|Ω 2εo V

. Thus the corresponding SR of the interaction Hamiltonian

HI = −iΞ0 P· u (R) σ+ + P ∗ · u (R) σ −

a ˆ−a ˆ† .

Writing P· u (R) = P· u (R) eiϕ , P ∗ · u (R) = P· u (R) e−iϕ , we thus obtained in the SR, HI = −i|g (R) σ+ eiϕ + σ− e−iϕ

a ˆ−a ˆ†

= |g (R) σ+ ei(ϕ− 2 ) + σ − e−i(ϕ+ 2 ) π

π

a ˆ−a ˆ† ,

(47.7)

|P· u(R)| Ξ0 is the position-dependent vacuum Rabi frequency. We where g (R) = | can choose the phases of the atomic basis states so that ϕ = π2 . We end up with SR of HI in Eq. (47.7) to be given as HI = |g (R) σ+ − σ−

a ˆ−a ˆ† .

This interaction Hamiltonian can be further simplified by going over to the interaction representation (IR). The Hamiltonian of free atom and free field, i.e., in the absence of interaction is 1 Ho = |Ωˆ a† a ˆ + |ωo σ z . (47.8) 2 Since the atomic and field operators commute, we obtained the IR of the interaction Hamiltonian as Ho Ho t HI exp −i t | | Ho Ho ˆ−a ˆ† exp −i t t |g (R) σ+ − σ− a = exp i | | 1 1 = |g (R) exp i ωo σ z t σ+ − σ− exp −i ωo σ z t 2 2

HIIR = exp i

× exp iΩˆ a† a ˆt

a ˆ−a ˆ† exp −iΩˆ ˆt . a† a

Consider the atomic part 1 exp i ωo σ z t = 2

ei

ωo t 2

0

0 ωo t e−i 2

,

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ˆ2n = I, ˆ and which follows from the relation exp iθAˆ = cos (θ) Iˆ + iAˆ sin θ if A ˆ Thus we have for the atomic part Aˆ2n+1 = A. 1 exp i ωo σ z t 2

1 σ+ − σ− exp −i ωo σ z t = σ+ eiωo t − σ − e−iωo t . 2

(47.9)

The radiation field part can be determined directly from the Heisenberg equation of motion i ˆ] a ˆ (t) = [Ho , a | = iΩ a ˆ† a ˆ, a ˆ , by using the commutation relation a ˆ, a ˆ† = 1, [ˆ a, a ˆ] = 0, ˆ† = 0, a ˆ† , a which yield a ˆ (t) = a ˆ (0) e−iΩt , a ˆ† (t) = a ˆ† (0) eiΩt . Thus, we have IR of the interaction Hamiltonian given as HIIR = |g (R) σ+ eiωo t − σ − e−iωo t = |g (R)

ˆ† eiΩt a ˆe−iΩt − a

ˆei(ωo −Ω)t + σ − a ˆ† e−i(ωo −Ω)t σ+ a + † i(ωo +Ω)t −σ a ˆ e − σ−a ˆe−i(ωo +Ω)t

.

(47.10)

This give way for further simplification by noting that the terms the sum of frequencies does not conserve energy and the terms containing the frequency diﬀerence, i.e., the detuning, ∆ = Ω − ω o are energy conserving and are slowly varying in time. For times of interest the high frequency terms average to zero, and we may write HIR I

|g (R) σ+ a ˆei(ωo −Ω)t + σ− a ˆ† e−i(ωo −Ω)t ˆe−i∆t + σ− a ˆ† ei∆t . = |g (R) σ+ a

(47.11)

This approximation is the so-called rotating wave approximation. Since we are interested in a single two-level system coupled to a reservoir consisting of infinitely many field modes, we modify Eq. (47.11) by including all modes of the radiation field to yield HIR I

|ω R σ+ A (t) + σ − A† (t) ,

(47.12)

where A (t) are made up of all the modes, A (t) =

1 ωR

gl (R) a ˆl e−i∆l t , l

(47.13)

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and we have introduced ω R which has units of frequency. Note that [A (ti ) , A (tj )] =

1 ωR

2

gl (R) gl (R) [ˆ al , a ˆl ] e−i(∆l ti +∆l

tj )

l,l †

= 0 = A (ti ) , A† (tj ) , A (ti ) , A† (tj ) = =

=

1 ωR

2

1 ωR

2

1 ωR

2

(47.14)

gl (R) gl∗ (R) a ˆl , a ˆ†l e−i(∆l ti −∆l l,l

gl (R) gl∗ (R) δ ll e−i(∆l ti −∆l

tj )

l,l

l,l

|gl (R)|2 e−i∆l (ti −tj )

= G (ti − tj ) . 47.2

tj )

(47.15)

Perturbation Expansion of Density Operator

In perturbation theory, the interaction picture is convenient when considering the eﬀect of a small interaction term, H1 , being added to the Hamiltonian of a solved system, H0 . The purpose of the interaction picture is to shunt all the time dependence due to H0 onto the operators, leaving only H1 , aﬀecting the time-dependence of the state vectors. By switching into the interaction picture, we can use timedependent perturbation theory to find the eﬀect of H1 on the supposedly known H0 . Let H = H0 + H1 , where 1 H0 = |Ωˆ a† a ˆ + |ωo σ z , 2 HI = |g (R) σ+ − σ−

a ˆ−a ˆ† .

The density matrix operator in the interaction picture or Heisenberg representation is i

i

ρI (t) = e | H0 (t−to ) ρ (t) e− | H0 (t−to ) ,

(47.16)

or equivalently i

i

ρ (t) = e− | H0 (t−to ) ρI (t) e | H0 (t−to ) ,

(47.17)

where ρ (t) is in the Schrödinger representation. Indeed, upon diﬀerentiating Eq. (47.17), we obtain the time evolution of the density matrix in the Schrödinger

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picture as i|

i i ∂ ∂ ρ (t) = [H0 , ρ (t)] + e− | H0 (t−to ) i| ρI (t) e | H0 (t−to ) , ∂t ∂t

which we can compare with the usual expression for the time evolution in the Schrödinger picture, i|

∂ ρ (t) = H0 + H1I , ρ (t) ∂t = [H0 , ρ (t)] + HI1 , ρ (t) .

Therefore, i

H1I , ρ (t) = e− | H0 (t−to ) i|

∂ I i ρ (t) e | H0 (t−to ) , ∂t

which yields the time evolution equation of the density matrix operator in the interaction picture as i|

i i ∂ I ρ (t) = e | H0 (t−to ) [H1 , ρ (t)] e− | H0 (t−to ) ∂t

= H1 , ρI (t) . Assuming that the interaction Hamiltonian is weak, we can resort to perturbation theory, ρI (t) = ρI (to ) +

∞ n=1

n

i − |

tn

t

dtn to

t2

dtn−1 ... to

to

× dt1 H1I (tn ) , H1I (tn−1 ) , .., HI1 (t1 ) , ρI (to ) .. . Expanding up to second order, we have t I

ρ (t)

i ρ (to ) − | I

i + − |

to

dt1 H1I (t1 ) , ρI (to ) t2

2 t

dt2 to

to

dt1 H1I (t2 ) , H1I (t1 ) , ρI (to ) .

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Substituting for HI1 (t1 ) the expression in Eq. (47.12), including all radiation modes, and taking the trace over the reservoir (radiation field), we obtained T rf ρI (t) − ρI (to ) t

i − |

to

dt1 T rf |ωR σ + A (t1 ) + σ − A† (t1 ) , ρI (to ) 2

+ (|ω R ) −

i − |

†

t2

t

2

dt2 to

to

dt1 T rf [{σ + A(t2 )

+ σ A (t2 )}, [{σ A(t1 ) + σ − A† (t1 )}, ρI (to )]] +

= ρI(1) (t) + ρI(2) (t) , s s

(47.18)

where A (t) is defined by Eq. (47.13) to account for all field modes, and ρIs indicates the reduced density operator pertaining to the subsystem.

47.2.1

First-order Contribution I(1)

Consider the first order term, ρs

(t),

t

(t) ρI(1) s

i =− |

to

dt1 T rf |ωR σ+ A (t1 ) + σ− A† (t1 ) , ρI (to ) .

Now since the two-level system and reservoir are independent before coupling, their respective Hamiltonian commutes in the SP, they must also commute whenever they are in the same picture or representation. We can decompose the density operator ρI (t) = ρIs (t) ⊗ ρIf (t), where ρIf (t) is the density operator of the radiation-field reservoir. Then we have,

t

ρI(1) (t) s

i =− |

=−

i |

to

     σ + ρIs (to ) |ω R T rf A (t1 ) ρIf (to )       − I † I    +σ ρs (to ) |ω R T rf A (t1 ) ρf (to )   dt1      ρI (t ) σ+ |ω T r A (t ) ρI (t )     R f 1 s o f o  −  +ρI (to ) σ − |ωR T rf A† (t1 ) ρI (to )  s f

σ+ ρIs (to ) |ΩR + σ − ρIs (to ) |Ω∗R

− ρIs (to ) σ+ |ΩR + ρIs (to ) σ− |Ω∗R = −i ΩR σ+ ρIs (to ) − ρIs (to ) σ + + Ω∗R σ− ρIs (to ) − ρIs (to ) σ−

,

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where t

ΩR = to t

dt1 ωR T rf A (t1 ) ρIf (to ) gl (R) e−i∆l t1 a ˆl .

dt1

=

l

to

where ∆l is the detuning from Eq. (47.10). Let us consider the time evolution of the system due to interaction with the reservoir for a time τ , as what would happen within a resonant cavity. We take the interaction to start at time t and ends at time t + τ (i.e., to ⇒ t and t ⇒ t + τ in the last equation. We shall also see that taking the limiting case τ ⇒ 0 will allow us to deduce the time evolution equations in the so-called coarse-grain approximation of the time derivative. Therefore, t+τ

ΩR = t

dt1 ω R T rf A (t1 ) ρIf (t) t+τ

=

dt1 e−i∆l t1

gl (R) a ˆl l

t −i∆l τ −i∆l t e

=

−1 −i∆l

gl (R) a ˆl e l

gl (R) a ˆl e−i∆l t e−i

=

∆l τ 2

ei

∆l τ 2

l −i∆l t −i

=

τ gl (R) a ˆl e

e

∆l τ 2

l

gl (R) a ˆl e−i∆l t e−iθl

=τ l

47.2.2

ei

− e−i i∆l

∆l τ 2

∆l τ 2

− e−i 2i ∆2l τ

sin θl . θl

∆l τ 2

(47.19)

Resonance Approximation

Equation (47.19) for the eﬀective Rabi frequency clearly shows the following properties: (i) only modes with nonvanishing values of the annihilation operator a ˆl contributes, and (ii) only modes which are near resonant with the two-level system are important, by virtue of the appearance of sinθlθl , hence only those for which θl 1. Therefore, we can approximate ΩR = τ go (R) a ˆo e−i∆o t = τ go (R) a ˆo e−i(Ωo −ωo )t = τ go (R) a ˆo e−iΩo t eiωo t = τ g (t) eiωo t ,

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where g (t) = go (R) a ˆo e−iΩo t is the eﬀective Rabi frequency, , |ω o is the energy diﬀerence between the two energy levels of the system, and we have denoted the resonant mode by the subscript ‘o’. Bloch Equation

47.2.3

Equation (47.18) is given to first order as ρIs (t + τ ) − ρIs (t) = −i ΩR σ+ ρIs (t) − ρIs (t) σ+ + Ω∗R σ− ρIs (t) − ρIs (t) σ−

= −iτ g (t) eiωo t σ+ ρIs (t) − ρIs (t) σ + + g ∗ (t) e−iωo t σ− ρIs (t) − ρIs (t) σ −

,

which yields Limτ ⇒0 =

ρIs (t + τ) − ρIs (t) τ

dρIs (t) dt

= −i g (t) eiωo t σ+ ρIs (t) − ρIs (t) σ+ + g∗ (t) e−iωo t σ− ρIs (t) − ρIs (t) σ−

.

Transforming to the Schrödinger picture, i 1 2 |ω o σ z t

ρI (t) = e |

i 1 2 |ω o σ z t

ρs (t) e− |

1

1

= ei 2 ωo σz t ρs (t) e−i 2 ωo σz t ,

gives 1 dρIs (t) 1 1 1 dρ (t) 1 = i ω o σz ei 2 ωo σz t ρs (t) e−i 2 ωo σz t + ei 2 ωo σz t s e−i 2 ωo σz t dt 2 dt 1 1 1 − iei 2 ωo σz t ρs (t) e−i 2 ωo σz t ω o σz 2 1

= ei 2 ωo σz t

1 dρs (t) + i ω o σz , ρs (t) dt 2

1

e−i 2 ωo σz t .

Using Eq. (47.9) 1 exp i ωo σ z t 2

1 σ+ − σ− exp −i ωo σ z t = σ+ eiωo t − σ − e−iωo t . 2

Therefore, the Bloch equations in the Schrödinger picture is i|

1 dρs (t) = [|ω o σz , ρs (t)] + |g (t) σ+ , ρs (t) + |g ∗ (t) σ− , ρs (t) , (47.20) dt 2

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or in matrix form i|

d dt

ρ11 ρ12 ρ21 ρ22

= |ω o =

0 ρ12 −ρ21 0

+ |g (t)

ρ21 ρ22 − ρ11 0 −ρ21

+ |g∗ (t)

−ρ12 0 ρ11 − ρ22 ρ12

|g (t) ρ21 − |g∗ (t) ρ12 |ω o ρ12 + |g (t) (ρ22 − ρ11 ) −|ωo ρ21 + |g∗ (t) (ρ11 − ρ22 ) |g ∗ (t) ρ12 − |g (t) ρ21

,

(47.21)

which is identical to Eq. (47.2) by replacing |g (t) ⇒ HI12 , |g ∗ ⇒ HI21 and changing the sign, |ω o ⇒ −|ω o , in the expression for the two-level Hamiltonian of Eq. (47.1). Thus to first order, we have recovered the Bloch equation where the radiation field were treated as parameters, not as quantum operators. Separating the real and imaginary part of g (t), we can rewrite Eq. (47.21) as i|

ρ11 ρ12 ρ21 ρ22   −i|gr (t) v |ω o ρ12 + |gr (t) w   +i|gi (t) u +i|gi (t) w  =   −|ωo ρ21 − |gr (t) w i|gr (t) v +i|gi (t) w −i|gi (t) u   −i|Bx Sy −2|Bz ρ12 + |Bx Sz  1 +i|By Sx +i|By Sz  =   i|Bx Sy 2  2|Bz ρ21 − |Bx Sz +i|By Sz −i|By Sx   −2|Bz ρ12 + |Bx Sz −i| B × S  1 z +i|By Sz , =    2 2|Bz ρ21 − |Bx Sz i| B × S +i|By Sz z

d dt

where we have define the following variables S = u v w = Sx Sy Sz , B = 2gr (t) 2gi (t) −ω o = Bx By Bz , where u = 2 Re ρ12 = (ρ12 + ρ21 ) , v = 2 Im ρ12 = −i (ρ12 − ρ21 ) , w = ρ22 − ρ11 .

(47.22)

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Therefore in terms of Bloch vector S and B, we have   u d dS = i|  v  i| dt dt w   (ρ12 + ρ21 ) d   = i|  −i (ρ12 − ρ21 )  . dt ρ22 − ρ11

Substituting the expressions for the evolution equations of the matrix elements from Eq. (47.22), we have   −i|Bz Sy + i|By Sz dS   =  i|Bz Sx − i|Bx Sz  i| dt i| B × S z   i| B × S x    =  i| B × S  , y  i| B × S z

so that the Bloch equations takes the form

d S = B × S, dt which is a precision of the vector S about the B vector, similar to the familiar precision of angular momentum S in a magnetic field B. 47.3

Second Order Contribution

The second order contribution from Eq. (47.18) is (t) ρI(2) s

i − |

2

= (|ωR ) −

2

t2

t

dt2 to

†

to

dt1 T rf [{σ + A(t2 )

+ σ A (t2 )}, [{σ A(t1 ) + σ − A† (t1 )}, ρI (to )]] 2

= (|ωR )

2

= (|ωR ) I

+

i − | i − |

t2

2 t

dt2 2

to

to

t

t2

dt2 to

to I

dt1 T rf V2 , V1 , ρI (to ) dt1 T rf [(V2 V1 ρI (to )

+ ρ (to )V1 V2 − V2 ρ (to )V1 ) − V1 ρI (to )V2 ],

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where we define, V1 ≡ σ+ A (t1 ) + σ− A† (t1 ) ,

V2 ≡ σ+ A (t2 ) + σ− A† (t2 ) . Hence, we have T rf V2 V1 ρI (to ) = σ+ σ − ρIs (t) T rf A (t2 ) A† (t1 ) ρIf (to )

+σ − σ+ ρIs (t) T rf A† (t2 ) A (t1 ) ρIf (to )

= σ+ σ − ρIs (t) A (t2 ) A† (t1 ) + σ− σ + ρIs (t) A† (t2 ) A (t1 ) ,

T rf ρI (to ) V1 V2 = A (t1 ) A† (t2 ) ρIs (t) σ + σ− + A† (t1 ) A (t2 ) ρIs (t) σ− σ+ . Consider the term, T rf V2 ρI (to ) V1 = T rf σ + A (t2 ) + σ− A† (t2 ) ρI (to ) σ + A (t1 ) + σ− A† (t1 ) = T rf = T rf =

A (t2 ) σ + ρI (to ) σ + A (t1 ) + A† (t2 ) σ − ρI (to ) σ − A† (t1 ) +A (t2 ) σ + ρI (to ) σ− A† (t1 ) + A† (t2 ) σ − ρI (to ) σ + A (t1 )

σ + ρIs (to ) σ + ρIf (to ) A (t1 ) A (t2 ) + σ − ρIs (to ) σ− ρIf (to ) A† (t1 ) A† (t2 ) +σ + ρIs (to ) σ− ρIf (to ) A† (t1 ) A (t2 ) + σ − ρIs (to ) σ+ ρIf (to ) A (t1 ) A† (t2 )

σ+ ρIs (to ) σ+ A (t1 ) A (t2 ) + σ− ρIs (to ) σ− A† (t1 ) A† (t2 ) +σ + ρIs (to ) σ − A† (t1 ) A (t2 ) + σ − ρIs (to ) σ + A (t1 ) A† (t2 )

,

and T rf V1 ρI (to ) V2 = T rf σ+ A (t1 ) + σ − A† (t1 ) ρI (to ) σ + A (t2 ) + σ− A† (t2 ) = T rf = T rf =

A (t1 ) σ + ρI (to ) σ + A (t2 ) + A† (t1 ) σ − ρI (to ) σ− A† (t2 ) +A (t1 ) σ + ρI (to ) σ − A† (t2 ) + A† (t1 ) σ − ρI (to ) σ+ A (t2 )

σ + ρIs (to ) σ + ρIf (to ) A (t2 ) A (t1 ) + σ − ρIs (to ) σ− ρIf (to ) A† (t2 ) A† (t1 ) +σ + ρIs (to ) σ − ρIs (to ) A† (t2 ) A (t1 ) + σ − ρIs (to ) σ + ρIf (to ) A (t2 ) A† (t1 )

σ + ρIs (to ) σ + A (t2 ) A (t1 ) + σ− ρIs (to ) σ − A† (t2 ) A† (t1 ) +σ+ ρIs (to ) σ − A† (t2 ) A (t1 ) + σ − ρIs (to ) σ+ A (t2 ) A† (t1 )

.

Combining all terms, we have T rf

V2 V1 ρI (to ) + ρI (to ) V1 V2 − V2 ρI (to ) V1 − V1 ρI (to ) V2

= σ+ σ − ρIs (t) A (t2 ) A† (t1 ) + σ − σ+ ρIs (t) A† (t2 ) A (t1 )

+ A (t1 ) A† (t2 ) ρIs (t) σ + σ− + A† (t1 ) A (t2 ) ρIs (t) σ− σ +

− −

σ+ ρIs (to ) σ + A (t1 ) A (t2 ) + σ − ρIs (to ) σ− A† (t1 ) A† (t2 ) +σ+ ρIs (to ) σ− A† (t1 ) A (t2 ) + σ − ρIs (to ) σ+ A (t1 ) A† (t2 ) σ+ ρIs (to ) σ + A (t2 ) A (t1 ) + σ − ρIs (to ) σ− A† (t2 ) A† (t1 ) +σ+ ρIs (to ) σ− A† (t2 ) A (t1 ) + σ − ρIs (to ) σ+ A (t2 ) A† (t1 )

.

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Using Eqs. (47.14) and (47.15), we can also write T rf

V2 V1 ρI (to ) + ρI (to ) V1 V2 − V2 ρI (to ) V1 − V1 ρI (to ) V2

= σ+ σ − ρIs (t) − σ− ρIs (to ) σ +

A (t2 ) A† (t1 )

+ σ− σ+ ρIs (t) − σ + ρIs (to ) σ −

A† (t2 ) A (t1 )

+ ρIs (t) σ+ σ − − σ − ρIs (to ) σ+

A (t1 ) A† (t2 )

+ ρIs (t) σ− σ + − σ + ρIs (to ) σ−

A† (t1 ) A (t2 )

− 2σ+ ρIs (to ) σ+ A (t1 ) A (t2 ) − 2σ− ρIs (to ) σ− A† (t1 ) A† (t2 ) . 47.4

(47.23)

Master Equation to Second Order

Using the result of Eq. (47.23), the second-order contribution to the master equaI(2) I(2) tion maybe written as [using ρ ˆA (t) instead of ρs (t) and using the Redfield 1 approximation for short-time memory ] 1 d I(2) (t) = − 2 ρ ˆ dt A | −

1 |2

−

1 |2

−

1 |2

+

2 |2

+

2 |2

∞

σ − t−s ρIA (t) σ + (t)−σ+ (t) σ − t−s ρIA (t)

A (t) A† t−s

ds

σ + t−s ρIA (t) σ− (t)−σ − (t) σ + t−s ρIA (t)

A† (t) A t−s

ds

σ − (t) ρIA (t) σ + t−s −ρIA (t) σ + t−s σ − (t)

A t−s A† (t)

ds

σ + (t) ρIA (t) σ − t−s −ρIA (t) σ − t−s σ+ (t)

A† t−s A (t)

ds

0 ∞ 0 ∞ 0 ∞ 0 ∞ 0 ∞ 0

ds σ+ t−s ρIA (t) σ+ (t) A t−s A (t) ds σ− (t) ρIA (t) σ − t−s

A† t−s A† (t) ,

(47.24)

where A (t) is given by Eq. (47.13). Here, we have assumed that the modes of the reservoir are not entangled so a ˆ†l a ˆ†l a ˆl = T rf a ˆl ρl = nl , ˆl a ˆ†l a

= T rf a ˆ†l ρl T rf (ˆ al ρl ) = a ˆ†l

a ˆl ,

1 Refer to Eq. (48.36). This amounts to substituting ρI (t ) under the integral sign by ρI (t), A 0 A this approximation is not necessary in the following calculations, however, the substitution is physically meaningful for short memory eﬀects to be compared with results from using nonequilibrium quantum superfield correlations in the next chapter.

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627

and t2

t+τ

I (∆l , ∆m ) =

dt1 ei(∆l t2 −∆m t1 )

dt2 t t+τ

t t2

dt1 e−i∆m t1

i∆l t2

=

dt2 e t

t t+τ

1 = −i∆m =τ

i(∆l −∆m )t2

dt2 e t

i(∆l −∆m )t

e

−i∆m

+ τ2

i

e

(∆l −∆m ) 2

τ

i

e

e−i∆m t + i∆m (∆l −∆m ) 2

2i

−i(∆m −∆l )t

e

τ

t+τ

dt2 ei∆l t2 t

− e−i

(∆l −∆m ) 2

τ

(∆l −∆m )τ 2

i∆l τ

e

i∆m τ

−1 i∆l τ

(∆l −∆m ) τ 2 ei(∆l −∆m )t i (∆l −∆m ) τ i sin 2 =τ e (∆l −∆m )τ ∆m 2

−τ

e−i(∆m −∆l )t ei∆l τ − 1 . ∆m ∆l τ

(47.25)

Therefore I (∆l , ∆l ) = = I (∆l , ∆l ) = τ

i 1 − τ2 ei∆l τ − 1 ∆l (∆l τ )2 i 1 − τ2 ∆l (∆l τ )2 2

−2 sin2

2 sin2 ∆2l τ i + ∆2l τ ∆l

∆l τ 2

1−

+ i sin (∆l τ)

sin (∆l τ) ∆l τ

.

Note that the parameters Γ and β involves the expectation values of the field annihilation and creation operators, and hence depends on the statistics of the reservoir. On the other hand, the parameter G, which involves the c-number commutation relation of the field operators, does not depend on the state or statistics of the reservoir. Let us consider two reservoir statistics, one in thermal state and one in coherent state. In thermal state, the reservoir is described by a diagonal density matrix operator ρth f = n

p (n) |n n|

n

1 − e− kT

|ω

=

e−

| ωn kT

|n n| ,

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where n =

n p (n) = e kT − 1

|ω

n

−1

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is the well-known Planck distribution for

radiation in thermal equilibrium. For thermal state of the radiation, we have for each mode a ˆl = T rf

a ˆl ρth = f

p (n) n| a ˆl |n

n

√ p (n) n n |n − 1 = 0,

= n

ˆl = T rf a ˆl a

ˆl ρth a ˆl a = f n

p (n)

= n

p (n) n| a ˆl a ˆl |n

n (n − 1) n |n − 2 = 0,

ˆ†l = 0, a ˆ†l a a ˆ†l ρth = f

p (n) n| a ˆ†l |n

n

n

ˆl a ˆ†l a

(47.27) (47.28)

a ˆ†l = T rf =

(47.26)

√ p (n) n + 1 n |n + 1 = 0,

= nl δ ll ,

(47.29) (47.30)

where we assumed that the modes, l and l , of the reservoir are not entangled with each other. Thus, the parameter β is zero for thermal reservoir. However, for squeezed vacuum the expectation value of a ˆl a ˆl is nonzero leading to the nonzero value of β. Indeed, for a reservoir initially in coherent state, |α , the density matrix operator ρ a, a† , to = |α α| is no longer diagonal in the number representation. From the expression, n 2 α 1 n |α = e− 2 |α| √ n!

we have 1

2

n| ρ a, a† , to |n = e− 2 |α|

αn − 12 |α|2 α∗n e n! n!

leading to nonzero average values of the creation and annihilation operators, a = α | a |α = |α | eiϑ , a† = α | a† |α = |α | e−iϑ .

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47.4.1

629

Thermal Reservoir

By virtue of Eqs. (47.27) and (47.28), the last two terms of Eq. (47.24) do not contribute, and so we end up with d I(2) ρ ˆ (t) dt A =−

1 |2 −

1 |2

−

1 |2

−

1 |2

∞

ds σ− (t−s ) ρIA (t) σ+ (t)−σ+ (t) σ − (t−s ) ρIA (t)

0 ∞

ds σ+ (t−s ) ρIA (t) σ− (t)−σ − (t) σ + (t−s ) ρIA (t)

A† (t) A (t−s )

ds σ− (t) ρIA (t) σ+ (t−s )−ρIA (t) σ+ (t−s ) σ − (t)

A (t−s ) A† (t)

ds σ+ (t) ρIA (t) σ− (t−s )−ρIA (t) σ− (t−s ) σ + (t)

A† (t−s ) A (t) .

0 ∞ 0 ∞

A (t) A† (t−s )

0

(47.31) Using Eqs. (47.10) and (47.13) we let the time-variable dependence in the interaction representation to be absorbed by reservoir operators, so we can write d I(2) 1 ρ ˆA (t) = − 2 σ − ρIA (t) σ + − σ + σ− ρIA (t) dt | −

1 σ + ρIA (t) σ − − σ − σ+ ρIA (t) |2

−

1 σ − ρIA (t) σ + − ρIA (t) σ + σ− |2

−

1 σ + ρIA (t) σ − − ρIA (t) σ − σ+ |2

∞ 0 ∞ 0 ∞ 0 ∞ 0

d1 A (2) A† (1) d1 A† (2) A (1) d1 A (1) A† (2) d1 A† (1) A (2) . (47.32)

Let us introduce the following variables A† (2) A (1) = −iS < (12) = Γ,

(47.33)

A (1) A† (2) = −iS > (12) = Γ + G,

(47.34)

A (2) A† (1) = Γ∗ ,

(47.35)

A† (1) A (2) = Γ∗ + G∗ .

(47.36)

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Note that from Eqs. (48.22) and (48.23), A (1) A† (2) − A† (2) A (1) = G = 2 Im S r bears the dissipative characteristics. Then we have d I(2) ρ ˆ (2) dt A =−

1 |2

−

1 |2

−

1 |2

−

1 |2

∞ 0 ∞ 0 ∞ 0 ∞ 0

d1 σ− (1) ρIA (2) σ+ (2) − σ+ (2) σ − (1) ρIA (2) Γ∗ d1 σ + (1) ρIA (2) σ − (2) − σ− (2) σ + (1) ρIA (2) Γ d1 σ − (2) ρIA (2) σ + (1) − ρIA (2) σ+ (1) σ − (2) (Γ + G) d1 σ + (2) ρIA (2) σ − (1) − ρIA (2) σ− (1) σ+ (2) (Γ∗ + G∗ ) . (47.37)

Consider the combination of the third and fourth lines in Eq. (47.37), we have the following grouping of terms, σ+ (1) ρIA (2) σ − (2) − σ− (2) σ + (1) ρIA (2) (Γr + iΓi ) ,

(47.38)

σ+ (2) ρIA (2) σ − (1) − ρIA (2) σ− (1) σ+ (2) (Γr − iΓi ) ,

(47.39)

σ+ (2) ρIA (2) σ− (1) − ρIA (2) σ− (1) σ+ (2) (Gr − iGi ) .

(47.40)

Adding Eqs. (47.38) and (47.39), yields 2σ+ (1) ρIA (2) σ− (2) − σ − (2) σ + (1) , ρIA (2)

Γr ,

(47.41)

Subtracting Eqs. (47.39) from (47.38), yields − σ− (2) σ + (1) , ρIA (2) iΓi .

(47.42)

where [x, y] stnads for the commutator of x and y, {x, y} is for the anti-commutator of x and y. Now consider the first and third lines in Eq. (47.37), we have the following group of terms, σ− (1) ρIA (2) σ+ (2) − σ+ (2) σ − (1) ρIA (2) (Γr − iΓi ) ,

(47.43)

σ− (2) ρIA (2) σ+ (1) − ρIA (2) σ+ (1) σ− (2) (Γr + iΓi ) ,

(47.44)

σ− (2) ρIA (2) σ+ (1) − ρIA (2) σ+ (1) σ− (2) (Gr + iGi ) .

(47.45)

Adding Eqs. (47.43) and (47.44) results in the following expressions, 2σ− (1) ρIA (2) σ+ (2) − σ + (2) σ− (1) , ρIA (2)

Γr ,

(47.46)

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631

Subtracting Eqs. (47.43) from (47.44) results in the following expressions, − ρIA (2) , σ + (1) σ− (2) iΓi = σ+ (1) σ − (2) , ρIA (2) iΓi .

(47.47)

Then we are left with combining Eqs. (47.40) and (47.45). Upon adding and subtracting, we have σ+ (2) ρIA (2) σ− (1) − ρIA (2) σ− (1) σ+ (2) (Gr − iGi ) ,

(47.48)

σ− (2) ρIA (2) σ+ (1) − ρIA (2) σ+ (1) σ− (2) (Gr + iGi ) ,

(47.49)

Regrouping the real and imaginary factors of Eqs. (47.48) and (47.49), we have for the real factor, Gr , σ+ (2) ρIA (2) σ− (1) + σ− (2) ρIA (2) σ+ (1)

Gr

−ρIA (2) σ− (1) σ+ (2) − ρIA (2) σ+ (1) σ− (2)

σ + (2) ρIA (2) σ − (1) + σ − (2) ρIA (2) σ + (1)

=

−ρIA (2) 12 (I − σz ) − ρIA (2) 12 (I + σz )

Gr ,

= σ+ (2) ρIA (2) σ− (1) + σ− (2) ρIA (2) σ+ (1) − ρIA (2) Gr ,

(47.50)

and for the imaginary factor, iGi , σ− (2) ρIA (2) σ+ (1) − σ+ (2) ρIA (2) σ− (1) −ρIA (2) σ+ (1) σ− (2) + ρIA (2) σ− (1) σ+ (2) =

iGi

σ − (2) ρIA (2) σ + (1) − σ+ (2) ρIA (2) σ − (1) −ρIA (2) 12 (I + σz ) + ρIA (2) 12 (I − σz )

iGi ,

= σ− (2) ρIA (2) σ+ (1) − σ+ (2) ρIA (2) σ− (1) − ρIA (2) σz iGi , =

σ − (2) ρIA (2) σ + (1) − σ+ (2) ρIA (2) σ − (1) + 12 σ z , ρIA (2) −

1 2

σ z ρIA (2)

iGi .

(47.51)

Collecting Eqs. (47.41) and (47.46) yields 2σ+ (1) ρIA (2) σ − (2) + 2σ − (1) ρIA (2) σ + (2) − σ− (2) σ + (1) , ρIA (2) − σ+ (2) σ− (1) , ρIA (2)

Γr .

Collecting Eqs. (47.42) and (47.47) yields − σ− (2) σ+ (1) , ρIA (2) + σ+ (1) σ− (2) , ρIA (2) =

−

1 1 (I − σ z ) , ρIA (2) + (I + σ z ) , ρIA (2) 2 2

= iΓi σ z , ρIA (2) .

iΓi iΓi ,

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Collecting Eqs. (47.50) and (47.51) yields σ + (2) ρIA (2) σ − (1) + σ− (2) ρIA (2) σ+ (1) − ρIA (2) Gr , σ − (2) ρIA (2) σ + (1) − σ+ (2) ρIA (2) σ − (1) − ρIA (2) σz iGi . Therefore collecting everything and substituting in the master equation to second order, we have d I(2) ˆρ (2) dt A =−

i |2

−

1 |2

−

1 |2

=−

i |2

∞ 0 ∞

1 d1 Γi + Gi 2

2σ+ (1) ρIA (2) σ− (2) + 2σ − (1) ρIA (2) σ + (2) − σ− (2) σ + (1) , ρIA (2) − σ+ (2) σ− (1) , ρIA (2)

d1Γr

0 ∞ 0 ∞ 0

σ z , ρIA (2)

d1Gr σ + (2) ρIA (2) σ − (1) + σ− (2) ρIA (2) σ+ (1) − ρIA (2) d1Gi σ − (2) ρIA (2) σ + (1) − σ + (2) ρIA (2) σ − (1) − σz , ρIA (2)

,

which simplifies to d I(2) ρ ˆ (2) dt A =−

i |2

−

2 |2

−

i |2

∞ 0 ∞

1 d1 Γi + Gi 2 d1 Γr +

0 ∞

d1Gi

0

Gr 2

σz , ρIA (2) σ+ (1) ρIA (2) σ− (2) +σ− (1) ρIA (2) σ+ (2) − ρIA (2)

σ− (2) ρIA (2) σ+ (1) −σ+ (2) ρIA (2) σ− (1) − σz , ρIA (2)

.

(47.52)

The contribution given by the term containing Γi + 12 Gi σ z corresponds to the correction of the atomic Hamiltonian, 1 Hatomic = |ωσ z , 2 by the correction term given by 1 HLS = 2 Γi + Gi σ z , 2 resulting in a ‘repulsive’ action between the energy levels.

(47.53)

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633

The two levels are shifted relative to each other. The first shift is due to Γi is proportional to the number of photons in the modes of the reservoir, this is analogous to the second order stark eﬀect of atoms in static electric field. The second shift due to Gi arises from the commutation of the fields and hence is a pure quantum eﬀect. Towards the end of the following chapter, we will demonstrate that the combined eﬀects represent the ‘mass’ kernel in the quantum superfield theoritical formulation of nonequilibrium physics. The meaning of Γr + G2r and iGi will become clear when we discussed the nonequilibrium superfield theoretical treatrment of the correlation functions of Eqs. (47.33) - (47.36) in the next chapter.

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Chapter 48

Stochastic Meaning of Nonequilibrium Quantum Superfield Theory

To gain more insights into the nature of nonequilibrium quantum superfield theory discussed in Part 4, we give here a discussion of the stochastic meaning of the correlation functions entering in the theory. A deeper understanding of the meaning of the correlation functions involved is very helpful, particularly in phenomenologically dealing with problems of describing a system interacting with a heat bath. Let us recall the following correlation functions of the theory [Eqs. (30.14) (30.29) of Part 4], ˆ (1) ψ ˆ † (2) = θ (12) i|G> (1, 2) + θ (21) i|G< (1, 2) = i|Gc (12) , T ψ L L

(48.1)

˜ (2) = T rρ ψ† (2) ψ (1) = −i|G< (1, 2) , ˆ L (1) ψ T ψ L H H H

(48.2)

˜ † (1) ψ ˆ † (2) = T rρ ψ (1) ψ † (2) = i|G> (1, 2) , T ψ L H H L H

(48.3)

˜ (2) = − θ (12) i|G< (1, 2) + θ (21) i|G> (1, 2) ˜ † (1) ψ T ψ L L = −i|Gac (12) ,

(48.4)

and the retarded and advanced correlation functions i|Gr (12) = θ (t1 − t2 ) T r ρH ψH (1) , ψ†H (2) Gr (12) = G> (1, 2) − G< (1, 2) θ (t1 − t2 ) , † (Gr (12)) = −G> (2, 1) + G< (2, 1) θ (t1 − t2 )

= − G> (1, 2) − G< (1, 2) θ (t2 − t1 )

= Ga (12)

634

,

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635

where Ga (12) = −θ (t2 − t1 ) T r ρH ψH (1) , ψ†H (2)

,

= − G> (1, 2) − G< (1, 2) θ (t2 − t1 ) . Now we also have the chronological and anti-chronological correlation functions, from Eqs. (48.1) and (48.4), Gc (12) = θ (12) G> (1, 2) + θ (21) G< (1, 2) , Gac (12) = θ (21) G> (1, 2) + θ (12) G< (1, 2) . Therefore, we obtain the following relations among these quantities            

Gr (12)† ⇒ Ga (12), Gc (12)† = −Gac (21), G> (12)† = −G> (21), G< (12)† = −G< (21), c ac gee (12)† = −ghh (21), > > † gee (12) = −ghh (21), < < † gee (12) = −ghh (21).

[iGc (12)]† = iGac (21), † [iG> (12)] = iG> (21), † [iG< (12)] = iG< (21), c ac [igee (12)]† = ighh (21), † > > [igee (12)] = ighh (21), † < < [igee (12)] = ighh (21).

We have further relations,

           

Gc (12) − G< (12) = G> (12) − Gac (12) = Gr (12) , Gc (12) − G> (12) = G< (12) − Gac (12) = Ga (12) , Gr (12) − Ga (12) = G> (1, 2) − G< (1, 2) = 2i Im Gr (12) , Im Gr (12) =

1 G> (1, 2) − G< (1, 2) , 2i

Gr (12) + Ga (12) = G> (1, 2) − G< (1, 2) [θ (t1 − t2 ) − θ (t2 − t1 )] = G> (1, 2) − G< (1, 2) [2θ (t1 − t2 ) − 1] = 2 Re Gr (12) ,

1 [2θ (t1 − t2 ) − 1] G> (1, 2) − G< (1, 2) 2 1 = sgn (t1 − t2 ) G> (1, 2) − G< (1, 2) , 2

Re Gr (12) =

(48.5)

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Gc (12) + (Gc (12)) = 2 Re Gc (12) , Gc (12) − Gac (12) = Gr (12) + Ga (12) = 2 Re Gr (12) , Gc (12) + Gac (12) = G> (1, 2) + G< (1, 2) = 2i Im Gc (12) . Similarly, we have gc (12) − g < (12) = g > (12) − gac (12) = gr (12) , g c (12) − g> (12) = g < (12) − g ac (12) = ga (12) , g c (12) − gac (12) = g r (12) + ga (12) , g c (12) + g ac (12) = g> (1, 2) + g < (1, 2) , α α and ghh of Eq. (48.5). which is obeyed by both gee

48.1

Kubo-Martin-Schwinger Condition

The Bose-Einstein distribution function, nBE , and Fermi-Dirac distribution function, fF D , are given by nBE (Ei ) =

gi , exp (β (Ei − µ)) − 1

fF D (E) =

1 , exp (β (E − µ)) + 1

respectively, where gi is the degeneracy of the bose systems. From these distribution functions, we have the following relations fF D (E) = exp (β (E − µ)) [1 − fF D (E)] ,

(48.6)

nBE (Ei ) = exp (β (Ei − µ)) [gi + nBE (Ei )] .

(48.7)

The above relations are important in making approximation to the correlation functions at steady state or at equilibrium conditions. At equilibrium the following expressions for the following fermionic field correlation functions holds −ihG< (p, E) = fF D (p, E) A (p, E) ,

(48.8)

i|G> (p, E) = [1 − fF D (p, E)] A (p, E) ,

(48.9)

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where A (p, E) is the spectral function, which is independent of temperature. For bosonic field correlation functions, the corresponding relations are −ihSi< (p, E) = nBE (Ei ) A (p, Ei ) ,

(48.10)

−i|Si> (p, E) = [gi + nBE (Ei )] A (p, Ei ) .

(48.11)

−G< (p, E) = exp (β (E − µ)) G> (p, E) ,

(48.12)

Therefore

which can also be written, by taking the inverse Fourier transform on the position variables only, as −G< (q, q , E) = exp (β (E − µ)) G> (q, q , E) .

(48.13)

This result can also be extended to the ‘self-energy’ correlation functions due to electron-electron interactions, namely, −Σ< (q, q , E) = exp (β (E − µ)) Σ> (q, q , E) .

(48.14)

Equations (48.12) - (48.14) are the expressions of the so-called fermionic KuboMartin_Schwinger (KMS) condition. One can deduce from these correlation functions the general space-time relations, expressed by −F < (q, q , t, t ) = F > (q, q , t, t + iβ) . For the boson field correlation functions at equilibrium, we have −iS < (p, E) = A (p, E) nBE (E) , −iS > (p, E) = A (p, E) [1 + nBE (E)] . Therefore −iS < (p, E) = −i exp (β (E − µ)) S > (p, E) , S < (p, E) = exp (β (E − µ)) S > (p, E) , hence, following the same procedure used for the fermions, we obtain for boson S < (q, q , t, t ) = exp (β (E − µ)) S > (q, q , t, t + iβ) ,

(48.15)

which is the bosonic KMS condition. Likewise this bosonic KMS condition holds for the bosonic self-energy, Σ< (q, q , E) = exp (β (E − µ)) Σ> (q, q , E) .

(48.16)

Note that bosonic KMS condition diﬀers from the fermionic KMS condition by a sign. We also have the following relations for interacting fermions, Σ> (q, q , E) + Σ< (q, q , E) = Σ> (q, q , E) − exp (β (E − µ)) Σ> (q, q , E) = [1 − exp (β (E − µ))] Σ> (q, q , E) ,

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Σ> (q, q , E) − Σ< (q, q , E) = Σ> (q, q , E) + exp (β (E − µ)) Σ> (q, q , E) = [1 + exp (β (E − µ))] Σ> (q, q , E) . Therefore, [1 − exp (β (E − µ))] Σ> (q, q , E) + Σ< (q, q , E) = Σ> (q, q , E) − Σ< (q, q , E) [1 + exp (β (E − µ))] =

− exp exp − β(E−µ) 2

β(E−µ) 2

exp − β(E−µ) + exp 2

β(E−µ) 2

=−

exp

β(E−µ) 2

− exp − β(E−µ) 2

exp

β(E−µ) 2

+ exp − β(E−µ) 2

= − tanh

β (E − µ) 2

=−

1 coth

β(E−µ) 2

.

For bosonic self-energy, we have [1 + exp (β (E − µ))] Σ> (q, q , E) + Σ< (q, q , E) = > < Σ (q, q , E) − Σ (q, q , E) [1 − exp (β (E − µ))] β (E − µ) = coth . 2 Therefore, we can write for both fermionic and bosonic self-energies the following relations, Σ> (q, q , E) + Σ< (q, q , E) = ± coth Σ> (q, q , E) − Σ< (q, q , E)

β (E − µ) 2

±1

,

where the ‘+’ sign refers to bosonic self-energies and the ‘−’ sign to fermionic selfenergies. Note that these relations are quite general irrespective of the exact form of the self-energies at steady state or equilibrium conditions. The relation Σ> (q, q , E) + Σ< (q, q , E) = coth

β (E − µ) 2

Σ> (q, q , E) − Σ< (q, q , E) ,

hence Σ> (q, q , t, t ) + Σ< (q, q , t, t ) i 1 β (E − µ) dE e | E(t−t ) coth = 2π 2

Σ> (q, q , E) − Σ< (q, q , E) ,

has been referred to in the literature as the fluctuation-dissipation theorem for bosons. For fermionic systems this would take the form of Σ> (q, q , t, t ) + Σ< (q, q , t, t ) =−

1 2π

dE e | E(t−t ) coth i

β (E − µ) 2

−1

Σ> (q, q , E) − Σ< (q, q , E) .

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All these self-energy relations translate to the relations for G< and G> for fermions and S < and S > for bosons, respectively. In the fluctuation-dissipation viewpoint, the term [Σ> (q, q , t, t ) + Σ< (q, q , t, t )] characterizes the fluctuations or ‘noise’ and hence [Σ> (q, q , E) − Σ< (q, q , E)] is responsible for the dissipation. 48.1.1

Mass, Dissipation, and Noise Kernels in Nonequilibrium Quantum Superfield Theory

To gain further insights into the transport equations of the quantum superfield theory of nonequilibrium physics discussed in Part 4, we will cast the equation in terms of mass, dissipation and noise kernels. Let us now introduce the following quantities for the self-energies Re Σr (12) = Σr (12) + Σa (12) = Σ> (1, 2) − Σ< (1, 2) [θ (t1 − t2 ) − θ (t2 − t1 )] = Σ> (1, 2) − Σ< (1, 2) [2θ (t1 − t2 ) − 1] , 1 r [Σ (12) − Σa (12)] 2i 1 = Σ> (1, 2) − Σ< (1, 2) , 2i

(48.17)

Im Σr (12) =

Im Σc (12) = Σc (12) + Σac (12) 1 = Σ> (1, 2) + Σ< (1, 2) . 2i

(48.18)

(48.19)

Let us now consider Eq. (31.23) of Part 4. For simplicity in the following considerations, we will omit the ‘pairing’ terms and write Eq. (31.23) simply as i|

∂ ∂ + ∂t1 ∂t2

G< = v, G< + Σr G< − G< Σa + Σ< Ga − Gr Σ< ,

(48.20)

where v ≡ |ω o is the local (diagonal) single-particle energy. We wish to express Eq. (48.20) in terms of the quantities defined in Eqs. (48.17) - (48.19). First we recast Eq. (48.20) as i|

∂ ∂ + ∂t1 ∂t2

G< = v + Re Σr , G< + i Im Σr , G< + Σ< Ga − Gr Σ< .

(48.21) From Eqs. (48.18) and (48.19), we can express Σ< in terms of Im Σr (12) and Im Σc (12), Σ< = i (Im Σc + Im Σr ) , Σ> = i (Im Σc − Im Σr ) .

(48.22) (48.23)

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Upon substituting in Eq. (48.21), we have i|

∂ ∂ + ∂t1 ∂t2

G<

= v + Re Σr , G< + i Im Σr , G< + i [(Im Σr + Im Σc ) Ga − Gr (Im Σr + Im Σc )] = v + Re Σr , G< + i Im Σr , G< + i [(Im Σr ) Ga − Gr (Im Σr )] + i [(Im Σc ) Ga − Gr (Im Σc )] , i|

∂ ∂ + ∂t1 ∂t2

G< = v + Re Σr , G< + i Im Σr , G<

− Gr (i Im Σr ) + [Gr (i Im Σr )]† − Gr (i Im Σc ) + [Gr (i Im Σc )]† , i|

∂ ∂ + G< ∂t1 ∂t2 = v + Re Σr , G< + i Im Σr , G< − 2 Re [Gr (i Im Σr )] − 2 Re [Gr (i Im Σc )] = v + Re Σr , G< + i Im Σr , G< + 2 [Im Gr Im Σr ] + 2 [Im Gr Im Σc ] .

Writing ∆vΣ (12) = Re Σr (12) , A (12) = −2 Im Gr , Γ (12) = −2 Im Σr , N (12) = −2 Im Σc ,

(48.24)

we therefore have i|

∂ ∂ + ∂t1 ∂t2

G< = v + ∆vΣ , G< −

i 1 1 Γ, G< + A Γ + AN . 2 2 2

(48.25)

We note that Eq. (48.25), which is related to the particle number density, contains all the quantities defined in Eq. (48.24), notably including the last term involving the noise correlator N . In contrast, the equations for the retarded and advanced correlation functions, Gr and Ga , respectively, do not involve the correlator N, i|

∂ ∂ + ∂t1 ∂t2

1 Gr = v + ∆vΣ − i Γ, Gr , 2

i|

∂ ∂ + ∂t1 ∂t2

1 Ga = v + ∆vΣ + i Γ, Ga . 2

These Gr and Ga correlation functions essentially describe the energetics of the propagation of particles or excitations, and are determined by the ‘mass shift’ term,

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∆vΣ , and dissipation term, Γ. A well-known approximation to the spectral function A = −2 Im Gr is given by the Lorentzian shape function, A (p, E) ⇒

Γ 1 2 π (E − (v + ∆vΣ ))2 +

, Γ 2 2

which goes into a Dirac-delta function, δ (E − (v + ∆vΣ )), in the limit that Γ ⇒ 0. For bosons the corresponding approximation is A (p, E) ⇒

1 π

Γ 2

|2 ω 2 − (v + ∆vΣ )2

2

+

Γ 2 2

,

which goes into a Dirac-delta function, δ |2 ω 2 − (v + ∆vΣ )2 , in the limit that Γ ⇒ 0. Remarkably, the particle number density of Eq. (48.25) is additionally influenced by the fluctuations through the noise correlator N . This can be understood simply by the following argument. Associating Σ< with scattering-in process and Σ> with the scattering-out interaction process, it is clear that Σ> − Σ< is associated with the net loss in phase space, i.e., dissipation. In contrast, for the number density, both the scattering-in process and the scattering-out interaction process produces fluctuations in the number density, precisely the reason why Σ> + Σ< only enters in the equation for the number density matrix equation. 48.2

A Two-State System Interacting with a Heat Bath

To further discuss the stochastic interpretation of the correlation functions which entered in quantum superfield theory of nonequilibrium physics, it is helpful to consider the perturbative treatrment of the density operator of a whole system consisting of a subsystem, A, interacting with the environmental heat bath or reservoir, B, described by the Hamiltonian in the Schrödinger representation as ˆ =H ˆ0 + H ˆ1. H

(48.26)

ˆ 0 = HA + HB , and H ˆ1 = H ˆ AB . H ˆ A is the Hamiltonian of system A and H ˆB where H ˆI = H ˆ AB involves both the is the Hamiltonian of the environment system, B, only H system and environment degrees of freedom. Remark 48.1 The two-state problem is ubiquitous in quantum physical systems. This sort of problem usually possesses eigenvalues that can assume only two diﬀerent values. Some familiar example are the system of spin- 12 , the polarization of photons, and ‘two-site’ system and the emitter and quantum-well interacting system in a resonant tunneling diode operating as a THz source (discussed in Part 3). A more common situation in chemistry and biological systems is when these generally multistate systems only two states are relevant for certain purposes. In practice, any twostate system is coupled to the environment. In the two-site, two-well, or two groundstate problems, the ‘tunneling dynamics’ between the two ‘eigenposition’ occurs in the

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presence of phonons. The quantum mechanics of these two-state systems is really a ‘subset’ of the discrete quantum mechanics of Bloch electrons in crystals based on finite prime-number of lattice points obeying the Born-von Karmann boundary condition, derived from discrete quantum mechanics discussed in Part 1 and Part 5. We assume the interaction Hamiltonian of a qubit, A, coupled to a reservoir, B, consisting of infinitely many field modes is given by Eq. (47.12), which we rewrite here for phonon modes as HIIR

σ+ A (t) + σ− A† (t) ,

(48.27)

where A (t) are made up of all the modes, A (t) =

gl (R) a ˆl e−i∆l t , l

HIR I

and is in the interaction representation. The corresponding wavefunction in the interaction representation obeys ∂t |ΨI (t) =

1 ˆ HI (t)|ΨI (t) , ı|

whose solution is formally given by ˆI (t, 0)|Ψ(0) , |ΨI (t) = U t ˆ I (t ) . ˆI (t, 0) = Tˆ← exp −ı dt H U | 0

(48.28) (48.29)

Hence the density matrix obeys ˆ I (t), ˆ ı|∂t ˆ ρI (t) = [H ρI (t)],

(48.30)

here ρ ˆI (t) is in the interaction representation, ˆI (t, 0)ˆ ˆ † (t, 0). ρ(0)U ρ ˆI (t) = U I

(48.31)

which is in the form of a Kraus representation of quantum operation. Equation (48.30) has formal solution ˆρI (t) = ρ ˆ(0) +

1 ı|

t

ˆ I (t ), ρ dt [H ˆI (t )],

(48.32)

0

which gives 1 ˆ 1 d ρ ˆ (t) = [H ˆ(0)] − 2 I (t), ρ dt I ı| |

t

ˆ I (t), [H ˆ I (t ), ˆ dt [H ρI (t )]].

(48.33)

0

Tracing over the environment on both sides of Eq. (48.33) gives 1 1 d ˆ I (t), ρ ρ ˆ (t) = T rB [H ˆ(0)] − 2 dt A ı| |

t

ˆ I (t), [H ˆ I (t ), ρ dt T rB [H ˆI (t )]]. 0

(48.34)

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The first term on the RHS of (48.34) is zero by virtue of Eqs. (47.26) and (47.29) ˆ I (t)ˆ T rB [H ρI (0)] = 0. Thus, assuming weak coupling of the A and B systems, the density matrix factors approximately at all times into ρI (t) = ρ ˆ ˆA (t) ⊗ ρ ˆB , where ρ ˆB is independent of time. Therefore, write can write Eq. (48.34) as t

1 d ρ ˆA (t) = − 2 dt |

0

ˆ I (t), [H ˆ I (t ), ρ dt T rB [H ˆA (t ) ⊗ ρ ˆB ]].

(48.35)

Consistent with the weak coupling limit, we will assume that the ‘memory’ kernel in Eq. (48.35) is suﬃciently short that the subsystem density matrix involved can be substituted by its current value, ˆ ρA (t ) → ρ ˆA (t). This corresponds to the memory timescale of environment B, by virtue of its interaction with subsystem A, that is very short and the environment is approximately stationary or very-fast relaxing. Hence, we have 1 d ρ ˆ (t) = − 2 dt A |

t 0

ˆ I (t), [H ˆ I (t ), ˆ dt T rB [H ρA (t) ⊗ ˆ ρB ]].

(48.36)

Equation (48.36) is known as the Redfield equation. It only involves ˆ ρA (t), but the resulting kernel still contains an explicit reference to the ‘starting time’ at t = 0. This dependence on the past can be made explicit by substituting t = t − s , in terms of which t

d 1 ρ ˆ (t) = − 2 dt A |

0

ˆ I (t), [H ˆ I (t − s ), ρ ds T rB [H ˆA (t) ⊗ ρ ˆB ]].

(48.37)

By virtue of the restrictive range of the resulting integral kernel, we can extend the integral on the RHS of equation (48.37) to infinity without significantly altering the results,1 ∞

1 d ρ ˆ (t) = − 2 dt A |

0

ˆ I (t), [H ˆ I (t − s ), ρ ds T rB [H ˆA (t) ⊗ ρ ˆB ]].

(48.38)

1 A exemplary situation motivating the assumptions leading to Eqs. (48.37) and (48.38) occurs in which there is a complete lack of memory as typified by the following integral t

t

ˆ t dt δ t − t ρ

ρ (t) dt δ t − t ˆ

=

0

0 ∞

dt δ t − t ˆ ρ (t)

= 0

=ˆ ρ (t)

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Equation (48.38) depends only on the current density matrix ˆ ρA (t) and contains no explicit reference to any other time, hence it is fully Markovian. Indeed, we apρA (0) d proximately have dt ρ ˆA (t) ≈ Limτ ⇒0 ρˆA (t)−ˆ , where τ is of the order of memory τ timescale. We make use of Eq. (47.23) to evaluate the right hand side of Eq. (48.38). Substituting t1 = (t − s ) and t2 = t, the dependence on the past can be written explicitly, and the result isLet 1 ≡ t − s and 2 ≡ t, with ds = −d1. Then we have d ρ ˆ (2) dt A

48.3

=−

1 |2

−

1 |2

−

1 |2

−

1 |2

∞ 0 ∞ 0 ∞ 0 ∞ 0

d1 σ− (1) ρIA (2) σ+ (2) − σ+ (2) σ − (1) ρIA (2)

A (2) A† (1)

d1 σ + (1) ρIA (2) σ − (2) − σ − (2) σ + (1) ρIA (2)

A† (2) A (1)

d1 σ − (2) ρIA (2) σ + (1) − ρIA (2) σ+ (1) σ − (2)

A (1) A† (2)

d1 σ + (2) ρIA (2) σ − (1) − ρIA (2) σ− (1) σ + (2)

A† (1) A (2) .

Nonequilibrium Quantum Superfield Theory Correlations

To gain more insight concerning the noise and dissipation kernel, let us rewrite the master equation to the second order using the correlation function of quantum superfield theory of nonequilibrium physics treated in Part 4. Rewriting the master equation in second order, Eq. (47.31), we have d 1 ρ ˆ (t) = − σ + σ− ρIs (t) − σ − ρIA (t) σ+ 2 dt A |

−

σ − σ+ ρIs

(t) −

σ + ρIA (t) σ−

− ρIs (t) σ+ σ− − σ− ρIA (t) σ+

−

ρIs

− +

(t) σ σ −

σ+ ρIA (t) σ−

∞

1 |2 1 |2 1 |2

ds A (t) A† (t − s)

0 ∞

ds A† (t) A (t − s)

0 ∞

ds A (t − s) A† (t)

0 ∞

ds A† (t − s) A (t) . (48.39)

0

In order to express the correlations in terms of the correlations of nonequilibrium quantum superfield theory, we make a transformation from the correlations defined

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over positive times onto correlations defined over the entire time axis. Therefore, we now define the following correlation functions, ∞

0

ds A (t) A† (t − s) ∞

= −∞

ds A (t) A† (t − s) θ (s) ∞

= −i

−∞

−i = 2

dsS > (t, t − s) θ (s)

∞

−∞

ds S > (t, t − s) (θ (s) + θ (−s))

∞

−i + 2

−∞

−i = 2

∞

−∞

ds S > (t, t − s) (θ (s) − θ (−s))

ds S > (s) + S > (s) (θ (s) − θ (−s)) ,

Similarly, we have for the complex conjugate correlation the following transformation, ∞

ds A† (t) A (t − s)

0 ∞

= −∞

ds A† (t) A (t − s) θ (s) ∞

= −i

=

−i 2

−∞

dsS < (t − s, t) θ (s)

∞

−∞

−i = 2

∞

−∞

ds S < (t − s, t) (θ (s) + θ (−s)) + S < (t − s, t) (θ (s) − θ (−s))

ds S < (−s) + S < (−s) (θ (s) − θ (−s)) ,

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Using similar procedure, we have for the adjoint correlation ∞

0

ds A (t − s) A† (t) ∞

= −∞

= −i =

=

−i 2

ds A (t − s) A† (t) θ (s) ∞

−∞ ∞

dsS > (t − s, t) θ (s)

−∞ ∞

−i 2

−∞

ds S > (t − s, t) (θ (s) + θ (−s)) + S > (t − s, t) (θ (s) − θ (−s)) ds S > (−s) + S > (−s) (θ (s) − θ (−s)) ,

and also for the corresponding complex conjugate correlation, ∞

ds A† (t − s) A (t)

0 ∞

= −∞

= −i =

=

−i 2 −i 2

ds A† (t − s) A (t) θ (s) ∞

−∞ ∞

dsS < (t, t − s) θ (s)

−∞ ∞

−∞

ds S < (t, t − s) (θ (s) + θ (−s)) + S < (t, t − s) (θ (s) − θ (−s)) ds S < (s) + S < (s) (θ (s) − θ (−s)) .

Summarizing, we have the following transformations, ∞

ds A (t) A† (t − s) =

−i 2

0 ∞

0

−i ds A (t) A (t − s) = 2 †

∞

−∞ ∞

−∞

ds S > (s) + S > (s) (θ (s) − θ (−s)) ,

(48.40)

ds S < (−s) + S < (−s) (θ (s) − θ (−s)) , (48.41)

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0

−i ds A (t − s) A (t) = 2 †

∞

∞

−∞

−i ds A (t − s) A (t) = 2 †

0

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ds S > (−s) + S > (−s) (θ (s) − θ (−s)) , (48.42)

∞

−∞

ds S < (s) + S < (s) (θ (s) − θ (−s)) .

(48.43)

We will later make use of the following identity for any function, F (s), ∞

∞

dsF (s) ≡ 0

dsF (s) θ (s) , −∞

where θ (s) is the step function. Moreover, we have for any integral the identity ∞

∞

ds F (s) = −∞

ds F (−s) . −∞

We will also make use of the relations, σ + σ− = (1 + σz ) , σ − σ+ = (1 − σz ) . Moreover, we will also make us of the following relations, which assigns mass kernel, M (12), the dissipative kernel, D (12), and the noise kernel, N (12), 1 S > (1, 2) − S < (1, 2) [2θ (t1 − t2 ) − 1] ⇒ M (12) 2 1 Re S r (s) = (48.44) S > (s) − S < (−s) [(θ (s) − θ (−s))] ⇒ M (s) , 2

Re S r (12) =

1 S > (1, 2) − S < (1, 2) ⇒ D (12) 2i 1 S > (s) − S < (−s) ⇒ D (s) , Im S r (s) = 2i

Im S r (12) =

1 > S (1, 2) + S < (1, 2) ⇒ N (12) 2i 1 Im S c (s) = S > (s) + S < (−s) ⇒ N (s) . 2i

(48.45)

Im S c (12) =

(48.46)

It is more convenient to establish useful relations by adding and taking the diﬀerence between pair of equations in Eqs. (48.40) - (48.43).

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Adding and subtracting Eqs. (48.40) and (48.41) yields ∞

∞

†

ds A (t) A (t − s) + 0

ds A† (t) A (t − s)

0

=

−i 2

∞

ds S > (s) + S > (s) (θ (s) − θ (−s))

−∞ ∞

−i 2

+

∞

= −∞

−∞

ds S < (−s) + S < (−s) (θ (s) − θ (−s)) ∞

ds (N (s) + N (s) (θ (s) − θ (−s))) = 2

ds N (s) (θ (s)) , −∞

and ∞

∞

†

ds A (t) A (t − s) − 0

ds A† (t) A (t − s)

0

=

−i 2

∞

ds S > (s) + S > (s) (θ (s) − θ (−s))

−∞ ∞

−

−i 2

∞

= −∞

−∞

ds S < (−s) + S < (−s) (θ (s) − θ (−s))

ds [D (s) − iM (s)] .

Adding and subtracting Eqs. (48.42) and (48.43) yields ∞

†

∞

ds A (t − s) A (t) + 0

ds A† (t − s) A (t)

0

=

−i 2

∞

ds S > (−s) + S > (−s) (θ (s) − θ (−s))

−∞ ∞

+

=

−i 2

−i 2 ∞

= −∞

−∞ ∞

−∞

ds S < (s) + S < (s) (θ (s) − θ (−s))

ds S > (s) + S < (−s) −

−i 2

∞

−∞

ds S > (s) + S < (−s) (θ (s) − θ (−s))

ds {N (s) − N (s) (θ (s) − θ (−s))} = 2

∞

−∞

dsN (s) {θ (−s)} ,

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and ∞

∞

†

ds A (t − s) A (t) − 0

ds A† (t − s) A (t)

0

=

−i 2

∞

ds S > (−s) + S > (−s) (θ (s) − θ (−s))

−∞ ∞

− =

−i 2

−i 2 ∞

= −∞

−∞ ∞

ds S < (s) + S < (s) (θ (s) − θ (−s))

ds −∞

S > (s) − S < (−s) − S > (s) − S < (−s) (θ (s) − θ (−s))

ds {D (s) + iM (s)} .

Summarizing, we have the correlations defined over positive times in terms of the mass, dissipation, and noise kernels, defined over the entire time axis, ∞

∞

ds A (t) A† (t − s) +

∞

ds A† (t) A (t − s) = 2

0

0

∞

∞ †

∞

ds A (t) A (t − s) =

0

0

∞

∞ †

∞

ds A (t − s) A (t) = 2 0

∞

∞ †

−∞

dsN (s) {θ (−s)} , (48.49)

∞ †

ds A (t − s) A (t) − 0

−∞

ds [D (s) − iM (s)] , (48.48)

†

ds A (t − s) A (t) +

(48.47)

−∞

†

ds A (t) A (t − s) −

0

ds N (s) (θ (s)) ,

ds A (t − s) A (t) = 0

−∞

ds {D (s) + iM (s)} . (48.50)

Therefore we can express each ‘+’ time correlations in terms of the nonequilibrium quantum superfield theoretical correlations, ∞

2

ds A (t) A† (t − s) =

0 ∞

2 0

†

ds A (t) A (t − s) =

∞

−∞

ds {2N (s) (θ (s)) + D (s) − iM (s)} ,

∞

−∞

ds {2N (s) (θ (s)) − D (s) + iM (s)} ,

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Nonequilibrium Quantum Transport Physics in Nanosystems ∞

2

∞

2

∞

†

ds A (t − s) A (t) =

0

−∞

ds {2N (s) (θ (−s)) + D (s) + iM (s)} ,

∞

†

ds A (t − s) A (t) = 0

48.4

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−∞

ds {2N (s) (θ (−s)) − D (s) − iM (s)} .

Lamb Shift, Dissipation Kernel, and Noise Kernel

Consider the ‘mass term’ (M (s)) in the master equation, Eq. (48.39), we have d ρ ˆ (t) dt A

M

=−

1 2|2

=−

1 2|2

=−

1 2|2

=−

1 |2

∞

ds iM (s)

σ+ σ − ρIA (t) − ρIA (t) σ+ σ − − σ− σ + ρIA (t) − ρIA (t) σ− σ +

ds iM (s)

(I + σ z ) ρIA (t) − ρIA (t) (I + σz ) − (I − σz ) ρIA (t) − ρIA (t) (I − σ z )

ds iM (s)

σ z ρIA (t) − ρIA (t) σz + (σ z ) ρIA (t) − ρIA (t) (σ z )

−∞ ∞

−∞ ∞

−∞ ∞

ds iM (s) σz , ρIA (t) .

(48.51)

−∞

Therefore the ‘mass term’ corresponds to the Lamb Shift in atomic physics, with the Lamb shift Hamiltonian, HLS = M (s) σz . Consider the dissipative, (D (s)), term which gives d ρ ˆ (t) dt A

D

1 =− 2 2| −

1 2|2

−

1 2|2

−

1 2|2

∞

−∞ ∞

−∞ ∞

−∞ ∞

−∞

ds σ− ρIA (t) σ + − σ + σ− ρIA (t) D (s) ds σ+ ρIA (t) σ − − σ − σ+ ρIA (t) (−D (s)) ds σ− ρIA (t) σ + − ρIA (t) σ+ σ− (D (s)) ds σ+ ρIA (t) σ − − ρIA (t) σ− σ+ (−D (s)) ,

(48.52)

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d ρ ˆ (t) dt A

∞

1 =− 2 2|

D

2σ− ρIA (t) σ + + σ− σ + , ρIA (t) − 2σ+ ρIA (t) σ− + σ+ σ− , ρIA (t)

ds −∞

651

D (s) ,

Now consider the noise, N (s), terms d ρ ˆ (t) dt A

N

1 =− 2 2| 1 − 2 2| 1 − 2 2| 1 − 2 2|

∞

−∞ ∞

−∞ ∞

−∞ ∞

−∞

ds σ − ρIA (t) σ+ − σ+ σ − ρIA (t) 2N (s) (θ (s))

ds σ + ρIA (t) σ− − σ− σ + ρIA (t) 2N (s) (θ (s))

ds σ − ρIA (t) σ+ − ρIA (t) σ+ σ− 2N (s) (θ (−s))

ds σ + ρIA (t) σ− − ρIA (t) σ− σ+ 2N (s) (θ (−s)) ,

which reduces to the following expression, d ˆρ (t) dt A

N

1 =− 2 | 1 − 2 | 1 − 2 | 1 − 2 |

∞

ds

σ− ρIA (t) σ + N (s) [θ (s) + θ (−s)]

ds

σ+ ρIA (t) σ − N (s) [θ (s) + θ (−s)]

−∞ ∞

−∞ ∞

−∞ ∞

−∞

ds −σ+ σ − ρIA (t) N (s) θ (s) − ρIA (t) σ + σ− N (s) θ (−s)

ds −σ− σ + ρIA (t) N (s) θ (s) − ρIA (t) σ − σ+ N (s) θ (−s) ,

which further simplifies as d ρ ˆ (t) dt A

N

1 =− 2 |

∞

ds

σ − ρIA (t) σ + + σ + ρIA (t) σ− N (s)

−∞

1 + 2 |

∞

−∞

ds ρIA (t) N (s)

(48.53)

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Collecting our results yields d d ρ ˆA (t) = ρ ˆ (t) dt dt A =−

1 |2

∞

+ M

d ˆρ (t) dt A

+ D

d ρ (t) ˆ dt A

N

ds iM (s) σ z , ρIA (t)

−∞

1 − 2 2|

∞

−∞

ds D(s)([2σ− ρIA (t)σ+ + {σ− σ+ , ρIA (t)}]

− [2σ + ρIA (t)σ − + {σ+ σ − , ρIA (t)}]) 1 − 2 |

∞

ds N (s)

σ− ρIA (t) σ +

+ σ + ρIA (t) σ−

−∞

1 + 2 |

∞

ds N (s) ρIA (t) ,

−∞

which can be further reduced to the expression, d d ρ ˆA (t) = ρ ˆ (t) dt dt A =−

−

1 |2

∞

M

d ρ ˆ (t) dt A

+ D

d ρ ˆ (t) dt A

N

ds iM (s) σz , ρIA (t)

−∞

1 2|2

1 − 2 |

+

∞

−∞ ∞

−∞

ds D (s) 2σ − ρIA (t) σ+ − 2σ + ρIA (t) σ − − 2 σz , ρIA (t)

ds N (s) σ − ρIA (t) σ + + σ + ρIA (t) σ− − ρIA (t) .

(48.54)

where the first term corresponds mass kernel, M (s), giving the Lamb shift, the second term involves the dissipation kernel, D (s), and the last term involves the noise kernel, N (s). If the heat bath is in thermal equilibrium, it is easy to see that all the temperature-dependence is contained in the noise kernel N (s), by virtue of Eqs. (48.6) - (48.14). Therefore, N (s) is appropriately referred to as the noise kernel. 48.4.1

Comparison with the Master Equation of Sec. 47.4

We now compare Lamb shift result of Eq. (48.54) with our previous result given by Eq. (47.52) given in Sec. 47.4. First, we establish the relations between the correlations defined for positive times only in Sec. 47.4, Eqs. (47.33) - (47.36), and the quantum superfield correlation functions of Eqs. (48.44) -(48.46) defined for the

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whole time axis. We will transform the positive time correlation functions to the correlation fucntions defined over the whole time axis. Thus A (1) A† (2) − A† (2) A (1) †

A (2) A (1)

= G = Gr + iGi =⇒ −i S > (s) − S < (s) θ (s) ,

= Γ = Γi + Γr =⇒ −iS < (s) θ (s) ,

(48.55) (48.56)

where θ (s) is the Heaveside-step function. These have to be compared with the correlations functions of Eqs. (48.44) -(48.46), namely, M (s) =

1 > {S (s) − S < (−s)} [(θ (s) − θ (−s))] , 2

(48.57)

D (s) =

1 > [S (s) − S < (−s)] [(θ (s) + θ (−s))] 2i

(48.58)

1 > S (s) + S < (−s) [(θ (s) + θ (−s))] . (48.59) 2i in terms of mass kernel, dissipation kernel, and noise kernel. We have to show that the Lamb shift has the following correspondence, N (s) =

∞

i 0

1 ds Γi (s) + Gi (s) 2

∞

=⇒

ds iM (s) . −∞

First, we write ∞

dsΓi (s) =

0

=

=

=

1 2i 1 2i 1 2i i 2i

∞ 0 ∞

−∞ ∞

−∞ ∞

−∞

ds (Γ − Γ∗ )

ds −iS < (−s) θ (s) − −iS < (−s) θ (s)

∗

ds −iS < (−s) θ (s) + iS > (s) θ (s) ds S > (s) − S < (−s) θ (s) .

Thus, we can also write ∞ 0

1 dsΓi (s) = 4

∞

−∞

1 + 4

ds S > (s) − S < (−s) [θ (s) − θ (−s)] ∞

−∞

ds S > (s) − S < (−s) [θ (s) + θ (−s)] .

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Using the definition for the mass kernel and dissipation kernel, Eq. (48.57) and Eq. (48.58), respectively, we therefore have ∞ 0 ∞

i

1 dsΓi (s) = 2 dsΓi (s) =

0

1 2

∞

−∞ ∞

−∞

∞

i dsM (s) + 2

dsD (s) , −∞

ds [iM (s) − D (s)] ,

(48.60)

where the last line agrees with Eq. (48.48). Now consider the expression of Gi (s). We have from Eq. (48.55) ∞

2i

∞

dsGi = −i 0

= −i = −i = −i

−∞ ∞

−∞ ∞

−∞ ∞

−∞

>

∞

<

ds S (s) − S (s) θ (s) + i ds S > (s) − S < (s) θ (s) − i ds S > (s) − S < (s) θ (s) − i

−∞ ∞

−∞ ∞

−∞

ds S < (−s) − S > (−s) θ (s) ds S > (−s) − S < (−s) θ (s) ds S > (s) − S < (s) θ (−s)

ds S > (s) − S < (s) [θ (s) + θ (−s)] .

Therefore, we obtain ∞

i 0

1 i ds Gi = − 2 4 =

1 2

∞

−∞ ∞

ds S > (s) − S < (s) [θ (s) + θ (−s)]

dsD (s) .

(48.61)

−∞

Combining Eqs. (48.60) and (48.61) we finally obtain ∞

i 0

1 ds Γi (s) + Gi (s) 2

∞

=

ds iM (s) , −∞

which shows that indeed the Lambda shift term of Eq. (47.52) is the mass kernel of Eq. (48.54). The virtue of the quantum superfield nonequilibrium technique is the systematic grouping of the transport kernel into the physically meaningful mass kernel, dissipation kernel, and noise kernel. Finally comparing Eq. (47.52) with Eq. (48.54), we have the following relations between the rest of the correlations defined for positive time and the dissipation

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and noise kernels. We have as before, ∞

∞

ds M (s) = 0

−∞

1 d1 Γi + Gi , 2

We also identify ∞

∞

ds iGi =

0

ds D (s) , −∞

as given by Eq. (48.61), and ∞

2 0

Gr ds Γr + = 2

∞

ds N (s) , −∞

so that 2 Γr + G2r corresponds to the noise kernel of the quantum superfield theory. This noise correlation relation can also be determined and verified by obtaining the expression of Gr and Γr from Eqs. (48.47) and (48.49). Subtracting Eq. (48.49) from Eq. (48.47), we have ∞

0

∞

†

ds A (t) A (t − s) + ∞

−

0

ds A† (t) A (t − s)

ds A (t − s) A† (t) +

∞



ds A† (t − s) A (t) 

0  ∞0  ∞   = ds A† (t) A (t − s) − ds A (t − s) A† (t)   0 0  ∞ ∞   ds A (t) A† (t − s) − ds A† (t − s) A (t) +   0

∞

=−

0

ds {G (s) + G∗ (s)} = −2Gr

0 ∞

=2 −∞

ds N (s) (θ (s) − θ (−s)) .

where the last equality makes use of Eq. (48.47) and (48.49). Therefore 1 1 Gr = − 2 2

∞

−∞

ds N (s) (θ (s) − θ (−s)) .

(48.62)

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Nonequilibrium Quantum Transport Physics in Nanosystems

Equation (48.47) immediately give us the expression for Γr , i.e., ∞

∞

†

ds A (t) A (t − s) + 0

∞

†

ds A (t) A (t − s) = 2

ds N (s) (θ (s))

−∞ ∞

0

ds (Γ + Γ∗ )

= 0

∞

ds Γr .

=2 0

We write ∞

0

1 ds Γr = 2

∞

∞

1 ds N (s) (θ (s) + θ (−s))+ 2

−∞

−∞

ds N (s) (θ (s) − θ (−s)) . (48.63)

Therefore, by combining Eq. (48.62) and Eq. (48.63), we end up with ∞

1 ds Γr + Gr 2

1 = 2

∞

ds N (s) (θ (s) + θ (−s)) , −∞

0

or ∞

2 0

1 ds Γr + Gr 2

∞

ds N (s) ,

= −∞

which verified the identification made from comparing the master equations.

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PART 8

Quantum Computing and Quantum Information: Discrete Phase Space Viewpoint

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Chapter 49

Discrete Phase Space Viewpoint

We continue the the language of Wannier states (q-states) and Bloch states (pstates), by viewing quantum computing through a discrete quantum mechanical point of view.

49.1

Quantum Teleportation

Suppose Alice has a qubit that she wants to teleport to Bob. This qubit can be written generally as: |ψ = α |0 U + β |1 U . Our quantum teleportation scheme requires Alice and Bob to share a maximally entangled state beforehand, for instance one of the four Bell states1 Φ+

AB

1 = √ (|0 2 1 = √ N

Φ−

AB

Ψ+

AB

1 = √ (|0 2 1 = √ N

1 Bell

|j

j

1 = √ (|0 2 1 = √ N

A |0 B

+ |1

A |j B

A |0 B

e

2πi N j

j

A |1 B

j=0,m=0

+ |1

|j

,

− |1 |j

A |j

A |1 B )

A |1 B )

A |j B

,

A |0 B )

+m

B

(mod N ) ,

states are complete orthonormal entanglement basis states for a two-qubit system. 659

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Nonequilibrium Quantum Transport Physics in Nanosystems

Ψ−

AB

1 = √ (|0 2 1 = √ N

A |1 B

e

− |1 2πi N j

j=0,m=0

A |0 B )

|j

A |j

+m

B

(mod N) .

Note that the Bell basis is obtained from standard basis by a general expression, 1 |ψnm = √ N ψp q

1 = √ N

1

e

2πi N jn

|j

A |j

+m

e | p ·q |q

A |q

+q

j=0

B

(mod 2) ,

1 i

q=0

B

(mod N ) ,

which yields 1 |ψ00 = √ N 1 |ψ01 = √ N 1 |ψ10 = √ N 1 |ψ11 = √ N

1

j

|j

A |j B

|j

A |j

⇒ Φ+

AB

,

(49.1)

1 j

+1

B

(mod 2) ⇒ Ψ+

AB

,

(49.2)

1

e

2πi N jn

j

|j

A |j B

|j

A |j

⇒ Φ−

AB

,

(49.3)

1

e j

2πi N jn

+1

B

(mod 2) ⇒ Ψ−

AB

.

(49.4)

Therefore, 

  + |ψ00 |Φ  |ψ01   |Ψ+     |ψ10  =  |Φ− |Ψ− |ψ11 

AB



   AB AB

AB

 |0 10 0 1   |0 1  0 1 1 0  = √  2  1 0 0 −1   |1 0 1 −1 0 |1

A |0 B A |1 B A |0 B A |1 B

   

Now let Alice takes one of the particles in the pair, and Bob keeps the other one. The subscripts A and B in the entangled state refer to Alice’s or Bob’s particle. We will assume that Alice and Bob share the entangled state |Φ+ , Eq. (49.1). So, Alice has two particles (U , the one she wants to teleport, and A, one of the entangled pair), and Bob has one particle, B. In the total system, the state of these three

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particles is given by |ψ ⊗ Φ+ = (α |0

U

α = √ (|0 2

+ β |1 U

|0

β + √ (|1 2

U)

1 ⊗ √ (|0 2

A |0 B

U

|0

+ |0

U

A |0 B

|1

+ |1

A |1 B )

A |1 B )

A |0 B

+ |1

U

|1

A |1 B ) .

U

|0

A

+ |1

U

|1

A) ,

U

|0

A

− |1

U

|1

A) ,

U

|1

A

+ |1

U

|0

A) ,

U

|1

A

− |1

U

|0

A) ,

In terms of the Bell basis Φ+

UA

Φ−

UA

Ψ+

UA

Ψ−

UA

1 = √ (|0 2 1 = √ (|0 2 1 = √ (|0 2 1 = √ (|0 2

where 1 = √ 2 1 = √ 2 1 = √ 2 1 = √ 2

|0

U

|0

A

|0

U

|1

A

|1

U

|0

A

|1

U

|1

A

U

1 ⊗ √ |0 2

Φ+

UA

+ Φ−

UA

,

Ψ+

UA

+ Ψ−

UA

,

Ψ+

UA

− Ψ−

UA

,

Φ+

UA

− Φ−

UA

,

we can write |ψ ⊗ Φ+ = α |0 α = √ 2 β +√ 2

U

+ β |1 1 √ 2 1 √ 2

Φ+

UA

Ψ+

+ Φ−

UA

A

|0

B

UA

− Ψ−

UA

+ |1

|0

A

|1

B

B

1 + √ 2

|0

B

1 + √ 2

Ψ+

UA

Φ+

+ Ψ−

UA

UA

− Φ−

UA

|1

B

|1

B

.

We can factor the qubit belonging to Bob in the above expression by factoring out Alice’s Bell basis, which are the complete orthonormal basis of her system, consisting of the unknown particle state, U, to be teleported and her EPR

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particle, A, α − α + α − α + Φ U A |0 B + Φ U A |0 B + Ψ UA |1 B + Ψ U A |1 B 2 2 2 2 β + β − β + β − Ψ U A |0 B − Ψ UA |0 B + Φ UA |1 B − Φ UA |1 + 2 2 2 2 1 + 1 Φ U A (α |0 B + β |1 B ) + Φ− UA (α |0 B − β |1 B ) = 2 2 1 1 + + Ψ UA (α |1 B + β |0 B ) + Ψ− UA (α |1 B − β |0 B ) 2 2 1 − 1 + α α Φ UA + Φ UA = β −β 2 2 B B 1 + 1 β −β + Ψ UA + Ψ− UA α B 2 α B 2

|ψ ⊗ Φ+ =

=

1 + 1 α 1 0 α Φ UA + Φ− UA β 0 −1 β 2 2 1 + 1 01 0 −1 α + Ψ UA + Ψ− UA 10 1 0 β 2 2

α β

.

Notice all we have done so far is a change of basis on Alice’s part of the system. No operation has been performed and the three particles are still in the same state. The actual teleportation starts when Alice measures her two qubits in the Bell basis. Given the above expression, evidently the results of her (local) measurement of |Φ± U A and |Ψ± U A is that the three-particle state would collapse to one of the following four states (with equal probability = 14 of obtaining each): Φ+

Φ−

Ψ+

Ψ−

UA

UA

UA

α β

UA

1 0 0 −1

, B

α β

01 10

α β

0 −1 1 0

α β

, B

, B

, B

where the state inside the square bracket belongs to Bob’ particle. Alice’s two particles are now entangled to each other, in one of the four Bell states. The entanglement originally shared between Alice’s and Bob’s is now broken. Bob’s particle takes on one of the four superposition state vector, namely, α β

,

1 0 0 −1

α β

,

01 10

α β

, and

0 −1 1 0

α β

.

B

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The four possible states for Bob’s qubit are unitary images of the state to be teleported. The local measurement done by Alice on the Bell basis is the crucial step. Experimentally, the projective Bell-basis measurement done by Alice may be achieved via a series of laser pulses directed at the two particles, which Alice possesses. The result of her Bell measurement tells her which of the four states the system is in. She simply has to send her results to Bob through a classical channel. Two classical bits can encode or carry which of the four results she obtained. After Bob receives the message from Alice, he will know which of the four states his particle is in. Using this information, he performs an appropriate unitary operation on his particle to transform it to the desired state, α |0 + β |1 : (i) If Alice indicates her result is |Φ+ , Bob simply perform an identity unitary operation, i.e., do nothing. (ii) If the message indicates |Φ− , Bob would send his qubit through the unitary gate given by the Pauli matrix σz =

1 0 0 −1

to recover the state. (iii) If Alice’s message corresponds to |Ψ+ , Bob applies the gate σx =

01 10

to his qubit. (iv) Finally, for the remaining case, the appropriate gate is given by σ z σ x = 0 1 0 −1 , which is the inverse to . iσy = −1 0 1 0 (v) Teleportation is thus achieved. (vi) After this operation, Bob’s qubit will take on the state |ψ = α |0 + β |1 , and Alice’s qubit becomes undefined, part of an entangled state (a sort of information conservation principle). Teleportation does not result in the copying of qubits, and hence is consistent with the no cloning theorem. (vii) There is no transfer of matter or energy involved, that there is no need by virtue of the indistinguishability of quantum mechanical particles. Alice’s particle has not been physically moved to Bob; only its state has been transferred. (viii) The remarkable feat achieved is that full quantum information is transferred by using only two classical bits. Recall that in the Bloch sphere formalism the state of the particle transported can be described by a continuous point on a unit sphere, hence by two real numbers, not just by a finite two bits. Entanglement has provided an important quantum channel for the transmission of the full quantum information. (xi) The teleportation scheme needs shared entangled state from Alice and Bob, and two-bit classical channel. If the classical channel is removed, then it becomes an attempt to achieve superluminal communication, which is impossible. (x) For every qubit teleported, Alice needs to send Bob two classical bits of

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information. These two classical bits do not carry complete information about the qubit being teleported. If an eavesdropper intercepts the two bits, she may know exactly what Bob needs to do in order to recover the desired state. However, this information is useless if she cannot interact with the entangled particle in Bob’s possession. 49.1.1

Unified Teleportation Procedure

Note that Bob’s unitary transformation corresponding to Alice’s results corresponds to the well-known expression 1

Uq p = Xq Zp =

i p · q δq |

exp q ,q=0

1

=

exp q=0

i p · q |q + q |

,q+q

|q

q| (mod 2)

q| (mod 2) .

which diﬀer from the Pauli-matrix operator of Eq. (39.16) by only a phase fac2π|1 , tor. Substituting the possible values of q and p, namely, {0, 1} and 2π|0 2 , 2 respectively, and using modular arithmetic, we have 10 01

U0,0 =

U01 = exp =

U10 = δ q

i p · q δq |

1 0 0 −1

,q+q

= Z1 = σ z , 01 10

=

,q+q

= I,

= X1 = σx ,

U1,1 = exp

i p · q δq |

,q+q

= exp

i p · q δq |

,q+q

=

0 1 −1 0

= iσ y .

Therefore, using the unified expressions above, when Alice communicate to Bob her measurement result to be |ψnm , 1 |ψnm = √ 2

1

e j=0

2πi 2 jn

|j

A |j

+m

B

(mod 2) ,

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then Bob perform the corresponding unitary transformation on his particle using the expression, Umn =

exp j

49.2

2π| i n · j |j + m j| (mod 2) . 2 |

N-State Particles

One can imagine how the teleportation scheme given above might be extended to N -state particles, i.e. particles whose states lie in the N dimensional Hilbert space. The combined system of the three particles now has a N 3 dimensional state space. To teleport, Alice makes a partial measurement on the two particles in her possession in some entangled basis on the N 2 dimensional subsystem 1 |ψnm = √ N

N−1

e

2πi N jn

j=0

|j

A |j

+m

B

(mod N ) .

This measurement has N 2 equally probable outcomes, which are then communicated to Bob classically using 2 log2 N bits. Bob recovers the desired state by sending his particle through an appropriate unitary gate N−1

Umn =

exp j=0

2π| i n · j |j + m j| (mod N) . N |

This transformation brings Bob’s particle to the original state of Alice unknown particle and the teleportation is complete.

49.3

Formal Derivation of Entangled Basis States

Consider the identity |p =

q

q| p |q ,

where the q| p is the transformation function. For discrete quantum mechanics, this is given by the discrete Fourier transform function, 1 i q| p = √ exp − p · q . | N Therefore, 1 |p = √ N

q

i exp − p · q |

|q .

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The product state in the ‘momentum’ basis is expanded in terms of the ‘position’ basis |p |p

1 = √ N 1 N

=

q

q ,q

i exp − p · q | i exp − p · q |

1 |q √ N

q

i exp − p · q |

i exp − p · q | |q

|q

|q

.

We can expressed the right hand side in terms of ‘correlated’ (entangled) product states by writing, q = q + m, where m is the quantum-correlation ‘distance’, |p |p

=

1 N

=

1 N

i exp − p · q | q ,m

q ,m

i exp − p · (q + m) |

i exp − (p + p ) · q |

i exp − p · m |

|q |q

|q + m |q + m .

Now writing p + p = p, we have |p |p − p =

1 N

i exp − (p) · q | q ,m

1 =√ N 1 =√ N

m

m

i exp − (p − p ) · m |q |q + m |     i 1 i √ exp − (p−p ) · m exp − p · q |q |q +m  N  | | q i exp − (p − p ) · m |

The inverse transformation gives ψp,m products 1 ψp,m = √ N

exp p

ψp,m .

in terms of the the ‘momentum’ basis

i (p − p ) · m |p |p − p . |

(49.5)

Clearly, the correlated basis defined by ψp,m forms orthonormal and complete set. 49.3.1

Bell Basis

Thus, for N = 2, we have the standard momentum product states in terms of the so-called Bell basis, ψp,m , for example, 1

1 exp (−πi (k − k ) · m) ψk,m , |k |k − k = √ 2 m=0

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which yields 1

1 |0 |k = √ exp (−πik · m) ψk,m , 2 m=0 and using the identities |ψ00 = Φ+ ,

|ψ01 = Ψ+ ,

|ψ10 = Φ− ,

|ψ11 = Ψ− ,

we have, 1

1 exp (−πik · m) ψ0,m |0 |0 = √ 2 m=0 1 = √ (|ψ00 + |ψ01 ) 2 1 Φ+ + Ψ+ , = √ 2 1

1 exp (−πi1 · m) ψ1,m |0 |1 = √ 2 m=0 1 = √ 2 1 = √ 2

ψ1,0 − ψ1,1 Φ− − Ψ−

,

We also have 1

1 exp (−πi (k + 1) · m) ψk,m (mod 2) , |1 |k + 1 = √ 2 m=0 which yields 1

1 |1 |1 = √ exp (−πi (1) · m) ψ0,m 2 m=0 1 = √ 2 1 = √ 2

ψ0,0 − ψ 0,1 Φ+ − Ψ+

,

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1

1 |1 |1 + 1 = √ exp (−πi (1 + 1) · m) ψ1,m (mod 2) , 2 m=0 1 ψ1,0 + ψ1,1 |1 |0 = √ 2 1 Φ− + Ψ− . = √ 2 Therefore, for N = 2, we have the following transformation from the maximally entangled Bell basis to the standard ‘momentum’-state product basis given by 

|0  |0   |1 |1

 +   |Φ 1 1 0 0 |0  0 0 1 −1   |Ψ+ 1 |1   −  =√  |0  2  0 0 1 1   |Φ 1 −1 0 0 |1 p |Ψ−

   

.

Bell

Hence, −1     |Φ+ |0 |0 1 1 0 0  |Ψ+   1  0 0 1 −1   |0 |1    −   =  |Φ   √2  0 0 1 1   |1 |0 |1 |1 1 −1 0 0 |Ψ− Bell    |0 |0 1 0 0 1   1  1 0 0 −1    |0 |1  . = √      |1 |0 0 1 1 0 2 |1 |1 p 0 −1 1 0 

   

p

(49.6)

The above result also follows from the general inverse expression, Eq. (49.5), in terms of ‘momentum’ product basis, 1 ψp,m = √ N

exp p

i (p − p ) · m |p |p − p , |

so that for N = 2, 1 ψk,m = √ 2

k

exp (iπ (k − k ) · m) |k |k − k ,

which yields 1 |ψ00 = Φ+ = √ 2

k

exp (iπ (0 − k ) · 0) |k |0 − k

1 = √ (|0 |0 + |1 |1 ) , 2

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1 |ψ01 = Ψ+ = √ 2

k

669

exp (iπ (k ) · 1) |k |k

1 = √ (|0 |0 − |1 |1 ) , 2 1 |ψ10 = Φ− = √ 2

exp (iπ (1 − k ) · 0) |k |1 − k

k

1 = √ (|0 |1 + |1 |0 ) , 2 1 ψ1,1 = ψk,m = √ 2

k

exp (iπ (1 − k ) · 1) |k |1 − k

1 = − √ (|0 |1 − |1 |0 ) . 2 We also have    |0 |Φ+ 10 0 1  |Ψ+   0 1 1 0   |0 1   −  =√   |Φ  2  1 0 0 −1   |1 |1 0 1 −1 0 |Ψ− Bell 

Therefore, we have the identity   |0 1 0 0 1   |0 1  1 0 0 −1  √  2  0 1 1 0   |1 |1 0 −1 1 0

 |0 |1   . |0  |1 q

   |0 |0 10 0 1  0 1 1 0   |0 1 |1   =√   |0  2  1 0 0 −1   |1 |1 p |1 0 1 −1 0

(49.7)

 |0 |1   , |0  |1 q

which gives 

|0  |0   |1 |1

  −1     |0 |0 10 0 1 1 0 0 1 |0       |1   =  √1  1 0 0 −1   0 1 1 0   |0 |1          |1 |0  1 0 0 −1 |0 2 0 1 1 0 |1 |1 q 0 1 −1 0 0 −1 1 0 |1 p      |0 |0 1 1 0 0 10 0 1    1  0 0 1 −1   √1  0 1 1 0   |0 |1  = √       |1 |0  2 0 0 1 1 2 1 0 0 −1 |1 |1 q 1 −1 0 0 0 1 −1 0     |0 |0 10 0 1 1 1 0 0    1  0 0 1 −1    0 1 1 0   |0 |1  . =        |1 |0 1 0 0 −1 0 0 1 1 2 |1 |1 q 0 1 −1 0 1 −1 0 0

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Therefore, ‘momentum’ basis is given by the transformation of the ‘position’ basis as      |0 |0 1 1 1 1 |0 |0     |0 |1   = 1  1 −1 1 −1   |0 |1  .   |1 |0  2  1 1 −1 −1   |1 |0  |1 |1 q 1 −1 −1 1 |1 |1 p 49.3.2

Three-Qubit Entangled Basis

As in the two-qubit, the product state in the ‘momentum’ basis is expanded in terms of the ‘position’ basis |p |p

1 = 3 N2

=

1 3 N2

|p q

i exp − p · q |

q ,q ,q

|q

q

i exp − p · q |

i exp − p · q |

i exp − p · q |

|q

q

i exp − p · q |

i exp − p · q |

|q |q

|q |q

.

We can expressed the right hand side in terms of ‘correlated’ (entangled) product states by writing, q = q +m, q = q +l where m and l are the quantum-correlation ‘distances’, |p |p

|p

=

1 3 N2

q ,m,l

i exp − p · q |

i exp − p · (q + m) |

i × exp − p · (q + l) |q |q + m |q + l | i 1 exp − (p + p + p ) · q = 3 | N 2 q ,m,l i i × exp − p · m exp − p · l | | Now writing p + p + p |p |p − p − p

|p

|q + m |q + l .

= p, we have 1

=

(N ) × =

|q

1 N

q

3 2

m,l

i i exp − (p − p ) · m exp − p · (l − m) | |

i exp − p · q |

m,l

|q

|q + m |q + l

i i exp − (p − p ) · m exp − p · (l − m) | |

× ψp,m,l ,

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where 1 ψp,m,l = √ N

q

i exp − p · q |

|q

|q + m |q + l ,

is the three-qubit correlated (entangled) state. Substituting N = 2, and p = π|k, we have 1

1 ψk,m,l = √ exp (−iπk · q ) |q 2 q =0

|q + m |q + l ,

and obtain 1 ψ0,0,0 = √ (|0 |0 |0 + |1 |1 |1 ) = Θ+ 3, 2 which is the well-known Greenberger-Horne-Zeilinger state, |GHZ , which is the maximally entangled state of three qubits since m = 0, and l = 0 (a complete correlation). We give the expressions for the rest of ψk,m,l , 1 ψ0,0,1 = √ (|0 |0 |1 + |1 |1 |0 ) = Γ+ , 2 1 ψ0,1,0 = √ (|0 |1 |0 + |1 |0 |1 ) = Ω+ , 2 1 ψ0,1,1 = √ (|0 |1 |1 + |1 |0 |0 ) = Ξ+ , 2 1 ψ1,0,0 = √ (|0 |0 |0 − |1 |1 |1 ) = Θ− 3, 2 1 ψ1,0,1 = √ (|0 |0 |1 − |1 |1 |0 ) = Γ− , 2 1 ψ1,1,0 = √ (|0 |1 |0 − |1 |0 |1 ) = Ω− , 2 1 ψ1,1,1 = √ (|0 |1 |1 − |1 |0 |0 ) = Ξ− . 2 Just like the Bell basis, the set of three-qubit correlated states, ψk,m,l , is a complete and orthonormal basis set, since each element is connected to the complete orthonormal product states by unitary transformations. In what follows, we will to ψk,m,l as the three-particle correlated basis (TPCB). Remark 49.1 There is another three-qubit entangled state, the so-called W state. A W state is a collective spin state with one ‘excitation’ as compared to a coherent

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spin state which is fully aligned. With all but one spin aligned in the x-direction, the state can be written: 1 |W = √ (|↓↑↑ ... ↑ N

1

+ |↑↓↑ ... ↑

2

+ .... |↑↑↑ ... ↓

N)

The |W maybe viewed as a ‘zero momentum’ superposition of ‘position coordinates’ indicated by the position of an excitation in an array of fully-aligned spin 1 |p = √ N 1 |W = √ N

N i

i=1

e | p·Ri |Ri

N

e

i |

0·Ri

i=1

1 |Ri = √ N

N

i=1

|Ri

where the ‘position’ coordinate, |Ri , correspond to the diﬀerent position of oneexcitation, ↓, in a spin array |spin ↓ array i as indicated above (more appropriately in a spin ring). However, the ‘position’ coordinates used is only a small subspace of the complete and orthonormal product Hilbert space of an array of spin, moreover |W corresponds only to |p = |p = 0 , i.e., zero-phase superposition. In general there are also N values of p producing other |W -type entangled states of array of spins with one excitation. Furthermore, corresponding to the superposition of position eigenstates to produce a Schroedinger wavefunction, one can also form an arbitrary superposition of spin array |Ri to form other correlated (entangled) state, for example, N

|Ψ =

i=1

ψ (Ri ) |Ri

where ψ (Ri ) may no longer be obtained from Schroedinger equation, but from some optimization criteria, say for optimal condition of a universal cloning machine, which for three-particle entangled state, in the form (Phys. Rev. A 57, 2368 (1998) |Ψ =

2 |100 − 3

1 |010 − 6

1 |001 6

It is worth pointing out that the method of generating entangled states using the product of ‘momentum’ basis states, |p , in the manner given above automatically yield complete orthonormal entangled basis states. 49.3.3

A Qubit Teleportation Using Three-Particle Entanglement

Here we will used as the quantum channel shared by Alice and Bob to be the GHZ state 1 |GHZ = Θ+ 3 = √ (|0 1 |0 2 |0 2

3

+ |1 1 |1 2 |1 3 ) .

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As before Alice wants to send to Bob the unknown state, |ψ = α |0 U + β |1 U . To set up the teleportation, let us assume that particles 1 and 2 are kept by Alice and particle 3 is sent to Bob. So, Alice has three particles (U , the one she wants to teleport, and particles 1 and 2, two of the entangled three qubits), and Bob has one particle 3. In the total system, the state of these four particles is given by |ψ ⊗ Φ+ 3 = (α |0

U

α = √ ([|0 2

+ β |1 U

1 √ (|0 1 |0 2 |0 2

U) ⊗

|0 1 |0 2 ] |0

β + √ ([|1 2

3

+ [|0

|0 1 |0 2 ] |0

U

3

U

3

+ |1 1 |1 2 |1 3 )

|1 1 |1 2 ] |1 3 )

+ [|1

U

|1 1 |1 2 ] |1 3 ) ,

where we have enclosed in square bracket the particles belonging to Alice. We will now express Alice particle states in terms of the TPCB basis. We have |0

U

|0 1 |0

2

1 − = √ Θ+ 3 + Θ3 2

|1

U

|1 1 |1

2

1 − = √ Θ+ 3 − Θ3 2

|0

U

|1 1 |1

2

1 = √ Ξ+ + Ξ− 2

|1

U

|0 1 |0

2

1 = √ Ξ+ − Ξ− 2

U 12

,

U 12

,

U12

,

U12

.

Therefore we can write 1 − √ Θ+ 3 + Θ3 2

α |ψ ⊗ Φ+ 3 = √ 2 β +√ 2 |ψ ⊗ Φ+ 3 =

=

U 12

1 √ Ξ+ − Ξ− 2

|0

U 12

3

|0

1 + √ Ξ+ + Ξ− 2 3

U12

1 − + √ Θ+ 3 − Θ3 2

|1

U 12

3

|1

3

,

1 + 1 + Θ3 U 12 (α |0 3 + β |1 3 ) + Ξ U12 (α |1 3 + β |0 3 ) 2 2 1 − 1 + Θ− Ξ U12 (α |1 3 − β |0 3 ) 3 U 12 (α |0 3 − β |1 3 ) + 2 2 1 Θ+ 3 2 +

U 12

1 Θ− 3 2

I

U 12

α δ

+

1 + Ξ 2

σz

α δ

+

U 12

1 − Ξ 2

σx

U 12

α δ (−iσy )

α δ

.

Hence, it follows that regardless of the unknown state |ψ , by using the maximallyentangled quantum channel, |GHZ , Alice can still perform only four measurements with outcomes equally likely with probability equal to 14 . After Alice measurement,

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Bob’s particle 3 will have been projected to one of the four pure states, after which Bob has to perform the necessary transformation to regain the original state that Alice has, as indicated in the following table, Alice measurement Θ+ 3 U12 Θ− 3 U 12 (Ξ+ )U12 (Ξ− )U12

Bob’s particle state (α |0 3 + β |1 3 ) (α |0 3 − β |1 3 ) (α |1 3 + β |0 3 ) (α |1 3 − β |0 3 )

Bob’s transformation I σz σx iσy

To transmit Alice’s classical (measurement) result to Bob, Alice needs a two-bit classical channel to transmit which one of the four equally likely results . This is in contrast with other proposed more complex teleportation scheme using the same maximally-entangled |GHZ quantum channel which employ Bob as an ancilla to transmit the unknown state to Cliﬀ, and use two classical bit between Alice and Bob and one classical bit between Bob and Cliﬀ. Here, whether we use the Bell basis or TPCB basis to transmit an unknown quantum state, only two-bit classical channel is simply needed directly between the sending and receiving parties, and without the use of intermediary ancilla and only one local measurement is involved. 49.4

Teleportation Using Three-Particle Entanglement and an Ancilla

Instead of keeping the particles 1 and 2 as done above, Alice only keep particle 1 and send particle 2 to Bob and particle 3 to Cliﬀ. The idea here is that Alice will locally only measure one of the orthonormal two-particle entangled Bell states, |Φ± , |Ψ± , instead of one of the complete orthonormal three-particle correlated (entangled) states given above. The price to pay in this scheme is that there is a need to have two bit of classical communication channel between Alice and Bob and one more bit of classical communication channel between Bob and Cliﬀ. Moreover, two local measurements has to be done, one local measurement by Alice on her Bell states and one local measurement by Bob on his qubit, in order to implement the teleportation of one qubit from Alice to Cliﬀ. Thus Bob becomes an ancilla in this teleportation scheme. Again, using the Alice wants to send to Bob the unknown state, |ψ = α |0 U + β |1 U , using the |GHZ shared quantum channel. To set up the teleportation, Alice keep particle 1 and send particle 2 to Bob, and particle 3 to Cliﬀ. So, Alice has two particles (U , the unknown qubit she wants to teleport and particle 1, one of the entangled three |GHZ qubits). In the total system, the state of these four particles is given by |ψ ⊗ |GHZ = (α |0

U

+ β |1

U)

1 ⊗ √ (|0 1 |0 2 |0 2

3

+ |1 1 |1 2 |1 3 )

α = √ ([|0 U |0 1 ] |0 2 |0 3 + [|0 U |1 1 ] |1 2 |1 3 ) 2 β + √ ([|1 U |0 1 ] |0 2 |0 3 + [|1 U |1 1 ] |1 2 |1 3 ) , 2

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where the particles enclosed in square brackets belongs to Alice. Upon changing the Alice product basis states to Bell basis, we have |ψ ⊗ |GHZ = (α |0 α =√ 2 β +√ 2

U

+ β |1 1 √ 2 1 √ 2

1 √ (|0 1 |0 2 |0 2

U) ⊗

Φ+

U1

Ψ+

+ Φ−

U1

− Ψ−

1 2 +

+ |1 1 |1 2 |1 3 )

|0 2 |0

U1

3

1 +√ 2

1 |0 2 |0 3 + √ 2

Ψ+

U1

Φ+

+ Ψ−

U1

− Φ−

U1

U1

|1 2 |1

3

|1 2 |1

3

1 ⊗ √ (|0 1 |0 2 |0 3 + |1 1 |1 2 |1 3 ) 2 + |Φ U 1 (α |0 2 |0 3 + β |1 2 |1 3 ) + |Ψ+ U1 (α |1 2 |1 3 + β |0 2 |0 3 )

|ψ ⊗ |GHZ = (α |0 =

U1

3

U

+ β |1

U)

|Φ− U1 (α |0 2 |0 3 − β |1 2 |1 3 ) + |Ψ− U 1 (α |1 2 |1 3 − β |0 2 |0 3 )

1 2

.

A local measurement of one of the four Bell states, which have equal probability equal to 14 , will project the joint state of the particles 2 and 3 possessed by Bob and Cliﬀ, respectively, into one of the entangled states shown above. Assume that Alice local measurement yields |Φ+ U 1 , which is communicated to Bob through a classical two-bit channel. Then the state of particles 2 and 3 is |ψ23 = α |0 2 |0

3

+ β |1 2 |1 3 .

In order to eﬀect a transfer of the unknown qubit to Cliﬀ, Bob must now unentangle his qubit from that of Cliﬀ. To do this Bob has to make a local measurement on his qubit 2. Suppose Bob’s measurement apparatus collapses his qubit state to two possible outcomes, namely k1 and k2 . Then Bob can decompose his incoming state to the new basis states, namely, |k1 and |k2 , and write the unitary/orthogonal transformation from the old basis to the new basis as |0 |1

2 2

= sin θ |k1 + cos θ |k2 , = cos θ |k1 − sin θ |k2 .

Then |ψ23 in terms of Bob’s new basis is |ψ23 = (sin θ |k1 + cos θ |k2 ) α |0 3 + (cos θ |k1 − sin θ |k2 ) β |1 3 = sin θ α |0 3 |k1 + cos θ α |0 3 |k2 + cos θ β |1 3 |k1 − sin θβ |1 3 |k2 = (sin θ α |0 3 + cos θ β |1 3 ) |k1 + (cos θ α |0 3 − sin θβ |1 3 ) |k2 .

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In general, the new basis and the old basis are mutually-unbiased basis2 , which means sin θ = cos θ or θ = π4 . Then |ψ23 becomes |ψ23 = ( α |0

3

1 + β |1 3 ) √ |k1 + ( α |0 2

3

1 − β |1 3 ) √ |k2 . 2

Whatever Bob’s measurement outcome is, Bob has to communicate to Cliﬀ his result through a one-bit classical channel. If the outcome of Bob’s local measurement is k1 then Cliﬀ has the unknown qubit from Alice and do nothing. On the other hand if Bob’s measurement yields k2 , then Cliﬀ has to perform a σz transformation on his qubit to obtain the unknown qubit from Alice. 49.5

Two-Qubit Teleportation Using Three-Particle Entanglement

We will now show that a full use of the capability of the GHZ quantum channel is achieved when teleporting two qubit of information. As before, the quantum channel shared by Alice and Bob to be the GHZ state 1 |GHZ = Θ+ 3 = √ (|0 1 |0 2 |0 2

3

+ |1 1 |1 2 |1 3 ) .

But now Alice wants to send an unknown two-particle bits (triplet) to Bob, namely, |ψ = α |0

U1 |0 U 2

+ δ |1

U 1 |1 U2 ,

which we may write in vector form as |ψ =

α |0 δ |1

U 1 |0 U2

U1 |1

U2

⇒

α δ

,

2 2 where |α| + |δ| = 1. To set up the two-bit teleportation, particle 1 is kept by Alice and particle 2 are3 are sent to Bob. In the total system, the state of these three particles is given by

|ψ ⊗ |GHZ = (α |0 U1 |0 U 2 + δ |1 U 1 |1 U 2 ) 1 ⊗ √ (|0 1 |0 2 |0 3 + |1 1 |1 2 |1 3 ) 2 1 α (|0 U 1 |0 U2 |0 1 |0 2 |0 3 + |0 U1 |0 U2 |1 1 |1 2 |1 3 ) = √ +δ (|1 U1 |1 U 2 |0 1 |0 2 |0 3 + |1 U 1 |1 U2 |1 1 |1 2 |1 3 ) 2

.

Now we change basis using the complete orthonormal set of three-particle correlated (entangled) states 1 ψ0,0,0 = √ (|0 |0 |0 + |1 |1 |1 ) = Θ+ 3, 2 1 ψ0,0,1 = √ (|0 |0 |1 + |1 |1 |0 ) = Γ+ , 2 2 Example

of mutually-unbiased bases are the position basis states and momentum basis states.

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Discrete Phase Space Viewpoint

1 ψ0,1,0 = √ (|0 |1 |0 + |1 |0 |1 ) = Ω+ , 2 1 ψ0,1,1 = √ (|0 |1 |1 + |1 |0 |0 ) = Ξ+ , 2 1 ψ1,0,0 = √ (|0 |0 |0 − |1 |1 |1 ) = Θ− 3, 2 1 ψ1,0,1 = √ (|0 |0 |1 − |1 |1 |0 ) = Γ− , 2 1 ψ1,1,0 = √ (|0 |1 |0 − |1 |0 |1 ) = Ω− , 2 1 ψ1,1,1 = √ (|0 |1 |1 − |1 |0 |0 ) = Ξ− , 2 to obtain |0

U 1 |0 U2 |0 1

1 − = √ Θ+ 3 + Θ3 , 2

|0

U 1 |0 U2 |1 1

1 = √ Γ+ + Γ− , 2

|0

U 1 |1 U 2 |0 1

1 = √ Ω+ + Ω− , 2

|0

U1 |1 U 2 |1 1

1 = √ Ξ+ + Ξ− , 2

|1

U1 |0 U 2 |0 1

1 = √ Ξ+ − Ξ− , 2

|1

U 1 |0 U 2 |1 1

1 = √ Ω+ − Ω− , 2

|1

U 1 |1 U2 |0 1

1 = √ Γ+ − Γ− , 2

|1

U 1 |1 U2 |1 1

1 − = √ Θ+ 3 − Θ3 . 2

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Upon changing the basis of Alice particles, we have   + Θ− α √12 Θ+ |0 2 |0 3 + √12 (Γ+ + Γ− ) |1 2 |1 3 3 3 1   |ψ ⊗ |GHZ = √  , 2 +δ √1 (Γ+ − Γ− ) |0 |0 + √1 Θ+ − Θ− |1 |1 3 3 2 3 2 3 2 2 

Θ+ 3 (α |0 2 |0

3

+ δ |1 2 |1 3 )

 +Θ− 1 3 (α |0 2 |0 3 − δ |1 2 |1 3 ) |ψ ⊗ |GHZ =   + 2  +Γ (δ |0 2 |0 3 + α |1 2 |1 3 ) 

      1 =  2      

+Γ− (−δ |0 2 |0 α δ

Θ+ 3I +Θ− 3 σz

α δ

+Γ+ σ x

α δ

+Γ− (−iσy )

α δ

3

+ α |1 2 |1 3 ) 

     

       .       

Thus, depending on the result of Alice measurement using the equi-probable, with − + − probability= 14 , three-particle entanglement basis, Θ+ 3 , Θ3 , Γ , and Γ , which is communicated to Bob via classical two-bit channel, Bob will then use the inverse transformation to recover the original triplet sent by Alice. Alice can also send an unknown singlet to Bob, namely, |ψ = α |0

U1 |1 U 2

+ δ |1

U 1 |0 U2 .

However, the quantum channel shared by Alice and Bob must now be chosen to be given by 1 |Qch = √ (|0 1 |1 2 |0 2

3

+ |1 1 |0 2 |1 3 )

instead of |GHZ used above. To set up the two-bit teleportation, particle 1 is kept by Alice and particle 2 are3 are sent to Bob. In the total system, the state of these three particles is given by |ψ ⊗ |Qch 1 √ (|0 1 |1 2 |0 2

= {α |0

U 1 |1 U2

1 =√ 2

α (|0 U 1 |1 U2 |0 1 |1 2 |0 3 + |0 U1 |1 U 2 |1 1 |0 2 |1 3 ) +δ (|1 U 1 |0 U 2 |0 1 |1 2 |0 3 + |1 U 1 |0 U2 |1 1 |0 2 |1 3 )

+ δ |1

U1 |0 U 2 }

⊗

3

+ |1 1 |0 2 |1 3 ) .

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Upon changing the basis to three-particle entanglement basis, we have |ψ ⊗ |Qch 

α

1  =√  2 +δ 

√1 2

(Ω+ + Ω− ) |1 2 |0

√1 2

(Ξ+ − Ξ− ) |1 2 |0

3+

√1 2

(Ξ+ + Ξ− ) |0 2 |1

+

√1 2

(Ω+ − Ω− ) |0 2 |1

3



Ω+ (δ |0 2 |1 3 + α |1 2 |0 3 )  + 1  +Ξ (α |0 2 |1 3 + δ |1 2 |0 3 )  =   2  +Ω− (−δ |0 2 |1 3 + α |1 2 |0 3 )  +Ξ− (α |0 2 |1 3 − δ |1 2 |0 3 )   α + Ω σx   δ         +Ξ+ I α   δ   1 . =    2 α    +Ω− (−iσ y )   δ       α +Ξ− σ z δ

3 3

  

Thus, depending on the result of Alice measurement using the equi-probable, with probability= 14 , three-particle entanglement basis, Ω+ , Ω− , Ξ+ , and Ξ− , which is communicated to Bob via classical two-bit channel, Bob will then use the inverse transformation to recover the original singlet sent by Alice.

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Chapter 50

Superdense Coding

The use of entanglement resource also plays a crucial role in dense coding. The question addressed here is: how many bits of classical information can be transmitted per qubit. With the assumption that Alice and Bob share an entangled state, two classical bits per qubit can be transmitted. Superdense refers to this doubling of information capacity in communication channels. The phenomenon of “superdense coding” is based on the observation that given some state belonging to the Bell basis, Eq. (49.7) there are local unitaries1 belonging to Alice which will map it onto any of the other states belonging to the basis, apart from an overall phase. There are similar unitaries belonging to Bob. Based on the two-particle product basis states, (|0 2 |0 1 , |0 2 |1 1 , |1 2 |0 1 , |1 2 |1 1 ), these unitaries belonging to Alice (designated here by index 1) are X1 ⊗ I2 

0

 1 |0 |1 0| 0|  2 1 1 2 =  0 0

1 |0 2 |0

1

1|1 0|2

0 0 0

0 0 0 1 |1 2 |1

1

0 0 0|1 1|2

which can be written for short as

1 |1 2 |0

1

0



  , 1|1 1|2 

 0100 1 0 0 0  X1 ⊗ I2 =  0 0 0 1. 0010 

Similarly, we have

 1 0 0 0  0 −1 0 0   Z1 ⊗ I2 =  0 0 1 0 , 0 0 0 −1 

1 Local unitary operations should be contrasted with local measurements, the later will destroy entanglement. The former are just transformations of Bell entangled basis.

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and Y1 ⊗ I2 ,

 0 −i 0 0 i 0 0 0   Y1 ⊗ I2 =   0 0 0 −i  . 0 0 i 0 

Thus if we start with

Φ+

   |0 |0 1  0 1 1  0 ⇒ √   = √  2 0  2 0 |1 |1 1 1 = √ (|0 |0 + |1 |1 ) , 2 

then it is mapped to |Ψ+ by X1 ⊗ I2 Ψ+ =

=

= = to |Φ− by Z1 ⊗ I2 , Φ−

    0 1  0 1 1  1  = X1 ⊗ I2 √   √  2 1 2 0 0 1    1 0100 0 1  1 0 0 0   √  2 0 0 0 10 1 0010 1 √ (|0 2 ⊗ σx |0 1 + |1 2 ⊗ σ x |1 1 ) 2 1 √ (|1 |0 + |0 |1 ) , 2

   1 1 0  0  1    =  0  = Z1 ⊗ I2 √2  0  1 −1    1 1 0 0 0 0 1  0 −1 0 0   = √  2 0 0 1 0 0 1 0 0 0 −1 1 = √ (|0 2 ⊗ σz |0 1 + |1 2 ⊗ σ z |1 1 ) 2 1 = √ (|0 ⊗ |0 − |1 ⊗ |1 ) , 2 

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and to i |Ψ− by Y ⊗ I,

Ψ− =

=

= =

   0 1  0 1  1 1  = −iY1 ⊗ I2 √   √  2  −1  2 0 0 1    1 0 −1 0 0 0 1  1 0 0 0   √  2  0 0 0 −1   0  1 0 0 1 0 1 √ (|0 2 ⊗ −iY |0 1 + |1 2 ⊗ −iY |1 1 ) 2 1 √ (|0 2 |1 1 − |1 2 |0 1 ) . 2 

Crucial to this procedure is the shared entangled state between Alice and Bob, and the property of entangled states that a (maximally) entangled state can be transformed into another Bell-entangled basis apart from an overall phase (note that entanglement is conserved in local unitary operations) via local manipulation of either Alice or Bob. Suppose parts of a Bell state, say 1 Ψ+ = √ (|0 2

A |1 B

+ |1

A |0 B )

are distributed to Alice and Bob. The first subsystem, denoted by subscript A, belongs to Alice and the second, B, system to Bob. By only manipulating her particle locally, Alice can transform the composite system into any one of the Bell states (entanglement cannot be broken using local operations) as follows. (a) Alice does nothing, the system remains in the state |Ψ+ , (b) Alice sends her particle through the unitary gate,

XA = σ x =

01 10

,

(σ x , one of the Pauli matrices), the total two-particle system now is in state 1 XA ⊗ IB Ψ+ = √ (σ x |0 2 1 = √ (σ x |0 2 = Φ+ .

A

⊗ |1

B

+ σ x |1

A

⊗ |0

B)

A

⊗ |1

B

+ σ x |1

A

⊗ |0

B)

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If X is replaced by Z, the initial state is transformed into |Ψ− , 1 ZA ⊗ IB Ψ+ = √ (σz |0 A ⊗ |1 B + σ z |1 A ⊗ |0 2 1 = √ (|0 A ⊗ |1 B − |1 A ⊗ |0 B ) 2 = Ψ− .

B)

Similarly, if Alice applies iY ⊗ I to the system, the resultant state is |Φ− , 1 iYA ⊗ IB Ψ+ = √ (iY |0 A ⊗ |1 B + iY |1 A ⊗ |0 2 1 = √ (− |1 A ⊗ |1 B + |0 A ⊗ |0 B ) 2 1 = √ (|0 A ⊗ |0 B − |1 A ⊗ |1 B ) 2 = Φ− .

B)

Therefore, a superdense coding scheme goes as follows. Assuming that Alice receives a two bit of classical information from Charlie, Alice can relay this two bit of classical information to Bob by encoding on the four distinct Bell basis. Depending on the message she would like to send, Alice performs one of the four local unitary operations given above and sends her qubit to Bob. By performing a projective measurement, Pj , in the Bell basis on the two particle system, Pj |ψi = δ ij ψj , where ψi denotes any one of the Bell basis, Bob can decode Alice message. Assuming that Alice performed the transformation σ x on her qubit, or XA ⊗ IB to the shared entangled state, in order to achieve the desired transmission, i.e., for Bob to decode the desired message, it is required that the result of Bob’s measurement satisfies δ ij

Φ+ · ψj

= δ ij .

Thus only one qubit is needed by Alice to send the two-bit classical information to Bob. On the other hand, two classical bits are needed to identify(encode) one out of four possible messages to be transmitted between Alice and Bob. Actually the quantum information resides (stored) in the entanglements (correlations) between Alice and Bob’s qubits. This again is another instance where nonlocality in a quantum communication channel is put to practical use. Notice, however, that if an eavesdropper, Eve, intercepts Alice’s qubit en route to Bob, all that is obtained by Eve is part of an entangled state. In other words, Eve would not be able to perform a Bell basis measurement on the two particle system. Thus, no proper decoding is possible and no relevant information is gained by Eve unless she have access to Bob’s qubit.

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General Dense Coding Scheme

General dense coding schemes can be formulated in the language used to describe quantum channels for ensembles. First let us introduced the concept of entanglement measure for ensembles. For a bipartite composite system, mixed states are just density matrices on Hilbert space HA ⊗ HB . Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as B wi ρA i ⊗ ρi ,

ρT = i

(50.1)

where ρA and ρB are they themselves states on the subsystems A and B respectively. The physical idea is that a separable state can always be constructed by two independent parties with the aid of classical communication only. Hence, a state is separable if it is probability distribution over uncorrelated states, or product states. B We can assume without loss of generality that ρA i and ρi are pure ensembles. A state is then said to be entangled if it is not separable. In general, finding out whether or not a mixed state is entangled is considered diﬃcult. A way to find out is to look at the reduced density matrix of the subsystem.

50.2

Reduced Density Matrices

Consider as above systems A and B each with a Hilbert space HA and HB , respectively. Let the state, |Ψ , of the composite system be |Ψ ∈ HA ⊗ HB . In general there is no way to associate a pure state to the component system A. However, it is still possible to associate a density matrix to subsystem A. Let the total density matrix operator be ρT ρT = |Ψ Ψ| , which is a projector operator on state |Ψ . The state of A is the partial trace of ρT over the basis of system B: ρA = T rB ρT = j

j|B (|Ψ Ψ|) |j

B

,

(50.2)

ρA is called the reduced density matrix of ρT on subsystem A. Therefore, we “trace out” system B to obtain the reduced density matrix on A. For example, the density

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matrix for the entangled state |Φ+ of a bipartite system is Φ+

1 (|0 A |0 B + |1 A |1 B ) ( 0|B 0|A + 1|B 1|A ) 2 1 1 = (|0 A |0 B ) ( 0|B 0|A ) + (|1 A |1 B ) ( 1|B 1|A ) 2 2 1 1 + (|0 A |0 B ) ( 1|B 1|A ) + (|1 A |1 B ) ( 0|B 0|A ) , 2 2

Φ+ =

which is not of the form of Eq. (50.1) for separable state. Using Eq. (50.2), the reduced density matrix of A for the entangled state |Φ+ is 1 (|0 A 0|A + |1 A 1|A ) , 2 a maximally mixed state of A, which can be represented by a diagonal matrix ρA =

ρA =

1 2

0

0 1 2

.

(50.3)

On the other hand if ρT is separable, by virtue of Eq. (50.1), ρT = i

B wi ρA i ⊗ ρi

1 = |0 2

A

0|A ⊗ |0

B

0|B +

1 |1 2

A

1|A ⊗ |1

B

1|B .

As a generalization of Eq. (50.3), a bipartite pure state ρ ∈ HA ⊗ HB is entangled if it is not of the form of Eq. (50.3) and is said to be a maximally entangled state if there exists some local bases on H such that the reduced state of ρ is the diagonal matrix, or maximally mixed state,  1 n 0 . . 0 0 0 1 . . 0 0   n 0 0 . . 0 0   (50.4)  0 0 0 . 0 0 .   1 0 0 00 0 n 0 0 0 0 0 n1 50.3

Quantum Channel, Generalized Dense Coding

Let Alice and Bob share a maximally entangled state ρω , i.e. has the maximally mixed state given by Eq. (50.4) as its partial trace. Let the subsystems initially possessed by Alice and Bob be labeled A and B, respectively. To transmit the message x, Alice applies an appropriate channel (encoding), which we denote by ∆x on subsystem A. On the combined system, this is eﬀected by ρω ⇒ (∆x ⊗ IB ) ρω ,

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where IB denotes the identity map on subsystem B. Alice then sends her subsystem to Bob, who performs a measurement on the combined system to recover the message. Let the eﬀects of Bob’s measurement on the bipartite system be Φy . The probability that Bob’s measuring apparatus registers the message y is T r ((∆x ⊗ IB ) ρω ) · Φy . Therefore, to achieve the desired transmission, we require that T r ((∆x ⊗ IB ) ρω ) · Φy = δ xy , where δ xy is the Kronecker delta, i.e., Φy = (∆x ⊗ IB ) ρω , and hence message x is the same as message y.

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Chapter 51

Quantum Algorithm

Quantum computing often makes use of the fact that a quantum register of n qubits contains all the numbers from 0 to 2n − 1 simultaneously, whereas classical register of n bits can contain only one of these numbers at a time. By virtue of this inherently massive parallelism, several quantum-computing algorithms have been proposed intended to demonstrate the power of quantum computing.

51.1

Quantum Fourier Transform

The one quantum algorithm that has demonstrated exponential speedup compared to the classical algorithms is the so-called Shor’s algorithm for factoring a number. It is worth pointing out that all quantum computer algorithms, such as the Shor’s algorithm, is quantum probabilistic. However, the correct answer is obtained with high probability, and the probability of failure can be significantly decreased by repeating the algorithm. The overpowering consideration is the massive parallelism that quantum computing possesses. The problem of factoring a number N can be simplified by finding even a single factor since then the whole problem is reduced to a simpler ones. First select a number a, where the number a and N are coprime, i.e., the integers a and N have no common factor other than 1 and −1, or equivalently, their greatest common divisor is 1. Then form the periodic function f (x) = ax (mod N ). The sequence as a function of x has the respective form x ⇒ 0, 1, 2, .....r, r + 1, ax ⇒ a0 , a1 , ....ar , ar+1 ...., f (x) ⇒ 1, a1 ..ar−1 , 1, a1 ..ar−1 , 1, a1 ..ar−1 ...., where the number r is the first nontrivial integer exponent where ar (mod N ) = 1. Having chosen a such that ar (mod N) has an even period r, we can rewrite ar = 1 (mod N) as

a

r 2

2

+1

a

r 2

ar − 1 = 0 (mod N ) ,

2

− 1 = 0 (mod N ) ,

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where the last line shows that the product of the two factors gives a multiple of the number N we wish to factor. Therefore either one or the other of this factor must also have a factor in common with the number N . The final step is to calculate the r 2 r 2 greatest common divisor, gcd, of a 2 − 1 or a 2 + 1 with the number N , r

2

i.e., the factors of N are gcd( a 2 ± 1, N). Since r is the smallest positive integer r 2 such that ar = 1 (mod N ), so N cannot divide a 2 − 1. If N also does not divide r 2 r 2 a 2 + 1, then N must have a nontrivial common factor with each of a 2 − 1 r 2 and a 2 + 1. This nontrivial gcd will be the factor we are looking for. If N is the product of two primes, this is the only possible factorization. Example: Let N = 15. Then 42 ≡ 1(mod N ) and 4 = ±1(mod N ). Both gcd(4 − 1, 15) = 3 and gcd(4 + 1, 15) = 5 are indeed nontrivial factors of 15. The one part of Shor’s algorithm for factoring a number N = pq, where p and q are prime, relies in finding the period r of the function f (x) = ax (mod N ) where a, chosen at random, is some integer less than and coprime to N . Given a and N , we have to find the order of a, i.e., the least integer r such that ar ≡ 1 (mod N ) or f (x + r) = fa (x) = ax (mod N ) . Classically, all known algorithms are unable to find the period r in length of time polynomial in log N , the length of the number being factorized. To obtain exponential speedup in getting the period out, Shor makes use of quantum register which contains all the numbers simultaneously, coupled with the so-called quantum Fourier transform, or QFT. Shor’s period-finding algorithm makes use of the ability of a quantum computer to be in many states simultaneously, i.e., in a “superposition” of states. To compute the period of a function f (x), we evaluate the function at all values of x simultaneously. Quantum physics does not allow access to all this information directly. A measurement will yield only one of all possible values, while destroying all others. To reversibly transform the superposition to another state that will return the correct answer with high probability upon measurement, one has to make use of the unitary quantum Fourier transform. The following unitary operator is a useful basis for QFT 2n

Iˆ = x=0

2n

|x x| =

y=0

|y y| ,

where the summation over x or y represent sum over the complete product Hilbert space. QFT in 2n -dimensional Hilbert space is a unitary operation on n qubits where the unitary operator is defined as a transition operator between mutuallyunbiased bases through the relations, Iˆ |y ≡ |y =

2n −1 x=0

|x x| y ,

(51.1)

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Quantum Algorithm 2n −1

Iˆ |x = |x =

x=0,y=0

689

|y y| x ,

(51.2)

where by virtue of being mutually-unbiased |x and |y bases, the transition matrix − 1 2πixy element is y| x = (2n ) 2 e 2n and therefore 1 −2πixy x| y = √ n e 2n , 2

(51.3)

with the inverse transition matrix element defined by 1 2πixy y| x = √ n e 2n 2

(51.4)

so that 2n −1

1 |x = √ 2n

1 |y = √ n 2

e

2πixy 2n

y=0

2n −1

e

|y ,

−2πixy 2n

x=0

|x .

The number xy is the multiplication of numbers x and y, x = xo 20 + x1 21 + x2 22 + x3 23 ...xn−1 2n−1 , y = yo 20 + y1 21 + y2 22 + y3 23 ...xyn−1 2n−1 , which are represented by arrays of qubits (product states) in the quantum registers |x and |y |x = |xn−1 ⊗ |xn−2 ⊗ ...... ⊗ |x0 , |y = |yn−1 ⊗ |yn−2 ⊗ ...... ⊗ |y0 , where |xk and |yk are individual qubits, with values of xk and yk either 0 or 1. Writing 2n = N , we have N−1

Iˆ x=0

N−1

ψ (x) |x

= x=0

1 = √ N

ψ (x) |x N−1

N−1

ψ (x) e y=0

x=0

2πixy N

|y ,

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Nonequilibrium Quantum Transport Physics in Nanosystems

yielding the ‘covariant’ quantum Fourier transform, ψ (y), of the old component ψ (x) given by N−1

ψ (y) =

e

2πixy N

ψ (x) .

(51.5)

x=0

This is the y th component of the coeﬃcient of expansion in terms of the |y basis, which looks like an ordinary discrete Fourier transform not involving quantum states but only integers. 2πixy 2πixy Now consider the expression for the transformation function, e N = e 2n . This is periodic in xy with period N = 2n . The invertible discrete Fourier transform 2πixy uses the terms in e N that corresponds to the ‘first circle’, namely, xy N < 1. Decomposing x and y into their binary components, we have xy 1 = n xo 20 + x1 21 + x2 22 + x3 23 ...xn−1 2n−1 2n 2 × yo 20 + y1 21 + y2 22 + y3 23 ...xyn−1 2n−1 x1 x2 x3 xn−1 xo = n + n−1 + n−2 + n−3 ... 2 2 2 2 2 × yo 20 + y1 21 + y2 22 + y3 23 ...yn−1 2n−1 x1 x2 x3 xn−1 xo = yo n + n−1 + n−2 + n−3 ... 2 2 2 2 2 x1 x2 x3 xn−1 xo + y1 n−1 + n−2 + n−3 + n−4 ... 0 2 2 2 2 2 x1 x2 x3 xn−1 xo + y2 n−2 + n−3 + n−4 + n−5 ... −1 2 2 2 2 2 x1 x2 x3 xn−1 xo + y3 n−3 + n−4 + n−5 + n−6 ... −2 2 2 2 2 2 + ...... x1 x2 x3 xn−1 xo + yn−1 n−(n−1) + n−1−(n−1) + n−2−(n−1) + n−3−(n−1) ... −(n−1) , 2 2 2 2 2 where xk and yk are either 0 or 1. This reduces to simpler expression by eliminating 2πixy yl > 1 since this simply yield unity factor in the expression for e N terms with xm 2k in Eq. (51.5). Thus we have xy = yo χ012...n−1 + y1 χ012...n−2 + y2 χ012...n−3 + yn−1 χ0 , 2n where we have used the notations symbol χ0 χ01 χ012 ..... χ012...n−1

value xo 2 xo 22 xo 23

+ x21 + x221 + x22 ...... x1 x2 xo 2n + 2n−1 + 2n−2 + .. +

xn−1 2

binary fraction 0.xo 0.x1 xo 0.x2 x1 xo ....... 0.xn−1 xn−2 xn−3 ...x2 x1 xo

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Quantum Algorithm

691

Therefore, 1 F (|x ) = √ n 2 1 = √ 2n

2n −1

e

2πixy 2n

y=0 2n −1 y=0

|y

e2πi(yo χ012...n−1 +y1 χ012...n−2 +y2 χ012...n−3 +....+yn−1 χ0 ) |y

 2πiy χ n−1 0 |yn−1 e    2n −1  ⊗...............  1 ⊗e2πiy2 χ012...n−3 |y2 = √ 2n y=0   ⊗e2πiy1 χ012...n−2 |y1    2πiyo χ012...n−1 |yo ⊗e

          

.

(51.6)

The summation over the number y from 0 to 2n − 1 will eventually assign both 0 and 1 to each yk . Thus, F (|x ) is equivalent to the following superposition of qubits 1 1 F (|x ) = √ |0 + e2πiχ0 |1 √ |0 + e2πiχ01 |1 2 2 1 2πiχ012...n−1 ... ⊗ √ |0 + e |1 . 2

1 ⊗ √ |0 + e2πiχ012. |1 2 (51.7)

Thus, the quantum Fourier transform as a unitary operation on n qubits, can be factored into tensor product of n single-qubit operations, suggesting that it can be represented as a quantum circuit, analogous to the use of parallel Hadamard transformation of each qubit in a quantum circuit. In fact, the QFT is a generalization of the Hadamard transformation of two-state systems to multi-state systems. In other words, these n single-qubit operations make up the quantum transform F (|x ), involving all the 2n numbers in one single massively parallel operation. That this is so is because we have calculated the QFT as a massively parallel single-operation on each qubit. In the expression above, the rotation depends on the values of the other bits. So we should expect to be able to build the Fourier transform out of Hadamard and controlled phase rotation gates. In fact, each single qubit operation is made up of Hadamard transformation followed by a controlled rotation about the z-axis of the Bloch sphere. For example, consider the derivation expression of the last term in Eq. (51.7) in terms of Hadamard and controlled rotation about the z-axis of the (n − 1) th qubit. We have 1 √ |0 + e2πiχ012...n−1 |1 2

xn−1 x1 x2 xo 1 = √ |0 + e2πi( 2n + 2n−1 + 2n−2 +..+ 2 ) |1 2

The Hadamard transform (gate) of the (n − 1) th qubit gives 1

2πixn−1 y 2πixn−1 1 1 2 e |y = √ |0 + e 2 |1 H (|x )n−1 = √ 2 y=0 2

.

.

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Nonequilibrium Quantum Transport Physics in Nanosystems

We now apply a series of controlled rotation on this qubit defined by 1 0 2πi 0 e 2∆k +1

R∆k =

,

(51.8)

where ∆k is the diﬀerence of the indices the qubit under consideration and that 1 0 of the controlling qubit (compare Eq. (51.8) with Pauli matrix σz = ,a 0 −1 rotation of Bloch sphere about z-axis by angle π). With xn−2 qubit as the control, the first rotation is CR∆1 =

1 0e

0 2πixn−2 22

=

1 0e

0 2πixn−2 4

and therefore 2πixn−2 2πixn−1 1 = √ |0 + e 4 e 2 |1 2 x ix 1 2πi n−2 + n−1 4 2 |0 + e = √ |1 2

2πixn−1 1 |1 CR∆1 √ |0 + e 2 2

.

The second rotation with xn−3 qubit as the control yields 1 CR∆2 √ 2

x ix 2πi n−2 + n−1 4 2

|0 + e

|1

1 =√ 2

x x ix 2πi n−3 + n−2 + n−1 8 4 2

|0 + e

|1

and so on. Finally the last rotation with xo qubit as the control yields

x ix x1 x2 1 2πi 2n−1 + 2n−2 +..+ n−2 + n−1 4 2 |1 CR∆n−1 √ |0 + e 2 xn−2 xn−1 x1 x2 xo 1 = √ |0 + e2πi( 2n + 2n−1 + 2n−2 +... 4 + 2 ) |1 2 1 = √ |0 + e2πiχ012...n−1 |1 . 2

Considering the other qubits, namely, (n − 2) th qubit, and so on, and applying the same procedure as done above for the (n − 1) th qubit, will yield all the terms in Eq. (51.7). Note that for the last xo -qubit line, there is no more qubit-line with lower indices and therefore only a Hadamard transformation can be applied to this qubit resulting in the first term of Eq. (51.7). The QFT circuit is shown in Fig. 51.1. The complexity is determined by the n Hadamard gates, n(n−1) controlled2 R∆k gates, and lastly the n2 swap gates, clearly an O n2 in complexity dominated by the controlled-R∆k gates. Note the the best classical algorithms for computing discrete Fourier transform on 2n elements, such as the Fast Fourier Transform (FFT) algorithm, compute the discrete Fourier transform using O (n2n ) gates. This means that in principle it requires exponentially more operations on a classical computer than on a quantum computer to compute the discrete Fourier transform.

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Quantum Algorithm

Fig. 51.1 QFT circuit with the bits of the transformed state in reverse order. After operations (not shown), the state of the qubits is as given in Eq. (51.7).

51.1.1

693

n 2

swap

Order-Finding Algorithm

The algorithm for finding the order of x (mod N) on a quantum computer will use two quantum registers which hold integers, 0 to N − 1, represented in binary and hence of length n = log2 N. There will also be some amount of workspace. This workspace gets reset to |0 after each subroutine of our algorithm, so there is no need to include it in writing down the state of the machine. The initial ‘blank’ state of the two quantum registers, each containing n qubits is represented by |0 1 ⊗ |0 2 . The first register is then initialized to a superposition of all integers from 0 to 2n − 1 by applying the Hadamard transform to the n qubits of the first register. The resulting prepared state is given by 1 √ 2n

2n −1 x=0

|x

1

⊗ |0 2 .

(51.9)

To continue to the next step, first, we note that the function f (x) = ax (mod N ) yields all integers from 1 to N −1 = 2n −1, so that this function can also be viewed as made up of n parallel single-qubit operations similar to that of the quantum Fourier transform in Eq. (51.7), but simpler in the sense that each eﬀective single-qubit f (x) operation, which we denote by Uk k , carries the state |0 of the output register into |fk (x) . To see this note that since the output of f (x) = ax (mod N ) amounts to a permutation of the binary representation of all the numbers in the first register |x , then it is easy to see that for every individual output-register |0 line (qubit) if its corresponding fk (x) = 1 the qubit is flipped to |1 , if fk (x) = 0 then the qubit in the output register stays |0 . Thus, in any case, the qubit simply becomes |fk (x) = |0 + fk (x) (mod 2) . This also amount to saying that the output f (x) = f (x) ax (mod N ) is placed in the output register. Thus, Uk k is a controlled gate, where the control is provided by fk (x). Moreover, entanglement is formed between f (x) gates entangles the the input and output register, i.e., the use of controlled Uk k whole computer. As we shall see, the power of quantum computing derives from superposition and non-locality.

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Nonequilibrium Quantum Transport Physics in Nanosystems f (x)

Thus, the application of the Uk k gates at the next step converts the second register from state |0 2 to |fk (x) 2 , and the state of the two registers is given by 1 √ 2n

2n −1 x=0

|x

1

⊗ |f (x)

(51.10)

2

Next, we apply the quantum Fourier transform on the first register. The quantum Fourier transform on N points is defined by Eq. (51.5). This leaves us in the following state 1 √ 2n

e x

2πixy 2n

y

|y

1

|f (x) 2 .

Next, we perform a measurement on the first register to obtain some number y. Since the two registers are entangled, this put the output register in some value |f (xo ) 2 . Since f (xo ) is periodic, the value f (xo ) in the second register corresponds to several values of x, namely, xo + br. Therefore, the probability to measure y is given by 2

1 2n

e

2πixy 2n

x:f (x)=f (xo )

=

1 2n

2

e

2πi(xo +br)y 2n

.

b

This probability is higher, the closer yr N is to an integer (b being an integer). Thus, turn Ny into an irreducible fraction, and extract the denominator r˜, which is a candidate for r. Then check if f(x) = f(x + r˜). If so, we are done. Otherwise, obtain more candidates for r by using values near y, or multiples of r˜. If any candidate works, we are done. Otherwise, go back to step 1 of the subroutine, Eq, (51.9). A variation of the above subroutine is as follows. First we consider and easy case in which r divides N . Before applying the QFT on the first register, as was done before, we instead measure the second register to obtain some value f(xo ) for some 0 ≤ xo ≤ r − 1. Then all superposed states inconsistent with the measured value must disappear. Because r divides N , and by virtue of the entanglement, the first register is collapsed into superposition of K = Nr states, determined by the periodicity of f(x) 1 √ K

K−1

j=0

|xo + jr

1,

where 0 ≤ j < Nr , N − r ≤ xo + (K − 1)r ≤ N , where xo + jr is in binary representation. Thus we have set up a periodic superposition of period r in register 1. Now we can drop the second register from consideration. The first register has a periodic superposition whose period is the value we wanted to compute. However, we have no assurance to obtain the period by measuring the first register, because all we will get is a random point, with no correlation across independent trials (because xo is random).

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Quantum Algorithm

695

Now we invoke the use of QFT. Applying QFT to register 1, we obtain 1 √ K

K−1 N−1

e

2πi(xo +jr)y N

j=0 y=0

|y

1

1 =√ K

We evaluate the interference of the sum of e K−1 j=0

2πi (jr) y = e N

=

2πiry

K−1

e j=0

    

N−1

K−1

e

2πixo y N

y=0

|y

1

.

over index j. We obtain

j

K = Nr , N 2πiry r 1− e N 1−e 2πiry N

2πi(jr)y N

j=0

2πi(jr)y N

2πiry N

e

if e

2πiry N

=1

= 0, if e

2πiry N

. =1

Since the equality, e N = 1, holds only iﬀ ry N is an integer, i.e., iﬀ y is a multiple of Nr (in group theoretical language this means that the terms generated by f (x) = ax (mod N ) form elements of a subgroup of order r of a larger finite group of order N), we are left with a superposition where only multiples of Nr have nonzero amplitude. Therefore the superposition upon measurement gives a random multiple y = c Nr , with c a random number in 0 ≤ c < r. Therefore, we have 1 √ K

N−1

K−1

e

2πixo y N

y=0

e j=0

2πi(jr)y N

|y

1

1 ⇒√ r

r

e c=0

2πixo c r

c

N r

. 1

We thus have the equality c y = . N r

(51.11)

Each repetition of the procedure will have a big probability that the random variable c will be coprime to r or gcd c, Nr = 1. An expected number of repetitions of the y in above procedure will yield y = c Nr where c is coprime to r. Thus by writing N Eq. (51.11) as irreducible fraction, we obtain r as the denominator. We will not consider the case where r does not divide N as this is more involved and will lead us to consider continued-fraction expansion to estimate r. This problem essentially uses the similar principle as the case where r does divide N. 51.1.2

Phase Estimation Algorithm

The period finding algorithm is a special case of an algorithm for generating arbitrary interference patterns through some function evaluations. The problem conˆ on n qubits, which sidered here is this: Suppose we are given a unitary operator U 2πiφ . We wish to find the has a known eigenstate |Ψ with an unknown eigenvalue e phase, φ, with some precision (say, t binary bits). By itself, the phase estimation algorithm gives an estimate of the eigenvalue associated with a given eigenvector of a unitary operator. However, the resulting solution algorithm turns out to be useful subroutine of several other algorithms, to solve other important problems.

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Nonequilibrium Quantum Transport Physics in Nanosystems

We require two registers, namely, the t-qubit control register |x and a target register for storing |Ψ , which contains as many qubit necessary to store |Ψ . Moreˆxα . All these over, we require that for each t-qubit string |x there is an operator U α ˆ operators Ux have common eigenvector |Ψ , which is the state of the target register. Obviously all these operators only give a global phase to |Ψ and hence do not change the state of the target register. Decomposing x into its binary components x = xo 20 + x1 21 + x2 22 + x3 23 ...xt−1 2t−1 ,

(51.12)

ˆxα for each control qubit string is a controlled by the corresponding the operator U ˆ 2k (here we drop the subscript x since the upperscript qubit, these are denoted by U is associated with each qubit line in the control register) with k = 0, 1, 2, ....t − 1. In quantum circuit language, we have assumed that in the expression of Eq. (51.12), the Hadamard gate has been applied to each of the qubit line of the control register originally in state |0 . Thus, 1 1 1 |x = √ (|0 + |1 )t−1 ⊗ √ (|0 + |1 )t−2 ... ⊗ √ (|0 + |1 )0 . 2 2 2 Note in quantum circuits all qubit lines are arranged in parallel by virtue of parallel processing. It is natural to assume that the k = t − 1 qubit line is the top line and the k = 0 qubit line is at the bottom of the ‘stack’ of a t-bit register. And so after ˆ 20 is applied to |Ψ , the combined state of the 0-line of control the first controlled U register and target register is ˆ 20 √1 (|0 + |1 ) |Ψ = √1 |0 |Ψ + |1 e2πiφ20 |Ψ U o 2 2 1 = √ |0 + e2πiφ |1 o |Ψ . 2

o

ˆ 21 is applied to |Ψ , the combined state of the 0-line and 1-line When the controlled U of control register and target register is 1 1 1 √ |0 + e2πiφ2 |1 √ |0 + e2πiφ |1 o |Ψ 1 2 2 1 1 2πiφ21 √ |0 + e2πiφ |1 = √ |0 + e |1 1 2 2

o

|Ψ .

ˆ 22 is applied to |Ψ , In similar manner, we have after U 2 1 1 1 1 √ |0 + e2πiφ2 |1 √ |0 + e2πiφ2 |1 √ |0 + e2πiφ |1 o |Ψ 2 1 2 2 2 1 1 1 = √ |0 + e8πiφ |1 2 √ |0 + e4πiφ |1 1 √ |0 + e2πiφ |1 o |Ψ . 2 2 2

ˆ 2k are applied, the n target bits are still left in the state |Ψ , and the After all the U

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Quantum Algorithm

697

t control bits are left in the state t

t 1 √ |0 + e2 πiφ |1 2 ⊗ ... ⊗ |0 + e8πiφ |1

t−1

t−1

⊗ |0 + e2

πiφ

⊗ |0 + e4πiφ |1 2

|1

t−2

⊗ |0 + e2πiφ |1 1

o

.

Let us suppose that there is an exact t-bit expression for the phase, φ = 0.φt φt−1 ...φo φ φ φ1 φ2 = ot + t−1 + t−2 + .. + t−1 . 2 2 2 2 We therefore have 2πi

φo

+

φ1

+

φ2

+..+

φt−1

2 2t 2t−1 2t−2 e2πiφ = e φ1 φ2 φo 2πi( 2t−1 + 2t−2 + 2t−3 +..+φt−1 ) 4πiφ e =e

2πi

=e .....

2t πiφ

e

φo 2t−1

+

φ1 2t−2

+

φ2 2t−3

+..+

φt−2 2

φ φ1 φ2 φ 2t πi 2to + 2t−1 + 2t−2 +..+ t−1 2

=e φo = e2πi 2 ,

and therefore, 1 √ 2

t

|0 + e2πi(0.φo ) |1

t−1

⊗ |0 + e2πi(0.φo φ1 ) |1

2πi(0.φo φ1 φ2 ..φt−3 )

⊗ ... ⊗ |0 + e 2πiφ

⊗ |0 + e

|1

o

|1

t−2

2πi(0.φo φ1 φ2 ..φt−2 )

2

⊗ |0 + e

|1

1

.

Dropping the subscript of each qubit by virtue of indistinguishability of particles, we end up with 1 √ 2

t

|0 + e2πi(0.φo ) |1

⊗ |0 + e2πi(0.φo φ1 ) |1

⊗ ... ⊗ |0 + e2πi(0.φo φ1 φ2 ..φt−3 ) |1

⊗ |0 + e2πi(0.φo φ1 φ2 ..φt−2 ) |1

⊗ |0 + e2πiφ |1 . The quantum circuit is shown in Fig. 51.2. Comparing with expression for the QFT of Eq. (51.7), the last result is the Fourier transform of the basis state φt φt−1 ...φo . Rewriting our expression for the state of the t control bits as 1 √ 2t

2t −1 y=0

e

2πi φ y 2t

|y .

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Fig. 51.2 Quantum circuit for the phase estimation algorithm, illustrating the estimation of the phase φ with a t-bit precision after the t-qubit register output passes through the quantum gate of Fig. 51.3.

This has the form of a Fourier-transformed state. Since the inverse relation is 2t −1

1 |y = √ 2t

e

−2πiφy 2n

x=0

|φ

by applying the inverse Fourier transform gate, we get ⇒

1 2t

1 = t 2 =

1 2t

2t −1

e

2πi (φ) y 2t

y=0

2t −1

e

−2πiφ y 2n

φ =0

2t −1

2πi

e

(φ−φ )

|φ

y

2t

φ

y=0,φ =0 2t −1 φ =0

2t δ φ − φ

φ

= |φ . It is now clear that after the control register is passed through an inverse QFT gate, the output upon measurement of this register yields the state φt φt−1 ...φo , as shown in Fig. 51.3. If φ has an exact t-bit expression, then this estimate will also be exact. If not, it will be quite close by repeating the computation and measurement. It has been shown that if we define to be our desired accuracy bound,i.e., φcorrect − φmeasured ≤ , where is a positive integer characterizing the desired error tolerance, then the probability, P , of being outside this bound is φcorrect − φmeasured >

P

If we suppose to approximate 1 2n ,

t−n

we may choose = 2

φcorrect 2t

=

1 . 2 ( − 1)

to accuracy equal to

φcorrect 2t

−

φmeasured 2t

=

−1. Then by making use of t = n+p qubits in the phase

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Quantum Algorithm

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Fig. 51.3 After the control register of Fig. 51.2 is passed through an inverse QFT gate, the output upon measurement of this register yields φt−1 ....φ0 .

estimation algorithm, we have = 2n+p−n − 1 = 2p − 1, and (2( 1−1)) = (2(2p1−2)) . Moreover, the probability of approximation correct to this accuracy is at least 1−

1 . − 2))

(2 (2p

Thus if we want φ accurate to n bits with probability of at least 1 − ε, then we need extra bits p determined by 1 = ε, (2 (2p − 2)) which yields p = log2 2 +

1 , 2ε

and hence the total number of qubits needed is t = n + p = n + log2 2 + 51.1.3

1 2ε

.

Connection Between Root Finding and Phase Estimation

Recall that the first steps of the period-finding algorithm involved preparing an input and output register in the state given by Eq. (51.10), where f (x) is a periodic function. There, we applied a QFT to the input register, and then measured it. To bring the connection to the phase estimation algorithm, we will now look at an almost identical procedure in an alternative way. Let us take the period of f (x) to be such that f (x + r) = f (x). Then the pertinent Hilbert space is r-dimensional. We define the inverse Fourier transform state of the output register as r−1

−2πix l 1 f˜ (l) = √ e r |f (x) , r x=0

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where r is the unknown period of f (x). Thus the state of the output register can be written as 1 |f (x) = √ r

r−1

e

2πix l r

f˜ (l) .

l=0

Upon substituting the last expression in Eq. (51.10), we obtain 1 √ 2n

2n −1 x=0

|x

1

1 =√ n 2 r

⊗ |f (x)

2

2n −1

r−1

e

2πix l r

x=0

l=0

|x

1

⊗ f˜ (l)

2

.

The first register has the form of a Fourier transformed state. In the present case, the inverse is |x

1

1 =√ n 2

2n −1

e

−2πixl 2n

|l

l =0

1.

n

l 2 l Writing 2πix = 2πix (here we need the nearest whole number for r 2n r apply the inverse Fourier transform gate to obtain r−1

1 √ 2n r l,l =0 =

2n −1

2πix 2n

x=0

1 √ 2n r 1 √ r

e

⊗ f˜ (l)

2n l −l r

|l ⊗ f˜ (l)

r−1

2n δ l,l =0

r−1

l=0

2n l r

n

( 2r l −l ) |l

⊗ f˜ (l)

2

,

2n r

) we

2

2

(51.13)

where the approximation holds when 2n is notn an exact multiple of r. Therefore, If we measure the first register, we will get 2r l for some value of l, chosen at n random from 0, ..., r − 1. We may identify 2r l as integer index of the corresponding 2n l ‘momentum’ p eigenvalue, with p = 2π 2n r . Let us now see the connection between period-finding and phase estimation via some function evaluations. First a brief digression. Let Tˆ be the unitary transformation that translates the argument of a periodic function |f (x) by one, i.e., Tˆ |f(x) = |f(x + 1) mod r . The eigenfunction |f(x) corresponds to the ‘Wannier function’ in band theory of crystalline solid. In general, the family of unitary translation operators which simply add a constant integer k (mod r) to the argument of |f (x) share the eigenstates given by the ’Bloch function’ given by 1 f˜ (l) = √ r

r−1

e x=0

−2πix l r

|f (x)

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since operating Tˆ on f˜ (l) gives r−1

−2πix l 1 Tˆk f˜ (l) = √ e r |f (x + k) mod r r x=0

=e =e

2πik l 2n

2πik l r

1 √ 2n

r−1

e

−2πi(x+k) l 2n

x=0

f˜ (l) ,

|f (x + k) mod r

k = 1, 2, .....r − 1,

(51.14)

2πik l where e r is the eigenvalue of Tˆk , Tˆ operator applied k times. Observe also that the equal superposition of the eigenfunctions, f˜ (l) , yields

1 √ r

r−1

l

1 f˜ (l) = r

r−1

e

−2πix l r

l,x=0

|f (x)

r−1

=

1 rδ (x) |f (x) r x=0

= |f (0) = a0 mod r = |1 .

(51.15)

We can build the state |f (x) of the output register in a manner of the phasek estimation algorithm (PEA), by using some unitary operator Tˆ2 ,except that in ˆ 2k . Let Tˆ be the PEA the “output register” is prepared in an eigenstate |Ψ of U unitary transformation that translates the argument of f(x) by one, Tˆ |f (x) = |f(x + 1) mod 2n . We can rewrite the transformation from |x 1 |0 2 ⇒ |x 1 |f (x) 2 in terms of controlled-Tˆ operations. The procedure is as follows: (i) . Shift the output register to |f (0) . (ii) . Apply a controlled-Tˆ operation from the least significant qubit of the input register to the output register: |x 1 |f (0) 2 ⇒ |x 1 |f (xo ) 2 . (iii) . Apply a controlled-Tˆ2 operation from the second qubit: |x 1 |f (xo ) 2 ⇒ |x 1 |f (2x1 + xo ) 2 . j (iv) . Successively apply controlled-Tˆ2 from qubits j = 2, ..., n − 1. The final state is |x

1

f 2n−1 xn−1 + ......2x1 + xo

2

≡ |x 1 |f (x) 2 .

Thus, in contrast with PEA the state of the output register is not an eigenstate j j of Tˆ2 . As shown in Eq. (51.14), this eigenstate of Tˆ2 is f˜ (l) . Since we don’t know the value r, we can’t actually prepare the second register in the state f˜ (l) . However, Eq. (51.15) has allowed us to circumvent the problem, since by preparing

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the output register in state |0 2 or |f (0) 2 , we are eﬀectively preparing a superposition of all the eigenstates f˜ (l) . When we make the final measurement of the first register in Eq. (51.13), one of these states is picked out at random; but our ability to solve the problem doesn’t depend on which l is selected. Hence the period-finding algorithm treated here becomes phase estimation algorithm. To estimate the phase φ ≡ rl accurate to m bits with probability (1−ε) r we need 1 qubits in the first register. t = m + log2 2 + 2ε 51.2

Quantum Search Algorithm

We have here a problem of searching through a space of N elements, indexed by a number x in the range 0 to N − 1. Again, we take N = 2n so that all indices can be simultaneously stored in n qubits, this is the input index register. We also need an output register, a single qubit, also known as oracle qubit, which is flipped if the search solution is found, and unchanged otherwise. It is useful to prepare the |0 −|1 oracle qubit of the second register in the state 2√2 2 . The initial state of the two registers is assumed to be 1 √ N

N−1 x=0

|x 1 |q

2

1 =√ N

N−1 x=0

|x

1

|0

− |1 √ 2

2

2

.

We suppose we are given a state marker, represented by a function fω which takes as input an integer index 0 ≤ x ≤ N − 1, which by definition fω (x) = 1 if x = ω is a solution to the search problem (oracle qubit is flipped), and fω (x) = 0 if x = ω is not a solution to the search problem (oracle qubit unchanged). Thus we write the transformation 1 Ufω : √ N

N−1

|x

x=0

1

|0

− 2√

|1 2

1 ⇒√ N

2

N−1 fω (x)

(−1) x=0

|x

|0

1

− 2√

|1 2

2

.

Since the oracle qubit is eﬀectively not changed throughout the search algorithm, it is more convenient to omit this in what follows. With this convention, the action of the oracle qubit on the first register gives N−1

1 Ufω : √ N

x=0

|x

1

1 ⇒√ N

N−1 fω (x)

(−1) x=0

|x 1 .

In eﬀect, the oracle qubit marks the solutions to the search problem by shifting the phase of the solution, if x = ω, then (−1)1 |ω 1 = − |ω 1 . Given that |ω 1 is one of the orthonormal basis states, one can write 1 √ N

N−1 fω (x)

(−1) x=0

|x

1

1 = (1 − 2 |ω ω|) √ N

N−1 x=0

|x 1 .

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It would be helpful to visualize the transformation Ufω by grouping the index register into a state |α , the sum over all N − M number of indices x which are not (orthogonal to) solutions to the search problem and state |β , the sum over all M number of indices x which are (or parallel to) solutions to the search problem. We define the normalized states, 1 |α ≡ √ N −M 1 |β ≡ √ M

x⊥ω

x ω

|x ,

|x ,

where x

|x =

x⊥ω

|x +

x ω

|x

and the initial state of the first register is thus given as  N−1 1  1 |x 1 = √ |x + Ψ= √ N x=0 N x⊥ω x =

N −M |α + N

M |β . N

ω



|x 

Since |α and |β are orthogonal, we can also write the initial state as N −M M |α + |β N N θ θ = cos |α + sin |β , 2 2

|Ψ =

where θ2 is the angle between |α and the initial state |Ψ . The idea of the quantum search algorithm is to rotate the initial state |Ψ towards the |β ‘axis’ (|α and |β correspond to the horizontal and vertical axis, respectively, in the geometric visualization), which is the superposition of the solution space. In general neither M nor the superposition space |α and |β are known. However, the search presupposes that we have an oracle that identify the solution, and this is embodied in the state marker given above. A search iteration, as a series of operation starting with the initial state, is the so-called Grover’s iteration. This iteration combines the state marker which carries information about the superposition space |α and |β with the rotation towards the solution space |β . The operation that mark the solution, which we denote by the symbol O (to designate an oracle) as derived above is O = (1 − 2 |ω ω|) = (1 − 2 |β β|)

if M = 1 if M > 1.

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This oracle has the eﬀect of reflecting the search space, |Ψ , about the |α axis (note the factor 2). In Grover’s iteration this oracle is followed a reflection of O |Ψ about |Ψ , which we designate by a symbol R R =2 |Ψ Ψ| − 1. The result of this combined operation, RO |Ψ = (2 |Ψ Ψ| − 1) (1 − 2 |β β|) |Ψ is a rotation of |Ψ towards the solution space |β by an angle θ, where sin θ2 = M N.

This eﬀective rotation step is known as the Grover’s iteration, and maybe represented as cos θ − sin θ sin θ cos θ

G= where sin θ = 2 sin θ2 cos θ2 = 2 51.2.1

√

M(N−M) . N

For M

, N, θ ≈ 2

M N.

Performance of the Search Algorithm π 2

The initial angle between solution space |β and initial search space |Ψ is From the identity, cos through ar cos

M N

π 2

−

θ 2

≡ sin

N M

=

M N,

N , sin θ

θ 2

.

we therefore need to rotate |Ψ

radians to align |Ψ with the solution space |β . Thus

the metric of the search iteration. If M we require O

θ 2

−

θ

2

M N,

M N

is

then it is clear that

applications of the Grover’s iteration to rotate |Ψ close to

N the solution space |β . On the classical computer it will require O M operations th to solve this search problem. To see this, note that at the l step of the Grover’s l

iteration the search space |Ψ has been rotated by an angle φ =

(2j + 1) θ = j=0 θ 2 , based

on the l2 + l θ towards the solution space |β . Upon equating φ = π2 − assumption that M N , we have to solve a quadratic equation for the number of iteration l. The result is that l

π 4

N M.

The foregoing analysis is very good for

M N (needle-in-a-haystack problem) but is no longer valid for M ≥ the classical methods will do better. 51.3

N 2,

where

Discrete Logarithms

Discrete logarithms problem is related to the period-finding problem in integer factorization, both problems are diﬃcult (also known as N P -hard). Algorithms from one are often adapted to the other, and the diﬃculty of both problems has been exploited to construct various cryptographic codes. Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u ∈ GF(q) is that integer k, 1 ≤ k ≤ q − 1, for which u = gk . Discrete logarithm algorithms has also been formulated for the finite fields which are power of primes, notably the finite

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field, GF(2n ). The well-known problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. In a discrete logarithm problem, we are given an element which generates a cyclic group of some finite order, p, here a prime. Thus αp = e (identity element of the group). Then another element β of the group is given and we want to know which power of α it is; that is, the integer a for which β = αa . This is also written as a = logα β . In the quantum solution, we prepare two registers, each in a uniform amplitude superposition of p (prime number) basis states: 1 N

p−1 p−1

x=0 y=0

|x |y .

Then we compute the function f (x, y) = αx β y in a third register and measure it. This will leave the two registers in a superposition of the form

y

|xo − a · y |y ,

where all arithmetic operations are understood to be modulo p, xo is random and y runs over 0...p − 1. By Fourier transforming each register with a QFFT we get a similar state but without the oﬀset xo , namely an equally weighted superposition of all states of the form |x |a.x with x = 0...p − 1. A measurement will now allow to compute a in all cases except when x = 0. Thus, we have the known and instance-independent success probability of 1 − 1/p, which allows to easily make the algorithm exact by using amplitude amplification. 51.3.1

Quantum Solution

The diﬀerence between the period-finding algorithm and the discrete logarithms algorithm (DLOGA) is that the function in DLOGA is more complex, in that there are now two domains stored in two registers, |x1 |x2 . The function is f (x1 , x2 ) = bx1 ax2 = asx1 +x2 , a periodic function, i.e., f (x1 + l, x2 − sl) = f (x1 , x2 ) for some integer l since f (x1 + l, x2 − sl) = as(x1 +l)+(x2 −sl) = asx1 +x2 . Here determining s allows to solve the discrete logarithm problem: given a and b = as , find the exponent s. Moreover, here knowledge of the minimum r > 0 such that ar mod N = 1 is assumed to be known (r serves as the order of the finite field). The DLOGA is described as follows. First we require the following registers: two t = O [log r] + log 1ε qubit registers and one function-evaluation register, all initialized to |0 |0 |0 . The output of the algorithm is the least positive integer s such that b = as . The DLOGA is described as follows.

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(i) Register initialization ⇒ |0 |0 |0 . (ii) Create superposition of the first two register using Hadamard gates ⇒

1 2t

2t −1 2t −1

x1 =0 x2 =0

|x1 |x2 |0 .

(iii) Apply unitary operator U , a controlled unitary operation such that U |x1 |x2 |y = |x1 |x2 |y ⊕ f (x1 , x2 ) , where y ⊕ f (x1 , x2 ) a bit by bit operation, i.e., ⊕ denotes bitwise addition modulo 2. Therefore in this step we end up with ⇒

1 2t

2t −1 2t −1

x1 =0 x2 =0

|x1 |x2 |f (x1 , x2 ) .

(iv) Since |f (x1 , x2 ) has a 2-tuple period (l, −sl), we can apply the QFT: r−1

|f (x1 , x2 ) ⇒

l e2πi(sx1 +x2 ) r f˜ (sl, l) . We can therefore write the state of

√1 r l=0

the computer as  r−1 1  √1 ⇒√ r l=0 2t

2t −1

2πisx1 rl

e

x1 =0



1 |x1   √ 2t

2t −1

2πix2 rl

e

x2 =0

(v) Applying inverse QFT to the first two registers yields 1 ⇒√ r

r−1

l=0

sl r

l r



|x2  f˜ (sl, l) .

f˜ (sl, l)

l (vi) Measuring the first two registers will return a pair ⇒ sl r , r , (vii) finally, apply generalized continued fractions algorithm to determine s.

51.4

Hidden Subgroup Problem

Viewed in terms of group theoretical viewpoint, almost all the quantum algorithms discovered so far that realize an exponential speed-up with respect to the best known classical algorithms can be seen as instances of the Hidden Subgroup Problem (HSP), a problem that asks to find a subgroup H hidden inside a group G. In particular the integer factoring problem and the discrete logarithm problem, and the periodicity finding problem, are instances of the special case of the HSP where the group G is Abelian. Indeed, the DLOGA discussed previously shows how to reduce the problem to a slightly more general kind of periodicity, on additive group of pairs of integers modulo N. More generally, a polynomial-time quantum algorithm solving the HSP over any Abelian group G is known, using as its main tool the Fourier transform over Abelian groups. However, no solution is known for the general case of G non-Abelian. The case of non-Abelian groups is indeed of paramount importance because a polynomial-time solution for the HSP when G is the symmetric group (the group of all the permutations over a given set) would give an eﬃcient quantum algorithm solving the graph isomorphism problem, a well known problem for which no

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polynomial-time classical algorithm is known. However, the symmetric HSP seems diﬃcult, even for quantum computers. Another fundamental instance of the nonAbelian HSP is the case where G is the dihedral group1 . It has been shown that an eﬃcient algorithm solving the HSP over the dihedral group by the coset sampling technique would enable a quantum computer to find, in polynomial time, the shortest vector in a lattice, at least for a class of lattices for which no eﬃcient classical algorithm is known. Since most of exponentially fast quantum algorithms fit in the framework of HSP, it is considered to be paradigm for the development of quantum algorithms. The HSP in the finite group G can be described as follows. Let X be a finite set and f : G → X a function such that f(g1 ) = f(g2 ) if and only if g1 and g2 are in the same coset (i.e., f is constant on each coset) of some subgroup H in G. The problem consists in determining generators for H by querying the function f. If the computational complexity of the algorithm is O(log|G|)2 considering that each query takes one unit of time, we say that the HSP is solved eﬃciently. It is usual to say that the function f ‘hides’ the subgroup H in G, and thus the term Hidden Subgroup Problem. First we give some definitions. For any subgroup H of a group G, we can speak of the left cosets and the right cosets of H. The notation gH for an element g in G means the set of all products g times an element h of H; this set is a left coset of H, because we multiplied by g on the left of H. One could do the same on the right, and form a right coset of H. The collection of all gH for every g in G is the collection of left cosets of H; similarly for right cosets. 1 Dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The dihedral group can actually be defined as the semi-direct product Dn = Zn m Z2 . There are two competing notations for the dihedral group associated to a polygon with n sides. In geometry the group is denoted Dn , while in algebra the same group is denoted by D2n to indicate the number of elements. Definition for semi-direct produc: Let G be a group, N a normal subgroup of G (i.e., N G) and H a subgroup of G. The following statements are equivalent:

• G = NH and N ∩ H = {e} (with e being the identity element of G) • G = HN and N ∩ H = {e} • Every element of G can be written as a unique product of an element of N and an element of H • Every element of G can be written as a unique product of an element of H and an element of N • The natural embedding H ⇒ G, composed with the natural projection G =⇒ G/N (G mod N), yields an isomorphism between H and G/N • There exists a homomorphism G ⇒ H which is the identity on H and whose kernel is N If one (and therefore all) of these statements hold, we say that G is a semidirect product of N and H, or that G splits over N. If both N and H are finite, then the order of G equals the product of the orders of N and H. 2 We will denote the order or the number of elements in a group G by |G|, and similarly other groups and subgroups.

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More specifically, if G is a group of order |G|, H, of order |H|, a subgroup of G, and g an element of G but not in H, then the following cosets containing |H| number of elements, gH ={ gh : h an element of H } is a left coset of H in G, and Hg ={ hg : h an element of H } is a right coset of H in G. Cosets cannot be subgroups because they cannot include the identity element. Two left (or right) cosets of a subgroup H either are identical or have no elements in common, i.e., cosets of either type partition the elements of the group G, and |G| hence |H| = l is an integer. Only when H is normal will the right and left cosets of H coincide, which is one definition of normality of a subgroup3 . A coset is a left or right coset of some subgroup in G. Moreover, gH = H if and only if g is an element of H, since as H is a subgroup. If H is normal, Hg = gH , hence it is not meaningful to speak of a coset as being left or right unless one first specifies the underlying subgroup. In the order-finding algorithm, the subgroup H consists of the set of elements given by H = {0, r, 2r, .....}, where from the additive group operation, f (x + r) = f (x) = f (x + 2r) = .. = ax , where x ∈ / H. Thus, f (x) is constant on each coset of subgroup H. The problem is to find r.

51.4.1

Quantum Hidden Subgroup Algorithm

In what follows, we demonstrate how the quantum factoring algorithm (QFA) fits the framework of quantum hidden subgroup algorithm (QHSA). For QFA, we take N = 21 to be factored, assuming a = 2 has been randomly chosen. For better accuracy of QFA, we choose 9 qubits for the input register, where N 2 ≤ 29 < 2N 2 . For the Abelian group of order |G| in the QHSA, we have the group-element representation (also the character) given by the |G|th root of unity indexed by χ, 2πiχg i.e., e |G| , with g ∈ G. A variation of the above subroutine is as follows. We assume r divides N . Before applying the QFT on the first register, Step 3, we instead measure the second register to obtain some value f(xo ) for some 0 ≤ xo ≤ r − 1. Then all superposed states inconsistent with the measured value must disappear. Because r divides N , and by virtue of the entanglement, the first register is collapsed into superposition of Nr states, determined by the periodicity of f(x)

1

N r

−1

N r j=0

|xo + jr

1,

where 0 ≤ j < Nr , N − r ≤ xo + ( Nr − 1)r ≤ N , where xo + jr is in binary representation. We give below the alternative QFA and QHSA algorithms. 3 A subgroup H of G is normal or self-conjugate, if every conjugate subgroup of H is equal to H, that is, if gHg −1 = H for every g in G.

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QFA

QHSA

Step 0: Initialize |ψ 0 = |0 |1

Step 0: Initialize |ψ0 = |0 |s0

Step 1: Apply inverse QFT

Step 1: Apply inverse QFT

511

|u |1 ⇒

√1 512

e

−2πiux 512

x=0

|x |1

|χ |s0 ⇒ √1

e

|G|

−2πiχg |G|

g∈G

|g |s0

511 √1 512

|0 |1 ⇒ |ψ1 =

x=0

|x |1

|0 |s0 ⇒ |ψ 1 = √1

|G|

g∈G

|g |s0

ˆf (x) Step 2: Apply unitary function U

ˆf(g) Step 2: Apply unitary function U

ˆf(x) : |x |1 ⇒ |x |2x mod 21 U

ˆf (g) : |g |s0 ⇒ |g |f (g) U

511

⇒ |ψ2 =

√1 512 x=0

|x |2x mod 21

Step 3: Apply QFT, left register 511

|x ⇒

√1 512

e

2πixy 512

y=0

1 512

=

y=0

511

|y

2πixy e 512

x=0

|y |2x mod 21

|2x

mod 21

g∈G

|g |f (g)

Step 3: Apply QFT, left register

|G|

2πixy 512

x=0 y=0

511 1 512

e

|G|

|g ⇒ √1

|y

511 511

⇒ |ψ3 =

⇒ |ψ 2 = √1

=

˜ χ∈G

2πigχ |G|

˜ χ∈G

1 |G|

⇒ |ψ 3 = 1 |G|

e

|χ

e

2πigχ |G|

˜ g∈G χ∈G



|χ 

2πigχ e |G|

g∈G

|χ |f (g) 

|f (g) 

Step 4: Measure the left register

Step 4: Measure the left register

0 ≤ y < 512 is the measured value, probability

χ is the measured value, with probability

=

511 2πixy e 512 |2x mod 21 x=0 (512)2

511

⇒ |ψ4 = |y

x=0 511

x=0

2

and the system goes

=

2πigχ e |G| |f (g) g∈G |G|2

and the system goes

2πigχ

2πixy

e 512 |2x mod 21

2 2πixy e 512 |2x mod 21

2

⇒ |ψ 4 = |χ

g∈G

e |G| |f (g)

2πigχ e |G| |f (g) g∈G

2

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QFA Step 0: Initialize |ψ0 = |0 |1

QHSA Step 0: Initialize |ψ_0 = |0 |s0

Step 1: Apply inverse QFT N−1 [

|u |1 ⇒ √1

N

e

−2πiux N

x=0

|0 |1 ⇒ |ψ1 = √1N

N−1 [ x=0

Step 1: Apply inverse QFT |χ |s0 ⇒ √1

|x |1

|G|

[

e

−2πiχg |G|

g∈G

|0 |s0 ⇒ |ψ1 = √1

|x |1

|G|

[

g∈G

|g |s0

|g |s0

ˆf (x) Step 2: Apply unitary function U

ˆf (g) Step 2: Apply unitary function U

ˆf (x) : |x |1 ⇒ |x |f (x) U

ˆf (g) : |g |s0 ⇒ |g |f (g) U

⇒ |ψ2 = √1N

511 [

x=0

⇒ |ψ 2 = √1

|x |f (x)

|G|

[

g∈G

|g |f (g)

Step 3: Measure the right register

Step 3: Measure the right register

f (xo ) is the measured value,

f (go ) is the measured value,

and the system goes to

and the system goes to

N −1 r [

|ψ3 ⇒ t1N r

j=0

|G| r j∈ G r

Step 4: Apply QFT, left register |xo + jr ⇒ √1N

N−1 [

e

2πi(xo +jr) y N

y=0

u 1 N N r

|go + jr ⇒ √1

|y

|G|

= √1

= u

|G|

N −1 r [ N−1 [ j=0

e

2πi(xo +jr)y N

y=0

N −1 [ 2πixo y 1 e N N N y=0 r

N −1 r [

e

j=0

|go + jr |f (go )

Step 4: Apply QFT, left register

⇒ |ψ4 =

=

[

|ψ3 ⇒ u 1

|xo + jr |f (xo )

|y |f (xo )

2πi jy N r

|y |f (xo )

[

e

[

˜ χ∈G

χ (go + jr) |χ

2πi(go +jr)χ |G|

˜ χ∈G

[ [

=u

1 e |G| |G| ˜ r j∈ G χ∈G r

=u

1 e |G| |G| χ∈G ˜ r

[

|χ ⇒ |ψ4 =

2πi(go +jr)χ |G|

2πigo χ |G|

[

e

|χ

2πi jrχ |G|

j∈ G r

|f (go )

|χ |f (go )

Step 5: Measure left register

Step 5: Measure left register

yo is the measured value, probability is N −1 2πi j yo 2 [ 2πixo yo r N u 1 r N = e e N N j=0 r 2 N −1 2πi j yo r N 1 [ r = 1r N e with peaks at r j=0 N N N yo ∈ r , 2 r , 3 r , ....N from which

χo is the measured value, probability is 2 2πigo χo [ 2πi jrχo u 1 = e |G| e |G| |G| |G| r j∈ G r 2 2πi jχo |G| 1 [ r = r1 |G| e with peaks at r G j∈ r q r |G| |G| |G| χo ∈ r , 2 r , 3 r , ... |G|

obtained ⇒

N r

⇒obtained r =

N N r

obtained ⇒

|G| r

⇒obtained r =

|G| |G| r

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Appendix A

Commutation Relation between Components of π (x, t) and A (x , t)

The reason that the commutation relation between the canonically conjugate momenta and the fields is a bit complicated is due to the supplementary condition, ∇ · A = 0, that must be satisfied. Thus we have to be careful when using the Cartesian system (i, j = x, y, z). If we tentatively write [πi (x, t) , Aj (x , t)] =

| δ ij δ (x − x ) i

(A.1)

then it follows that the left side would give

j

∂ [πi (x, t) , Aj (x , t)] = [πi (x, t) , ∇ · Aj (x , t)] = 0 ∂xj

by virtue of the supplementary condition. However, the right-hand side would not yield a zero value since

j

| ∂ ∂ | δ ij δ (x − x ) = δ (x − x ) = 0 ∂xj i i ∂xi

which evaluates to the following expression

j

1 ∂ | δ ij δ (x − x ) = ∂xj i (2π)3

dk3 (−ki ) eik(x−x ) = 0

contradicting the results of the left-hand side. Therefore to satisfy the supplementary condition, ∇ · A = 0, we need to ‘correct’ the right hand side of Eq. (A.1). For this purpose, let us replace the right hand of Eq. (A.1) by, as yet undetermined, δ⊥ ij (x − x ), i.e., [πi (x, t) , Aj (x , t)] =

| ⊥ δ (x − x ) i ij

(A.2)

To determine δ ⊥ ij (x − x ), first we need to consider the two diﬀerent types of basis vectors in reciprocal space used to define vector fields. Indeed, the vectors in reciprocal space are used for the identification of longitudinal (wave propagating direction) and tranverse-polarization directions of the electromagnetic fields. The first basis vectors to consider is based on the Cartesian system, namely, the set 711

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{ei } (i = x, y, z), used in reciprocal space of the Fourier transformation, and the second basis vectors is based on the set of longitudinal and transverse system of unit vectors ε, ε , κ , where ε and ε are the transverse-polarization unit vectors and κ =

k

|k|

is the longitudinal unit vector along the propagating wave direction. ε·ε =ε·κ=κ·ε =0 ε·ε=1=κ·κ=1

The following products of the components in reciprocal space of the transverse polarization unit vectors will be useful, (ε)i (ε)j = εi εj + εi εj (ε)⊥k

= (ei · ε) (ε · ej ) + ei · ε

ε · ej + (ei · κ) (κ · ej ) − (ei · κ) (κ · ej )

= ei · ej − (ei · κ) (κ · ej ) = δ ij − κi κj = δ ij −

ki kj

(A.3)

2

k

Clearly, we have κ×ε = ε κ × ε = −ε from which we also have

ε⊥k

(κ × ε)i (κ × ε)j = δ ij − κi κj = δ ij −

ki kj 2

k

and

ε⊥k

εi (κ × ε)j = εi εj + εi (−εj ) = (ei · ε) ε · ej − ε · ei (ej · ε) = ε · ei ε · ej − ej ε · ei = ε · ε × (ei × ej ) = ε × ε · (ei × ej ) =

εijl κl l

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where εijl is the anti-symmetric tensor and κl is the component of the longitudinal unit vector in the Cartesian basis. The second-order tensor (ε ) (ε) = ε ε + ε ε (ε)⊥k

is clearly also a projection operator, ∆⊥ , on the subspace of transverse fields, with the (i, j) components given by ∆⊥ ij = (ei · ε) (ε · ej ) + (ei · ε ) ε · ej + (ei · κ) (κ · ej ) − (ei · κ) (κ · ej ) = δ ij − κi κj ki kj = δ ij − (A.4) 2 k Thus, given a vector field Λ (r) and its Fourier transform Λ (k), Λ⊥ (k) is obtained from Λ (k) by projecting Λ (k) onto the plane normal to k , namely Λ⊥ (k) =

Λ⊥ (k)

i

(ε)⊥k

= j

(ε ) (ε) · Λ (k)





ki kj   δ ij − 2  Λ (k) k ∆⊥ ij Λ (k)

= j

j

j

A general projector in the subspace of transverse fields in reciprocal space may thus be written as Λ⊥ (k)

i

=

∆⊥ ij (k, k ) Λ (k )

dk j

j

⊥ where ∆⊥ ij (k, k ) is viewed as a matrix element of the operator ∆ in the Cartesian basis system,  

ki kj   ∆⊥ ij (k, k ) = δ ij − 2  δ (k − k ) k

(A.5)

Now we take the Fourier transform on both sides of Eq. (A.2) to yield 1 (2π)3 =

eik·x e−ik ·x dx dx [π i (x, t) , Aj (x , t)] | 1 i (2π)3

e−ik·x eik ·x dx dx δ ⊥ ij (x − x )

(A.6)

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The left side will give us 1 (2π)3

eik·x e−ik ·x dx dx [πi (x, t) , Aj (x , t)] = π†i (k, t) , Aj (k , t)

For transverse fields, we can express π (k, t) and A (k , t) in terms of the two orthogonal unit transverse vectors ε and ε , namely, π † (k, t) = ε ε · π† (k, t) + ε ε · π† (k, t) = ε ε + ε ε · π † (k, t)

= ε π †ε (k, t) + ε π†ε (k, t) π†i (k, t) = (ei · ε) π†ε (k, t) + (ei · ε ) π†ε (k, t) A (k , t) = ε (ε · A (k , t)) + ε (ε · A (k , t)) = ε ε + ε ε · A (k , t) = ε Aε (k , t) + ε Aε (k , t) Aj (k , t) = (ej · ε) Aε (k , t) + (ej · ε ) Aε (k , t) Therefore the commutator of the transverse conjugate fields is given by π†i (k, t) , Aj (k , t) =

(ei · ε) π†ε (k, t) + (ei · ε ) π†ε (k, t) , {(ej · ε) Aε (k , t) + (ej · ε ) Aε (k , t)}

= (ei · ε ε · ej ) π†ε (k, t) , Aε (k , t)

+ (ei · ε ε · ej ) π†ε (k, t) , {Aε (k , t)}

= (ei · ε ε · ej ) δ (k − k ) + (ei · ε ε · ej ) δ (k − k ) = [(ei · ε ε · ej ) + (ei · ε ε · ej )] δ (k − k ) From Eq. (A.4)for the expression of the projection operator in the subspace of transverse fields, we thus have   ki kj   π†i (k, t) , Aj (k , t) = δ ij − 2  δ (k − k ) k = ∆⊥ ij (k, k )

where the last line makes use of Eq. (A.5). Thus, the right-hand side of Eq. (A.6) is given by | 1 i (2π)3

⊥ eik·x e−ik ·x dx dx δ ⊥ ij (x − x ) = ∆ij (k, k )

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Appendix B

Lattice Weyl Transform of One-Particle Eﬀective Hamiltonian in Magnetic Field

The lattice Weyl transform of the one-particle eﬀective-Hamiltonian operator is defined by (10.14) or (10.16) with the arbitrary operator A replaced by Ho + Σ(z), Eqs. (11.30)-(11.31). It is convenient to introduce the magnetic translation operator defined by T (q) = exp

i |

| ∇r + (e/c)A(r) · q i

(B.1)

where A(r) is the vector potential, A(r) = 12 B × r, and q is the crystal lattice vector. In terms of this operator the following relation holds for the magnetic Wannier functions: |q, λ = T (q) |0, λ

(B.2)

In other words, T (q) generates all the magnetic Wannier functions belonging to a band index λ from a given magnetic Wannier function centered at the origin. Using notation in Appendix A we have wλ (r, −q) = T (−q) wλ (r, 0) = exp −

ie A (r) · q wλ (r − q) |c

(B.3)

T (q) commutes with Ho and moreover T (q) T (ρ) = exp

ie A (q) · ρ T (q + ρ) |c

(B.4)

By virtue of the last equation we can write the matrix element of Ho + Σ(z) between two magnetic Wannier functions as q + ρ, λ| Ho + Σ(z) q + ρ , λ = exp

ie A (q) · ρ − ρ |c

ρ, λ| Ho ρ , λ

+ ρ, λ| T (q)† Σ(z) T (q) ρ , λ 715

(B.5)

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The second term within the curly brackets can be written ρ, λ| T (q)† Σ(z) T (q) ρ , λ =

ie ie A (r) · ρ wλ∗ (r + ρ) exp − A (r) · q Σ(r − q, r , z) |c |c ie ie A (r ) · q exp − A (r + q ) · ρ wλ r + q + ρ (B.6) |c |c

dr dr exp − × exp

Changing the variable of integration r ⇒ r − q, with Jacobian unity, we obtain ρ, λ| T (q)† Σ(z) T (q) ρ , λ ie ie = dr dr exp − A (r) · ρ wλ∗ (r + ρ) exp − A (r) · q Σ(r − q, r − q, z) |c |c ie ie (B.7) A (r ) · q exp − A (r ) · ρ wλ r + ρ × exp |c |c We now make use of the forma of Σ(r, r , z) given by Eq. (11.31). Substituting in (B.7), we obtain the result ρ, λ| T (q)† Σ(z) T (q) ρ , λ

= ρ, λ| Σ(z) ρ , λ

(B.8)

and thus we can write (B5) as = exp

q + ρ, λ| Ho + Σ(z) q + ρ , λ

ie A (q) · ρ − ρ |c

ρ, λ| Ho + Σ(z) ρ , λ (B.9)

The last equation implies ρ, λ| Ho + Σ(z) ρ , λ

= exp

ie A ρ · ρ Fλλ ρ − ρ |c

(B.10)

where Fλλ ρ − ρ is a function which depends on ρ − ρ, Fλλ ρ − ρ = ρ − ρ , λ Ho + Σ(z) 0, λ

(B.11)

Fλλ ρ − ρ = 0, λ| Ho + Σ(z) ρ − ρ, λ

(B.12)

The Hamiltonian Ho + Σ(z) operating on magnetic Wannier function is therefore given by H |q, λ =

exp q ,λ

ie A (q) · q |c

Fλ λ (q − q ) q , λ

(B.13)

Taking the lattice Fourier transform of (B.13), we obtain the eﬀect of operating Ho + Σ(z) on the magnetic Bloch function H |ρ, λ =

exp q,λ

e ie ρ · q Fλ λ (q) ρ + A (q) , λ |c c

(B.14)

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It is now a trivial task to take the lattice Weyl transform of Ho + Σ(z), defined by (10.14) and (10.16). Using (B.13) and the identity (q + v) × (q − v) = −q × v, we obtain Hλλ (ρ, q) =

exp v

e ie ρ − A (q) · 2v Fλλ (2v) |c c

(B.15)

Writing Fλλ (2v) =

1 (N |3 )

ρ

i Hλλ ρ exp − ρ · 2v |

(B.16)

we end up with Hλλ (ρ, q) = Hλλ

e ρ − A (q) ; B, z c

(B.17)

Equation (B.17) will of course be diagonal in band index if biorthogonal magnetic Wannier functions or biorthogonal magnetic Bloch functions are used. This will result in an eﬀective Schrödinger equation in p-q space for each band index λ, following Eq. (11.49). A WKB solution of this equation has indeed been employed in Ref. [224] in discussing the influence of scattering or magnetic breakdown in zinc alloys, assuming for simplicity that the imaginary part of Wλ (π) δ λλ is a constant.

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Appendix C

Second Quantization Operators in Solid-State Band Theory (1)

(2)

Let Aop and Aop denote one-body and two-body quantum-mechanical operators, respectively. Following the usual prescription of second quantization procedure we (1) (2) write Aop and Aop in terms of field operator ψ and its adjoint ψ† as ψ† (x) A(1) op ψ (x) dx

A(1) op =

A(2) op =

(C.1)

ψ† (x) ψ† (x ) A(2) op ψ (x ) ψ (x) dx dx

(C.2)

We decompose the field operators in terms of the Wannier functions: ψ (x) = q,λ,σ

ψλσ |q, λ, σ

(C.3)

ψ†λσ q, λ, σ|

(C.4)

ψ† (x) = q,λ,σ

ψ and ψ† satisfy the following anticommutation relations: {ψλσ (q) , ψλ σ (q )}+ = ψ†λσ (q) , ψ†λ σ (q ) ψλσ (q) , ψ†λ σ (q ) (1)

+

= δ q,q δ λλ δ σσ

+

=0

(C.5)

(C.6)

(2)

Substituting the expression for Aop and Aop given by Eqs. (11.1) and (11.9), respectively, we obtain the following: 3 A(1) op = N |

−1

(1)

p,q,λ,λ ,σ,σ ,v

Aλλ σσ (p, q) e(2i/|)p·v ψ†λσ (q + v) ψλ σ (q − v) (C.7) 718

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Second Quantization Operators in Solid-State Band Theory

3 A(2) op = N|

−2

(2)

A[λσ] λ˜ (p, p˜, q, q˜) exp [ ˜ σ] p,q,λ,λ ,σ,σ ,v ˜λ ˜ ,˜ p,˜ ˜ q,λ, σ ,˜ σ ,˜ v

2i |

719

(p · v + p˜ · v˜)

q + v˜) ψλ˜ σ˜ (˜ q − v˜) ψλ σ (q − v) × ψ†λσ (q + v) ψ†λ˜ ˜ σ (˜

(C.8)

We can cast these equations in terms of the distribution-function operator by rewrit(1) (2) ing Aop and Aop as 3 A(1) op = N |

−1

(1) (1) Aλλ σσ (p, q) fˆλλ σσ (p, q)

(C.9)

p,q,λ,λ ,σ,σ

3 A(2) op = N|

−2

(2) (2) A[λσ] λ˜ (p, p˜, q, q˜) fˆ[λσ] λ˜ (p, p˜, q, q˜) [ ˜σ] [ ˜σ]

(C.10)

p,q,λ,λ ,σ,σ , ˜λ ˜ ,˜ p,˜ ˜ q ,λ, σ ,˜ σ

where the phase-space distribution function operators are (1) fˆλλ σσ (p, q) = v

(2) fˆ[λσ] λ˜ (p, p˜, q, q˜) = [ ˜σ]

exp v,˜ v

e(2i/|)p·v ψ†λσ (q + v) ψλ σ (q − v) 2i |

(C.11)

(p · v + p˜ · v˜)

q + v˜) ψλ˜ σ˜ (˜ q − v˜) ψλ σ (q − v) × ψ †λσ (q + v) ψ†λ˜ ˜ σ (˜

(C.12)

(1) (2) We see that the quantities fˆλλ σσ (p, q) and fˆ[λσ] λ˜ (p, p˜, q, q˜) given above for [ ˜ σ] solid-state band theory correspond to the second quantization operator for one- and two-particle [interacting Bloch particles] phase-space distribution functions, respectively. Our band-theory result essentially diﬀers from that of previous continuum formulation by replacement of integration by summation and the absence of “half(1) displacements" of the field operators. The expectation value of fˆλλ σσ (p, q) which we will denote by fˆ(1) (p, q) is equal to the Wigner distribution function, which

is shown here to be the lattice Weyl transform, Eq. (11.2), of the density-matrix operator. In the solid-state band-theory case, one can easily verify that indeed, (1) (1) fˆλλ σσ (p, q) = ρλλ σσ (p, q)

(C.13)

where the right-hand side of Eq. (C.13) is the lattice Weyl transform [Eq. (11.2)] of the density-matrix operator ˆ ρ(1) , (1)

e(2i/|)p·v

ρλλ σσ (p, q) = v

ˆ(1) |q + v, λ, σ q − v, λ , σ ρ

(C.14)

and ρ ˆ(1) is the one-particle density-matrix operator. Similarly, we have (2) fˆ[λσ] λ˜ (p, p˜, q, q˜) [ ˜σ]

(2)

=ρ

˜ σ ]T [λσ][λ˜

(p, p˜, q, q˜)

(C.15)

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˜σ where [λσ] λ˜

T

˜σ ˜σ stands for λ σ λσ, λ ˜λ ˜ and the right-hand side of Eq. (C.15)

is the lattice Weyl transform of the two-particle density-matrix operator ρ ˆ(2) , (2)

ρ

T

˜σ] [λσ][λ˜

=

(p, p˜, q, q˜) exp

v,˜ v

2i |

(p · v + p˜ · v˜) ˜ σ |q + v, λσ ˜σ ˆ(2) q˜ + v˜, λ˜ q˜ − v˜, λ ˜ ρ

× q − v, λ σ (1)

(C.16)

(1)

The expectation value of Aop and Aop , Eqs. (C.9) and (C.10) is thus given by A(1) = N |3 op

−1

(1)

(1)

Aλλ σσ (p, q) ρλλ σσ (p, q)

(C.17)

p,q,λ,λ ,σ,σ

A(2) = N |3 op

−2

(2)

p,q,λ,λ ,σ,σ , ˜λ ˜ ,˜ p,˜ ˜ q ,λ, σ,˜ σ

(2)

(p, p˜, q, q˜) ρ A[λσ] λ˜ (p, p˜, q, q˜) ˜ σ ]T [ ˜σ] [λσ][λ˜

(C.18)

Equations (C.17) and (C.18), calculate quantum-mechanical averages using the phase-space distribution functions in exactly the same manner as in calculating classical-mechanical averages. However, some diﬀerences in the quantities involved (1) (2) (p, p˜, q, q˜) which must be noted. For multiband particles, Aλλ σσ (p, q) and A[λσ] λ˜ [ ˜σ] (1) (2) are the lattice Weyl transforms of Aop and Aop respectively, do not in general resemble any of the dynamical expressions in classical mechanics since they are in general functions of |, Planck’s constant divided by 2π. Solutions to problems in solid-state physics often start by writing down the effective Hamiltonian. However, the writing down of a correct (good approximation to the problem at hand) eﬀective Hamiltonian usually hinges on one’s intuition and experience. We will give a rigorous derivation of a universal eﬀective Hamiltonian for Bloch particles. We will cast the eﬀective Hamiltonian of a many-body problem which we assumed to be a sum of one- and two-body operators, Eqs. (C.9) and (C.10), respectively, in a form that is more familiar and most amenable to fieldtheoretical perturbation techniques. First we derive the eﬀective second quantization one- and two-body operators in lattice coordinate space. This is immediately done by carrying out the summation over the (momentum) variables in Eqs. (C.7) and (C.8). The result is 3 A(1) op = N |

−1

(1)

q,λ,λ ,σ,σ ,v

Wλλ σσ (2v, 2q) ψ†λσ (q + v) ψλ σ (q − v)

(C.19)

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3 A(2) op = N |

−2

721

(2)

W[λσ] λ˜ (2v, 2 v˜, 2q, 2 q˜) [ ˜σ] q,λ,λ ,σ,σ ,v ˜λ ˜ ,˜ q˜,λ, σ,˜ σ ,˜ v

q + v˜) ψλ˜ σ˜ (˜ q − v˜) ψλ σ (q − v) × ψ †λσ (q + v) ψ†λ˜ ˜ σ (˜

(C.20)

where we have defined (1)

Wλλ σσ (2v, 2q) = N |3

−1

(1)

Aλλ σσ (p, q) e(2i/|)p·v

(C.21)

p

(2)

W[λσ] λ˜ (2v, 2 v˜, 2q, 2 q˜) = N |3 [ ˜σ] × exp

−2

(2)

p,p˜

A[λσ] λ˜ (p, p˜, q, q˜) [ ˜σ]

2i (p · v + p˜ · v˜) |

(C.22)

A more transparent form is obtained by transforming the lattice coordinate variables involved in the summation and writing 3 A(1) op = N |

−1

(1)

q,λ,λ ,σ,σ ,v

Wλλ σσ (R1 − R2 , R1 + R2 ) ψ†λσ (R1 ) ψλ σ (R2 ) (C.23)

3 A(2) op = N |

−2

(2)

q,λ,λ ,σ,σ ,v ˜λ ˜ ,˜ q˜,λ, σ,˜ σ ,˜ v

× ψ †λσ (R1 ) ψ†λ˜ ˜σ

˜1 − R ˜ 2 , R1 + R2 , R ˜1 + R ˜2 R1 − R2 , R W[λσ] λ˜ [ ˜σ]

˜1 R

ψλ˜ σ˜

˜2 ψ R λ σ (R2 )

(C.24)

It is easy to show by using Eqs. (11.2) and (C.21), Eqs. (11.10) and (C.22), that indeed (1)

Wλλ σσ (R1 − R2 , R1 + R2 ) = R1 , λ, σ| A(1) op R2 , λ , σ

(C.25)

(2)

˜1 − R ˜ 2 , R1 + R2 , R ˜1 + R ˜2 R1 − R2 , R W[λσ] λ˜ [ ˜ σ] ˜ σ ˜ ˜ ˜ ˜ 1 , λ, ˜ A(1) = R1 , λ, σ| R op R2 , λ , σ

R2 , λ , σ

(C.26)

as one would expect from the form of Eqs. (C.23) and (C.24). Our result also shows that W (1) and W (2) are functions of the sum and diﬀerence of lattice-point coordinates only.4 To derive the eﬀective one- and two-body operators in p (momentum) space we lattice-Fourier-transform the field operator in q (lattice coordinate) space to the p 4 This reflects the fact that the corresponding lattice Weyl transform is a function of the canonical position and momentum variables.

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(momentum) space: − 12

ψ†α (q + v) = N|3

p − 12

ψβ (q − v) = N |3

p

p exp −i · (q + v) a†α (p) |

(C.27)

p exp i · (q − v) aβ (p) |

(C.28)

The equivalent expression for the one- and two-particle distribution-function operators fˆ(1) and fˆ(2) can thus be written as (1) fˆλλ σσ (p, q) =

2i u · q a†λσ (p − u) aλ σ (p + u) |

exp u

(2) fˆ[λσ] λ˜ (p, p˜, q, q˜) = [ ˜σ]

2i |

exp u,˜ u

(C.29)

(u · q + u ˜ · q˜)

p−u ˜) aλ˜ σ˜ (˜ p+u ˜) aλ σ (p + u) (C.30) × a†λσ (p − u) a†λ˜ ˜ σ (˜ Substituting the last two equations in Eqs. (C.9) and (C.10), we obtain 3 A(1) op = N |

−1

a†λσ (p−u) aλ σ (p+u)

A[λσ] λ˜ (p, p˜, q, q˜) exp [ ˜ σ]

(C.31)   (u · q + u ˜ · q˜) 

q

p,u,λ,λ ,σ,σ

3 A(2) op = N|

2i u·q |

(1)

Aλλ σσ (p, q) exp

−2 p,u,λ,λ ,σ,σ , ˜λ ˜ ,˜ p,˜ ˜ u,λ, σ,˜ σ

  

2i |

(2)

q,˜ q

p−u ˜) aλ˜ σ˜ (˜ p+u ˜) aλ σ (p + u) × a†λσ (p − u) a†λ˜ ˜ σ (˜

(C.32)

where (1)

Wλλ σσ (2p, 2u) = N |3

−1

(1)

Aλλ σσ (p, q) exp q

2i u·q |

(C.33)

(u · q + u ˜ · q˜)

(C.34)

(2)

W[λσ] λ˜ (2p, 2 p˜, 2u, 2 u ˜) [ ˜ σ] = N |3

−2

(2)

q,˜ q

A[λσ] λ˜ (p, p˜, q, q˜) exp [ ˜ σ]

2i |

A familiar second quantization form in p (momentum or wave vector) space is again obtained by transforming the momentum variables involved in the summation to yield (1)

A(1) op = k1 ,k2 ,λ,λ ,σ,σ

q

Wλλ σσ (k1 + k2 , k1 − k2 ) a†λσ (k1 ) aλ σ (k2 ) (C.35)

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Second Quantization Operators in Solid-State Band Theory

A(2) op = k1 ,k2 ,λ,λ ,σ,σ , ˜1 ,k ˜2 ,λ, ˜λ ˜ ,˜ k σ ,˜ σ

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723

(2) k + k2 , k˜1 + k˜2 , k1 − k2 , k˜1 − k˜2 W[λσ] λ˜ [ ˜ σ] 1

˜ × a†λσ (k1 ) a†λ˜ ˜ σ k1

aλ˜ σ˜

k˜2 aλ σ (k2 )

(C.36)

where again one can easily show (1)

Wλλ σσ (k1 + k2 , k1 − k2 ) = k1 , λ, σ| A(1) op k2 , λ , σ

(C.37)

(2) k + k2 , k˜1 + k˜2 , k1 − k2 , k˜1 − k˜2 W[λσ] λ˜ [ ˜σ] 1

˜ ˜ ˜ ˜ σ ˜ A(2) = k1 , λ, σ| k˜1 , λ, op k2 , λ , σ

k2 , λ , σ

(C.38)

as expected. The virtue of the calculation presented here is that it explicitly shows the functional dependence of the matrix elements, entirely quantum-mechanical quantities, and their exact relation to the lattice Weyl transform of the corresponding operators through the defining relations, Eqs. (11.2), (11.4), (11.10), and its equivalent expression (C.21), (C.22), (C.33), and (C.34).

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Appendix D

Direct Construction of Fermionic Path Integral

We will generalize the method and unify the results of Soper [21] and Berezin [22] to construct the path-integral forms given in the text. We make use of the eigenvectors construction given by Eqs. (12.22)-(12.25). The reader is referred to Ref. [20] for the notations used here. The basic idea is to formulate the Weyl correspondence in the external representation of fermions. We will need the following theorems. Theorem D.1

Resolution of identity in the external representation is given by 1 = exp ψ† · = eψ

†

·q

∂ ∂p

|q = 0 p = 0| ep ·ψ

|q = 0 p = 0|

← − ∂T ·ψ ∂q

† ∂ · ψ |p = 0 q = 0| e−ψ ·q ∂q ← − ∂T = e−p ·ψ |p = 0 q = 0| exp −ψ† · ∂p

= exp −

(D.1)

The proof can be shown by comparison with [225] 1 = |p = 0 q = 0| + ψ r |p = 0 q = 0| ψ†r + ........... = |q = 0 p = 0| + ψ †r |q = 0 p = 0| ψr + ...........

(D.2) (D.3)

Corollary D.1 In terms of the q eigenvectors only or in terms of p eigenvectors only, we have the following expression for completeness: 1 = |q = |p

← − ∂T ∂ ∂q ∂q ← − ∂T ∂ ∂p ∂p

724

q |

(D.4)

p |

(D.5)

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Corollary D.2

725

The scalar product in the q representation is given by ← − ∂T ∂ ψ (q ) φ| ψ = φ (q ) ∂q ∂q ← − T ∂T ∂ ψ (q ) − = φ (q ) ∂q ∂q ← − T ∂ ∂T φ (q ) ψ (q ) − = − ∂q ∂q T

= =

−

∂ φ (q) ψ (q) = ∂q

T

∂ φ (q) ψ (q) ∂q

d [q] φ (q) ψ (q)

(D.6)

and similarly in the p representation. For any operator A, we have − ← − ← ∂T ∂T = d [q ] d [p ] ep ·q p | Aop |q ∂q ∂p

Theorem D.2

p | Aop |q

(D.7)

Proof:

T r Aop = p | Aop =

∂ ∂q

← − ∂T ∂p

∂ ∂p

= p | Aop |q

− ← − ← ∂T ∂T ∂q ∂p

p | Aop |q

(D.8)

But T r Aop is also given by [20] T r Aop = =

d [q ] d [p ] ep ·q

p | Aop |q

d [q ] d [p ] ep ·q p = 0| [1+(p · ψ)+....] Aop 1+ ψ† · q +... |q = 0

= p = 0| Aop |q = 0 + p = 0| [(p · ψ)] Aop

ψ† · q

|q = 0 + .....

= p = 0| Aop |q = 0 + p = 0| ψr Aop ψ†r |q = 0 + .....

(D.9)

and therefore the theorem is proved. Corollary D.3 The expression for completeness in the p−q mixed representation is given symbolically by 1=

d [q ] d [p ] e−p ·q |q

p|

(D.10)

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Nonequilibrium Quantum Transport Physics in Nanosystems

Proof: ← − ∂T ∂ ∂q ∂p

1 = |q =

p|=

−

d [q ] d [p ] e−p ·q |q

∂ ∂p

∂ ∂q

|q

p|

p|

(D.11)

The last result is the completeness relation used by Halpern, Jevicki, and Senjanovic [225] for constructing fermionic path integrals, which was later generalized by Soper [21]. We will now establish the Weyl correspondence. We make use of the following identity for an arbitrary operator A:

A = |q

← − ∂T ∂q

=

d [q ] d [p ]

∂ ∂p

p | A |p

← − ∂T ∂p

d [q ] d [p ] e−p

∂ ∂q

·q +p ·q

q| |q

(D.12) p | A |p

q|

(D.13)

where the last line is obtained using D.3. Introducing the change of variables p = p − u, p = p + u,

q =q−v q =q+v

(D.14)

we obtain A=

d [q] d [p] A (p, q) ∆ (p, q)

(D.15)

where A (p, q) =

∆ (p, q) =

d [u] e−2u·q p + u| A |p − u

(D.16)

d [v] e−2p·v |q + v q − v|

(D.17)

A(p, q), which is a c-number function, represents the Weyl transform of an arbitrary operator A and Eq. (D.16) is the realization of Weyl correspondence in the external algebra representation. One can show using the diﬀerential operator realization of Eqs. (12.30)-(12.33) and the properties of the delta function in the external representation that indeed A(p, q) can be obtained from the operator A(ψ† , ψ) by replacement as ψ† ⇒ p and ψ ⇒ q. One can easily check the consistency of the

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Direct Construction of Fermionic Path Integral

727

expression (D.15) by calculating the trace of A. From (D.9) we have T rA =

d [q ]

d [p ] ep ·q

=

d [q]

d [p] A (p, q) δ (p) δ (q) e−2p·q

=

d [u] u| Aop |−u

p | Aop |q

(D.18)

The last line of Eq. (D.18) can indeed be shown to be the variant expression for the trace of an arbitrary operator specific to the external algebra representation [20]. To show this one makes use of the form of Eq. (D.7) using only q variables. We note that an arbitrary operator A can also be expressed in the form which is a variant of Eq. (D.12) ← − ← − ∂T ∂ ∂T ∂ A = |p q | A |q p| ∂p ∂q ∂q ∂p =

˜ (p, q) d [q] d [p] A (p, q) ∆

(D.19)

where A (p, q) =

˜ (p, q) = ∆

d [v] e2p·v q + v| A |q − v

(D.20)

d [u] e2u·q |p + u p − u|

(D.21)

We are now in the position to calculate the path-integral expression of the evolution operator n+1 =t U (tj , tj−1 )|ttn+1 o =to

U (t, to ) = e−(i/|)(t−to )H =

(D.22)

j=1

Using the resolution of identity (D.1) we have  ← − n+1 ∂ ∂T  qj | U (tj , tj−1 ) |qj−1 U (t, to ) = |qn+1 ∂qn+1 j=1 ∂qj

 ← − ∂T  ∂ ∂qj−1 ∂qo

qo | (D.23)

or the equivalent expression U (t, to ) = |pn+1

 ← − n+1 ∂ ∂T  ∂pn+1 j=1 ∂pj

pj | U (tj , tj−1 ) |pj−1

 ← − ∂T  ∂ ∂pj−1 ∂po

po | (D.24)

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Nonequilibrium Quantum Transport Physics in Nanosystems

It will be convenient to use the representation of an arbitrary operator given by Eqs. (D.15)-(D.17) in evaluating the products in Eq. (D.23) and similarly the representation of Eqs. (D.19)-(D.21) in evaluating the products of Eq. (D.24). We have from (D.17) and (D.23), making use of Eq. (12.36), the following: d [v] e−2p·v qj | |q + v

qj | ∆ (p, q) |qj−1 =

d [v] e−2p·v

=

α

q α + vα − qjα

α

α 2vα − qjα − qj−1

d [v] e−2p·v

= ×

γ

γ

q −

−p·(qj −qj−1 )

=e

q − v| |qj−1

γ qj−1

γ

γ − qγ + vγ qj−1

+ qjγ /2

δ [(qj−1 + qj ) /2 − q]

(D.25)

By virtue of Eqs. (D.15) and (D.22) we may then write, correct to order (tj −tj−1 )2 , qj | U (tj , tj−1 ) |qj−1 i (qj + qj−1 ) d [p] e−p·(qj −qj−1 ) exp − (tj − tj−1 )H p, | 2

=

(D.26)

With the aid of corollaries D.1 and D.2, we can symbolically write n+1

U (t, to ) =

n+1

d [pj ] j=1

  × exp − 

j=0

d [qj ] |qn+1

n+1 j=1

pj ·

(qj − qj−1 )

1 (qj + qj−1 ) + H pj , | 2

  

qo | (D.27)

Similarly the use of Eq. (D.24) together with Eqs. (D.19)-(D.21) yield the equivalent expression U (t, to ) n+1

n+1

d [pj ]

= j=0

× exp

  

j=0 n+1 j=1

d [qj ] |pn+1

(pj − pj−1 )

1 · qj−1 − H |

(pj + pj−1 ) , qj−1 2

  

po | (D.28)

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Direct Construction of Fermionic Path Integral

729

Using the expression for the trace given by Eq. (D.18), we have from Eq. (D.27) T rU (t, to ) n+1

n+1

d [pj ]

=

d [qj ] δ (qo + qn+1 )

j=1

  × exp − 

(D.29)

j=0 n+1

j=1

pj ·

(qj − qj−1 )

1 (qj + qj−1 ) + H pj , | 2

  

(D.30)

which upon integrating over d[qo ] leads to Eq. (12.44) for the grand partition function, with antiperiodic boundary conditions on q resulting from the Kronecker delta function in Eq. (D.30). Equation (12.44) corresponds to the result given by Soper [21]. Similarly taking the trace of the expression in Eq. (D.28) yields the equivalent expression for the trace of U (t, to ) , n+1

T rU (t, to ) =

n+1

d [pj ] j=0

× exp

  

d [qj ] δ (po + pn+1 ) j=0

n+1

j=1

(pj − pj−1 )

1 · qj−1 − H |

(pj + pj−1 ) , qj−1 2

  

(D.31)

which upon integrating over d [pn+1 ] leads to Eq. (12.45) for the grand partition function, with antiperiodic boundary conditions on p resulting from the Kronecker delta function in Eq. (D.31). Equation (12.45) corresponds to the result given by Berezin [22]. In closing, we note that one can combine Eqs. (12.44) and (12.45) to obtain a properly symmetrized form of the “Lagrangian.”

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Appendix E

Hot-Electron Green’s Function

We take a simple Hamiltonian of free electrons subjected to a very intense electric field. This model is of some relevance to the formulation of high-field carrier transport in submicron devices; H=

2 ˆ † (x) − | ∇2 − x · F ψ ˆ (x) dx ψ 2m

(E.1)

where F is the electric field. The one-particle Green’s function is defined by G (x, t, x , t ) =

1 ˆ (x) ψ ˆ † (x) . T ψ i|

(E.2)

For hot-electron physics, we may take our ground state to be the vacuum or empty band. The Heisenberg equation of motion is i|

∂ ˆ ˆ (x) , H e(−i/|)H ψ (x, t) = e(i/|)H t ψ ∂t ˆ (x, t) , H = ψ

t

(E.3)

so that the time dependence of the field operation is in general given by ˆ (x, 0) e(−i/|)H ˆ (x, t) = e(i/|)H t ψ ψ

t

(E.4)

However, for the Hamiltonian of Eq. (E.1), the right-hand side of Eq. (E.3) is linear in the field operator and becomes i|

|2 2 ∂ ˆ ˆ (x, t) = H ˆ (x, t) ˜ψ ψ (x, t) = − ∇ −x·F ψ ∂t 2m

(E.5)

˜ is independent of the field operators. Therefore, we can also write the field where H ˆ (x, t) as dependence of ψ ˆ (x, 0) ˆ (x, t) = e(−i/|)H˜ t ψ ψ 730

(E.6)

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Hot-Electron Green’s Function

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731

Equation (E.2) for the Green’s function can therefore be written as 1 ˆ † (x , 0) |0 ˆ (x, 0) e(−i/|)H˜ (t−t ) ψ Θ (t − t ) 0| ψ i| 1 ˜ = Θ (t − t ) x| e(−i/|)H (t−t ) |x i|

G (x, t, x , t ) =

(E.7)

where ket and bra denote the eigenvectors of position operator [226]. Therefore, we have reduced the problem to finding the kernel or propagator of a Schrödinger ˜ We can immediately equation to a Feynman path integral with Hamiltonian H. [ ] write 17 ˜ x| e(−i/|)H (t−t ) |x = x| U (t, t ) |x n

d3 qi

= lim

n→∞

i=1 n+1

im 2|

× exp

i=1

m i2π|

3(n+1) 2

(qi − qi−1 )2 +

2

m

F · (qi − qi−1 )

(E.8)

where qn+1 = x and qo = x . Equation (E.8) represents a set of Gaussian integrals. The integration may be carried out on one variable after the other. In this manner, a recursion process [17] can be established. We have after 3n integration in the limit n⇒∞ x| U (t, t ) |x =

2πi| (t − t ) m

− 32

3

× exp

im 2 2 F · (x + x ) |x − x | + (t − t ) 2| (t − t ) m

exp

−im (t − t ) 24|

F m

where t − t = (n + 1) , n ⇒ ∞, Eq. (E.7), using Eq. (E.9), G (k, k , t, t ) =

1 (2π)3

2

(E.9) ⇒ 0. We now take the Fourier transform of G (x, t; x , t ) e−ik·x eik ·x dxdx

By making a change of variables, (x − x ) = w, and integrating over w and x variables, where the integrals are all Gaussian, we obtain    t−t 2 1 F i ||k−τ F | dτ  G (k, k , t, t )= − Θ (t−t ) δ k− k + (t−t ) exp − | | | 2m 0

(E.10) Equation (E.10) was also obtained by Jauho and Wilkins [227], in their discussion of nonlinear transport properties, by solving the equation for the Green’s function. The virtue of the calculation presented here is that path-integral formulation allows for a generalization to a wider class of problems.

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Appendix F

Derivation of Generalized Semiconductor Bloch Equations

We make use of the following Tables which give the mapping between the electron picture and electron-hole picture. electron picture < −i{ghh < −i{ghh < −i{ghh < −i{ghh > i{ghh > i{ghh > i{ghh > i{ghh < −i{gee < −i{gee < −i{gee < −i{gee > i{gee > i{gee > i{gee > i{gee

12 vv 12 vc 12 cv 12 c c 12 vv 12 vc 12 cv 12 cc 12 vv 12 vc 12 cv 12 cc 12 vv 12 vc 12 cv 12 cc

e − f ield

e − h f ield

ψ v (2) ψv (1)

φ†v (2) φ†v (1)

ψc (2) ψ v (1)

ψ c (2) φ†v (1)

i{Ge−h,>T (12) vc

ψv (2) ψ c (1)

φ†v (2) ψ c (1)

−i{Ge−h,< (12) cv

ψ c (2) ψc (1)

ψ c (2) ψ c (1)

e−h,< −i{ghh

ψ v (1) ψv (2)

φ†v (1) φ†v (2)

e−h,> i{gee

ψv (1) ψ c (2)

φ†v (1) ψ c (2)

−i{Ge−h,
ψc (1) ψ v (2)

ψ c (1) φ†v (2)

i{Ge−h,> (12) cv

ψ c (1) ψc (2)

ψ c (1) ψ c (2)

e−h,> i{ghh

ψ †v (2) ψ†v (1)

φv (2) φv (1)

e−h,< −i{ghh

ψ †c (2) ψ †v (1)

ψ †c (2) φv (1)

−i{Ge−h,< (12) vc

ψ †v (2) ψ †c (1)

φv (2) ψ †c (1)

i{Ge−h,>T (12) cv

ψ †c (2) ψ†c (1)

ψ†c (2) ψ †c (1)

e−h,< −i{gee

ψ †v (1) ψ†v (2)

φv (1)φv (2)

e−h,> i{ghh

ψ †v (1) ψ †c (2)

φv (1) ψ †c (2)

i{Ge−h,> (12) vc

ψ †c (1) ψ †v (2)

ψ †c (1) φv (2)

−i{Ge−h,
ψ †c (1) ψ†c (2)

ψ†c (1) ψ †c (2)

732

e − h picture e−h,< −i{gee

e−h> i{gee

12 vv

12 cc 12 vv

12 cc 12 vv

12 cc 12 vv

12 cc

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Derivation of Generalized Semiconductor Bloch Equations

electron picture 12 −i{G< vv 12 −i{G< vc 12 −i{G< cv 12 −i{G< cc 12 −i{G i{G vv 12 > i{G vc 12 > i{G cv 12 > i{G cc 12 >T i{G vv 12 >T i{G vc 12 >T i{G cv 12 >T i{G cc electron picture 12 c i|ghh vv 12 c i|ghh vc 12 c i|ghh cv 12 c i|ghh cc

733

e − f ield

e − h f ield

e − h picture

ψ†v (2) ψ v (1)

φv (2) φ†v (1)

i{Gh,>T (12) vv

ψ†c (2) ψv (1)

ψ†c (2) φ†v (1)

e−h,< −i{gee

ψ†v (2) ψc (1)

φv (2) ψc (1)

e−h,< −i{ghh

ψ†c (2) ψc (1)

ψ†c (2) ψc (1)

−i{Ge−h,<

ψ†v (1) ψ v (2)

φv (1) φ†v (2)

ψ†v (1) ψc (2)

φv (1) ψc (2)

e−h,> i{ghh

ψ†c (1) ψv (2)

ψ†c (1) φ†v (2)

e−h,> i{gee

ψ†c (1) ψc (2)

ψ†c (1) ψc (2)

−i{G
ψv (1) ψ †v (2)

φ†v (1) φv (2)

−i{Gh,
ψv (1) ψ†c (2)

φ†v (1) ψ†c (2)

e−h,> i{gee

ψc (1) ψ†v (2)

ψc (1) φv (2)

e−h,> i{ghh

ψc (1) ψ†c (2)

ψc (1) ψ†c (2)

i{Ge−h,>

ψv (2) ψ †v (1)

φ†v (2) φv (1)

ψc (2) ψ†v (1)

ψc (2) φv (1)

ψv (2) ψ†c (1)

φ†v (2) ψ†c (1)

ψc (2) ψ†c (1)

ψc (2) ψ†c (1)

e − h picture 12 e−h,c i|gee vv e−h,cT (12) −i|Gvc e−h,c (12) i|Gcv e−h,c i|ghh

12 cc

electron picture 12 ac i|ghh vv 12 ac i|ghh vc 12 ac i|ghh cv 12 ac i|ghh cc

12 vc 12 cv 12 cc

i{Gh,> vv (12) 12 vc 12 cv 12 cc

12 vc 12 cv 12 cc

−i{Gh,< vv (12) 12 vc 12 e−h,< −i{gee cv 12 >T i{G cc

e−h,< −i{ghh

e − h picture 12 e−h,ac i|gee vv (12) −i|Ge−h,acT vc e−h,ac (12) i|Gcv e−h,ac i|ghh

12 cc

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Nonequilibrium Quantum Transport Physics in Nanosystems

electron picture 12 c i|gee vv 12 c i|gee vc 12 c i|gee cv 12 c i|gee cc electron picture i|Gcvv (12) i|Gcvc (12) i|Gccv (12) i|Gccc (12) electron picture Grvv (12) Grvc (12) Grcv (12) Grcc (12) electron picture 12 r ghh vv 12 r ghh vc 12 r ghh cv 12 r ghh cc electron picture 12 r gee vv 12 r gee vc 12 r gee cv 12 r gee cc

e − h picture 12 e−h,c i|ghh vv e−h,c (12) i|Gvc e−h,cT (12) −i|Gcv e−h,c i|gee

12 cc

electron picture 12 ac i|gee vv 1 2 ac i|gee vc 12 ac i|gee cv 12 ac i|gee cc

e − h picture e−h,cT −i|Gvv (12) 12 e−h,c i|gee vc 12 e−h,c i|ghh cv e−h,c i|Gcc (12)

electron picture i|Gac vv (12)

e − h picture −Ge−h,aT (12) vv 1 2 e−h,r gee vc 12 e−h,r ghh cv e−h,r Gcc (12)

electron picture Gavv (12)

e − h picture 12 e−h,r gee vv

electron picture 12 a ghh vv 12 a ghh vc 12 a ghh cv 12 a ghh cc

(12) −Ge−h,aT vc e−h,r (12) Gcv e−h,r ghh

12 cc

e − h picture 12 r ghh vv e−h,r (12) Gvc

(12) −Ge−h,aT cv e−h,r gee

12 cc

i|Gac vc (12) i|Gac cv (12) i|Gac cc (12)

Gavc (12) Gacv (12) Gacc (12)

electron picture 12 a gee vv 12 a gee vc 12 a gee cv 12 a gee cc

e − h picture 12 ac i|ghh vv e−h,ac (12) i|Gvc

(12) −i|Ge−h,acT cv e−h,ac i|gee

12 cc

e − h picture −i|Ge−h,acT (12) vv 1 2 e−h,ac i|gee vc 12 e−h,ac i|ghh cv e−h,ac i|Gcc (12) e − h picture

−Ge−h,rT (12) vv 1 2 e−h,a gee vc 12 e−h,a ghh cv e−h,a Gcc (12) e − h picture 12 e−h,a gee vv (12) −Ge−h,rT vc e−h,a (12) Gcv e−h,a ghh

12 cc

e − h picture 12 e−h,a ghh vv e−h,a (12) Gvc

(12) −Ge−h,rT cv e−h,a gee

12 cc

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735

e−h,< Thus, to obtain the equation for the hole density given by −i{Gvv (12) , 1 2 in the general nonequilibrium one may look for the equation for i{G>T vv 12 formulation of electrons. The equation for i{G>T can be obtained from the vv equation for G> vv ,by taking the complex conjugate of the above equation and making use of the relation

G>† (12) = −G>T (12) . We have ∂ ∂ + ∂t1 ∂t2

i|

G> vv (12)

T > r > > a (1ξ) G> = vvv vv (ξ2)−Gvv (1ξ) vvv (ξ2) + Σvv (1ξ) Gvv (ξ2)−Gvv (1ξ) Σvv (ξ2) > a > a r > + Σrvc (1ξ) G> cv (ξ2)−Gvc (1ξ) Σcv (ξ2) + Σvv (1ξ) Gvv (ξ2)−Gvv (1ξ) Σvv (ξ2) a r > + Σ> vc (1ξ) Gcv (ξ2)−Gvc (1ξ) Σcv (ξ2) +

> (vv; ξ2) ∆rhh (vv; 1ξ) gee > a −ghh (vv; 1ξ) ∆ee (vv; ξ2)

+

> a (cv; ξ2) ∆> ∆rhh (vc; 1ξ) gee hh (vv; 1ξ) gee (vv; ξ2) + > r −ghh −ghh (vc; 1ξ) ∆aee (cv; ξ2) (vv; 1ξ) ∆> ee (vv; ξ2)

+

a ∆> hh (vc; 1ξ) gee (cv; ξ2) . r −ghh (vc; 1ξ) ∆> ee (cv; ξ2)

Taking the complex conjugate, we obtain ∂ ∂ + ∂t1 ∂t2

i|

G>T vv (12)

T T >T > a r > = −G> vv (2ξ) vvv (ξ1)+vvv (ξ2) Gvv (1ξ) + −Gvv (2ξ) Σvv (ξ1)+Σvv (2ξ) Gvv (ξ1) a r > r > > a + −G> vc (2ξ) Σcv (ξ1)+Σvc (2ξ) Gcv (ξ1) + −Gvv (2ξ) Σvv (ξ1)+Σvv (2ξ) Gvv (ξ1) > a + −Grvc (2ξ) Σ> cv (ξ1)+Σvc (2ξ) Gcv (ξ1) +

> (vv; 2ξ) ∆aee (vv; ξ1) −ghh r > +∆hh (vv; 2ξ) gee (vv; ξ1)

+

> r (vc; 2ξ) ∆aee (cv; ξ1) (vv; 2ξ) ∆> −ghh −ghh ee (vv; ξ1) + > r > a +∆hh (vc; 2ξ) gee (cv; ξ1) +∆hh (vv; 2ξ) gee (vv; ξ1)

+

r (vc; 2ξ) ∆> −ghh ee (cv; ξ1) > a +∆hh (vc; 2ξ) gee (cv; ξ1)

=−i|

∂ ∂ + ∂t1 ∂t2

(12) Ge−h,< vv

(in e-h picture).

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Nonequilibrium Quantum Transport Physics in Nanosystems

> e−h,< Applying the relation −G>T (12) , and going entirely vv (12) = −Gvv (21) = Gvv to the defect representation, we obtain for the time evolution of the hole density the following relation

−i|

∂ ∂ + ∂t1 ∂t2

(12) Ge−h,< vv

e−h,< T (ξ2) vvv (1ξ) − vvv (ξ2) Ge−h,< (1ξ) + = Gvv vv



e−h,>  −gee

 + 

e−h,r +∆ee



e−h,r  −gee

 + 

+∆e−h,> ee

2ξ e−h,a ∆hh vc 2ξ e−h,> ghh vc

ξ c ξ c

1 v 1 v

2ξ ∆e−h,> hh vc 2ξ e−h,a ghh vc

ξ c ξ c

1 v 1 v

(1ξ) Ge−h,< (ξ2) −Σe−h,r vv vv

e−h,a (1ξ) Σvv (ξ2) +Ge−h,< vv



e−h,< e−h,a  (1ξ) Gvv (ξ2) −Σvv  + e−h,r e−h,<  +Gvv (1ξ) Σvv (ξ2)

 



2ξ e−h,a e−h,> ∆hh   −gee vv   + 2ξ  e−h,> e−h,r +∆ee ghh vv

2ξ e−h,r ∆e−h,>  −gee hh vv  + + e−h,< e−h,a 2ξ  +Gcv (1ξ) Σvc (ξ2) e−h,a +∆e−h,> ghh ee vv e−h,r −Σcv (1ξ) Ge−h,< (ξ2) vc

+

e−h,< −Σvc (1ξ) Ge−h,a (ξ2) cv e−h,r +Gvc (1ξ) Σe−h,< (ξ2) cv

ξ v ξ v

1 v 1 v

ξ v ξ v 

1 v 1 v

   

   

.

In the absence of real pairing between holes, the above equation reduces to

i|

∂ ∂ + ∂t1 ∂t2

e−h,< (12) Gvv

e−h,< T = − vvv (1ξ) Gvv (ξ2) − Ge−h,< (1ξ) vvv (ξ2) vv e−h,< e−h,< Σvv (1ξ) Ge−h,a (ξ2) (1ξ) Gvv (ξ2) Σe−h,r vv vv + e−h,< e−h,a e−h,r e−h,< −Gvv (1ξ) Σvv (ξ2) −Gvv (1ξ) Σvv (ξ2)    2ξ ξ1 2ξ ξ1 e−h,> e−h,r gee ∆e−h,a ∆e−h,>   gee  hh hh vc cv   vc cv  + + 2 ξ ξ 1 2 ξ ξ1    e−h,> e−h,a e−h,r −∆ee −∆e−h,> ghh ghh ee vc cv vc cv

+



  , 

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737

which may also be expressed as i|

∂ ∂ + ∂t1 ∂t2

=−

Ge−h,< (12) vv

vvv − Re Σe−h,r , Re Ge−h,r , Ge−h,< + Σe−h,< vv vv vv vv

+ i Im Σe−h,r − i Σe−h,< , Ge−h,< , Im Ge−h,r vv vv vv vv     2 ξ ξ 1 2ξ ξ1 e−h,a e−h,< e−h,< e−h,r Re ∆ ∆ g Re g ee ee hh hh   vc cv  vc cv  +  +  2ξ ξ1   2ξ ξ1  e−h,< e−h,a e−h,r e−h,< − Re ∆ee −∆ee ghh Re ghh vc cv vc cv     2 ξ ξ 1 2 ξ ξ1 e−h,a e−h,< e−h,< e−h,r Im ∆ ∆ g Im g ee ee hh hh   vc cv  vc cv   +i . + i  2ξ ξ1   2ξ ξ1  e−h,< e−h,a e−h,r e−h,< − Im ∆ee −∆ee ghh Im ghh vc cv vc cv

The equation for the conduction electron density is given by ∂ ∂ + ∂t1 ∂t2

i|

G< cc (12)

< T < a = v G< + Σrcv (1ξ) G< cc (12) − Gcc (12) v vc (ξ2) − Gcv (1ξ) Σvc (ξ2) < a < a r < + Σrcc (1ξ) G< cc (ξ2) − Gcc (1ξ) Σcc (ξ2) + Σcv (1ξ) Gvc (ξ2) − Gcv (1ξ) Σvc (ξ2) a r < + Σ< cc (1ξ) Gcc (ξ2) − Gcc (1ξ) Σcc (ξ2) +

+ +

< (vc; ξ2) ∆rhh (cv; 1ξ) gee < −ghh (cv; 1ξ) ∆aee (vc; ξ2)

< a (cc; ξ2) ∆< ∆rhh (cc; 1ξ) gee hh (cv; 1ξ) gee (vc; ξ2) + < r −ghh −ghh (cc; 1ξ) ∆aee (cc; ξ2) (cv; 1ξ) ∆< ee (vc; ξ2) a ∆< hh (cc; 1ξ) gee (cc; ξ2) . r −ghh (cc; 1ξ) ∆< ee (cc; ξ2)

Transforming to the defect representation using the mapping table, we have i|

∂ ∂ + ∂t1 ∂t2

G< cc (12)

< T = v G< cc (12) − Gcc (12) v

< a + Σrcc (1ξ) G< cc (ξ2) − Gcc (1ξ) Σcc (ξ2)

a r < + Σ< cc (1ξ) Gcc (ξ2) − Gcc (1ξ) Σcc (ξ2)    1ξ ξ2 1ξ e−h,r e−h,< e−h,< ∆ g ee hh    ∆hh c v v c cv + +  e−h,< 1 ξ ξ 2   e−h,r 1 ξ e−h,a −ghh −ghh ∆ee cv vc cv

+

+

e−h,a gee e−h,< ∆ee

e−h,r < (1ξ) Ge−h,< (ξ2) (cc; ξ2) ∆rhh (cc; 1ξ) gee Σcv vc + < e−h,< e−h,a −Gcv −ghh (cc; 1ξ) ∆aee (cc; ξ2) (1ξ) Σvc (ξ2)

a (1ξ) Ge−h,a (ξ2) ∆< Σe−h,< cv vc hh (cc; 1ξ) gee (cc; ξ2) + . r e−h,r e−h,< −Gcv −ghh (cc; 1ξ) ∆< (1ξ) Σvc (ξ2) ee (cc; ξ2)

ξ2 vc ξ2 vc

   

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Nonequilibrium Quantum Transport Physics in Nanosystems

In the absence of real pairing between conduction electrons, this reduces to i|

∂ ∂ + ∂t1 ∂t2

G< cc (12)

< T = vc G< cc (12) − Gcc (12) vc

< a + Σrcc (1ξ) G< cc (ξ2) − Gcc (1ξ) Σcc (ξ2)

a r < + Σ< cc (1ξ) Gcc (ξ2) − Gcc (1ξ) Σcc (ξ2)    1ξ ξ2 1ξ e−h,r e−h,< e−h,< gee  ∆hh   ∆hh c v v c cv    + + ξ 2   e−h,r 1 ξ  e−h,< 1 ξ −ghh −ghh ∆e−h,a ee cv vc cv

e−h,a gee

∆e−h,< ee

which may also be expressed as, i|

∂ ∂ + ∂t1 ∂t2

ξ2 vc ξ2 vc



  , 

G< cc (12)

< r = (vc + Re Σrcc ) , G< cc + Σcc , Re Gcc < r + i Im Σrcc , G< cc − i Σcc , Im Gcc  1ξ ξ2 e−h,r e−h,< gee  Re ∆hh cv vc  + 1 ξ ξ2  e−h,< e−h,a −ghh Re ∆ee cv vc  1ξ ξ2 e−h,< e−h,a Re gee  ∆hh cv vc  + 1 ξ ξ2  e−h,r e−h,< − Re ghh ∆ee cv vc  1ξ ξ2 e−h,r e−h,< gee  Im ∆hh cv vc  + i ξ2  e−h,< 1 ξ e−h,a −ghh Im ∆ee cv vc  1ξ ξ2 e−h,< e−h,a Im gee  ∆hh cv vc  + i 1 ξ ξ2  e−h,r − Im ghh ∆e−h,< ee cv vc

    

    

    



  . 

One observes that since the summation of bands is between valence and conduction bands only, the rate of change of conduction electrons and that of holes are equal in the absence of particle kinematics within the respective bands. The equations e−h,< e−h,< , gee , etc. can be obtained from for the interband or polarization terms, ghh < the equations for G (12) , where the interband indices are absorbed in (12) indices.

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Derivation of Generalized Semiconductor Bloch Equations

739

The equations for G< (12) is given by (writing band indices explicitly), i|

∂ ∂ + ∂t1 ∂t2

G< vc (12)

< T r < < a = vv G< vc (12) − Gvc vc + Σvv (1ξ) Gvc (ξ2) − Gvv (1ξ) Σvc (ξ2)

< a < a r < + Σrvc (1ξ) G< cc (ξ2) − Gvc (1ξ) Σcc (ξ2) + Σvv (1ξ) Gvc (ξ2) − Gvv (1ξ) Σvc (ξ2)

a r < + Σ< vc (1ξ) Gcc (ξ2) − Gvc (1ξ) Σcc (ξ2)    1ξ ξ2 r < r  ∆hh v v gee v c   ∆hh    + + 1ξ ξ2   < < ∆aee −ghh −ghh vv vc    1ξ ξ2 < < a  ∆hh v v gee v c   ∆hh    + + 1ξ ξ2   r r −ghh −ghh ∆< ee vv vc

1ξ vc 1ξ vc 1ξ vc 1ξ vc

< gee

∆aee a gee

∆< ee

ξ2 cc ξ2 cc ξ2 cc ξ2 cc

   



 . 

Upon going over to the defect representation, the change of polarization due to the creation of electron-hole pairs by impact ionization, optical excitation, Zener tunneling, and other excitation processes is therefore given by ∂ ∂ 12 e−h,< + gee vc ∂t1 ∂t2   ξ2 e−h,<  vvv (1ξ) gee vc   =   e−h,< 1 ξ −gee vcT (ξ2) vc     ξ2 1ξ e−h,aT e−h,< e−h,r e−h,< (1ξ) g (ξ2) −Σ ∆ G vv ee cc    ee vc  vc  + +     ξ2 1ξ e−h,>T e−h,a e−h,< e−h,a (1ξ) ∆ee (ξ2) −gee Σcc +Gvv vc vc     ξ2 1ξ e−h,>T e−h,a e−h,< e−h,a (1ξ) g (ξ2) −Σ G ∆ vv ee cc    ee vc  vc  + +     ξ2 1ξ e−h,aT e−h,< e−h,r e−h,< −gee (1ξ) ∆ee (ξ2) +Gvv Σcc vc vc     1ξ ξ2 e−h,r e−h,< e−h,aT e−h,< (ξ2) (1ξ) g −Σ G ∆ vc vc ee    ee vv cc  +  +    e−h,< 1 ξ ξ2  e−h,>T e−h,a −gee (ξ2) (1ξ) ∆ +G Σe−h,a vc vc ee vv cc     1ξ ξ2 e−h,< e−h,a e−h,>T e−h,a −Σ (ξ2) (1ξ) g ∆ G vc vc ee    ee vv cc  + . +    e−h,r 1 ξ ξ2  e−h,< e−h,aT (ξ2) (1ξ) ∆e−h,< −gee Σvc +Gvc ee vv cc

i|

The last four bracketed terms arise from the pairing between conduction electrons.

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e−h,< Similarly, ghh is determined from the general formalism given by the following equation, which yield the change of polarization due to the destruction of electronhole pairs brought about by Auger recombination, optical de-excitation, and other recombination processes,

i|

∂ ∂ + ∂t1 ∂t2

G< cv (12)

< T r < < a = vc G< cv (12)−Gcv vv + Σcv (1ξ) Gvv (ξ2)−Gcv (1ξ) Σvv (ξ2) < a < a r < + Σrcc (1ξ) G< cv (ξ2)−Gcc (1ξ) Σcv (ξ2) + Σcv (1ξ) Gvv (ξ2)−Gcv (1ξ) Σvv (ξ2) a r < + Σ< cc (1ξ) Gcv (ξ2)−Gcc (1ξ) Σcv (ξ2)     1ξ ξ2 1ξ ξ2 r < r <  ∆hh c v gee v v   ∆hh c c gee c v  +  +  < 1ξ ξ 2   < 1ξ ξ2  a a −ghh −ghh ∆ee ∆ee cv vv cc cv    1ξ ξ2 1ξ ξ2 < < a a  ∆hh c v gee v v   ∆hh c c gee c v + +  r 1ξ ξ2   r 1ξ ξ2 < −ghh −ghh ∆ee ∆< ee cv vv cc cv

Transforming to the defect representation, we obtain i|



 . 

∂ ∂ 12 e−h,< + ghh cv ∂t1 ∂t2   1ξ e−h,<   vcc (ξ2) ghh cv  =   e−h,< ξ 2 T vvv (1ξ) −ghh cv     1ξ ξ2 e−h,r e−h,< e−h,>T e−h,r (ξ2)   Σcc (1ξ) ghh Gvv  −∆hh  cv cv +  +     1ξ ξ 2 e−h,< e−h,a e−h,rT e−h,< +ghh (ξ2) (1ξ) ∆hh −Gcc Σvv cv cv     1ξ ξ2 e−h,< e−h,a e−h,rT e−h,< (ξ2)   Σcc (1ξ) ghh Gvv   −∆hh cv cv +  +     1ξ ξ 2 e−h,r e−h,< e−h,>T e−h,r +ghh (ξ2) (1ξ) ∆hh −Gcc Σvv cv cv     ξ2 1ξ e−h,< e−h,r e−h,r e−h,>T (1ξ) ghh (ξ2)  Gcv   −∆hh  Σcv vv cc  + +   ξ2   1ξ e−h,a e−h,< e−h,< e−h,rT +ghh (1ξ) ∆hh (ξ2) −Gcv Σcv vv cc     ξ2 1ξ e−h,a e−h,< e−h,< e−h,rT (1ξ) ghh (ξ2)}  Gcv   −∆hh  Σcv vv cc . + +     ξ2 1ξ e−h,< e−h,r e−h,r e−h,>T (1ξ) ∆hh (ξ2) +ghh Σcv −Gcv vv cc

The last four bracketed terms originate from the pairing between holes.

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Appendix G

Calculation of Nonequilibrium Self-Energies

G.1

G.1.1

Nonequilibrium Self-Energy due to Electron-Electron Interaction First-Order Contribution to the Electron Self-Energy

In the superfield representation of fermi particles, the self-energy from Eq. (30.77) is given by Σ (1w) =

V (1234) 3!

K (23) δ (4w) − K (24) δ (3w) + K (34) δ (2w) +K (2ξ) D (ξηνw) K (η3) K (ν4)

,

where 4

V (1234) =

v (1234) =

D (ξηµw) =

i=1

δ (t1 − ti )

(−1)P v (1234) , P

1 [δ α1 ,iii δ α1 ,iii δ α1 ,i δ α1 ,i − δ α1 ,iv δ α1 ,iv δ α1 ,ii δ α1 ,ii ] 2 × {v (|r1 − r2 |) δ (r2 − r3 ) δ (r1 − r4 )} ,

δΣ (ξη) i|δK (wµ)

+ G(i)

δΣ (ξη) i|δK (56)

G(i)

K 55 D 5 6 µw K 6 6 ,

where the Roman indices pertain to the components of the superfield, Eq. (30.1). In what follows, we make use of the matrix elements of K (1, 2) given by   < c ghh (1, 2) ghh (1, 2) Gc (1, 2) −G< (1, 2)  g > (1, 2) g ac (1, 2) G> (1, 2) −Gac (1, 2)  hh hh  K (1, 2) =  c <  −GcT (1, 2) −G>T (1, 2) gee (1, 2) −gee (1, 2)  > ac (1, 2) gee (1, 2) G
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with the resulting electron-electron self-energy given by the following expression   c cT Σ>T ∆ee ∆< ee Σ ac ΣacT  ∆ (1, 2) ΣT (1, 2) ee ∆ee Σ ˘ (1, 2) =  (1, 2) , Σ = c < c   Σ Σ ∆hh ∆< Σ (1, 2) ∆ (1, 2) hh ac Σ> Σac ∆> ∆ hh hh

where = −1 for fermions. Carrying out the indicated operations, we obtain the first-order contribution to ˘ (1, 2) given by Σ †

∆ (1w) = i|

∆ (1w) = i| 

  −Σ (1w) = i|    + T

c (34)} 0 − {gee ac 0 {gee (34)}

× {v (|r3 − r4 |) δ r3 r1 δ r4 rw } ,

c (24)} 0 − {ghh ac 0 {ghh (24)}

× {v (|r1 − rw |) δ r3 r1 δ r4 rw } ,

GcT (34) 0

× {v (|rw − r3 |) δ r4 r3 δ r1 rw }

GacT (34)

0 − GcT (34) 0



0 − GacT (34)

× v (|r3 − rw |) δ r3 r1 δ r4 rw

  ,  



 − {Gc (34)} 0 × {v (|r1 − r4 |) δ r4 r3 δ r1 rw }   0  − {Gac (34)}   Σ (1w) = i|  .   c {G (34)} 0   × {v (|r − r |) δ δ } + 4 1 r r r r 3 1 4 w {Gac (34)} 0

The Feynman diagrams corresponding to the expressions given above are given below. G.1.2

Four-Point Vertex Function to Second Order

We now investigate the second-order contribution from the 4-point vertex function given by the equation, D (ξηµw) =

δΣ (ξη) i|δK (wµ)

+ G(i)

δΣ (ξη) i|δK (56)

G(i)

K 55 D 5 6 µw K 6 6 , (G.1)

to the electron nonequilibrium self-energy. This contribution is given by Σ (1w)4−point =

V (1234) {K (2ξ) D (ξηνw) K (η3) K (ν4)} . 3!

(G.2)

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We will approximate D (ξηµw) in Eq. (G.1) by using the first three terms of Σ (1w) , namely, D (ξηµw) ≈

δΣ (ξη) i|δK (wµ)

G(i)

V (ξ234) δ {K (23) δ (4η)−K (24) δ (3η)+K (34) δ (2η)} 3! i|δK (wµ) V (ξ234) = {δ (2w) δ (3µ) δ (4η)−δ (2w) δ (4µ) δ (3η)+δ (3w) δ (4µ) δ (2η)} i|3! 1 {V (ξwµη)−V (ξwηµ)+V (ξηwµ)} = i|3! 1 1 {V (ξηwµ)}= − {V (ξηµw)} . = i|2! 2i| ≈

Hence, we can approximate Σ (1w)4−point in Eq. (G.2) as Σ 14

4−point

≈−

V (1234) K 21 3!2i|

V 1234

K 23 K 34

,

(G.3)

where 4

V (1234) = p

v (1234) =

i=1

δ (t1 − ti ) v (1234) ,

v (1234)

p

  v (1234) − v (2134) + v (2314) − v (2341)        −v (1324) + v (3124) − v (3214) + v (3241)        +v (1342) − v (3142) + v (3412) − v (3421) = .   −v (1243) + v (2143) − v (2413) + v (2431)         +v (1423) − v (4123) + v (4213) − v (4231)     −v (1432) + v (4132) − v (4312) + v (4321)

Now the expression for v (1234) is v (1234) =

1 [δ α1 ,iii δ α2 ,iii δ α3 ,i δ α4 ,i − δ α1 ,iv δ α2 ,iv δ α3 ,ii δ α4 ,ii ] 2 × {v (|r1 − r2 |) δ (r2 − r3 ) δ (r1 − r4 )} ,

where the subscripted α indices labels the components of the superfield Ψ, Eq. (30.1). Note that in the permutation of indices in v (1234), the functional form of v relative to the order of it indices is unaﬀected, for example, we have v (1234) =

1 [δ α1 ,iii δ α2 ,iii δ α3 ,i δ α4 ,i − δ α1 ,iv δ α2 ,iv δ α3 ,ii δ α4 ,ii ] 2 × {v (|r1 − r2 |) δ (r2 − r3 ) δ (r1 − r4 )} ,

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v (3421) =

1 [δ α3 ,iii δ α4 ,iii δ α2 ,i δ α1 ,i − δ α3 ,iv δ α4 ,iv δ α2 ,ii δ α1 ,ii ] 2 × {v (|r3 − r4 |) δ (r4 − r2 ) δ (r3 − r1 )} .

Consider the following expression: V (1234) K 21

K 23 K 34

V 1234

.

The product of v (1234) × v (1 2 3 4 ) consists of the following expression, v (1234) v (1 2 3 4 )  δ α1 ,iii δ α2 ,iii δ α3 ,i δ α4 ,i × δ α1 ,iii δ α2 ,iii δ α3 ,i δ α4 ,i 1 −δ α1 ,iii δ α2 ,iii δ α3 ,i δ α4 ,i × δ α1 ,iv δ α2 ,iv δ α3 ,ii δ α4 ,ii =   −δ 4 α1 ,iv δ α2 ,iv δ α3 ,ii δ α4 ,ii × δ α1 ,iii δ α2 ,iii δ α3 ,i δ α4 ,i +δ α1 ,iv δ α2 ,iv δ α3 ,ii δ α4 ,ii × δ α1 ,iv δ α2 ,iv δ α3 ,ii δ α4 ,ii × {v (|r1 − r2 |) δ (r2 − r3 ) δ (r1 − r4 )} × {v (|r1 − r2 |) δ (r2 − r3 ) δ (r1 − r4 )} .

   

˘ (1, 4 ) for Based on the above product of v (1234) × v (1 2 3 4 ), we may write Σ convenience in what follows as   Σi,i −Σi,ii Σi,iii −Σi,iv   ˘ (1, 4 ) =  −Σii,i Σii,ii −Σii,iii Σii,iv  Σ  Σiii,i −Σiii,ii Σiii,iii −Σiii,iv  −Σiv,i Σiv,ii −Σiv,iii Σiv,iv

to explicitly account for the sign of the terms in the product of v (1234)×v (1 2 3 4 ). Now consider the following permutation of indices with reference to the potential. There are 24 permutations of indices in both v (1234) and v (1 2 3 4 ). These can be arranged into equivalent groups, for example, the permutation pair (1234, 2143) have equivalent members. Therefore the products of the permutation of v (1234) × v (1 2 3 4 ) can be counted in terms of product of pairs, such as (2314, 3241) × (1 2 3 4 , 2 1 4 3 ), etc. We note that with the exception of 1 and 4 , the rest of the indices are ’integrated’ and hence act as dummy variables. ˘ (1, 4 ), we can evaluate the product of V s and Thus, for the first two rows of Σ K s entering in the expression for the electron-electron self-energy, Eq. (G.3). The (i, i) terms (Σi,i ) involve the matrix elements of K 21 K 2 3 K 3 4 from the following combinations, (1, 2) = i, (3, 4) = iii; (1, 3) = i, (2, 4) = iii; (1, 4) = i, (2, 3) = iii (3 , 4 ) = i, (1 , 2 ) = iii; (2 , 4 ) = i, (1 , 3 ) = iii; (1 , 4 ) = i, (2 , 3 ) = iii

This combination gives nine terms. In what follows, we will schematically indicate the combinatorics involved in the calculation of Σ (1w)4−point . Referring to the tables given above, the terms belonging to the (i, i) terms of ˘ (1, 4 ) are (the factor of 4 comes from the pairs entry in the table in view of the Σ

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symmetry of the Coulomb potential between electrons) (1) (1, 2) = i, (3, 4) = iii; (4 , 3 ) = i, (1 , 2 ) = iii ⇒ Ki,iii 21

Kiii,iii 2 3 Ki,iii 3 4

⇒ Gc 21

c gee 2 3 Gc 3  c     +G   + (3412, 4321) × (1 2 3 4 , 2 1 4 3 )        −Gc − (3412, 4321) × (2 1 3 4 , 1 2 4 3 )  ⇒4  − (3421, 4312) × (1 2 3 4 , 2 1 4 3 )    −Gc       + (3421, 4312) × (2 1 3 4 , 1 2 4 3 )    +Gc

4  c gee 3 2 Gc 3 1        c c  g 42 G 31 

24 23

ee

c  gee 3 1 Gc 3 2        c c gee 4 1 G 3 2 

24 23

× 14 v (|r1 − r2 |) v (|r3 − r4 |)

(2) (1, 2) = i, (3, 4) = iii; (4 , 2 ) = i, (1 , 3 ) = iii ⇒ Ki,iii 21

Ki,iii 2 3 Kiii,iii 3 4

⇒ Gc 21

c Gc 2 3 gee 3   −Gc      − (3412, 4321) × (1 3 2 4 , 3 1 4 2 )        +Gc + (3412, 4321) × (3 1 2 4 , 1 3 4 2 )  ⇒4  + (3421, 4312) × (1 3 2 4 , 3 1 4 2 )    +Gc       − (3421, 4312) × (3 1 2 4 , 1 3 4 2 )    −Gc

4  c Gc 2 2 gee 21        c c  G 2 2 gee 4 1 

24 22

c  Gc 2 1 gee 22        c c G 2 1 gee 4 2 

24 22

× 14 v (|r1 − r2 |) v (|r2 − r4 |)

(3) (1, 2) = i, (3, 4) = iii; (4 , 1 ) = i, (2 , 3 ) = iii ⇒ Ki,i 21

Kiii,iii 2 3 Kiii,iii 3 4

c 21 ⇒ ghh

c c gee 2 3 gee 34  c  21 −ghh      − (3412, 4321) × (3 2 1 4 , 2 3 4 1 )     c    +ghh 21 + (3412, 4321) × (2 3 1 4 , 3 2 4 1 )  ⇒4  + (3421, 4312) × (3 2 1 4 , 2 3 4 1 )  c     +ghh 21     − (3421, 4312) × (2 3 1 4 , 3 2 4 1 )    −g c 21 hh

× 14 v (|r1 − r2 |) v (|r1 − r4 |)

 c c  gee 1 2 gee 41       c c  g 42 g 11  ee

ee

c c  gee 1 1 gee 42        c c gee 4 1 gee 1 2 

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(4) (1, 3) = i, (2, 4) = iii; (4 , 3 ) = i, (1 , 2 ) = iii ⇒ Kiii,iii 21

Kiii,i 2 3 Ki,iii 3 4

c 21 ⇒ −gee

GcT 2 3 Gc  c     +gee   − (2413, 4231) × (1 2 3 4 , 2 1 4 3 )     c    −gee + (2413, 4231) × (2 1 3 4 , 1 2 4 3 )  ⇒4  c + (2431, 4213) × (1 2 3 4 , 2 1 4 3 )    −gee       − (2431, 4213) × (2 1 3 4 , 1 2 4 3 )   c  +gee

34 34 33 14 13

× 14 v (|r1 − r3 |) v (|r3 − r4 |)

 GcT 3 3 Gc 3 1        cT c  G 43 G 31   GcT 3 3 Gc 3 3        cT c G 43 G 33 

(5) (1, 3) = i, (2, 4) = iii; (4 , 2 ) = i, (1 , 3 ) = iii ⇒ Kiii,iii 21 c 21 ⇒ gee

Ki,i 2 3 Kiii,iii 3 4

c c ghh 2 3 gee  c  −gee      − (2413, 4231) × (3 1 2 4 , 1 3 4 2 )       +g c  ee + (2413, 4231) × (1 3 2 4 , 3 1 4 2 )  ⇒4  c + (2431, 4213) × (3 1 2 4 , 1 3 4 2 )    +gee       − (2431, 4213) × (1 3 2 4 , 3 1 4 2 )   c  −gee

34 32 34 12 14

× 14 v (|r1 − r3 |) v (|r2 − r4 |)

 c c ghh 2 3 gee 41         gc 2 3 gc 2 1  hh

ee

c c  ghh 2 3 gee 43        c g c 2 3 gee 23  hh

(6) (1, 3) = i, (2, 4) = iii; (4 , 1 ) = i, (2 , 3 ) = iii ⇒ Kiii,i 21 ⇒ GcT 21

Kiii,i 2 3 Kiii,iii 3 4

c GcT 2 3 gee   −GcT      − (2413, 4231) × (2 3 1 4 , 3 2 4 1 )        +GcT + (2413, 4231) × (3 2 1 4 , 2 3 4 1 )  ⇒4  cT + (2431, 4213) × (2 3 1 4 , 3 2 4 1 )      +G     − (2431, 4213) × (3 2 1 4 , 2 3 4 1 )    −GcT

34 31 31 11 11

× 14 v (|r1 − r3 |) v (|r1 − r4 |)

 c  GcT 4 3 gee 11       cT c  G 13 g 41  ee

c  GcT 4 3 gee 13        cT c G 1 3 gee 4 3 

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(7) (1, 4) = i, (2, 3) = iii; (4 , 3 ) = i, (1 , 2 ) = iii ⇒ Kiii,iii 21

Kiii,iii 2 3 Ki,i 3 4

c c gee 2 3 ghh 3  c  −gee      − (2341, 3214) × (1 2 3 4 , 2 1 4 3 )     c   +gee + (2341, 3214) × (2 1 3 4 , 1 2 4 3 ) ⇒4 + (2314, 3241) × (1 2 3 4 , 2 1 4 3 )   c   +gee    − (2314, 3241) × (2 1 3 4 , 1 2 4 3 )     −g c ee

⇒

c gee

21

4  c c gee 3 4 ghh 34       c c  gee 4 4 ghh 3 4 

14 13

c c  gee 3 1 ghh 34       c c gee 4 1 ghh 3 4 

44 43

× 14 v (|r1 − r4 |) v (|r3 − r4 |)

(8) (1, 4) = i, (2, 3) = iii; (4 , 2 ) = i, (1 , 3 ) = iii ⇒ Kiii,iii 21

c 21 ⇒ −gee

Ki,iii 2 3 Kiii,i 3 4

Gc 2 3 GcT  c  −gee      + (2341, 3214) × (1 3 2 4 , 3 1 4 2 )     c   +gee − (2341, 3214) × (3 1 2 4 , 1 3 4 2 ) ⇒4 − (2314, 3241) × (1 3 2 4 , 3 1 4 2 )   c   +gee    + (2314, 3241) × (3 1 2 4 , 1 3 4 2 )     −g c ee ×

1 4 v (|r1

34 14 12 44 42

− r4 |) v (|r2 − r4 |)

 Gc 2 4 GcT 2 4       c cT  G 24 G 44   Gc 2 1 GcT 2 4       c cT G 21 G 44 

(9) (1, 4) = i, (2, 3) = iii; (4 , 1 ) = i, (2 , 3 ) = iii ⇒ Kiii,i 21

⇒ GcT 21

Kiii,iii 2 3 Kiii,i 3 4

c gee 2 3 GcT   GcT      − (2341, 3214) × (2 3 1 4 , 3 2 4 1 )       GcT + (2341, 3214) × (3 2 1 4 , 2 3 4 1 ) ⇒4 + (2314, 3241) × (2 3 1 4 , 3 2 4 1 )   cT    G  − (2314, 3241) × (3 2 1 4 , 2 3 4 1 )     GcT

×

1 4 v (|r1

34 11 11 41 41

− r4 |) v (|r1 − r4 |)

 c gee 4 4 GcT 1 4       c cT  gee 1 4 G 44  c  gee 4 1 GcT 1 4       c cT g 11 G 44  ee

Similar calculations can be carried out for (i, ii), (ii, i), (ii, ii), (i, iii), (i, iv), ˘ (1, 4 ) are determined by the product (ii, iii), and (ii, iv). The last two rows of Σ of permutation pairs such as (1234, 2143) × (1 2 3 4 , 2 1 4 3 ). We can reduce the number of terms by making use of the following facts, namely, (a) that only the indices 1 and 4 are not dummy indices, and (b) transposing or interchanging the

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Nonequilibrium Quantum Transport Physics in Nanosystems

indices follows the following relations: > < ghh (2, 1) = −ghh (1, 2) , c c (2, 1) = −ghh (1, 2) , ghh ac ac (2, 1) = −ghh (1, 2) , ghh > < (2, 1) = −gee (1, 2) , gee c c (2, 1) = −gee (1, 2) , gee ac ac (2, 1) = −gee (1, 2) . gee

Upon combining identical graphs, the e − e Coulomb interaction contributions to the self-energy are †

∆ (14 ) =

=

∆cee (14 ) −∆< ee (14 ) > ac −∆ee (14 ) ∆ee (14 )   2   − 12 (i|)   Σi,i −Σi,ii = v (|r1 − r2 |)   −Σii,i Σii,ii   × v (|r3 − r4 |)

  cT c 13 gee 3 2 Gc 24     +G         cT cT c  +G 2 g 13 G 3 24    ee          c c c    +gee 13 G 3 2 G 24   c c c      −gee 14 G 23 G 3 2           c c c   +g 2 14 g 23 g 3  ee ee hh           c c c  −gee 13 ghh 3 2 gee 24  ×   <   +G 24         > 24 +GT 3 2 gee         >  +gee   13 G< 3 2 G> 24 −  > > <    −gee 14 G 23 G 3 2         > < > 13 ghh 3 2 gee 24  −gee         > > < +gee 14 ghh 23 gee 32

                      −  < < >     2 −g 14 G 23 G 3  ee            < > <    −g 2 g 13 g 3 24   ee ee hh             < < >  +gee 14 ghh 23 gee 3 2  ,    acT ac ac  +G 2 G 13 g 3 24   ee              ac  +GacT 13 GacT 3 2 gee 24                 ac ac ac  +gee 13 G   32 G 24   ac ac ac      14 G 23 G 3 2 −g ee              ac ac ac   2 g −g 13 g 3 24    ee ee hh            ac ac ac +gee 14 ghh 23 gee 32  > +G>T 13 gee  3 2 G< 24      >T <  13 G 3 2 G< 24

                            

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∆ 14

=

∆chh (14 ) ∆< hh (14 ) ac (14 ) ∆> (14 ) ∆ hh hh

=

Σiii,iii −Σiii,iv −Σiv,iii Σiv,iv

  c   +Gc 13 ghh 3 2 GcT 24            c 13 c 3 2 gc    G 24 +G   hh          c cT cT   32 G 24   +ghh 13 G     −g c 14 GcT 23 GcT 3 2    hh         c c c    −ghh 13 gee 3 2 ghh 24            c c c  +ghh 14 gee 23 ghh 3 2    ×    −G> 13 g < 3 2 G >    13 G< 3 2 ghh 24  −G        −g > 13 G>T 3 2 G   +ghh 14 GT 3 2        > > <    +ghh 13 gee 3 2 ghh 24         −g > 14 g > 23 g < 3 2 ee hh hh −Σ 14

T

=

=

  

− 12 (i|)2 = v (|r1 − r2 |)   × v (|r3 − r4 |)

749

    

 >  −G< 13 ghh 3 2 G>T 24       <  24 −G< 13 G> 3 2 ghh       <  −ghh 13 GT 24

                              

                      −  <   >T 3 2 g<      13 gee 24 +ghh    hh           −g < 14 g < 23 g > 3 2   ee hh hh    ,  ac 13 ac 3 2 GacT 24   +G g    hh             ac 13 ac 3 2 g ac 24   G +G    hh             ac 13 acT acT   +ghh  G 32 G 24      ac acT acT    2 −g 14 G 23 G 3   hh              ac ac ac   −g 13 g 3 24 2 g    ee hh hh           +g ac 14 g ac 23 g ac 3 2  ee hh hh

−Σc (14 )T −Σ> (14 )T

Σ< (14 )T Σi,iii −Σii,iii

Σac (14 )T    −Σi,iv =  Σii,iv 

  c c   −gee 13 Gc 3 2 ghh 24             c c cT  13 g 3 24 −g 2 G    ee hh           cT c c   −G  2 g 13 g 3 24 ee hh    cT 14 c c  ghh 23 gee 32    +G          cT 13 cT 3 2 GcT 24     +G G            −GcT 14 Gc 23 Gc 3 2     ×  > < 3 2 g>   hh 24  −gee 13 G         < >  −gee 13 ghh 3 2 G  −G    +GT 3 2 G 23 G< 3 2

 − 12 (i|)2   v (|r1 − r2 |)   ×v (|r3 − r4 |)

 < < 13  +gee G> 3 2 ghh 24       > < >T  24 +gee 13 ghh 3 2 G       < >T >  +G 13 gee 3 2 ghh 24

                              

                     −  <    >T 14 > 3 2   −G g 23 g    ee hh          >T 13 T 24      −G G             +G>T 14 G< 23 G> 3 2       , ac ac ac    +gee 13 G 3 2 ghh 24                ac ac acT   +g 2 G 13 g 3 24    ee hh             acT ac ac  +G   13 gee 3 2 ghh 24      acT 14 ac 23 ac 3 2   g g −G    ee hh           acT 13 acT 3 2 GacT 24      G −G             +GacT 14 Gac 23 Gac 3 2 

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Σ 14

=

=

Σc (14 ) −Σ< (14 )

Σ> (14 ) −Σac (14 ) Σiii,i −Σiii,ii

−Σiv,i Σiv,ii

 c  c 24   +ghh 13 GcT 3 2 gee           c c c   +g 2 G 13 g 3 24   ee hh           c c   +Gc 13 ghh  2 g 3 24 ee    c c c     −G 14 gee 23 ghh 3 2          c c c   −G 13 G 3 2 G 24             +Gc 14 Gc 23 Gc 3 2   ×     −g > 13 G>T 3 2 g> 24 ee    hh      > < >     −ghh 13 gee 3 2 G 24      >  −G> 13 g< 3 2 gee   24 hh −  > > <    +G 14 gee 23 ghh 3 2       +G> 13 G< 3 2 G> 24            −G> 14 G> 23 G< 3 2

=

    

− 12 (i|)2

v (|r1 − r2 |)

× v (|r3 − r4 |)

    

                    −  < < >    −G 14 gee 23 ghh 3 2                 −G< 13 G> 3 2 G< 24              +G< 14 G< 23 G> 3 2       ac  . acT ac  −g 13 G 3 24 2 g   ee hh              ac ac  13 gee 3 2 Gac 24  −ghh                ac ac ac  −G 13 ghh 3 2 gee 24     ac ac ac      +G 2 14 g 23 g 3 ee hh               +Gac 3 2 Gac 24 Gac 13                ac ac ac −G 14 G 23 G 32  < <  +g 13 G <    +ghh 13 gee 3 2 G 24     > <  +G< 13 ghh 3 2 gee 24

                            

We make the following check on the identities that the matrix elements must obey, namely, the following relations must hold, > ac r ∆cee (14 ) − ∆< ee (14 ) = ∆ee (14 ) − ∆ee (14 ) = ∆ee (14 ) ,

(G.4)

> ac r ∆chh (14 ) − ∆< hh (14 ) = ∆hh (14 ) − ∆hh (14 ) = ∆hh (14 ) ,

(G.5)

T

T

T

T

T

Σc (14 ) − Σ> (14 ) = Σ< (14 ) − Σac (14 ) = Σa (14 ) ,

(G.6)

Σc (14 ) − Σ< (14 ) = Σ> (14 ) − Σac (14 ) = Σr (14 ) .

(G.7)

Upon substituting and carrying out the above identity relations, indeed we obtain the following identities (dropping the factor − 12 (i|)2 v (|r1 − r2 |) v (|r3 − r4 |)). For

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the 2 × 2 matrix † ∆ (14 ), we have ∆cee (14 ) − ∆< ee (14 )  cT c >  13 gee 3 2 Gc 24 − G>T 13 gee 3 2 G< 24   +G    a  3 2 Gr 24 = GaT 13 gee      c <  24 − G>T 13 G 3 2 G< 24 +gee      r  13 Ga 3 2 Gr 24 = gee = c <  14 Gc 23 Gc 3 2 − −gee 14 G< 23 G> 3 2 −gee      r  14 Gr 23 Ga 3 2 = −gee      < c c c < >  14 ghh 21 gee 1 2 − gee 14 ghh 21 gee 12 +gee      r r a  14 ghh 21 gee 12 = gee      > c c c < <  11 ghh 1 2 gee 24 − −gee 11 ghh 1 2 gee 24  −gee     r r  11 ga 1 2 gee 24 = −gee hh

=

∆ree

(14 ) ,

ac ∆> ee (14 ) − ∆ee (14 )  T            >  13 G< gee       = > >    −gee 14 G          < >  11 ghh −gee            > >  14 ghh gee      

= ∆ree (14 ) .

3 2 G> 24 = GaT 13 > 3 2 gee 24

= GaT 13 3 2 G> 24 r 13 = gee

23

− GacT 13

ac gee 3 2 Gac 24

− GacT 13

ac GacT 3 2 gee 24

a gee 3 2 Gr 24

r GrT 3 2 gee 24 ac − gee 13

Gac 3 2 Gac 24

Ga 3 2 Gr 24

ac G< 3 2 − −gee 14

Gac 23

r 14 = −gee

Gr 23

r 11 = −gee

a r ghh 1 2 gee 24

> 1 2 gee 24

21

Ga 3 2

ac − −gee 11

< ac gee 1 2 − gee 14

r 14 = gee

r ghh 21

Gac 3 2

ac ac ghh 1 2 gee 24

ac ghh 21

a gee 12

ac gee 12

                                                              

                                                              

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Likewise, for the 2 × 2 matrix ∆ (14 ), we have ∆chh (14 ) − ∆< hh (14 )  > c  3 2 GcT 24 − G< 13 ghh 3 2 G>T 24 +Gc 13 ghh      a  3 2 GaT 24 = Gr 13 ghh      < c  24 − G< 13 G> 3 2 ghh 24 +Gc 13 Gc 3 2 ghh      r  24 = Gr 13 Ga 3 2 ghh      < c  13 GcT 3 2 GcT 24 − ghh 13 GT 24  +ghh     r  13 GrT 3 2 GaT 24 = ghh = < c  14 GcT 23 GcT 3 2 − −ghh 14 G>T 23 G  11 gee 1 2 ghh 24 − −ghh 11 gee 1 2 ghh 24 −ghh      r a r  11 gee 1 2 ghh 24 = −ghh      < > c c c <  21 ghh 12 +ghh 14 gee 21 ghh 1 2 − ghh 14 gee      r r a  14 gee 21 ghh 12 = ghh = ∆rhh (14 ) ,

                                                              

ac ∆> hh (14 ) − ∆hh (14 )   < > > < ac ac ac   13 G 3 24 − G 13 G 3 24 G 2 g 2 g   hh hh         r a r   13 G 3 24 2 g = G   hh      >    >T T 3 2 − −ghh 14 GacT 23 GacT 3 2     −ghh 14 G        r aT rT   14 G 23 G 3 2 = −g   hh         > > < ac ac ac   11 g 1 24 − −g 11 g 1 24 2 g 2 g −g   ee ee hh hh hh hh         r a r   11 g 1 24 = −g 2 g   ee hh hh         > < > ac ac ac   14 g 21 g 1 14 g 21 g 1 2 − g 2 g   ee ee hh hh hh hh         r r a   = ghh 14 gee 21 ghh 1 2

= ∆rhh (14 ) .

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T

For the 2 × 2 matrix Σ (14 ) , we have T

T

Σc (14 ) − Σ> (14 )  < c c <  13 Gc 3 2 ghh 24 − gee 13 G> 3 2 ghh 24 gee      r a r  = gee 13 G 3 2 ghh 24      > c c <  13 ghh 3 2 GcT 24 − gee 13 ghh 3 2 G>T 24 gee      r a  13 ghh 3 2 GaT 24 = gee      < c c >  3 2 ghh 24 − G>T 13 gee 3 2 ghh 24 GcT 13 gee      aT a r  13 gee 3 2 ghh 24 =G = < c c >  23 gee 3 2 − −G>T 14 ghh 23 gee 32 −GcT 14 ghh      aT r a  14 ghh 23 gee 3 2 = −G        −GcT 11 GcT 1 2 GcT 24 − −G>T 11 GT 24      = −GaT 11 GrT 1 2 GaT 24       GcT 14 Gc 21 Gc 1 2 − G>T 14 G< 21 G> 1 2       = GaT 14 Gr 21 Ga 1 2 T

= Σa (14 ) ,

T

                                                              

T

Σ< (14 ) − Σac (14 )  > > ac ac  13 G< 3 2 ghh 24 − gee 13 Gac 3 2 ghh 24 gee      r r  13 Ga 3 2 ghh 24 = gee      < > < ac ac  3 2 ghh 24 − GacT 13 gee 3 2 ghh 24 G T 21 G< 1 2 − GacT 14 Gac 21 Gac 1 2       = GaT 14 Gr 21 Ga 1 2 T

= Σa (14 ) .

                                                              

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Similarly, for the 2 × 2 matrix Σ (14 ), we have Σc (14 ) − Σ< (14 )  < c cT c <  3 2 gee 24 − ghh 13 G  13 gee 3 2 Gc 24 − ghh 13 gee 3 2 G< 24 +ghh      r a  13 gee 3 2 Gr 24 = ghh      > c c <  3 2 gee 24 − G< 13 ghh 3 2 gee 24 +Gc 13 ghh      r a r  = G 13 ghh 3 2 gee 24 = > c c c < <    −G 14 gee 23 ghh 3 2 − −G 14 gee 23 ghh 3 2    r a  23 ghh 32 = −Gr 14 gee       −Gc 11 Gc 1 2 Gc 24 − −G< 11 G> 1 2 G< 24       = −Gr 11 Ga 1 2 Gr 24      c c c < < >    +G 14 G 21 G 1 2 − G 14 G 21 G 1 2     = Gr 14 Gr 21 Ga 1 2 = Σr (14 ) ,

Σ> (14 ) − Σac (14 )  >  13 G>T ghh            > <  13 gee ghh            <   G> 13 ghh      = >  −G> 14 gee             −G> 11 G<             G> 14 G>      

> 3 2 gee 24 r 13 = ghh

3 2 G> 24 r 13 = ghh > 3 2 gee 24

= Gr 13 23

ac − ghh 13

ac GacT 3 2 gee 24

ac − ghh 13

ac gee 3 2 Gac 24

− Gac 13

ac ac ghh 3 2 gee 24

r GrT 3 2 gee 24

a gee 3 2 Gr 24

a r ghh 3 2 gee 24

< ghh 3 2 − −Gac 14

ac gee 23

= −Gr 14

r gee 23

= −Gr 11

Ga 1 2 Gr 24

1 2 G> 24

21

a ghh 32

− −Gac 1 2 Gac 24

G< 1 2 − Gac 14

= Gr 14

Gr 21

ac ghh 32

Gac 21

Ga 1 2

= Σr (14 ) .

Therefore the required identities, Eqs. (G.4)-(G.5) are verified.

Gac 11 Gac 1 2

                                                                                                                             

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Nonequilibrium Self-Energy without Coulomb Pairing In the limit when the Coulomb pairing is neglected, the expression for the contribution of the 4-point vertex function to the electron-electron self-energy simplifies considerably. We have †

∆ (1w) =

∆cee (12) −∆< ee (12) > ac −∆ee (12) ∆ee (12)

∆ (14 ) =

∆chh (14 ) ∆< hh (14 ) > ∆hh (14 ) ∆ac hh (14 ) T

−Σ 14

T

=

=

−Σc (14 ) −Σ> (14 ) Σ< (14 ) Σi,iii −Σii,iii

T

=

=

00 00

00 00

,

,

T

T

Σac (14 )    −Σi,iv =  Σii,iv 

− 12 (i|)2

v (|r1 − r2 |)

× v (|r1 − r4 |)

    

   +GcT 11 GcT 1 2 GcT 24  −G>T 11 GT 24  −   +G>T 14 G< 21 G> 1 2  −GcT 14 Gc 21 Gc 1 2    ×    +GT 1 2 G 21 G< 1 2   +GacT 14 Gac 21 Gac 1 2 Σ 14

=

=

        ,     

Σc (14 ) −Σ< (14 )

Σ> (14 ) −Σac (14 ) Σiii,i −Σiii,ii

=

−Σiv,i Σiv,ii

  −Gc 11 Gc 1 2 Gc    +Gc 14 Gc 21 Gc  ×     +G> 11 G< 1 2  −  −G> 14 G> 21

24

 

12 

G> 24

−

    

− 12 (i|)2

v (|r1 − r2 |)

× v (|r1 − r4 |)

  −G< 11  +G< 14

 

      12    . 11      12

G> 1 2 G< 24 G< 21

   +Gac 1 2 Gac 24

G< 1 2   −Gac 14

  

Gac 21

G> Gac Gac

The resulting expressions for the self-energy matrix elements can be interpreted in terms of the usual ‘direct’ (first term) and ‘exchange’ process (second term).

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Feynman Diagrams The first-order contribution to the electron Coulomb self-energy is indicated by the following diagrams.

∆cee ⇒

−∆< ee (1, 4 ) ⇒ 0 −∆> ee (1, 4 ) ⇒ 0

∆ac ee (1, 4 ) ⇒

−ΣcT (1, 4 ) ⇒

                                     −Σ>T (1, 4 ) ⇒ 0 Σ
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ΣacT (1, 4 ) ⇒

                                  

Σc (1, 4 ) ⇒

                                 −Σ< (1, 4 ) ⇒ 0

Σ> (1, 4 ) ⇒ 0

−Σac (1, 4 ) ⇒

                                  

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∆chh (1, 4 ) ⇒

∆< hh (1, 4 ) ⇒ 0

∆> hh (1, 4 ) ⇒ 0

∆ac hh (1, 4 ) ⇒

The general expressions of the Coulomb self-energy to second order is represented by the following diagrams

∆cee (1, 4 ) ⇒

                                                        

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−∆< ee (1, 4 ) ⇒

−∆> ee (1, 4 ) ⇒

                                                                                                     

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∆ac ee (1, 4 ) ⇒

Nonequilibrium Quantum Transport Physics in Nanosystems

                                                  

−ΣcT (1, 4 ) ⇒

                                                  

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−Σ>T (1, 4 ) ⇒

Σ
                                                

                                                

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ΣacT (1, 4 ) ⇒

−Σc (1, 4 ) ⇒

                                                

                                                

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Calculation of Nonequilibrium Self-Energies

−Σ< (1, 4 ) ⇒

Σ> (1, 4 ) ⇒

                                                                                                 

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−Σac (1, 4 ) ⇒

∆chh (1, 4 ) ⇒

                                                

                                                

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∆< hh (1, 4 ) ⇒

                                                

∆> hh (1, 4 ) ⇒

                                              

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∆ac hh (1, 4 ) ⇒

                                              

Electron Self-Energy Due to Electron-Phonon Interaction Consider the electron self-energy contribution due to electron-phonon interaction given by the following expression ΣX (12) = Xio (12; ξ) Ui (ξ) + Xlo 12; ξ K 2s Xi s2; ξ Sli ξ ξ

(G.8)

where ξ

Xio (12; ξ) = P (12)

eP δ α1 ,iii δ α2 ,i δ αξ ,i − δ α1 ,iv δ α2 ,ii δ αξ ,ii

χi (12; ξ) δ (tn − t) √ mkξ n=1

(G.9) is the e-symmetrized ‘bare’ super-electron-phonon interaction vertex, involving the permutation of the electron indices 1, 2. The ‘dressed’ super electron-phonon interaction vertex, Xi (12; ξ), is given by the iterative Dyson equation, Xi (12; ξ) = −

δK (12)−1 δ Ui (ξ)

= Xio (12; ξ) +

δΣ (12) δK (34)

U

K (3¯ 3) K (¯ 44) Xi (¯ 3¯ 4; ξ)

The electron-phonon interaction, χi (12; ξ), is given by χi (12; ξ) = 1| ∇irξ

zξ∗ e |2 |x − rξ |

here zξ∗ is the eﬀective charge of the nucleus located at ξ, 1| and |2 are electron basis vectors. The Kronecker deltas within the square bracket of Eq. (G.9), such as δ α1 ,iii and δ αξ ,i , pertain to the component of the electron and phonon superfields, respectively. We will neglect the first term in Eq. (G.8) since this does not contribute in the limit that the Schwinger source term goes to zero.

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Using the following definition for K and Sli  c < ghh (12) ghh (12)  > ac  ghh (12) ghh (12) K (12) = i|   −Gc (12)T −G> (12)T  T T G< (12) Gac (12) 

767

given by  Gc (12) −G< (12)  G> (12) −Gac (12)   c < gee (12) −gee (12)   > ac −gee (12) gee (12)

 Ki,i (1, 2) Ki,ii (1, 2) Ki,iii (1, 2) Ki,iv (1, 2)  K (1, 2) K  ii,ii (1, 2) Kii,iii (1, 2) Kii,iv (1, 2)   ii,i =   Kiii,i (1, 2) Kiii,ii (1, 2) Kiii,iii (1, 2) Kiii,iv (1, 2)  Kiv,i (1, 2) Kiv,ii (1, 2) Kiv,iii (1, 2) Kiv,iv (1, 2)

Sli ξξ ; tt



 = −i|  =

Slic ξξ ; tt

−Sli< ξξ ; tt

Sli>

−Sliac

ξξ ; tt

Slii,i ξ ξ

ξξ ; tt

Slii,ii ξ ξ

  

Sliii.i ξ ξ Sliii,ii ξ ξ

carrying out the calculation, and collecting the various matrix elements, we have the contribution of the electron-phonon interaction, to second-order in χi (12; ξ), to the filling of the electron self-energy matrix given by, †

∆ (12) =

=

∆ (12) =

=

Σi,i (12) Σi,ii (12) Σii,i (12) Σii,ii (12)

=

∆cee (12) −∆< ee (12) ac (12) ∆ −∆> ee ee (12)

c < 2s Slic ξ ξ −gee 2s Sli> ξ ξ −gee > gee 2s Sli< ξ ξ

ac gee 2s Sliac ξ ξ  ξ  χi s2; ξ l χ 21; ξ × |2 √ δ (tn − t)  √mkξ mkξ n=1

Σiii,iii (12) Σiii,iv (12) Σiv,iii (12) Σiv,iv (12) c 2s Slic ξ ξ − ghh

=

ξ

n=s

δ tn − t

  

∆chh (12) ∆< hh (12) ac ∆> (12) ∆ hh (12) hh

< ghh 2s Sli> ξ ξ

> − ghh 2s Sli< ξ ξ

ac ghh 2s Sliac ξ ξ  ξ  χi 2s; ξ l χ ξ 12; |2 √ δ (tn − t)  √mkξ mkξ n=1

ξ

n=s

δ tn − t

  

,

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Σi,iii (12) Σi,iv (12)

−Σ (12)T =

Σii,iii (12) Σii,iv (12) −GcT 2s Slic ξ ξ

=

Sli<

ξξ

χ 21; ξ |2 √ mkξ

ξ

G
×

Σ (12) =

=

=

n=1

Σiii,i (12) Σiii,ii (12) Σiv,i (12) Σiv,ii (12) Gc 2s Slic ξ ξ

−Σc (12)T −Σ> (12)T T T Σ< (12) Σac (12)

G>T 2s Sli> ξ ξ −GacT 2s Sliac ξ ξ   χi 2s; ξ δ (tn − t)  √mkξ =

n=s

δ tn − t

Σc (12) −Σ< (12) Σ> (12) −Σac (12)

  

,

− G< 2s Sli> ξ ξ

G> 2s Sli< ξ ξ Gac 2s Sliac ξ ξ  ξ  χi s2; ξ l 12; χ ξ δ (tn − t) × |2 √  √mkξ mkξ n=1

Feynman Diagrams

ξ

ξ

n=s

δ tn − t

  

.

It is worth pointing out that the following electron-phonon interaction Feynman diagrams mimics5 the first-order exchange diagrams of the self-energy due to electronelectron Coulomb interaction, with the Coulomb potential replaced by the phonon correlation function, the only diﬀerence is that whereas there the Coulomb interaction is instantaneous (i.e., exchange of virtual photons) here the phonon Green’s function is nonlocal in time and hence we have exchange of real phonons,

∆cee (1, 2) ⇒

−∆< ee (1, 2) ⇒ 5 The resemblance is striking if the electron-phonon interaction vertex is reduced to a point (Dirac-delta function in the arguments). This invites others to start with an eﬀective Hamiltonian for the electron quantum field operators whereby the electron-electron interaction is renormalized by adding the phonon propagator (due to exchange of phonons) to the Coulomb potential.

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−∆> ee (1, 2) ⇒

∆ac ee (1, 2) ⇒

−ΣcT (1, 2) ⇒

−Σ>T (1, 2) ⇒

Σ
ΣacT (1, 2) ⇒

Σc (1, 2) ⇒

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−Σ< (1, 2) ⇒

Σ> (1, 2) ⇒

−Σac (1, 2) ⇒

∆chh (1, 2) ⇒

∆< hh (1, 2) ⇒

∆> hh (1, 2) ⇒

∆ac hh (1, 2) ⇒

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Phonon Self-Energy Due to Phonon-Electron Interaction Consider the electron self-energy contribution due to electron-phonon interaction given by the following expression phn−el Πij ξξ

1¯ 2; ¯ ξ K (¯ 22) . 1) Xj ¯ = Xio (12; ξ) K (1¯

Now we evaluate the the following product of sums to obtain the contribution to electron self-energy, namely, 1¯ 2; ¯ξ K (¯ 22) − Xjo ¯ 2¯ 1; ¯ξ K (¯ 22) [Xio (12; ξ) K (1¯1) − Xio (21; ξ) K (1¯ 1)] Xjo ¯

= Xio (12; ξ) K (1¯1) Xjo ¯1¯2; ¯ξ K (¯ 22) + Xio (21; ξ) K (1¯ 2¯ 1; ¯ ξ K (¯ 22) 1) Xjo ¯

2¯ 1; ¯ξ K (¯ 22) − Xio (21; ξ) K (1¯ 1¯ 2; ¯ ξ K (¯ 22) . − Xio (12; ξ) K (1¯1) Xjo ¯ 1) Xjo ¯

The phonon self-energy is of the form Πlj ξ ¯ξ =

¯ Πlj ξ¯ξ Πlj i,i ξ ξ i,ii ¯ξ Πlj ξ ξ¯ Πlj ξ ii,i ii,ii

=

¯ Πclj ξ¯ξ −Π< lj ξ ξ > ac ¯ Πlj ξ ξ −Πlj ξ¯ξ

.

Carrying out the calculation by collecting the various matrix elements of the electron self-energy due to phonon-electron interaction, we have c c ¯ ¯ ¯ c ¯ Πlj i,i ξ ξ = Πlj ξ ξ = gee (11) ghh (22) δ αξ ,i δ α¯ξ ,i ξ

1¯ 2; ¯ξ χj ¯ χl (12; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ n=¯ 1

δ (tn − t)

c c ¯ (1¯1) gee (22) δ αξ ,i δ α¯ξ ,i + ghh ξ

2¯ 1; ¯ξ χj ¯ χl (21; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ n=¯ 1

δ (tn − t)

− GcT (1¯ 1) GcT (¯ 22) δ αξ ,i δ α¯ξ ,i ξ 2¯ 1; ¯ξ χj ¯ χl (12; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ n=¯ 1

δ (tn − t)

22) δ αξ ,i δ α¯ξ ,i − Gc (1¯1) Gc (¯ ξ 1¯ 2; ¯ξ χj ¯ χl (21; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ

n=s

δ (tn − t) ,

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Nonequilibrium Quantum Transport Physics in Nanosystems < < ¯ ¯ ¯ > ¯ Πlj i,ii ξ ξ = −Πlj ξ ξ = +gee (11) ghh (22) δ αξ ,i δ α¯ξ ,ii ξ

1¯ 2; ¯ξ χj ¯ χl (12; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ n=¯ 1

δ (tn − t)

< > ¯ (1¯1) gee (22) δ αξ ,i δ α¯ξ ,ii + ghh ξ 2¯ 1; ¯ξ χj ¯ χl (21; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ n=¯ 1

δ (tn − t)

− G>T (1¯ 1) G
ξ n=¯ 1

δ (tn − t)

22) δ αξ ,i δ α¯ξ ,ii − G< (1¯1) G> (¯ ξ

1¯ 2; ¯ξ χj ¯ χl (21; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ

n=s

δ (tn − t) ,

> > ¯ ¯ ¯ < ¯ Πlj ii,i ξ ξ = Πlj ξ ξ = +gee (11) ghh (22) δ αξ ,ii δ α¯ξ ,i ξ 1¯ 2; ¯ξ χj ¯ χl (12; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ n=¯ 1

δ (tn − t)

> < ¯ (1¯1) gee (22) δ αξ ,ii δ α¯ξ ,i + ghh ξ

2¯ 1; ¯ξ χj ¯ χl (21; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ n=¯ 1

δ (tn − t)

− GT (¯ 22) δ αξ ,ii δ α¯ξ ,i ξ 2¯ 1; ¯ξ χj ¯ χl (12; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ n=¯ 1

δ (tn − t)

22) δ αξ ,ii δ α¯ξ ,i − G> (1¯1) G< (¯ ξ 1¯ 2; ¯ξ χj ¯ χl (21; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ

n=s

δ (tn − t) ,

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ac ac ¯ ¯ ¯ ac ¯ Πlj ii,ii ξ ξ = −Πlj ξ ξ = gee (11) ghh (22) δ αξ ,ii δ α¯ξ ,ii ξ 1¯ 2; ¯ξ χj ¯ χl (12; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ n=¯ 1

δ (tn − t)

ac ac ¯ + ghh (1¯ 1) gee (22) δ αξ ,ii δ α¯ξ ,ii ξ 2¯ 1; ¯ξ χj ¯ χl (21; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ n=¯ 1

δ (tn − t)

1) GacT (¯ 22) δ αξ ,ii δ α¯ξ ,ii − GacT (1¯ ξ 2¯ 1; ¯ξ χj ¯ χl (12; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

ξ n=¯ 1

δ (tn − t)

1) Gac (¯ 22) δ αξ ,ii δ α¯ξ ,ii − Gac (1¯ ξ 1¯ 2; ¯ξ χj ¯ χl (21; ξ) × √ δ (tn − t) √ mkξ n=1 mkξ

Feynman Diagrams

Πcij (1, 2) ⇒

                                                                    

ξ

n=s

δ (tn − t) .

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Nonequilibrium Quantum Transport Physics in Nanosystems

−Π< ij (1, 2) ⇒

Π> ij (1, 2) ⇒

                                                                      

                                                                      

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−Πac ij (1, 2) ⇒

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                                                                      

where the symbol 2× means that the diagram below this symbol is counted twice, similar to the quantum field theory results for zero-temperature [228] phonon selfenergy.

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Appendix H

Radon Transformation of Phase Space Functions

The Radon transform W (u, v; c) of the Wigner quasiprobability W (p, q) can be defined in the following canonical form W (u, v; c) = (2π|)−1

dp dq W (p, q) δ (q.u + p.v − c)

(H.1)

The normalization of the Radon transform is closely connected to the normalization of the Wigner quasiprobability in the following way: dc W (u, v; c) = (2π|)−1

dp dq W (p, q) = 1

Owing to the relation, Eq. (H.1), we have W (µu, µv; µc) = (2π|)−1 −1

= (2π|) =

dp dq W (p, q) δ (q.µu + p.µv − µc) dp dq W (p, q)

1 W (u, v; c) , |µ|

µ∈R

δ (q.u + p.v − c) |µ| (H.2)

where µ is an arbitrary real number, this Radon transform depends eﬀectively only on two continuous variables, φ and r, Eq. (37.10). The presence of the delta function under the two-dimensional integral in Eq. (H.1) restricts the integrations to the one-dimensional objects q.u + p.v = c These are the equations for straight lines with (u, v) as a normal vector to the lines and r = √u2c+v2 as a measure for the orthogonal (nearest) distance of the line to the coordinate origin. However, this ‘oriented distance’ can take on all real values contrary to positively definite distances. This is because for fixed values of u and v (i.e., constant slope), varying the values of c in the range −∞ ≤ c ≤ ∞ will map 776

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777

all the parallel lines in the whole p-q plane. By writing u + v2 v sin φ = √ u2 + v2

cos φ = √

u2

(H.3)

the angle φ in the range − π2 ≤ φ ≤ π2 covers all the values of slope of parallel lines. The coordinates (u, v; c) are homogeneous line coordinates in the dual plane to the p-q plane. In the case of the generalization of the Radon transform to Ndimensional spaces the corresponding N −1-dimensional objects over which the field is integrated are the (N − 1)-dimensional hyperplanes. Owing to Eq. (H.1) the full information of the Radon transform is already contained in W (cos φ, sin φ; r) where (u; v) = (cos φ, sin φ) is now the normal unit vector to the straight lines. The Radon transform W (u, v; c) is closely related to the Fourier transform ˜ (u, v) of the Wigner quasiprobability W (p, q) which can be defined by W i dp dq W (p, q) e( | )(q.u

˜ (u, v) = (2π|)−3 W

+ p.v)

with the inversion W (p, q) =

˜ (u, v) e(− |i )(q.u du dv W −1

∞

= (2π|)

−1

−∞ ∞

= (2π|)

dc du dv −∞

= (2π|)−1

∞

du dv −∞

−1

= (2π|)

u v , ;c b b

dc du dv W ∞

+ p.v)

db e−i

ib u e(− | )(q. b

b c |

+ p. vb −c)

˜ (u, v) e(− ib| )(q. ub W

+ p. vb −c)

−∞ i

˜ (u, v) e− | (q.u db 2π|δ (b − b) W

˜ (u, v) e− |i (q.u du dv W

+ p.v)

+ p.v)

(H.4)

and with the normalization ˜ (0, 0) = (2π|)−3 W

dp dq W (p, q) = 1

The relation of the Radon transform to the Fourier transform is given by ˜ (u, v) = (2π|)−1 W

∞

−∞

bc

dc ei | W

u v , ;c b b

(H.5)

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with arbitrary real numbers b = 0 and its inversion is given by ∞

W (u, v; c) =

bc ˜ (bu, bv) db e−i | W

−∞

The full inversion of the two-dimensional Radon transform can be made in two steps. The first step is the transition from the Radon transform to the Fourier transform and the second step is the inversion of the Fourier transform. If we introduce polar coordinates (r, φ) instead of (u, v), then π 2

du dv dc ⇒

dφ −π 2

1 = 2

∞

∞

ro dro −∞

0 π 2

dr ∞

∞

dφ −π 2

ro dro −∞

dr −∞

where q.u + p.v = c ⇒ q cos φ + p sin φ = ro =

c =r ro

u2 + v2

The following integral is useful in what follows, ∞

−∞

|r| exp (−ixr) dr

∂ =i ∂x

∞

0

∂ dr exp (−ixr) − i ∂x

0

dr exp (−ixr) −∞

 ∞ 0  ∂  dr exp (−ixr) − dr exp (−ixr) =i  ∂x  −∞ 0  ∞ 0  ∂  dr exp (−i (x − iε) r) − dr exp (−i (x + iε) r) =i  ∂x  0

1 ∂ 1 + =i ∂x i (x − i0) i (x + i0) 1 1 ∂ + = ∂x (x − i0) (x + i0) =2

∂ 1 P ∂x x

−∞

ε→0

(H.6)

where P means that the integral over the singularity of x1 must be taken in the ∂ P x1 is that of a gensense of Cauchy principal value. Note that the meaning of ∂x

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779

eralized function or linear functional. The symbol R is often used for the canonical ∂ P x1 = −R x12 , and we have regularization of singular function, P x1 ≡ R x1 . Thus ∂x ∞

−∞

∞

1 dx R 2 x

dx

ϕ (x) =

ϕ (x) + ϕ (−x) − 2ϕ (0) x2

0

which can be derived by the following transformations including partial integrations as ∞

−∞

1 dx R 2 x

∞

ϕ (x) = − ∞

dx −R

−∞

∂

dx ∂x

= 0 ∞

=

dx

1 x

∂ ϕ (x) ∂x

∂ ϕ (x) − ∂x ϕ (−x) x

1 ∂ [ϕ (x) + ϕ (−x) − 2ϕ (0)] x ∂x

0 ∞

=−

dx 0

1 [ϕ (x) + ϕ (−x) − 2ϕ (0)] −x2

[ϕ (x) + ϕ (−x) − 2ϕ (0)] − x ∞

=−

dx 0

x=0

1 [ϕ (x) + ϕ (−x) − 2ϕ (0)] − 0 −x2

The following example clarifies that R x12 is not a positive definite generalized function. Example H.1

Let ϕ (x) = exp(−x2 ), a really positive definite function. We have

1 R 2 , exp(−x2 ) x

∞

=

dx

1 exp(−x2 ) + exp(−x2 ) − 2 exp(0) x2

dx

2 exp(−x2 ) − 1 x2

0 ∞

= 0

∞

=−

dx 0 ∞

= −4

0

4x exp(−x2 ) x

√ dx exp(−x2 ) = −2 π

which give a negative number as shown. Thus, R x12 is not positive definite.

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Applying the mathematical results, Eqs. (H.4) and (H.5), we have

W (p, q) =

 

du dv



−1

= (2π|)

−∞

dφ

∞

ro dro −∞

0

cos φ sin φ , ;r b b

×

π 2

= (4π|)

−∞

|ro | dro

−∞

dr W

i cos φ sin φ , ; r e(− | ro )(q cos φ b b

×

π 2

+ p sin φ−br)

∞

−1

= (4π|)

dφ −π 2

−∞

dr W ∞

cos φ sin φ , ;r b b

×

+ pro sin φ−bc)

∞

dφ −π 2

+ p.v)

dr W

i e(− | )(qro cos φ

∞

−1

i u v , ;c e(− | )(q.u  b b

ei | dc W

∞

−π 2

 

bc

(2π|)

π 2

−1

∞

−∞

i |ro | dro e(− | ro )(q cos φ

+ p sin φ−br)

Now applying Eq. (H.6), we obtain π 2

∞

−1

dφ

W (p, q) = (4π|)

−π 2

×2

1 |

∂

−∞

∞

−1

dφ

= (4π|)

−π 2

∂

1 |

−∞

dr W

−1

= | (4π|)

1 (q cos φ + p sin φ − br)

1 |

1 (br − q cos φ + p sin φ)

∞

dφ −π 2

1 |

cos φ sin φ , ;r b b

∂ P (br − q cos φ + p sin φ) π 2

2

cos φ sin φ , ;r b b

∂ P (q cos φ + p sin φ − br) π 2

×2

dr W

−∞

dr W (cos φ, sin φ; r) 2

1 ∂ P ∂r (r − q cos φ + p sin φ)

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Radon Transformation of Phase Space Functions π 2

∞

−1

2

= | (2π|)

dφ −π 2

× r¯o2

781

W (cos φ, sin φ; r) r¯o

dc −∞

1 ∂ P ∂c c − r¯o (q cos φ + p sin φ) π 2

=

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∞

−1

r¯o2 | (2π)

dφ −π 2

1 ∂ P ∂c c − r¯o (q cos φ + p sin φ)

dc −∞

× W (¯ ro cos φ, r¯o sin φ; c) π 2

−1

2

W (p, q) = | (2π|)

∞

dφ −π 2

× r¯o2

−∞

W (cos φ, sin φ; r) r¯o

1 ∂ P ∂c c − r¯o (q cos φ + p sin φ) π 2

=

dc

−1

r¯o2 | (2π)

∞

dφ −π 2

dc −∞

1 ∂ P ∂c c − r¯o (q cos φ + p sin φ)

× W (¯ ro cos φ, r¯o sin φ; c) where r¯o is now an arbitrary fixed number, used above to restore the variable c. A more symmetrical representation of the inversion formula can be obtained by integrating with respect to the variable c   ∞   i bc u v −1 , ;c e(− | )(q.u + p.v) ei | dc W W (p, q) = du dv (2π|)   b b −∞  ∞   ib u v v u , ;c e(− | )(q. b + p. b −c) = (2π|)−1 du dv dc W   b b −∞

Let

u c v v = c c =c

u =

then 2 du dv = |c| du dv b u v , ;c = W (u , v ; b) W b b c

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where the last line was obtained using Eq. (H.2). We have   ∞   i b W (u , v ; b) dc e(− | c)(q.u W (p, q) = (2π|)−1 |c|2 du dv   c −∞

= (2π|)

−1

= (2π|)

−1

×

−1

−1

−∞

+ p.v −b)

du dv W (u , v ; b) |b| 2|2

c˜ b

du dv W (u , v ; b) |b| 2|2

2

|˜ c|2 du dv W (u , v ; b) |b| 2|2

1 ∂ P u ∂˜ c c˜ − q. b − p. vb

= (2π|)−1 ×

i dc |c| e(− | c)(q.u

1 ∂ P ∂ c˜ − q. ub − p. vb c˜ − q. ub − p. vb

= (2π|) ×

du dv W (u , v ; b) |b|

1 ∂ P ∂ (q.u + p.v − b) (q.u + p.v − b)

= (2π|) ×

∞

+ p.v −b)

dudv W (u , v ; b) |b| 2|2

1 b

2

1 b

2

1 ∂ P ∂˜ c c˜ − q. ub − p. vb

Therefore, we also have ˆ (p, q) = (2π|)−1 ∆

1 b

ψu ,v ;b |b| 2|2

dudv ψu ,v ;b

2

1 ∂ P u ∂˜ c c˜ − q. b − p. vb

From Eq. (H.2), we also write |˜ c| u v W , ; c˜ |b| b b

W (u , v ; b) = ψu ,v ;b

ψu ,v ;b =

|˜ c| ψu v |b| b , b ;˜c

ψ ub , vb ;˜c

dudv |˜ c| W

u v , ; c˜ 2|2 b b

therefore we arrived at W (p, q) = (2π|)−1 ×

1 b

2

1 ∂ P u ∂˜ c c˜ − q. b − p. vb

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Radon Transformation of Phase Space Functions

ˆ (p, q) = (2π|)−1 ∆ × 2|2

1 b

783

2

dudv |˜ c| ψ ub , vb ;˜c

ψ ub , vb ;˜c

1 ∂ P u ∂˜ c c˜ − q. b − p. vb

Now let u b v v˜ = b

u ˜=

the we can write W (p, q) = 2|2 (2π|)−1 ×

1 ∂ P ∂˜ c (˜ c − q.˜ u − p.˜ v)

| = − |˜ c| π

ˆ (p, q) = − | |˜ ∆ c| π

d˜ ud˜ v |˜ c| W (˜ u, v˜; c˜)

d˜ ud˜ v R

d˜ ud˜ v R

(H.7) 1

(˜ c − q.˜ u − p.˜ v )2

1

W (˜ u, v˜; c˜)

ψu˜,˜v;˜c

(˜ c − q.˜ u − p.˜ v )2

ψu˜,˜v;˜c

(H.8)

(H.9)

Note that if we substitute the expression for W (˜ u, v˜; c˜) from Eq. (H.1), we should arrive at an identity which serves to check the last results, W (p, q) = −2|2 (2π|)−2 ×R =

d˜ ud˜ v |˜ c|

dp dq W (p , q ) δ (q .˜ u + p .˜ v − c˜)

1 (˜ c − q.˜ u − p.˜ v )2

dp dq W (p , q ) −

|˜ c| 2π2

d˜ ud˜ vR

1 (˜ c − q.˜ u − p.˜ v)2

δ (q .˜ u + p .˜ v − c˜)

To obtain the identity, the integral within the square bracket must obey the relation −

|˜ c| 2π 2

d˜ ud˜ v R

1

(˜ c − q.˜ u − p.˜ v )2 = δ (q − q ) δ (p − p )

δ (q .˜ u + p .˜ v − c˜)

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Indeed, comparing Eq. (38.4) with Eq. (H.8), we must have

W (p, q) = (2π|) | c| = − |˜ π

i du dv dc e(− | )(q.u+p.v−c ) W (u, v; c )

−1

d˜ ud˜ v

R

1 (˜ c − q.˜ u − p.˜ v)2

W (˜ u, v˜; c˜)

(H.10)

Upon retracing the derivation given above for the right hand side, we have | − |˜ c| π

d˜ ud˜ v

= (2π|)

= (2π|)

−1

−1

= (2π|)−1

1

R

W (˜ u, v˜; c˜) (˜ c − q.˜ u − p.˜ v )2   ∞   i u v , ;c e(− | c)(q.u +p.v −b) |c|2 du dv dc W   b b dudv

dudv

     

−∞

∞

−∞ ∞

−∞

  i |b| dc W (u, v; bc) e(− | )(q.u+p.v−bc)  dc W (u, v; c )

  

i e(− | )(q.u+p.v−c )

thus proving the identity of Eq. (H.10). Remark H.1

Write the representation of the delta-function in the form

1 δ (q .˜ u + p .˜ v − c˜) = lim √ exp a⇒0 a π

u + p .˜ v − c˜)2 − (q .˜ 2 a

Then we have

−

|˜ c| 2π 2

d˜ ud˜ v

R

1

(˜ c − q.˜ u − p.˜ v)2  |˜ c|   u d˜ v  2π2 d˜

1 = − lim √ a⇒0 a π   

× exp

δ (q .˜ u + p .˜ v − c˜) R (˜c−q.˜u1−p.˜v)2

−(q .˜ u+p .˜ v−˜ c) a2

2

      

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Now d˜ v

R

1 (˜ c − q.˜ u − p.˜ v )2

∞

=− =

p p

exp

u + p .˜ v − c˜)2 − (q .˜ a2

1 1 R p (q.˜ u + p.˜ v − c˜)

d˜ v −∞ ∞

∂ exp ∂˜ v

2 (q .˜ u + p .˜ v − c˜) 2 a (q.˜ u + p.˜ v − c˜)

d˜ v −∞

u + p .˜ v − c˜)2 − (q .˜ a2

u + p .˜ v − c˜)2 − (q .˜ a2

exp

Upon integrating with respect to u ˜ this gives p pq

∞

∞

d˜ v −∞

p =− pq =

pq pq

−∞ ∞

∞

d˜ v −∞ ∞

d˜ u −∞ ∞

d˜ v

pq pq

−∞

d˜ v d˜ u

R

− (q .˜ u + p .˜ v − c˜)2 ∂ exp ∂u ˜ a2

1 (q.˜ u + p.˜ v − c˜)

1 (q.˜ u + p.˜ v − c˜)2

u + p .˜ v − c˜)2 − (q .˜ a2

exp

1 (q.˜ u + p.˜ v − c˜)2

d˜ u

−∞

=

2q (q .˜ u + p .˜ v − c˜) a2 (q.˜ u + p.˜ v − c˜)

d˜ u

exp

exp

− (q .˜ u + p .˜ v − c˜)2 a2

− (q .˜ u + p .˜ v − c˜)2 a2

Therefore we have

d˜ u =

d˜ v R pq pq

d˜ v

1 (˜ c − q.˜ u − p.˜ v)2 d˜ u

R

exp

u + p .˜ v − c˜)2 − (q .˜ a2

1 (q.˜ u + p.˜ v − c˜)2

exp

− (q .˜ u + p .˜ v − c˜)2 a2

The above equality only holds if p p

q q

=1

In the generalized function sense, and for dimensional reasons, we may write −

|˜ c| 2π2

d˜ ud˜ v

R

1 (˜ c − q.˜ u − p.˜ v )2

δ (q .˜ u + p .˜ v − c˜) = δ (p − p ) δ (q − q )

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and therefore arrive at the following expression W (p, q) |˜ c| 2π 2

d˜ ud˜ vR

1

=

dp dq W (p , q ) −

=

dp dq W (p , q ) δ (p − p ) δ (q − q ) = W (p, q)

(˜ c − q.˜ u − p.˜ v )2

δ (q .˜ u + p .˜ v − c˜)

which is the identity we are seeking. Using the expression for ψu˜,˜v;˜c in Eq. (38.6), one can perform the integration indicated in Eq. (H.9) to obtain ˆ (p, q) |q q |∆ | = − |c| π

dudv

R

| = − |c| π

dudv

R

× =− × =− × =− ×

1 (c − q.u − p.v)2

dudv

(c − q.u − p.v)2

R

dudv

R

2

c 1 u q − exp i 2|v u 2π| |v|

2

1 (c − q.u − p.v)2

u 1 exp −i q2−q |v| 2|v 1 |c| 2π 2

ψu,v;c q

1

1 c u q − exp −i 2|v u 2π| |v| 1 |c| 2π 2

q ψu,v;c

2

exp i

c (q − q ) |v

1 (c − q.u − p.v)2

u c 1 exp −i (q − q ) (q + q ) exp i (q − q ) |v| 2|v |v 1 |c| 2π 2

dudv

R

1 (c − q.u − p.v)2

1 1 exp −i (q − q ) [(q + q ) u − 2c] |v| 2|v

Recall the Jacobian of a coordinate transformation, x = x(u, v) and y = y(u, v), is the determinant given by J(u, v) =

∂x ∂x ∂u ∂v ∂y ∂y ∂u ∂v

and with this the change of variables in the integration becomes dx dy f(x, y) =

|J(u, v)| f (x(u, v), y(u, v)) du dv

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Let x 2 x q =q+ 2 q =q−

(H.11)

then we have ˆ (p, q) |q q |∆

=−

1 |c| 2π2

R

1 (c − q.u − p.v)2

1 1 exp −i (−x) [(2q) u − 2c] |v| 2|v

× =−

dudv

1 |c| 2π2

dudv

R

1 (c − q.u − p.v)2

1 1 exp i x [qu − c] |v| |v

× Let y=

|v| (q.u + p.v − c) |c|

z=v

v c

y+

c

z

|v| |v| |c| q |c| p

=

0

u v

1

Then u v

  |z| 1  1 − |c| p  = |z| 0 |z| |c| q |c| q 

=

y+| zc |c |z| q |c|

−

z c

y+

c

z



pz q



z

The Jacobian is J (y, z) =

1

1

|z| q |c|

|z| q |c|

0

−

1

p q

=

1 |z| |c| q

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then we have ˆ (p, q) |q q |∆

=−

1 |c| 2π 2

× =−

dydz

1 |v| |c|

2

1 (c − q.u − p.v)2

exp i

R

1 (c − q.u − p.v)2 c y − pz z

1 x |z

1 1 1 exp −i px 2π 2 q |

× exp i =−

R

1 1 exp i x [qu − c] |v| |v

1 1 2π 2 q

×

=−

dudv

1 c 1 x | z z

dz

|c| x 2π| q

R

1 y2

y

1 1 1 exp −i px 2π 2 q |

1 = exp −i px |

dy

dz −π dz

1 c 1 x | z z

1 z2

Now let z=

πcx z 2|q

Then ˆ (p, q) |q q |∆

1 = exp −i px |

|c| x 2π| q

1 = exp −i px |

1 π2

dz

1 = exp −i px |

1 π2

dz

1 = exp −i px |

π2 π2

dz δ (z)

1 = exp −i px |

πcx 1 dz 2 2|q z 1 2|q πcx

1 z2

2

z

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The last two lines were obtain by writing 1 z − iε

1 = z2

1 z + iε

1 + iπδ (z) z

=

P

=P

1 + π2 δ (z) z2

P

1 − iπδ (z) z

Then we have, taking into account the transformation of Eq. (H.11), the expression ˆ (p, q) |q q |∆

1 = exp i p (q − q ) δ | 1 = exp i p (q − q ) | =

q +q −q 2 1 dv q q + v 2

1 q q+ v 2

1 dv exp i pv |

1 q− v q 2

1 q− v q 2

The Wigner distribution function, in terms of the density operator, follows from the expression in Eqs. (37.5) and transformation of Eq. (H.11), ˆ (p, q) W (p, q) = T r ρ ˆ∆ =

ˆ (p, q) |q ˆ∆ dq q | ρ

=

dq

dq

q |ρ ˆ |q

=

dq

dq

q |ρ ˆ |q

=

1 dx exp i px |

ˆ (p, q) |q q |∆ 1 exp i p (q − q ) δ | q−

x x ρ ˆ q+ 2 2

q +q −q 2 (H.12)

The last expression is the well-known expression of the Wigner distribution function, also known more generally as the Weyl transform of the density matrix.

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Appendix I

Introduction to Finite Fields

An arithmetic system with addition and multiplication, such that the operations are commutative, associative, distributive, and invertible (except that the number zero does not have a multiplicative inverse) is a defintion of a field6 . Note that the integers mod n, for n composite(factorable) or not a prime number, do not form a field since this would allow zero divisors a × b = n = 0(mod n). If xy = 0, and x and y are nonzero, x is a left zero divisor and y is a right zero divisor. Suppose x has inverse w, and x is a zero divisor. Write 0 = xy = w(xy) = (wx)y = 1y = y, which contradicts x being a zero divisor. Therefore, Invertible and zero divisor are mutually exclusive. The real number system is an example of a field with infinite number of elements. Every finite field has pn elements for some prime p and some integer n ≥ 1. This p is the characteristic of the field, which also meant that any element in the field sum up n times yield zero (mod n). For every prime p and integer n ≥ 1, there exists a finite field with pn elements. Only for n = 1 that the finite field has modular arithmetic with modulo p. On the other hand, the elements of GF(pn ) for n > 1 may be represented as polynomials of degree strictly less than n over the prime field, GF(p). Operations are then performed modulo R where R is an irreducible polynomial of degree n over GF(p), for instance using polynomial long division. The addition of two polynomials P and Q is done as usual. The multiplication may be simply understood by computing W = P Q as usual, then compute the remainder modulo R. Hence multiplication in a finite field is multiplication modulo an irreducible reducing polynomial used to define the finite field (i.e., it is multiplication followed by division using the reducing polynomial as the divisor–the remainder is the product). All fields with pn elements are isomorphic (that is, their addition tables are essentially the same, and their multiplication tables are essentially the same). This justifies using the same name for all of them; they are denoted by GF(pn ). The letters “GF" stand for “Galois field". For quantum information science and quantum computing applications, we are interested here in the number N of field elements which is a power of the prime number 2, by considering the two-state quantum bit or qubit. However, a number N of field elements which is a power of arbitrary prime number is also of interest for multi-level qubit. 6 We shall follow more or less the introduction to finite field found in the following website: http://www-math.cudenver.edu/~wcherowi/courses/finflds.html

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To generate GF(N = rn ), one would use a polynomial of degree n. When n > 1, GF(pn ) can be represented as the field of equivalence classes of polynomials whose coeﬃcients belong to GF(p). Any irreducible polynomial of degree n yields the same field up to an isomorphism. For example, for GF(23 ), the modulus can be taken as x3 + x2 + 1 or x3 + x + 1. Using the modulus x3 + x + 1, the elements of GF(23 )—written 0, x0 , x1 , ... can be represented as polynomials with degree less than 3. For instance, x3 = −x − 1 = x + 1 x4 = x x3 = x (x + 1) = x2 + x x5 = x x2 + x = x3 + x2 = x2 − x − 1 = x2 + x + 1

x6 = x x2 + x + 1 = x3 + x2 + x = −x − 1 + x2 + x = x2 − 1 = x2 + 1

x7 = x x2 + 1 = x3 + x = −x − 1 + x = −1 = 1 = x0

x8 = x1 = x x9 = x2 x10 = x3

We see that the set of elements 1, x, x2 do serve as finite-field basis elements. Consider the following several diﬀerent representations of the elements of a finite field. The columns are the power, polynomial representation, triples of polynomial representation coeﬃcients (the vector representation), and the binary integer corresponding to the vector representation (the regular integer representation). Power

Polynomial

Vector

Regular Integer

0 x0 x1 x2 x3 x4 x5 x6

0 1 x x2 x+1 x2 + x x2 + x + 1 x2 + 1

(000) (001) (010) (100) (011) (110) (111) (101)

0 1 2 4 3 6 7 5

The set of polynomials in the second column is closed under addition and multiplication modulo x3 + x + 1, and these operations on the set satisfy the axioms of finite field. This particular finite field is said to be an extension field of degree 3 of GF(2), written GF(23 ), and the field GF(2) is called the base field of GF(23 ). If an irreducible polynomial generates all elements in this way, it is called a primitive polynomial. For any prime or prime power q and any positive integer n, there exists a primitive irreducible polynomial of degree n over GF(q). On the other hand, using modulus x3 +x2 +1 for for GF(23 ), we will demonstrate that the two irreducible polynomials, x3 + x2 + 1 or x3 + x + 1, yield two isomorphic

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fields of GF(23 ). We have using modulus x3 + x2 + 1, x3 = −x2 − 1 = x2 + 1 x4 = x x3 = x x2 + 1 = x3 + x = −x2 − 1 + x = x2 + x + 1

x5 = x x2 + x + 1 = x3 + x2 + x = −x2 − 1 + x2 + x = x + 1

x6 = x (x + 1) = x2 + x x7 = x x2 + x = x3 + x2 = −x2 − 1 + x2 = −1 = 1 = x0

x8 = x1 = x x9 = x2 x10 = x3

We see again that the set of elements 1, x, x2 do serve as finite-field basis elements. The table which contains several diﬀerent representations of the elements of the corresponding finite field is as follows: Power 0 x0 x1 x2 x3 x4 x5 x6

Polynomial 0 1 x x2 x2 + 1 x2 + x + 1 x+1 x2 + x

Vector (000) (001) (010) (100) (101) (111) (011) (110)

Regular Integer 0 1 2 4 5 7 3 6

For any element c of GF(q), cq = c, i.e., the observation that their multiplicative groups are cyclic, and for any nonzero element d of GF(q), dq−1 = 1. Thus for GF(23 ), x8 = x1 = x, and x7 = 1 = x0 . There is a smallest positive integer p satisfying the sum condition e + e + e + ......e = 0 p times

for some element e in GF(q). This number is called the field characteristic of the finite field GF(q). The field characteristic is a prime number for every finite field, and it is true that (x + y)p = xp + yp over a finite field with characteristic p. This can easily be seen using the binomial theorem with the coeﬃcients evaluated modulo p. One can prove that for every prime p and integer n ≥ 1, there exists a finite field with pn elements. Consider the polynomial f(T ) = T q − T , where q = pn . It is possible to construct a field F (called the splitting field of f ), which contains Z/pZ, and which is large enough for f (T ) to split completely into linear factors: f(T ) = (T − r1 ) (T − r2 ) (T − r3 ) (T − r4 ) ... (T − rq )

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where each ri is an element of F . (The existence of splitting fields in general is discussed in construction of splitting fields.) These q roots are distinct, because f (T ) is a polynomial of degree q, and it has no repeated roots because its derivative is qT q−1 − 1 = −1

mod (p)

which has no roots in common with f (T ). Furthermore, setting R to be the set of these roots, one sees that R itself forms a field, as follows. Both 0 and 1 are in R, because 0q = 0 and 1q = 1. If r and s are in R, then (r + s)q = rq + sq = r + s, so that r +s is in R; the first equality above follows from the binomial theorem and the fact that F has characteristic p. Therefore R is closed under addition. Similarly, R is closed under multiplication and taking inverses, because (rs)q = rq sq = rs and (r−1 )q = (rq )−1 = r−1 . Therefore R is a field with q elements, proving the statement. I.1 I.1.1

Constructing Finite Fields GF(9)

Since 9 = 32 , the prime field must be GF(3) whose elements we will represent by 0, 1 and 2, and where addition and multiplication are done modulo 3. We seek an extension of degree 2 over the prime field, so our first task is to find a monic irreducible polynomial of degree 2 in GF (3)[x]. First, we shall employ the brute force procedure if the prime field is small, since we can in fact easily list all of the monic quadratics in this ring, they are: (1) (2) (3) (4) (5) (6) (7) (8) (9)

x2 x2 + 1 x2 + 2 x2 + x x2 + x + 1 x2 + x + 2 x2 + 2x x2 + 2x + 1 x2 + 2x + 2

The problem is to find the irreducible ones in this list. Clearly, any polynomial without a constant term is factorable (x is a factor), so the first, fourth and seventh can immediately be crossed out. For the remaining six polynomials, we can follow one of two procedures. We could take each in turn and substitute all the field elements of GF(3) for x, if none of these substitutions evaluates to zero, the polynomial is irreducible (i.e., it has no root in the field GF(3)). So, for example, substituting in x2 + 2 gives the values 02 + 2 = 2, 12 + 2 = 0 and 22 + 2 = 0, thus x2 + 2 factors, in fact x2 + 2 = (x + 1)(x + 2)(mod 3). On the other hand, the same

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procedure for x2 + 1 gives 02 + 1 = 1, 12 + 1 = 2 and 22 + 1 = 2 and so x2 + 1 is irreducible, i.e., no root in GF(3). The second possible procedure is to take all the linear factors (since in this case, we only want quadratic products) and multiply them in all possible pairs to get a list of all the factorable quadratics, removing these from our list leaves all the irreducible quadratics. So, (x + 1)(x + 1) = x2 + 2x + 1 (x + 1)(x + 2) = x2 + 2 (mod 3) (x + 2)(x + 2) = x2 + x + 1 (mod 3) implying that x2 + 1, x2 + x + 2 and x2 + 2x + 2 are the only irreducible monic quadratic polynomials in GF(3)[x]. We could now choose any one of these letting r be a zero of the chosen polynomial and write out the elements of GF(9) in its vector form representation using the basis {1, r}. This however does not give us the most useful representation of the field. Rather, we will use the fact that the multiplicative group of the field is cyclic, so if we can find a primitive element (i.e., a generator of the cyclic group) we will have a handy representation of the elements. The primitive elements are to be found among the roots of the irreducible polynomials (they cannot be elements of the prime field). The cyclic group we are after has order 8, so not every root need be primitive. For example, letting r be a root of x2 + 1, i.e., r2 + 1 = 0, so r2 = 2, we can write out the powers of r. r1 r2 r3 r4

= r, = 2, = 2r, = 2r(r) = 2r2 = 2(2) = 1

and so r has order 4 and does not generate the cyclic group of order 8, i.e., r is not a primitive element. On the other hand, consider µ a root of the polynomial x2 + x + 2, so that µ2 + µ + 2 = 0 or µ2 = 2µ + 1. Now the powers of µ give us: µ1 µ2 µ3 µ4 µ5 µ6 µ7 µ8

=µ = 2µ + 1 = µ(2µ + 1) = 2µ2 + µ = 2(2µ + 1) + µ = 2µ + 2 = 2µ2 + 2µ = µ + 2 + 2µ = 2 = 2µ = 2µ2 = µ + 2 = µ2 + 2µ = 2µ + 1 + 2µ = µ + 1 = µ2 + µ = 2µ + 1 + µ = 1

So µ is a primitive element and so we have represented the elements of GF(9) as the 8 powers of µ together with 0. Notice also that the bolded terms on the right are all the possible terms that can be written as linear combinations of the finite-field basis {1, µ} over GF(3). When working with finite fields it is convenient to have both

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of the above representations, since the terms on the left are easy to multiply and the terms on the right are easy to add. So for instance, if we wanted to calculate (2µ + 2)3 + µ + 2, we would do so in this way, (2µ + 2)3 = (µ3 )3 = µ9 = µ and so (2µ + 2)3 + µ + 2 = µ + µ + 2 = 2µ + 2 = µ3 . I.1.2

GF(8)

Since 8 = 23 , the prime field is GF(2) and we need to find a monic irreducible cubic polynomial over that field. Since the coeﬃcients can only be 0 and 1, the list of irreducible candidates is easily obtained. x3 + 1 x3 + x + 1 x3 + x2 + 1 x3 + x2 + x + 1 Now substituting 0 gives 1 in all cases, and substituting 1 will give 0 only if there are an odd number of x terms, so the irreducible cubics are just x3 + x + 1 and x3 + x2 + 1. Now the multiplicative group of this field is a cyclic group of order 7 and so every nonidentity element is a generator. Letting µ be a root of the first polynomial, we have µ3 + µ + 1 = 0, or µ3 = µ + 1, so the powers of µ are: µ1 µ2 µ3 µ4 µ5 µ6 µ7 µ8

=µ = µ2 = µ + 1 = µ + µ0 = µ2 + µ = µ2 + µ1 = µ2 + µ + 1 = µ2 + µ1 + µ0 = µ2 + 1 = µ2 + µ0 =1 =µ

which also shows that all the elements of GF 23 can be expanded in terms of the basis: µ0 , µ1 , and µ2 . Now suppose we had chosen a root of the second polynomial, x3 + x2 + 1, say, ω. We would then have ω 3 = ω2 + 1 and the representation would be given by ω1 ω2 ω3 ω4 ω5 ω6 ω7

=ω = ω2 = ω2 + 1 = ω2 + ω + 1 = ω +1 = ω2 + ω =1

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We know that these two representations must be isomorphic, the isomorphism is induced by µ ⇒ ω 6 . Since µ3 + µ + 1 ⇒ ω18 + ω 6 + 1 = ω3 + ω + ω2 + ω + 1 = ω3 + ω 2 + 1 then we have ω6

1

ω6

2

ω

= ω6 = ω

6

2

4

= ω6

2

ω6

5

= ω6

2

ω6

6

= ω6

2

6 7

⇒ω+1

+ 1 ⇒ ω2 + ω + 1

ω6

ω

I.2

6 3

= ω6 ⇒ ω2 + ω

+ ω6 ⇒ (ω + 1) + ω2 + ω = ω 2 + 1 + ω6 + 1 ⇒ ω + 1 + ω 2 + ω + 1 = ω2 +1⇒ω

=1

Constructing Bases of Finite Field

We will pedagogically build a basis and show that the finite field can be expressed in terms of this basis. Pick any finite field F of characteristic p. There are two possibilities. Either the field is equal to GF(p), or it has something besides GF(p). In the latter case, pick an element b2 not in GF(p). (We begin the index from 2 and not 1 for two reasons, namely, b2 not in GF(p) and for ease in the enumeration process in what follows.) Now form all possible combinations using GF(p) and b2 . They are 0 0 + b2 0 + 2b2 0 + 3b2 ....... 0 + p − 1b2

1 1 + b2 1 + 2b2 1 + 3b2 ....... 1 + p − 1b2

2 2 + b2 2 + 2b2 2 + 3b2 ....... 2 + p − 1b2

3 3 + b2 3 + 2b2 3 + 3b2 ....... 3 + p − 1b2

....... ....... ....... ....... ....... .......

p−1 p − 1 + b2 p − 1 + 2b2 p − 1 + 3b2 ....... p − 1 + p − 1b2

There are p rows, and each row has p entries, thus there are p2 elements in this table. All of the elements in the table are diﬀerent. For suppose otherwise. Then we would have two elements in the table equal to each other, say r1 + r2 b2 = s1 + s2 b2 or if we rearrange terms, r1 − s1 = (s2 − r2 )b2

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Either the diﬀerence s2 − r2 is zero or it is not. If it is zero, then s2 = r2 and it follows that also s1 = r1 so that we have not found diﬀerent elements in the table at all. Otherwise, the diﬀerence s2 − r2 is not zero and we can divide by it, because we are working with a field. In that case, we can solve for b2 to be b2 =

r1 − s1 s2 − r2

so we have succeeded in expressing b2 in terms of elements of GF(p). But we explicitly postulated that b2 was not in GF(p), so this expression for b2 cannot occur and we are led to conclude that no two elements of the table are alike. There are therefore p2 distinct elements. Again, there are two possibilities. Either this table with p2 is all of the finite field F , or there are more elements that do not appear anywhere in the table above. In the latter case, pick an element b3 that we haven’t listed yet, and consider all possible combinations c1 + c2 b2 + c3 b3 where the coeﬃcients c1 , c2 , and c3 are allowed to range over all possible values in GF(p). Since there are p choices for each coeﬃcient, independently of all the others, and there are three such choices to make, there are p3 possibilities we can form here. In a similar manner as above, we can prove that all the p3 possibilities are diﬀerent. If two are the same, we do the same of writing down what those two possibilities are, say r1 + r2 b2 + r3 b3 = s1 + s2 b2 + s3 b3 If the coeﬃcients r3 and s3 of b3 are the same in those two equal possibilities, then we can eliminate b3 from the equation, and conclude that all the other coeﬃcients are the same too, because we already know that that all expressions involving only GF(p) and b2 are diﬀerent if the coeﬃcients are diﬀerent (so we have not succeeded in writing an element of F in two diﬀerent ways). Otherwise, coeﬃcients r3 and s3 are diﬀerent and we can divide by their diﬀerence and express b3 in terms of GF(p) and b2 ; in other words b3 is in the previous table with p2 , which also cannot happen, as it goes contrary to what we postulated about b3 . So we have therefore succeeded in listing p3 diﬀerent elements. Either these p3 elements are all of the finite field (only for q = p3 ), or there are more elements we haven’t accounted for yet ( hence, q > p3 ). In the latter case, we pick another element b4 that we have missed, and use it to form all p4 possible combinations, which will all be diﬀerent by the exact same argument. This procedure is bound to stop, say upon finding bn (for q = pn ), because each time we are taking into account more and more elements of F , first p, then p2 , and so on, and finally pn since F has finitely many elements! At each step we have a power of a prime, and F must be of this size. Indeed, for counting purposes, the p elements of GF(p) correspond to digits and the number n corresponds to the number of bits. The exponent of the size of a finite field is known as the degree of the field.

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Thus it is possible to view F as a vector space over GF(p) with b1 , b2 , b3 , ..bn as the basis, where we may take b1 = 1 ∈ GF(p). In most applications, only primal bases are considered which is of the form 1, α, α2 , α3 , ...., αn−1 where α is the primitive element of GF(2n ). A primitive element is an element which does not lie in the subfield GF(p), i.e., such that 1, α, α2 , α3 , ...., αn−1 is the basis for GF (2n ) /GF (2). A finite field is considered a primitive finite field if the element x is a generator for the finite field. In other words, if the powers of x assume every nonzero value in the field, it is a primitive finite field. As it turns out, the GF (28 ) finite field with the reducing polynomial x8 + x4 + x3 + x + 1 is not primitive, although x + 1 is a generator in this field. The GF (28 ) finite field with the reducing primitive polynomial x8 + x4 + x3 + x2 + 1, however, is a primitive field. So every finite field must contain a prime subfield, and must have size a power of a prime. We now turn to the question of how to build finite fields on top of prime fields. I.3

Trace Operation on Elements of Finite Field

Let F = GF (p), K = GF (pn ). The trace operation on elements on finite field is a linear map from larger field, K, to the smaller field, F . In general, we don’t know any good way of enumerating all the elements of F other than by checking each element of K to see if it is a fixed point of the map α = αp . But there are eﬃcient maps from K to F through the trace operation defined as follows. Definition I.1

Trace: Let T rFK : K ⇒ F be the map defined by 2

n−1

T rFK (x) = x + xp + xp + .... + xp Claim I.1

Then,

• For all α ∈ K, T rK (α) ∈ F • For all a; b ∈ F and α; β ∈ K, T rFK (aα + bβ) = a · T rFK (α) + b · T rFK (β) Proof. Let us prove the second part first. First of all, note that for any x i i i and y in K and for any positive integer i, (x + y)p = xp + y p ; this fact is used very frequently in finite field calculations. In what follows we will omit superscript and subscript of T r when there is no confusion with respect to the subfield F . So, n−1

i

(aα + bβ)p

T r (aα + bβ) = i=0 n−1

i

i

i

ap αp + bp β p

= i=0 n−1

i

i

aαp + bβ p

= i=0

= aT r (α) + bT r (β)

i

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To see that T r(α) ∈ F for all α ∈ K, note that 2

n

2

n−1

(T r(α))p = αp + αp + .... + αp

= αp + αp + .... + αp +α 2 n−1 = α + αp + αp + .... + αp = T r (α)

(I.1)

Therefore T r(α) ∈ F . Example I.2 Let K = GF 24 , F = GF (2). Let the extension to a larger field K be via the irreducible polynomial f (x) = x4 + x3 + x2 + x + 1. Let α be a root of f (x) = 0. Every element β ∈ K can be expressed uniquely as β = b0 + b1 α + b2 α2 + b3 α3 . Then, we have T r (β) = b0 T r (1) + b1 T r (α) + b2 T r α2 + b3 T r α3 2

3

T r (1) = T rFK (x) = 1 + 12 + 12 + x2 = 1+1+1+1=0+0=0 T r (α) = α + α2 + α4 + α8 But α8 = α3 , since α5 = 1 mod (f (α)). Therefore T r (α) = α + α2 + α4 + α3

Now using the formula for the trace function, T r (α) = 1, since α is the root of f (x) = 0. We also have T r α2 = T r (α) = 1, using Eq. (I.1). To evaluate T r α3 , we note that α6 = α, which means α3 3

T r (α) = T r α

2

= α and hence T r

α3

2

=

= 1. Therefore, we end up with T r (β) = b1 + b2 + b3

On the other hand, if we have chosen to build K = GF 24 with the irreducible polynomial f (x) = x4 + x + 1 we will have T r (1) = 0 T r (α) = α + α2 + α4 + α8 = 1 + α2 + α8 = 1 + α2 + 1 + α2 =0 T r α2 = T r (α) = 0 This leaves only T r (β) = T r α3 ∈ F . Thus T r α3 is either 0 or 1. But if T r α3 = 0 then T r (β) = 0 identically, which contradicts the mapping theorem,

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namely the trace function maps K onto F . Hence T r α3 = 1 and therefore in this new coordinate system T r (β) = b3 ∈ F . I.4

Dual Basis

Let {α0 , α1 , ......αm−1 } be a basis for GF (2m ) over GF (2). For {α0 , α1 , ......αm−1 } we may consider primal bases of the form {1, α1 , α2 ......αm−1 } where α does not lie in GF (2). The corresponding dual basis is defined to be a unique set of elements β 0 , β 1 , ......β m−1 ⊆ GF (2m ) such that T r αi β j = δ ij By expanding the dual basis in terms of the primary basis in the form m−1

βj =

αk bk,j k=0

then m−1

T r αi β j = T r αi

αk bk,j

k=0 m−1

αi αk bk,j

= Tr k=0 m−1

T r (αi αk ) bk,j

= k=0

I.4.1

Construction of Dual Basis

Define a matrix m−1

A = (aij )i,j=0 aij = T r (αi αj ) and let B = A−1 = (bk,j )m−1 k,j=0 Then the dual basis β 0 , β 1 , ......β m−1 is given by m−1

βj =

αk bk,j k=0

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Proof.

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We may now write m−1

T r αi β j = T r αi

αk bk,j k=0

m−1

= Tr

αi αk bk,j k=0

m−1

=

T r (αi αk ) bk,j k=0 m−1

=

aik bk,j k=0

= AA−1

ij

= δ ij

Example I.3 Consider GF 23 with the basis 1, α, α2 where α is a primitive root which satisfies α3 = α + 1. We have T r (α) = α + α2 + α4 = α + α2 + α α3 = α + α2 + α (α + 1) = 2 α + α2 = 0, T r (1) = 1 + 1 + 1 = 1 T r (α) = T r α2 = 0 2

Thus if x ∈ GF 23 and x =

xi αi , then T r (x) = x0 . The matrix A is given by i=0

 α0 α0 α0 α1 α0 α2 A = T r  α1 α0 α1 α1 α1 α2  α2 α0 α2 α1 α2 α2   1 α α2 = T r  α α2 α3  α2 α3 α4   100 = 0 0 1 010 

The matrix B = A−1   100 = 0 0 1 010

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Therefore the dual basis is given by β0 = 1 β 1 = α2 β2 = α

I.5

Transformation of Coordinates

The matrix A and B can be used to change coordinates from the α-basis (the m−1

primal basis) to the β-basis (the dual basis). Let x =

xi αi be the primal i=0

m−1

basis expansion of x ∈ GF (2m ), and let x =

xj β j be the dual basis expansion, i=0

where xi and xj ∈ GF (2) or the set{0, 1}. Writing the components of the respective expansion as column vectors, x = (x0 , x1 , ........xm−1 )T x = x0 , x1 , ........xm−1

T

and since m−1

x=

m−1

xi αi = i=0

xj β j i=0

then we have, by virtue of the property of dual basis, m−1

xj = T r (xαj ) =

xi T r (αi αj ) i=0

which yields x T = x0 , x1 , ........xm−1 and m−1

xi = T r (xβ i ) = T r x

m−1

αk bk,i

=

k=0

m−1

T r (xαk ) bk,i = k=0

xk bk,i k=0

which yields xT = (x0 , x1 , ........xm−1 ). In terms of the column vectors, we have x = Ax x = Bx

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Bibliography

[1] E. I. Blount, in Solid State Physics, edited by F. Seitz and D. Turnball, volume 13, Academic, New York, 1961, See pp. 335-343, and 371. [2] G. H. Wannier, Rev. Mod. Phys. 34, 645 (1962). [3] B. Leaf, J. Math. Phys. 9, 65 (1968), For more comprehensive account of Leaf’s work, see the book by S. R. de Groot and L. G. Suttorp, Foundation of Electrodynamics (North Holland, 1972), pp.341-364. For other works which make use of the Weyl transform, although not concerned with multiband dynamics, see P. N. Argyres, Physics 2, 131 (1965); W. R. Theis, Z. Phys. 142, 503 (1955). [4] F. A. Buot, Phys. Rev. A 8, 1570 (1973), Equation (A3) of this paper contains an omission which is corrected in Phys. Rev. A 9, 2811 (1974). [5] G. H. Wannier and U. N. Upadhyaya, Phys. Rev. A 136, 803 (1964), The operator method of Roth and Blount [L.M. Roth, J. Phys. Chem. Solids 23, 433 (1962) and Phys. Rev. 126, 1636 (1962) accomplishes basically the same thing as that of Wannier and Upadhyaya in that they were able to calculate the Trace of F(H) to second order in the magnetic field. However, the use of Wannier functions as basis states generalizes and greatly simplifies the calculational technique. [6] C. P. Flynn and J. A. Rigert, Phys. Rev. B 7, 3656 (1973), Interest in impurity in bismuth is mentioned in this paper. [7] L. M. Roth, J. Phys. Chem. Solids 23, 433 (1962). [8] W. G. Henry and J. L. Rogers, Philos. Mag. 1, 237 (1956), See also, Can. J. Phys. 38, 908 (1960). [9] F. A. Buot and J. W. McClure, Phys. Rev. B 6, 4525 (1972). [10] E. I. Blount, Phys. Rev. 128, 2454 (1962). [11] L. G. Suttorp and S. R. de Groot, Nuovo Cimento A 65, 245 (1970). [12] N. H. March, Adv. Phys. 6, 1 (1957). [13] A. K. Das and E. de Alba, J. Phys. C 2, 852 (1969). [14] D. C. Mattis, The Theory of Magnetism, Harper and Row, New York, 1965, p. 186. [15] D. C. Mattis, The Theory of Magnetism I, Springer-Verlag, New York, 1981, p.219. [16] R. W. Zwanzig, Quantum Statistical Mechanics, Gordon and Breach, New York, 1966, pp. 139-172. [17] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integral, McGrawHill, New York, 1965. [18] W. Tobocman, Nuovo Cimento 3, 1213 (1952). [19] D. J. Candlin, Nuovo Cimento 4, 231 (1952). [20] J. Schwinger, Quantum Kinematics and Dynamics, Benjamin, New York, 1970. [21] D. E. Soper, Phys. Rev. D 18, 4590 (1978).

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[163] K. K. Gullapalli, A. J. Tsao, and D. P. Neikirk, Appl. Phys. Lett. 62, 2856 (1993). [164] K. K. Gullapalli, A. J. Tsao, and D. P. Neikirk, Appl. Phys. Lett. 62, 2971 (1993). [165] F. A. Buot, P. Zhao, H. L. Cui, D. Woolard, K. L. Jensen, and C. M. Krowne, Phys. Rev. B 61, 5644 (2000). [166] W. Kohn and L. J. Sham, Phys. Rev. A 40, 1133 (1965). [167] A. K. Rajagopal and F. A. Buot, Int. J. Quantum Chem. 56, 389 (1995). [168] A. K. Rajagopal and F. A. Buot, Generalized Functional Theory of Interacting Coupled Liouvillean Fields in Condensed Matter, in Density Functional Theory II: Relativistic and Time-Dependent Extensions, edited by R. F. Nalewajski, volume 181 of Topics in Current Chemistry, Springer-Verlag, New York, 1996. [169] H. Haken, Quantum Theory of Solids, Elsevier Science Publishing Company, New York, 1983. [170] F. A. Buot and C. M. Krowne, J. Appl. Phys. 86, 5215 (1999), See also, J. Appl. Phys. 87, 3189 (2000). [171] D. L. Woolard, H. L. Cui, B. L. Gelmont, F. A. Buot, and C. M. Krowne, Int. J. High-Speed Elec. Systems 13, 1149 (2003), See Sec. 3 of this paper. [172] D. A. McLaughlin, J. Math. Phys. 13, 1099 (1972). [173] V. A. Fedirko and V. V. Vyurkov, Phys. Stat. Sol. (b) 221, 447 (2000). [174] M. Kira, I. Tittonen, and W. K. L. andS. Stenholm, Phys. Rev. A 51, 2826 (1995). [175] A. H. Nayfeh, Introduction to Perturbation Techiques, Wiley, New York, 1981. [176] F. A. Buot, Phys. Reports 234(2 and 3), 73 (1993). [177] J. Schwinger, J. Math. Phys. 2, 1018 (1961). [178] L. V. Keldysh, Zh. Eksp. Theor. Phys. 47, 1515 (1964), [Sov. Phys.- JETP 20, 1018 (1965). [179] L. P. Kadanoﬀ and G. Baym, Quantum Statistical Mechanics, Benjamin, New York, 1962. [180] V. Korenman, Annals Phys. 39, 72 (1966), See also, P. Danielewicz, Annals Phys. 152, 239 (1984). [181] G. D. Mahan, Phys. Report 145, 251 (1987). [182] D. S. Chemla, Nature 411(6837), 549 (2001). [183] F. A. Buot, Phys. Rev. B 10, 3700 (1974), For crystals with inversion symmetry, there is an odd number of lattice points obeying the Born-von Karman boundary condition. [184] F. A. Buot, La Rivista del Nuovo Cimento 20(9), 1 (1997). [185] M. Schmutz, Z. Physik B 30, 97 (1978). [186] T. Arimatsu and H. Umezawa, Prog. Theor. Phys. 77, 53 (1987). [187] J. Frenkel, Wave Mechanics, Advanced General Theory, Clarendon, Oxford, 1934, p. 253, 435. [188] E. Gerjouy, A. R. Rao, and L. Spruch, Rev. Mod. Phys. 55, 725 (1983). [189] R. Balian and M. Veneroni, Annals Phys. 187, 29 (1988). [190] R. Jackiw and A. Kerman, Phys. Lett. A 71, 158 (1979). [191] A. K. Rajagopal and F. A. Buot, Phys. Rev. B 51, 1883 (1995). [192] T. Kato, T. Kobayashi, and M. Namiki, Suppl. Prog. Theor. Phys. 15, 3 (1960). [193] C. D. Dominicis and P. Martin, J. Math. Phys. 5, 14 (1964). [194] Y. Nambu, Phys. Rev. 117, 648 (1960). [195] T. Morita and K. Hiroike, Prog. Theor. Phys. 25, 537 (1961). [196] R. L. Stratonovich, Topics in the Theory of Random Noise I, Gordon and Brach, New York, 1963. [197] G. Lindblad, Commun. Math. Phys. 48, 119 (1976). [198] T. R. Kirkpatrick and J. R. Dorfman, J. Low Temp. Phys. 58, 301 (1985).

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[199] A. G. Aronov, Y. M. Gal’perin, V. L. Gurevich, and V. I. Kosub, Nonequilibrium Properties of Superconductors (Transport Equation Approach), in Nonequilibrium Superconductivity, edited by D. N. Langenberg and A. I. Larkin, North Holland, New York, 1986. [200] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw Hill, New York, 1971, See, for example, p. 183 for a discussion on Gorkov equations. [201] K. L. Jensen and F. A. Buot, Phys. Rev. Lett. 66, 1078 (1991). [202] S. Datta and R. Lake, Phys. Rev. B 44, 6538 (1991). [203] S. Datta, J. Phys. Condens. Matter 2, 8023 (1990). [204] F. A. Buot, J. Comp. Theor. Nanoscience 1, 144 (2004). [205] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, New York, 1995. [206] A. Stahl and I. Balslev, Electrodynamics of Semiconductor Band Edge, SpringerVerlag, New York, 1987. [207] L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover Publications, Inc., 1975. [208] H. C. Torrey, Phys. Rev. 76, 1059 (1949). [209] S. L. McCall and E. L. Hahn, Phys. Rev. 183, 457 (1969). [210] W. Quade, E. Scholl, and F. R. andC. Jacobini, Phys. Rev. B 50, 7398 (1994). [211] J. R. Madureira, D. Semkat, M. Bonitz, and R. Redmer, J. Appl. Phys. 90, 829 (2001). [212] D. Kremp, T. Bornath, M. Bonitz, and M. Schlanges, Phys. Rev. A 60, 4725 (1999). [213] E. I. Blount, Phys. Rev. 126, 1636 (1962). [214] E. N. Adams, Phys. Rev. 85, 41 (1952). [215] E. O. Kane and E. I. Blount, Interband Tunneling, in Tunneling Phenomena in Solids, edited by E. Burstein and S. Lundqvist, Plenum Press, New York, 1969. [216] D. R. Fredkin and G. H. Wannier, Phys. Rev. 128, 2054 (1962). [217] S. M. Sze, Physics of Semiconductor Devices, John Wiley & Sons, New York, 1969. [218] F. A. Buot, J. Phys. D: Appl. Phys. 30, 3016. [219] J. M. Ziman, Electrons and Phonons, Clarendon Press, Oxford, 1979. [220] C. Horie and J. Krumhansl, Phys. Rev. A 136, 1397 (1964). [221] F. A. Buot, J. Phys. C: Solid State Phys. 5, 5 (1972). [222] D. Benin, Phys. Rev. B 11, 145 (1975). [223] B. Perrin, J. Chemie Phys. 82, 191 (1985). [224] F. A. Buot, P. L. Li, and J. . Strom-Olsen, J. Low Temp. Phys. 22, 535 (1976). [225] M. B. Halpern, A. Jevicki, and P. Senjanovi’c, Phys. Rev. D 16, 2476 (1977), See Appendix. [226] S. S. Schweber, An Introduction to Relativistic Quantum Field Theory, Row, Peterson, New York, 1961, p.134. [227] A. P. Jauho and J. W. Wilkins, in Proceedings of the Third International Conference on Hot Carriers in Semiconductors, 1981, [J. Phys. (Paris) Colloq. 42, C7-301 (1981)]. [228] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, Englewood Cliﬀs, New Jersey, 1963.

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Index

geometric phase bosonization, 181

acoustic modes optic modes, 43 action eﬀective super-action, 371 algorithm discrete logarithms, 705 hidden subgroup problem , 707 mapping of quantum hidden subgroup algorithm, 708 period finding, 702 phase estimation algorithm, 695 quantum search algorithm, 703 analytic continuation virtue of over-completeness, 79

canonical operators, 6 characteristic functions, 98 Christoﬀel symbol, 159 coherent states diagonal representation, density operator, 82 overcomplete basis states, 68 commutation relation discrete translation operators, 534 generalized Pauli-matrix operators, 481 line projector and translation operator, 479 connected diagrams graph theoretical approach, 379 Cooper pairing, 389 correlation function supercorrelation function, 379 correlation functions retarded and advanced Green’s function, 237 covariant derivative, 158

Baker-Campbell-Hausdorf operator identity, 15 beam splitters and Hadamard matrix, 570 Berry’s phase, 169 binary-string encoding points in discrete phase space dual basis, 533 biorthogonal bases, 65 biorthogonal basis, 20 Bloch equations RTD THz sources, 343 two-state system, 401 Bloch sphere, 491 experimental quantum control, 507 quantum control for a qubit, 498 rotation, 497 Boltzmann equation phonon Boltzmann equation, 442 Boltzmann transport equation, 299 Born-Oppenheimer approximation, see

D-branes, 190 decoupling energy bands, 14 hierarchy, 382 renormalization procedure, 385 density operator line projector expansion, 465 point projector expansion, 447 discrete phase space generalized Pauli-matrix operator basis, 483 811

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Nonequilibrium Quantum Transport Physics in Nanosystems

discrete phase space viewpoint Bell basis, 660 Bell basis states, 667 correlated basis states entangled basis, 666 multi-state particle entangled basis, 665 quantum computing, 659 teleportation, 662 three-qubit entangled basis, 670 Greenberger-Horne-Zeilinger state, 671 discrete Wigner function single qubit, 520 two qubits, 528, 556 discrete Wigner function reconstruction use of intersecting line states, 517 discreteness, Planck scale, 189 displacement operator generator of coherent states, 74 symmetric, 16 dissipation kernel noise kernel mass kernel, 652 doubling of Hilbert-space operators, 363 dual basis construction of dual field basis, 533 e-symmetrized potential, 378 eﬀects of measurements create entanglement, 574 decoherence, 573 destroy entanglement, 574 electrodynamics, 51 classical gauge theory, 182 covariant formulation, 54 quantization, gauge fixing, 187 virtual photon exchange, 53 electron and ‘defect’ representation tables, 732 electron-phonon supervertex, 432 energy-band dynamics eﬀective Hamiltonian, 135 eﬀective Hamiltonian in a magnetic field, 145 lattice Weyl transform, 124 second quantization scheme, 138 entanglement fidelity, 592 Feynman diagrams Coulomb pairing, 758

direct and exchange, 756 electron-phonon, 768 phonon-electron diagrams, 773 finite field basis expansion, 531 fluctuation-dissipation theorem, 638 four probe measurement, 204 gauge theory classical, 182 quantization gauge fixing, 187 Yang-Mills action, 186 general quantum transport equations, 390 generalized Bloch equations, 396 generalized optical Bloch equations, 397 generalized semiconductor Bloch equations, 417 generalized coherent states squeezing, 110 generalized coherent wave theory, 415 generalized dense coding, 685 generalized four-probe conductance, 209 generalized measurements, 579 generalized quantum distribution, 379 generation of a striation eigenstates, 542 geometric phase holonomy on fiber bundle, 164 Green’s function in mesoscopic physics multi-probe current formula, 268 tight-binding recursive technique, 244 guiding principle informatics in mechanics, 4 Hadamard matrix transformation function, 486 harmonic oscillator ladder operators, 29, 64 second quantization, 25 wavefunction, 62 holonomy on fiber bundle, 170 Husimi distribution, see Wigner distribution smoothing, 105 Husimi trajectories, 323 hyperbolic secant pulse self-induced transparency, 413 inelastic scatterers, 212 instanton, 176 interference ands measurements

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Index

beam splitter and Hadamard matrix, 569 in quantum teleportation, 574 intersecting lines at a phase space point, 457 Joule heating, 219 Kraus representation theorem, 576 Kubo-Martin-Schwinger condition, 637 Lagrangian linear chain, 26 Lamb shift, 633, 650 Landauer formula barrier conductance, 203 contact resistance, 203 lattice dynamics Hamiltonian, 33 normal coordinates, 32 lattice finite fields discrete phase space dynamics, 477 Lindblad equation harmonic oscillator, 606 phase damped harmonic oscillator, 609 line states two qubits , 535 line states for a two-state system diagonal striation, 512 horizontal striation, 512 vertical striation, 512 Majorana fermions, 180 master equation, 598 Lindblad master equation, 599 memdiode memristor, 324 memristor, 325 mesoscopic conductance formulas applications integral and fractional quantum Hall eﬀect, 228 persistent current in normal metal loop, 230 universal conductance fluctuations, 230 combined elastic and inelastic scatterers, 220 multichannel, 204 mesoscopic physics

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inelastic coherence length Lφ , 198 scattering channels, asymptotes, 200 mixed q-p representation coherent states, 21 coherent-state basis, 66 mutually-unbiased bases, 512 intersection of line projectors, 472 nanodevice physics time-dependent correlations lattice Weyl transform, 295 nanotransistors, 281 no cloning theorem, 506 noise correlator, 641 non-Hermitian operators harmonic oscillator, 24 in quantum field theory, see canonical operators nonconserving fermion processes, 421 nonequilibrium self-energies electron-phonon, 766 first order contribution, 742 four-point vertex function, 742 phonon-electron interaction, 771 normal coordinates Hamiltonian, 48 operator indentities, 87 operator ordering antinormal ordering, 92 normal ordering, 91 symmetric ordering, 98 operator space generalized Pauli projector Pp ,q ,c , 454 operator basis ˆ (p, q), 447 ∆ point projector as sum of intersecting lines, 456 transformation function ˘ (p, q) , 456 T r Yu,v ∆ optic modes, see acoustic modes P-function, 104 pairing dynamics, 396 parallel transport, see geometric phase connection, 160 particle/hole creation vacuum condensate, 176

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Nonequilibrium Quantum Transport Physics in Nanosystems

path integral quantum field theory, 148 Schroedinger wave mechanics, 146 with Grassmann canonical field variables, 724 Pauli master equation, 606 Pauli spin matrices Wannier function and Bloch function, 485 perturbation expansion density operator, 619 first order, 620 second order, 624 phase randomization, 216 phase space transport equation lattice Weyl transformation, 392 phonon super-Green’s function, 435 phonons, 35 points in a striation, 529 points in discrete phase space, 532 POVM, 582 distinguishing non-orthogonal states, 584 power-of-prime finite fields generalized Pauli-matrix operators dual basis, 509 Q-function, 104 quantized radiation field in two-level system, 613 quantum channels, 589 amplitude-damping channel, 595 depolarizing channel, 589 phase damping channel, 593 quantum control, 498 Bloch sphere, 507 quantum nets, 564 quantum operation, 575 quantum search algorithm, 702 quantum superfield theory ‘scattering channels’, asymptotes, 369 characteristic functional, 378 nonequilibrium supercorrelation functions, 375 quantum field superoperator, 372 Rabi frequency, 405 Rabi oscillation, 409 Radon transform, 461 transformation property, 470

rectangular exciting light pulse, 410 Redfield equation, 643 relaxation-time approximation, 301 resonant tunneling device hysteresis loop, 302 intrinsic bistability, 301 intrinsic oscillating current, 303 numerical results coupled rate equations, 352 RTD memory device, 325 RTD Tera-Hertz sources, 338 RTD THz sources limit-cycle solutions, 354 unified stability analysis for type I & II, 351 rotating wave approximation, 406 Schrödinger equation, 11 Schwinger equation eﬀective Schrödinger equation Kohn-Sham density functional theory, 381 Gross-Pitaevskii equation, 381 Shor’s algorithm order-finding algorithm, 693 quantum Fourier transform, 687 single-electron eﬀects, 233 small-ring structures, 206 striation construction two qubits, 529 string theory discreteness, Planck scale, 190 superdense coding, 680 superfield theory quantum superfield theory, 366 superoperator, 363 superposition principle, 5 symplectic transformation, 116, 120 teleportation schemes GHZ channel, 673 GHZ channel and ancilla, 674 two qubit teleportation, 676 three-terminal devices, 273 trace formulas T rAn , 130 translational covariance point projector, 467 Wigner function, 469 two-probe measurement, 204

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Index

uncertainty principle canonical variables, 7 minimum uncertainty, 69, 79 vacuum ‘ray’ line projectors single qubit, 520 vacuum ‘ray’ states ‘Beau’ striation, 552 ‘Belle’ striation, 550 diagonal striation, 547 horizontal striation, 544 vertical striation, 541

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van Hove singularity, 45 variational principle Liouville space, 368 von Neumann measurements projective measurement, 571 wave function coherent state, 78, 79 wavefunction guided nanostructures, 233 Wigner distribution, 100 Wigner trajectories, 319