Physics Reports vol.304

Physics Reports 304 (1998) 1—87 Finite-temperature excitations in a dilute Bose-condensed gas Hua Shi, Allan Griffin* D...

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Physics Reports 304 (1998) 1—87

Finite-temperature excitations in a dilute Bose-condensed gas Hua Shi, Allan Griffin* Department of Physics, University of Toronto, Toronto, Ont., Canada M5S 1A7 Received July 1997; editor: A.A. Maradudin

Contents 1. Introduction 1.1. History of BEC studies 1.2. Theories of weakly interacting Bose gases 1.3. Summary 2. Green’s function formalism for a Bosecondensed gas 2.1. Imaginary frequency Green’s functions 2.2. Bogoliubov prescription for a Bosecondensed system 2.3. Dyson—Beliaev equations 2.4. Real-time Green’s functions 2.5. The Bogoliubov approximation 3. The first-order Popov approximation 3.1. The HFB self-energies 3.2. The Popov self-energies in terms of the t-matrix 3.3. Normal phase (¹'¹ ) # 3.4. Bose-condensed phase (¹(¹ ) # 4. The many-body t-matrix approximation 4.1. The many-body t-matrix 4.2. Analytical results for the t-matrix 4.3. Numerical results for the t-matrix 4.4. Assessment of the many-body t-matrix approximation

4 4 7 9 10 11 13 15 19 21 22 23 25 28 29 36 36 40 42 42

5. The second-order Beliaev—Popov (B—P) approximation 5.1. The B—P self-energy diagrams 5.2. Connections to other theories 5.3. Evaluation of the B—P self-energies 5.4. Energy of excitations in the B—P approximation 6. Calculations at ¹"0 using the B—P approximation 6.1. The B—P self-energies at ¹"0 6.2. Quasiparticle spectrum 7. Discussion of the ¹O0 B—P approximation 7.1. The k"0 limit 7.2. Expansions of RT(k, u"E #d) for k small k 7.3. Damping of phonons at high temperature 8. Concluding remarks Appendix A. Scattering theory and the t-matrix for Bose systems A.1. Multiple scattering in vacuum A.2. Ladder diagrams A.3. Connections between fI and C References Note added in proof

* Corresponding author. Tel.: #1 416 978 5199; fax: #1 416 978 2537; e-mail: [email protected]. 0370-1573/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 1 5 - 5

44 45 50 54 59 62 62 64 68 70 71 76 78 79 80 81 83 85 87

FINITE-TEMPERATURE EXCITATIONS IN A DILUTE BOSE-CONDENSED GAS

Hua SHI, Allan GRIFFIN Department of Physics, University of Toronto, Toronto, Ont., Canada M5S 1A7

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

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Abstract We present a systematic account of several approximations for the Beliaev self-energies for a uniform dilute Bose gas at finite temperature. We discuss the first-order Popov approximation, which gives a phonon velocity which vanishes as Jn (¹), where n (¹) is the Bose condensate density. We generalize this Popov approximation using a t-matrix calculated 0 0 with self-consistent ladder diagrams. This analysis shows that this t-matrix becomes very temperature-dependent and vanishes at ¹ , in agreement with Bijlsma and Stoof (1995). Finally, we make a detailed study of the Beliaev—Popov (B—P) # approximation for the self-energies which are second order in the t-matrix. We work out the contribution from each individual diagram and give formal expressions for the self-energies and the excitation energy spectrum valid at arbitrary temperature. Careful attention is given to all infrared-divergent contributions. We rederive the well-known ¹"0 result of Beliaev (1958) for long-wavelength excitations. The analogous evaluation of the finite-temperature B—P self-energies is given in the low-frequency and long-wavelength limit. The corrections to the chemical potential, excitation energy and damping are all found to be proportional to the temperature near ¹ . All infrared divergent terms are shown to cancel out # in physical quantities in the long-wavelength limit at arbitrary temperatures. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 05.30.Jp; 03.75F; 67.40.Db Keywords: Bose—Einstein condensation; Excitations; Dilute gas

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

1. Introduction The final achievement of Bose—Einstein condensation (BEC) in a trapped gas of Rb atoms in 1995 [1] was the result of several decades of research. It also marks the beginning of a whole new area of research into the properties of atomic Bose condensates. This subject is presently undergoing a world-wide explosion of research activity, both experimental and theoretical. As a result, the theory of a weakly interacting Bose-condensed gas has suddenly become the subject of intense research interest. The present paper is motivated by these exciting new developments, even though it deals entirely with uniform weakly interacting Bose gases while all the recent experimental studies are concerned with Bose gases in an anisotropic parabolic potential well (or trap), i.e. a non-uniform Bose condensate (For reviews, see Refs. [2,3]). Our emphasis is on developing an understanding of the elementary excitations of a uniform Bose-condensed gas at finite temperatures, where a substantial fraction of the atoms are no longer in the lowest energy single-particle state (the Bose condensate). In contrast to the well-understood situation at ¹"0, where all but a few percent of the atoms are in the condensate, very few theoretical studies have been concerned with weakly interacting gas with a strongly depleted condensate. We believe our results in this paper are the first step to understanding the analogous questions in the trapped atomic gases of current interest. 1.1. History of BEC studies It is useful to first give a brief history of BEC studies, starting with the initial prediction of Einstein in 1925 [4] for the case of a non-interacting gas of Bose atoms (A Bose atom has a net integer spin. All atoms with an even number of neutrons satisfy Bose statistics, which accounts for about 75% of the atoms in the periodic table). Bose—Einstein condensation in an ideal gas is a standard topic in all textbooks on statistical mechanics [5]. The momentum distribution of an ideal uniform Bose gas is given by f (ek)"1/(eb(ek~k)!1) , (1.1) B where ek"+2k2/2m, b"1/k ¹, and k is the chemical potential determined by the condition that B the sum over all possible states equals the total number of particles N. In a system of macroscopic size, the sum can be converted into an integral

P

N"+ f (ek)" D(e) f (e) de , k

(1.2)

where the density of states D(e) goes as &Je. If the volume » and the particle number N are fixed, the (negative) chemical potential k increases as the temperature ¹ decreases. At a certain temperature ¹ , k reaches zero, the maximum value it can reach in an ideal Bose gas, and thus the integral # in Eq. (1.2) gives N"2.612 »/j3(¹ ) , # where the thermal de Broglie wavelength j(¹) is j(¹),(2p+2/mk ¹)1@2 . B

(1.3)

(1.4)

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

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One may view ¹ defined by Eq. (1.3) as equivalent to the condition that the thermal de Broglie # wavelength j(¹) becomes comparable to the average separation d between particles (d&n~1@3). In this case, the quantum mechanical wave nature of the particles leads to all atoms becoming increasingly correlated. This is the essential physics behind BEC in an ideal Bose gas. As ¹ goes lower than ¹ , k is “pinned” at zero, and the right-hand side of Eq. (1.3) becomes # smaller than N. As Einstein [4] first pointed out, the additional particles enter the ground state to form what is now called the Bose—Einstein condensate. The number of particles in this lowest state is given by N "N!2.612 »/j3(¹)'0 if ¹(¹ . 0 #

(1.5)

From Eqs. (1.3) and (1.5), one obtains the temperature-dependent condensate fraction N (¹)/N"1!(¹/¹ )3@2 . 0 #

(1.6)

It is immediately seen that N "N at ¹"0, that is, all the atoms are in the lowest single-particle 0 state (k"0). To summarize, Bose—Einstein statistics described by Eq. (1.1) predicts that at a finite temperature given by Eq. (1.3), the lowest state becomes macroscopically occupied and a transition occurs to a new phase of matter characterized by the condensate fraction. It is the only phase transition in condensed matter physics that can occur in the absence of interactions, being entirely due to quantum mechanical effects related to Bose statistics. At the time of Einstein’s prediction, the nature of second-order phase transitions was not yet understood, and Uhlenbeck criticized Einstein’s analysis in his thesis in 1927. It was only in 1937 that it was generally agreed that Uhlenbeck’s objections to a phase transition in an ideal Bose gas were incorrect. Around the same time, superfluidity had been just discovered in liquid 4He below 2.17 K. Fritz London [6] immediately made the bold conjecture that this superfluidity was related to the form of Bose—Einstein condensation, suitably generalized to a liquid. The basis of London’s argument was that 4He was a Bose atom (S"0) and moreover if liquid 4He was treated as an ideal gas, with the density 2.20]1022 atoms/cm3, its BEC transition temperature given by Eq. (1.3) would be 3.15 K. This is remarkably close to the observed superfluid transition temperature. Essentially, London and Tisza argued that the superfluid characteristics were related to the motion of the Bose condensate, moving as a whole. The development and confirmation of these ideas about superfluid 4He took many years. Indeed, it has been controversial since the very successful Landau theory [7] of the low-temperature properties of superfluid 4He (based on a phenomenological model of a weakly interacting gas of phonon—roton excitations) made no explicit reference to any role of the Bose statistics, let alone a Bose condensate. Our current understanding of superfluid 4He as a Bose-condensed liquid has been extensively reviewed in a monograph by Griffin [8], to which we refer the reader. Some key theoretical papers (all at ¹"0) are: 1. The work of Bogoliubov [9], who showed the connection between a Bose condensate and a phonon-like dispersion relation for long-wavelength elementary excitations. This paper was restricted to a weakly interacting Bose gas but later it was realized that it already captured many general features of Bose-condensed systems.

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2. The work of Penrose and Onsager [10], who showed how to formulate in a precise way the concept of Bose-condensation in any interacting Bose system (gas or liquid), even when the condensate fraction was small (strongly depleted). 3. The general field-theoretical formulation of Beliaev [11] of how to work out the properties of a Bose-condensed system in the presence of a Bose condensate. Beliaev’s work also first emphasized the key role of the phase of the Bose-order parameter and its relation to the superfluid velocity. These papers laid the foundation for intensive theoretical work on the general theory of interacting Bose-condensed system in the period 1958—1965. The motivation was always to understand the excitations and other properties of superfluid 4He [8], but the hypothetical case of a dilute weakly interacting Bose gas was often considered as an illustrative example of the very complex calculations. Since that period, the Green’s function treatment of a weakly interacting Bose-condensed gas at ¹"0 has become one of the standard topics in textbooks on many-body theory [12—14]. However, to a large extent, this topic has only been of research interest to a small number of many-body theorists. One of the reasons is that, apart from some general relations, it is very difficult to carry out quantitative calculations for superfluid 4He. In a sense, the characteristic features of the Bose condensate and Bose broken symmetry were hard to extract from the complicated dynamics which arise in any liquid. In view of the difficulties in dealing with a strongly interacting, dense system like superfluid 4He, it is not surprising that since 1960s, there has been an increasing experimental effort to achieve BEC in a dilute Bose gas. At the low temperature needed for BEC, most systems will form condensed phases (liquid or solid) due to the attractive inter-atomic interactions. Thus, one requires conditions that allow Bose condensation to occur rapidly, relative to the longer time scales needed for the competing phase changes. One of the most extensively studied candidates for BEC has been spin-aligned hydrogen (For an authoritative review, see Greytak in [3]. By magnetically aligning the spins of the hydrogen atoms, one obtains a system that remains a gas at even ¹"0. Moreover, its density can be varied over several orders of magnitude, ranging from a weakly to strongly interacting Bose gas, and with the possibility of BEC. So far (1998), experiments have not produced a Bose condensate in this system, due to the increasing tendency towards recombination to molecular hydrogen H (due to three2 body collisions involving spin flips) as the density needed for BEC is approached. However, in the last decade or so, the study of spin-polarized hydrogen stimulated much theoretical and experimental interest in achieving BEC in an atomic gas. Another system as a candidate for BEC has been excitons in optically pumped semiconductors (For a review, see Wolfe et al. in [3]). Excitons (a bound state of an electron and a hole) move like a weakly interacting gas in the crystal. Because of the small exciton mass (on the order of the mass of an electron), the BEC transition temperature of exciton systems can be as high as &100 K if the density of the exciton gas is 1019 cm~3 (easily achievable with laser pulses). Excitons have a finite lifetime due to recombination, but in high purity Cu O crystals, these lifetimes can be as long as 2 10~3 s, which are enormous compared to other relaxation times (&10~8 s) involved. While this finite exciton lifetime is a complication, the resulting decay luminescence turns out to be a very direct measure of the kinetic energy of the decaying exciton and hence can be used to detect the presence of an exciton Bose condensate (excitons with zero kinetic energy). The theory of this decay

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

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spectrum has been worked out by Shi et al. [15]. In 1993, Lin and Wolfe [16] announced the first convincing evidence of BEC transition of paraexcitons (S"0) in a stressed high-quality Cu O 2 crystal. However, due to the current poor knowledge of exciton—exciton interactions, and the complications due to various recombination processes and exciton—phonon interactions, the theoretical analysis of the decay luminescence data is quite complicated and much work remains to be done [15,17]. In passing, we note that the work of Shi et al. [15] shows that the decay luminescence is a direct probe of the single-particle Green’s function of a Bose gas. Very few experimental techniques do this. The search for BEC in a dilute gas of alkali atoms has also been actively pursued for about a decade [2]. In these experiments, alkali atoms are first cooled and trapped by laser beams, then loaded into a magnetic trap, where a selection of hyperfine spin state is made so that all the atoms in the trap have the same spin. The gas is then further cooled by an evaporation technique, involving selective rf pulses which flip the spins of atoms with relatively high kinetic energy, which are then expelled from the trap. Due to a large scattering cross section, the remaining atoms quickly re-thermalize at a lower temperature, populating the lower levels of the parabolic well. Therefore, one obtains a lower temperature and higher density (i.e. at the center of the trap) at the same time. The first successful observation of BEC was made with 87Rb atoms [1] (which have 87 nucleons and 37 electrons, and thus obey Bose statistics) at temperatures of the order of 10~7 K. Shortly after, BEC condensates were also produced using 23Na atoms [18] with much larger numbers of atoms (&106) and hence a higher transition temperature (&10~6 K). This field of research is expanding very rapidly at the present time. 1.2. Theories of weakly interacting Bose gases With this brief history in mind, one can understand the special excitement among theorists when BEC was finally achieved in a gas of atoms. Suddenly all the theoretical calculations on dilute Bose gases over the last 40 years had a direct significance, rather than as tentative steps to understanding liquid 4He. Moreover, the experiments involve lowering the temperature of the trapped atomic gases through the BEC transition to well below ¹ . As a result, these new experimental studies have # also emphasized the need to understand the finite temperature properties of Bose-condensed systems. With few exceptions, most of the available theoretical literature on Bose gases is restricted to ¹"0. As we have noted earlier, the theory of excitations at ¹"0 is simple because one can assume that all atoms are in the condensate and hence the single-particle excitations only involve atoms going in and out of this condensate. This problem was first solved by Bogoliubov [9] for a uniform gas (It is straightforward to extend this theory to a trapped Bose gas [19]. The latter is usually referred to as the time-dependent Gross—Pitaevskii theory [20,21]. In his classic paper, Bogoliubov [9] presented a theory of a dilute weakly interacting Bose gas at temperatures far below the transition temperature. A major result of Bogoliubov’s calculation was that only a small fraction of the atoms were removed from the condensate at ¹"0 due to interactions. Specifically, the number of particle in the condensate, N , is given by 0 N "N[1!8(na3/p)1@2] , (1.7) 0 3

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where n"N/» is the density and a is the s-wave scattering length. For a dilute Bose gas, we have a;n~1@3 or na3;1. In spite of this small interaction-induced “depletion”, the Bogoliubov model shows the interactions in the presence of a condensate lead to an acoustic (or phonon) excitation spectrum at long-wavelengths. More precisely, Bogoliubov found that for wavevector k;mc/+, the excitation energy is given by E K+ck , k with the phonon velocity c"(4p+2a/m (n/m))1@2 .

(1.8)

(1.9)

Obviously, this phonon spectrum collapses if a"0. All of the low-temperature thermodynamics of a Bose-condensed system follow from this phonon dispersion relation. Gavoret and Nozie`res [22] were the first to show at ¹"0 that to all orders in perturbation theory, a Bose-condensed system always exhibited such a phonon spectrum with a speed given by usual thermodynamic derivative. Bogoliubov’s calculation also introduced the seminal idea of treating the atoms in the condensate classically. He noted that for atoms in the k"0 state, the Bose commutation relation gave aL aL s !aL s aL "1 . (1.10) 0 0 0 0 However, since the expectation values of the two terms on the left-hand side are each of the order of N , which is a very large number. Thus to an extremely good approximation, the operators of the 0 k"0 state commute with each other (as well as with all other kO0 operators). Therefore they can be treated as c-numbers, with aL "aL s "JN . In this Bogoliubov prescription, the condensate acts 0 0 0 like a classical particle reservoir, which non-condensate atoms can enter and leave via scattering [11]. Thus, the number of atoms is no longer a constant of motion. As an immediate consequence, one needs to include anomalous propagators (Green’s functions) representing two particles going into or out of the condensate. This approach forms the basis of the systematic application of quantum field theory to an interacting system of bosons due to Beliaev [11] in 1958, which was developed further by Hugenholtz and Pines [23], and by Gavoret and Nozie`res [22]. This led to a generalized Green’s function formalism which built in the crucial role of the Bose condensate, and allowed one to determine the general characteristics of the system, such as the excitation spectrum, the momentum distribution of atoms, etc. After this pioneer work, this field of study has been extensively developed and extended (for a review, see Ref. [8]). The first finite-temperature calculations for an interacting Bose gas were made by Lee, Huang and Yang [24], by means of a pseudopotential method for a gas of hard spheres. For the low-energy scattering, the details of the potential are not important and can be shown to be described by s-wave. The potential can be treated as an effective zero-range interaction v(x!x@)"(4p+2a/m)d(x!x@) .

(1.11)

In a hard-sphere gas, the scattering length a equals to the diameter of the atoms. Lee, Huang and Yang carried out calculations near ¹ to first order in the interaction assuming # a;j(¹), a;n~1@3 , (1.12) where j(¹) is the thermal de Broglie wavelength given by Eq. (1.4). For a review of the thermodynamic properties of a Bose gas near ¹ to first order in a, see Ref. [25]. #

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Following the work at ¹"0, many people also developed the analogous field-theoretic approach applicable at finite temperatures. We mention here specifically the papers of Hohenberg and Martin [26], Popov and Faddeev [27], Singh [28], and Cheung and Griffin [29]. In principle, these allow one to calculate the properties of an interacting Bose system, such as the thermodynamic potential, specific heat, condensate density, etc. As at ¹"0, such finite ¹ calculations are complicated by the subtle role of correlations induced by the Bose-broken symmetry. Moreover, even in a dilute gas, the finite ¹ case is difficult because the thermally induced depletion fraction is now large. One has to be careful in treating the condensate and excited atoms in a consistent fashion. Another well-known difficulty is the fact that even for regular repulsive interactions, perturbation theory for Bose-condensed systems diverges at small momenta [22,30]. That is to say, certain terms in the perturbation series are singular for k, uP0, a result which can be traced back to the fact that the single-particle excitations are phonon-like. These singularities have to be handled with care in order to obtain correct final results. Fortunately, the infrared divergences appear to cancel out in all physical quantities [22,30]. One major omission of the present paper is any discussion of the relation between the single-particle excitations (which we calculate) and the density fluctuation spectrum. The Bose broken symmetry is known (see Section 5 of Ref. [8]) to lead to the equivalence of these two kinds of excitations and this requirement imposes a very strong constraint on approximations [26]. 1.3. Summary The present paper is devoted to a systematic study of the finite-temperature excitations, using the Beliaev—Green’s function formalism. Our emphasis is on how to include the effects of the noncondensate atoms and is based on the second-order Beliaev—Popov self-energy diagrams. For clarity, we should note that while our calculations are based on second-order Dyson—Beliaev self-energies, we use renormalized propagators, and thus these self-energies really involve terms to all orders in the interaction. As with any many body calculation based on a selection of self-energy diagrams which involve self-consistent propagators, it is sometimes difficult to give convincing arguments about their relative importance. We have tried to do this as best as we are able to at the present time, following mainly the arguments given by Popov [31]. The thermal Green’s function method is briefly reviewed in Section 2, where we give the Dyson—Beliaev equations expressing the 2]2 matrix Green’s function in terms of the two kinds of self-energies. For illustration, we use this formalism to discuss the simple ¹"0 Bogoliubov model (this was first done by Beliaev [11] and is discussed in most many-body textbooks [12,13]). These techniques are then applied to a dilute Bose gas at finite temperature in Section 3, using the first-order Hartree—Fock approximation to deal with the interactions between the excited atoms plus the Bogoliubov approximation to account for interactions with the condensate, as first suggested by Popov [32]. Calculating the poles of the single-particle diagonal and off-diagonal Green’s functions, we find the quasiparticle energy spectrum has a structure similar to that of the ¹"0 Bogoliubov model, but now with a temperature-dependent condensate. This leads to the collapse of the low k phonon-spectrum to free particles as ¹P¹ . Like many other non-self# consistent first-order calculations, the Popov approximation leads to a jump in the condensate density at the transition point, and thus predicts a first-order phase transition. Following recent

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

work of Bijlsma and Stoof [33], in Section 4 we modify the Popov approximation by using a self-consistent t-matrix, rather than the ordinary one using unperturbed propagators. This self-consistent t-matrix is found to be very temperature-dependent, and vanishes at ¹ . It leads to # a smooth change of the condensate density at the BEC transition point. In Section 5, we discuss the more complicated second-order Beliaev—Popov approximation, and work out the formal expressions of the self-energies and their relations to the single-particle excitation energy. Using these results, Beliaev’s second-order calculations at ¹"0 are re-derived in Section 6. We evaluate the self-energies near the poles u&E (where E is the Bogoliubov k k excitation energy of Section 2) and expand our results in powers of k in the long-wavelength limit. Our explicit expressions of the self-energies involve infrared divergent terms. We show that these divergent terms cancel out exactly (as expected), leading to well-defined physical quantities, such as the quasiparticle energy and the single-particle spectral density. As expected, the energy spectrum of excitations in the Beliaev approximation is phonon-like in the long-wavelength limit. It also contains an imaginary part (damping) which goes as &k5. These second-order Beliaev selfenergies are the lowest approximation which involves damping of excitations. While all the final results of Section 6 are known, we hope our very detailed analysis will make Beliaev’s work [34] more accessible. The analogous evaluation of our expressions for the finite-temperature Beliaev—Popov selfenergies at small k and u is much more difficult than in the ¹"0 case. By expanding the self-energies in powers of k at u&E (where E is now the Popov excitation energy of Section 3) as k k in the zero-temperature case, we are taking the limit uP0 and kP0 at the same time. This is not appropriate at finite temperature because the additional thermal scattering terms involved in the self-energies require that one take the limit kP0 first. To overcome this difficulty, in Section 7 we introduce a small gap between the frequency and the Popov quasiparticle energy and expand the self-energy at u"E #d, rather than at u"E . This gap d is kept finite until the expansion in k k powers of k is done. We thus obtain the expansion for the self-energies in powers of k at finite ¹. These results allow us to calculate the single-particle excitation energy and damping, as well as the chemical potential by means of the Hugenholtz—Pines theorem [23]. We believe the results will be of considerable interest to the growing BEC community, who are interested in the analogous problems for trapped Bose gases. We have made an attempt, in an admittedly highly mathematical development, to discuss the physics involved in the diagrammatic analysis as well as to describe the intermediate steps in some detail. Without denying the brilliance of the papers by Beliaev at ¹"0 [34] and Popov at finite temperatures close to the transition [31,32], their accounts are written in a very terse style with many important features left for the reader to figure out. The present paper is also a status report of our current understanding of excitations in uniform Bose gases at finite temperature. We hope that it will provide the basis for future theoretical studies which will clarify several questions raised in the paper, as well as in generalizations to deal with excitations in trapped Bose gases. (For a recent review, see Ref. [60].)

2. Green’s function formalism for a Bose-condensed gas In this section, we briefly review the finite-temperature (or thermal) Green’s function formalism. This technique is the most effective way of calculating the equilibrium thermodynamic properties,

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as well as single-particle excitations of the system. Although this section does not contain anything new, it is included for the convenience of the reader and to introduce notation. In the Bosecondensed phase characterized by a macroscopic number of atoms in the zero-momentum single-particle state, it is convenient to regard the operators aL and aL s for this state as c-numbers. 0 0 This procedure, called the Bogoliubov prescription, is discussed in Section 2.2. Within this broken symmetry phase, one deals with anomalous averages and it is also necessary to introduce (Section 2.3) a 2]2 matrix Green’s function or propagator. Its diagonal elements are the normal Green’s functions, and off-diagonal elements are called the anomalous Green’s functions. We write down the Dyson—Beliaev equations for a Bose-condensed system [11,28], which give different kinds of Green’s function in terms of the 2]2 matrix self-energy. In Section 2.4, we review how the thermal Green’s function at imaginary frequencies are related to the real-time Green’s function by a simple analytic continuation, in which the single-particle spectral density function plays a central role. Finally, in Section 2.5, we illustrate this formalism by considering the Bogoliubov model approximation at ¹"0. 2.1. Imaginary frequency Green’s functions The system we are concerned with consists of N atoms obeying Bose statistics, enclosed in a box of volume » and interacting through the two-body potential v(x!x@). For simplicity, the atoms are assumed to have zero net spin. For a Bose-condensed system, it is particularly convenient to use the grand canonical ensemble, which allows for the possibility of a variable number of particles. In the second quantized formalism, the grand canonical “Hamiltonian” of the system is given by

P

C

D

+2 KK ,HK !kNK " d xtK s(x) ! + 2!k tK (x) 2m

P P

1 # dx dx@ tK s(x)tK s(x@)v(x!x@)tK (x@)tK (x) . 2

(2.1)

Here tK (x), tK s(x) are the boson field operators, satisfying the usual Bose commutation relations [tK (x), tK (x@)]"0, [tK (x), tK s(x@)]"d(x!x@) .

(2.2)

The chemical potential k is chosen so that SNK T"N. With the grand canonical Hamiltonian KK , we introduce the modified q-dependent Heisenberg picture for any Schro¨dinger operator OK (x) (see standard texts on many-body systems such as Refs. [12,13,35]: OK (xq),eKK qOK (x)e~KK q .

(2.3)

In particular, the field operators have a q-dependence given by tK (xq)"eKK qtK (x)e~KK q ,

(2.4)

tK s(xq)"eKK qtK s(x)e~KK q .

(2.5)

Comparing this representation with the standard real-time representation, the variable q can be viewed as an “imaginary” time q"it/+.

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The single-particle thermal (or imaginary time) Green’s function is defined as G(xq, x@q@),!Tr[eb(X~KK )¹ tK (xq)tK s(x@q@)]"!S¹ tK (xq)tK s(x@q@)T , q q

(2.6)

where b"1/k ¹ and ¹ is a q ordering operator, which arranges the smallest q to the right. The B q grand canonical ensemble average is denoted by S2T,Tr[eb(X~KK )2]

(2.7)

where X is the thermodynamic potential, defined as e~bX,Tr(e~bKK ) .

(2.8)

For most purposes, it is more convenient to work with the single-particle momentum representation. Assuming periodic boundary conditions, the operators tK (x), tK s(x) can be written in the single-particle momentum representation as tK (x)"1/J»+ aL ke*k > x, k

tK s(x)"1/J»+ aL kse~*k > x ,

(2.9)

k

where aL k, aL ks are the Bose annihilation and creation operators, respectively, for the single-particle state of momentum k. From Eq. (2.2) follows the Bose commutation rules for aL k and aL ks: [aL k, aL k ]"0, {

[aL k, aL ks ]"dk k . { ,{

(2.10)

In the single-particle momentum representation, we may write G(xq, x@q@) in Eq. (2.6) as 1 G(xq, x@q@),! + e*(k > x~k{> x{)S¹ aL k(q)aL ks (q@)T . { q »k k ,{

(2.11)

In the case of a uniform system governed by a time-independent Hamiltonian, G depends only on the differences x!x@ and q!q@ (see standard many-body texts such as Ref. [13]), with 1 G(x,q)" + e*k > xG(k, q) , » k

(2.12)

where G(k, q),!S¹ aL k(q)aL ks(0)T . q

(2.13)

It is easy to verify that for bosons, G(k,q) satisfies the famous periodic condition G(k, q)"G(k, q#b)

(2.14)

for !b(q(0, where the right-hand side of the equation gives the Green’s function for 0(q#b(b. This periodicity of G(k, q) in the variable q with period b leads immediately to the

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

13

following Fourier expansion: 1 G(k, q)" + e~*unqG(k, iu ) , n b *un b G(k, iu )" dq e*unqG(k, q) , n 0 pn u " , n"0,$2,$4,2 . n b

P

(2.15) (2.16) (2.17)

Here u is a Bose Matsubara frequency (which has the dimension of energy in our notation), and n because of Bose statistics, the integer n is restricted to even values. The Fourier component G(k, iu ) n is a function of the discrete Matsubara frequencies that are evenly distributed on the imaginary frequency axis. At ¹P0, we note that the spacing between the discrete Matsubara frequencies goes to zero. The thermal Green’s function defined in Eq. (2.6) can be used to evaluate the equilibrium thermodynamic properties of the system, such as the number of particles with a certain momentum. By definition (see, for example, p. 229 of [13]), 1 SnL kT,SaL ksaL kT"S¹ aL k(q)aL ks(q`)T"! + e*ungG(k, iu ) , (2.18) q n b *un where q` denotes the limiting value q#g as g approaches zero from positive values. The mean number of particles in the system is then given by

P

» dk + e*ungG(k, iu ) . (2.19) N(¹, », k)"N ! n 0 b (2p)3 n *u The ensemble average of any single-particle operator, as well as the mean potential energy is expressible in terms of G. Moreover, one can also relate the thermodynamic potential X(¹, »k) to the single-particle thermal Green’s function G [13,28], which enables us to calculate the general equilibrium thermodynamic properties of the system. The excited states of the system containing N$1 particles are related to the poles of the real-time Green’s function [13,36]. At finite temperature, however, the real-time Green’s function involves a very complicated perturbation expansion. In contrast, the thermal Green’s function as defined here has a simpler perturbation expansion, similar to that for the ¹"0 Green’s function. Fortunately, as we shall discuss in Section 2.4, the Fourier transforms of the thermal Green’s function and the real-time Green’s function can be uniquely related to each other through a simple analytic continuation. Thus, the imaginary frequency (or thermal) Green’s function can be used to calculate the single-particle excitations of the system. 2.2. Bogoliubov prescription for a Bose-condensed system In the preceding subsection, we have defined the single-particle temperature Green’s function. The remaining task is to calculate the Green’s function G. For a Bose system, this task is complicated by the possible phase transition to a Bose-condensed state, in other words, by the

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

spontaneous symmetry-breaking below a certain temperature ¹ . A more rigorous way of dealing # with the new phase is to explicitly include a symmetry-breaking term in the Hamiltonian (2.1),

P

1 ! dx[l*tK (x)#ltK s(x)] 2

(2.20)

where l is the Bose symmetry breaking field. The spontaneous symmetry breaking and a new condensed phase is signaled by the non-vanishing of the average [11,26] StK (x)T"U (x)O0 , 0

(2.21)

even in the limit of a zero off-diagonal field (lP0). We refer to the thesis of Talbot [37] for a detailed discussion of Green’s function in a Bose system where the symmetry breaking field is kept throughout. In this paper, however, we will assume the limit of lP0 and leave this term implicit. U (x) in Eq. (2.21) is often referred to as the macroscopic wavefunction of the condensate and in 0 general has an amplitude and a phase. In a uniform system and in the absence of any supercurrent, we can take U (x) to be real and independent of position. In this case, U equals to square root of 0 0 the condensate density n . We are then led to separate the Boson field operator into two parts: 0 tK (x),aL /»1@2#tI (x) , 0 tK s(x),aL s /»1@2#tI s(x) . 0

(2.22) (2.23)

Since the commutator of aL and aL s is unity, and is small compared with their product which is of 0 0 order N, we can replace operators aL and aL s by the c-number N1@2 in Eqs. (2.22) and (2.23). This 0 0 0 procedure, known as the Bogoliubov prescription, is appropriate when the number of particles in the zero-momentum state is a finite fraction of N. The error introduced is of the order O(»~1), and thus vanishes in the thermodynamic limit (NPR and »PR, with fixed density n"N/»). Clearly, the new field operators tI (x), tI s(x) describe the non-condensate atoms, and satisfy the Bose commutation relations (2.2) in the thermodynamic limit. The average of tI and tI s now vanishes as in the usual case of a normal system, namely StI (x)T"StI s(x)T"0 .

(2.24)

With the above Bogoliubov broken-symmetry prescription, the thermal Green’s function defined in Eq. (2.6) now separates into two parts: G(xq, x@q@)"!n #GI (xq, x@q@) 0

(2.25)

with the non-condensate Green’s function defined by GI (xq, x@q@),!S¹ tI (xq)tI s(x@q@)T . q

(2.26)

In momentum representation for kO0, we have GI (k, q)"!S¹ aL k(q)aL ks(0)T . q

(2.27)

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15

The separation of the condensate using Eqs. (2.22) and (2.23) and the Bogoliubov prescription modify the Hamiltonian in a fundamental way. The grand canonical Hamiltonian is now given by 1 7 KK "E !kN # + (e !k)aL ksaL k# + »K 0 0 »k k j E0 j/1 7 ,KK # + »K , 0 j j/1 where the interaction Hamiltonian »K separates into eight distinct parts: E "1n2»v(0) , 0 2 0 1 »K " n + v(k)aL kaL k , 1 2 0k ~

(2.28) (2.29)

(2.30) (2.31)

1 »K " n + v(k)aL ksaL s k , ~ 2 2 0k

(2.32)

»K "n + v(k)aL ksaL k , 3 0k

(2.33)

»K "n + v(0)aL ksaL k , 4 0k

(2.34)

n1@2 »K " 0 + v(q)aL ks qaL kaL q , ` 5 »1@2k q , n1@2 »K " 0 + v(q)aL ksaL qsaL k q , 6 »1@2k q ` , 1 + v(q)aL ks qaL ks qaL k aL k , »K " ` {~ { 7 2»k k q , {,

(2.35) (2.36) (2.37)

with

P

v(q), dx e*q > xv(x) .

(2.38)

These parts are represented by the vertex diagrams in Fig. 1. In a normal system (n "0), only 0 »K is present. We also note that KK has no term containing a single aL †k or aL k because these would 7 violate momentum conservation. This is in agreement with Eq. (2.24), which gives in momentum representation SaL kT"SaL ksT"0, (kO0) .

(2.39)

2.3. Dyson—Beliaev equations The fact that there exist interaction terms such as » in Eq. (2.31) implies the non-conservation 1 of particles in the system described by non-condensate operators aL k and aL †k, due to exchanges

16

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

Fig. 1. Eight distinct processes involved in the interaction »K . A solid line denotes aL or aL s, a wiggly line corresponds to a condensate atom or a factor n1@2, and a dashed line denotes the interaction v. 0

Fig. 2. Proper self-energies for a Bose-condensed system.

between non-condensate atoms and the condensate atoms. Nevertheless, if a proper self-energy is defined as a part of a Feynman diagram connected to the rest of the diagram by two noncondensate particle lines, then it is still possible to analyze the contributions to the Green’s function in a form similar to Dyson’s equation for interacting fermions. However, there are now three distinct proper self-energies, as indicated in Fig. 2. One type of self-energy has one particle line going in and one coming out (R ), similar to that for fermions. The other ones have two particle 11 lines either coming out (R ) or going in (R ), and reflect the new features associated with the 12 21 existence of a Bose condensate reservoir. Correspondingly, we must also introduce two new Green’s functions, G (k, q),!S¹ aL k(q)aL k(0)T , 12 q ~ G (k, q),!S¹ aL ks(q)aL s k(0)T . ~ 21 q

(2.40) (2.41)

G and G are usually called the anomalous Green’s functions, representing the disappearance 12 21 and appearance of two non-condensate particles, respectively. The normal Green’s function GI defined in Eq. (2.26) is denoted as G , representing the propagation of a single particle 11 G (k, q),!S¹ aL k(q)aL ks(0)T . (2.42) 11 q

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17

These three Green’s functions are shown in Fig. 3, where the arrows indicate the direction of momentum of the atoms involved. The Dyson equations for this system were first derived by Beliaev [11], which in frequency—momentum space are given by G (p)"G(0)(p)#G(0)(p)R (p)G (p)#G(0)R (p)G (p) , 11 11 11 21 21

(2.43)

G (p)"G(0)(p)R (p)G (!p)#G(0)(p)R (p)G (p) , 12 12 11 11 12

(2.44)

G (p)"G(0)(!p)R (p)G (p)#G(0)(!p)R (!p)G (p) . 21 21 11 11 21

(2.45)

For simplicity, we use the letter p to represent the four-dimensional vector (k,iu ). Unless otherwise n noted, we follow this convention in the rest of the paper. The Dyson—Beliaev equations Eqs. (2.43), (2.44) and (2.45) are illustrated diagrammatically in Fig. 4. The structure of these equations can be simplified by introducing a matrix operator [38]

CD

AK k,

aL k , aL sk

(2.46)

and, correspondingly, a 2]2 matrix Green’s function G(k,q),!S¹ AK k(q)AK ks(0)T . q

Fig. 3. Green’s functions for a Bose-condensed system.

Fig. 4. Dyson—Beliaev equations for a Bose-condensed system.

(2.47)

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

A single matrix equation can represent the three Dyson’s equations of Eqs. (2.43), (2.44) and (2.45), namely G(p)"G(0)(p)#G(0)(p)R(p)G(p) ,

(2.48)

with

C

G (p) 11 G (p) 21 G(0)(p) G(0)(p)" 0 G(p)"

C

C

D

G (p) 12 , G (p) 22 0

(2.49)

D

G(0)(!p)

,

D

R (p) R (p) 11 12 . R (p) R (p) 21 22 Here the unperturbed Green’s function is given by R(p)"

G(0)(p),G(0)(k, iu )"1/(iu !e #k) . n n k The matrix elements are not independent, with the following useful identities:

(2.50)

(2.51)

(2.52)

G (p)"G (!p), G (p)"G (!p) ; 22 11 12 21 R (p)"R (!p), R (p)"R (!p) . 22 11 12 21 Eq. (2.48) can be solved to give iu #e !k#R (!p) n k 11 , (2.53) G (p)" 11 [iu #e !k#R (!p)][iu !e #k!R (p)]#[R (p)]2 n k 11 n k 11 12 !R (p) 12 . (2.54) G (p)" 12 [iu #e !k#R (!p)][iu !e #k!R (p)]#[R (p)]2 n k 11 n k 11 12 These last two equations express the normal and anomalous Green’s functions in terms of the exact proper self-energies, and therefore are entirely general. They apply to any uniform Bose-condensed fluid, liquid or gas. We see that in Eqs. (2.53) and (2.54), both Green’s functions share the same poles. Moreover, it is easy to check that in the kP0 limit, this pole occurs at iu "0 if n k"R (0, 0)!R (0, 0) . (2.55) 11 12 In fact, Eq. (2.55) can be shown to be true to all orders in perturbation theory and is known as the Hugenholtz—Pines theorem [23]. It was first derived by Hugenholtz and Pines for the ¹"0 case by a direct diagrammatic analysis, and generalized to finite temperature by Hohenberg and Martin [26]. This theorem is very important because it shows, quite generally, the energy E(k) of single particle excitations (which are related to the poles of G ) vanishes at k"0; in other words, the ab excitations have no energy gap in the long-wavelength limit. The Hugenholtz—Pine theorem thus provides a criterion for ensuring such a gapless approximation [26].

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2.4. Real-time Green’s functions In Section 2.2, we pointed out that the thermal Green’s functions G can be used to calculate thermodynamic properties. Through its relation to the real-time Green’s function, it also can be used to determine the energy and lifetime of the states of a system when one adds or subtracts a single atom. As usual, we restrict ourselves to the case of a homogeneous system with a time-independent Hamiltonian. Then the eigenstates DmT of the grand canonical Hamiltonian KK and the momentum operator PK are the same, with KK DmT"E DmT , (2.56) m PK DmT"P DmT . (2.57) m In such a basis, the imaginary-frequency Green’s function G(k, iu ) can be written in a Lehmann n representation (see, for example, Ch. 9 of [13]), which yields the following exact spectral representation:

P

= du@ A(k, u@) , 2p iu !u@ ~= n where A(k, u) is the single-particle spectral density function defined by G(k, iu )" n

(2.58)

A(k, u),ebX+ Me~bEm(2p)3d[k!+~1(P !P )] n m mn ]2pd[u!(E !E )](1!e~bu)DSmDtK (x"0)DnTD2N . (2.59) n m Following directly from its definition, one can show quite generally that for a Bose system, A(k, u) has the following properties (see, for example, Ch. 9 of [13]: sgn(u)A(k, u)50 ,

(2.60)

P P

= du A(k, u)"1 , (2.61) 2p ~= = du f (u)A(k, u)"nk , (2.62) 2p B ~= where f (u) is the Bose distribution function and n is the momentum distribution of atoms. In B k Eq. (2.60), the equality holds if u"0. Eq. (2.61) is an example of a frequency—moment sum rule which is useful as a check on specific approximations for A(k, u). For other frequency—moment sum rule see Ch. 8 of [8]. The spectral density function A(k, u) and its relation to G in Eq. (2.58) plays a central role in the finite-temperature formalism, since one can show that the same A(k, u) also determines the various real-time Green’s functions. At finite temperature, the retarded and advanced Green’s functions involving real-time t are defined as G3%5(xt, x@t@),!ih(t!t@)S[tK (xt)tK s(x@t@)!tK s(x@t@)tK (xt)]T ,

(2.63)

G!$7(xt, x@t@),!ih(t@!t)S[tK (xt)tK s(x@t@)!tK s(x@t@)tK (xt)]T .

(2.64)

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

Here S ) ) ) T denotes the grand canonical ensemble average; and the real-time Heisenberg operator tK (xt) are defined by tK (xt)"e*KK t@+tK (x)e~*KK t@+ .

(2.65)

G3%5 and G!$7 may be also written in the Lehmann representation, and expressed in terms of the spectral density function A(k,u) defined in Eq. (2.59) as [13]

P P

= du@ A(k, u@) , (2.66) 2p u!u@#ig ~= = du@ A(k, u@) . (2.67) G!$7(k, u)" 2p u!u@!ig ~= Eqs. (2.58), (2.66) and (2.67) show that the function A(k, u) determines the thermal Green’s function as well as G3%5 and G!$7, suggesting the following analytic continuation: G3%5(k, u)"

G(k, iu Pu#ig)"G3%5(k, u) , (2.68) n G(k, iu Pu!ig)"G!$7(k, u) . (2.69) n Although G(k, iu ) is a function only at the discrete set of points iu along the imaginary frequency n n axis, it can give G3%5 and G!$7 through the analytic continuation to the real axis by using Eqs. (2.68) and (2.69). Since the sum rule (2.61) requires that both G3%5(k, u) and G!$7(k, u)&u~1 as DuDPR, the above analytic continuation is guaranteed to be unique [39]. Using these results, A(k, u) can be also written as A(k, u)"!2Im G3%5(k, u)"!2Im G(k, iu Pu#ig) . (2.70) n Therefore, any approximation for the thermal Green’s function immediately yields a corresponding A(k, u), as well as an approximation for G3%5 and G!$7. For completeness, we discuss the relation between the thermal Green’s function and the real-time Green’s function used at ¹"0. For this purpose, we define the following real-time Green’s function at finite temperature, in direct analogy to that at ¹"0: GM (xt, x@t@),(!i)TrMeb(X~KK )¹ [tK (xt)tK s(x@t@)]N . (2.71) t Here ¹ is the real-time order operator. Again, by means of a Lehmann representation, the Fourier t transform of GM can be shown to be related to G3%5 and G!$7 by [13]

P

GM (k, u),

=

dt e*utGM (k, t)"[1#f (u)]G3%5(k, u)!f (u)G!$7(k, u) . B B

(2.72)

~= In the limit of ¹P0, the last equation reduces to

G

G(k, iu Pu#ig), u'0 , n (2.73) G(k, iu Pu!ig), u(0 . n Eq. (2.73) shows explicitly how one obtains the ¹"0 Green’s function from the imaginary frequency thermal Green’s function using the analytical continuation. GM T/0(k, u)"h(u)G3%5(k, u)#h(!u)G!$7(k, u)"

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2.5. The Bogoliubov approximation To illustrate the formalism we have briefly reviewed in this section, we use it to solve for G and 11 G in the famous Bogoliubov approximation [9]. 12 One expects that in a weakly interacting Bose gas at low temperatures, most of the atoms will still remain in the k"0 state, only a few of them being “kicked” out of the condensate by the interactions. The number of non-condensate atoms is usually called the “depletion”. If the depletion is small, the interaction between two non-condensate atoms will not be important. This is the basic assumption that leads to the so-called Bogoliubov model, in which the terms »K , »K and 5 6 »K in Eq. (2.29) are neglected. The remaining terms are quadratic, diagonalizable by a canonical 7 transformation (see, for example, Ch. 25 of [36]. This Bogoliubov model can be described in terms of Green’s function language, as first discussed by Beliaev [11]. In Green’s function language, the Bogoliubov approximation consists of keeping the lowest-order self-energy diagrams that contain condensate lines, as shown in Fig. 5. These can be written as R (p)"n [v(0)#v(k)] , (2.74) 11 0 R (p)"n v(k) , (2.75) 12 0 where n ,N /» is called the condensate density. An important feature of these self-energies is that 0 0 they are frequency independent. With the Hugenholtz—Pines theorem [23], the lowest-order contribution to the chemical potential is given by k"n v(0) . 0 Using these results in Eq. (2.53) and Eq. (2.54), we obtain iu #e #n v(k) n k 0 G (p)" , 11 (iu )2!e2!2n v(k)e n k 0 k !n v(k) 0 G (p)" . 12 (iu )2!e2!2n v(k)e n k 0 k

Fig. 5. Self-energy diagrams in the Bogoliubov approximation.

(2.76)

(2.77) (2.78)

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One most striking feature of the Bogoliubov model is the form of the excitation spectrum Ek, given by the poles of G and G , 11 12 Ek"Je2#2n v(k)e , (2.79) k 0 k which agrees with the result from a direct canonical diagonalization. In the long-wavelength limit (kP0), Ek reduces to the phonon-like dispersion relation Ek&+ck, DkDP0

(2.80)

with the characteristic velocity c"[n v(0)/m]1@2 . (2.81) 0 A detailed calculation shows that the ground-state energy (see Section 2.2 of [13]) is given by E "1(N2/»)v(0) , 0 2 which yields the pressure

(2.82)

P"!(E /») "1n2v(0) , 0 N 2 and hence the macroscopic speed of sound

(2.83)

s,[1/m P/n]1@2"[nv(0)/m]1@2 .

(2.84)

Comparison of Eqs. (2.81) and (2.84) shows that c equals the macroscopic speed of sound s given by the usual thermodynamic derivative. Indeed, it has been proved to all orders in perturbation theory [22,26] for the ¹"0 case that the single-particle excitation spectrum vanishes linearly as DkDP0, with a slope equal to the macroscopic speed of sound. This linear dependence of excitation spectrum at long wavelengths is the essential piece of physics that already emerges from the simple Bogoliubov approximation. At low temperatures, the phonon spectrum will make the dominant contribution to thermodynamic properties and thus the Bogoliubov results gives the low-temperature behavior characteristic of Bose-condensed fluids. In this section, we have given a brief review of the finite-temperature Green’s function formalism for a Bose-condensed system. Due to the macroscopic occupation of the lowest energy state, one must introduce the anomalous Green’s functions G and G in addition to the normal Green’s 12 21 function G , and correspondingly, the self-energies R , R and R . Approximate solutions are 11 12 21 11 determined by our choice of self-energies. In the succeeding sections, we discuss finite-temperature approximations for a Bose condensed system, namely, the first-order Popov approximation and the second-order Beliaev—Popov approximation (in which the self-energies are explicitly first order and second order in the interaction, respectively).

3. The first-order Popov approximation In this section, we discuss a first-order self-energy approximation for a weakly interacting Bose gas at finite temperatures. It was first discussed in 1965 by Popov [32], and thus we call it the Popov approximation [15]. The well-known Bogoliubov approximation discussed in Section 2.5 only describes the physics of a low-temperature Bose gas. Clearly, as the number NI of excited

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atoms (those not in the zero-momentum state) increases with temperature, the Bogoliubov approximation becomes inappropriate since it assumes NI "N!N ;N and consequently 0 ignores the effect of interactions between the excited atoms. At finite temperature when NI is not small compared to N, the interactions between the excited atoms must be taken into account. This is done in the Popov approximation by treating the excited atoms in the Hartree—Fock approximation. Apart from Popov’s work, most theories of temperature-dependent properties (for example, Ref. [40]) of a weakly interacting Bose gas have been based on the Bogoliubov model, which means they are limited to the low temperature (¹;¹ ) region. The Popov approximation, as noted in # Refs. [15,41], gives a reasonable first approximation for a weakly interacting Bose gas at much higher temperatures. In the present section, we will give explicit expressions of the self-energies in the Popov approximation, and calculate the Green’s functions and some physical quantities at finite temperatures. We will also discuss the reason why the Popov approximation is not valid in the very small temperature interval near ¹ , # D¹!¹ D[(n1@3a)¹ . (3.1) # # The Popov approximation was largely unknown until very recently, when it was first applied to discuss the BEC of paraexcitons in Cu O by Shi et al. [15]. In this work, a general expression for 2 decay luminescence spectrum from a gas of excitons was derived. This was evaluated at finite temperatures using the Popov approximation as a simple first-order theory of the effects of interaction in a dilute weakly interacting gas of excitons at finite temperature. 3.1. The HFB self-energies In his original paper, Popov [32] expressed the self-energies in terms of the t-matrix (more precisely, the ladder approximation of the t-matrix), instead of the bare potential v(r). The t-matrix is a more effective description of the effective interactions between the atoms than the interatomic potential itself. However, in order to emphasize the physics instead of going into the details of many-body scattering theory, we shall first proceed assuming that the bare potential v(r) and its Fourier component v(k) is weak and well-defined. After we have discussed the basic physics of the Popov approximation, we will incorporate the t-matrix into our results. The self-energy diagrams in the Popov approximation include all possible first-order diagrams of a Bose-condensed system. Among the diagrams in Fig. 6, only those (a, b, e) containing two condensate lines are included in the Bogoliubov approximation of Section 2.5. The new ones (c and d) are the Hartree—Fock diagrams including excited atoms. The self-energy R and R of Fig. 6 11 12 can be explicitly written as 1 R (k)"n [v(0)#v(k)]! + G(0)(q, iu )[v(0)#v(k!q)] , 11 0 n b»q ,*un R (k)"n v(k) . 12 0 Here G(0)(q, iu ) is the unperturbed Green’s function of non-interacting atoms, given by l G(0)(q, iu )"1/(iu !e #k(0)) , l l q with k(0) denoting the chemical potential of an ideal Bose gas.

(3.2) (3.3) (3.4)

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

Fig. 6. Self-energy diagrams in the Popov approximation involving the bare potential v. A solid line with arrow denotes an ideal gas propagator G(0), a wiggly line denotes a condensate atom, and a dash line denotes the interaction v.

Since v(k) does not depend on the Matsubara frequency iu , one sees that neither do R (k) and n 11 R (k). One can then perform the frequency sum in Eq. (3.2) to obtain 12 dq v(0)#v(k!q) R (k)"n [v(0)#v(k)]# , (3.5) 11 0 (2p)3 eb(eq~k(0))!1

P

R (k)"n v(k) . (3.6) 12 0 For ¹P0, the Bose distribution function [exp b(e !k(0))!1]~1 becomes very small, and thus q the contribution due to interactions between two excited atoms (the second term in R ) is 11 negligible compared to that due to interactions between one excited atom and one condensate atom (the first term in R ). In this limit, the Popov approximation reduces to the Bogoliubov 11 approximation [see Eqs. (2.74) and (2.75)], as expected. On the other hand, the condensate density n (¹) vanishes when ¹'¹ (normal phase). In this 0 # case, we are only left with diagrams (c) and (d) in Fig. 6, i.e. the usual Hartree [(c)] and Fock [(d)] diagrams of a normal system. The Hartree—Fock approximation is basically a static mean-field theory, which treats the motion of single particles in an average static field generated by all the other particles. In the Popov approximation, the average Hartree—Fock field is determined by non-interacting particles as represented by the ideal gas propagator G(0) in diagrams (c) and (d). Taking into account the fact that the background particles also move in the average field coming from the presence of all the other particles, one can improve the Hartree—Fock theory by replacing the free propagator G(0) by the full renormalized propagator G. This is the well-known selfconsistent Hartree—Fock (SCHF) approximation [13,39]. For a Bose-condensed system, the analogue of this self-consistent approach is called the self-consistent Hartree—Fock—Bogoliubov (HFB) approximation (for further discussion and earlier

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25

references, see Ref. [41]. In this self-consistent HFB approximation, one replaces the free propagator G(0) in Fig. 6 by the matrix propagator G given by (2.47), which includes both the diagonal (normal) Green’s functions (G and G ) and the off-diagonal (anomalous) Green’s functions (G 11 22 12 and G ). The self-consistent HFB approximation, however, has well-known problems in the 21 Bose-condensed phase, such as the violation of various conservation laws and the presence of an energy gap in the excitation spectrum [26,41]. As Griffin [41,42] has pointed out, these problems with the self-consistent HFB approximation arise because the condensate atoms and excited atoms are not treated in an equal manner: the condensate atoms are treated in a collective way while the excited atoms are only treated as renormalized single particles. The Popov approximation corresponds to a simplified version of the HFB which does not introduce an energy gap in the excitation spectrum because it omits the off-diagonal self-energies. With the self-energies given by Eqs. (3.5) and (3.6), one can, in principle, calculate the singleparticle properties related to single-particle Green’s function G for a given interparticle potential ab v(r), assuming the interparticle potential has a well-defined Fourier transform v(k). However, the typical inter-atomic potential has a weak long-range attractive tail and a strong short-range repulsive core, as sketched in Fig. 7. The hard core means that the Fourier transform v(k) in Eqs. (3.5) and (3.6) is not well-defined; indeed, it is singular. We now discuss how this problem can be solved. 3.2. The Popov self-energies in terms of the t-matrix Fortunately, for a low-density or dilute gas of atoms, one can make use of the ladder approximation. The word “dilute” has the meaning that the average interatomic distance, d&n~1@3, is much larger than the s-wave scattering length a, which is the characteristic length representing the influence of the interatomic potential (see Appendix A, where the basic elements of scattering theory are reviewed). This condition implies that a/d&n1@3a;1 .

(3.7)

Fig. 7. Typical inter-atomic potential.

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

As discussed by Beliaev [34], in a dilute gas, the ladder diagrams are all of equal importance. This sum over all the ladder diagrams yields the so-called t-matrix C, as illustrated by the hatched square in Fig. 8. The diagrammatic definition of C in Fig. 8 can be written explicitly as 1 + v(q)G(0)(k !q, iu !iu ) C(p , p ; p , p ),v(k !k )! 1 1 l 1 2 3 4 1 3 b»q ,*ul ]G(0)(k #q, iu #iu )C(p !q, p #q; p , p ) , (3.8) 2 2 l 1 2 3 4 where the four-dimensional vector p ,(k , iu ) represents the momentum k and Matsubara j j j j frequency iu of a particle before ( j"1, 2) or after ( j"3, 4) scattering; similarly, q,(q, iu ). It is j l often more convenient to write C in the center-of-momentum frame of the scattering pair of atoms (see Appendix A)

P

dq F (K, k!q) C(k, k@, K; z),C(p , p ; p , p )"v(k!k@)# v(q) ` C(k!q, k@, K; z) , 1 2 3 4 (2p)3 z!2(ek q!k) ~ where we have introduced the center-of-mass variables k,(k !k )/2, k@,(k !k )/2 , 1 2 3 4 K,k #k "k #k , 1 2 3 4 u ,u #u "u #u , N 1 2 3 4 z,iu !m ,iu !+2K2/4m , N K N F (K, k!q),1#f (e )#f (e ). ` B (1@2)K`k~q B (1@2)K~k`q

Fig. 8. Diagrammatic definition of the t-matrix C.

(3.9)

(3.10) (3.11) (3.12) (3.13) (3.14)

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

27

Fig. 9. Self-energy diagrams in the Popov approximation involving t-matrix.

Here the factor F (K, k!q)(z!2ek q#2k)~1 comes from the frequency sum of the product ~ ` G(0)G(0) over iu . The function F (K, k!q) incorporates the effect of the Bose statistics obeyed by l ` the atoms involved in the intermediate scattering states. With this definition of C, the lowest-order diagrams for a dilute Bose-condensed gas in the ladder approximation are shown in Fig. 9, which are precisely the diagrams that Popov included in his original paper [32]. The self-energies represented by the diagrams in Fig. 9 can be written explicitly as

CA

B A CA

BD

k k k k R (k, iu )"n C , ; k, iu !m #C ! , ; k, iu !m 11 n 0 n k n k 2 2 2 2

B

1 k!q k!q ! + G(0)(q, iu ) C , ; k#q, iu #iu !mk q l n l ` b»q 2 2 l ,*u

A

BD

k!q k!q #C ! , ; k#q, iu #u !mk q n l ` 2 2 R (k, iu )"n C(k, 0; 0, iu ) . 12 n 0 n

,

(3.15) (3.16)

In comparing this with the self-consistent HFB discussed in Section 3.1, we note there are two differences: 1. The ladder diagrams are included, replacing the bare interatomic potential v by the t-matrix C; 2. Ideal Bose gas propagators are used in the self-energies, rather than the renormalized ones.

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As shown in Appendix A, the t-matrix C can be expressed in terms of the vacuum scattering amplitude fI defined by

P

1 dq v(k!q) fI (k, k@)"v(k!k@)# fI (q, k@) . (3.17) 2 (2p)3 e !e #ig k{ q We note that our definition of fI differs from the ordinary definition of the vacuum scattering amplitude f by a factor of !4p+2/m, that is fI (k, k@),!(4p+2/m) f (k, k@) .

(3.18)

In the long-wavelength limit (DkD"Dk@DP0), fI can be shown to reduce to fI (k, k@)"(4p+2a/m)#O(ka2) ,

(3.19)

where a is the s-wave scattering length. The vacuum scattering amplitude fI describes the effect of the potential on the wavefunction of two atoms in free space, while the t-matrix C describes the similar effect in a medium, that is, in the presence of other atoms in an ideal Bose gas. It is shown in Appendix A that the momentum dependence of C(k, k@; K, z) in Eq. (3.9) is at least the order of fI 2 ; in terms of a, this means of order a2 [see Eq. (3.19)]. In the first-order approximation we are discussing in this section, it thus seems reasonable to neglect the momentum dependence of C in Eqs. (3.15) and (3.16), and keep the self-energies only to first order in a. Thus, we arrive at R "2(n #nJ (0)) fI 11 0 0 (3.20) R "n fI , 12 0 0 where C has been replaced by its value in the limit of DkD"Dk@DP0, that is C(k, k@; K, z)KC(0, 0; 0, 0)"fI (0, 0)"4p+2a/m,fI . (3.21) 0 The quantity nJ (0) in Eq. (3.20) is the (temperature-dependent) density of excited atoms in a noninteracting Bose gas, given by

P

dk 1 1 + e*ungG(0)(k, iu )" . (3.22) nJ (0),! n (2p)3 eb(ek~k(0))!1 b»k E0,*un Here, and in the rest of this paper, we use the superscript “(0)” as a reminder that the quantity is for a non-interacting Bose gas. In Eq. (3.20), we have dropped the (k, iu ) arguments of R and R because they are now both n 11 12 frequency and momentum-independent. The self-energies in Eq. (3.20) will be referred to as the Popov first-order approximation. With it, one can calculate the normal and anomalous Green’s functions through Eq. (2.53) and Eq. (2.54) and use these to evaluate various physical quantities. We now discuss the normal phase (¹'¹ ) and the condensed phase (¹(¹ ) separately. # # 3.3. Normal phase (¹'¹ ) # In the normal phase, we have n "0 and then Eq. (3.20) reduces to 0 R "2nJ (0)fI , 11 0 R "0 . 12

(3.23)

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29

Using Eq. (3.23) in Eq. (2.53) and Eq. (2.54), the single-particle Green’s functions are 1 G (p)" , 11 iu !(e !k#2nJ (0)fI ) n k 0 G (p)"0 . 12 With Eq. (3.24), the density of the excited atoms as a function of k and ¹ is given by

(3.24) (3.25)

P

1 dk 1 + e*ungG (k, iu )" . (3.26) nJ (k,¹),! 11 n b»k (2p)3 eb(ek~k`2nJ (0)fI 0)!1 ,*un If the total density n and the temperature ¹ are given, the chemical potential is determined by the usual condition nJ (k,¹)"n (normal system) .

(3.27)

Similarly, the chemical potential k(0) of a non-interacting Bose gas at ¹'¹ is determined by # nJ (0)(k(0),¹)"n . (3.28) Comparing Eqs. (3.22) and (3.26), one sees that for a given n and ¹, we have k"k(0)#2nfI

(¹'¹ ) , (3.29) 0 # i.e., the chemical potential is increased by 2nfI due to the effectively repulsive interaction (a'0) 0 between the atoms. Eq. (3.29) was derived in Ref. [15] in a discussion of BEC in an exciton gas. The energy spectrum Ek of single-particle excitations (or quasiparticles) in the normal phase is determined by the poles of the Green’s function, that is, Ek"e !k#2nfI "e !k(0) . (3.30) k 0 k This shows that in the present first-order approximation, the quasiparticles above ¹ act just like # free particles in a non-interacting Bose gas. Because the quasiparticles determine the thermodynamic properties of a system, we conclude that in Popov approximation, the normal properties of a dilute Bose gas are unchanged compared to that of a non-interacting Bose gas and in particular, the BEC transition temperature ¹ is the same. # 3.4. Bose-condensed phase (¹(¹ ) # In the Bose-condensed phase, G and G are obtained by substituting the self-energies (3.20) 11 12 into the Dyson—Beliaev expressions in Eq. (2.53) and Eq. (2.54): iu #e #D n k G (p)" , 11 (iu )2!e2!2De n k k D G (p)"! . 12 (iu )2!e2!2De n k k

(3.31) (3.32)

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

Here the quantity D is defined by D,k!2nJ (0)fI , (3.33) 0 where the chemical potential k has been shown by Popov [31] to satisfy the finite-temperature version of Hugenholtz—Pines theorem given by Hohenberg and Martin [26] R (0, 0)!R (0, 0)"k . 11 12 We find using Eqs. (3.20) and (3.34) that

(3.34)

k"2nJ (0)fI #n fI (¹(¹ ) . 0 0 0 # Combining the last equation with the definition of D in Eq. (3.33), we arrive at

(3.35)

D(¹)"n (¹) fI . (3.36) 0 0 Here D is written as an explicit function of ¹ to emphasize the dependence of n on ¹. We note that 0 D is positive (for a'0) and approaches zero as n P0 (¹P¹ ). 0 # Both G and G in Eqs. (3.31) and (3.32) have identical poles at u"$E , where 11 12 k (3.37) E "Je2#2e D(¹)"Je2#2e n (¹) fI . k k 0 0 k k k This gives the energy spectrum of elementary excitations for ¹(¹ . This spectrum is formally # identical to that of the ¹"0 Bogoliubov approximation in Eq. (2.79) if one replaces v by fI (the 0 t-matrix). In contrast to the Bogoliubov model, where n Kn, the condensate density n (¹) in 0 0 the Popov model is strongly dependent on temperature and becomes very small (n (¹);n) in the 0 vicinity of the BEC phase transition. On the other hand, one should not over-interpret the formal similarity between the excitation spectrum in the Popov and Bogoliubov models. At the first glance, it seems to imply that one can generalize the Bogoliubov model to finite temperatures simply by taking into account the temperature dependence of n of an ideal gas. Although this gives the correct excitation spectrum, it 0 gives an incorrect expression for chemical potential, namely k"n (¹) fI instead of the correct 0 0 result in Eq. (3.35). This difference was missed in many early attempts to generalize the Bogoliubov theory to finite temperatures. We see that the Popov—Bogoliubov spectra in Eq. (3.37) has quite different limits: E &e #D(¹), k<+~1JmD , (3.38) k k E &+ck, k;+~1JmD . (3.39) k E is free-particle like in the short-wavelength limit, and phonon-like in the long-wavelength limit, k the phonon velocity c being given by (3.40) c,JD(¹)/m"Jn (¹) fI /m . 0 0 The single-particle excitation spectrum in Eq. (3.39) is proportional to k in the long-wavelength limit. As we noted earlier, this linear dependence is expected to be true in any Bose-condensed fluid to all orders in perturbation theory [22,26]. Although the Popov approximation is only the lowest-order approximation, we see that it gives gapless phonon excitations in the low-energy limit

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

31

in a Bose gas at finite temperatures. Moreover, c(¹)P0 as ¹P¹ , i.e., the phonons in the Popov # theory are the “broken-symmetry” soft modes of the Bose-condensed phase. However, unlike at ¹"0, there is no established correspondence between the excitation phonon velocity and the compressional speed of sound at finite temperature. As a side remark, we note that the finite temperature behavior of a Bose liquid like superfluid 4He is quite different from that of a Bose gas (for a review, see Ch. 7 of Ref. [8]). In particular, the observed excitation phonon velocity in liquid 4He is found to be essentially temperature-independent right through the superfluid transition. With the Green’s function Eq. (3.31), one can now calculate the density of particles in the condensate. For a given total density n, we have n"n #nJ , 0

(3.41)

with

C

D P C

D

1 bE 1 iu #e #D dk e #D n k k nJ ,! + e*ung " coth k! (¹(¹ ) . (3.42) # b»k 2 2 (iu !E )(iu #E ) (2p)3 2E n k n k k ,*un The two Eqs. (3.41) and (3.42) are coupled equations for n , since nJ depends on E and D, which are 0 k themselves functions of n [see Eqs. (3.37) and (3.36)]. In general, such non-linear equations must 0 be solved for numerically. However, in two important special cases, analytical results can be obtained, as we now discuss. We first consider the low-temperature limit (k ¹;D"n fI ). In this case, the integral in B 0 0 Eq. (3.42) can be approximated as

P A A B

B P AB

dk e #D 1 dk e #D k k ! # f (E ) B k 2 (2p)3 2E (2p)3 E k k 1 mD 3@2 1 m 3@2 K # (D)~1@2(k ¹)2 , B 3p2 +2 12 +2

nJ "

(3.43)

where in the second step, we have approximated f (E )&e~bEk. With Eqs. (3.41) and (3.43), we B k obtain

A B

8n n a3 1@2 1 m (k ¹)2 , n Kn! 0 0 ! 0 3 p 12 +3c B

(3.44)

where c is the phonon velocity defined in Eq. (3.40). For a low-density gas at low temperatures, the second and third terms in Eq. (3.44) are small, implying n &n. Therefore, when calculating the depletion nJ ,n!n at ¹"0, we can drop the 0 0 third term and replace n by n on the right-hand side of Eq. (3.44) to obtain 0 8 na3 1@2 nJ /n,[n!n (0)]/n" ;1 . (3.45) 0 3 p

A B

The error introduced by the replacement of n by n in obtaining (Eq. (3.45)) is of higher order in a. 0 Eq. (3.45) is the well-known result for the fractional depletion of the condensate in a dilute gas at absolute zero, as first obtained by Bogoliubov [9] in 1947. As a result of interparticle interactions,

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

a small number of particles are exited out of the condensate even at ¹"0. For a weakly interacting Bose gas, the fractional depletion is proportional to (na3)1@2, showing that the number of excited atoms increases with density. One expects that a large fraction of the total atoms would occupy the excited states in a strongly interacting Bose system such as liquid 4He. In fact, more than 90% of the atoms in 4He are in excited states at ¹"0, as shown by inelastic neutron scattering experiments and by direct computer simulation studies (for a review, see Ch. 4 of Ref. [8]). The ¹2 term in Eq. (3.44) was first exhibited by Glassgold et al. [40] in 1960. Their calculation of finite-temperature corrections was based on the use of the ¹"0 Bogoliubov spectrum, and thus is clearly limited to the region ¹&0. Kehr [43] has also derived this ¹2 correction, with an extra factor n (¹"0)/n: 0 n (¹)"n (0)! 1 n (0)/n (m/+3c)(k ¹)2 . 0 0 12 0 B

(3.46)

Kehr’s rigorous derivation was based on the structure of the low-lying single-particle excitations of a Bose-condensed system, which are known in all orders of perturbation theory at ¹"0 [22]. Therefore, the result Eq. (3.46) was not limited to low density or weakly interacting Bose gas, but valid for any Bose fluid at low temperatures, even for liquid 4He where n (0)+0.1n. In a dilute 0 Bose gas, we have n (0)Kn, and then Eq. (3.46) reduces to Eq. (3.44). We should emphasize that in 0 the limit of vanishing interaction ( fI P0), the results in Eqs. (3.44) and (3.46) are limited to ¹"0 0 because these results are derived under the condition k ¹;n fI . B 0 0 The ¹2 law for an interacting Bose gas is quite different from the ¹3@2 law of an ideal gas as given by [5] n (¹)"n[1!(¹/¹ )3@2] . 0 #

(3.47)

This reminds us that the introduction of even a small interaction between the particles can drastically change the properties of a Bose gas at low temperatures, where the phonon part of the spectrum [see Eq. (3.39)] completely determines the thermodynamics. As seen from Eq. (3.43), the ¹-dependent part of nJ tends to decrease with increasing value of D given by Eq. (3.36), in contrast to the ¹"0 part, which increases with D. As a result, one expects a “crossover” between n (¹) of an interacting gas and n(0)(¹) of a non-interacting gas, as sketched 0 0 in Fig. 10. This crossover means that at sufficiently high temperature, the effect of repulsive interactions between the atoms in a dilute weakly interacting Bose gas is to put more atoms in the condensate, in contrast to the ¹"0 case. It seems to imply that the thermal effect is not as significant in a dilute Bose gas as in an ideal gas. This “crossover” temperature ¹ can be estimated 9 by comparing nJ for a weakly interacting gas in Eq. (3.43) with nJ (0) for an ideal gas [the second term at the right-hand side of Eq. (3.47)], yielding k ¹ &0.46D&0.46(4p+2a/m)n B 9

(3.48)

¹ /¹ &1.74(n1@3a);1 , 9 #

(3.49)

or

where the last step follows from Eq. (3.7). For example, for n"2.6]1012 cm~3 and a"53 A_ (taken from Ref. [1] for 87Rb atoms), we estimate that n1@3a"0.0073 and thus ¹ &0.013¹ ;¹ . 9 # #

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

33

Fig. 10. Crossover between n (¹) of an interacting gas (solid line) and n(0)(¹) of an ideal Bose gas (dashed line). The 0 0 changes are exaggerated for clarity.

We next discuss the region close to the transition, ¹[¹ . It is convenient to calculate the # difference

P C

D

bE 1 be dk e #D k nJ !n " coth k! coth k , #3 2 2 2 (2p)3 2E k where n is defined by #3 dk 1 mk ¹ 3@2 B n , "2.612 . #3 (2p)3 ebek!1 2p+2

P

A B

(3.50)

(3.51)

This is the density of excited atoms (or critical density for the BEC phase transition) of an ideal gas at a given temperature ¹ [5]. The dominant contribution to the difference nJ !n comes from the #3 energy region where the excitation spectrum E differs significantly from the free-particle energy e , k k which is the region e &D in our case [see Eq. (3.37)]. We note that D"n fI is very small near ¹ , k 0 0 # namely D/k ¹&D/k ¹ ;1 . (3.52) B B # Therefore, we can approximate coth(bE /2) by (2/bE ) and coth(be /2) by 2/be in Eq. (3.50), and k k k k carry out the integral analytically to obtain

P C A B

D

P

1 k ¹ = dk e #D D k nJ !n K ! "! B dk k2 #3 be 2p2 (2p)3 bE2 e (e #2D) k k k k 0 1 2m 3@2 "! k ¹(2D)1@2 . B 8p +2

(3.53)

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Combining this result with Eq. (3.41), we find nKn #n !(1/8p)(2m/+2)3@2k ¹(2n fI )1@2 , (3.54) 0 #3 B 0 0 in agreement with Popov’s result [31]. For given n and ¹, Eq. (3.54) is a quadratic equation for n which has two solutions: 0 Jn "1[Jn (¹)$Jn (¹)#4(n!n )] . 0,B 2 ' ' #3

(3.55)

Here Jn (¹) is the coefficient of Jn in the third term of Eq. (3.54), ' 0 Jn (¹),(1/8p)(2m/+2)3@2k ¹(2 fI )1@2 B 0 '

(3.56)

n (¹)"(4p/(2.612)4@3)(n1@3a)n . '

(3.57)

or

in Eq. (3.55) is negative, Since n'n below the phase transition temperature, the solution Jn #3 0,~ and thus unphysical. Therefore, the condensate density is given by the other solution n . which 0,` yields a finite value at ¹P¹~, namely # n (¹ ), ¹P¹~ , # n " ' # (3.58) 0 0, ¹P¹` . # This clearly indicates a finite jump in the condensate density n at the transition point [44]. This 0 jump is small compared to the total density n in a dilute Bose gas (n1@3a;1), and can be estimated for the 87Rb experiment in Ref. [1] as 6.6]106 cm~3 (compared to n"2.6]1012 cm~3). Nevertheless, the finite jump in the order parameter (Jn , in a Bose fluid with no supercurrent) is the 0 characteristic of a first-order phase transition. This, however, is in contradiction to the fact that BEC in an interacting Bose system involves a second-order phase transition. This unphysical behavior near the phase transition point implies that the Popov approximation breaks down in this region. As emphasized by Popov [31], the self-energies in Eq. (3.20) are linear in D"n fI , while the largest diagrams dropped in the Popov approximation are of the order 0 0 (m/+2)3@2k ¹fI D1@2 (see Section 6). As ¹ approaches ¹ from below, the neglected diagrams become B 0 # larger than the first-order diagrams, since the former (proportional to D1@2) decrease at a slower rate than D. Therefore, the Popov approximation is valid if

G

(m/+3)3@2k ¹fI D1@2;D , B 0 or equivalently,

(3.59)

¹ !¹<(n1@3a)¹ . (3.60) # # The temperature region defined by (n1@3a)¹ is very small in a dilute Bose gas. For example, for # the case of 87Rb atoms using densities at the center of the trap given in Ref. [1], the Popov approximation should be applicable when (¹ !¹)<0.0073¹ . # # We should note that a result very similar to Eq. (3.53), and the resulting discontinuity at ¹ , was # originally obtained by Lee, Huang and Luttinger (see second paper in Ref. [24]. In a discussion

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35

valid for ¹ close to ¹ , these authors evaluated nJ in Eq. (3.42) using the approximate quasiparticle # energy E Ke #D . k k

(3.61)

This is justified by the fact that at finite temperatures, the thermally dominant excitations are at finite k, where e ;D and then the spectrum in Eq. (3.37) can be approximated by Eq. (3.61). Using k Eq. (3.61), the expression in Eq. (3.42) reduces to

P

nJ "

dk 1 . (2p)3 eb(Ek`D)!1

(3.62)

This can be evaluated to give [5,24] nJ "(1/j3)g (z"e~bD) , 3@2

(3.63)

where j is the thermal de Broglie wavelength Eq. (1.4) and the Bose—Einstein function is = zl g (z), + . n ln l/1

(3.64)

For zK1, one had the well-known asymptotic expression g

(z)"2.612!3.54Ja#2 , 3@2

(3.65)

where a,!ln z"D/k ¹;1. Using Eq. (3.65) to evaluate Eq. (3.63), one obtains B

A B

3.54 1 2m 3@2 (2D)1@2k ¹ . nJ "n ! B #3 J2p 8p +2

(3.66)

We have written the second term for easy comparison with Eq. (3.54). We see that these second terms agree, up to a numerical factor of 3.54/J2pK1.4. This slight difference is not unexpected since Eqs. (3.54) and (3.66) are based on different approximations for the integrand of Eq. (3.42) at finite temperatures. The properties of a physical system in the neighborhood of the phase transition point is always an interesting (and complicated!) problem. However, our major interest in this paper is not the critical region but the finite-temperature properties of a weakly interacting Bose gas in general. The critical region requires the renomalization group (RG) techniques developed to study the behavior of second-order phase transitions. In their recent work, Bijlsma and Stoof [45] have presented a detailed study of a uniform Bose gas using the ingenious application of the RG approach. Using the knowledge of the microscopic details of the interatomic interaction, they were able to calculate the non-universal properties of a dilute Bose gas. In particular, they found that due to interaction effects, the BEC critical temperature could be raised as much as 10% compared to the ideal gas value for the Rb and Na gases used in recent experimental studies [1,18].

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4. The many-body t-matrix approximation In Section 3, we discussed the Popov approximation as the basis of a simple theory which included the thermally induced depletion of the condensate. The self-energy diagrams which are included in the Popov approximation (see Fig. 9) were built out of the unperturbed propagator G(0) for a non-interacting Bose gas. As we noted in Section 3, the Popov approximation is not valid very close to the phase transition, where it leads to an unphysical discontinuity in the condensate density n . In an attempt to solve this well-known problem, Bijlsma and Stoof [33] used 0 a many-body t-matrix approximation, based on the use of the self-consistent propagator G in the 11 Popov self-energy diagrams. In the present section, we examine this many-body t-matrix approximation. In agreement with Bijlsma and Stoof, we show that this t-matrix becomes very temperature-dependent and, as one approaches ¹ , it vanishes at the transition point. This leads to # a smooth change of n at ¹ and thus to the correct order of the phase transition. However, it also 0 # shows that ¹ of an ideal Bose gas is unchanged in the many-body t-matrix approximation. The # required Green’s function formalism for this analysis is the same as we used in Section 3. We emphasize only the features that are different from those in the simpler Popov approximation. 4.1. The many-body t-matrix The self-energy diagrams in the many-body t-matrix approximation are shown in Fig. 11. They look the same as those in Fig. 9. However, the gray-filled square now represents the multiple scattering of two “dressed” particles, whose propagators G are to be self-consistently determined. 11

Fig. 11. Self-energy diagrams in the many-body t-matrix approximation.

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37

The modified t-matrix, denoted as CI to distinguish it from the original C, is defined by the following integral equation [compare with Eq. (3.8)]: 1 CI (p , p ; p , p ),v(k !k )! + v(q)G (p !q) 1 2 3 4 1 3 11 1 b»q ,*uq ]G (p #q)CI (p !q, p #q; p , p ) . (4.1) 11 2 1 2 3 4 This equation is illustrated by the ladder diagrams in Fig. 12. In the center-of-mass frame, the many-body t-matrix can be written as 1 CI (k, k@, K; iu !m )"v(k!k@)! + v(q)G (1P#p!q) N K 11 2 b» q (4.2) ]G (1P!p#q)CI (k!q, k@, K; iu !m ) . N K 11 2 Here k, k@, K, and u have the same meaning as in the definition of C in Eq. (3.8). We recall that N m "+2K2/4m represents the center-of-mass kinetic energy of the scattering pair whose center-ofK mass momentum is K. With the expression of CI in Eq. (4.2), the self-energies in the many-body t-matrix approximation take the following form [compare with Eq. (3.15) and Eq. (3.16)]:

CA

B A CA

BD

kk kk R (k, iu )"n CI , , k; iu !m #CI ! , , k; iu !m 11 n 0 n k n k 22 22

k!k k!k 1, 1, k#k ; iu #iu !m # + G (k , iu ) CI 11 1 l 1 n l k`k1 2 2 k 1, *ul k!k k!k 1, 1, k#k ; iu #iu !m #CI ! , 1 n l k`k1 2 2

A

BD

B (4.3)

R (k, iu )"n CI (k, 0, 0; iu ) . (4.4) 12 n 0 n We are interested in the temperature range where ¹ is not too high compared to ¹ ; in other words, # we do not consider the case of ¹<¹ . As is well-known, the energies of important excitations are # of the order k ¹, which is of the order k ¹ or smaller in the system of interest. We recall that B B # (4.5) k ¹ "(2p+2/m)(n/2.612)2@3&(+2/m)n2@3 , B #

Fig. 12. Integral equation for the many-body t-matrix CI . A gray-filled square with four “legs” represents CI , a solid line with two arrows represents a “dressed” particle or G , and a dashed line represents the interaction potential. 11

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which means that the momenta of important excitations are of the order k ,(2mk ¹ /+2)1@2&n1@3 . (4.6) b B # One sees that k is roughly the order of d~1, where d is the average distance between the atoms. For b a low-density gas such as we are interested in, the average distance between the atoms are much greater than the s-wave scattering length a, which implies that k ;a~1; in other words, only the b low momentum region is important. Under the above condition, we can neglect the energy and momentum dependence of CI in Eqs. (4.3) and (4.4), arriving at R "2(n #nJ (1))CI , 11 0 0 Here CI is defined as 0

R "n CI . 12 0 0

1 CI ,CI (0, 0, 0; 0)"v(0)! + v(q)G (q, iu )G (!q,!iu )CI (!q, 0, 0; 0) ; 0 11 l 11 l b»q ,*ul and nJ (1) is the density of non-condensate atoms,

(4.7)

(4.8)

1 + e*ungG (k, iu ) . (4.9) nJ (1),! 11 n b»k E0,*un Comparing Eq. (4.7) with Eq. (3.20), one sees that the many-body t-matrix CI now replaces the 0 free-space scattering amplitude fI "4p+2a/m in the Popov approximation in Section 3. While fI is 0 0 a constant, CI depends on the temperature through the many-body propagator G as shown by 0 11 Eq. (4.8). Using Eq. (4.7) in the Dyson’s equation Eq. (2.53) and Eq. (2.54) for G and G , we obtain for 11 12 ¹'¹ , # G "1/(iu !e #DI ), G "0 . (4.10) 11 n k 12 For ¹(¹ , we have # iu #e #DI n k G "! 11 u2#e2#2DI e n k k DI G " . (4.11) 12 u2#e2#2DI e n k k In both cases, the quantity DI is defined by DI ,k!2nJ (1)CI . (4.12) 0 Using the Hugenholtz—Pines theorem which is valid for ¹(¹ , the chemical potential and the # quantity DI are given by

H

k"2nJ (1)CI #n CI , 0 0 0 DI "n CI 0 0

(¹(¹ ) . #

(4.13)

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The quasiparticle excitation energies are determined by the poles of Green’s functions given by Eqs. (4.10) and (4.11). We find

G

e !DI , (¹'¹ ) , # E" k k Je2#2De , (¹(¹ ) . k k # The density of non-condensate atoms is given by

P P A

nJ (1)"

(4.14)

dk 1 (¹'¹ ) # (2p)3 eb(ek~DI )!1

(4.15)

B

bE 1 dk e #DI k! k coth (¹(¹ ) . (4.16) # 2 2 (2p)3 2E k With the structure of G known, we can now examine CI in more detail. Eq. (4.8) implies that to 11 0 find CI requires the knowledge of CI (!q, 0, 0; 0) for all q2'0, not just for q"0. Since we are 0 interested in a low-density Bose gas at temperature &¹ or lower, the main contribution to the # sum over q in Eq. (4.8) comes from the region q;a~1 [see discussion following Eq. (4.4)]. In this case, v(q) and CI (!q, 0, 0; 0) can be approximated by constants v(0) and CI , respectively. We thus 0 arrive at nJ (1)"

(4.17) CI Kv(0)!(1/b»)v(0)CI + @ G (q, iu )G (!q,!iu ) 11 n 11 n 0 0q ,un Here the prime means that the summation is performed only up to a cut-off momentum to avoid the divergence at large q. The ultraviolet divergence in Eq. (4.17) is artificially caused by assuming that v(q) and CI (!q, 0, 0; 0) are constant for all q in Eq. (4.8), while in fact these quantities vanish at large q. The uncertainty of the cut-off momentum can be removed by expressing CI in terms of fI (0, 0) given by 0 Eq. (3.17). A similar procedure leads to 1 , (4.18) e q q where the sum over q is also divergent if performed over the entire q-space. Fortunately, the underlying source of the divergence at large q is the same in both Eq. (4.17) and Eq. (4.18), and thus the q-cutoffs are expected to be the same. Using this and fI ,fI (0, 0), we 0 combine Eq. (4.17) and Eq. (4.18) to find fI (0, 0)Kv(0)!(1/2»)v(0) fI (0, 0)+@

CI "fI !a(¹, k) fI CI 0 0 0 0

(4.19)

or CI "fI /(1#a(¹) fI ) , 0 0 0 where we have defined the function

C

(4.20)

D

1 1 1 a(¹), +@ + G (q, iu )G (!q,!iu )! . 11 l 11 l »q b 2e q *ul

(4.21)

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Using Eq. (4.10) for ¹'¹ and Eq. (4.11) for ¹(¹ in Eq. (4.21), we can perform the frequency # # sum to obtain

G

P A P CA

B

bE 1 dk 1 coth k! (¹'¹ ) , # 2 2e (2n)3 2E k k a(¹)" (4.22) dk 1 DI 2 bE 1 # coth k! (¹(¹ ) , # (2p)3 2E E3 2 2e k k k where E is given by Eq. (4.14). The function a is now free of the large-q divergence if the integration k is taken over the entire q-space. This justifies our removal of the cutoff in Eq. (4.21). The term of order DI 2 in the integrand of Eq. (4.22) for the case of ¹(¹ is higher order in CI . This term is # 0 omitted (see further remarks at end of Section 4.4). Eq. (4.20) is still a rather complicated integral equation of CI , since a depends on E which in 0 k turn involves CI . To solve for CI , we also need to know DI , which involves the unknown k (at 0 0 ¹'¹ ) or n (at ¹(¹ ). Therefore, Eq. (4.20) must be solved in conjunction with # 0 # (4.23) n(1)(k,¹)#n "n , 0 Of course, for ¹'¹ , n in Eq. (4.23) vanishes. # 0

B

D

4.2. Analytical results for the t-matrix We next consider cases where analytical results for the function a in Eq. (4.22) can be obtained. We first consider the low-temperature limit. At ¹P0, the function coth(bE /2)& k 1#2 exp(!bE ). The integral in Eq. (4.22) can be carried out to the lowest order in DI ("n CI ) k 0 0 and ¹:

P A AB

B P C C A BD

D

dk 1 dk 1 bE 1 1 coth k! K (1#2e~bEk)! (2p)3 E (2p)3 2E 2 2e e k k k k 1 m 3@2 1 k ¹ 2 B K! . (4.24) (n CI )1@2 1! 0 0 p2 +2 2 n CI 0 0 Using this expression for a in Eq. (4.19), the low-temperature limit of CI is given by 0 1 m 3@2 1 k ¹ 2 B CI "fI # . (4.25) fI (n CI 3)1@2 1! 0 0 p2 +2 0 0 0 2 n CI 0 0 The solution of Eq. (4.25) requires knowledge of n (¹). Using Eqs. (4.23) and (4.16), we obtain 0 (4.26) nKn #(1/p2)(m/+2)3@2n (n CI 3)1@2[1# 1 (k ¹/n CI )2] , 0 0 0 0 0 0 3 12 B which is valid at low temperature k ¹;n CI . B 0 0 For a low-density gas at low temperature, the second term in both Eqs. (4.25) and (4.26) is a small correction to the first term. Therefore, to first order we obtain CI KfI and n Kn. At the next level 0 0 0 of approximation, the quantity n CI in the higher-order terms of Eqs. (4.25) and (4.26) can be 0 0 a"

AB

C A BD

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41

approximated by nfI . Within this approximation, the depletion given by the second term in 0 Eq. (4.26) equals that given by Eq. (3.44); and Eq. (4.25) becomes

C

D

1 CI "fI 1#8(na3/p)1@2! (na3)1@2(k ¹/nfI )2 , 0 0 B 0 p3@2

(4.27)

Eq. (4.27) shows that in the low-density and low-temperature limit [(na3)1@2;1 and k ¹;nfI ], B 0 the t-matrix CI differs very little from fI . In this limit, the many-body t-matrix approximation 0 0 reduces to the simpler Popov approximation in Section 3. We next turn to the finite-temperature region just below ¹ . For ¹[¹ , the function coth(bE /2) # # k in Eq. (4.22) can be approximated as 2k ¹/E , yielding B k CI "fI [1!(1/2p)(m/+2)3@2[CI /(n CI )1@2]k ¹] . (4.28) 0 0 0 0 0 B If we multiply both sides by n , Eq. (4.28) becomes an equation of n CI , which yields the following 0 0 0 solution just below ¹ (where n is small) # 0 +2 3 n (¹) +2 3 4p2n ¹ 3@2 ¹ 3@2 0.286 0 CI "4p2 " 1! " fI 1! , (4.29) 0 m (k ¹ )2 m (k ¹ )2 ¹ ¹ (n1@3a) 0 B # B # # # where the coefficient 0.286 comes from (2.612)4@3/4p. In the second line of Eq. (4.29), we have made use of the ideal gas condensate density value, n (¹)"n[1!(¹/¹ )3@2]. Eq. (4.29) shows that the 0 # t-matrix CI goes to zero in the same way as n does. This result is confirmed by numerical 0 0 calculations, in which the self-consistent condensate density n (¹) is used (see Section 4.3). 0 Finally, we consider the region just above ¹ . In this case, we can approximate Eq. (4.15) and # Eq. (4.22) by taking coth bE /2&2k ¹/E , arriving at k B k 1 2m 3@2 n"n (¹)! k ¹DDI D1@2 (4.30) #3 B 4p +2

AB

C

AB

A B A B

C A B D

D

2m 3@2 k ¹ B CI "fI 1!CI . 0 0 0 +2 8pDDI D1@2

C A B D

(4.31)

Here n ,2.612(mk ¹/2p+2)3@2 is the critical density of an ideal Bose gas at a given ¹. Using #3 B n"2.612(mk ¹ /2p+2)3@2, we can rewrite Eq. (4.30) in the form B # (2.612)2 DI "! k ¹ [(¹/¹ )3@2!1]2 . (4.32) B # # 4p We see that DI goes to zero as ¹P¹ from above. Using this result for DI in Eq. (4.31), we obtain # 0.286 +2 3@2 8pDDI D1@2 ¹ 3@2 CI K " fI (4.33) !1 . 0 0 (n1@3a) 2m k ¹ ¹ B # # Combining this with Eq. (4.29), we see that CI vanishes as ¹ approaches ¹ from above or below in 0 # a systematic way. How fast CI approaches zero depends on the value of (n1@3a)~1 ; the higher the 0 density, the slower it goes to zero. This result is in agreement with that obtained by numerical calculations (see Section 4.3 and Ref. [33]). We should note that Eq. (4.29) as well as Eq. (4.33) are only valid in a small temperature region near ¹ given by # D¹!¹ D;(n1@3a)¹ . (4.34) # #

A B

CA B

D

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4.3. Numerical results for the t-matrix At intermediate temperatures, we must solve the coupled equation for CI and E numerically, 0 k using Eqs. (4.13), (4.14), (4.15) and (4.16) and Eqs. (4.19), (4.20) and (4.21). In Fig. 13, we plot CI (¹) as a function of temperature for a given n. It goes to zero at the 0 transition temperature and approaches the constant fI at temperatures far away from ¹ . This is in 0 # agreement with the analysis given in Section 4.2. CI /fI also is seen to be slightly greater than unity 0 0 near ¹"0, due to the correction term 8(na3/p)1@2 in Eq. (4.29). We also plot n (¹) in Fig. 14 for different gases, including a comparison with result for an ideal 0 Bose-condensed gas. We find, at a given temperature, that the condensate fraction is higher in a dilute gas with repulsive interactions than in an ideal Bose gas. However, the situation is opposite near ¹"0, where an ideal Bose gas has a higher condensate fraction. The crossover happens at ¹ , 9 as discussed at the end of Section 3. As expected, the corrections due to interactions are very small. It can be also seen from Fig. 14 that in the present approximation, n (¹) goes to zero at the BEC 0 transition temperature of an ideal gas. 4.4. Assessment of the many-body t-matrix approximation In this section, we have presented a detailed investigation of a dilute Bose gas using the many-body t-matrix approximation. We solved the problem analytically at low temperatures as well as near the phase transition point. As far as we know, it is the first time that these analytical results have been obtained. In the limit of ¹P0, the results of the simpler Popov approximation

Fig. 13. Many-body t-matrix vs. temperature.

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43

Fig. 14. Condensate density vs. temperature in the many-body t-matrix approximation.

were obtained. In contrast, as ¹P¹ , the t-matrix CI goes to zero, as explicitly shown by # 0 Eq. (4.29) and Eq. (4.33). Our numerical results for the many-body t-matrix approximation shown in Figs. 13 and 14 are in agreement with those recently obtained by Bijlsma and Stoof [33]. Their work was part of a detailed study of thermodynamic properties of a dilute Bose gas, within the framework of a many-body variational approach. Our present calculation expressed in terms of the Popov approximation with a many-body t-matrix is less ambitious but perhaps more direct. In a separate investigation, Bijlsma and Stoof [45] have reported a detailed study of how ¹ of # a dilute Bose gas is affected by interactions using a renormalization group (RG) approach. This is based on RG recursion relations resulting from a systematic integration over the large k (short distance) correlations, which removes the infrared divergent terms in a systematic way. This more general analysis shows that many-body t-matrix approximation is not sufficient near ¹ . This is not # unexpected since in using the many-body t-matrix in the Popov approximation, one is clearly including a class of higher order self-energy diagrams built out of the normal propagator G , but 11 still omitting any contribution of other higher-order diagrams, especially those built out of the anomalous propagators G and G . 12 21 In the succeeding sections, we shall give a systematic study of the second-order self-energy diagrams first studied by Beliaev [34] at ¹"0 and later by Popov [31] at ¹&¹ . In particular, # we show that these second-order diagrams have contributions which are infrared divergent due to the phonon region of the first-order Popov propagators, as given in Section 3. It turns out that the second-order ladder diagram is one of these divergent terms and this is associated with a(¹) defined

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in Eq. (4.21). It turns out that the term involving DI 2/E3 in the integrand of Eq. (4.22) for ¹(¹ , k # which we have neglected in Sections 4.2 and 4.3, is one such infrared-divergent contribution. Our neglect of this is partially justified by the results of Sections 6 and 7, where we show that all such divergent terms cancel out in physical quantities. Still, it is worrisome that the remaining contribution to a(¹) is divergent at ¹"¹ . One might wonder if this divergent term is also cancelled out in # a more complete theory. In fact, the RG analysis of Bijlsma and Stoof [45] shows that results such as shown in Fig. 13 based on the many-body t-matrix approximation are qualitatively similar to what one obtains in a more complete study.

5. The second-order Beliaev—Popov (B—P) approximation The Beliaev—Popov approximation is the next step beyond the first-order Popov approximation discussed in Section 3. Beliaev [34] was the first one who gave a consistent second-order selfenergy approximation for a weakly interacting Bose gas at zero temperature. In this Beliaev approximation, the Bogoliubov quasiparticles interact with each other and are damped. It is well-known [26] that the complete HFB approximation has a gap in the excitation energy spectrum, i.e. E O0 as kP0. This energy gap in the HFB approximation arises from the use of the k off-diagonal propagator in the exchange self-energy diagram and is second order in the interaction. It turns out that the second-order self-energy diagrams included by Beliaev remove the energy gap found in the HFB approximation. This shows that the use of the self-consistent HFB approximation picks up contributions which are higher order in the interaction. The Beliaev approximation is the lowest-order approximation which: (i) leads to an imaginary part to the energy E , which corresponds to damping of the quasiparticles; (ii) exhibits infraredk divergent self-energy contributions. Beliaev did not comment on the infrared divergent terms or even explicitly include them in his equations. However, it seems he was aware of their existence and knew that these divergent terms exactly cancel out of long-wavelength properties. However, these infrared divergent terms have physical importance, since they are a consequence of microscopic occupation of a quantum state, i.e. the presence of a Bose condensate. Therefore, it is useful to retain these infrared-divergent terms in the intermediate steps, which is what we do in our discussion of the second-order approximation at both zero (Section 6) and non-zero (Section 7) temperatures. The analogous work at finite (nonzero) temperature was done by Popov [31]. In his analysis, he argued that the second-order self-energy diagrams considered by Beliaev at zero temperature were still the dominant ones at finite temperatures. We shall call this class of diagrams the Beliaev—Popov (B—P) self-energy approximation. In his original work, Popov evaluated the B—P self-energy R (p) and R (p) for p,(k, u)"0. In this section, we give a detailed evaluation of each 11 12 of the self-energy diagrams in the B—P approximation at finite temperatures. Following this, we assess the B—P approximation and compare it with simpler approximations. We also derive formal expressions of R (p) and R (p) valid for arbitrary p and temperature, and use this to arrive at 11 12 a general expression for the excitation energy spectrum in the B—P approximation. In Sections 6 and 7, we use the results of the present section to consider various properties of a weakly interacting Bose gas in the low frequency, long-wavelength limit.

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5.1. The B—P self-energy diagrams The second-order B—P approximation involves a large set of self-energy diagrams. Beliaev wrote down these diagrams with little explanation, and his original 1957 notation was somewhat different from modern conventions. In order to give the reader (especially a non-expert in this field) a clear picture of what the B—P approximation entails, we will comment on the different self-energy diagrams before we evaluate them. The diagrams for the diagonal self-energy R in the B—P approximation are shown in 11 Fig. 15. The basic “building block” of these diagrams is the hatched line, representing the t-matrix C defined in Eq. (3.8). In order to make it easier to compare with conventional diagrams, we draw C (represented by a hatched square in Section 3) in a way that it resembles an ordinary interaction “line”. Each diagram in Fig. 15 contains at most two C’s, connected to each other by elements of the 2]2 matrix propagator G, which includes the normal diagonal propagators G (p) and G (p)("G (!p)) and the anomalous off-diagonal propagators G (p) and 11 22 11 12 G (p). 21 Following Popov’s notation, we divide the diagrams of R in Fig. 15 into 7 groups: 11 a , a , a ,2, a . The first diagram in each group is the “main” diagram; others in the same group 0 1 2 6 are exchange diagrams that can be generated from the main one by exchanging two outgoing arrows or two ingoing arrows (but not both!) associated with interaction vertices. An interaction vertex, shown in Fig. 16, consists of four “arrows” connecting to an interaction “line” (actually a hatched square representing C), two of the four “arrows” going in and the other two going out. We now examine each group in R in turn: 11 1. a : the diagrams in this group are exactly the Bogoliubov diagrams (see Fig. 9). 0 2. a : these are the Hartree—Fock diagrams, but now the free propagator G(0) used in the first-order 1 Popov approximation (see Section 3) has been replaced by the first-order Popov propagator G . 11 The diagrams in the other groups (a ,2, a ) all involve two C’s and one pair of condensate 2 6 (wavy) lines. 3. a : these are built out of two G ’s, both propagating forward. We note that if both of the G ’s 2 11 11 are replaced by the lowest-order approximation G(0), then a reduces to a , because by 2 0 definition, two G(0)’s connecting two C’s is already included in C. To avoid this double-counting, we must subtract the contribution of G(0) from a . This subtraction is represented in Fig. 15 by 2 a stroke across the G line (following Beliaev’s notation [34]) in the a diagram. 11 2 4. a : these are built out of a G and a G . 3 11 21 5. a : these are built out of a G and a G . 4 11 12 6. a : these are built out of two G ’s, one propagating forward, the other propagating backward. 5 11 7. a : these are built out of a G and a G . 6 12 21 The diagrams for R in the B—P approximation are shown in Fig. 17. Again following Popov’s 12 notation, we have divided these diagrams into seven groups: b , b , b , 2, b . For simplicity, only 0 1 2 6 the main diagram in each group (as defined above) is shown in Fig. 17. One can easily work out the exchange diagrams in each group, analogous to those shown in Fig. 15. The total number of

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Fig. 15. Diagrams for the diagonal self-energy R in the B—P approximation. 11

diagrams (main one plus exchange diagrams) in each group is denoted by the number in the bracket. Unlike the diagonal self-energy R , the off-diagonal self-energy diagrams for R always 11 12 contain at least one pair of condensate lines, even though this is implicit in b . As a result, all these 1 diagrams vanish in the normal phase when the condensate density n vanishes. We list some 0

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

47

Fig. 16. An interaction vertex involving a t-matrix.

Fig. 17. Diagrams for the off-diagonal self-energy R in the B—P approximation. 12

features of each group in R 1. 2.

3. 4. 5. 6. 7.

in Fig. 17: 12 b : the diagrams in this group are simply the Bogoliubov diagrams included in the first-order 0 approximation (see Fig. 9). b : this involves the anomalous propagator G . As before, the contributions from the lowest1 12 order term of G , given by n fI G(0)G(0), must be subtracted, because they are already included 12 0 0 in b . This subtraction is denoted by a bold stroke across the G line. 0 12 b : these are built out of two G ’s, and there is one exchange diagram in this group. 2 12 b : these are built out of a G and a G which is propagating forward, and there are three 3 12 11 exchange diagrams in this group. b : these are built out of a G and a G which is propagating backward, and there are three 4 12 11 exchange diagrams in this group. b : these are built out of two G ’s, one propagating forward and the other propagating 5 11 backward, and there are three exchange diagrams in this group. b : these are built out of a G and a G , and there are three exchange diagrams in this group. 6 12 21

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It should be noted that Beliaev [34] and Popov [31] draw their self-energy diagrams using a different definition of vertex functions. Figs. 15 and 17 give the self-energy diagrams in a more conventional, direct manner. We now write down the explicit mathematical expressions for each of the diagrams in Figs. 15 and 17. For convenience, we call the diagrams built with a single C type-I diagrams (a , a , b , b ), 0 1 0 1 and denote their contributions to the self-energies as RI and RI . Similarly, the diagrams built 11 12 with two C’s are called type-II diagrams (a ,2, a , b ,2, b ) and their contributions are denoted 2 6 2 6 as RII and RII . 11 12 We first examine the type-II diagrams a ,2, a , and b ,2, b . From Appendix A, we recall that 2 6 2 6 the t-matrix C(k, k@, K; z) in the limit DkD"Dk@DP0 takes the following value:

AB

P

i m dk 1 1 1 C(k, k@, K; z)"fI ! kfI 2# fI 2 0 4p +2 0 2 0 (2p)3 e !e !ig k{ k1 dk 1#f (ek1)#f (eK k1) 1 B B ~ #O( fI 3) . #fI 2 (5.1) 0 (2p)3 z!2e 0 K k #2k (1@2) ~ 1 The integral in the last term in Eq. (5.1) comes from the frequency sum over the product of a pair of free propagators. It is convenient for subsequent discussion to give this contribution a name, namely

P

P A B

P

1 dk dk 1#f (ek1)#f (eK k1) 1 ! + G(0)(p )G(0)(P!p )" 1 B B ~ . (5.2) 1 1 b (2p)3 (2p)3 z!2e K k #2k 1 (1@2) ~ *u1 Here P,(K, iu ) represents the total four-dimensional “momentum” of the two propagators. The N terms that are order of fI 3 or higher are neglected in the B—P self-energies. Therefore, in all type-II 0 diagrams which involve the factor C]C, we set C"fI , neglecting the other terms of order fI 2 or 0 0 higher in Eq. (5.1). Within this approximation, the contribution of each exchange diagram equals that of the main diagrams in the same group. Therefore, we can simply evaluate the main diagram and multiply the result by the total number of diagrams in each group. The contributions of a ,2, a in Fig. 15 are given by 2 6 J(P),

P P P P P P

a (p)"2n fI 2 d4p [G (p )G (p!p )!G(0)(p )G(0)(p!p )] 2 0 0 1 11 1 11 1 1 1 "2n fI 2 d4p G (p )G (p!p )!2n fI 2J(p) 1 11 1 11 1 0 0 0 0

(5.3)

a (p)"4n fI 2 d4p G (p )G (p!p ) , 3 0 0 1 12 1 11 1

(5.4)

a (p)"4n fI 2 d4p G (p )G (p!p ) , 1 12 1 11 1 4 0 0

(5.5)

a (p)"4n fI 2 d4p G (p )G (p !p) , 5 0 0 1 11 1 11 1

(5.6)

a (p)"4n fI 2 d4p G (p )G (p!p ) , 1 12 1 12 1 6 0 0

(5.7)

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and those of b ,2, b in Fig. 17 are given by 2 6

P P P P P

b (p)"2n fI 2 d4p G (p )G (p!p ) , 1 12 1 12 1 2 0 0

(5.8)

b (p)"4n fI 2 d4p G (p )G (p!p ) , 1 12 1 11 1 3 0 0

(5.9)

b (p)"4n fI 2 d4p G (p )G (p !p) , 4 0 0 1 12 1 11 1

(5.10)

b (p)"4n fI 2 d4p G (p )G (p !p) , 5 0 0 1 11 1 11 1

(5.11)

b (p)"4n fI 2 d4p G (p )G (p !p) . 6 0 0 1 12 1 12 1

(5.12)

Here the integral over the four-dimensional momentum p is defined as usual, 1 1 d3k 1. d4p ,! + 1 b (2p)3 *u1 We next turn to the type-I diagrams containing a single C. The contribution from the momentum-dependent part of C is of order fI 2, as shown in Eq. (5.1). Thus, we must take into account 0 the momentum-dependence of C in the type-I diagrams. The contributions of the type-I diagrams are given by

P

P

a (p)"n [C(p, 0; p, 0)#C(p, 0; 0, p)] , 0 0

(5.13)

a (p)" d4p G (p )[C(p, p ; p, p )#C(p, p ; p , p)] , 1 11 1 1 1 1 1 1

(5.14)

b (p)"n C(p,!p; 0, 0) , 0 0

(5.15)

P P P

b (p)" d4p [G (p )!n fI G(0)(p )G(0)(!p )]C(p,!p; p ,!p ) 1 12 1 0 0 1 1 1 1 1 " d4p G C(p,!p; p ,!p )!n fI 2J(0) . 1 12 1 1 0 0

(5.16)

Eqs. (5.3), (5.4), (5.5), (5.6), (5.7), (5.8), (5.9), (5.10), (5.11) and (5.12) plus Eqs. (5.13), (5.14), (5.15) and (5.16) include explicit expressions of all the contributions to the self-energies in the B—P approximation. These expressions are slightly different in form from those given by Popov in Ch. 6 of Ref. [31], although the sum of the various diagrams yields the same self-energy in the end. The first-order contributions considered in the Popov approximation [see Eqs. (3.15) and (3.16)] are included in the lowest-order part of a , b and a . The other diagrams, a ,2, a , b ,2, b and the 0 0 1 2 6 1 6 higher-order contributions (proportional to fI 2) of a , b and a , are the new contributions, not 0 0 0 1 included in the Popov approximation of Section 3. When comparing these higher-order diagrams with those considered by Beliaev [34] at ¹"0, we find that they are exactly the same, apart for the finite-temperature formalism being used here.

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5.2. Connections to other theories Up to now in this section, we have concentrated on what the B—P approximation involves. We now make some additional remarks on its physical significance and connections to some other standard approximations. It is crucial to understand that the B—P theory is different from simple second-order perturbation theory, in which the diagrams are constructed using non-interacting (free particle) propagators. In contrast, the B—P self-energy diagrams involve the 2]2 matrix propagators which we obtained in the first-order approximation in Section 3. These propagators describe a gas of non-interacting quasiparticles in the Popov model. Correlations between these quasiparticles are taken into account in the B—P approximation. Although it is explicitly second order in C, the B—P self-energies actually involve a selected set of higher-order diagrams, due to the fact that the first-order propagators G and G are also functions of C. In fact, as we shall see in 11 12 Section 6, the physical quantities such as the quasiparticle energy E in the B—P approximation k contain non-perturbative corrections that are the order of (na3)1@2 (or C3@2), relative to those in the first-order Popov approximation of Section 3. We are now in the position to compare the Bogoliubov, HFB, Popov and many-body t-matrix approximation with the Beliaev—Popov approximation. The following discussion is strongly influenced by the recent analysis of Proukakis et al. [46]. 5.2.1. The Bogoliubov approximation In the Bogoliubov approximation, one retains in the interaction Hamiltonian (see Eqs. (2.29), (2.30), (2.31), (2.32), (2.33), (2.34), (2.35), (2.36) and (2.37)) all parts that contain two operators (aL s or aL ), which represents the creation or annihilation of an excited atom. In terms of vertex diagrams, this means that one keeps, in Fig. 1, E and » — but drops » — . In other words, one only considers 0 14 57 collisions between condensate—condensate and condensate—excited atoms, but ignores those between two excited atoms. Since the Hamiltonian now involves quadratic terms, it can be diagonalized by a canonical transformation, namely the Bogoliubov transformation, yielding a gas of non-interacting quasiparticles. The Bogoliubov approximation gives a good description of a weakly interacting dilute Bose gas at zero temperature, since the depletion of the condensate (population of the excited states) is small. However, as the gas becomes denser or the temperature becomes higher, it is not appropriate to neglect the interactions between excited atoms. Therefore, the Bogoliubov model lose its validity in these limits. In connection with the Beliaev—Popov approximation, the Bogoliubov self-energy diagrams only include a and b in Figs. 15 and 17, evaluated 0 0 to first order in fI . 0 5.2.2. The HFB approximation The HFB approximation upgrades the Bogoliubov interactions by taking into account the contributions of » — . These parts contain three or four-operator terms, such as aL saL aL and aL saL saL aL , 57 which can be approximated into various terms containing the normal average SaL saL T, and the anomalous averages SaL saL sT and SaL aL T. For example, » and » can be approximated as 5 7 »K "2aL saL aL +2[2SaL saL TaL #SaL aL TaL s] , (5.17) 5 »K "2aL saL saL aL +2[4SaL saL TaL saL #SaL saL sTaL aL #SaL aL TaL saL s] . (5.18) 7

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

51

The average denoted by S2T is to be carried out self-consistently. Any average that includes three or more operators is neglected in the HFB approximation. With the above treatment, the Hamiltonian can again be diagonalized, defining a new set of quasiparticles. These quasiparticles are no longer Bogoliubov quasiparticles, since they are “dressed” by the mean-field effects on excited atoms. The self-energy diagrams of the HFB approximation are shown in Fig. 18. The dashed lines in Fig. 18 represent the actual interatomic potential. It has been generally assumed that the HFB diagrams can be generalized to include the effect of repeated collisions by simply replacing the bare interaction with a t-matrix. However, Proukakis et al. [46] have recently pointed out that it is inconsistent to do so. Their argument is equivalent to the following: the iteration of the selfconsistent anomalous propagator G in part (f) of Fig. 18 yields precisely the t-matrix in part (e), 12 which means that there exists an over-counting of diagrams in keeping both part (e) and part (f) if one uses the t-matrix instead of the actual potential. In dilute gases, the actual interatomic potential is usually approximated in the literature by a contact potential v(r!r@)"gd(r!r@)

(5.19)

where g"4p+2a/m"fI . (5.20) 0 Here a is the s-wave scattering length. Such a contact potential is equivalent to a t-matrix, approximated to first-order in a. Thus, the use of a short-ranged potential like v(r!r@) in Eq. (5.19) in the HFB theory is not really consistent since it involves double-counting [46]. Beliaev [34] was presumably the first to realize the problem of diagram over-counting when introducing a t-matrix. As we described earlier, he took care of this by adding strokes to the

Fig. 18. The HFB self-energy diagrams in terms of the bare interatomic potential.

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

self-energy diagram a in Fig. 15 and b in Fig. 17, representing the subtraction of contributions 2 1 that are proportional to the quantity n fI 2J, where J is defined in Eq. (5.2). As can be seen from 0 0 Eq. (5.1), the t-matrix C also contain the quantity J to second order in fI . The precise cancellation 0 of those terms containing J in the final expressions of R and R given by Eqs. (5.33) and (5.36) 11 12 shows that the subtraction is just sufficient to get rid of the over-counted diagrams. In fact, a , a of 0 1 Fig. 15, and b , b (the latter with a stroke) of Fig. 17 are equivalent to the correct HFB self-energy 0 1 diagrams to second order in the t-matrix. This new modified HFB approximation is not usually discussed in the literature and is not subject to the criticism made by Proukakis et al. [46] of the usual HFB. As we mentioned in Section 3, the self-consistent HFB approximation has an energy gap in the excitation spectrum. It may not be clear at the first-order level of approximation why such problems arise in the HFB approximation but not the simpler first-order Popov approximation. In view of the above discussion, one can easily see that the gap arises because some of the diagrams are double-counted in R . In other words, the naive HFB approximation is not “consistent” to second 12 order in fI . The Popov approximation, on the other hand, is not only simpler but consistent to first 0 order. It includes all self-energies computed to first order in the interaction [41]. Finally, we would like to make an important comment concerning the use of a contact potential such as in Eq. (5.19). Strictly speaking, the use of a contact potential makes sense only in a first-order approximation, in which the physical quantities are evaluated to first order in the interaction. In a higher-order approximation, explicit or implicit, one must take into account other higher-order terms contained in the t-matrix as shown in Eq. (5.1). In other words, it is not consistent to drop terms which are proportional to fI 2 in the t-matrix, but keep such terms in the 0 evaluation of other physical quantities. The problem of double-counting diagrams in the naive HFB approximation (as discussed above) is just one example. 5.2.3. The first-order Popov approximation The Popov approximation may be viewed as a simplified HFB approximation. The only difference between the Popov and HFB approximation is the way that the averages are performed in the Hartree—Fock term. Instead of doing them self-consistently, the averages in the Popov approximation are calculated in a non-interacting gas basis, that is, one only includes the “bare” mean-field effects on the propagation of excited atoms. The anomalous averages SaL aL T in the HFB are omitted since they are second-order in fI . 0 5.2.4. The many-body t-matrix approximation It is worthwhile to compare the present B—P approximation with the many-body t-matrix approximation discussed in Section 4. The similarities and differences between these two approaches are schematically shown in Fig. 19. To first order in the vertex function C, the self-energy diagrams included in these two approaches are exactly the same. However, the higher order self-energy diagrams are different. For R , the many-body t-matrix approximation includes only 11 a , a and a , but not a , a , a , and a ; and for R , the many-body t-matrix approximation include 0 1 2 3 4 5 6 12 only b and b , but not b , b ,2, b . On the other hand, the many-body t-matrix approximation 0 1 2 3 6 also includes the set of diagrams shown in Fig. 20, which are not considered in the B—P approximation. Among these extra diagrams in Fig. 20, only a and b are of order C2. 7 7

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

53

Fig. 19. Overlap and difference between the self-energies kept in the many-body t-matrix and the B—P approximations.

Fig. 20. Extra self-energy diagrams included in the many-body t-matrix approximation.

It is clear that a is built out of three G ’s, two of them propagating forward, the other 7 11 propagating backward. The explicit expression for a is 7 1 a (p)" fI 2+ G (q)+ [G (p )G (p#q!p )!G(0)(p )G(0)(p#q!p )] , 7 11 11 1 11 1 1 1 (b»)2 0 q p1

(5.21)

where to avoid double counting, one has to subtract the contribution of G(0)G(0) from a (p). Using 7 the definition 1 C/%8(p, q; p, q),! fI 2+ G (p )G (p#q!p ) , 11 1 11 1 b» 0 p1

(5.22)

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

a is given by 7 1 (5.23) a (p)"! + G (q)[C/%8(p, q; p, q)!C(p, q; p, q)] . 11 7 b» q Comparing Eq. (5.23) with Eq. (5.14), we find that the structure of a is similar to that of a , with 7 1 only C/%8 replacing C. The frequency sum in Eq. (5.22) cuts off the momentum integral at k ;a~1, b where the difference between C/%8 and C is expected to be small. This is why a is not important in 7 the B—P approximation. A similar analysis applies to b for R . 7 12 5.2.5. Further remarks on the B—P approximation We note that above ¹ (where n "0), all diagrams containing one or more condensate lines # 0 vanish, and thus only the Hartree—Fock diagrams a of Fig. 15 survive in the B—P approximation. 1 This means in the non-condensate phase, the B—P theory is equivalent to the familiar normal H—F theory. However, the “loop” in a involves the first-order propagator G , not the free propagator 1 11 G(0). Thus, the collisions between excited atoms are treated in a “dressed” H—F mean-field manner, unlike in the first-order Popov approximation in which the excited atoms move in the H—F mean field of a non-interacting gas. In contrast, below ¹ , the B—P approximation is much more # complicated than the HFB approximation. This can be seen from the extra diagrams (the type-II diagrams, which are labeled a — for R and b — for R ) which are kept in the B—P approxima26 11 26 12 tion. The collisions between condensate—condensate and condensate—excited atoms are treated in an improved manner, as compared to the HFB approximation. This leads to a gapless approximation, as proved [23] by the fact that the chemical potential satisfies the Hugenholtz—Pines theorem. Compared to all the other approximations mentioned above, there is a key difference in the self-energy diagrams of the B—P theory. It contains vertices like » and » , which involves three 5 6 non-condensate operators, which we will refer to as a “triplet”. In fact, the entire set of type-II diagrams are made of all possible combinations of the triplet terms » and » . This means that in 5 6 the B—P approximation, one takes into account the contributions of three-operator averages such as SaL saL aL T, etc. which are completely neglected in the Bogoliubov, Popov, HFB as well as in the many-body t-matrix approximation. These contributions are crucial and thus the B—P theory represents a significant improvement over the other approximations mentioned above. The crucial role of these three-body contributions is discussed within the “dielectric formalism” approach to Bose-condensed systems [8]. It is always difficult to justify why one includes a certain set of higher order diagrams but not the other, and especially so in Bose-condensed systems. We have concentrated here on what is done in the B—P approximation, without going into a thorough analysis as whether it is justifiable or not. With a clear understanding of what is included and the consequences, it is then possible to examine such questions. However, we hope our present analysis of the Beliaev—Popov approximation indicates why it gives the first consistent “upgraded” approximation past the first-order Popov approximation. 5.3. Evaluation of the B—P self-energies In Section 5.1, we have given formal expressions for all the self-energy diagrams in the B—P approximation. We now use these results to obtain more explicit expressions for the self-energies.

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

55

It is convenient to write the Green’s functions G and G , given by Eqs. (3.31) and (3.32), in the 11 12 following form: Bk Ak ! , G (p)" 11 iu !E iu #E n k n k Ck Ck G (p)" ! , 12 iu !E iu #E n k n k where

(5.24) (5.25)

E #e #D k , Ak,u2k " k 2E k !E #e #D k k , Bk,v2k " 2E k

(5.26) (5.27)

(5.28) Ck,!JAkBk"!D/2E , k With these expressions, we perform the Matsubara frequency sums in Eqs. (5.3), (5.4), (5.5), (5.6) and (5.7) and add a to a to obtain 2 6 R (p)II,a (p)#a (p)#a (p)#a (p)#a (p) 11 2 3 4 5 6

P

"n fI 2 d4p [2G (p )G (p!p )#8G (p )G (p!p )#4G (p )G (p !p) 0 0 1 11 1 11 1 12 1 11 1 11 1 11 1 #4G (p )G (p!p )]!2n fI 2J(p) , (5.29) 12 1 12 1 0 0 dk 2A A #8C A #4A B #4C C 1 1 2 1 2 1 2 1 2 "!2n fI 2J(p)#n fI 2 0 0 0 0 (2p)2 iu !E !E n 1 2 2B B #8C B #4B A #4C C 1 2 1 2 1 2 F (E , E ) ! 1 2 ` 1 2 iu #E #E n 1 2 2B A #8C A #4B B #4C C 1 2 1 2 1 2 1 2 # iu #E !E n 1 2 2A B #8C B #4A A #4C C 1 2 1 2 1 2 F (E , E ) , ! 1 2 (5.30) ~ 1 2 iu !E #E n 1 2 where the two F thermal functions are defined as B F (E , E ),f (E )#f (E )#1 , (5.31) ` 1 2 B 1 B 2 F (E , E ),f (E )!f (E ) , (5.32) ~ 1 2 B 1 B 2 f (E) is the Bose distribution function. Here the subscript “1” denotes k and “2” denotes B 1 k ,k!k . On the right-hand side of Eq. (5.30), a variable change of k to k interchanges “1” and 2 1 1 2

P GC

C

D

D

H

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

“2” only, but has no effect on the integral. Taking advantage of this, we can use Eqs. (5.26), (5.27) and (5.28) to further simplify Eq. (5.30) to

P

C

D

g(!E ,!E ) 1 dk 1 g(E , E ) 1 2 1 1 2 ! R (p)II" n fI 2 0 0 11 iu #E #E 2 (2p)3 E E iu !E !E n 1 2 1 2 n 1 2 g(!E ,!E ) dk 1 g(E , E ) 1 2 1 1 2 #n fI 2 ! 0 0 (2p)3 E E iu !E !E iu #E #E n 1 2 1 2 n 1 2 g(!E , E ) g(E ,!E ) 1 2 ! 1 2 # f (E )!2n fI 2J(p) . (5.33) 0 0 iu #E !E iu !E #E B 1 n 1 2 n 1 2 The function g(E , E ) is defined as 1 2 g(E , E ),3e e !E E #D(e #e )#D2!D(E #E )#e E #e E . (5.34) 1 2 12 1 2 1 2 1 2 1 2 2 1 We emphasize that in Eq. (5.30), the terms involving F and F contain different physics. The ` ~ F terms involve creation or destruction of two quasiparticles [poles at $(E #E )] while the ` 1 2 F terms involve scattering of a quasiparticle from another quasiparticle [poles at $(E !E )]. ~ 1 2 The latter thermal scattering terms vanish at ¹"0. This distinction between these two kinds of terms is less obvious in the expression (5.33). Similarly, the type-II diagrams of R can be added up to give 12

P

C

D

P

R (p)II,b (p)#b (p)#b (p)#b (p)#b (p)"n fI 2 d4p [4G (p )G (p!p ) 12 2 3 4 5 6 0 0 1 12 1 11 1 #4G (p )G (p !p)#4G (p )G (p !p)#6G (p )G (p!p )] 12 1 11 1 11 1 11 1 12 1 12 1 dk 4C B #4C A #4A B #6C C 1 1 2 1 2 1 2 1 2 "n fI 2 0 0 (2p)3 iu !E !E n 1 2 4C B #4C A #4B A #6C C 1 2 1 2 1 2 F (E , E ) ! 1 2 ` 1 2 iu #E #E n 1 2 4C B #4C A #4B B #6C C 1 2 1 2 1 2 1 2 # iu #E !E n 1 2 4C A #4C B #4A A #6C C 1 2 1 2 1 2 F (E , E ) . ! 1 2 ~ 1 2 iu !E #E n 1 2 A further simplification [compare with Eq. (5.33)] leads to

P GC

D

C

D

P

(5.35)

C

H

D

h(!E ,!E ) dk 1 h(E , E ) 1 1 2 1 1 2 ! R (p)II" n fI 2 0 0 12 iu #E #E (2p)3 E E iu !E !E 2 n 1 2 1 2 n 1 2 dk 1 h(E , E ) h(!E ,!E ) 1 1 2 1 2 #n fI 2 ! 0 0 (2p)3 E E iu !E !E iu #E #E 1 2 n 1 2 n 1 2 h(!E , E ) h(E ,!E ) 1 2 ! 1 2 # f (E ) , iu #E !E iu !E #E B 1 n 1 2 n 1 2

P

C

(5.36)

D

(5.37)

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

57

where the function h(E , E ) is defined as 1 2 h(E , E ),2e e !2E E #D2 . (5.38) 1 2 12 1 2 The contributions of type-II diagrams are given by Eqs. (5.33), (5.34), (5.37) and (5.38). We next calculate the contribution of the type-I diagrams. Using Eq. (5.1), a (p) in Eq. (5.13) can be 0 expressed in terms of fI as 0 k kk k a (p),a (k, iu )"n C , , k; iu !m #C ,! , k; iu !m 0 0 n 0 n k n k 2 22 2

CA AB

P

B A

BD

i m dk 1 1 "2n fI ! kn fI 2#n fI 2 #2n fI 2J(p) . 0 0 4p +2 0 0 0 0 (2p)3 e !e !ig 0 0 k1 k@2 It is easy to check that the imaginary part of the integral in Eq. (5.39) is given by

P

AB

i m dk n fI 2 ipd(e !e )" kn fI 2 , 0 0 (2p)3 q k@2 0 0 4p +2

(5.39)

(5.40)

and it exactly cancels the second term of Eq. (5.39). We are thus left with

P

dq P #2n fI 2J(p) , (5.41) a (p)"2n fI #n fI 2 0 0 0 0 0 0 0 (2p)3 e !e q k@2 where P means the principle value of the integral in Eq. (5.41). We note that the term containing J(p) in Eq. (5.41) precisely cancels the last term in Eq. (5.33) for reasons discussed in Section 5.2. Clearly, the remaining integral in Eq. (5.41) is divergent at large q. However, this does not cause any trouble because, as we shall see shortly, this term is canceled by another divergent contribution to R (p)II in Eq. (5.33). 11 For the evaluation of a (p) in Eq. (5.14), we use Eq. (5.1) and obtain 1 k!k k!k 1, 1, k#k ; iu #iu !mk k a (p)" d4p G (p ) C 1 1 11 1 1 n 1 `1 2 2

P

CA

A

BD

k!k k!k 1, 1, k#k ; iu #iu !mk k #C ! 1 n 1 `1 2 2

B

.

(5.42)

The main contribution from the four-dimensional momentum integral comes from the region Dk D&k ,(2mk ¹/+2)1@2 . (5.43) 1 b B Since we are only interested in the temperature region that are not too high compared with ¹ , we # have [31]

A

Dk D& 1

B

2mk ¹ 1@2 B # &n1@3;a~1 . +2

(5.44)

In the small momentum region given by Eq. (5.44), the t-matrix can be treated as the constant fI . 0 The integral of G (p ) over p gives the density nJ (1) of excited atoms in the Popov approximation 11 1 1

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[see Eq. (3.42)], namely

P P C

P

P

dk dk 1 [A f (E )!B f (!E )]" 1 [(A #B ) f (E )#B ] nJ (1), d4p G (p )" 1 11 1 1 B 1 1 B 1 1 (2p)3 1 B 1 (2p)3 1

D

bE 1 dk e #D 1 1 coth 1! . 2 2 (2p)3 2E 1 With Eq. (5.45) in Eq. (5.42), a reduces to 1 dk 1 [(A #B ) f (E )#B ] . a (p)"2nJ (1)fI "2fI 1 0 0 (2p)3 1 1 B 1 1 "

(5.45)

P

(5.46)

Adding a and a to give the contribution from type-I diagrams of R , Eqs. (5.41) and (5.46) give 0 1 11 dk P 1 RI ,a (p)#a (p)"2(n #nJ (1)) fI #n fI 2 #2n fI 2J(p) . (5.47) 11 0 1 0 0 0 0 (2p)3 e !e 0 0 k@2 k1 The two anomalous type-I diagrams are b and b . With Eqs. (5.1) and (5.15), b is given by 0 1 0 1 dk P 1 b (p)"n C(k, 0, 0; 0)"n fI # n fI 2 #n fI 2J(0) . (5.48) 0 0 0 0 2 0 0 (2p)3 e !e 0 0 k@2 k1 In evaluating b , we recall the fact that the lowest-order of G is proportional to n fI G(0)G(0). 1 12 0 0 From Eq. (5.16), it is clear that b is at least the order of fI 2, therefore we can neglect the momentum 1 0 dependence of C in Eq. (5.1) and set it equal to fI . Within this approximation, b is given by 0 1

P

P

P P A B P

b (p)"fI d4p [G (p )!n fI G(0)(p )G(0)(!p )] 1 0 1 12 1 0 0 1 1 1 dk n fI 1 ! + 0 0 !n fI 2J(0) 0 (2p)3 0 0 b (iu !E )(iu #E ) 1 1 1 1 *u1 1 dk 1 bE 1 "! n fI 2 coth 1!n fI 2J(0) . (5.49) 0 0 2 0 0 (2p)3 E 2 1 Using Eqs. (5.48) and (5.49), the sum of b and b gives the type-I diagram contribution to R : 0 1 12 1 1 bE dk P 1 RI ,b (p)#b (p)"n fI # n fI 2 ! coth 1 . (5.50) 12 0 1 0 0 2 0 0 (2p)3 e !e E 2 1 k1 k@2 We note that the terms involving J(0) in Eqs. (5.48) and (5.49) do not show up in the sum b #b 0 1 because they cancel each other exactly (see discussion of over-counting in Section 5.2). This completes our evaluation of type-I and type-II diagrams. Now, we combine the contributions of the two types to find R and R . By combining Eqs. (5.33) and (5.47), we obtain the 11 12 following expression for R , 11 dk dk P 1 B #n fI 2 1 R (p)"RI (p)#RII (p)"2n fI #2fI 11 11 11 0 0 0 (2p)3 1 0 0 (2p)3 e !e 1 k@2 1 g(!E ,!E ) dk 1 g(E , E ) 1 2 1 1 2 # n fI 2 ! 2 0 0 (2p)3 E E iu !E !E iu #E #E n 1 2 1 2 n 1 2 "fI

P C

P

C

P

D

P

D

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

P

C

59

g(!E ,!E ) g(!E , E ) dk 1 g(E , E ) 1 2# 1 2 1 1 2 #n fI 2 ! 0 0 (2p)3 E E iu !E !E iu #E #E iu #E !E n 1 2 n 1 2 1 2 n 1 2 g(E ,!E ) dk 1 2 1 (A #B ) f (E ) , ! f (E )#2 fI (5.51) B 1 0 (2p)3 1 1 B 1 iu !E #E n 1 2 where the function g is defined in Eq. (5.34). Similarly, combining Eqs. (5.37) and (5.50), we obtain

D

P

P C

1 dq R (p)"RI (p)#RII (p)"n fI # n fI 2 12 11 12 0 0 2 0 0 (2p)3

D D

1 P ! E e !e q q k@2 1 h(!E ,!E ) dk 1 h(E , E ) 1 2 1 1 2 # n fI 2 ! 2 0 0 (2p)3 E E iu !E !E iu #E #E n 1 2 1 2 n 1 2 h(!E ,!E ) h(!E , E ) dk 1 h(E , E ) 1 2 # 1 2 1 1 2 #n fI 2 ! 0 0 (2p)3 E E iu !E !E iu #E #E iu #E !E n 1 2 n 1 2 1 2 n 1 2 h(E ,!E ) dq f (E ) 1 2 B q , ! f (E )!n fI 2 (5.52) B 1 0 0 (2p)3 E iu !E #E n 1 2 q where the function h is defined in Eq. (5.38). In Eqs. (5.51) and (5.52), we have deliberately separated out the part that does not involve the Bose factor and the part that is explicitly temperature-dependent. The first two lines in each expression gives the former contribution, which we define as R0,

P

P

D

P

C C

P

P

dk dk P 1 B #n fI 2 1 R0 (p),2n fI #2fI 11 0 0 0 (2p)3 1 0 0 (2p)3 e !e 1 k@2 1 g(!E ,!E ) dk 1 g(E , E ) 1 2 , 1 1 2 # n fI 2 ! (5.53) 2 0 0 (2p)3 E E iu !E !E iu #E #E n 1 2 1 2 n 1 2 dq P 1 1 ! R0 (p),n fI # n fI 2 12 0 0 2 0 0 (2p)3 e !e E q k@2 q 1 h(!E ,!E ) dk 1 h(E , E ) 1 2 . 1 1 2 # n fI 2 ! (5.54) 0 0 2 iu #E #E (2p)3 E E iu !E !E n 1 2 1 2 n 1 2 These expressions correspond to Beliaev’s results at ¹ " 0, as given in Eqs. (5.9) and (5.10) of Ref. [34]. However, when used at finite temperatures, R0 is implicitly temperature-dependent through its dependence on the condensate density n (¹). 0

P

C P C P C

D

D

D

5.4. Energy of excitations in the B—P approximation Once we have obtained a specific approximation for R and R , we can substitute these into 11 12 Eqs. (2.53) and (2.54) to calculate the Green’s functions at finite temperatures. Due to the lengthy expressions in Eqs. (5.51) and (5.52), the results for G and G are very complicated. However, 11 12 there is a way [34] to simplify such results so that the poles (and their residues) of G and G are 11 12 more clearly exhibited, and the second-order corrections to the energy of excitations are shown in

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an explicit manner. To do so, it is convenient to separate out contributions which are explicitly linear and quadratic in fI , 0 R (p) " R(1)(p)#R(2)(p) , (5.55) 11 11 11 R (p) " R(1)(p)#R(2)(p) , (5.56) 12 12 12 k " k(1)#k(2) . (5.57) Here the linear expressions R(1), R(1), and k(1) have been given earlier in Section 3 [see Eqs. (3.20) 11 12 and (3.35)], while the quadratic contributions R(2) and R(2) can be deduced from Eqs. (5.51) and 11 12 (5.52) in Section 5.3. The “second-order” correction to k, denoted by k(2), can be obtained using Hugenholtz—Pines theorem [23,26], namely k(2) " R(2)(0)!R(2)(0) . (5.58) 11 12 With a little algebra, one can show, after dropping all terms of the order O( fI 4) and higher, that 0 the denominator of G (p) and G (p) in Eqs. (2.53) and (2.54) is given by 11 12 DK(iu )2!iu (R` !R~ )!E2#(e #D)(2k(2)!R` !R~ )#2DR(2) , (5.59) n n 11 11 k k 11 11 12 where R` ,R(2)(p) , (5.60) 11 11 R~ ,R(2)(!p) . (5.61) 11 11 Solving D(k, iu ) " 0 for iu (which can be analytically continued to real frequencies u$ig), we n n obtain 1 uK (R` !R~ )$JE2!(e #D)(2k(2)!R` !R~ )!2DR(2) 11 k k 11 11 12 2 11

G

H

1 1 1 K$ E # (e #D)(R` #R~ !2k(2))! DR(2)$ (R` !R~ ) k 2E k 11 11 12 2 11 11 E k k ,$(E #KB (u)) . (5.62) k k Here u"$E are the two poles of the Green’s functions in the first-order Popov approximation k [see Eq. (3.37)] and the new term in Eq. (5.62) is the correction from the second-order B—P approximation, KB (u),(e /2E )(R` #R~ !2k(2))#(D/2E )(R` #R~ !2k(2)!2R(2)) k k k 11 11 k 11 11 12 $1(R` !R~ ) . 2 11 11 Using Eq. (5.62), the denominator D(k, iu ) of G and G is given by n 11 12 D(k, iu ) " (iu !E !K`)(iu #E #K~) . n n k k n k k

(5.63)

(5.64)

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

61

Consequently, the diagonal Green’s function in the B—P approximation can be rewritten in the form iu #e !k(1)!k(2)#R(1)#R~ k 11 11 . G (k, iu ) " n 11 n (iu !E !K`)(iu #E #K~) n k k n k k This can be expressed as the sum of two poles,

(5.65)

B@ A@ k k ! , (5.66) G (p) " 11 u!E !K` u#E #K~ k k k k where A@ and B@ satisfy k k A@ (iu #E #K~)!B@ (iu !E !K`),iu #e #D!k(2)#R~ . (5.67) k n k k k n k k n k 11 Since Eq. (5.67) must hold for arbitrary values of iu , we can solve Eq. (5.67) for A@ and B@ to give n k k (5.68) A@ " A #a , k k k B@ " B #a . (5.69) k k k Here A and B , defined in Eq. (5.26) and Eq. (5.27), are the corresponding weights for the two poles k k $E of G in the first-order Popov approximation; and a is the “second-order” correction k 11 k defined by a ,(D/4E3)[2e R(2)!D(R` #R~ !2k(2)!2R(2))] . k k k 12 11 11 12 Using Eqs. (5.68) and (5.69) in Eq. (5.66), we finally arrive at

(5.70)

B #a A #a k k k k ! . (5.71) G (k,iu ) " 11 n iu !E !K` iu #E #K~ n k k n k k This type of expression was first derived by Beliaev [34] at ¹ " 0 and the above analysis formally extends his approach to finite temperature. Following the analysis in obtaining Eq. (5.71), it is straightforward to also express the anomalous Green’s function G in a similar way, namely 12 C #c C #c k k k k G (k, iu ) " ! . (5.72) 12 n iu !E !K` iu #E #K~ n k k n k k Here the “second-order” correction to C is given by k c ,(D/2E3)[e (R` #R~ !2k(2))#D(R` #R~ !2k(2)!2R(2))]!R(2)/2E . (5.73) k k k 11 11 11 12 12 12 k We can use Eqs. (5.71) and (5.72) to calculate the properties of the Green’s function within the B—P approximation. This will be done in Sections 6 and 7. The energy of single-particle excitations (poles of G and G ) are given by 11 12 u"$(E #KB (u)) (5.74) k k

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which may contain a finite imaginary part, describing the damping of the single-particle excitations. We emphasis that these approximate expressions for G , G and EBP are only valid within 11 12 k the B—P approximation for the self-energies. They should not be used in other approximations without further discussion.

6. Calculations at T 5 0 using the B—P approximation In Section 5, we worked out the single-particle Green’s functions in the B—P approximation for a weakly interacting Bose gas. The formalism developed was valid for arbitrary temperature. In this section, we use the results obtained in Section 5 to study the special case of ¹ " 0, reproducing the famous “second-order” calculations worked out by Beliaev [34] some 40 years ago. We have tried to give a more detailed treatment of some of the intermediate steps and hope that our discussion will be useful to many readers who have become interested in Beliaev’s work in connection with calculating the properties of the recently discovered trapped atomic Bose gases. In particular, we give, for the first time, explicit expressions for the self-energies in the B—P approximation which contain infrared divergent terms. As well-known [22,30], these divergent contributions cancel out in the final expressions for physical quantities, such as the long-wavelength excitation spectrum. However, they have a physical basis in the Bose broken symmetry and are of interest in this more general context. 6.1. The B—P self-energies at ¹"0 At zero temperature, the diagonal Green’s function G in Eq. (5.71) is given by 11 A #a B #a k k k k G (k, u) " ! , (6.1) 11 u!E !K`#ig u#E #K~!ig k k k k where KB and a are defined in Eqs. (5.63) and (5.70). The functions a and KB are combinations of k k k k R(2)($p) (or RB ), R(2) and k(2), which are “second-order” corrections to the corresponding 11 11 12 quantity R ($p), R (p), and k, respectively. Using Eqs. (5.53) and (5.54), leaving out the 11 12 first-order terms and carrying out the following integrals:

P

P

AB

2 m 3@2 dk dk !E #e #D 1 B " 2fI 1 1 1 " Jn f 3 D 0 (2p)3 1 0 (2p)3 0 0 3p2 +2 2E 1 1 2 mD 3@2 dk P 1 n fI 2 ! " Jn f 3D , 0 0 (2p)3 e !e 0 0 E p2 +2 1 1 k@2 we obtain the ¹"0 “second-order” terms 2fI

P A

P AB

B A B C

(6.2)

D

g(!E ,!E ) dk 1 g(E , E ) 1 1 2 #E #E 1 1 2 ! R(2)(p)" n fI 2 1 2 11 2 0 0 (2p)3 E E u!E !E #ig u#E #E !ig 1 2 1 2 1 2 8 m 3@2 # Jn f 3D , 0 0 3p2 +2

(6.3)

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

P AB

C

1 h(!E ,!E ) dk 1 h(E , E ) 1 2 1 1 2 R(2)(p)" n fI 2 ! 12 0 0 2 (2p)3 E E u!E !E #ig u#E #E !ig 1 2 1 2 1 2

63

D

1 m 3@2 Jn f 3D . # 0 0 p2 +2

(6.4)

Here the functions g and h are defined by Eqs. (5.34) and (5.38), respectively; and D,n fI [see 0 0 Eq. (3.36)]. The results in Eqs. (6.3) and (6.4) agree precisely with those given by Eqs. (5.9) and (5.10) of Beliaev [34]. The form of Eq. (6.1) makes it convenient to study the behavior of the Green’s function near the poles at uK$E . For this purpose, we first do a Taylor expansion of R(2)(k, u) and R(2)(k, u) k 11 12 about the pole u"E , k R` (k, u) " R(2)(k, E )#U` (k, E )(u!E )#O[(u!E )2] , 11 11 k 11 k k k

(6.5)

R~ (k, u),R(2)(k,!u)"R(2)(k,!E )#U~ (k, E )(u!E )#O[(u!E )2] , 11 11 11 k 11 k k k

(6.6)

R(2)(k, u) " R(2)(k, E )#U (k, E )(u!E )#O[(u!E )2] , 12 12 k 12 k k k

(6.7)

where the first-order coefficients are defined by

K K K

K

U` (k, E ), 11 k

R(2)(k, u) 11 u

U~ (k, E ), 11 k

R(2)(k,!u) 11 u

U (k, E ), 12 k

R(2)(k, u) 12 u

,

K

K

(6.8)

u/Ek ,

(6.9)

u/Ek .

(6.10)

u/Ek

In this Taylor expansion, both the zeroth and the first-order coefficients are complex quantities. The real and imaginary parts of these quantities are denoted by “Re” and “Im”, respectively, in the following discussion. In the limit of small momentum (compared with JmD/+), the integrals involved in Eqs. (6.5), (6.6) and (6.7) can be worked out to give explicit expressions for R` , R~ and R(2) in powers of E . 11 11 12 k The expansion, which is extremely tedious, is conveniently done using MAPLE (a computer software that can do symbolic computations). However, one has to separate out the divergent terms from well-behaved terms before carrying out the final integration. Our final results for the zeroth-order terms in Eqs. (6.5), (6.6) and (6.7) are: Re R(2)(k,$E )"8(n a3/p)1@2D[(14!2D )$(2D !1)l #( 161 !1D ! 3 D )l2] , (6.11) 11 k 0 3 1 1 2 k 1440 2 1 32 3 k

A B

Re R(2)(k, E )"8 12 k

n a3 1@2 0 D[(3!2D )#( ~79 #1D ! 3 D )l2] , 1 1440 2 1 32 3 k p

(6.12)

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

A

B

13l4 193l5 307l6 k# k , (6.13) Im R(2)(k,$E ) " !Jpn a3D 1Gl !l2/24$7l3/24# k G 11 k 0 k k k 1920 1920 35840

A

B

13l2 293l4 1255l6 k! k . Im R(2)(k, E ) " !Jpn a3D 1! k # 12 k 0 24 1920 21504

(6.14)

Here a (from fI "4p+2a/m) is the s-wave scattering length; the quantity (n a3)1@2 is a small 0 0 dimensionless parameter in a low-density gas [see Eq. (3.7)]. The dimensionless ratio l defined as k l ,E /D , (6.15) k k is also a small quantity in the limit of k;JmD/+. The other two quantities D and D are 1 3 dimensionless integrals:

P P

D , 1 D , 3

=

0 =

dx , x(x2#4)3@2

(6.16)

dx . x3(x2#4)7@2

(6.17) 0 One sees that as xP0, both integrals D and D are divergent: D goes to infinity as ln x and D as 1 3 1 3 x~2, which shows the infrared divergence inherent in the quantum field analysis of a Bosecondensed system [22,30]. As far as we know, this is the first time that infrared divergence associated with these higher-order self-energy expressions have been written down explicitly. Comparing Eqs. (6.11) and (6.12) for the real parts Re R(2) and Re R(2) with those given by Cheung 11 12 and Griffin [29], we note that our coefficient of the l2 term is different, and furthermore the infrared k divergent terms were left out of their results. Similarly, coefficients of the linear terms in the Taylor expansion of RB (k, u) and R(2)(k, u) in 11 12 Eqs. (6.5), (6.6) and (6.7) can be obtained by (a) taking their derivatives with respect to u, (b) evaluating the derivatives at u " E , and (c) in the limit of small k, expanding the integrands in k powers of E . This procedure gives (again with the aid of MAPLE) k Re UB (k, E )"8(n a3/p)1@2[G(1!2D )!( 1 #1D #16D )l ] , (6.18) 11 k 0 2 1 15 4 1 3 k (6.19) Re U (k, E )"8(n a3/p)1@2(7D !16D ! 1 )l , 3 15 k 12 k 0 4 1 Im UB (k, E ) " !Jpn a3(G1!l /2$25l2/24#13l3/48 11 k 0 k k k G1133l4/1920!97l5/768) (6.20) k (6.21) Im U (k, E ) " !Jpn a3(!3l /2#13l3/16!613l5/1280) . k k k 12 k 0 Here the imaginary terms have to be evaluated up to order E5 because the lower-order terms all k cancel out when we calculate the quasiparticle energy spectrum. 6.2. Quasiparticle spectrum In the previous section, we have obtained all the zero-order and first-order terms in the Taylor expansion of RB and R(2) around the pole u " E . The “second-order” correction to the chemical 11 12 k

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

65

potential at ¹ " 0 can now be easily calculated using Eqs. (5.58), (6.11), (6.12), (6.13) and (6.14), to give k(2)"40(n a3/p)1@2D . (6.22) 3 0 With these results for RB , R(2), and k(2), it is straightforward to calculate a and K`, defined by 11 12 k k Eqs. (5.70) and (5.63). We obtain a " (n a3/p)1@2(16D/E #i p/8 E /D) , (6.23) k 0 3 k k K` (u)"(n a3/p)1@2(28E !i(3p/80)E5/D4)!(n a3/p)1@2(4#(i p/4)E2/D2)(u!E ) k 0 3 k k 0 k k ,N !j (u!E ) , (6.24) k k k where Eq. (6.24) also defines the new functions N and j . k k The calculation of K~ requires expansions of RB and R(2) near the negative frequency pole k 11 12 u"!E . Using the following equality for any differentiable function y(x) of x: k dy(!x) dy(x) "! , (6.25) dx dx x/x0 x/~x0 we have

K

K

U` (k,!E ) " !U~ (k, E ) , (6.26) 11 k 11 k U~ (k,!E ) " !U` (k, E ) , (6.27) 11 k 11 k U (k,!E ) " !U (k, E ) . (6.28) 12 k 12 k With these last three equalities, we can calculate the quantity K~ near the negative pole u " !E k k to obtain K~(u)" (n a3/p)1@2(28E !i(3p/80)E5/D4)#(n a3/p)1@2(4#i(p/4)E2/D2)(u#E ) k 0 3 k k 0 k k ,N #j (u#E ) . (6.29) k k k The results of Eqs. (6.23), (6.24) and (6.29) were first obtained by Beliaev in Ref. [34], if we recall that in his notation a"f /4p, and D"n f . However, as we have noted earlier, Beliaev did not 0 0 0 write down the equivalent of Eqs. (6.11), (6.12), (6.13) and (6.14) and Eqs. (6.18), (6.19), (6.20) and (6.21), nor did he mention the infrared divergent terms (which cancel out in the final results). With Eqs. (6.24), (6.29) and (6.1), the Green’s function near to the poles at small momenta may be written in the form

C

D

B #a A #a k k k k ! G (k, u) " (1!j ) . (6.30) 11 k u!E !N u#E #N k k k k Comparing with the first-order diagonal Green’s function given by Eq. (3.31), one sees that the quantities j , a and N are “second-order” corrections. These corrections are small for a lowk k k density gas, since they are all proportional to Jn a3(Jna3;1 under condition (3.7). 0 The quasiparticle energy is determined by the pole of G (k, u). In the momentum range 11 DkD;JmD/+, Eqs. (6.1) and (6.24) together yield E(2)"E #N "E [1#28 (n a3/p)1@2]!i 3 Jpn a3E5/D4 . 80 0 k k k k k 3 0

(6.31)

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

We recall that E "+ck for small momenta k;JmD/+, where c " JD/m " +J4pn a/m is the k 0 sound velocity in the first-order approximation. Eq. (6.31) shows that for small momentum, the quasiparticles are phonon-like. The “second-order” approximation gives a small correction to the sound velocity c(2) " c[1#28 (n a3/p)1@2] . (6.32) 3 0 In addition, E(2) contains an imaginary part proportional to E5 or k5. This corresponds to a finite k k lifetime for the phonons. This ¹ " 0 long wavelength damping is due to the decay of a phonon into two single excitations. This is clear from the structure of the self-energies in Eqs. (6.3) and (6.4) with poles at u"$(E #E ). 1 2 The mean number of the atoms NM with a given momentum k at ¹"0 can be calculated directly k from the Green’s function, using

P

`= du NM "i lim e*ugG (k, u) k 11 2p ` g?0 ~= A #a B #a dz k k ! k k "! e*zg(1!j ) . (6.33) k z!E !N z#E #N 2pi k k k k C Here the path of integration C is a contour consisting of the real axis from !R to #R, together with a semicircle in the upper half-plane. We emphasize that NM is the momentum distribution of k atoms, not the quasiparticle momentum distribution. Because the imaginary part of N is negative k [see Eq. (6.24)], the only contribution is from the negative energy pole of G at u"!E !N . 11 k k Therefore using Eqs. (5.27) and (6.33), we obtain

Q

C

D

(6.34) NM "(1!j )(B #a )"D/2E [1#20(n a3/p)1@2] . k k k k k 3 0 The imaginary parts of a and j cancel in Eq. (6.34), and thus NM is real, as it should be. To k k k calculate the total number of particles with non-zero momentum, we need to know NM for all k momenta. Unfortunately, no general analytic form is available for arbitrary momentum. Numerical methods are required to calculate the number of particles NI with non-zero momentum, from which, one can find the number N of atoms in the condensate, for a fixed total number of atoms N. 0 We can also calculate the ground-state energy from the chemical potential k. The first-order contribution to k is k(1)"D"n fI given by Eq. (3.35) and the “second-order” contribution k(2) is 0 0 given by Eq. (6.22). We add these two parts to obtain k"(4p+2n a/m)[1#40 (n a3/p)1@2] . (6.35) 0 3 0 Here we may use the result for n obtained in the first-order Popov approximation, as given by 0 Eq. (3.44) of Section 3. Expressing n in terms of n by means of Eq. (3.44), we finally obtain 0 k " (4p+2na/m)[1#32 (na3/p)1@2] . (6.36) 3 By definition, at ¹"0 we have k"(E /»)/n. Therefore, integrating Eq. (6.36) with respect to n, 0 one obtains the ground-state energy

C

A B D

2p+2n2a 128 na3 1@2 E 0" 1# . m 15 p »

(6.37)

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

67

This result coincides with the result of Lee et al. [24] for a hard-sphere gas, and later re-derived by Beliaev [34] and by Hugenholtz and Pines [23]. The pressure P is given by

A B

P"!

E 0 »

" N

C

A B D

64 na3 1@2 2p+2n2a 1# . 5 p m

(6.38)

The derivative of P with respect to n yields the usual compressional speed of sound s"

A B

A

B C A B D A

B C

A B D

1 P 1@2 4p+2na 1@2 na3 1@2 4p+2n a 1@2 28 n a3 1@2 0 0 " 1#8 " 1# . (6.39) m n m p m 3 p

Comparison of Eq. (6.39) with Eq. (6.32) shows that the phonon velocity c(2) is equal to the compressional speed of sound s in the second-order approximation. This is in agreement with Gavoret and Nozie`res’s famous result that, to arbitrary order in the perturbation, the phonon velocity equals the macroscopic speed of sound at ¹"0. We now turn to the calculation of the spectral density function A(k, u), which we recall from Section 2 is given by A(k, u) " !2 Im G (k, u) . 11

(6.40)

At the end of the previous section, we obtained an expression for the Green’s function G (k, u). 11 Using the results for a , j and N in Eqs. (6.23) and (6.24), we find after some calculation that the k k k spectral density function is given by (l ,E /D " +ck/D; D,n 4p+2a/m) k k 0

C

p1@2 u(20#10l )!E (20#7l ) k k k A(k, u)" (n a3)1@2l2 k 80 0 (u!EI )2#c2 k k

D

u(!20#10l )!E (20!7l ) k k k ! (u#EI )2#c2 k k

C A B D A B

28 n a3 1@2 0 EI ,+ck 1# k 3 p

(6.41)

(6.42)

AB

3 +2 k5 3p n a3 1@2 (+ck)5 0 " . c " k 640p m n 80 p D4 0

(6.43)

Somewhat surprisingly, c in Eq. (6.43) does not explicitly depend on the parameter a (except for an k implicit, weak dependence through n , the condensate density). One recalls that the collision rate 0 goes as a2 for a dilute classical gas. The result in Eq. (6.43), therefore, is puzzling at first glance. It is a consequence of the fact that at ¹ " 0, the quasiparticles are phonons rather than free atoms. The excitation energy depends on Ja in the long-wavelength limit, and therefore the s-wave scattering length a enters the calculation of one phonon decaying into two in a complicated way. The final result is that a cancels out in the expression for c in Eq. (6.43). The explicit expression given in k Eq. (6.41) has not been given in the previous literature.

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Fig. 21. Sketch of spectral density function A(k, u) for a given k in the B—P approximation at ¹"0.

We recall that in the Bogoliubov approximation, the spectral density function is given by A (k, u) " A d(u!E )!B d(u#E ) , (6.44) B k k k k which consists of two peaks at E and !E , respectively. The Beliaev spectral density function k k given by Eq. (6.41) also consists of a positive-weight resonance at u"E and a negative-weight k resonance at u"!E , as sketched in Fig. 21. Associated with the damping of the excitations, the k peaks in Fig. 21 or Eq. (6.41) have a finite width, given by Eq. (6.43). It is easy to check that at u"0, Eq. (6.41) reduces to A(k, u"0)K!(7p1@2/40)(na3)1@2E2/D3 (6.45) k for small k (E ;D). By definition, A(k, u"0) should vanish for any value of k [see Eq. (2.60)]. k However, if we remember the Green’s function G (k, u) in Eq. (6.30) is only valid at near the poles 11 u&$E , the small but finite zero frequency value in Eq. (6.45) is not unexpected. k 7. Discussion of the TO0 B—P approximation In this section, we consider the case of finite (¹O0) temperature, using the general expressions for the self-energy R and R obtained in Eqs. (5.51) and (5.52) in Section 5. These self-energies 11 12 can be separated into two parts: one part denoted by R0 which does not depend explicitly on the Bose factor; and another temperature-dependent part denoted by RT which does depend on the Bose factor. In Section 6, we worked out the expressions for R0 near the poles u&$E in the k small momentum limit. Those expressions now have a new meaning, since they involve the condensate density n (¹) which is temperature-dependent at finite temperature. 0

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

69

We now turn our attention to the temperature dependent parts of the self-energies associated with the Bose factors, namely

P

C

dk 1 g(E , E ) g(!E ,!E ) 1 1 2 1 2 RT (p)"n fI 2 ! 11 0 0 (2p)3 E E u!E !E #ig u#E #E #ig 1 2 1 2 1 2

D

P

g(!E , E ) g(E ,!E ) dk e #D 1 1 1 2 1 1 # ! f (E )# 2 fI f (E ) 0 (2p)3 E B 1 u#E !E #ig u!E #E #ig B 1 1 2 1 2 1

P

(7.1)

C

h(!E ,!E ) h(!E , E ) dk 1 h(E , E ) 1 2 # 1 1 1 1 2 ! RT (p)"n fI 2 12 0 0 (2p)3 E E u!E !E #ig u#E #E #ig u#E !E #ig 1 2 1 2 1 2 1 2

D

P

h(E ,!E ) dk 1 1 2 1 ! f (E )!n fI 2 f (E ) . (7.2) B 1 0 0 u!E #E #ig (2p)3 E B 1 1 2 1 Here we remind the reader that the functions g and h are as defined in Eqs. (5.34) and (5.38), respectively: g(E , E )"3e e !E E #D(e #e )#D2!D(E #E )#e E #e E 1 2 12 1 2 1 2 1 2 1 2 2 1 h(E , E )"2e e !2E E #D2 , 1 2 12 1 2 and the subscript “1” refers to k and “2” refers to k!k . We have made the analytical 1 1 continuation iu Pu#ig in the expressions of RT and RT , implying that we are now dealing n 11 12 with the self-energies for the retarded Green’s function G3%5 (see Section 2.5). We recall that at ¹"0, the self-energies in Eqs. (6.3) and (6.4) involve poles at u"$(E #E ), 1 2 which describes the decay of one phonon into two phonons. At finite temperature, the self-energies in Eqs. (7.1) and (7.2) involve additional poles at u"$(E !E ), representing the process of 1 2 a phonon scattering with a thermally excited phonon. Due to these additional thermal scattering processes (present only at finite temperatures), it is difficult to carry out the equivalent calculations to those in Section 6. One difficulty is associated with the fact that it now matters what order one takes the limits uP0 and kP0. For example, if one takes the limit uP0 first and then the limit kP0, one immediately sees that the thermal terms of the diagonal self-energy RT in Eq. (7.1) 11 involving poles at u"$(E !E ) are singular: 1 2

G C

DH

g(!E , E ) g(E ,!E ) 1 2 ! 1 2 lim lim u#E !E u!E #E k 1 2 1 2 ?0 u?0 g(!E , E )#g(E ,!E ) 1 2 1 2 "lim E !E 1 2 2?1 6e e #2E E #2D(e #e )#2D2 1 2 1 2 "lim 1 2 "R . (7.3) E !E 1 2 2?1 The limit of kP0 means k ("k!k ) P!k [which is equivalent to k because the integrands 2 1 1 1 of Eqs. (7.1) and (7.2) depend only on Dk D and Dk D]. This is denoted by “2” P “1”. Since the 1 2

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

numerator of Eq. (7.3) is finite, we see the expression is singular for any value of k . On the other 1 hand, if we reverse the order of the two limits in Eq. (7.3), we find

G C

DH

g(E , !E ) g(!E , E ) g(!E ,E )!g(E ,!E ) 1 2 1 2 ! 1 1 1 1 "0 . lim lim " lim (7.4) u!E #E u#E !E u 1 2 1 2 u?0 2?1 u?0 The above limit vanishes because g(!E , E )"g(E ,!E ). The similar kind of analysis also 1 1 1 1 applies to the thermal terms of RT in Eq. (7.2). Due to this dependence on the order of limits, we 12 cannot expand RT(k, E ) in the small momentum limit following the procedure used in Section 6, k which effectively took the limits uP0, kP0 at the same time. This kind of dependence on the ratio k/u in the limit of (k, u)P0 is well-known in the theory of Fermi liquids. 7.1. The k"0 limit We now take the limit kP0 first and then the limit uP0. In this case, the thermal terms in RT and RT vanish. Using Eqs. (5.51) and (5.52) for the expressions of the total self-energies R (p) 11 12 11 and R (p), we find that in the above limit, these expressions can be simplified to 12 bE 1 dk e #D 1 1 R (0, 0)"2n fI #2 fI coth 1! 11 0 0 0 (2p)3 2E 2 2 1 dk 1 2De D2 bE 1 1 1# ! n fI 2 1! coth 1! , (7.5) 0 0 (2p)3 E E2 2E2 2 e 1 1 1 1 dk 1 1 2De D2 bE 1 1 1# ! coth 1! . (7.6) R (0, 0)"n fI !n fI 2 12 0 0 0 0 (2p)3 E 2 E2 2E2 2 2e 1 1 1 1 The second term on the right-hand side of Eq. (7.5) equals the density nJ (1) of non-condensate atoms in the first approximation, given by Eq. (3.42). It is very important to note that n which appears in 0 these results is the temperature-dependent condensate n (¹), to be calculated self-consistently. We 0 note that both R (0, 0) and R (0, 0) contain the term 11 12 bE dk D2 1 !n fI 2 coth 1 , (7.7) 0 0 (2p)3 E3 2 1 which is divergent at k P0, namely 1 1 k2 dk bE 1. dk coth 1J dk 1 J (7.8) 1E3 1 E4 k2 2 1 1 1 The chemical potential for the condensed phase can be calculated using the Hugenholtz—Pines theorem (we recall that nJ (1) is given by Eq. (3.42)):

P A P CA P CA

B

B

B

D

D

P

P

P

P

P A

B

1 dk 1 bE 1 1 k"R (0, 0)!R (0, 0)"n fI #2nJ (1)fI ! n fI 2 coth 1! 11 12 0 0 0 2 0 0 (2p)3 E 2 e 1 1 5 m 3@2 dk 2e #D 1 1 "n fI # f (E ) . Jn fI 3 n fI #2fI 0 0 3p3 +2 0 (2p)3 2E B 1 0 0 0 0 1 It is seen that the infrared-divergent terms in Eq. (7.7) cancel out in the expression for k.

AB

P

(7.9)

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

71

We next estimate k. The last term in Eq. (7.9) is temperature dependent, and can be written as

P

P C

D

dk 2e #D dk 2e #D 1 1 1 1 f (E )"2 fI f (E )!f (e ) #2n (¹) fI . (7.10) 0 (2p)3 2E B 1 0 (2p)3 2E B 1 B 1 #3 0 1 1 Here n (¹) ["2.612(mk ¹/2p+2)3@2] is the critical density of an ideal Bose gas at a given ¹, as #3 B given by the integral of f (e ) over k . The main contribution to the integral over k in the first term B 1 1 1 of Eq. (7.10) comes from the region e [D, where E differs significantly from e . For temperature 1 1 1 ¹ which satisfies kT,2 fI

k ¹
P C A B

D

(7.11)

AB

(7.12)

where in the last step, we have used fI "4p+2a/m. The result in Eq. (7.12) is in agreement with that 0 of Popov (see Eq. (7.35) of Ref. [31]) if we remember that he used the units +"2m"k "1. B We now calculate the total chemical potential by inserting the above result for kT into (7.9), and find

C

A B D

A B

4p+2n a n a3 1@2 40 n a3 1@2 8p+2an (¹) 0 1# 0 #3 !12p 0 k(¹)" k ¹. # B m p 3 p m

(7.13)

The first term is the ¹"0 result already derived in (6.35), except that n (¹) is now temperature 0 dependent. The new temperature-dependent parts consist of a Hartree—Fock term [8p+2an (¹)/m] #3 and another correction term which is linear in both ¹ and Jn a3. The second-order B—P 0 approximation introduces a small correction (about 0.3% for 87Rb atoms under the conditions of Ref. [1]) to the chemical potential with respect to that in the first-order Popov approximation. We emphasize that Eq. (7.13) has been derived assuming n fI ;k ¹, which is valid as long as the 0 0 B temperature ¹ is not too low compared to ¹ , namely # ¹<3.8(n1@3a)¹ . (7.14) # It can be estimated that for a"53 A_ and n"2.6]1012 cm~3 (taken from [1] for 87Rb in the center of the trap), n1@3a"0.0073 and the condition (7.14) implies that ¹<0.027¹ . # 7.2. Expansions of RT(k, u"E #d) for small k k We are most interested in the behavior of the Green’s function near the single-particle poles. However, as mentioned earlier, we cannot set u"E and expand R(k, E ) in the small momentum k k limit, due to the difficulty associated with the order of limits taken. In an attempt to overcome this problem, we introduce a gap and expand the self-energies around u"E #d instead of u"E . k k After this expansion is done, we take the dP0 limit. In this approach, we are not taking the limits

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

kP0, uP0 at the same time, but taking the limit kP0 first. See the end of this section for some further comments on this procedure. Following this procedure, we arrive at the following results (again using MAPLE): lim RT (k,$E $d)"a $a k#[a $a (d~1)#a (d~2)]k2 , 11 k 0 1 2 3 4 d?0

(7.15)

lim RT (k, E #d)"b #b k#[b #b (d~1)#b (d~2)]k2 , 12 k 0 1 2 3 4 d?0

(7.16)

with

P

P

n fI 2 = 2e2#D2 e #D fI = a "! 0 0 dk k2 1 dk k2 1 f (E )# 0 f (E ) , 0 1 1 E3 1 1 E B 1 B 1 2p2 p2 1 1 0 0 = 1 a "!J2Dn fI 2/2p2 +/J2m dk k2 (!2e D2#De #e3) f (E ) 1 1 1 1 1 1 E5 B 1 0 0 0 1 n fI 2 +2 = a "! 0 0 dk k2 (!2e9!24e8D!47e7D2#48e6D3# 156e5D4 2 1 1 1 1 1 1 1 12 2m 0 #80e4D5#32e3D6)(1/E11) f (E ) 1 1 1 B 1 n fI 2 +2 = dk k2 (2e5D!e4D2!10e3D3)(1/E7) f (E ) a " 0 0 1 1 1 1 1 1 B 1 3 3p2 2m 0 n fI 2 +2 = a " 0 0 dk k2 (12e7#80e6D#171e5D2# 135e4D3#34e3D4)(1/E7)f (E ) , 4 1 1 1 1 1 1 1 1 B 1 3p2 2m 0

P

P

P P

(7.17) (7.18)

(7.19) (7.20) (7.21)

and

P

P

n fI 2 = n fI 2 = !4e D#D2 1 1 b "! 0 0 f (E )! 0 0 (7.22) dk k2 dk k2 f (E ) 0 B 1 1 1 1 1E B 1 2p2 2p2 E3 1 1 0 0 b "0 (7.23) 1 n fI 2 +2 = dk k2 (!8e8D!11e7D2#40e6D3#44e5D4 (7.24) b "! 0 0 1 1 1 1 1 1 2 12p2 2m 0 1 !16e4D5#32e3D6) f (E ) (7.25) 1 1 E11 B 1 1 b "0 (7.26) 3 b "a . (7.27) 4 4 We note that kT"a !b agrees with Eqs. (7.9) and (7.12). We can approximate f (E ) for ¹&¹ 0 0 B 1 # by k ¹/E for E ;k ¹, which limits the following results to high temperatures. In this approxiB 1 1 B mation, we can evaluate the a and b’s:

P

AB

C

D

m 3@2 3J2p a "2n fI !J2n f 3/p2 #D k ¹ 0 #3 0 0 0 2 B +2 4

(7.28)

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

AB C AB A B AB

D

J2p J2 m a "! fI k ¹ !D 1 2 p2 +2 0 B 16

73

(7.29)

C

J2 m 1@2 403J2p k ¹ a "! #5D #32D Jn f 3 B ! 2 2 4 0 0 D 4096 12p2 +2

D

J2n f 3 m 3@2 0 0 k ¹[0#D ] b "! B 2 0 p2 +2

C

(7.30) (7.31)

D

J2 m 1@2 k ¹ 685J2p b "! !D #32D , Jn f 3 B 2 2 4 0 0 D 4096 12p2 +2

(7.32)

where D and D denote two infrared-divergent dimensionless integrals: 2 4 = 1 D " dx , (7.33) 2 x2(x2#2)2 0 = 1 . (7.34) D " dx 4 x4(x2#2)6 0 We recall that in Section 5, the quantity KB near the poles u"$E can be expressed in terms k k of RB (k,$E ) and R (k, E ) as 11 k 12 k e D 1 (R` #R~ !2k(2)!2R(2)) . (7.35) KB"$ (R` !R~ )# k (R` #R~ !2k(2))# 11 11 11 12 k 2E 11 2E 11 2 11 k k The Bose-factor-independent part has been already calculated in Section 6, where we find [see Eqs. (6.24) and (6.29)]

P P

K$ (u"$E ),K "28(n a3/p)1@2E . (7.36) k k k 3 0 k The temperature-dependent part (associated with the Bose factors) of KT can be calculated using k the corresponding temperature-dependent self-energies RT (k,$E ) and RT (k, E ) given by 11 k 12 k Eqs. (7.15), (7.16), (7.17), (7.18), (7.19), (7.20), (7.21), (7.22), (7.23), (7.24), (7.25), (7.26), (7.27), (7.28), (7.29), (7.30), (7.31) and (7.32) and the chemical potential given by Eq. (7.12). To first order in k, we obtain

AB

AB

1 1 +2 1@2 JD m 1@2 KT" (2a )k# (2a !2kT) k# [2a #2a (d~2) k 2 1 0 2 4 2 +2 4JD m

AB

31 m fI (k ¹)k"!31(k ¹)ak . !2b !2b (d~2)] k"! 96 B 2 4 384p +2 0 B

(7.37)

We note that in KT, the infrared divergent terms D , D , O(d~1) and O(d~2) all cancel out exactly to k 2 4 lowest order in k, just as we showed in the ¹"0 case in Section 6. Combination of Eqs. (7.36) and (7.37) yields the following result of K : k (7.38) K "K0#KT"28(n a3/p)1@2+ck!31(k ¹)ak . 96 B k k k 3 0

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

This immediately leads to the following result for the energy spectrum of excitations in the second-order B—P approximation:

C

A

B D

28 n (¹)a3 1@2 0 E(2)(¹)"E(1)(¹)#KT"+c(1)(¹)k 1# !31(k ¹)ak . k k k 96 B p 3

(7.39)

Here E(1)(¹)"Je2#2e n (¹) fI K+c(1)(¹)k (for small k) k k k 0 0 n (¹) fI 4p+2an (¹) 0" 0 c(1)(¹)" 0 m m2

S

S

(7.40) (7.41)

are the first-order excitation energy and phonon velocity, respectively, obtained in the Popov approximation of Section 3. Eq. (7.39) shows that for small k, the quasiparticles are still phonons, with a velocity given by

A

B

31 a 28 n (¹)a3 1@2 0 k ¹. c(1)(¹)! c(2)(¹)"c(1)(¹)# p 96 + B 3

(7.42)

The second term in Eq. (7.42) corresponds to the ¹"0 result derived in Section 6, but now is a temperature-dependent quantity through the condensate density n (¹). The new correction term 0 is proportional to k ¹, but independent of n (¹). B 0 The relative importance of the correction terms can be estimated for 87Rb atoms (a"53 A_ and n"2.6]1012 cm~3 [1]) as follows. Denoting the second and third terms in Eq. (7.42) as dc and dc@, respectively, we have

A

B S

C A B D

28 n (¹)a3 1@2 4p+2an (¹) ¹ 3@2 0 0 K0.0033 1! dc, c(1)(0) , 3 p m2 ¹ #

(7.43)

and 31 a ¹ dc@,! k ¹K0.026 c(1)(0) , B 96 + ¹ # where c(1)(0) is given by c(1)(0)"J4p+2an/mK0.031(cm/s) .

(7.44)

(7.45)

As seen from Eqs. (7.43) and (7.44), the second-order corrections are very small for a dilute 87Rb atomic gas. Since c(1)(¹)P0 as ¹P¹ , one sees that the phonon velocity takes a small negative value near # ¹ . This unphysical result is due to the fact that Eq. (7.42) is derived assuming the correction terms # are small, namely (4p+2n a/m2)1@2<31(a/+)k ¹ . 0 96 B This condition is equivalent to ¹ !¹<(n1@3a)¹ . # #

(7.46)

(7.47)

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

75

The last inequality also defines the temperature range where the first-order Popov approximation is valid [see Eq. (3.59)]. Such a coincidence is not unexpected because in the second-order B—P approximation, we use the first-order Popov propagators, which are valid only if Eq. (7.47) is true. Therefore the results in the B—P approximation are subject to the same limitation near ¹ . We # should also emphasize that in deriving Eq. (7.42), we have used the high-temperature approximation, which really means that it is subject to the condition Eq. (7.14). Together with Eq. (7.47), this limits the validity of our results to temperatures which are not too close to either ¹"0 or ¹"¹ . # As far as we know, Eq. (7.42) is a new result. Moreover, we have shown for the first time that the infrared-divergent terms cancel out in physical quantities in the long-wavelength limit at finite temperature, just as in the case of zero temperature. However, it is not clear at present whether these infrared-divergent terms cancel out in physical quantities in general, i.e., not just in the long-wavelength limit. This could be checked by using our method to calculate the self-energies to higher order in k. As pointed out by Stoof [47], the infrared divergence we found in the second-order approximation might be closely related to the vanishing of R (kP0, u"0), a rigorous result (i.e. to all order 12 in perturbation expansion) first shown by Nepomnyashchii et al. in 1974 [48]. More precisely, one finds that R (k, u"0) approaches zero as 12 1 at ¹"0 , R (k, u"0)P -/ k (7.48) 12 k at ¹O0 .

G

Eq. (7.48) suggests that in the long-wavelength limit, the self-energies, in a rigorous theory, take the following asymptotic form:

G

1 at ¹"0 , (7.49) R(kP0, u"0)J 1`-/ k 1 at ¹O0 . 1`1@k These contain the singular infrared-divergent part in the denominator. What we found in this paper, however, is that, if expanded to first order, the self-energies R(kP0, u"0) are finite. On the other hand, if we go to second order, the self-energies R(kP0, u"0) are infinite, due to divergent terms D at ¹"0 [see Eq. (6.16)] and D at ¹O0 1 2 [see Eq. (7.33)]. Eqs. (6.16) and (7.33) suggest that the types of divergent terms we found in the self-energies are given by

G

R(2)(kP0, u"0)P

ln k

at ¹"0 ,

(7.50) at ¹O0 . 1 k A comparison of Eqs. (7.48) and (7.49) suggests the origin of the divergent terms found in our second-order calculations. In the perturbative calculation we have developed, one may be effectively expanding Eq. (7.49) in powers of ln k or 1/k, arriving at results consistent with Eq. (7.50),

G

R(kP0, u"0)J

1#O(ln k)#O(ln2 k)#2

at ¹"0 ,

(7.51) at ¹O0 . 1#O(1)#O(12)#2 k k Here the first term corresponds to the first-order self-energy result and the second term corresponds to the second-order self-energy result, and so on. This scenario gives an explanation of why our second-order self-energies contain the types of divergence shown in Eq. (7.50).

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

The above analysis puts a question mark on the traditional perturbation approach we have used. However, the fact that these divergent terms all cancel out in the end seems to suggest that the singular parts of the self-energies do not contribute to physical quantities, at least to second-order in fI , and therefore, the results we obtained in Section 6 and in the present section still remain meaningful. The exact cancellation also implies that the problem associated with infrared divergence may be solved by some kind of re-formulation. Recently, Bijlsma and Stoof [45] have studied the problem using the renormalization group (RG) techniques, and shown how, in principle, the infrared divergence problems can be solved using such techniques. As we remarked at the beginning of this section, we changed the order of limits kP0 and uP0 by introducing a gap d with respect to the Popov quasiparticle energy E in the frequency u. As one k can easily see from Eqs. (7.15) and (7.16), had we taken the limit uP0 (or dP0) together with kP0, we would get singular results, due to the presence of k2/d and k2/d2 terms. The change of the order of limits might have changed the physics contained in a and b (such as the type of 0 0 contributions that go like k/u, etc.). Further work on this topic is needed to clarify these questions. 7.3. Damping of phonons at high temperature As first shown by Beliaev [34] and rederived in Section 6, at zero temperature, the damping of low-momentum phonons is proportional to k5 [see Eq. (6.31)]. We can use the finite-temperature results of the present section to calculate the imaginary part of the self-energies. From these, we can calculate the phonon damping at high temperatures, close to ¹ . We briefly sketch these calcu# lations. We first calculate the imaginary parts of R and R at the pole u"E , starting from the results 11 12 k given in Eqs. (7.1) and (7.2). The imaginary part of RT (k, E #ig) is given by 11 k

P

dq f (E ) B 1 (!p)[g(E ,E )d(E !E !E ) Im R` ,Im R (k, E #ig)"n fI 2 11 11 k 0 0 (2p)3 E E 1 2 k 1 2 1 2 !g(!E ,!E )d(E #E #E )#g(!E , E )d(E #E !E ) 1 2 k 1 2 1 2 k 1 2 !g(E ,!E )d(E !E #E )] 1 2 k 1 2

A B CP

n fI 2 m 2 1 "! 0 0 4p +2 k

P P

#

= =

Ek

K

Ek

g(E , E ) 1 2 f (E ) dE 1 (e #D)(e #D) B 1 1 2 0 E2 / Ek~E1

K

dE

g(E ,!E ) 1 2 f (E ) 1 (e #D)(e #D) B 1 1 2 E2 / Ek`E1

dE

g(E ,!E ) 1 2 f (E ) . 1 (e #D)(e #D) B 1 1 2 E2 / E1~Ek

0

!

(7.52)

K

D

(7.53)

The function g(E , E ) is defined in Eq. (5.34) as well as at the beginning of the present section 1 2 following Eq. (7.2). Im R` in Eq. (7.53) consists of three parts, which involve different lower and 11 upper limits to ensure the positiveness of E . In arriving at Eq. (7.53), we have made use of the 2

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

77

following property of the d-function:

P

C K K D

df (x) ~1 dx f (x)d[ f (x)] " f (x) 2 1 2 1 dx

,

x / x0

where x is the solution of f (x) " 0. In the high-temperature limit, we approximate f (E ) by 0 2 B 1 k ¹/E , and after some manipulations we finally arrive at the following Taylor expansion of B 1 Im R` (k, E ) for small k: 11 k

AB C

A

B

n fI 2 m 2k ¹ p E2 E B 2DM # 4! !2DM k#(p!2) k Im R` " ! 0 0 11 k k D 4p +2 2 D2 k

A

#

B

D

75p!400 E3 k #O(E4) . #DM k D3 k 96

(7.54)

Here DM is a dimensionless divergent integral defined by k

P

D2 1 Ek dE f (E ) . DM , 1 E2#D2 B 1 k k ¹ 1 B 0

(7.55)

Following a similar procedure, we can also work out the Taylor expansion of Im R (!k,!E ) 11 k and Im R (k, E ) for small k, with the following results: 12 k

AB C A B B D AB C A B

n fI 2 m 2k ¹ p E B 2DM # 4! #2DM k Im R~ ,Im R (!k,!E !ig)"! 0 0 11 11 k k k D 4p +2 2 k

A

E2 75p!112 E3 k #O(E4) , !(p!2) k # !DM k D3 k D2 96

(7.56)

p E E2 n fI 2 m 2k ¹ k B 2DM # 4! k!DM Im R ,Im R ($k,$E $ig)"! 0 0 k k 12 12 k 2 D D2 4p +2 k

D

111p!256 E3 k #O(E4) . # k 96 D3

(7.57)

Using Eqs. (7.54), (7.56) and (7.57) in Eq. (7.37), we find that the imaginary part of the quantity K` as well as the imaginary part of the excitation energy E in the low-momentum limit is given by k k the very simple expression 3pa k ¹k . Im EBP " Im K " ! k k 8 B

(7.58)

We remark that, unlike the ¹ " 0 case discussed in Section 6, the imaginary parts of the self-energies exhibit divergent terms at finite temperature. This can be seen from the quantity DM which appears in various terms in the expressions for Im RB and Im R given above. However, k 11 12 these divergent terms all cancel out exactly in the final expression (7.58) for the damping. This

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H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

shows, once again, that the singular parts of the self-energies do not contribute to long-wavelength physical quantities in the Beliaev—Popov approximation at finite temperatures. The result in Eq. (7.58) shows that the damping of phonons is proportional to ¹k in the high-temperature region near ¹ . Sze´pfalusy and Kondor [49] also worked out the damping of # phonons near the phase transition temperature. Their result of damping is !ak ¹k, only a factor B of 3p/8 different from our result in Eq. (7.58) [50—52]. This paper was based on the dielectric formalism. Within the shielded potential approximation (See Ch. 5 of [8] that SK use, the only self-energy diagrams, to second order in the interaction, consist of (a) the third diagram in group a in Fig. 15 for R ; (b) the main diagram in group b in Fig. 17 for R . Since these second-order 5 11 5 12 self-energy diagrams are included in the more complete Beliaev—Popov approximation on which Eq. (7.58) is based, it is not surprising that the damping that SK found was similar, apart from an overall numerical factor of order unity. We also call attention to the fact that temperature-dependent correction to the real part of the phonon energy given by the last term in Eq. (7.39) is very similar to the damping given by Eq. (7.58). The phonon width in the low-temperature limit has been calculated by Popov [31] using a hydrodynamic Hamiltonian approach (see also Ref. [50]). In this limit, the phonon width is proportional to ¹4k.

8. Concluding remarks Most of this paper has concentrated on the technical details of evaluating the diagonal and off-diagonal Beliaev self-energies to second order in the interaction. In this brief concluding section, we would like to put some of our results into a larger context by discussing the connections to recent field-theoretic literature on uniform interacting Bose-condensed gases. The first-order Popov approximation worked out in detail in Section 3 is a simplified version of the complete self-consistent Hartree—Fock—Bogoliubov (HFB) approximation for the singleparticle self-energies. The latter is known to give rise to an energy gap in the long wavelength excitation spectrum. The reason for this gap, as we mentioned earlier, is that some of the anomalous self-energy diagrams are double-counted. However, the HFB approximation is of special interest since it can be used to generate a density-response function (by functional differentiation with respect to an auxiliary field) which exhibits the same spectrum as the singleparticle Green’s function computed in the Beliaev—Popov approximation. This is proven in detail by Cheung and Griffin [29] at finite temperatures and has been recently reviewed by Griffin [41] in the context of trapped atomic gases. We should also note that the Popov approximation to the self-consistent HFB has been recently formulated for a Bose gas trapped in an external potential well [53], extending the work in Section 3. In Sections 6 and 7, the second-order self-energies exhibited infrared-divergent terms both at ¹ " 0 (Beliaev) and ¹ O 0 (Popov). We showed explicitly that these divergent contributions cancel out in thermodynamic properties like the chemical potential, as well as in the low-frequency excitation energy and damping. In this paper, we have not discussed the physics of such divergent terms (see Section 6.3 of Ref. [8] and also Ref. [30]), or alternative formulations in which they do not explicitly appear (see however the remarks at the end of Section 7.2). There is an extensive literature on this topic, going back to the pioneer work of Gavoret and Nozie`res [22] at ¹ " 0. We call

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attention to the renormalization group type of calculation given by Popov [31] which derives a renormalized “quantum hydrodynamic” description of long-wavelength modes which is free of such divergent terms. Related recent studies using formal renormalization group (RG) techniques at ¹ " 0 [54] and near ¹ [45] have addressed this problem in a systematic way. # As we noted in the introduction of this paper, we have concentrated on calculating the single-particle Green’s functions at finite temperatures. However, it is known that in the presence of a Bose broken symmetry, the single-particle and density fluctuations are hybridized, leading to the same poles for both single-particle Green’s functions and the density-response function. The so-called “dielectric formalism” developed by Ma and Woo [55] (for a review, see Section 5 of Ref. [8]) is a diagrammatic approach which enables one to develop approximations for “regular” functions (the “regular” self-energy diagrams, by definition, cannot be split into two by cutting a single interaction line) which will lead to G (k, u) and s (k, u) having the same spectrum (but ab nn with different weights) in a Bose-condensed system. This dielectric formalism has been used in the so-called one-loop approximation to discuss the excitations of a dilute Bose gas at ¹ " 0 by Wong and Gould [56] and at ¹ O 0 by Talbot and Griffin [57]. At ¹ " 0, this one-loop approximation has been shown to include the second-order Beliaev spectrum (see Section 6 of this paper). We hope that the explicit calculations given in this paper (and especially Sections 6 and 7) will be useful in the on-going effort to provide a more complete and satisfactory understanding of excitations in a Bose-condensed gas. In principle, the thermal Green’s functions can be used to calculate the thermodynamic quantities, such as the pressure, the specific heat, etc. This was done for the first-order Popov approximation (as discussed in Section 3) by Popov [32]. We have not carried out these calculations for the second-order Beliaev—Popov approximation at finite temperatures in this paper. One of the reasons is that to do this, one needs the Green’s functions for all momenta and frequencies, which is difficult to obtain in analytical form. In this paper, we have given analytical expressions for the Green’s functions near the poles u " $E , which allowed us to determine the k explicit excitation energy spectrum. Nevertheless, it would be interesting to calculate some thermodynamic quantities by numerical methods, using the formalism developed in this paper.

Acknowledgements This paper is based on the Ph.D. Thesis of H. Shi (University of Toronto, 1997). We would like to acknowledge useful discussions concerning the infrared divergent terms with Dr. H.T.C. Stoof (University of Utrecht). This work was partially supported by funds from a research grant from NSERC of Canada.

Appendix A. Scattering theory and the t-matrix for Bose systems The typical inter-atomic potential involves a hard core. This poses a problem for the perturbation theory in terms of the bare potential v. Since v can be large, the first few terms in such perturbation expansion are no longer sufficient. Indeed, one has to sum over an infinite number of terms. This appendix reviews the so-called ladder approximation for the t-matrix C in a Bose gas.

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In this approximation, C involves the sum over all the ladder diagrams to infinite order in v, taking into account the repeated scattering of two particles in a gas. To understand this multiple scattering of two atoms in the presence of many other atoms, one has to first understand the simpler problem of multiple scattering of two atoms in a vacuum, since it can be shown that the t-matrix C is related to the free space or vacuum scattering amplitude (see Ch. 4 of Ref. [13]). A.1. Multiple scattering in vacuum We consider two particles of mass m interacting in vacuum via potential v(x). The Schro¨dinger equation in the center-of-mass coordinate system is given by (+2/2m@)(+ 2#k2)/(x)"v(x)/(x) .

(A.1)

Here m@"m/2 is the reduced mass and /(x) describes the wavefunction of the two particles with a separation x between them. In scattering problems, it is generally convenient to rewrite Eq. (A.1) as an integral equation representing an incident plane wave with wavevector k plus an outgoing scattered wave:

P

/k(x)"e*k > x! dy

m e*k@x~y@ m v( y)/k( y) . 4p+2 Dx!yD +2

(A.2)

At large x, i.e. far away from the influence of the potential, the scattered wave-function take the following asymptotic form: e*k{> x#f (k, k@)

e*k{x , x

xPR .

(A.3)

This defines the t-matrix for a transition from an incident wave vector k to a final wave vector k@, namely

P

P

m m dq f (k, k@) , ! dy e~*k{> yv( y)/k( y)"! v(q)/k(k@!q) , 4p+2 4p+2 (2p)3

(A.4)

where

P

/k(k@!q)" dy e~*(k{~q)> y/k( y) ,

P

v(q)" dy e~*q > yv( y) .

(A.5)

The Schro¨dinger equation (A.2) may be rewritten in momentum space

P

dq 1 v(q)/k(k@!q) 2e !2e !ig (2p)3 k{ k fI (k@, k) "(2p)3d(k@!k)! , 2e !2e !ig k{ k where we have defined the quantity fI (k, k@) /k(k@)"(2p)3d(k@!k)!

P

4p+2 fI (k, k@),! f (k, k@)" dqv(q)/k(k@!q) . m

(A.6)

(A.7)

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81

We note that this definition of fI differs from the usual definition of the free-space scattering amplitude f by a factor of !4p+2/m. Combining Eqs. (A.6) and (A.7), one obtains

P

dq v(k@!q) fI (k, q) , (A.8) (2p)3 2e !2e #ig k q which is an integral equation for fI in terms of the bare potential v. If the potential has no bound states, as will be assumed throughout this section, then the exact scattering solutions with a given boundary condition form a complete set of states and satisfy the following completeness relation in momentum space [58]: fI (k, k@) " v(k@!k)#

P

dk /k(k )/*k(k ) " (2p)3d(k !k ) . 1 2 1 2 (2p)3

(A.9)

The significance of Eq. (A.8) is that fI (k, k@) is well-defined even if v(r) has a hard core. The solution of Eq. (A.8) is discussed in textbooks on scattering theory. In particular in the long-wavelength limit (DkD"Dk@DP0), the solution of Eq. (A.8) is given by fI (k, k@) "

4p+2a #O(ka2) , m

(A.10)

where a is the s-wave scattering length. A.2. Ladder diagrams The previous discussion has been restricted to the scattering of two atoms in vacuum. We now turn to the much more complicated problem of two interacting atoms in a gas. As discussed in [34,59,13], in a dilute gas, the ladder diagrams are all of equal importance. For example, there exists a set of ladder diagrams, shown in Fig. 22, all of which are the same order in the gas density as the lowest-order Hartree—Fock diagrams are. Therefore, in a dilute gas, we need to sum over all the ladder diagrams, yielding the so-called t-matrix C, as illustrated by the hatched square in Fig. 8. The diagrammatic definition of C in Fig. 8 which can be written explicitly as 1 C(p , p ; p , p ) , v(k !k )! + v(q)G(0)(k !q, iu !iu ) 1 2 3 4 1 3 1 1 q b»q ,*uq ]G(0)(k #q, iu #iu )C(p !q, p #q; p , p ) , 2 2 q 1 2 3 4 dq "v(k !k )# v(q)C(p !q, p #q; p , p ) 1 3 1 2 3 4 (2p)3

(A.11)

P

1#f (ek1 q)#f (ek2 q) B ~ B ` . (A.12) ] iu #iu !(ek1 q!k)!(ek2 q!k) 1 2 ~ ` The four-dimensional vector p , (k , iu ) represents the momentum k and Matsubara frequency j j j j iu of a particle before ( j " 3, 4) or after ( j " 1, 2) scattering, similarly, q , (q, iu ). Since the j q interatomic potential v(r) is not time-dependent, v(q) does not depend on the fourth component iu q

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Fig. 22. Ladder diagrams that are equally important in the Hartree—Fock approximation.

of q, and neither does C(p !q, p #q; p , p ). One can thus perform the frequency sum 1 2 3 4 (A.13) + G(0)(k !q, iu !iu )G(0)(k #q, iu #iu ) 1 1 q 2 2 q *uq in Eq. (A.11), yielding the last factor in Eq. (A.12). Eq. (A.11) is known as the Bethe—Salpeter equation (more precisely, the ladder approximation to the Bethe—Salpeter equation). If Eq. (A.12) is iterated in powers of v (perturbation theory), we obtain the sum of all ladder diagrams. It is often more convenient to write C in the center-of-momentum frame of the scattering pair of atoms. Defining the total momentum and total Matsubara frequency of the scattering pair as K,k #k " k #k , 1 2 3 4 iu ,iu #iu "iu #iu , N 1 2 3 4 the denominator in the integral of Eq. (A.12) can be rewritten as

(A.14)

z!2(ek q!k) , ~ where the complex variable z is defined by

(A.16)

z,iu !m ,iu !+2K2/4m . N K N Writing C in terms of variables in the center-of-mass frame, we have

(A.17)

C(k, k@, K; z) , C(p , p ; p , p ) " v(k!k@) 1 2 3 4 F (K, k!q) dq ` v(q) C(k!q, k@, K; z) , # z!2(ek q!k) (2p)3 ~

P

(A.15)

(A.18)

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83

where k !k 2, k, 1 2

(A.19)

k !k 4, k@, 3 2

(A.20)

)#f (e ). (A.21) F (K, k!q),1#f (e B (1@2)K~k`q ` B (1@2)K`k~q Here k@ and k are the relative momenta of the scattering pair before and after the scattering, respectively; and the function F (k, q, K) incorporates the effect of the Bose statistics obeyed by the ` atoms involved in the intermediate scattering states. Eq. (A.18) can be compared with the integral equation for the free-space scattering amplitude fI given by Eq. (A.8). The free-space scattering amplitude fI describes the effect of the potential on the wavefunction of two atoms in a vacuum, while the t-matrix C describes the similar effect in a medium, that is, in the presence of other atoms in a Bose gas. The diagrams in Fig. 22 are redrawn in Fig. 23 in terms of the new quantity C defined in Fig. 8. These diagrams in Fig. 23 can be obtained by simply “replacing” v (the dashed lines) in the Hartree—Fock diagrams (c and d in Fig. 6) with C (the hatched squares). A.3. Connections between fI and C We next show the connection between the free-space scattering amplitude fI defined in Eq. (A.7) and the t-matrix C. The following analysis at ¹ O 0 is a generalization of the ¹ " 0 discussion given by Beliaev (for Bose systems) and Fetter and Walecka (for Fermi systems) [34,13]. Introducing the quantity s defined by C(k, k@, K; z) ,

P

dq v(q)s(k!q, k@, K; z) , (2p)3

(A.22)

one can verify that s satisfies the following integral equation:

P

dq F (K, k) ` v(q)s(k!q, k@, K; z) . (A.23) s(k, k@, K; z)"(2p)3d(k!k@)# z!2e #2k (2p)3 k The equation for s may be compared with the similar Eq. (A.6) for /k(k@), which suggests that s can be interpreted as an effective wavefunction of two particles in the Bose gas medium.

Fig. 23. Hartree—Fock self-energy diagrams in the t-matrix approximation.

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The connection of s to fI can be made by first considering the analogous function s0(k, k@, K; z), which denotes the solution of Eq. (A.23) with F (K, k) replaced by unity: `

P

1 s0(k, k@, K; z),(2p)3d(k!k@)# z!2e #2k k

dq v(q)s0(k!q, k@, K, z) . (2p)3

(A.24)

The corresponding quantity C0 is defined through s0 by [see Eq. (A.22)] C0(k, k@, K; z) ,

P

dq v(q)s0(k!q, k@, K; z) . (2p)3

(A.25)

We now rewrite Eq. (A.24) for s0 and Eq. (A.6) for /k1(k) as

P

dq v(q)s0(k!q, k@, K; z) (z!2e #2k)s0(k, k@, K; z)! k (2p)3 " (2p)3(z!2e #2k)d(k!k@) , k

P

dq (2e 1!2e #ig)/k1(k)! v(q)/k1(k!q) " 0 . k k (2p)3

(A.26) (A.27)

Multiplying Eq. (A.26) by /*k1(k) and integrating over k, we find that the second term on the left-hand side equals

P P P P

dk dq v(q)s0(k!q, k@, K; z)/*k1(k) (2p)3 (2p)3 "

"

P

dk dq s0(k, k@, K; z) v(q)/*k1(k!q) (2p)3 (2p)3 dk s0(k, k@, K; z)(2e 1!2e !ig)/*k1(k) . k k (2p)3

(A.28)

In the second line, we have used the variable change (k!q)Pk, and in the third line, we made use of the complex conjugate of Eq. (A.27). Using Eq. (A.28), Eq. (A.26) leads to the desired result:

P

dk [z!2e 1#2k#ig]s0(k, k@, K; z)/*k1(k) " (z!2e #2k)/*k1(k@) . k k{ (2p)3

(A.29)

Since the factor D,(z!2e 1#2k#ig) is independent of the integral variable k, we can take it k outside of the integral. We next multiply both side of Eq. (A.29) by D~1/k1(k ), and integrate over 2 k . Using the completeness relation Eq. (A.9), one arrives at 1

P

dk /*k1(k@)/k1(k ) 1 2 . s0(k , k@, K; z) " (z!2e #2k) 2 k{ (2p)3 z!2e 1#2k k

(A.30)

H. Shi, A. Griffin / Physics Reports 304 (1998) 1—87

85

Using the complex conjugate of Eq. (A.6) in Eq. (A.30), one finds

P

C

D

1 1 # fI *(k , k@) . (A.31) 1 z!2e 1#2k 2e 1!2e !ig k{ k k We now multiply Eq. (A.31) by v(k!k ) and integrating over k . The definition of s0 and C0 in 2 2 Eqs. (A.24) and (A.25) plus Eq. (A.6) leads to s0(k , k@, K; z)"/k (k )# 2 { 2

dk 1 /k (k ) (2p)3 1 2

P

C

D

1 dq 1 fI (k, q) # fI *(k@, q) . (A.32) (2p)3 z!2e #2k 2e !2e !ig q k{ q This last expression has the important feature that it gives C0 completely in terms of the free space (or vacuum) scattering amplitude fI . We are now in position to express the t-matrix C in the medium, as given by Eq. (A.18), in terms of fI . A slight rearrangement of Eq. (A.23) yields C0(k, k@, K; z)"fI (k, k@)#

P

1 dq s(k, k@, K; z)! s(k!q, k@, K; z) z!2e #2k (2p)3 k F (K, k)!1 "(2p)3d(k!k@)# ` C(k, k@, K; z) . (A.33) 2!2e # 2k k Comparing this with Eq. (A.26) divided by z!2e #2k shows that the operator on the left-hand k side of Eq. (A.33) is just the inverse of s , which means s can be expressed in terms of s as follows: 0 0 dq F (K, q)!1 s (k, q, K; z) ` C(q, k@, K; z) . (A.34) s(k, k@, K; z)"s (k, k@, K; z)# 0 (2p)3 0 z!2e #2k q Taking the convolution with v in Eq. (A.34), we find that

P

C

D

P

C

D

dq F (K, q)!1 C0(k, q, K; z) ` C(q, k@, K; z) . (A.35) (2p)3 z!2e #2k k Eq. (A.35) expresses C in term of the reduced t-matrix C0, while C0 in turn can be expressed in terms of the free-space scattering amplitude fI using Eq. (A.32). In other words, through the intermediate function, C0, we can relate C directly to fI . This result is important because it enables us to deal with an interatomic potential with a hard core. In that case, the iteration of the integral equation (Eq. (A.11)) for C in powers of v makes only formal sense, since each term is large. In contrast, Eqs. (A.32) and (A.35) show that C is well-defined as long as fI is. C(k, k@, K; z)"C0(k, k@, K, z)#

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[50] L.P. Pitaevskii, S. Stringari, Phys. Lett. A 235 (1997) 398. This reference derives the results in Eq. (7.58) in a simple manner. [51] S. Giorgini, Phys. Rev. A 57 (1998) 2949. [52] W.V. Liu, Phys. Rev. Lett. 79 (1998) 4056. [53] D.A.W. Hutchinson, E. Zaremba, A. Griffin, Phys. Rev. Lett. 78 (1997) 1842. [54] C. Castellani, C. Di Castro, F. Pistolesi, G.C. Strinati, Phys. Rev. Lett. 78 (1997) 1612. [55] S.K. Ma, C.W. Woo, Phys. Rev. 159 (1967) 165; P. Sze`pfalusy, Acta Physica Hungaria 19 (1965) 109. [56] V.K. Wong, H. Gould, Ann. Phys. (New York) 83 (1974) 252. [57] E. Talbot, A. Griffin, Ann. Phys. (New York) 151 (1983) 71. [58] L.I. Schiff, Quantum Mechanics, McGraw-Hill, New York, 1968. [59] V.M. Galitskii, A.B. Migdal, Soviet Phys. JETP 7 (1958) 96. [60] A.S. Parkins, D.F. Walls, Phys. Rep. 303 (1998) 1.

Note added in proof For a recent review of the theory of trapped Bose-condensed gases, with extensive references in the literature, see Ref. [60].

Physics Reports 304 (1998) 89—144

Computer simulation of time-resolved optical imaging of objects hidden in turbid media K. Michielsen!, H. De Raedt!,*, J. Przeslawski", N. Garcia" ! Institute for Theoretical Physics and Materials Science Centre, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands. [email protected] " Laboratorio de Fı& sica de Sistemas PequenJ os y Nanotecnologı& a, Consejo Superior de Investigaciones Cientı& ficas, Serrano 144, Madrid E-28006, Spain. [email protected] Received January 1998; editor: M.L. Klein

Contents 1. Introduction 1.1. Breast tissue imaging: Physical basis 1.2. Light propagation in turbid media 1.3. Outline 2. Model 2.1. Algorithm 2.2. Simulation software 3. Comparison with experiment 4. Direct imaging 5. Determination of tissue optical properties 5.1. Experimental setup 5.2. Estimation of the reduced scattering and absorption factor 5.3. Transillumination 5.4. Reflection 6. Data processing method: Theory 6.1. Instantaneous light-intensity detection

92 93 94 95 95 97 98 100 103 104 105 106 108 112 114 116

6.2. Integrated light-intensity detection 6.3. Bounds on the difference 6.4. Illustrative examples 7. Image processing technique: Simulation results 7.1. Effect of the source type and the time gate 7.2. Small objects 7.3. System size and mesh size 7.4. D(r) versus D 0 7.5. More than one object 7.6. Random fluctuations 7.7. 3D examples 8. Image processing technique: Experimental results 9. Summary References

* Corresponding author. 0370-1573/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 2 3 - 4

116 117 118 120 120 124 124 126 126 130 133 135 137 138

COMPUTER SIMULATION OF TIME-RESOLVED OPTICAL IMAGING OF OBJECTS HIDDEN IN TURBID MEDIA

K. MICHIELSENa, H. DE RAEDTa, J. PRZESLAWSKIb, N. GARCIAb a

Institute for Theoretical Physics and Materials Science Centre, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands. [email protected] b Laboratorio de Fı´ sica de Sistemas PequenJ os y Nanotecnologı´ a, Consejo Superior de Investigaciones Cientı´ ficas, Serrano 144, Madrid E-28006, Spain. [email protected]

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

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Abstract We review research on time-resolved optical imaging of objects hidden in strongly scattering media, with emphasis on the application to breast cancer detection. A method is presented to simulate the propagation of light in turbid media. Based on a numerical algorithm to solve the time-dependent diffusion equation, the method takes into account spatial variations of the reduced scattering and absorption factors of the medium due to the presence of objects as well as random fluctuations of these factors. It is shown that the simulation method reproduces, without fitting, experimental results on tissue-like phantoms. Using experimental and simulation results, an assessment is made of the reliability for extracting the reduced scattering and absorption coefficients of the medium from time-resolved reflection and transillumination data. The simulation technique is employed to study the conditions for locating mm-sized objects immersed in a turbid medium, by direct, time-resolved imaging. We discuss a simple method to enhance the imaging power of the time-resolved technique. The mathematical justification of the method, as well as some applications to simple problems, is given. The simulation technique is employed to demonstrate the effectiveness of the data processing technique. Results of time-resolved reflection experiments and simulations are presented, showing that the use of the latter allow us to locate 1 mm diameter objects under conditions which would prevent detection otherwise. Our results demonstrate that the combination of simulation and the appropriate processing of the diffusive part of the time-resolved reflected or transmitted light intensity may substantially increase the potential of the time-resolved near-infrared diffusive light imaging technique as a diagnostic tool for breast cancer detection. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 87.59.!e; 42.30.!d Keywords: Time-resolved imaging; Diffusion equation; Breast cancer

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1. Introduction Imaging through scattering media is a challenging problem and has motivated extensive scientific research. The fields of application vary widely, ranging from atmospheric science and oceanics to medicine. One potential application, breast-tissue imaging, has been drawing special attention. Breast cancer is the most common cancer in women and one of the leading causes of death in women. In many cases the cancer can be cured when it is small and localized in the breast. Hence, non-invasive diagnostic methods for detection of breast cancer at an early stage are of great importance. A successful screening method should be able to distinguish small tumors from surrounding healthy tissue before metastasis occurs [1]. The ultimate goal is to image and characterize millimeter-sized objects in 40—100 mm-thick human tissue [2]. The female breast is also prone to benign diseases. Therefore it is also very important to discriminate between benign and malignant lesions. Recent developments in breast imaging and their present clinical importance is reviewed in Ref. [3]. The most common imaging method at this moment, X-ray mammography, is not very sensitive to differences between normal fibrotic tissue and cancer, making it less suitable for imaging young dense breasts which usually are fibrotic [4,5]. Moreover both X-ray and radioisotope imaging techniques expose the body to potentially harmful, ionizing radiation and increase the risk of contracting cancer. Therefore, one would like to avoid using these techniques for extensive routine screening. Ultrasound lacks the resolution to detect objects with linear dimension smaller than a few millimeters and, like X-ray mammography, cannot distinguish between benign and malignant tumors [2,6]. Magnetic resonance imaging of the breast [7] is a very powerful technique with submillimeter resolution and the ability to detect specific chemicals but its high operating costs [6,8] make it less suited as a routine screening modality [2,5,8]. Several clinical experiments have been performed to study the (dis)advantages of each of the above-mentioned methods [9—13]. The need for diagnostic imaging equipment that is non-invasive, safe, compact and capable of monitoring tissue chemistry in vivo, together with the advent of picosecond pulse lasers in the near-infrared wavelength regime and fast optical detectors able to resolve such pulses, has increased the interest in optical techniques as a tool for early breast cancer detection [3,14—28]. Already in the Victorian age doctors conducted the first experiments in mammography using candle light [2]. In the mid-1800s, British physicians used tissue transillumination to detect scrotal cancer by holding a lamp behind the testes and observing the shadows resulting from the presence of tumors. Ewen [29] first investigated light transmission through the breast in the late 1920s by exploring various light sources and by visual observation of the transmitted light. Cutler improved this technique by using a special tungsten lamp as a light source [29]. The instrumentation used had problems in providing enough light intensity for the human eye to discern variations in the tissue. Although various attempts [4,30—32] were made to improve the instrumentation, the sensitivity of the human eye or photographic material appears to be insufficient to differentiate the minute variations in light intensities corresponding to healthy and cancerous tissue. Initial attempts to image the breast using continuous-wave near-infrared light intensity measurements demonstrated very poor spatial resolution and indicated that this breast transillumination methods had no utility as a screening method [33]. According to a more recent study [34] conventional light-scanning equipment does not add any value to physical examination and

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mammography in breast cancer diagnosis. Diagnostic equipment [35,36] based on continuous wave illumination and modified X-ray backprojection tomographic algorithms [35—37] has been developed. Time-resolved transillumination with short laser pulses has also been studied for the extraction of tissue parameters and tumor detection [21,38]. An alternative is to modulate the intensity of the light source at frequencies between some ten to several hundred megahertz and to measure the demodulation and phase shift of the transmitted light [39]. Recently, by the use of frequency-domain techniques, optical mammography was evaluated in a clinical study [40]. Within approximately 3 min transillumination images of a compressed mamma were obtained by scanning. Current time-resolved techniques require much longer times to record images [1,38,41]. At present, none of the techniques described above are in a clinically usable stage because most of them do not (yet) provide a reliable distinction factor between normal and cancerous tissue. 1.1. Breast tissue imaging: Physical basis For diagnostic optical imaging the wavelengths of interest are in the range 650—1300 nm, i.e. the so-called therapeutic window: Near-infrared light is not as strongly absorbed by human tissue than visible light [4,42] and so will have higher transmission and less likelihood of causing burns. Near-infrared transillumination can be used to look at the structure and function of biological systems [43,44] and is now used for various applications including the monitoring of blood oxygenation in tissue [45], muscle oxygenation [46,47] and physiology [48], brain oxygenation in new-born infants [2,49—52] and adults [53,54], and the flow of blood and specific chemicals, such as oxyhaemoglobin or glucose, in arteries close to the surface of the skin. The latter provides a noninvasive way for diabetics to monitor their blood glucose levels [2,55—57]. In contrast to optical mammography these applications may have less stringent technical requirements because either the tissue does not scatter very much, such as the brains of newborn babies [2], or the tissue is only a few millimeters thick. Breast imaging with red and near-infrared light relies on the differences in the optical properties of breast tissue and cancerous tissue. Tumors can be regarded as optical inhomogeneities in the sense that they have other absorption and scattering factors than the surrounding healthy tissue. In cancerous tissue the incident light is strongly absorbed due to the higher blood content in the neovascular zone surrounding most malignant tumors [58,59]. The absorption of light in tissues is due to chromophores such as the heme pigment of hemoglobin, myoglobin, and bilirubin, the cytochrome pigments of the respiratory chain in the mitochondria, and melanin pigment [60,61]. The light scattering in tissues is due to discontinuities in refractive index on the microscopic level, such as the aqueous-lipid membrane interfaces surrounding and within each cell or the collagen fibrils within the extracellular matrix [60,61]. Light propagation in breast tissue, which is a strong scattering medium, may be described by the time-dependent diffusion equation (see below) involving two optical properties: The absorption factor k and the reduced scattering factor k@ . For breast tissue there appears to be a large variation ! 4 in the values for the reduced scattering and absorption factor. For various in vitro breast tissue samples the reduced scattering factor k@ "0.7—1.4 mm~1 and the absorption factor k " 4 ! 0.02—0.07 mm~1, for wavelengths near 800 nm [62,63]. In vitro measurements on breast tissue for wavelengths of 653 nm suggest that k@ +0.4 mm~1 and k (0.02 mm~1 [64]. In vivo measurements 4 !

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at 800 nm on six volunteers suggested for k@ "0.72—1.22 mm~1 and for k "0.0017—0.0032 mm~1 4 ! [38]. A smaller study on two volunteers of 30 and 58 years old, yielded values of k@ "1.13 and 4 0.76 mm~1 and k "0.0068 and 0.0028 mm~1, respectively at a wavelength of 753 nm [65]. In vivo ! experiments on one volunteer having a rather large abdomen and back tumor show that tumor and normal tissue optical properties and physiological parameters differ substantially within the studied individual [66]. The absorption factors of tumor tissue are 2—3 times larger than the absorption factor of normal tissue and this at all studied wavelengths [66]. The scattering changes are less significant (the reduced scattering factors of tumor tissue are somewhat smaller than the reduced scattering factors of normal tissue), but exhibit consistent wavelength-dependent behavior. In time-resolved optical imaging experiments on breast phantoms one generally assumes that the reduced scattering factor of the breast tissue is of the order of 1 mm~1, whereas the absorption factor of the breast tissue is of the order of 0.01 mm~1, at a wavelength of 800 nm [24,67,68]. 1.2. Light propagation in turbid media The dilemma encountered with optical imaging of human breasts is that although the wavelength of the light can be chosen such as to minimize the absorption it is impossible to avoid the blurring of the images due to strong scattering by the tissue. In contrast to X-ray mammography in which the incident X-rays are usually scattered only once, most near-infrared light travelling through a few centimeters of tissue has been scattered several thousand times before it reaches the detector [62,64]. Light entering a strongly scattering medium (e.g. the woman breast) that contains one or more small objects (with reduced scattering and/or absorption coefficients different from that of the surrounding medium, e.g. malignant tumors) can arrive at a detector by two different routes: (1) By ballistic transport, in which the light travels without scattering through the sample to yield a projection of the object(s). This light arrives first at the detector, within a time interval [0, q ] 1 where q +l/v, l is the distance between the source and the detector and v is the velocity of light in 1 the medium. For l"6 cm and v"220 km/s, q +300 ps. The unscattered light (ballistic compon1 ent) can be selected by means of a very fast (ps) time gate. A large variety of time-gating techniques are developed based on the degree of polarization [69—72] coherent detection [73—75] and selection of the length of the photon paths [1,15,18,38,76—80]. A systematic study of the time-gating technique has shown that it is highly sensitive with respect to spatial variations in the absorption or reduced scattering factors [38], in particular under conditions that are similar to those of biological systems of interest [38]. Recent experiments have shown that the spatial resolution of the time-gating technique can also be obtained with the absorption method [81]. However, any technique based on the detection of unscattered light only for image formation is subject to intrinsic physical limitations. As the signal at the detector contains only an exponentially small fraction of the number of photons in the light pulse emitted by the source, the signal-to-noise ratio is low. In addition, the intensity of these photons decreases exponentially with l, effectively limiting the tissue sample size. For applications to optical breast imaging, the intensity of these photons seems to be too low to be of practical use (there obviously is a limitation on the amount of power of the incident light pulse) [38]. (2) By diffusion, in which the light is scattered many times before it reaches the detector. In this regime the strong scattering by the medium blurs the variations in the transmitted or reflected light

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that result from the local (small) variations of the absorption and diffusion factor induced by the objects immersed in the medium. Only if the diffusion factors of the object and the medium differ considerably, direct detection (i.e. without image processing) is possible [82]. Diffusive light imaging can be performed in the frequency domain [39,83—86] or in the time domain [23—25,38,87]. In the former case a continuous intensity-modulated light source is used to generate diffusive monochromatic light-intensity waves in the medium. In many respects these waves behave like damped propagating waves [83—97]. Inverse scattering techniques in combination with perturbation schemes have been employed to compute spatial variations in the absorption factor and the diffusion coefficient (and hence the location of the objects) from the measured phase and amplitude of these waves at various source and detector positions [83,98]. In the time domain, short light pulses are used to excite a spectrum of light-intensity waves, the frequency-range being determined by the scattering and absorption factors of the medium. This approach requires the use of a detection system with a time resolution that is sufficiently high to cover the relevant range of frequencies. In theory the time and frequency-domain data are intimately related through Fourier transformation. However, usually the former contains more information because frequencydomain measurements are carried out for at most a few selected modulation frequencies and this information is not sufficient to determine the time-dependent response of the medium to a light pulse. For this reason the emphasis of this paper is on simulations and experiments performed in the time-domain. In practice, the differences between frequency and time-domain methods are mainly technical, and the choice is a trade off between sensitivity, time resolution, complexity and cost. 1.3. Outline The organization of the paper is as follows. Section 2 introduces the model, i.e. the timedependent diffusion equation (TDDE), and describes the algorithm that we employ to solve the TDDE. In Section 3 the simulation results for tissue-like phantoms are compared to experimental data, establishing the validity of the TDDE model and the simulation approach. A discussion of the limitations of the direct imaging approach is given in Section 4. In Section 5 we assess the quality of the estimates of optical properties of the tissue, as obtained from the analysis of the time-resolved transillumination and reflection data. In Section 6 we give the theoretical justification of the time-resolved image processing method that facilitates the detection of mm-sized objects. Sections 7 and 8 are devoted to the assessment of the effectiveness of the image processing method. Section 9 summarizes our results.

2. Model Most of the light entering a turbid medium (possibly containing one or more small objects with reduced scattering and/or absorption factors different from those of the medium) is scattered many times before it reaches the detector. For weakly absorbing media the propagation of the light is, to a good approximation, described by the time-dependent diffusion equation [39,99,100] (TDDE) I(r, t)/t"+ ) D(r)+I(r, t)!vk (r)I(r, t)#S(r, t) , !

(1)

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where I(r, t) is the intensity of light at a point r and at time t, D(r)"v/3[k@ (r)#k (r)] , (2) 4 ! is the diffusion coefficient, k@ (r) is the reduced scattering factor, k (r) denotes the absorption factor, 4 ! and v is the velocity of light in the medium in the absence of objects. The light source is represented by S(r, t). The presence of objects in the medium is reflected by spatial variations in the absorption factor and/or the reduced scattering factor (and hence also in the diffusion coefficient). In general, both the absorption and the reduced scattering factor of the medium will fluctuate randomly around their spatial averages, denoted kN and kN @ , respectively. For weakly absorbing media, ! 4 kN ;kN @ . ! 4 For purposes of notation it is convenient to write the intensity distribution as a “vector” DI(t)T, i.e. I(r, t)"SrDI(t)T where S.D.T denotes the inner product of two vectors. The intensity at a point r and time t is given by SrDe~tHDsT, where DsT is the intensity distribution at time t"0. Obviously, since we are dealing with intensities, SrDe~tHDsT50 for any r and any intensity distribution s. The formal solution of Eq. (1) reads

C

P

DI(t)T"e~tH DI(0)T#

D

t

dt@ et{HDS(t@)T , 0 with the time evolution of the light intensity governed by the “Hamiltonian” H"!+ ) D(r)+#»(r) ,

(3)

(4a)

and where, for later convenience, we have introduced the “potential’ »(r)"vk (r) . (4b) ! According to Eq. (3) the time evolution of the light intensity at time t#q is related to the light intensity at time t through

C

P

DI(t#q)T"e~qH DI(t)T#

q

D

dq@ eq{HDS(t#q@)T ,

(5)

0 where q denotes the time step. From Eq. (5) it follows that all we need to solve the TDDE (1) is an algorithm to compute exp(!qH)A(r) for arbitrary A(r). Denoting the one-sided Fourier transform of DA(t)T by DA T":= dt e~*utDA(t)T, Eq. (1) can be u 0 written as DI(0)T#iuDI T"!HDI T#DS T , u u u

(6a)

or DI T"(H#iu)~1(DS T!DI(0)T) , (6b) u u showing that DI T is the solution of a time-independent diffusion equation, with a complex, u frequency-dependent “absorption” coefficient vk (r)#iu. Frequency-domain experiments provide ! direct information about DI T, for u corresponding to the frequency of the amplitude modulation of u the light source intensity [101—103]. The intensity DI(t)T can be retrieved from I by solving Eq. (6) u for several u and performing the inverse Fourier transform. However, in numerical work it is more efficient to solve Eq. (1) directly in the time domain. On the other hand, if only the solution of

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Eq. (6) for a few selected frequencies is desired, solving Eq. (6) for these frequencies may be the numerical method of choice. For practical applications the TDDE (1) and the formal solutions (3) and (5) have to be supplemented by boundary conditions that mimic those of the actual experimental set-up. For the applications we have in mind these boundary conditions take a non-trivial form: Usually, the surface of the medium (e.g. a breast) has an irregular shape and is partially absorbing and partially reflecting. Furthermore, the source and detectors are not in direct contact with the medium as the light enters and leaves the sample through the skin (surface). Except for the shape of the sample, incorporating all these elements into the boundary conditions is rather difficult and it is therefore expedient to assume that the surface is reflecting and/or absorbing. Comparison of simulation and experimental data should tell us if this is a reasonable assumption or not. 2.1. Algorithm We have developed an algorithm to compute exp(!qH)A(r) [104], based on the fractal decomposition of matrix exponentials proposed by Suzuki [105]. It is accurate to second order in the spatial mesh size d and to fourth order in the temporal mesh size q. Conceptually the algorithm is closely related to the one that we developed for the time-dependent Schro¨dinger equation [106]. The first step in setting up a numerical method to solve the TDDE (1) is to discretize the derivatives with respect to the spatial coordinates. The simplest approximation scheme having satisfactory properties is [107]

A

BK

K I(r, t),! D(r) I(r, t) x x x

D #D i,j,kI +! i`1,j,k i`1,j,k 2d2 r /(id,jd,kd)

D #2D #D D #D i,j,k i~1,j,k I ! i~1,j,k i,j,kI # i`1,j,k , i,j,k i~1,j,k 2d2 2d2

(7)

where I "I(r"(id, jd, kd)) and D "D(r"(id, jd, kd)). For the derivatives with respect to i,j,k i,j,k y and z we use expressions similar to Eq. (7). Proceeding as in the case of the time-dependent Schro¨dinger equation, the time-step operator e~qH is approximated by a product of matrix exponentials. An approximation correct to second order in the time step q is given by [108,109] e~qH+e~qKz@2e~qKy@2e~qKx@2e~qVe~qKx@2e~qKy@2e~qKz@2 .

(8)

Instead of using fast-fourier-transform techniques [110] to compute the quantities such as e~qKx@2A(r), we replace e~qKx@2 by a first-order product-formula approximation and obtain [111]

C A C A

1 1#e~qai,j,k e~qKx@2+X(q/2)"< < 2 1!e~qai,j,k j,k i|E 1 1#e~qai,j,k ] < 2 1!e~qai,j,k i|O

B D B D

1!e~qai,j,k (i,i`1) 1#e~qai,j,k

1!e~qai,j,k (i,i`1) , 1#e~qai,j,k

(9)

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where the triples (i, j, k) appearing in Eq. (9) represent a point on the lattice, E and O are the sets of even and odd numbers, respectively, and a "d~2(D #D )/2. The superscripts (i, i#1) i,j,k i,j,k i`1,j,k labeling the two-by-two matrices indicate that this matrix operates on the vector (I , I ) i,j,k i`1,j,k y z only. In Eq. (8) we replace e~qK @2 and e~qK @2 by similar approximations, ½(q/2) and Z(q/2), respectively. The resulting product formula remains correct to order q2. It is also of interest to note that all the matrix elements of both e~qKx@2 and X(q/2) are positive so that approximation (9) has the desirable feature that it will never lead to negative light intensities. For »(r)50 (the case of interest) the explicit, second-order algorithm defined by e~qH+S (q)"Z(q/2)½(q/2)X(q/2)e~qVX(q/2)½(q/2)Z(q/2) (10) 2 is unconditionally stable. Indeed, if EME denotes the eigenvalue of the matrix M of largest absolute value, we have EZ(q/2)½(q/2)X(q/2)e~qVX(q/2)½(q/2)Z(q/2)E41 ,

(11)

which is the condition expressing numerical stability of the scheme [107]. The accuracy of the second-order algorithm may be insufficient if we want to solve the TDDE for long times. In practice, this is only a minor complication because the second-order algorithm can be reused to build an algorithm that is correct to fourth-order in the time step. According to Suzuki’s fractal decomposition [105], S (q)"S (pq)S (pq)S ((1!4p)q)S (pq)S (pq) (12) 4 2 2 2 2 2 will be an approximation to the time-step operator that is correct to fourth-order in q provided p"(4!41@3)~1. In our simulations we have used both (10) and (12) and obtained quantitatively similar results. The contribution from the source S(r, t) is computed using the standard Simpson rule [112]

P

e~qH

q

q dq@ eq{HDS(t#q@)T+ (e~qHDS(t)T#4e~qH@2DS(t#q )T#DS(t#q)T) , 2 6

(13) 0 which is correct to fourth-order in q. From the structure of Eq. (9) and S (q) it is clear that the propagation of light over a time step 2 q has been reduced to elementary operations: Repeated multiplications of two elements of a vector by the corresponding two elements of another vector (in the case of e~qV), or of matrix-vector multiplications involving two-by-two matrices only. The resulting algorithm is fast, stable and flexible. For practical applications to the tumor detection problem it is important that the software can deal with irregularly shaped samples. As the Suzuki-product-formula-based algorithm presented above operates on numbers labeled by real-space indices only, it is as easy to solve the TDDE for a particular shape as it is to solve the TDDE for a rectangular box. 2.2. Simulation software Our current version of the software solves Eq. (1) in two and three dimensions subject to perfectly reflecting and/or perfectly absorbing boundary conditions. The intensity of light transmitted by the sample is collected by detectors located at r"(¸ ,y, z), where ¸ denotes the size of the x x simulation box in the direction of the incident light [113]. The reflected light intensity is recorded

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at r"(0, y, z). The light source is placed at x"0 [i.e. S(r, t)"0 unless r"(0, y, z)]. We have carried out simulations using sources of variable size, including the cases of a point source [S(r, t)"S (t)d(x)d(y!y )d(z!z )] and uniform illumination [S(r, t)"S (t)d(x)]. At t"0 the 0 0 0 0 source starts to illuminate the system, until t"t when it is turned off. Detection of the light 1 intensity starts at t (t 't ). Our simulation software allows the detectors to record the instan$ $ 1 taneous or the time-integrated light intensity. Our software generates real-time images when run on a PC or data files suitable for processing by IBM’s Data Explorer. The pictures are then collected into digital video’s (Autodesk Animator .FLC format, typically less than 1 Mb/video) suitable for playback on PC’s and X-window systems. These video’s give a good impression of how the light intensity diffuses through the medium. Included in these video’s are the (time-dependent) intensities recorded by the detectors. The digital video’s also serve as an archive for the simulation results. For a representative two-dimensional (2D) sample of 71 mm]71 mm, using a mesh size of d"1 mm (¸ "¸ "71), it takes about 1 min on a Pentium Pro 200 MHz system to carry out x y 1000 time steps, including data analysis and visualization. On a Pentium II 266 MHz system it takes about 2.8 s to carry out the same calculation but without data analysis and visualization. Calculation of the time evolution of the light intensity for a three-dimensional (3D) grid of 63]63]63"250047 points and 1000 time steps takes 390 s on the same machine. The number of (floating point) operations required to solve the TDDE by the algorithm based on the fractal decomposition of matrix exponentials [104] scales linearly with the dimensionality of the system, the number of time steps and the total number of grid points. Clearly the CPU times obtained for the (representative) 2D and 3D simulations do not seem to reflect this scaling behavior: On the basis of the CPU time for the 3D system we would expect the 2D simulation to take 2]390]71]71/(3]63]63]63)+5.2 s, almost a factor two more than actually measured. This extra speed-up of a factor two is due to the fact that the 2D problem fits in the cache memory of the Pentium II processor. Calculation of the time evolution of the light intensity for a 3D grid of 63]63]63"250 047 points and 1480 time steps takes 6 minutes on a Cray C98 (single processor), 42 minutes on a SGI Power Challenger (single processor), and 65 minutes on an IBM RS/6000 model 43P workstation, including data analysis and visualization processing. For exploratory purposes it is advantageous to simulate 2D systems: simulation and visualization only takes a few minutes. Then, for selected cases, we simulate the corresponding 3D system. On a qualitative level there seems to be little difference between the results obtained from 2D or 3D simulations. In fact, we never encountered a 3D simulation that forced us to change a conclusion drawn from 2D simulation data. As the effect of particular implementations, compilers and computer systems on the performance of specific simulation algorithms can be substantial, it is difficult to compare different approaches on a quantitative basis but it may be of interest to compare, on a conceptual level, our approach with others. Techniques which have been developed to calculate light propagation in turbid media can be classified into stochastic methods, such as the Monte Carlo (MC) simulations [114—123] and random walk models [124,125], or deterministic methods which are based on the solutions of the diffusion equation. MC simulations trace a large number of individual paths of photons propagating through a scattering medium. The scattering events are determined by a probability function that matches

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the scattering coefficient. At each scattering event part of the photons are absorbed. The amount of absorption is determined by the absorption coefficient. The scattering angle is given by the scattering phase function. A MC algorithm has the advantages of being conceptually simple and of being flexible with respect to geometry, phase functions and boundary conditions. A disadvantage of MC simulations is the large amount of computer time needed to obtain statistically good quality data. In general, the numerical methods based on the solution of the diffusion equation are much faster than the MC methods to describe the light propagation in tissue, if the scattering and absorption mechanisms allow the use of the diffusion approximation. The diffusion equation, including its time-dependent form, can be solved analytically for a homogeneous medium and rather simple geometries [88,126]. More flexible than analytical methods are numerical methods, such as the finite element method (FEM) [127—132], the finite difference method (FDM) [24], the method of exact enumeration [133] and our method [104], described above. The basic idea of the FEM is that a model that satisfies the time-dependent diffusion equation is divided into elements, and an approximate solution to the equation is calculated over each element. The solution for the whole model can be derived from the algebraic solutions for each element. The mean time of flight can be calculated directly from the first moment of the time-dependent light intensity [129]. In the FDM the diffusion equation is converted into a finite difference equation for finite steps of the spatial variables and the temporal variable. The method of exact enumeration is an iterative method that accounts for the spatial and temporal distribution of photons after a discrete pulse is applied to a scattering medium. The method is equivalent to performing MC calculations for movement on a lattice, but by use of the method of exact enumeration, all random-walk configurations are taken into account properly even when only a few photons would reach a site as realizations of MC experiments. The algorithm we use is identical to the FDM as far as the treatment of the spatial variables is concerned.

3. Comparison with experiment We have tested our software and the validity of the TDDE model by comparing simulation data [104,134] with the experimental results of Ref. [38], concerning a range of optical parameters which from the point of view of breast imaging range from realistic to unrealistic. We have simulated exactly the same systems as those studied in Ref. [38], i.e. systems of size 40 mm]127 mm, with a spatial mesh size d"1 mm, a time step q"1 ps, t "7 ps, t "350 ps and 1 $ the speed of light in the medium v"0.222 mm/ps [135]. We verified (by reducing the mesh size and time step) that the numerical results are, for all practical purposes, exact. The simulation itself is carried out in a manner identical to the procedure used in the time-gating technique [38]. The sample is illuminated uniformly by the pulsed light source. This corresponds to the experimental situation in which the phantom is located on an x—y stage and is moved in the horizontal plane under computer control. The detector accumulates the light intensity over the interval t (t(t#*t where *t denotes the time gate. In Fig. 1 we depict our simulation results $ (dashed curves) and the experimental data (solid curves) which are taken from Fig. 12a of Ref. [38]. The experimental light intensities are normalized to the continuous-wave case. For each of Figs. 1—5 we have multiplied the simulation data of each figure by a factor, determined visually, so

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Fig. 1. Comparison of experimental [38] (solid curves) and computer simulation (dashed curves) results for the time-resolved transilluminated diffusive light intensity. In experiment and simulation the turbid medium has a reduced scattering factor k@ "0.9 mm~1 and an absorption factor k (0.001 mm~1; the absorption factor of the 8 mm diameter 4 ! tube is k "0.014 mm~1 [38]. The reduced scattering factor inside the object is k@ "0. The tube is located in the center of ! 4 the sample. As in Ref. [38] we only show the centerpart of the detected light intensity. The experimental intensities are normalized to the continuous-wave case. We have multiplied the simulation data by a factor, determined visually, so that the scale of the experimental and the simulation data is roughly the same. No attempt to make a best fit was made. As in Fig. 12a of Ref. [38]: 1, continuous-wave case (experimental data only); 2, *t"960 ps; 3, *t"480 ps; 4, *t"240 ps; 5, *t"30 ps. The inset shows the light distribution inside the sample for *t"960 ps; the arrow indicates the direction of the incident light. See also Fig. 12a of Ref. [38]. Fig. 2. Same as Fig. 1 except that the absorption factor of the 8 mm diameter tube is k "0.13 mm~1 [38]. See also ! Fig. 12b of Ref. [38].

that the scale of the experimental and the simulation data is roughly the same. No attempt to make a best fit was made. The medium contains a plastic tube (8 mm diameter) filled with diluted ink, positioned in the centre of the sample. As in Ref. [38] we only show the centerpart of the detected light intensity. In the simulation both the absorption and the reduced scattering factors are allowed to fluctuate randomly within 10% of their values specified in Ref. [38]. Our numerical results are in remarkably good agreement with the experimental data. The inset shows the distribution of light inside the sample at *t"960 ps. The object is clearly visible. Increasing the absorption factor of the diluted ink by a factor of 9 (Fig. 12b of Ref. [38]) yields the results shown in Fig. 2. Again the overall agreement with the experimental [38] data is excellent. Simulation and experimental [38] data for a medium containing bead pairs are shown in Fig. 3. The agreement between experiment and theory is remarkable. As in the previous case no attempt has been made to make a best fit. Fig. 4 displays the results for a system containing a plastic tube filled with ink-tinted milk with the same reduced scattering factor as the medium but with a different absorption factor. Clearly, the simulation data display all the features observed in experiment. In particular, the full-width at half-maximum (FWHM) agrees well with the experimental value.

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Fig. 3. Same as Fig. 1 except that instead of one there are two 10 mm diameter objects, separated by 20 mm, with an absorption factor k "0.029 mm~1 [38]. See also Fig. 13a of Ref. [38]. ! Fig. 4. Same as Fig. 1 except that the turbid medium has a reduced scattering factor k@ "0.8 mm~1 and that the 4 absorption and reduced scattering factors of the 8 mm diameter tube are k "0.1 mm~1 and k@ "0.8 mm~1, respectively ! 4 [38]. See also Fig. 15 of Ref. [38]: 1, continuous-wave (experimental data only); 2, *t"30 ps.

Fig. 5. Same as Fig. 1 except that the turbid medium has a reduced scattering factor k@ "0.8 mm~1 and an absorption 4 factor k "0.0005 mm~1 and that the absorption and the reduced scattering factors of the 8 mm diameter tube are ! k "0.0005 mm~1 and k@ "2.6 mm~1, respectively [38]. See also Fig. 14 of Ref. [38]. ! 4

Our simulation results indicate that the time-resolved transilluminated intensity is highly sensitive to the choice of absorption and reduced scattering factors. This is illustrated in Fig. 5 where we show the experimental [38] and simulation results for the same sample as the one used for Fig. 4 except that instead of a difference in the absorption factor of the medium and the object,

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only the reduced scattering factors differ. Clearly, qualitatively similar signals can be obtained for different choices of absorption-reduced scattering factors of the object. Therefore, to determine the properties of an object the effect of the scattering and the absorption factors have to be taken into account simultaneously, in agreement with experiment [38]. The above simulation results and others (not shown) demonstrate that our simulation software reproduces all the features observed in the experiments on the tissue-like phantoms reported in Ref. [38].

4. Direct imaging Our simulation software reproduces, without fitting, the data obtained from time-resolved transillumination measurements on turbid media containing relatively large ($8 mm diameter) objects. Hence, it can be used to determine the conditions under which it is possible to detect a hidden object by direct (i.e. without further processing of the measured intensities) imaging. For instance, it is of interest to know the effect of changing the size of the sample and/or the object(s).

Fig. 6. Simulation of a time-resolved transillumination experiment on a turbid medium with a reduced scattering factor k@ "1.1 mm~1 and an absorption factor k (0.001 mm~1 containing a 4 mm radius object with absorption and reduced 4 ! scattering factors k "0.11 mm~1 and k@ "1.1 mm~1, respectively. The dimensions of the sample are ! 4 63 mm]63 mm]63 mm. The object is located in the middle of the sample. The transmitted intensity for *t"1.7 ns is projected onto the rightmost plane of the sample. Fig. 7. Simulation of a time-resolved reflection experiment on a turbid medium with a reduced scattering factor k@ "1.1 mm~1 and an absorption factor k "0.011 mm~1 containing a 1 mm-radius object with absorption and reduced 4 ! scattering factors k "0.11 mm~1 and k@ "1.1 mm~1, respectively. The dimensions of the sample are 63 mm]127 mm. ! 4 The object is located at (10, 64) mm. Curves bottom to top: *t"30 ps, *t"240 ps, *t"480 ps, *t"960 ps. The inset shows the light distribution inside the sample for *t"960 ps; the arrow indicates the direction of the incident light. Random noise ($10%) has been added to the absorption and scattering factor of the medium.

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In Fig. 6 we show a result of a simulation for a sample of size 63 mm]63 mm]63 mm, containing a sphere of 4 mm radius, positioned right at the middle of the sample. Although not as sharp as in the case of a 40 mm thick sample, the time-resolved transmitted intensity, projected onto the right most plane of the sample, contains a clear image of the object. In some source—detector arrangements transillumination of breasts would require compressing the breast tissue. This in turn makes the interpretation of the images more difficult as compression may have an effect on the scattering and absorption factors. Therefore, it may be advantageous to measure the reflected instead of the transmitted light, raising the interesting question to which extent a small object can be detected by performing a time-resolved reflection experiment. In Fig. 7 we show the result of a simulation that gives an indication of the limits of this technique. A 1 mm-radius object is placed 10 mm away from the plane where the light pulse enters the sample. The sample is 63 mm thick and 127 mm wide. In the turbid medium k@ "1.1 mm~1 and 4 k "0.011 mm~1. The object has k@ "1.1 mm~1 and k "0.11 mm~1. From our simulations (see ! 4 ! below) it follows that the blurring of the signal due to the random fluctuations can be reduced by increasing the time t at which the detectors start to record the reflected photons. The results shown $ in Fig. 7 were obtained for t "1000 ps; other parameters such as the time gate *t are identical to $ those in Figs. 1—5. As in Figs. 1—5 we only show the centerpart of the detected light intensity. We conclude that although the object is small (it has a 1 mm radius), it leaves a clear trace in the time-resolved reflected intensity. There seems to be no direct relation between the FWHM and the size of the object. The fingerprint of this small object in the reflected signal disappears if the object is more than 15 mm away from the entrance plane. In this case, where the object is located rather near to the entrance plane of the light, the time-gating method does not give a significant advantage in detecting the object. For both the transillumination and reflection method, using point sources instead of uniform illumination is much less favorable for the direct imaging of the objects hidden in the medium (see below).

5. Determination of tissue optical properties The non-invasive determination of the optical absorption and scattering properties of tissues is of great importance in both therapeutic and diagnostic applications of light in medicine, such as for example in photodynamic therapy [136], in laser surgery [137] and in monitoring changes in blood oxygenation and tissue metabolism [52]. Methods for measuring the optical properties of tissues include the pulsed photothermal radiometry method [138—140], the acoustical method [141], the time-resolved reflectance method [126], the frequency-domain reflectance method [89] and the spatially resolved steady-state diffuse reflectance method [142—145]. For a review see [146,147]. Time-resolved transillumination and reflection experiments have the possibility of measuring the optical properties of biological tissue in vivo, noninvasively. Noninvasive measurements imply that the optical properties of tissue must be deduced from measurements of the light emerging from the tissue. This requires a theoretical model to describe the photon propagation through the tissue and/or to provide an analytical expression of the transmittance and/or reflectance curve, depending on the optical parameters. Fitting the theoretical curve to the experimental data gives an estimate of the optical parameters of the tissue. The accuracy of the results achieved with

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time-resolved transmittance and/or reflectance techniques depends on how appropriate the model is for the description of the particular problem under study. Different ranges of optical parameters and experimental conditions may require different theoretical approaches [148]. Interpretation of the experimental data is generally based on the radiative transport theory [99]. Various models and approximations of the radiative transport equation are used to relate the measured light intensities to the optical parameters of the sample. Apart from the Kubelka—Munk theory [149] and the diffusion theory [99,150,151], there are the inverse adding-doubling method [152], which takes into account anisotropic scattering, and the Monte Carlo approach [114,121,122,153,154] to model light propagation in tissue. Closed-form expressions for the time-resolved reflectance and transmittance curves have been obtained by using a discrete lattice random walk model [124,155] and diffusion theory [126,156,157]. The object of this section is to explore in detail how the measured intensity profiles obtained from time-resolved transillumination and reflectance experiments can be related to the optical coefficients of the tissue under investigation. The reflectance and transmittance curves are obtained by simulating time-resolved experiments on tissue phantoms of known absorption and reduced scattering factors and by performing time-resolved experiments on various solutions of intralipid. 5.1. Experimental setup The experimental setup for time-resolved reflectance imaging is depicted in Fig. 8. The setup for time-resolved transillumination imaging is very similar. The laser system consists of a mode-locked Tsunami Ti : Sapphire laser (power: 1 W, wavelength: 800 nm, repetition frequency: 82 MHz and

Fig. 8. Setup for the time-resolved reflectance experiment. Inset: Scetch of time-resolved reflectance curves as obtained from the various detection fibers.

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pulse width: 0.06 ps) which is pumped with a BeamLok Ar ion-laser. The pulsed beam is splitted in two beams: a probe beam to illuminate the tissue phantom and a beam which on its turn is splitted in three beams, namely a beam for triggering a 2 ps resolution synchroscan streak camera (Hamamatsu C5680), a reference beam for setting an absolute time scale and a beam to visualize the pulse shape through an autocorrelator. The probe beam is attenuated by a factor of 103 with a neutral density filter, expanded and collimated to a diameter of 2 mm using two lenses (¸ and ¸ ) 1 2 in Keplerian configuration in combination with a small pinhole and is finally directed to the phantom. The phantom consists of a methacrylate vessel filled with various solutions of commercial intralipid-10%. For the time-resolved reflectance experiments the vessel has dimensions 70 mm]70 mm]70 mm. In the case of the time-resolved transillumination experiments the vessel has dimensions 30 mm]70 mm]70 mm or 10 mm]70 mm]70 mm. In all cases the vessel is filled with intralipid to a height of 55 mm. Light from the probe beam is scattered many times in the phantom before it reaches the detection side of the phantom. The detection side of the vessel is the front (back) surface of the vessel for reflection (transillumination) experiments. An array of nine optical fibers (96 lm core diameter and 0.29 numerical aperture) is glued into the detection side (to avoid direct reflections from the methacrylate) at a height of z"25 mm as shown in Fig. 8. The distance between the fibers in the y-direction is 5 mm (light propagates in the x-direction). The free end of the fibers, together with the reference beam, are set right in the input slit of the streak camera (SC). The slit dimension is adjusted to 125 lm ]6 mm (the photocathode of the streaktube has dimensions 100 lm ]8 mm) and coincides with the focal plane of the input optics of the SC. The detected light is temporally resolved in a single streak image. The SC working in synchroscan mode allows the integration of the signal over a large number of consecutive pulses. The intensity profiles (see Fig. 8) for each position (i.e. each fiber) are easily obtained from the individual streak image “traces”. Background subtraction is applied to the images to reduce camera dark-current noise. The effect of the non-uniform response of each channel in the SC, and the possible difference in sensitivity between the fibers is corrected for by dividing the streak-image by a shading image obtained by illumination through the fibers with uniform and constant light. The position of the beam, the sample and the detectors are kept fixed during the measurements. 5.2. Estimation of the reduced scattering and absorption factor In the light diffusion model the turbid medium is characterized by its reduced scattering and absorption factor. The question then arises whether these factors can be determined from timeresolved measurements. Pioneering work in this direction has shown that under certain conditions the answer to this question is affirmative [126]. Assuming that at t"0 the light intensity in the medium DI(0)T"0, the distribution of light at t"t , i.e. the time at which the source is turned off, is given by 1

P

DI(t )T" 1

t1 dt e~tHDST+t e~t1H@2DST , 1 0

(14)

where we have assumed that the intensity of the source is constant (i.e. S (t)"S ) during the pulse 0 0 time t and that t is small enough such that to a good approximation the integral in Eq. (14) can be 1 1

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evaluated by using the trapezium rule. From formal solution (3) it then follows that for t5t the 1 light intensity is given by (15) DI(t)T+t e~(t~t1@2)HDST, t5t . 1 1 The next step is to assume that the medium is uniform, i.e. it can be described by a constant diffusion coefficient DK "v/3(kL @ #kL ) and a constant absorption factor »K "vkL . Expression (15) 0 4 ! 0 ! then simplifies to (16) DI(t)T+t e~(t~t1@2)VK 0e(t~t1@2)DK 0+2DST, t5t . 1 1 Further evaluation of e(t~t1@2)DK 0+2DST requires a choice of the boundary conditions on DI(t)T. The desire to obtain for Eq. (16) an analytical expression suitable for fitting purposes severely limits this choice: No boundaries (free-space), a semi-infinite medium with Dirichlet boundary conditions per direction (e.g. I((0, y, z), t)"0), Dirichlet boundary conditions on both sides (e.g. I((0, y, z), t)"I((¸ , y, z), t)"0), and any combination of any of these. We have systematically tested x the various possibilities and found that adopting Dirichlet boundary conditions consistently gave superior fits, in concert with the results of Ref. [126]. In view of this we will in this paper not present results for boundary conditions other than the latter. In the case of Dirichlet boundary conditions (16) reads

P

I(r, t)+t e~(t~t1@2)VK 0 dr@ K(x, x@, (t!t /2)DK , ¸ )K(y, y@, (t!t /2)DK , ¸ ) 1 1 0 x 1 0 y ]K(z, z@, (t!t /2)DK ,¸ )S(r@), t5t , 1 0 z 1

(17)

where `= (18) K(u, u@, q, ¸)"(4nq)~1@2 + (e~(u~u{`2mL)2@4q!e~(u`u{`2mL)2@4q) , m/~= is the propagator for one-dimensional diffusion on a line segment of length ¸. In practice, +`= can be replaced by +`m0 0 where m is the integer nearest to 1#(!q ln e)1@2/¸, m/~= m/~m 0 e denoting the machine precision. Following Ref. [126] we assume that the source is positioned not at the boundary but inside the medium, at a distance x "1/k@ from the boundary. In the case of a point source the remaining 0 4 integral in Eq. (17) becomes trivial and Eq. (17) reduces to I(r, t)+S t e~(t~t1@2)VK 0K(x, x , (t!t /2)DK , ¸ ) 0 1 0 x 01 ]K(y, 0, (t!t /2)DK , ¸ )K(z, 0, (t!t /2)DK , ¸ ), t5t , (19) 1 0 y 1 0 z 1 where we have chosen for the light pulse to be incident along the x-axis. Clearly, Eq. (19) is sufficiently simple for fitting purposes. For particular choices of m , Eq. (19) leads to the formulae 0 given in Ref. [126]. Improvement of the theoretical description of time-resolved optical experiments, especially at early times, may be achieved by matching the temporal position of the theoretical expression (19) and the experimental data by means of an extra parameter, the time-shift

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t , introduced in Eq. (19) [148]. This yields 4 I(r, t)+S t e~(t`t4~t1@2)VK 0K(x, x , (t#t !t /2)DK , ¸ )K(y, 0, (t#t !t /2)DK , ¸ ) 01 0 4 1 0 x 4 1 0 y ]K(z, 0, (t#t !t /2)DK , ¸ ), t5t , 4 1 0 z 1 where t , DK and »K are fitting parameters. 4 0 0

(20)

5.3. Transillumination We are now in the position to assess the reliability of the procedure of extracting DK and »K (and 0 0 therefore also kL @ and kL ) from experimental or simulation data. Table 1 contains the values of 4 ! k@ and k used as input for the simulation with those (kL @ and kL , respectively) obtained by fitting 4 ! 4 ! Eq. (20) to the simulation data of the transmitted intensity. Because we are dealing with simulation data we know that t "0. From Table 1 it is clear that in all cases the fitting procedure works very 4 well, independent of the source type and detector position. Table 1 Comparison between the reduced scattering (k@ ) and the absorption (k ) factor and the corresponding estimates kL @ and 4 ! 4 kL obtained by fitting the simulated transmitted intensity to expression (20) with t "0. The dimensions of the sample are ! 4 71 mm]71 mm. The light velocity v"0.22 mm/ps. º(d, t ): Uniform illumination by a beam of diameter d (in mm), 1 centered around (0, 36 mm), during a time t (in ps). G(d, t ): Gaussian light beam of width d, centered around (0, 36 mm), 1 1 during a time t (in ps). R denotes the distance between the source and the detector 1 k (mm~1) !

k@ (mm~1) 4

Source

R (mm)

kL (mm~1) !

kL @ (mm~1) 4

0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040 0.040

0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.11 0.11 0.11 0.11 0.20 0.20 0.20 0.20 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30

U(1,100) U(1,100) U(1,100) U(1,100) G(2,100) G(2,100) G(2,100) G(2,100) U(1,6) U(1,6) U(1,6) U(1,6) G(5,6) G(5,6) G(5,6) G(5,6) G(5,6) G(5,6) G(5,6) G(5,6) G(5,100) G(5,100) G(5,100) G(5,100)

71.0 71.7 72.6 73.8 71.0 71.7 72.6 73.8 71.0 71.7 72.6 73.8 71.0 71.7 72.6 73.8 71.0 71.7 72.6 73.8 71.0 71.7 72.6 73.8

0.030 0.030 0.030 0.029 0.030 0.030 0.030 0.029 0.039 0.038 0.038 0.038 0.040 0.040 0.039 0.038 0.040 0.040 0.039 0.039 0.040 0.040 0.040 0.039

0.49 0.49 0.49 0.48 0.49 0.49 0.49 0.48 0.12 0.12 0.12 0.13 0.20 0.20 0.20 0.20 0.29 0.29 0.29 0.29 0.30 0.30 0.30 0.30

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In Tables 2 and 3 we show the results of fitting the simulation data for a particular set of model parameters used in the experiments of Ref. [38]. Also in this case there is good agreement between the values used as input to the simulations and the values obtained by fitting to Eq. (20), except for the case of uniform illumination over one whole side of the sample. This might have been expected because Eq. (20) has been obtained by assuming a point source. A representative example of a fit is depicted in Fig. 9. The two sets of curves correspond to the first and fourth entry of Table 3. It is clear that there is excellent agreement. Tables 4 and 5 show the results of fitting experimental data of the transmitted intensity for various intralipid solutions of 30 and 10 mm thick, respectively. The intralipid concentrations are measured with respect to pure intralipid and not to the commercially available intralipid-10% product, which is a 10% solution of pure intralipid. Our 1% solution corresponds to the commercial product when diluted 10 times. The values for kL and kL @ , measured at a wavelength of ! 4 j"800 nm, lie well within the range of values previously found in other studies: k@ " 4 0.935 mm~1]conc and k "0.95]10~3 mm~1]conc at j"630 nm; [158] k@ "1.119 mm~1] ! 4 conc and k "0.57]10~2 mm~1]conc at j"633 nm; [159] k@ "1.104 mm~1]conc and k " ! 4 ! 0.15]10~2 mm~1]conc at j"632.8 nm; [160] k@ "0.681 mm~1]conc and k "0.48] 4 ! 10~1 mm~1]conc at j"1064 nm; [160] k@ "1.44 mm~1]conc and k "0.027]10~2 mm~1] 4 ! conc at j"632.8 nm; [161] k@ "1.224 mm~1]conc and k "0.12 mm~1]conc at j"633 nm; 4 ! Table 2 Comparison between the scattering (k@ ) and the absorption (k ) factor and the corresponding estimates kL @ and kL obtained 4 ! 4 ! by fitting the simulated transmitted intensity to expression (20) with t "0. The dimensions of the sample are ¸ "40 mm 4 x and ¸ "127 mm. The light velocity v"0.22 mm/ps. º(d, t ): Uniform illumination by a beam of diameter d (in mm), y 1 centered around (0, 63) mm, during a time t (in ps). R denotes the distance between the source and the detector 1 k (mm~1) !

k@ (mm~1) 4

Source

R (mm)

kL (mm~1) !

kL @ (mm~1) 4

0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010

0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90

U(1,10) U(1,10) U(1,10) U(1,10) U(1000,10) U(1000,10) U(1000,10) U(1000,10)

40.0 41.2 42.7 44.7 40.0 41.2 42.7 44.7

0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010

0.86 0.86 0.86 0.86 0.94 0.90 0.84 0.77

Table 3 Same as Table 2, except that the dimensions of the sample are ¸ "40 mm, ¸ "¸ "63 mm. The centre of the x y z illuminating beam is at (0, 31, 31) mm k (mm~1) !

k@ (mm~1) 4

Source

R (mm)

kL (mm~1) !

kL @ (mm~1) 4

0.010 0.010 0.010 0.010

0.90 0.90 0.90 0.90

U(1,10) U(1,10) U(1,10) U(1,10)

40.0 41.2 42.7 44.7

0.010 0.010 0.010 0.010

0.86 0.86 0.87 0.89

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Fig. 9. Fit of transillumination intensity data to expression (20) with t "0. The dimensions of the sample are 4 40 mm]63 mm]63 mm. Solid curve: Simulation data for R"40 mm, k@ "0.9 mm~1, and k "0.01 mm~1. Diamonds: 4 ! Data obtained from Eq. (20) using R"40 mm, kL @ "0.86 mm~1 and kL "0.01 mm~1. Dashed curve: Simulation data for 4 ! R"44.7 mm, k@ "0.9 mm~1, and k "0.01 mm~1. Crosses: Data obtained from Eq. (20) using R"44.7 mm, 4 ! kL @ "0.89 mm~1 and kL "0.01 mm~1. All curves are normalized to their maximum. 4 ! Table 4 Estimates for the reduced scattering (kL @ ), the absorption (kL ) factor and the time-shift (t ) obtained by fitting the 4 ! 4 transmitted intensity for various solutions of intralipid to expression (20). The dimensions of the sample are ¸ "30 mm, x ¸ "70 mm and ¸ "55 mm. The light velocity v"0.22 mm/ps. The source emits a Gaussian light distribution of width y z 2 mm, centered around (0, 36, 25) mm during a time of 0.06 ps. R denotes the distance between the source and the detector Concentration (%)

R (mm)

kL (mm~1) !

kL @ (mm~1) 4

t (ps) 4

0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0

30.0 30.4 31.6 33.5 30.0 30.4 31.6 33.5

0.0008 0.0008 0.0012 0.0016 0.0025 0.0025 0.0029 0.0035

0.60 0.60 0.60 0.60 1.16 1.16 1.18 1.19

35 35 35 30 175 175 175 175

[23] k@ "1.128 mm~1]conc and k "0.12 mm~1]conc at j"780 nm; [23] k@ "1.4 mm~1] 4 ! 4 conc and k "0.5 mm~4]conc at j"633 nm; [67] k@ "0.888 mm~1]conc at j"632.8 nm ! 4 [162]. The abbreviation conc denotes the concentration of intralipid in percent.

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Differences in these results may at least in part be attributed to a difference in the composition of intralipid-10% [159,160]. Moreover, the absorption factor of intralipid seems difficult to determine accurately [163]. Fig. 10 depicts a typical example of a fit of experimental transillumination curves to theoretical expression (20). The two sets of curves correspond to the second and fourth entry of Table 4. Table 5 Same as Table 4, except that the dimensions of the sample are ¸ "10 mm, ¸ "70 mm and ¸ "55 mm x y z Concentration (%)

R (mm)

kL (mm~1) !

kL @ (mm~1) 4

t (ps) 4

1.0 1.0 1.0 1.0 2.0 2.0 2.0 2.0

10.0 11.2 14.1 18.0 10.0 11.2 14.1 18.0

0.0005 0.0005 0.0006 0.0006 0.0010 0.0015 0.0026 0.0024

1.34 1.34 1.30 1.24 2.20 2.20 2.14 1.90

0 2 5 8 0 6 10 20

Fig. 10. Fit of transillumination intensity data to expression (20). The dimensions of the sample are 30 mm]70 mm]55 mm. Solid curve: Experimental data for a 0.5% intralipid solution and R"30.4 mm. Long-dashed curve: Data obtained from (20) using R"30.4 mm, kL @ "0.6 mm~1, kL "0.0008 mm~1 and t "35 ps. Short-dashed 4 ! 4 curve: Experimental data for a 0.5% intralipid solution and R"33.5 mm. Dotted curve: Data obtained from (20) using R"33.5 mm, kL @ "0.6 mm~1 and kL "0.0016 mm~1. All curves are normalized to their maximum. 4 !

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Table 6 Comparison between the reduced scattering (k@ ) and the absorption (k ) factor and the corresponding estimates kL @ and 4 ! 4 kL obtained by fitting the simulated reflected intensity to expression (20) with t "0. The dimensions of the sample are ! 4 71 mm]71 mm. The light velocity v"0.22 mm/ps. º(d, t ): Uniform illumination by a beam of diameter d (in mm), 1 centered around (0, 36 mm), during a time t (in ps). G(d, t ): Gaussian light beam of width d, centered around (0, 36 mm), 1 1 during a time t (in ps). R denotes the distance between the source and the detector. A ‘‘x” indicates that the fitting 1 procedure failed to produce reasonable (i.e. positive) estimates k (mm~1) !

k@ (mm~1) 4

Source

R (mm)

kL (mm~1) !

kL @ (mm~1) 4

0.005 0.005 0.005 0.005 0.005 0.005 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.040 0.040 0.040 0.040 0.040 0.040

0.90 0.90 0.90 0.90 0.90 0.90 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.11 0.11 0.11 0.60 0.60 0.60

U(1,10) U(1,10) U(1,10) G(5,100) G(5,100) G(5,100) U(1,6) U(1,6) U(1,6) U(2,6) U(2,6) U(2,6) U(1,100) U(1,100) U(1,100) G(0.5,6) G(0.5,6) G(0.5,6) G(2,6) G(2,6) G(2,6) G(5,6) G(5,6) G(5,6) U(1,6) U(1,6) U(1,6) U(1,6) U(1,6) U(1,6)

10 15 20 10 15 20 10 15 20 10 15 20 10 15 20 10 15 20 10 15 20 10 15 20 10 15 20 10 15 20

0.005 0.005 0.005 x x x 0.010 0.010 0.010 0.010 0.010 0.010 x x x 0.010 0.010 0.010 0.010 0.010 0.010 x x x 0.040 0.040 0.040 0.040 0.040 0.040

0.88 0.88 0.88 x x x 0.59 0.59 0.59 0.56 0.57 0.58 x x x 0.57 0.58 0.59 0.52 0.55 0.57 x x x 0.11 0.11 0.12 0.57 0.57 0.58

5.4. Reflection We have seen that the values of k@ and k can be estimated rather accurately from fitting Eq. (20) 4 ! to the transillumination data. This unfortunately is not always the case when fitting to reflection data. In Table 6 we show a collection of fits to simulation data of the reflected intensity. The crosses in the fifth and sixth column indicate that it was impossible to let Eq. (20) resemble the simulation data. In general the fitting procedure fails if the light pulse is too long (t '10 ps) or if 1 the illumination area is too large (d'5 mm). This could be expected on the basis of the

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113

Table 7 Comparison between the reduced scattering (k@ ) and the absorption (k ) factor and the corresponding estimates kL @ and 4 ! 4 kL obtained by fitting the simulated reflected intensity to expression (20) with t "0. The dimensions of the sample are ! 4 ¸ "40 mm and ¸ "127 mm. The light velocity v"0.22 mm/ps. º(d, t ): Uniform illumination by a beam of diameter x y 1 d (in mm), centered around (0, 63) mm, during a time t (in ps). R denotes the distance between the source and the detector. 1 A ‘‘x” indicates that the fitting procedure failed to produce reasonable (i.e. positive) estimates k (mm~1) !

k@ (mm~1) 4

Source

R (mm)

kL (mm~1) !

kL @ (mm~1) 4

0.010 0.010 0.010 0.010 0.010 0.010

0.90 0.90 0.90 0.90 0.90 0.90

U(1,10) U(1,10) U(1,10) U(1000,10) U(1000,10) U(1000,10)

10 15 20 10 15 20

0.010 0.010 0.010 x x x

0.87 0.87 0.87 x x x

Table 8 Same as Table 7, except that the dimensions of the sample are ¸ "40 mm, ¸ "¸ "63 mm and the beam centre is at x y z (0, 31, 31) mm k (mm~1) !

k@ (mm~1) 4

Source

R (mm)

kL (mm~1) !

kL @ (mm~1) 4

0.010 0.010 0.010

0.90 0.90 0.90

U(1,10) U(1,10) U(1,10)

10 15 20

0.010 0.010 0.010

0.88 0.88 0.88

assumptions made in deriving Eq. (20), but it is of interest to note that, in contrast to the case of transillumination, the pulse time t affects the quality of the fit. In Tables 7 and 8 we show 1 the results of fitting the simulation data for a particular set of model parameters used in the experiments of Ref. [38]. There is good agreement between the values used as input to the simulations and the values obtained by fitting to Eq. (20) again except, for uniform illumination over the whole sample. A typical example of a fit to reflection data is shown in Fig. 11. The two sets of data used correspond to the first and third entries of Table 8. It is clear that Eq. (20) fits the data very well. Fig. 12 depicts a fit of an experimental reflection curve for a 1% intralipid solution to the curve obtained from Eq. (20). The source—detector distance (or R) is 7 mm. For higher R values the experimental data becomes rather noisy and fitting becomes more difficult. For the particular case shown in Fig. 5, kL "0.01 mm~1 and kL "1.16 mm~1. ! 4 In summary, in the regime where the diffusion equation can be used to model light transport through a turbid medium, the reduced scattering and absorption factor can be extracted from measured data by fitting a simple expression, derived from diffusion theory, to the data. In contrast to simulation data, for experimental data there is an additional ambiguity due to the uncertainty in defining time zero, which has to be accounted for by introducing a third fitting parameter. Evidently this has a negative impact on the reliability of the whole procedure of extracting the reduced scattering and absorption factors. Reliable estimates of both factors can be obtained from

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Fig. 11. Fit of reflected intensity data to expression (20) with t "0. The dimensions of the sample are 4 40 mm]63 mm]63 mm. Solid curve: Simulation data for R"10 mm, k@ "0.9 mm~1, and k "0.01 mm~1. Diamonds: 4 ! Data obtained from Eq. (20) using R"10 mm, kL @ "0.88 mm~1 and kL "0.01 mm~1. Dashed curve: Simulation data for 4 ! R"20 mm, k@ "0.9 mm~1, and k "0.01 mm~1. Crosses: Data obtained from Eq. (20) using R"20 mm, 4 ! kL @ "0.88 mm~1 and kL "0.01 mm~1. All curves are normalized to their maximum. 4 !

transillumination data, irrespective of the pulse time, the detector positions and the width of the beam (excluding illumination over a whole side of the sample). In general, the estimates obtained from reflection measurements are less reliable.

6. Data processing method: Theory Usually, the quality of the images rendered by an imaging technique can be substantially enhanced by the use of appropriate data processing. An obvious approach is to compare the data with other data obtained from a reference, or model, system. In the case at hand we know that the immense scattering of light is responsible for the blurring of the images of the objects. The variations of both the reduced scattering and absorption factor due to the objects are relatively small. Therefore, as a starting point, it is reasonable to take as a reference model a system with a constant diffusion coefficient D and a constant absorption factor » . We call this model 0 0 H "!D +2#» . We will prove some rigorous bounds on the ratio of the intensity distribu0 0 0 tions of models H and H "H #» with »"»(r)!» . These bounds give the maximum 0 1 0 0 enhancement of the image quality one can obtain by using model H and also provide a clue as to 0 the information that is contained in this ratio. In the next section we will use the simulation

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115

Fig. 12. Fit of reflected intensity data to expression (20). The dimensions of the sample are 70 mm]70 mm]55 mm. Solid curve: Experimental data for a 1% intralipid solution and R"7 mm. Dashed curve: Data obtained from Eq. (20) using R"7 mm, kL @ "1.16 mm~1, kL "0.01 mm~1 and t "!115 ps. Both curves are normalized to their maximum. 4 ! 4

technique to demonstrate that this image processing method can be very effective. In this section we will, for the sake of simplicity, restrict ourselves to the case where D(r)"D . The effect of 0 a non-uniform D(r) will be studied in the next section. A first step in proving these bounds is to generalize the exact inequalities [164]

C P

SrDe~t(H0`V)Dr@T5SrDe~tH0Dr@T exp !t and SrDe~t(H0`V)Dr@T4SrDe~tH0Dr@T exp

CP

1

1

0

D

SrDe~txH0»e~t(1~x)H0Dr@T dx SrDe~tH0Dr@T

dx ln

(21)

D

SrDe~txH0e~tVe~t(1~x)H0Dr@T , SrDe~tH0Dr@T

(22) 0 and prove that Dr@T can be replaced by an arbitrary distribution of light intensity to be denoted by DsT. The inequalities (21) and (22) hold if SrDe~tH0Dr@T50 for any r and r@ and »DrT"SrD»DrTDrT"»(r)DrT, conditions that are satisfied in the case of the TDDE. Making use of the fact that the light intensity is always positive, i.e. s(r)"SrDsT50, and repeating the steps of the proof that led to Eqs. (21) and (22) we find

C P

SrDe~t(H0`V)DsT5SrDe~tH0DsT exp !t

1

0

D

SrDe~txH0»e~t(1~x)H0DsT dx SrDe~tH0DsT

(23)

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and

CP

SrDe~t(H0`V)DsT4SrDe~tH0DsT exp

1

0

dx ln

D

SrD e~txH0 e~tVe~t(1~x)H0DsT . SrDe~tH0DsT

(24)

The above inequalities suggest that if we generate a reference signal I (r, s, t)"SrDe~tH0DsT, 0 “measure” the intensity I (r, s, t)"SrDe~tH1DsT and form the ratio I (r, s, t)/I (r, s, t) we will obtain 1 0 1 information on the potential ». We now proceed to set rigorous bounds on this ratio. 6.1. Instantaneous light-intensity detection From Eqs. (23) and (24) we obtain t

P

P

1

I (r, s, t) SrDe~txH0»e~t(1~x)H0DsT 1 SrDe~txH0e~tVe~t(1~x)H0DsT 5ln 0 5! dx ln , dx I (r, s, t) I (r, s, t) I (r, s, t) 1 0 0 0 0

(25)

or using ln x4x!1 for x'0 to weaken the lower bound in Eq. (25) t:1 dxSrDe~txH0»e~t(1~x)H0DsT I (r, s, t) :1 dxSrDe~txH0(1!e~tV)e~t(1~x)H0DsT 0 5ln 0 50 , I (r, s, t) I (r, s, t) I (r, s, t) 1 0 0

(26)

indicating that ln[I (r, s, t)/I (r, s, t)] contains information on ». From Eq. (26) it follows that to 0 1 first order in », inequalities (23) and (24) are equalities. 6.2. Integrated light-intensity detection In practice, measuring the instantaneous light intensity is impossible: Due to the finite precision of the instruments one measures the intensity integrated over a time interval of finite length. In this section, we generalize the above inequalities to the case of time-integrated light-intensity detection. We will denote the time-integrated intensities by

P

(27a)

P

(27b)

IM (r, s, t , t )" 1 0 1

t1 dt I (r, s, t) 1 t0

and IM (r, s, t , t )" 0 0 1

t1 dt I (r, s, t) , 0 t0

respectively. To obtain the time-integrated version of the bounds (26) we proceed as follows. Integration of Eq. (23) with respect to t yields IM (r, s, t , t ) :t10 dt I (r, s, t) exp[!t:1 dxSrDe~txH0»e~t(1~x)H0DsT/I (r, s, t)] 1 0 15 t 0 0 0 . I (r, s, t , t ) :t10 dt I (r, s, t) 0 0 1 0 t

(28)

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117

Application of Jensen’s inequality [165] : dx g(x)D f (x)D5exp[: dx g(x) lnD f (x)D] X X : dx g(x)"1 to the r.h.s. of Eq. (28) yields X IM (r, s, t , t ) :1 dx:t10 dt tSrDe~txH0»e~t(1~x)H0DsT 0 1 . 0 t 5ln 0 IM (r, s, t , t ) :t10 dt I (r, s, t) 1 0 1 0 t Introducing the symbol SrDe~txH0e~tVe~t(1~x)H0DsT K(x, t)" SrDe~tH0DsT

with

(29)

(30)

and integrating Eq. (24) from t to t yields 0 1 IM (r, s, t , t ) :t10 dt I (r, s, t) exp[:1 dx ln K(x, t)] 0 0 1 0 14 t . (31) :t10 dt I (r, s, t) IM (r, s, t , t ) 0 0 1 0 t The upper bound in (31) can be brought in a more convenient, albeit weaker, form by invoking a generalization of Ho¨lders inequality [165]

P

CP

D

GP

dy h(y) exp dx g(x) lnD f (x, y)D 4exp dx g(x) ln Y X X valid for h(y)50, g(x)50 and : g(x) dx"1. We obtain X 1 :t dt I (r, s, t)K(x, t) 1 IM (r, s, t , t ) 0 0 1 4 dx ln t0 , ln 1 :t10 dt I (r, s, t) IM (r, s, t , t ) 0 t 0 0 1 0 or, again using ln x4x!1 for x'0,

CP

DH

dy h(y)D f (x, y)D

Y

P

IM (r, s, t , t ) :1 dx :t10 dt SrDe~txH0(1!e~tV)e~t(1~x)H0DsT t 0 15 0 . ln 0 :t10 dt I (r, s, t) IM (r, s, t , t ) 0 t 0 0 1 Combining Eqs. (29) and (34) gives

,

(32)

(33)

(34)

IM (r, s, t , t ) :1 dx:t10 dt tSrDe~txH0»e~t(1~x)H0DsT 0 1 t 0 5ln 0 1 IM (r, s, t , t ) :t0 dt I (r, s, t) 1 0 1 0 t :1 dx:t10 dt SrDe~txH0(1!e~tV)e~t(1~x)H0DsT t 50 . (35) :t10 dt I (r, s, t) 0 t The similarity between Eq. (35) and the bounds (26) on the ratio of the instantaneous intensities is striking: Each quantity has simply been replaced by its time-integrated value. 6.3. Bounds on the difference Weaker bounds, not on the ratio but on the difference of the intensities, can be obtained as follows. Rewriting the upper bound in Eq. (35) as :1 dx:t10 dt tSrDe~txH0»e~t(1~x)H0DsT IM (r, s, t , t ) 0 1 5! 0 t , ln 1 IM (r, s, t , t ) :t10 dt I (r, s, t) 0 0 1 0 t

(36)

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and using ln x(x!1 for x'0 we find

P P 1

t1

dt tSrDe~txH0»e~t(1~x)H0DsT . (37) t0 0 An upper bound can be derived by starting from Eq. (24) and invoking Jensen’s inequality once more. We obtain IM (r, s, t , t )!IM (r, s, t , t )5! 1 0 1 0 0 1

P

SrDe~t(H0`V)DsT4

1

0

dx

dxSrDe~txH0e~tVe~t(1~x)H0DsT ,

(38)

or equivalently

P

IM (r, s, t , t )!IM (r, s, t , t )4! 1 0 1 0 0 1

1

0

dxSrDe~txH0(1!e~tV)e~t(1~x)H0DsT .

(39)

Combining Eqs. (37) and (39) yields

P P P P 1

dx

0

t1

t 1 0

dtSrDe~txH0(1!e~tV)e~t(1~x)H0DsT4IM (r, s, t , t )!IM (r, s, t , t ) 0 0 1 1 0 1 t1

dt tSrDe~txH0»e~t(1~x)H0DsT . (40) 0 t0 Comparing Eqs. (35) and (40) it is clear that the upper and lower bounds contain essentially the same information. 4

dx

6.4. Illustrative examples In this subsection we consider some simple cases for which the upper and lower bounds derived above can be worked out analytically. We only treat the case of instantaneous light detection and a point source, i.e. DsT"Dr@T where r@ denotes the position of the source. As a starter consider the case of uniform absorption by the medium. With H "!D + 2#» 1 0 1 and H "!D + 2, Eq. (22) reduces to 0 0 I (r, r@, t) t» 5ln 0 5t» 51!e~tV1 , (41) 1 1 I (r, r@, t) 1 as expected because I (r, r@, t)/I (r, r@, t)"etV1. 0 1 Next consider the case of a point absorber (i.e. »(r)"» d(r!r ) and 2 0 e~tV(r)"1#(e~tV2!1)d(r!r ), » '0) embedded in an homogeneous infinite medium. In this 0 2 case SrDe~tH0Dr@T"G (r!r@, D t) , d 0 where

A B

1 d@2 r2 e~ @4x , G (r, x)" d 4px

(42)

(43)

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119

describes the light diffusion in the infinite medium without the absorber. Here and in the following d denotes the dimension of the system. Straightforward algebra gives SrDe~txH0»e~t(1~x)H0Dr@T "» G (r !(1!x)r!xr@, x(1!x)D t) 2 d 0 0 SrDe~tH0Dr@T

(44a)

SrDe~txH0e~tVe~t(1~x)H0Dr@T "1#(e~tV2!1)G (r !(1!x)r!xr@, x(1!x)D t) . d 0 0 SrDe~tH0Dr@T

(44b)

and

The inequalities (26) become I (r, r@, t) 5(1!e~tV2)F(r , r, r@, t) , t» F(r , r, r@, t)5ln 0 0 2 0 I (r, r@, t) 1 where

P

F(r , r, r@, t)" 0

1

0

(45)

dx G (r !(1!x)r!xr@, x(1!x)D t) . d 0 0

(46)

For P point objects, the inequalities corresponding to Eq. (45) read P I (r, r, t) P t + »(p)F(r(p), r, r@, t)5ln 0 5 + (1!e~tV(p)2)F(r(p), r, r@, t) , (47) 2 0 0 I (r, r, t) 1 p/1 p/1 suggesting that the reflected signal is a superposition of intensities “emitted” by the P “point sources” located at r(p). 0 The above example can be analysed in more detail by considering the case of backscattering. Then r"r@ and

P

P

I (r, r, t) 1 dx G (r !r, x(1!x)D t)5 0 5(1!e~tV2) dx G (r !r, x(1!x)D t) , (48) d 0 0 d 0 0 I (r, r, t) 1 0 0 showing that for a point object, ln [I (r, r, t)/I (r, r, t)]/(1!e~tV2) is larger than the light intensity 0 1 resulting from a superposition of light pulses emitted by the point object and received by the detector. In this case there is a clear physical interpretation of the upper and lower bounds themselves. Our simulation results presented below, show that this interpretation is qualitatively correct, also for the more complicated case rOr@ and/or DsTODr@T. From the t-dependence of bounds such as Eqs. (45) and (48), we may expect that they contain useful information about the absorber if t» is sufficiently small (but q ;t of course). In general, 2 1 ln [I (r, r, t)/I (r, r, t)] or ln [IM (r, r, t , t )/IM (r, r, t , t )] will only show signatures of the object(s) if 0 1 0 0 1 1 0 1 q 4q 4t4q or q 4q 4t 4t 4q respectively, where q and q have to be determined by 1 2 3 1 2 0 1 3 2 3 (numerical) experiment. For the turbid media of interest, » +24 ns~1 and D +0.6 cm2/ns. 2 0 Putting for instance t"1.5 ns we find t» 2

1

AP

365

1

0

B

dx G (r !r, x(1!x)D t) d 0 0

~1 I (r, r, t) 51 , ln 0 I (r, r, t) 1

(49)

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demonstrating that the presence of a small object can have a substantial effect on the processed signal.

7. Image processing technique: Simulation results Application of the theory presented above is much more straightforward than the mathematics used might suggest. In practice, the procedure is as follows: Measure the intensity IM (r, s, t , t )":t10 dtSrDe~tHDsT and calculate or measure the reference signal IM (r, s, t , t ) correspond0 1 t 0 0 1 ing to a “test-model” H for various detector-positions r. The resulting ratio ln IM (r, s, t , t )/ 0 0 1 IM (r, s, t , t ) (or difference IM (r, s, t , t )!IM (r, s, t , t )) should reveal whether there are hidden 0 0 1 0 0 1 0 1 objects or not. The simulation results presented below demonstrate that this procedure works and in the next section we show that it also works in real experiment. We will use the term “processed” signal whenever we mean ln IM (r, s, t , t )/IM (r, s, t , t ). In the remainder of this work we set t "t 0 1 0 0 1 0 $ and t "t #*t and we confine ourselves to presenting results obtained from intensities integrated 1 0 over a finite interval (i.e. *t). Guided by the experiments on breast tissue (see the discussion in the introduction) in our simulations we will assume that the turbid medium is characterized by an absorption and reduced scattering factor k "0.01 mm~1 and k@ "0.9 mm~1, respectively. For tumor tissue we will take ! 4 k "0.1 mm~1 and k@ "0.9 mm~1, unless explicitly mentioned otherwise. ! 4 Our simulations show that on a qualitative level, there is little difference between the results obtained from 2D or 3D models. Printed images obtained from 2D model simulations are more easy to interpret than their 3D counterpart. Therefore, in the discussion that follows we will mainly present results of 2D model simulations. For some selected cases we also show images obtained from 3D model simulations. Unless explicitly mentioned otherwise, the spatial mesh size d"1 mm, the time step q"1 ps, the source pulse time t "10 ps, the time at which the detectors start to 1 record intensity t "t "500 ps, and the 2D and 3D samples have dimensions 71 mm]71 mm, $ 0 and 63 mm] 63 mm] 63 mm, respectively. The speed of light in the medium v"0.222 mm/ps. We will use the short-hand notation IM and IM for the transmitted and reflected intensity T R respectively, integrated over the corresponding detection area. All intensities given below are normalized with respect to the light intensity supplied by the source. 7.1. Effect of the source type and the time gate In Fig. 13 we show results of a simulation for a sample containing an object of 2.5 mm radius, positioned at (36, 26) mm. The sample is illuminated by a source of diameter 1 mm centered around (0, 36) mm. In the sequel we will call such a small source a point source. Fig. 13a and c show the integrated intensity while Fig. 13b and d show the corresponding processed signal. For *t"612 ps (Fig. 13a and b), it is clear that the object leaves no trace in the transmitted and reflected intensity whereas the processed data clearly indicate that there is an object inside the sample. For *t"1836 ps (Fig. 13c and d) the transmitted intensity shows an asymmetry. Since the light source is positioned at (0, 36) mm this asymmetry indicates that there is an object immersed in the sample. In the reflected intensity, however, the object leaves no trace. In the processed reflected intensity the object is clearly visible. Note that the position of the maximum of the processed intensity changes

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121

Fig. 13. Simulation of a time-resolved optical imaging experiment on a turbid medium with a reduced scattering factor k@ "0.9 mm~1 and an absorption factor k "0.01 mm~1 containing a 2.5 mm radius object with absorption and reduced 4 ! scattering factors k "0.1 mm~1 and k@ "0.9 mm~1, respectively. The dimensions of the sample are 71 mm]71 mm. ! 4 The object is located at (36, 26) mm. The sample is illuminated by a source of diameter 1 mm centered around (0, 36) mm during a time t "10 ps. (a) Time-integrated transmitted (right) and reflected (left) intensity for *t"612 ps. 1 IM +0.6]10~8 and IM +0.2]10~2; (b) processed signal corresponding to (a); (c) same as (a) except that *t"1836 ps. T R IM +0.5]10~6 and IM +0.3]10~2; (d) processed signal corresponding to (c). The (processed) light intensities inside the T R sample are also shown.

with *t (Fig. 13b and d). This effect is the largest for the processed reflected intensity. Qualitatively similar results are obtained by illuminating the sample with a light pulse of width t "100 ps or 1 t "300 ps (results not shown). 1 Starting the recording of light intensity at a different point of time or changing *t mainly affects the processed reflected intensity. An example is provided in Fig. 14 where we present simulation results for the same setup as the one of Fig. 13 except for the time t at which the detection of light 0 starts. Comparison of the results of Fig. 14a and b, corresponding to t "1000 ps and *t"612 ps 0 (t "t #*t"1612 ps), with those of Fig. 13a and b where t "500 ps and *t"612 ps 1 0 0 (t "1112 ps), shows that, *t being the same, there are only minor differences in the processed 1 reflected signal (the small changes in the background signal are due to the time integration and do not carry relevant information). Comparing the results shown in Fig. 14a and b with those of Fig. 14c and d where t "500 ps and *t"1112 ps (t "1612 ps) confirm that, the processed 0 1 reflected intensity depends on t and *t, t being the same. This is mainly due to the fact that for 0 1 illumination by a point source, the maximum of the processed reflected intensity changes with *t (see Fig. 13b and d). However, changing t or *t does not affect the detectability of the object. 0 Fig. 15a and b show simulation results for the same setup as the one used for Fig. 13c and d except that the light source is now positioned at (0, 16) mm. The transmitted and reflected

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Fig. 14. Same as Fig. 13 except for the time t at which the detection of light starts: (a) t "1000 ps and *t"612 ps. 0 0 IM +0.1]10~6 and IM +0.3]10~3; (b) processed signal corresponding to (a); (c) same as (a) except that t "500 ps and T R 0 *t"1112 ps. IM +0.1]10~6 and IM +0.2]10~2; (d) processed signal corresponding to (c). Medium: T R (k , k@ )"(0.01, 0.9) mm~1; object: (k , k@ )"(0.1, 0.9) mm~1. ! 4 ! 4

intensities in Fig. 13c, d and Fig. 15a, b look completely different. As the image of the point source itself resembles the image of an object it is as good as impossible to conclude from the transmitted or reflected intensities whether or not there is an object inside the sample. The processed intensities however, correctly signal the presence of an object. From Fig. 13 and Fig. 15a, b, it follows that there is no obvious relationship between the position of the object and the maximum of the processed intensity. Illumination of the sample by a source emitting a Gaussian light distribution of width 5 mm yields the results shown in Fig. 15c and d. These results can hardly be distinguished from those shown in Fig. 13c and d obtained by using a point source. Fig. 16 displays results for a system containing an object of 2.5 mm radius, positioned at (16, 26) mm. Fig. 16a and b show the time-integrated transmitted and reflected intensities for illumination by a point source while Fig. 16c and d show the intensities for uniform illumination of the whole left side of the sample. Using a point source yields no trace of the object in the reflected intensity and only a weak one in the transmitted intensity (Fig. 13a, c and Fig. 16a). In the processed signals, however, there is a clear signal of the object (Fig. 13b, d and Fig. 16b). Uniform illumination of the whole left side of the sample (Fig. 16c and d) yields a very weak signal of the object in the transmitted and reflected intensities and a clear signal in the processed intensities. The fingerprint of the object in the reflected intensity disappears completely if the object is more than 20 mm away from the entrance plane, while in the transmitted intensity the very weak signal is present, independent of the location of the object within the sample (results not shown). In the case of uniform illumination the position

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Fig. 15. Same as Fig. 13 except for the source illuminating the system. *t"1836 ps (a) Point source centered around (0,16) mm. IM +0.5]10~6 and IM +0.3]10~2; (b) processed signal corresponding to (a); (c) Source emitting a Gaussian T R light distribution of width 5 mm, centered around (0,36) mm. IM +0.5]10~6 and IM +0.3]10~2; (d) processed signal T R corresponding to (c). Medium: (k , k@ )"(0.01, 0.09) mm~1; object: (k , k@ )"(0.1, 0.9) mm~1. ! 4 ! 4

of the maximum of the processed intensity does no longer change as a function of time (not shown) and corresponds to the y-coordinate of the position of the object. Uniform illumination of the other sides of the sample should therefore allow the determination of the position of the object. Uniform illumination during a time t "100 ps or t "300 ps (results not shown) yields qualitatively the 1 1 same results as for t "10 ps, just as in the case of illumination by a point source. 1 If the distance between the object and the illuminated side of the sample is approximately 10 mm or less, the time-integrated reflected intensity clearly shows a signature of the object provided the source is sufficiently large. This is illustrated in Fig. 17, which displays results for a system containing an object of 2.5 mm radius, positioned at (10, 26) mm. Fig. 17a—d show the timeintegrated transmitted and reflected intensities for illumination of the sample by a source emitting circular light distributions of various size. Using a point source (Fig. 17a) yields no trace of the object in the reflected intensity and only a weak one in the transmitted intensity. Increasing the radius of the source improves the detectability of the object in the reflected and transmitted intensities, as shown in Fig. 17b and d. For a source of radius 30 mm (Fig. 17c) the ratio between the dip and maximum is approximately 0.6, which is nearly the same as for uniform illumination of the whole left side of the sample (Fig. 17d). As mentioned above, even with uniform illumination of the sample the fingerprint of the object in the reflected intensity disappears completely if the object is more than 20 mm away from the entrance plane (results not shown). Note that unlike Fig. 13c and d, Fig. 17 displays only the results of time-integrated intensities.

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Fig. 16. Same as Fig. 13 except for the source illuminating the system and the position of the object (16, 26) mm. *t"1836 ps. (a) Point source centered around (0.36) mm. IM +0.6]10~6 and IM +0.2]10~2; (b) processed signal T R corresponding to (a); (c) Uniform illumination of the left side of the sample. IM +0.6]10~6 and IM +0.2]10~2; (d) T R processed signal corresponding to (c). Medium: (k , k@ )"(0.01, 0.9) mm~1; object: (k , k@ )"(0.1, 0.9) mm~1. ! 4 ! 4

7.2. Small objects In Fig. 18 we present simulation results for a system containing an object of 0.5 mm radius. For comparison the setup is kept the same as the one used for Fig. 16. As seen from Fig. 18a, using a point source, it is no longer possible to detect the object in the transmitted intensity. Also for uniform illumination of the whole left side of the sample the fingerprint of the small object in the transmitted and reflected intensities (Fig. 18c) is much weaker than for an object of radius 2.5 mm (Fig. 16c). In the simulation the object always leaves a trace in the processed signals, independent of its location in the sample. Evidently, due to the signal-to-noise ratio, this will not necessarily be the case in experiment. From Figs. 16 and 18 it is clear that it is very difficult, not to say impossible, to infer the size of the object from the measured intensities.

7.3. System size and mesh size Repeating the simulations for the same set-up as the one of Fig. 13 but reducing the mesh size by a factor two and the time step by a factor four gives the same results as the ones shown in Fig. 13. Reducing the diameter of the object, hidden in this sample, to 0.5 mm yields the results displayed in Fig. 19. In the processed transmitted intensity the object is still clearly visible.

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Fig. 17. Same as Fig. 13 except for the source illuminating the system and the position of the object (10, 26) mm. Only time-integrated transmitted (right) and reflected (left) intensities are shown for *t"1836 ps: (a) point source centered around (0, 36) mm. IM +0.6]10~6 and IM +0.2]10~2; (b) source emitting a circular light distribution of radius 15 mm, T R centered around (0, 36) mm. IM +0.5]10~6 and IM +0.2]10~2; (c) source emitting a circular light distribution of radius T R 30 mm, centered around (0.36) mm. IM +0.5]10~6 and IM +0.2]10~2; (d) uniform illumination of the left side of the T R sample. IM +0.6]10~6 and IM +0.2]10~2. Medium: (k , k@ )"(0.01, 0.9) mm~1; object: (k , k@ )"(0.1, 0.9) mm~1. T R ! 4 ! 4

Fig. 18. Same as Fig. 16 except that the object has a radius of 0.5 mm. Medium: (k , k@ )"(0.01, 0.9) mm~1; object: ! 4 (k , k@ )"(0.1, 0.9) mm~1. ! 4

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Fig. 19. Same as Fig. 13 except that the object has a radius of 0.25 mm. (a) *t"614 ps. IM +0.7]10~8 and T IM +0.3]10~2; (b) processed signal corresponding to (a); (c) same as (a) except that *t"1843 ps. IM +0.8]10~6 R T and IM +0.3]10~2; (d) processed signal corresponding to (c). Medium: (k , k@ )"(0.01, 0.9) mm~1; object: R ! 4 (k , k@ )"(0.1, 0.9) mm~1. ! 4

7.4. D(r) versus D

0

An interesting question for theoretical developments is to what extent the propagation of light is affected by replacing, in the TDDE, the diffusion coefficient D(r) by its homogeneous counterpart D "v/3(kN @ #kN ). In Fig. 20 we present some results of a calculation in which we address that 0 4 ! question. Usually, we compute the light intensity in the full model H"!+D(r)+#»(r) and compare it to the light intensity in the approximate model H "!D + 2#» (r). The results are 0 0 0 depicted in Fig. 20a and b for the same sample as in Fig. 13 except that the radius of the object is 0.5 mm. Comparing the light intensity of the full model H with the light intensity in the approximate model H@ "!D + 2#»(r) yields the results shown in Fig. 20 c and d. The processed 0 0 intensities only show the effect of replacing D(r) by D . We find that this approximation changes the 0 processed intensities by no more than 2%. For most practical purposes, it will be safe to replace D(r) by D , unless the reduced scattering factors of the medium and objects differ significantly. 0 7.5. More than one object In Fig. 21 we show a result of a simulation for a sample containing two objects of 0.5 mm radius which are positioned 15 mm from each other. The sample is illuminated by a point source. The transmitted and reflected intensities (Fig. 21a and c) show no trace of the objects. For *t"612 ps the processed data give a weak indication that there are two objects hidden in the sample and that they are not positioned symmetrically with respect to the position of the light source. For *t"1836 ps (Fig. 21c and d) the reflected processed data signals two objects immersed in the system while the transmitted processed data suggests that there is only one object inside the

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Fig. 20. Same as Fig. 13 except that the object has a radius of 0.5 mm. (a) *t"1836 ps. IM +0.6]10~6 and T IM +0.3]10~2; (b) processed signal using reference model H and corresponding to (a); (c) same as (a); (d) processed R 0 signal using reference model H@ and corresponding to (c). Medium: (k , k@ )"(0.01, 0.9) mm~1; object: 0 ! 4 (k , k@ )"(0.1, 0.9) mm~1. ! 4

sample. Additional simulations (see Fig. 22) indicate that in reflection it is possible to resolve the two objects down to a separation distance of 2 mm by using the appropriate time gate. Note that, unlike Fig. 21c, Fig. 22c displays the results of the processed intensities. Fig. 23 displays the results for the same setup as the one of Fig. 21 but instead of a point source, uniform illumination is been used. Although there is a weak signal in the transmitted intensity (see Fig. 23a and c) the two objects cannot be distinguished. In the reflected intensity the objects leave no trace. For *t"767 ps the two objects appear in both the processed transmitted and reflected intensities (see Fig. 23b). For larger *t (e.g. Fig. 23d) the images of the objects merge and cannot be distinguished from an image of a single object. In general, we find that in order to resolve nearby objects it is expedient to employ a point source. In Fig. 24 we depict the simulation results for the same setup as before (Fig. 21) except that the sample contains three objects of 0.5 mm radius. The transmitted and reflected intensities (Fig. 24a and c) show no sign of an object but the processed intensities (Fig. 24b and d) do. As can be seen from Fig. 24b and d it is very difficult to determine, from the maxima of the processed intensities, the number and positions of the objects: The processed transmitted and reflected intensity suggest the presence of two objects and one object respectively. In Fig. 25 we show results of a simulation of a sample containing several 0.5 mm radius objects. The sample is uniformly illuminated on its left side. In Fig. 25a and b the three objects are located at exactly the same position as in Fig. 24c and d. In Fig. 25a there is a very weak trace of only two objects in the transmitted signal and no trace of objects in the reflected intensity. Previously, we

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Fig. 21. Simulation of a time-resolved optical imaging experiment on a sample containing two 0.5 mm-radius objects located at (36, 43) mm and at (36, 28) mm. The sample is illuminated by a point source centered around (0, 36) mm. (a) *t"612 ps, I +0.6]10~8 and IM +0.2]10~2; (b) processed signal corresponding to (a); (c) same as (a) but T R *t"1836 ps, IM +0.5]10~6 and IM +0.3]10~2; (d) processed signal corresponding to (c). Medium: T R (k , k@ )"(0.01, 0.9) mm~1; object: (k , k@ )"(0.1, 0.9) mm~1. ! 4 ! 4

Fig. 22. Same as Fig. 21 except that the two objects are located at (36, 37) mm and at (36, 35) mm. (a) *t"612 ps, I +0.6]10~8 and IM +0.2]10~2; (b) processed signal corresponding to (a); (c) processed signal corresponding to (a) T R but for *t"1224 ps, IM +0.1]10~6 and IM +0.2]10~2; (d) same as (c) but *t"1836 ps, IM +0.5]10~6 and T R T IM +0.3]10~2. The (processed) light intensities inside the sample are also shown. Medium: (k , k@ )"(0.01, 0.9) mm~1; R ! 4 object: (k , k@ )"(0.1, 0.9) mm~1. ! 4

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Fig. 23. Same as Fig. 21 except that the sample is illuminated uniformly on its whole left side. (a) *t"693 ps, IM +0.1]10~7 and IM +0.2]10~2; (b) processed signal corresponding to (a); (c) same as (a) except that *t"1881 ps. T R IM +0.6]10~6 and IM +0.3]10~2; (d) processed signal corresponding to (c). Medium: (k , k@ )"(0.01, 0.9) mm~1; object: T R ! 4 (k , k@ )"(0.1, 0.9) mm~1. ! 4

Fig. 24. Simulation of a sample containing three 0.5 mm radius objects located at (46, 50) mm, (36, 40) mm and (26, 15) mm. The sample is illuminated by a point source. (a) *t"816 ps, IM +0.3]10~7 and IM +0.2]10~2; (b) T R processed signal corresponding to (a); (c) same as (a) except that the objects are positioned at (26, 50) mm, (36, 40) mm and (46, 15) mm, IM +0.3]10~7 and IM +0.2]10~2; (d) processed signal corresponding to (c). Medium: T R (k , k@ )"(0.01, 0.9) mm~1; object: (k , k@ )"(0.1, 0.9) mm~1. ! 4 ! 4

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Fig. 25. Simulation of samples containing several 0.5 mm-radius objects. The sample is illuminated uniformly on its left side. (a) Three objects located at (26, 50) mm, (36, 40) mm and (46, 15) mm, *t"816 ps, IM +0.3]10~7 and T IM +0.2]10~2; (b) processed signal corresponding to (a); (c) Four objects located at (26, 50) mm, (36, 40) mm, (46, 15) mm R and (46, 50) mm, *t"816 ps, IM +0.3]10~7 and IM +0.2]10~2; (d) processed signal corresponding to (c). Medium: T R (k , k@ )"(0.01, 0.9) mm~1; object: (k , k@ )"(0.1, 0.9) mm~1. ! 4 ! 4

have seen that using uniform illumination it should be possible to determine the y-coordinate of a single object. This is also the case here. From the processed transmitted and reflected intensities in Fig. 25b we may conclude that there are at least three objects in the sample: The position of the maximum in the processed reflected data corresponds to the y-coordinate of the position of the centre of the uppermost object, the maxima of the smallest peak in the processed transmitted data corresponds to the y-coordinates of the downmost objects and the maximum of the largest peak in the processed transmitted data corresponds to the two upper objects. However, depending on the positions of the objects, the method cannot resolve all objects. An example is shown in Fig. 25d: four objects are hidden in the sample but the processed reflected and transmitted intensities suggest the presence of three objects only. 7.6. Random fluctuations For some categories of biological tissues, including human breasts, the hypothesis of an homogeneous medium containing evenly distributed light scatterers does not seem correct, indicating that the tissue heterogeneities have to be taken into account in order to model accurately the photon migration in biological media [61]. In this section we use our simulation technique to study the effect of random fluctuations in the absorption and reduced scattering factor of the medium. Unless mentioned otherwise the absorption and reduced scattering factor of the medium fluctuate randomly around their spatial averages by 2%.

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Fig. 26. Simulation of the same sample as in Fig. 20. (a) *t"612 ps, IM +0.6]10~8 and IM +0.2]10~2; (b) processed T R signal corresponding to (a); (c) same as (a) except that the reduced scattering and absorption factor of the medium fluctuate randomly by at most 2%. IM +0.6]10~8 and IM +0.2]10~2; (d) processed signal corresponding to (c). T R Medium: (k , k@ )"(0.01, 0.9) mm~1; object: (k , k@ )"(0.1, 0.9) mm~1. ! 4 ! 4

Fig. 26a and b show simulation results for the case that the medium is homogeneous, and Fig. 26c and d depict the results for the same sample with random fluctuations in the absorption and reduced scattering factor of the medium. The object of 0.5 mm radius leaves no trace in the transmitted and reflected intensity (Fig. 26a and c). The processed transmitted intensity indicates that there is an object inside the sample, in spite of the random fluctuations in the absorption and reduced scattering factor of the medium (Fig. 26b and d). In the processed reflected intensity the object is only visible if the medium is homogeneous. Increasing the gate time *t does not help to make the object appear in the processed reflected signal. An example is shown in Fig. 27a and b for *t"2610 ps. The processed signals do not change with time if the medium itself is not homogeneous. Increasing t can improve the signal-to-noise ratio intensity and can make it possible to find the 0 0.5 mm object, as illustrated in Fig. 27d for t "2500 ps. The object cannot be detected in the 0 processed reflected intensity if light detection starts at an earlier time. The optimal choice of t depends on the size of the object and on the position of the object inside the sample, an example 0 being provided in Fig. 28. Comparison of Fig. 28a and b and Fig. 16c and d demonstrates that except for the noise on the processed signals the results are qualitatively the same. Illuminating the same system with a point source (results not shown) yields qualitatively the same results as the ones displayed in Fig. 16a and b. In Fig. 28c and d we depict simulation results for the case where there is an object of 2.5 mm radius hidden inside the sample and the detection of light starts at t "2000 ps. The object leaves a trace in the processed transmitted as well as in the processed 0 reflected intensity. This trace disappears when the light detection is started at an earlier time. For

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Fig. 27. Same as Fig. 26 except for the choice of t . (a) t "500 ps and *t"2610 ps. IM +0.9]10~6 and IM +0.3]10~2; 0 0 T R (b) processed signal corresponding to (a); (c) same as (a) except that t "2500 ps and *t"612 ps. IM +0.3]10~6 0 T and IM +0.3]10~5; (d) processed signal corresponding to (c). Medium: (k , k@ )"(0.01, 0.9) mm~1; object: R ! 4 (k , k@ )"(0.1, 0.9) mm~1. ! 4

Fig. 28. Simulation of the same sample as in Fig. 16 (Fig. 28a and b) and Fig. 13 (Fig. 28c and d) except that the reduced scattering and absorption factor of the medium fluctuate randomly by at most 2%. (a) Uniform illumination of the left side of the sample, *t"1836 ps, IM +0.6]10~6 and IM +0.2]10~2; (b) processed signal corresponding to (a); (c) Point T R source centered around (0, 36) mm, *t"1836 ps, t "2000 ps, IM +0.6]10~6 and IM +0.2]10~4; (d) processed signal 0 T R corresponding to (c). Medium: (k , k@ )"(0.01, 0.9) mm~1; object: (k , k@ )"(0.1, 0.9) mm~1. ! 4 ! 4

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a fixed position of the object, the smaller the object, the larger t has to be in order to detect the 0 object in the processed reflected signal (see Fig. 27d and Fig. 28d). This is also the case for objects located deeper inside the medium, as shown in Fig. 28. The existence of an optimum delay time for visualizing an object embedded in a turbid medium has previously been discussed in Ref. [133]. 7.7. 3D examples In Fig. 29 we present results of a simulation for a sample of size 63 mm]63 mm]63 mm, containing a sphere of 2.5 mm radius, positioned at (32, 30, 36) mm. The processed signals for illumination of the sample by a point source centered around (0, 32, 32) mm and for uniform illumination of the sample on its whole left plane are shown in Fig. 29a, b and Fig. 29c, d respectively. Fig. 29a and c depict the results for *t"245 ps and Fig. 29b and d for *t"949 ps. The object is clearly visible in both the processed transmitted data and in the processed reflected

Fig. 29. Simulation of a time-resolved experiment on a turbid medium with a reduced scattering factor k@ "0.9 mm~1 4 and an absorption factor k "0.01 mm~1 containing a 2.5 mm radius object with absorption and reduced scattering ! factors k "0.1 mm~1 and k@ "0.9 mm~1, respectively. The dimensions of the sample are 63 mm ] 63 mm ] 63 mm and ! 4 t "500 ps. The object is located at (32, 30, 36) mm. The sphere inside the sample denotes the position of the object. (a) 0 Processed transmitted (right) and reflected (left) signal for *t"245 ps. The sample is illuminated by a source of diameter 1 mm centered around (0, 32, 32) mm during a time t "10 ps. (b) Same as (a) except that *t"949 ps; (c) same as (a) 1 except that the sample is uniformly illuminated on its whole left plane; (d) same as (c) except that *t"949 ps. Medium: (k , k@ )"(0.01, 0.9) mm~1; object: (k , k@ )"(0.1, 0.9) mm~1. ! 4 ! 4

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data. The position of the maximum of the processed intensity changes with *t if a point source is used, just as in the 2D case. Fig. 30 depicts simulation results for the same setup as the one used for Fig. 29. The sample is illuminated uniformly on its whole left plane. In Fig. 30a and b the sample contains a 2.5 mm diameter sphere, while in Fig. 30c and d the sample contains a 0.5 mm diameter sphere. Fig. 30a and b show the transmitted and reflected intensities and Fig. 30c and d show the corresponding processed signals. Only in the case of the 2.5 mm radius object there is a weak signal of the object in the transmitted intensity. In the processed signals, however, there is a clear signal of both the 2.5 mm radius object and the 0.5 mm object. Fig. 31a and c display processed signals for the case that the medium is homogeneous, and Fig. 31b and d depict the processed signals for the same sample with random fluctuations in the absorption and reduced scattering factor of the medium. Fig. 31a and b show results for illumination of the sample by a point source and Fig. 31c and d show the results for uniform illumination of the sample on its whole left plane. In the processed reflected intensity the sphere of 0.5 mm is only visible if the medium is homogeneous. In the case of illumination of the sample by a point source the processed transmitted intensity indicates that there is an object hidden in the sample, in spite of the random fluctuations in the absorption and reduced scattering factor of the medium (Fig. 31a

Fig. 30. Same as Fig. 29 except for the radius of the object. The sample is illuminated uniformly on its whole left plane. (a) Time-integrated transmitted (right) and reflected (left) intensity for *t"857 ps. The sample contains a 2.5 mm radius object. (b) processed signal corresponding to (a); (c) same as (a) except that the radius of the object is 0.5 mm; (d) processed signal corresponding to (c). Medium: (k , k@ )"(0.01, 0.9) mm~1; object: (k , k@ )"(0.1, 0.9) mm~1. ! 4 ! 4

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Fig. 31. Same as Fig. 29 except that the sample contains a 0.5 mm radius object. (a) Processed signal for *t"444 ps. The sample is illuminated by a source of diameter 1 mm centered around (0, 32, 32) mm during a time t "10 ps. (b) Same as 1 (a) except that the reduced scattering and absorption factor of the medium fluctuate randomly by at most 1%; (c) same as (a) except that the sample is illuminated uniformly on its whole left plane; (d) same as (c) except that the reduced scattering and absorption factor of the medium fluctuate randomly by at most 1%. Medium: (k , k@ )"(0.01, 0.9) mm~1; object: ! 4 (k , k@ )"(0.1, 0.9) mm~1. ! 4

and b). In the case of uniform illumination of the sample the sphere is only visible if the medium is homogeneous. Just as in 2D, increasing t may improve the signal-to-noise ratio intensity (results 0 not shown).

8. Image processing technique: Experimental results As shown in the previous section the data processing method can be very useful to detect objects in turbid media. The fact that our simulation method reproduces experimental results suggests that the data processing technique might be useful in real experiments too. Hence, it is used to explore its usefulness in time-resolved reflection experiments on tissue-like phantoms [166]. The experimental setup is, apart from some small changes which will be discussed below, described in Section 5.2 and is depicted in Fig. 11. In this experiment the probe beam is expanded and collimated to a diameter of 2 cm using a set of two lenses in Keplerian configuration. Object detection in turbid media using the time-resolved reflectance curves is done in the following way:

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First the measured reflectance curves are integrated numerically over a given time interval *t, which is usually taken to be around 50 ps, for which the noise-to-signal ratio is good. The integrated intensities are then plotted as a function of the spatial coordinate y, resulting in a one-dimensional image. This image contains information on the inner structure of the sample. The starting time t of the integration window is set by scanning t over the whole time range till 0 0 the contrast in the processed signal is maximal. The optimal choice for t is usually found to be in 0 the tail of the reflectance curve (t '200—300 ps). In order to have a higher signal-to-noise ratio, the 0 streak images are delayed in such a way that the maxima of the temporal intensity profiles fall out of the streak image and that larger measuring times can be used without having the streak camera saturated. Under these conditions the choice of *t has little effect on the spatial profiles. In the experiment we use a methacrylate vessel (70 mm]70 mm]70 mm) filled with a 10% of commercial intralipid-10% solution. At (x"d, y"25 mm) we place a 1 mm diameter tube filled with blood. We vary d from 5 to 15 mm. To compare with the experimental results we performed

Fig. 32. Comparison of experimental and computer simulation results [166] for the time-resolved reflected light intensity. In the experiment and simulation the turbid medium has a reduced scattering factor k@ "1 mm~1 and an 4 absorption factor k "0.01 mm~1. The 1 mm diameter tube, located at (x"d, y"35 mm), is filled with blood. In the ! simulations the tube has k "0.1 mm~1 and k@ "1. The sample is illuminated uniformly on the x"0 plane. Circle: ! 4 d"5 mm; up triangle: d"6 mm; down triangle: d"8 mm; diamond: d"9 mm, cross (#): d"10 mm; cross (]): d"12 mm; star: d"15 mm; square: No object. (a) Experimental results for the reflected intensity (left) and the processed signal (right) for t "395 ps and *t"50 ps; (b) same as (a) but for the simulation results. See also Figs. 3 and 5 of Ref. $ [166].

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simulations for exactly the same system. The simulation is carried out in a manner identical to the procedure used in the time-resolved reflection technique, as described above. In Fig. 32 we show the experimental and simulation data, taken from Figs. 3 and 5 of Ref. [166]. Fig. 32a depicts the experimental data and Fig. 32b depicts the simulation data. The reflected intensities are shown on the left while the processed reflected intensities are shown on the right. In the processed intensities the tube is clearly distinguishable to depths of 15 mm from the illuminated surface. The simulation results are in excellent agreement with the experimental data.

9. Summary We have described an algorithm that solves the time-dependent diffusion equation, taking into account spatial variations of the reduced scattering and absorption coefficient as well as random fluctuations in these quantities. The algorithm is unconditionally stable, accurate, efficient and unlike other stable algorithms, explicit. Futhermore, the algorithm can easily be modified to deal with samples of arbitrary shape. On the basis of this algorithm a software package has been developed to simulate time-resolved optical imaging of objects immersed in strongly scattering media. Our simulation technique reproduces the experimental results on tissue-like phantoms, without the need of adjusting parameters, strongly suggesting that the time-dependent diffusion equation adequately describes the light propagation in these systems and that our software is working properly. In the regime where the diffusion equation can be used to model the light transport in a turbid medium, the reduced scattering and the absorption factor can, in principle, be extracted from measured data by fitting to the measured data, simple expressions, derived from diffusion theory. We have used simulation data to assess the robustness of this approach and found that reliable estimates for both factors can be obtained from transillumination data, irrespective of the pulse time, the detector positions and the width of the beam (excluding illumination of a large area of the sample). In general, the estimates obtained from reflection measurements are less reliable. Much of the current interest in time-resolved imaging with diffuse near-infrared light results from its potential to develop into a non-invasive and safe tool for breast cancer diagnosis. An important issue thereby is the size of the smallest object that can be detected. We have used our simulation approach to determine the conditions under which 1 mm diameter breast-cancer-like objects surrounded by a normal-tissue-like, strongly scattering, medium can be seen directly, i.e. without further processing of the measured intensities. Our findings are corroborated by experimental results on blood tubes immersed in intralipid solutions and suggest that without appropriate data processing the technique is of limited use. As with many other imaging methods, the imaging power can be increased substantially by proper processing of the recorded data. For the case at hand this means that it is necessary to deal with the immense scattering, responsible for the blurring of the images of the object(s). In this paper we have described a simple procedure to enhance the image quality such that mm-sized objects can be located, under seemingly unfavourable conditions. We have given a mathematical justification of this procedure and have demonstrated that it works, not only in computer experiments but also in genuine experiments.

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An essential feature of our data processing method is that it makes use of the diffusive part of the time-resolved light intensity only. We have presented a systematic study of the virtues and limitations of the data processing approach. An important finding is that for a particular source position, the position of the maxima in the recorded transmitted or reflected intensity can change with time, except for the rather ideal case of uniform illumination of one side of the sample. Several measurements for different source—detector arrangements are required to determine the actual position of an object. Furthermore, it seems very difficult to infer the size of the object from the measured intensities. In general, we found that proper processing of the diffuse time-resolved light intensity significantly enhances the signal-to-noise ratio, making it possible to detect the presence of mm-sized tumor-like objects in breast-like tissue. For some categories of biological tissue, notably human breasts, it is unrealistic to assume that the sample can be modelled by a homogeneous strongly scattering medium. We have used our simulation method to study the effect of randomness in the absorption and reduced scattering factors of the medium on the detectability of the objects. We have found that the signal-to-noise ratio can be improved by increasing the delay between the time at which the source is turned off and the time at which the detectors start to record the light intensity. A characteristic feature of the images obtained from time-resolved reflection or transillumination measurements of the diffuse light is that the information contained in these images is scrambled and therefore difficult to interpret correctly. As with other imaging techniques (such as e.g. highresolution electron microscopy) that suffer from similar problems, it is expedient and sometimes also essential for the interpretation of the images to compare experimental results with model simulations. In this paper we have introduced a few theoretical concepts and then used these concepts to develop simulation and image processing algorithms specifically tuned to timeresolved imaging with diffuse light. Application of these ideas to actual experimental data has shown that this approach can be beneficial and suggests that further improvements to the imaging technique can be made by a more sophisticated processing of the measured time-resolved diffuse light intensity.

Acknowledgements This work is supported by EEC and Spanish research contracts and by a supercomputer grant of the “Stichting Nationale Computer Faciliteiten (NCF)”.

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[115] P. van der Zee, D.T. Delpy, Simulation of the point spread function for light in tissue by a Monte Carlo technique, Adv. Exp. Med. Biol. 215 (1987) 179—191. [116] S.L. Jacques, Time-resolved reflectance spectroscopy in turbid tissues, IEEE Trans. Biom. Eng. 36 (1989) 1155—1161. [117] S.T. Flock, M.S. Patterson, B.C. Wilson, D.R. Wyman, Monte Carlo modeling of light propagation in highly scattering tissues. I. Model predictions and comparison with diffusion theory, IEEE Trans. Biom. Eng. 36 (1989) 1162—1168. [118] S.T. Flock, B.C. Wilson, M.S. Patterson, Monte Carlo modeling of light propagation in highly scattering tissues. II. Comparison with measurements in phantoms, IEEE Trans. Biom. Eng. 36 (1989) 1169—1173. [119] Y. Hasegawa, Y. Yamada, M. Tamura, Y. Nomura, Monte Carlo simulation of light transmission through living tissues, Appl. Opt. 30 (1991) 4515—4520. * [120] J.C. Hebden, R.A. Kruger, Transillumination imaging performance: a time of flight imaging system, Med. Phys. 17 (1990) 351—356. [121] E.B. de Haller, C. Depeursinge, Simulation of time-resolved breast transillumination, Med. Biol. Eng. Comput. 31 (1993) 165—170. [122] R. Graaff, M.H. Koelink, F.F.M. de Mul, W.G. Zijlstra, A.C.M. Dassel, J.G. Aarnoudse, Condensed Monte Carlo simulations for the description of light transport, Appl. Opt. 32 (1993) 426—434. [123] E. Tinet, S. Avrillier, J.M. Tualle, Fast semianalytical Monte Carlo simulation for time-resolved light propagation in turbid media, J. Opt. Soc. Am. A 13 (1996) 1903—1915. [124] R.F. Bonner, R. Nossal, S. Havlin, G.H. Weiss, Model for photon migration in turbid biological media, J. Opt. Soc. Am. A 4 (1987) 423—432. [125] A.H. Gandjbakhche, H. Taitelbaum, G.H. Weiss, Random walk analysis of time-resolved transillumination measurements in optical imaging, Physica A 200 (1993) 212—221. [126] M.S. Patterson, B. Chance, B.C. Wilson, Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties, Appl. Opt. 28 (1989) 2331—2336. *** [127] S.R. Arridge, M. Schweiger, M. Hiraoka, D.T. Delpy, A finite element approach for modeling photon transport in tissue, Med. Phys. 20 (1993) 299—309. * [128] M. Schweiger, S.R. Arridge, M. Hiraoka, D.T. Delpy, The finite element method for the propagation of light in scattering media: Boundary and source conditions, Med. Phys. 22 (1995) 1779—1792. [129] S.R. Arridge, M. Schweiger, Direct calculation of the moments of the distribution of photon time of flight in tissue with a finite element method, Appl. Opt. 34 (1995) 2683—2687. [130] K.D. Paulsen, H. Jiang, Spatially varying optical property reconstruction using a finite-element diffusion equation approximation, Med. Phys. 22 (1995) 691—701. [131] R. Model, M. Orlt, M. Walzel, R. Hu¨mlich, Reconstruction algorithm for near-infrared imaging in turbid media by means of time domain data, J. Opt. Soc. Am. A 14 (1997) 313—324. * [132] M.R. Jones, S.G. Proskurin, Y. Yamada, Y. Tanikawa, Application of the zooming method in near-infrared imaging, Phys. Med. Biol. 42 (1997) 1993—2009. [133] S. Havlin, J.E. Kiefer, B. Trus, G.H. Weiss, R. Nossal, Numerical method for studying the detectability of inclusions hidden in optically turbid tissue, Appl. Opt. 32 (1993) 617—627. [134] K. Michielsen, H. De Raedt, N. Garcı´ a, Time-gated transillumination and reflection by biological tissues and tissuelike phantoms: Simulation versus experiment, J. Opt. Soc. Am. A 14 (1997) 1867—1871. [135] In the TDDE the speed of light v sets only the time scale. To rescale the results presented in this paper to the case where the speed of light in the medium is v@, multiply all times by v/v@. [136] B.C. Wilson, M.S. Patterson, The physics of photodynamic therapy, Phys. Med. Biol. 31 (1986) 327—360. * [137] S.L. Jacques, S.A. Prahl, Modeling optical and thermal distributions in tissue during laser irradiation, Lasers Surg. Med. 6 (1987) 494—503. [138] F.H. Long, R.R. Anderson, T.F. Deutsch, Pulsed photothermal radiometry for depth profiling of layered media, Appl. Phys. Lett. 51 (1987) 2076—2078. [139] R.R. Anderson, H. Beck, U. Bruggeman, W. Farinelli, S.L. Jacques, J.A. Parrish, Pulsed photothermal radiometry in turbid media: Internal reflection of backscattered radiation strongly influences optical dosimetry, Appl. Opt. 28 (1989) 2256—2262. [140] S.A. Prahl, I.A. Vitkin, Determination of optical properties of turbid media using pulsed photothermal radiometry, Phys. Med. Biol. 37 (1992) 1203—1217.

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[141] A.A. Oraevsky, S.L. Jacques, F.K. Tittel, Determination of tissue optical properties by piezoelectric detection of laser induced stress waves, in: S.L. Jacques (Ed.), Laser Tissue Interaction IV, Proc. Soc. Photo-Opt. Instrum. Eng. 1882 (1993) 86—101. [142] J. Langerholc, Beam broadening in dense scattering media, Appl. Opt. 21 (1982) 1593—1598. [143] R.A.J. Groenhuis, H.A. Ferwerda, J.J. Ten Bosch, Scattering and absorption of turbid materials determined from reflection measurements. 1. Theory, Appl. Opt. 22 (1983) 2456—2462. [144] R.A.J. Groenhuis, J.J. Ten Bosch, H.A. Ferwerda, Scattering and absorption of turbid materials determined from reflection measurements. 2. Measuring method and calibration, Appl. Opt. 22 (1983) 2463—2467. [145] R. Bays, G. Wagnie`res, D. Robert, D. Braichotte, J.-F. Savary, P. Monnier, H. van den Bergh, Clinical determination of tissue optical properties by endoscopic spatially resolved reflectometry, Appl. Opt. 35 (1996) 1756—1766. [146] W. Cheong, S.A. Prahl, A.J. Welch, A review of the optical properties of biological tissues, IEEE J. Quantum Electron. 26 (1990) 2166—2185. [147] B.C. Wilson, Measurement of tissue optical properties: methods and theory, in: A.J. Welch, M.J.C. van Gemert (eds.), Optical-Thermal Response of Laser-Irradiated Tissue, Plenum press, New York, 1995, pp. 233—274. [148] R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, Experimental test of theoretical models for time-resolved reflectance, Med. Phys. 23 (1996) 1625—1633. ** [149] P. Kubelka, New contributions to the optics of intensily light-scattering materials. Part I, J. Opt. Soc. Am. 38 (1948) 448—457. [150] D. Contini, F. Martelli, G. Zaccanti, Photon migration through a turbid slab described by a model on diffusion approximation. I. Theory, Appl. Opt. 36 (1997) 4587—4599. [151] F. Martelli, D. Contini, A. Taddeucci, G. Zaccanti, Photon migration through a turbid slab described by a model on diffusion approximation. II. Comparison with Monte Carlo results, Appl. Opt. 36 (1997) 4600—4612. [152] S.A. Prahl, M.J.C. van Gemert, A.J. Welch, Determining the optical properties of turbid media by using the adding-doubling method, Appl. Opt. 32 (1993) 559—568. [153] M.H. Eddowes, T.N. Mills, D.T. Delpy, Monte Carlo simulations of coherent backscatter for identification of the optical coefficients of biological tissues in vivo, Appl. Opt. 34 (1995) 2261—2267. [154] A. Kienle, M.S. Patterson, Determination of the optical properties of turbid media from a single Monte Carlo simulation, Phys. Med. Biol. 41 (1996) 2221—2227. [155] A.H. Gandjbakhche, R. Nossal, R.F. Bonner, Scaling relationships for theories of anisotropic random walks applied to tissue optics, Appl. Opt. 32 (1993) 504—516. [156] T.J. Farrell, M.S. Patterson, B. Wilson, A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the non-invasive determination of tissue optical properties in vivo, Med. Phys. 19 (1992) 879—888. [157] A.H. Hielscher, S.L. Jacques, L. Wang, F.K. Tittel, The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues, Phys. Med. Biol. 40 (1995) 1957—1975. [158] W.M. Star, J.P.A. Marijnissen, H. Jansen, M. Keijzer, M.J.C. van Gemert, Light dosimetry for photodynamic therapy by whole bladder wall irradiation, Photochem. Photobiol. 46 (1987) 619—624. [159] C.J.M. Moes, M.J.C. van Gemert, W.M. Star, J.P.A. Marijnissen, S.A. Prahl, Measurements and calculations of the energy fluence rate in a scattering and absorbing phantom at 633 nm, Appl. Opt. 28 (1989) 2292—2296. [160] H.J. van Staveren, C.J.M. Moes, J. van Marle, S.A. Prahl, M.J.C. van Gemert, Light scattering in intralipid-10% in the wavelength range 400—1100 nm, Appl. Opt. 30 (1991) 4507—4514. [161] S.T. Flock, S.L. Jacques, B.C. Wilson, W.M. Star, M.J.C. van Gemert, Optical properties of intralipid: a phantom medium for light propagation studies, Lasers Surg. Med. 12 (1992) 510—519. [162] A.H. Hielscher, J.R. Mourant, I.J. Bigio, Influence of particle size and concentration on the diffuse backscattering of polarized light from tissue phantoms and biological cell suspensions, Appl. Opt. 36 (1997) 125—135. [163] S.J. Madsen, B.C. Wilson, M.S. Patterson, Y.D. Park, S.L. Jacques, Y. Hefetz, Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffuse reflectance experiments, Appl. Opt. 31 (1992) 3509—3517. [164] K. Symanzik, Proof and refinements of an inequality of Feynman, J. Math. Phys. 6 (1965) 1155—1156. [165] N. Dunford, J.T. Schwartz, Linear Operators, Interscience, New York, 1958. [166] J. Przeslawski, J. Calsamiglia, N. Garcı´ a, H. De Raedt, K. Michielsen, M. Nieto-Vesperinas, Detection of 1 mm size hidden objects in tissue-like phantoms by backscattered diffused light, Appl. Opt. (in press).

GRAPHICAL ANALYSIS OF ANGULAR MOMENTUM FOR COLLISION PRODUCTS

Michael DANOS!, Ugo FANO" ! Enrico Fermi Institute, University of Chicago, 5640 South Ellis Ave., Chicago IL 60637, USA "James Franch Institute, University of Chicago, 5640 South Ellis Ave., Chicago IL 60637, USA

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

Physics Reports 304 (1998) 155—227

Graphical analysis of angular momentum for collision products Michael Danos!,1, Ugo Fano" ! Enrico Fermi Institute, University of Chicago, 5640 South Ellis Ave., Chicago IL 60637, USA " James Franck Institute, University of Chicago, 5640 South Ellis Ave., Chicago IL 60637, USA Received January 1998; editor: J. Eichler Contents 1. Introduction 1.1. Historical background 1.2. Novel analysis of reaction products 1.3. Overview 2. States, operators, notations 3. Absorption of an unpolarized photon by an unpolarized target 3.1. Evaluation in the m-summation scheme 3.2. Evaluation by the Racah calculus 4. Photon—particle correlation 4.1. Extracting angular distributions from the wave function 4.2. Angular distributions as expectation values of observable operators

158 159 163 166 167 175 176 177 183 183

5. 6. 7. 8. 9.

Inclusion of spin measurement Addition of a spectator core Two-body interaction Auger transitions following photo-ionization Processing of results 9.1. Correlation invariants 9.2. Radial integrals 9.3. Remarks Appendix A. General remarks on recoupling graphs Appendix B. Definitions and notation References

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Abstract The complexity of atomic and nuclear structures and their collision processes involves conservation laws, bearing mainly on angular momenta; indeed angular momentum treatments prove most laborious. The analytic treatments, preferably carried out in Racah’s calculus, combine initially independent elements stepwise into structures branching out into resulting products. Graphical procedures that ensure phase and amplitude control of their manifold elements, illustrate these sequential steps and provide their results. The present report should familiarize readers with these procedures through examples of reactions of increasing complexity, bearing of course on structure calculations as well. The report has thus two aims: (i) computing correlation functions for reactions yielding several emitted particles (hence of arbitrary complexity) in terms of a novel method of computation, and (ii), describing the mathematical techniques relevant to solve high-complexity angular momentum problems, including the computation of many-body systems’ bound states. The complexity reflects the symmetries of the reaction products, and, more generally, of many-body system. The basic mathematical tool for such treatments is the Racah calculus which employs recoupling transformations, thus 1 Visiting Scholar. Correspondence address: 4633 Q street NW, Washington DC 20007-2576, USA. E-mail: [email protected]. Tel.: (202) 338-2718; fax: (202) 338-1756. 0370-1573/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 2 0 - 9

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avoiding the many summations required by expansions in terms of vector coupling coefficients. The application of the Racah calculus is greatly aided by appropriate definitions and graphical procedures ensuring phase and amplitude control of their manifold elements, as well as illustrating the physical content. Beginning with photon absorption by discrete states, our examples progress to an Auger process yielding a correlation function with seven direction and polarization parameters. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 02.70.!c; 02.90.#p; 23.20.En; 25.20.!x; 25.20.Lj; 32.80.!t; 32.80.Fb Keywords: Angular momentum; Angular correlations; Racah calculus; Recoupling graphs

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1. Introduction The physics of multi-particle aggregates (atoms, molecules, nuclei) involves exchanges of energy and angular momentum between their constituents — often through tensorial interactions, i.e., with multi-index operators. Visualizing and organizing the relevant mechanisms has led for decades to utilize a variety of diagrams. The invariance of mechanical processes under rotation of space coordinates — underlying the conservation of angular momenta — affords expanding their treatment into spherical harmonic functions, thereby articulating complex processes into sequences of elementary steps. A principal device for their analytic treatment standardizes tensorial variables to transform under coordinate rotations as the sets of spherical harmonics ½ (h, u), whose label l, unchanged lm under rotations, represents an “orbital” quantum number, whereas its “magnetic” m depends on the orientation of a reference axis. Such sets are “irreducible”. Inclusion of spin variables extends the pair of integers (l, m) to integer or half-integer pairs ( j, m), indices playing the role of mechanical angular momenta. Direct products of two such variables can be expanded (“reduced”) into single standard sets of variables in accordance with the famous 19th century Clebsch—Gordan theorem.2 A comprehensive review of the analytical procedures for the analysis of atomic spectra by Judd has appeared recently [2]. The diagrammatic approach of this report might serve as an illustrative complement. The addition of two angular momenta,3 involving products of their components together with its reduction to single-pair ( j, m) form, is called their “coupling”. This term applies equally to the combination and reduction of operators or tensorial variables with multiple Cartesian indices. A major feature of the resulting extended manifold of variables lies in ordering of coupling in alternative sequences: The alternative triple products [(ab)c] and [a(bc)] generally differ, expressions of their alternative results being related by linear “recoupling” transformations. The analytic representation of two coupled “irreducible tensorial sets” indicated by ( j m Da), 1 1 ( j m Db), yielding the “irreducible tensorial set” ( j m Dab) reads 2 2 12 12 ( j m Dab)" + ( j m D j m j m )( j m Da)( j m Db) . (1.1) 12 12 12 12 1 1 2 2 1 1 2 2 m1m2 Multiple products arise by iteration of this pairwise coupling in alternative sequences. Details of the above example read thus: (( j j )j j , jmD[ab]c)" + ( jmD j m j m )( j m D j m j m )( j m Da)( j m Db)( j m Dc) 1 2 12 3 12 12 3 3 12 12 1 1 2 2 1 1 2 2 3 3 m1m2m3m12 (1.2)

2 The expansion coefficients for the group of physical space rotations were identified and evaluated algebraically by Wigner in the late 1920s, being thus properly referred to as “Wigner coefficients”. See Ref. [1], Chapter 17. 3 The two concepts, the angular momentum of a physical object, and the behavior of the components of a “tensorial set” under rotations of the coordinate system, are quite distinct, but commonly are used interchangeably. The present paper deals exclusively with tensorial sets.

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with alternative coupling ( jmD j m j m )( j m D j m j m )( j m Da)( j m Db)( j m Dc) . ( j ( j j )j , jmDa[bc])" + 1 1 23 23 23 23 2 2 3 3 1 1 2 2 3 3 1 2 3 23 1 2 3 23 mmmm (1.3) The recoupling transformation (( j j )j j , jmD[ab]c)"+ (( j j )j j , jDj ( j j )j , j)( j ( j j )j , jmDa[bc]) , 1 2 12 3 1 2 3 23 1 2 3 23 1 2 12 3 j23 is represented in terms of Wigner’s symmetrized 6j-coefficient [1, ch. 24] by

G

(1.4)

H

j j j (( j j )j j , jD j ( j j )j , j)"(!1)j1`j2`j3`jJ(2j #1)(2j #1) 1 2 12 . (1.5) 1 2 12 3 1 2 3 23 12 23 j j j 3 23 This 6j-coefficient belongs to a class of Wigner “3nj” coefficients [1], ch. 24, with n"2. The next member of this class, with n"3, i.e., the 9j-coefficient to be described in Section 2, serves as the symmetrized recoupling coefficient for four angular momenta. The value n"1, the 3j-coefficient, corresponds to a symmetrized Wigner coefficient. Coupling any number of sets leads often to a final invariant product, labelled by j "0. 1,2,... Invariants amount, of course, to sets with a single element, such as a system’s energy, and otherwise to scalar or hermitian products of two irreducible sets of equal degree, j "j , with j "0 in 1 2 12 Eq. (1.1), an elementary example being a vector’s squared modulus. Important examples of invariants are the state representatives of physical systems in a form independent of coordinate orientation, such as the + a t that contains amplitudes a specifying a system’s orientation m jm jm jm together with a basis set t of its states. Similarly, the energy of an atomic system arises as an jm invariant combination of matrix elements representing all the interactions of particle pairs (or triplets), whose evaluation often involves extensive recoupling operations. Quite generally, since many physical quantities are independent of coordinate orientation, results of calculation are mostly invariant. For example, an invariant factor P (cos h) may appear in the expression of a cross L section, with h representing the angle between two observable directions. 1.1. Historical background It was recognized long ago that the basic symmetry of physical space is embodied in the transformation of vectors, tensors and their combinations under coordinate rotations. Tensors with several Cartesian indices can be “reduced” [1], i.e., transformed and resolved into individual eigenvectors of the squared angular momentum, denoted “irreducible tensors”, being thus characterized by the index j of their eigenvalues and by its projection m on a z axis. These eigenvectors pertain usually to single-particle states, multi-particle states being built up as their products. An important starting point toward this reduction lies in the Clebsch—Gordan theorem, stating that a direct product of two irreducible tensors can be expanded into irreducible tensors:4 ¹*A+?¹*B+"+ K ¹*C+ . C C 4 Notation as in Fano and Racah, Refs. [3—5].

(1.6)

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Fig. 1.1. (a) Vector diagram for coupling of two angular momenta. (b) Yutsis symbol for the Wigner 3j-coefficient.

For the unitary group SU(2), the labels [A], etc., are the angular momentum quantum numbers, j, obeying the triangularity condition; in the example with A"1/2, B"2, K reduces to unity for C C"3/2 or 5/2, and vanishes otherwise. For other groups, e.g., SU(3), higher multiplicities occur, with values of K accordingly no longer limited to 0 and 1. Explicit reduction of a direct product is C accomplished as in Eq. (1.1), ¹*C+"+ (CMDA m B M!m)¹*A+¹*B+ . (1.7) M m M~m m The Wigner coefficients, (CMDA m B M!m), also called ‘‘vector coupling coefficients”, are abbreviated in the following as VC-s, frequently erroneously labelled as ‘‘Clebsch—Gordan” coefficients. Eq. (1.7) is written in compact notation [3]a, ch. 7 as (1.8) ¹*C+"[¹*A+¹*B+]*C+ . M M Alternative couplings occur when dealing with many particles, as in Eqs. (1.2) and (1.3), related by the transformation (1.4). Labelling such alternative couplings and recouplings gave rise to graphical procedures, beginning with the vector addition of angular momenta (Fig. 1.1(a)) and with the early introduction of symbols for the coefficients of pair-reduction (actually in their symmetrized form, the 3j coefficient, Fig. 1.1(b)), by Yutsis et al. [6]. (Connecting the lines of two or more such symbols represents summations over the m quantum numbers of the corresponding coefficients as in Eqs. (1.1), (1.2), (1.3), (1.4), (1.5), (1.6) and (1.7).) There soon emerged, however, difficulties in keeping track of ‘‘minus” factors, a drawback alleviated by modifications introduced in the 1960s by El-Baz [7] and others. Yutsis’ graphical method thus allowed solving many not too complex problems, essentially by m-summations, as described by Sandars [8], Briggs [9], and more extensively in textbooks (Lindgren and Morrisson [10]). In its modified form the coupling of two angular momenta is graphed as in Fig. 1.2.

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Fig. 1.2. Yutsis graph for coupling of two angular momenta.

The need for angular momentum recoupling arose early in the development of atomic physics, e.g. when evaluating the spin—orbit interaction with the relevant angular momentum quantum numbers, H&[p*1+ ) ¸*1+]*0+

(1.9)

and its expectation value for the single-particle states t*j+"[m*s+/*l+]*j+ . (1.10) m m Here m and t indicate the spin and the orbital functions with s"1/2 and l the orbital angular momentum quantum number. In Eq. (1.9) the spin and orbit operators have been coupled together, as are the spin and orbital wave functions in Eq. (1.10), with the understanding that each operator acts on its corresponding wave function. The necessary recoupling is readily performed in this instance: Writing out the angular momentum couplings in terms of the reduction formulas Eq. (1.1)ff, one can collect the functions and operators to form matrix elements. One must now perform sums over VCs, i.e., ‘‘m-sums”, readily carried out in the present example. This msummation technique, fully exploited in the textbook by Rose [11], served well for rather simple problems, being still widely employed even to solve problems for which more powerful methods would be advisable.5 This method can be, and in practice is being, substantially improved by recognizing groups of summations that can be collapsed into recoupling coefficients. On the other hand, Fano and Racah realized in the early 1950s that recoupling transformations can be carried out directly on operators and wave functions in their initial coupled form, without resolving them into coordinate-dependent components and thus bypassing m-summations entirely [3]a, ch. 13, 16; [3]b, ch. 7. Greater transparency of the procedure resulted, in addition to substantial savings of time and effort in performing calculations. Further steps in the evolving treatment of angular momentum problems were taken in the 1970-s by Danos [12] who recognized the importance of minimizing the number of basic elements and of their definitions, eliminating the occurrence of phase factors as fully as possible. This goal can be achieved by universal employment of the Biedenharn—Racah phase convention, introduced below in Eqs. (2.6) and (2.14) and of the 9j-type recoupling coefficients, Eq. (2.27). Further gain in transparency, with concomittant elimination of potential sources of error, ensues when formulating problems exclusively in terms of invariants. A relevant example is provided by the state invariant W &[a*j+t*j+]*0+, j

(1.11)

5 For example, an m-computation for the problem treated in Section 5 — the photo-ionization reaction including spin polarization — would require the handling of (2j#1)24 terms; already for rather small j-values, this number would be around 1015, large even for main-frame computers.

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Fig. 1.3. Graph symbol for the permutation of two coupled tensors.

where t*j+ represents a set of system’s wave functions according to Eq. (1.10), and a*j+ a corresponding set of amplitudes specifying the system’s orientation in space. This complete program, which actually emerged in the context of many-body structure calculations, was implemented by Danos, Gillet, and Cauvin, Refs. [4,13]. Identifying states by means of their amplitudes a*j+ in Eq. (1.11) m serves both in the presence of external fields that determine their values and in constructing symmetric particle structures where the a*j+’s combine into products with the relevant symmetry m under coordinate transformations. This formulation is particularly important in describing reactions, by identifying their essential geometric variables in the amplitudes a*j+ at the very outset. The graphical symbols introduced by Yutsis for the VCs in the 1950s and their extension to represent recoupling coefficients did facilitate the evaluation of not-too-complex problems within the framework of m-summations. One first represents an initial expression as a graph; by inspection one recognizes that a summation over certain indices, a ‘‘contraction”, can be carried out yielding recoupling coefficients; redrawing the graph now containing these recoupling coefficients may suggest further possible contractions yielding further recoupling coefficients. More complicated problems, however, in particular reactions involving correlation functions of their products, require more powerful methods. One should thus formulate any problem directly in the Racah—Fano philosophy, bypassing all VCs and working only with generic recoupling transformations. The graphical method designed to implement the Racah—Fano philosophy (in which no VCs arise), which will be employed throughout this paper, was introduced by Danos in the 1970s [12]. The basic elements of its graphical system are lines, each representing a tensorial set, i.e., a single parameter, like a set of basis function, an operator, an amplitude, or a coupled pair, with given angular momentum. One line forking into two, shown in Fig. 1.3, simply identifies individual elements of two coupled tensorial sets. Similarly, a tapered confluence of two lines into a single line indicates coupling of their angular momenta to the value labelling that line. In contrast to the Yutsis method, forking carries no significance. Changes of coupling are confined to ‘‘recoupling boxes”, the simplest one representing the permutation, also shown in Fig. 1.3, with algebraic value (!1)j1`j2~j12. This method depicts sequences of recoupling transformations forming a flowthrough graph, where single tensors — or, generally, tensorial sets — with initial coupling [e.g., as in Eqs. (1.9), (1.10) and (1.11)] are followed through several recoupling transformations until terminating in matrix elements or in symbols representing correlation functions. A complete transformation is thereby contained within a single overall graph; its corresponding algebraic expression, including all of its factors and phases, can be read directly off this graph. The recoupling operations being linear transformations, these diagrams amount to graphical representations of Dirac’s transformation theory, with recoupling blocks embodying Dirac’s transformation brackets, such as Eq. (1.4).

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1.2. Novel analysis of reaction products The directional distributions of collision products reflect much of collision dynamics through multiple fragments emerging in several directions, with observable excitations and spin orientations, thereby providing access to a greater wealth of physics than is available from bound-state spectra. A correspondingly larger computational effort results from the larger number of variables and parameters. The present paper aims at such large computations. To cope with this complexity, the analysis of reactions has utilized successfully seemingly abstract applications of angular momentum theory known as the Racah calculus6. These applications, on the other hand, flow naturally from the ‘‘overall graphs” introduced above, analogous to plans of industrial operations that culminate in a single master blueprint. Let us consider how such an opportunity arises: The probability amplitude of a collision output takes the generic form of a transition matrix element SBDDDAT, where SBD and DAT represent the process’ output and input and D the relevant Hamiltonian operator, yielding the corresponding probability P"DSBDDDATD2,SADDsDBTSBDDDAT .

(1.12)

The symbols A, D, B represent in turn state-representatives (wave functions or density matrices) and operators. The Racah calculus represents each of these matrix elements as the invariant (Hermitian) product of geometrical coefficients and of dynamical elements (base sets of wave functions or operators), Eq. (1.11), as displayed in Section 2. We aim at recasting Eq. (1.12) into the form Measurement"+ Dynamics]Experimental Layout ,

(1.13)

by disassembling the geometric and dynamical elements, and reassembling them into separate geometric and dynamic blocks by linear recoupling transformations. (The word ‘‘Measurement” in Eq. (1.13) amounts to the English word for the symbol P in Eq. (1.12).) Since, as mentioned above, the states are given in invariant form as shown in Eq. (1.11), where, e.g., the amplitudes a*j+ for the emitted particles are functions of the emission angles, the reassembled geometric elements, denoted ‘‘Experimental Layout” in Eq. (1.13), provide angular dependences of the correlation functions, with coefficients provided by the ‘‘Dynamics”, including matrix elements, overlap integrals, and recoupling transformations. The relevant linear transformations consist of universal elementary operations, chiefly exchanging (‘‘recoupling”) the partners of two pairs of angular momenta, or, more generally, of irreducible tensorial sets [3], labelled by angular momentum quantum numbers j.7 Hence, importantly: No vector coupling coefficients arise or have to be written explicitly throughout the calculation; they appear only in the specifications of particular experimental setups, for example to indicate that a counter is placed in the x-direction and is sensitive to a spin alignment along the y direction, etc. As an important corollary, no VCs arise in problems with a trivial ‘‘Experimental Layout”, e.g., in calculating the energy of an isolated atom or nucleus, or the total cross section of a reaction for unpolarized or unaligned systems.

6 frequently, imprecisely, called ‘‘Racah algebra”. 7 That is, identifying them as eigenstates of a squared angular momentum operator.

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Each step of an elaborate task will be standardized, as the whole operation resolves into extended sequences of recouplings that may be organized in alternative ways, equivalent in substance but different in computational effort. Several inherent circumstances require specific attention in designing details: (a) Multiple intermediate sets of angular momentum quantum numbers, i.e., j-numbers, occur, requiring summation as in Eq. (1.4). Minimizing the number of such intermediate sums, by changing the order of recouplings upon inspection of the whole graph, often saves much computational effort. (Two relevant examples, one of them demonstrating the Biedenharn—Elliot identity, are presented in Appendix A.) (b) Even the simplest recoupling of two tensorial sets, their permutation, yields a factor $1 dependent on the values of relevant quantum numbers. (c) Sums over observable quantum numbers yield interference effects hinging on consistent application of phase conventions in defining relevant quantum amplitudes. Appropriate consistent standardizations remove any need for seemingly arbitrary phase factors [3]a, ch. 5; [4], ch. 3. The graphical procedure presented in this review — and previously in Refs. [4,12] — takes into account the preceding considerations automatically, through chains of suitably designed graphical elements. Laying out the whole interlinked chain of symbols representing a calculation of interest affords an oversight of its development and ready comparison of the calculational effort required by alternative organizations of the graphs. Graphs thus incorporate all coefficients and signs relevant to calculations and thereby afford reading the complete final analytic expression of a transition probability directly from the ‘‘overall blueprint” in the form of Eq. (1.13). A general description of the construction and use of such graphs occurs in [4]. The task of determining the sequence and groupings of recoupling transformations, and of their corresponding ‘‘crossing boxes” — i.e., permutations of two coupled factors — appropriate to transforming any collision probability (1.12) into its desired form (1.13) may itself present appreciable difficulty. This task is facilitated by subsidiary graphs, introduced here under the name of ‘‘guide graphs”, which diagram linkages of geometric and dynamic elements from Eq. (1.12) to be properly coupled together into invariants or irreducible tensorial products. One or more such graphs will be presented at the outset of following Sections for guidance in drawing the relevant recoupling graphs. In the present formulation the correlation function arises in terms of amplitude products. These products — representing ‘‘density matrices” of relevant states — are generally ‘‘reducible”, meaning that they resolve into sums of irreducible components labelled by different ¸ indices [14]. These parameters thus are cast as ‘‘vectors” of the Liouville representation of quantum mechanics (see Refs. [15,16]), with components labelled by a single magnetic number in addition to their ¸ label. Each of these irreducible components, called here ‘‘directional polarization vector”, or simply ‘‘polarization vector”, etc., depending on the circumstances, represents the direction of a particle, of its spin, or the multipole moment of a particle group. The factors denoted ‘‘Experimental Layout” in Eq. (1.13) arise from reshuffling invariant products of polarization vectors, the desired correlation functions being generally cast as weighted sums over these invariant products, with weights denoted ‘‘Dynamics” in Eq. (1.13). For brevity we shall call the invariant ‘‘Experimental Layout” factors ‘‘correlation invariants”.

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The initial formalism is familiar to atomic or nuclear physicists, with two minor exceptions: (i) the relationship between complex and hermitian conjugation for spherical harmonics and operators (matrices), Eq. (2.5) below, is treated according to the Biedenharn—Racah convention [4] so as to hold equally for all tensors and spinors. Specifically, we shall use phase-renormalized spherical harmonics, denoted by ½*l+, Eq. (2.14). (ii) A relation equivalent to the Wigner—Eckart m theorem, [5], ch. 5; [3]a, ch. 14; [3]b, ch. 6, will emerge spontaneously in Section 3, Eq. (3.17), containing ‘‘invariant matrix elements” which are related to the ‘‘reduced matrix element” [1], ch. 21 by a phase. Similarly, phase and algebraic renormalizations will be avoided by employing a renormalized 9j-coefficient with the same symmetries as the corresponding recoupling coefficient. These two conventions eliminate the explicit occurrence of factors i or (!1). Examples, taken for simplicity from atomic physics, should familiarize the reader with properties of our tools, i.e., with the ‘‘small blueprints” in our analogy, which can then be collected into the ‘‘overall blueprint”, the overall graph. (The treatment of examples taken from the nuclear shell model would be identical except for replacing the radial parts of wave functions.) Working through the series of examples should provide readers with the opportunity to acquire the skills required to actually solve angular momentum problems with essentially arbitrary complexity and, more generally, to deal with systems obeying SU(N) symmetries as in particle physics. As a side effect, we shall also obtain solutions to examples, some of them with complexity making their solution difficult without our ‘‘heavy tools.” Our examples will only include photon-induced reactions. Particle-induced reactions involve the more complicated reaction dynamics of the ‘‘complex” (‘‘compound system” in nuclear physics) formed by combining projectile and target. The overall angular momentum problem is, however, no more complicated for particle-induced than for photon-induced reactions, except for requiring additional diagram(s) detailing the formation of the ‘‘collision complex” projectile and target. Each example will be formulated in terms of pure states, i.e., of wave functions of relevant quantum states of the incident radiation, target, and emitted particles. Final results, however, will be expressed in terms of ‘‘polarization vectors” representing both the input and the output of each reaction. These expressions apply equally to pure or impure states, since the label ‘‘polarization vector” pertains to an irreducible component of a density matrix, equally for pure or impure states [14,17]. Understanding the physics of a process serves to choose the specific form of the two factors in Eq. (1.12). We shall not address this question here, leaving generally explicit formulations to the reader thus affording him flexibility: Actual experimental setups determine appropriate coordinate systems, the specific coupling scheme of correlations with more than three factors; identities of trigonometry often allow casting multi-variable functions in alternative forms. Our procedure maps how physical processes transform a set of initial (‘‘input”) parameters into a corresponding set of resulting (‘‘output”) parameters through a sequence of elementary steps. Each step is embodied into a recoupling or analogous coefficient provided by angular momentum theory. The interlinking of steps may be viewed as a ‘‘geometric” (or ‘‘kinematic”) function, whereas the input and output parameters and the matrix elements provide ‘‘dynamical” information. The ‘‘dynamical” elements may not be known initially with sufficient precision; experimental verification of the ‘‘output” parameters’ relation to the ‘‘input” ones may serve to assess the assumed values of the several ‘‘dynamical” elements, accessible by inversion of the expressions obtained in our procedure.

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1.3. Overview Sections 2—4 introduce the essential features of our approach, the following Sections 5—7 adding complementary aspects, culminating in Section 8 in an extensive application to an Auger process. Section 9 provides contact with experiment for the explicit determination from experimental data of spin and directional polarization vectors and other parameters of interest. Readers may find it helpful to give this section an early cursory look. Section 2 begins our detailed development by introducing state functions in rotationally invariant form and evaluating their norm, as examples of graphical techniques. Parameter definitions are adjusted for general applicability and avoidance of unnecessary factors. The expectation value of a scalar operator, having the same geometry as a norm, is also introduced at this point. Section 3 computes the simplest transition matrix element, both by summing over m quantum numbers and in the Racah technique, still ignoring particle spins. Even in this elementary example, the Racah technique seems at least as easy as, perhaps even preferable to, explicit m-summations. The Racah technique alone will serve in further more complicated examples. Most details of our graphical procedures are first presented in Sections 2 and 3. Section 4 introduces the photo-emission process, including polarization vectors for the initial target, for the photon polarization, as well as the directional polarization vector for the ejected particle, again ignoring its spin. We calculate the correlation functions for this process in terms of these vectors, deriving results in the form of Eq. (1.13). Section 5 deals with measuring the spin polarization of a photo-emitted electron, treated in jj-coupling, as a prototype evaluation of observables’ expectation values. After introducing the treatments of a spectator core and of a two-body force in Sections 6 and 7, to complete our basic tools, Section 8 presents the full correlation function for an Auger process, including spins, which involves seven variables in its ‘‘Experimental Layout” factor of Eq. (1.13). Translating the output of experimental data into the values of tensorial or other parameters of interest in Section 9 will present the problem of ‘‘inverting” the theoretical developments. Supplementary remarks appear in Appendix A, together with the complete recoupling graph, the ‘‘overall blueprints”, for the problems treated in Sections 4 and 5. A list of the expressions used in this paper is given in Appendix B. All the tools needed for to evaluate the Hamiltonian in bound state computations are introduced in Sections 2, 3, 6 and 7. A more complete discussion of the details encountered in that context is contained in Refs. ([13]; [4], ch. 6; [3]b); it lies outside the frame of the present paper. Reading the present paper requires familiarity only with the basic theories of quantum mechanics and angular momentum. To render the paper self-contained we shall define relevant parameters as and when needed, mostly to set notation and conventions with a minimum of explanations. The reader will be referred to specific items of textbook literature for details and background, holding bibliographic references to a bare minimum. The present emphasis on graphical methods should afford the reader both familiarity with the substance of the Racah calculus and skills to apply the methods discussed in [4]. A computer program to optimize complete algebraic transformations has been constructed by Williams and Silbar [18]. Tables of 3-j and 6-j coefficients, together with extensive discussion of their properties are given in Ref. [19].

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2. States, operators, notations This section introduces the notation of angular momentum eigenvectors and of other parameters, including the corresponding graphical symbols, required to carry out calculations, i.e., the transformations leading from the starting form, Eq. (1.12) to the desired form Eq. (1.13). It also exhibits the prototype of these graphs, containing the ‘‘basic building block”, using as example the evaluation of the norm of a state. In general, our notation corresponds to the usual definitions, as given, e.g., in [3]a, 5, except for the definitions of Eqs. (2.14) and (3.18), designed to avoid unnecessary phases. Readers familiar with this subject may skip this section and proceed to Section 3. The state of any physical system is independent of arbitrarily selected coordinate systems. Thus, in mathematical terms, the state of a physical system with angular momentum j is represented as an invariant [3]a, ch. 2; [3]b, ch. 3, by a sum over terms with different magnetic quantum numbers m,8 yielding the Hermitian product [3]a, ch. 6; [3]b, ch. 3.2 W"+ a(j)t*j+ , (2.1) m m m whose sets of variables a(j) and t*j+ are contravariant9 to each other under coordinate rotations [3a, m m chs. 4, 5]. The coefficients a(j) labelled with round parentheses, ( j), are called ‘‘standard tensorial m elements”, and the basis functions t*j+, labelled with square brackets, [j], are called ‘‘contram standard tensorial elements” of irreducible sets with 2j#1 elements. The coefficients a(j) represent m probability amplitudes defining the orientation of the physical system with respect to specific coordinate axes, while the t*j+ represent elements of a basis set of eigenstate wave functions. m Operators will also be defined as in Eq. (2.1): A"+ a(j)A*j+ , (2.2) k k k whose amplitudes a(j) specify the operator’s orientation or alignment. The graph symbol for k tensors, be it amplitudes, basis functions, or operators, will be lines, as for example in Fig. 2.1. Each line will be labelled by its angular momentum quantum number, and sometimes by the tensor’s ‘‘name”. A more complete discussion of Fig. 2.1 is given below. Our task in transforming Eq. (1.12) to Eq. (1.13) will be to re-group parameters to combine the functions and operators into matrix elements independent of coordinate orientation, while amplitudes combine into scalar functions of the angles specifying the mutual orientations of reaction products and other devices. The structure of invariants (2.1) and (2.2) affords separating the geometrical — a(j) — and dynamical — t*j+ and A*L+ — elements essential to construct the right-hand m m M

8 The labelling by j and m implies irreducibility of the representation. 9 The two factors of the product in Eq. (2.1) would transform equally only under ‘‘orthogonal” coordinate rotations; in general they undergo reciprocal (i.e., contravariant) transformation, as encountered when dealing with oblique axes. The need for distinguishing these two representations reflects the context of the bra and ket symbols in matrix elements; cf. Eq. (2.18) below.

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Fig. 2.1. Graph symbol for the 9j-type recoupling transformation. Summation over the ‘‘new” quantum numbers, R, S, is implied. The occurrence of the same index K at the input and output ends of the graph reflects the conservation of the total angular momentum.

side of Eq. (1.13). Standard elements of a general irreducible tensorial set (e.g., A(L)), whatever their M physical significance, transform under rotation of the coordinate system as [3]a, ch. 5; [3]b, ch. 2; 5, ch. 4.1 A(L)"+ DL A(L) , M MM{ M{ M{ while the contrastandard elements transform as

(2.3)

A*L+"+ DL* A*L+"+ A*L+DL ; M MM{ M{ M{ M{M M{ M{ the two forms being equivalent owing to the symmetry

(2.4)

DL* "DL . (2.5) MM{ M{M Note how the standard tensor A(L) is transformed in Eq. (2.3) by applying the D matrix on its left, M{ whereas D operates on the right of the contrastandard tensor A*L+ in Eq. (2.4). In the present M{ treatment of reactions no use of rotation matrices will be required. The tensorial elements are defined to obey the conjugation property A(L)"(!)L`MA*L+ . (2.6) M ~M In this generally used Condon—Shortley convention the coupling of two tensors to form an eigenvector of angular momentum — i.e., the reduction of a product of two tensors — involves either a pair of standard tensors, or a pair of contrastandard tensors. We indicate this operation by the symbolic notations [a(A)b(B)](L)" + (¸MDAm, Bm@)a(A)b(B) , M m m{ m,m{

(2.7)

[a*A+b*B+]*L+" + a*A+b*B+(Am, Bm@D¸M) . (2.8) M m m{ m,m{ The values of vector coupling coefficients (¸MDAm, Bm@) coincide with those of their reciprocals, appearing in Eq. (2.8) [3]a, ch. 7; [5], ch. 3.4. With the standard abbreviation jK "J2j#1 ,

(2.9)

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to be employed throughout the paper, and with the vector coupling coefficient for ¸"0 (!1)A`m (A!m, Am@D00)" d , m,m{ AK

(2.10)

Eq. (2.6) allows re-casting Eqs. (2.1) and (2.2), in the form of [4], ch. 2 W"jK [a*j+ t*j+]*0+ ,

(2.11)

A"jK [a*j+ A*j+]*0+ ,

(2.12)

where both factors of each product are contrastandard. Note the order of the factors: the amplitude is on the left side. Eqs. (2.11) and (2.12), exhibit the invariance of the state function W and of the operator A explicitly through their zero total angular momentum; these expressions contain no arbitrary phase owing to the phase convention (2.6). The usual spherical harmonics do not obey the relation (2.6), being defined to obey ½* (rL )"(!1)m½ (rL ) , (2.13) l,m l,~m with rL ,(h, u). For consistency, and to avoid unnecessary and confusing phases, we introduce suitably phased angular functions, i.e., spherical harmonics that obey the conjugation property (2.6), by defining ½*l+(rL )"(!i)l½ (rL ) . (2.14) m l,m For simplicity, and in conformity with general practice, we shall designate them as in Eq. (2.14) by upper case ½. The m-indices of ‘‘ket” wave functions in quantum mechanics experience by general definition contrastandard transformations, Eq. (2.4), whereby the invariant state function (2.11) transforms as a ket function. The corresponding bra function, required to form matrix elements, must be taken as the Hermitian conjugate of the ket function (2.11), generated by complex conjugation of the numerical coefficients a*l+, and by complex conjugation plus transposition for matrices. We shall indicate transposition by a tilde, thus complementing Eq. (2.6) by F*A+s"FI (A)"(!1)(A`a)FI *A+ . a a ~a Herewith we obtain from Eq. (2.11) together with Eq. (2.10) Ws"jK [aJ *j+ tI *j+]*0+ .

(2.15)

(2.16)

Thus, spinors, e.g., t*j+, are represented by column vectors, while tI *j+ by row vectors. m m Consider now the state normalization, implied by the conditions + Da D2"1, m m

(2.17)

St*j+Dt*j+T" t*j+st*j+"d . m m{ m m{ mm{

(2.18)

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These normalization conditions will verify that WsW"

G

HG

H

+ aJ (j)tI *j+ + a(j)t*j+ "1 , (2.19) m m m{ m{ m m{ by applying Eq. (2.18) and then Eq. (2.17) to each term of the double sum over m and m@. [The sum-integral symbol in Eqs. (2.18) and (2.19), as, e.g., in Ref. [20], signifies integration over all continuous variables, e.g., the angular and radial variables, and summation over all discrete variables, e.g., spinor indices.] In preparation for the following developments we rewrite Eq. (2.18), using Eqs. (2.6) and (2.10), as a (ket-bra) combination 1 1 + St*j+Dt*j+T" m m jK jK m or, in symbolic form,

+ (!)j`mtI *j+ t*j+" [tI *j+t*j+]*0+"jK ~m m m

[t*j+Dt*j+], [tI *j+t*j+]*0+"jK ,

(2.20)

(2.21)

calling it ‘‘the invariant norm”. Note the omission of the tilde on the left-hand form of Eq. (2.21) in view of the ‘‘bra” notation, implied by placing St*j+D and [t*j+D on the left-hand side of the vertical m line. Similarly, we rewrite Eq. (2.17) to achieve [a*j+Da*j+]"1/jK ,

(2.22)

calling it ‘‘the invariant amplitude norm.” The expressions (2.21), (2.22) will be generally called ‘‘overlaps.” The graph symbols of both amplitude and basis-function overlaps will be end-boxes, as shown in Fig. 2.3. The norm box, Eq. (2.21), will be hatched, while the amplitude overlap box, Eq. (2.22), will be cross-hatched. Evaluating the normalization integral in Eq. (2.19) represents the simplest case of the transformation from the form of Eq. (1.12) into the form of Eq. (1.13), as it represents the evaluation of a matrix element composed of two invariants, Ws,W, rather than three, SBD, D, DAT, as in Eq. (1.12). It provides the basic building block for all computations of correlation functions requiring the separation of the ‘‘geometric” from the ‘‘dynamic” parts. Here the former are represented by the amplitudes a*j+, the latter by the functions t*j+. At the same time this evaluation affords a prototype of the general basic recoupling operation called ‘‘exchange of partners” in Section 1: performing the double sum in the integrand of Eq. (2.21) was followed by integration over the product of basis state functions tI *j+t*j+, one of these functions originating from each sum in the braces. The m m{ integration result, on the right-hand side of Eq. (2.18), allow the remaining factors (aJ (j)a(j)) — one of m m{ them also drawn from each brace of Eq. (2.19) — to be summed in Eq. (2.17). This procedure amounts to a special case of the general alternative reduction procedure for multiple products of irreducible tensorial set elements, a*A+b*B+c*C+d*D+ , a b c d

(2.23)

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which may be carried out either by pairing the sets according to Ma*A+, b*B+N and Mc*C+, d*D+N, or instead to Ma*A+, c*C+N and Mb*B+, d*D+N. The resulting irreducible sets are represented by [[a*A+b*B+]*M+[c*C+d*D+]*N+]*K+"+ [a*A+b*B+]*M+[c*C+d*D+]*N+(Mk, NlDKi) i k c d l k,l "+ + + a*A+b*B+c*C+d*D+(Aa, BbDMk)(Bb, DdDNl)(Mk, NlDKi) a b c d k,l ab cd or alternatively by

(2.24)

[[a*A+c*C+]*R+[b*B+d*D+]*S+]*K+"+ [a*A+c*C+]*R+[b*B+d*D+]*S+(Ro, SpDKi) i o p o,p "+ + + a*A+c*C+b*B+d*D+(Aa, CcDRo)(Bb, DdDSp)(Ro, SpDKi) . (2.25) a c b d o,p ac bd The relationship between these irreducible sets is described in standard references, [3]a, ch. 12; [3]b, ch. 7; [5], ch. 6, 4, as a basic operation of Racah calculus, namely as the recoupling of irreducible quadruple products of irreducible tensorial sets. We shall indicate this relationship in compact form as

C

A B

M

D

[[a*A+b*B+]*M+[c*C+d*D+]*N+]*K+"+ C D N [[a*A+c*C+]*R+[b*B+d*D+]*S+]*K+ RS R S K

(2.26)

whose recoupling coefficient, the square array of 9 indices, is defined in terms of the usual 9jcoefficients [3]a, ch. 12; [3]b, ch. 7; [5], ch. 6, 4 by [4], ch. 1

C

A B

M

D

G

A B

M

H

C D N "M K NK RK SK C D N . R S

K

R S

K

(2.27)

The recoupling transformation, Eq. (2.26), is represented in our graphical method by the basic block, or graph symbol, shown in Fig. 2.1. We shall express all recoupling transformations in terms of Eq. (2.26). The values of the recoupling coefficients, the l.h.s. of Eq. (2.27), can be calculated by appropriate computer programs to be obtained, for example, from the CERN program library. We stress the one-to-one correspondence between the recoupling graph Fig. 2.1 and Eq. (2.26): on the left of the graph the tensors entering the recoupling box are represented by lines beginning at the top, in the same order as beginning at the left in Eq. (2.26); they are labelled by their names and their angular momenta. The coupling of a and b to angular momentum M, indicated by the brackets in Eq. (2.26), is indicated in Fig. 2.1 as the combined input tensor, also labelled by M. The one-to-one correspondence of graphs and equations continues for the tensors c and d, coupled to N, while the overall coupling to total angular momentum K of Eq. (2.26) is indicated by the left-most line labelled by K. The recoupling takes place in Fig. 2.1 within the recoupling box, from which lines emerge in the coupling order and coupling scheme of the right-hand side of Eq. (2.26).

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Fig. 2.2. The guide graph for evaluating the state norm sorts out amplitudes and functions toward their respective overlaps, as a preliminary to a recoupling transformation. Fig. 2.3. Recoupling graph corresponding to the guide graph of Fig. 2.2 the initially invariant states are recoupled into invariant norms, represented as cross-hatched and hatched boxes for the amplitudes and basis functions, respectively.

In this graph the ‘‘new” quantum numbers, R and S, are understood to be summed over, as implied by Eq. (2.26). The ‘‘exchange of partners” transformation involved in reorganizing the integrand of Eq. (2.19) may thus be viewed as proceeding through a special case of the Eqs. (2.24), (2.25) and (2.26): In the integrand of Eq. (2.19) the four irreducible sets MaJ *j+, tI *j+; a*j+, t*j+N, correspond to Ma*A+, b*B+, c*C+, d*D+N, m m m{ m{ a b c d respectively. The integrand of Eq. (2.19) is ‘‘special” by having all four indices MA, B, C, DN equal to j. The further specification of invariance of each sum in the brackets of Eq. (2.19) implies setting M"N"0 in Eq. (2.24) and therefore necessarily K"0 in Eqs. (2.24), (2.25) and (2.26) Return now to compute the normalization, now written as N" IK 2[aJ *I+tI *I+]*0+[a*I+t*I+]*0+ .

(2.28)

The guide graph for the recoupling relevant to Eq. (2.28), shown in Fig. 2.2, indicates how to combine and overlap both the amplitudes and the functions. The corresponding recoupling graph in Fig. 2.3 serves to evaluate the matrix element (2.28) involving essentially three ingredients, viz. its input, recoupling, and output. The input consists of the two original invariants, made up of two tensors each; the output here is particularly simple, consisting of invariant overlaps. This recoupling illustrates the elementary 9j-type transformation (2.26): Each of the two initial invariants, on the left-hand side of Eq. (2.28) consists of a sum over products of amplitudes and wave functions, as in Eq. (2.11), coupled to zero angular momentum and multiplied by the normalization constant IK . The four lines emerging from these invariants with labels I enter the recoupling box of Fig. 2.3, converging at the exit into a pair of ‘‘overlap boxes” that represent the invariant expressions (2.21) and (2.22), respectively, corresponding to the ‘‘invariant (wave function) norm” and the ‘‘invariant amplitude norm”. The value of the recoupling transformation for (2.28) amounts thus to these three

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factors augmented by the normalization factors, all visible on Fig. 2.3, yielding

C D I

I

0

N"IK 2 I

I

0

0 0 0

1 IK "1 , IK

(2.29)

regardless of whether t represents a single-particle or a many-body wave function. The 9j-type recoupling coefficient’s value will appear below in Eq. (2.34). Note that the tilde lines, representing bra amplitudes and functions, enter the matrix element boxes in the top position. Major modifications of this procedure emerge upon its application to the generic recoupling relation (2.26), illustrated by Fig. 2.1. Whereas the lines exit from the recoupling box in Fig. 2.1 with their (implied) previous labels I, converging immediately into overlap boxes that pair them to zero angular momentum, a generic case allows in Fig. 2.1 the emerging pairs (ac) and (bd) to couple into unspecified angular momenta (R, S) with any value allowed by the triangularity conditions, DA!CD4R4A#C, DB!DD4S4B#D. The values of R and S, and their respective significance, are generally implied — and restricted — by successive elements of a complete graph. In our specific example of the normalization integral, the convergence of each line pair into an overlap box restricts both R and S to zero, as required by [tI *I+t*I+]*R+"IK d ; R,0

(2.30)

the vanishing of the overall angular momentum also enforces S"0 for the amplitudes. Here we note a main feature of recoupling graphs: The labels of lines emerging from recoupling boxes are set by following graph elements; specifically they should be developed from the graph’s right end. We shall use this technique systematically, by assigning values to the ‘‘new” quantum numbers only after completing a graph; then, beginning from the end, one can immediately take into account the selection rules imposed by the terminating boxes, whatever they may be. The geometrical structure of the normalization Eq. (2.28) has wide implications: For example, evaluating an energy matrix element, SWDHDWT" IK 2[aJ *I+tI *I+]*0+H[a*I+t*I+]*0+ ,

(2.31)

requires no geometrical development beyond that of Eq. (2.28) because the operator H is itself invariant. Quite generally all examples of the following sections will have the structure of Eq. (2.31) requiring the evaluation of invariant operators between initial and final states. The properties of 9j coefficients are fully described in the literature [3]a, ch. 11, 12; [5], ch. 6.2, 6.4; we only quote:

G H A B

I

G

H

1 A B I , C D I "(!)B`C`I`L IK ¸K D C ¸ ¸ ¸ 0

G

A B B

H

C

1 "(!)A`B`C . AK BK A 0

(2.32)

(2.33)

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In the present Eq. (2.29), five of the entries in the recoupling coefficient vanish, simplifying its algebraic expression, emerging from the slightly more general relation

C DC D A A 0

A B C

CK . 0 " A B C " AK BK C C 0 0 0 0 B

B

(2.34)

Recoupling three rather than four tensors requires only the 6j-coefficient of Eq. (2.32). Simplicity and ease in drawing the recoupling graphs, favors expressing all recouplings as special cases of four-tensor recouplings, regardless of vanishing index. As in the example

C

0

B

B

D

[[a*0+b*B+]*B+[c*C+d*D+]*N+]*K+"+ C D N [[a*0+c*C+]*C+[b*B+d*D+]*S+]*K+ S C S K

(2.35)

tensors with zero index can be factored out. Four cases are possible, corresponding to alternative locations of the zero-index element:

A"0:

C

0

C

A 0

B

B

D

(2.36a)

D

(2.36b)

[b*B+[c*C+d*D+]*N+]*K+"+ C D N [c*C+[b*B+d*D+]*S+]*K+ ; S C S K B"0:

[a*A+[c*C+d*D+]*N+]*K+"+ C D R R D C"0:

C

A

[[a*A+b*B+]*M+d*D+]*K+"+ 0 S A D"0:

C

A

[[a*A+b*B+]*M+c*C+]*K+"+ C R R

B

A

N [[a*A+c*C+]*R+d*D+]*K+ ; K

D

(2.36c)

D

(2.36d)

M

D D [a*A+[b*B+d*D+]*S+]*K+ ; S

K

B M 0

C [[a*A+c*C+]*R+b*B+]*K+ .

B K

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Fig. 2.4. Graph symbols for 6j-type recoupling transformations, with complementary dotted lines (whose implied ‘‘zero” label forms the standard 9j-type graphs), in correspondence to Eqs. (2.35a)—(2.35d).

In drawing the recoupling boxes for the Eq. (2.36) we supply the missing scalar by a dashed line showing where to set a zero in the algebraic expression, the corresponding graphs appearing in Fig. 2.4. (Expressing the 9j-type recoupling coefficients in terms of 6j-coefficients would introduce phases and square roots obscuring the transparency of the expressions (2.36a)—(2.36d).) A complete table of recoupling coefficients with one or more zeroes, implied by Eqs. (2.26), (2.32) and (2.33), appears in [4], ch. 10.

3. Absorption of an unpolarized long wave photon by an unpolarized target We treat here the simple process of photon absorption in a dipole transition by a hydrogen-like atom within its discrete spectrum, ignoring the electron spin, first in its m-scheme formulation, and then in the Racah technique, that is in terms of invariants. (A corresponding case in nuclear physics would be the photon absorption by the nucleus 5Li.) This example, requiring the reshuffling of the geometric and the dynamic parts of three invariants (two for the states, one representing the operator) provides the full prototype, the ‘‘full basic building block”, for all computations of reaction probabilities: All relevant graphs will exhibit the same basic structure, with varying complexity of detail.

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3.1. Evaluation in the m-summation scheme Consider an initial state DnlmT, a final state Dn@l@m@T, and light linearly polarized along the z-direction. We should evaluate the squared matrix element DM(m, m@)D2"SnlmDDDn@l@m@TSn@l@m@DDDnlmT ,

(3.1)

setting Sr0uDnlmT"f (r)½ (0, u),/ , (3.2) nl lm nl,m SnlmDr0uT"( f (r)½ (0, u))*"(!)mf (r)½ (0, u),(/ )*, (3.3) nl lm nl l~m nl,m Sr0uDn@l@m@T"f (n@, r)½ (0, u),t , (3.4) l{ l{m{ n{l{,m{ (0, u),(t )* . (3.5) Sn@l@m@Dr0uT"(f (r)½ (0, u))*"(!)m{f (r)½ n{l{ l{m{ n{l{ l{~m{ n{l{,m{ In a continuum final state, treated in the next section, the radial function f (r) is replaced by f (kr); n{l{ l{ thus the label n@ is replaced by the momentum k where k would be placed next to r in the parentheses. We indicate the dipole operator (for photons polarized along the z-axis) by

S

D"

3 z"r½ (0, u) . 1,0 4p

(3.6)

Common practice applies the Wigner—Eckart theorem [3]a, ch. 14; [3]b, ch. 6; [5], ch. 5.4, setting

A A

Sn@l@m@DDDn l m T"(!)(l{~m{)

l@

1

l

B B

!m@ 0 m l@

1

"(!)(l{~m{)

l

!m@ 0 m

Sn@l@Er½ (0/)En lT 1 S½ E½ E½ TSf (n@, r)DrD f (n, r)T l{ 1 l l{ l

(3.7)

with the reduced matrix element

A

B

l@K 1ªlK l@ 1 l . S½ E½ E½ T" l{ 1 l J4p 0 0 0

(3.8)

Note that l@"l$1, m@"m, and that the transitions to l@"l#1 and l@"l!1 combine incoherently when we are interested only in total probability, thus expecting no interference terms and hence no relevant phases. Dealing here with unpolarized photons and atoms, we average Eq. (3.1) over photon polarization and atomic orientation, i.e., over the values of m, and sum over the final states, i.e., over the values of m@ and l@. To this end we apply the formula [5], ch. 3.7

A

BA

l@ 1 l lK ~2+ !m 0 m m

l@

1 l

B

!m 0 m

"lK ~21ª~2 .

(3.9)

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Averaging over polarizations becomes unnecessary after summing over atomic orientations. Assembling the partial results yields 1 Sf (n@, r)DrDf (n, r)T2S½ E½ E½ T2 . ¼" + DM(m, m@)D2" l l{ 1 l lK 21ª2 l{ m, m{

(3.10)

3.2. Evaluation by the Racah calculus This calculation will now be repeated with the Racah calculus, that is, with the invariant states and operators introduced in Section 2, thus replacing Eqs. (3.2), (3.3), (3.4), (3.5) and (3.6) with DUT"lK [a*l+/*l+]*0+ ,

(3.11)

DWT"lK @[b*l{+t*l{+]*0+ ,

(3.12)

D"jK [d*j+D*j+]*0+

(3.13)

and their conjugates according to Eq. (2.16). For dipole transitions j"1. Adjusting the phases of the functions /*l+, etc. amounts to replacing ½ by ½*l+, Eq. (2.14), in Eqs. (3.2), (3.3), (3.4), (3.5) and l,m m (3.6). We thus have to evaluate the squared matrix element DMD2"SUDDDWTSWDDDUT ,SlK [a*l+/*l+]*0+D1ª[d*1+D*1+]*0+DlK @ [b*l{+t*l{+]*0+TSlK @ [b*l{+t*l{+]*0+D1ª[d*1+D*1+]*0+DlK [a*l+/*l+]*0+T , (3.14) with the notations of Section 2. The guide graph for evaluating (3.14), Fig. 3.1, differs from the previous Fig. 2.1, by inserting the dipole operator D between each pair of states, (W,U). Dealing here with fully unpolarized states, the amplitudes carry no information concerning W,U and D, being accordingly eliminated by overlapping as in Eq. (2.22). Note that the basis functions are integrated within each matrix element, whereas each amplitude pairs with its conjugate from the other matrix element. We now digress to discuss some technical points, meeting a new concept, viz. an integral over the invariant product of three tensors, specifically tensorial sets of base functions and operators. In fact, evaluation of an integral containing any triple product of tensors selects out the invariant (i.e., zero angular momentum) part of the product. A product of three tensors coupled to zero angular momentum, called an ‘‘invariant triple product” [3]a, ch. 10; [3]b, ch. 5.4, is independent of coupling scheme, as indicated by [A*I+B*J+C*K+]*0+"[A*I+[B*J+C*K+]*I+]*0+"[[A*I+B*J+]*K+C*K+]*0+ .

(3.15)

This property allows to ‘‘change the coupling order” in invariant triple products without using (explicitly) recoupling coefficients. The special case of I"J"K"1 reduces to the usual product of three vectors: J6[A*1+B*1+C*1+]*0+"A ) B]C"A]B ) C

(3.16)

without any sign or complex phase. The invariant triple product (3.15) introduces the class of the invariant matrix elements [4], ch. 4 [A*I+DB*J+DC*K+]" [AI *I+B*J+C*K+]*0+ .

(3.17)

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Fig. 3.1. Guide graph for the photon absorption process. Here, as in Fig. 2.2, the amplitudes are sorted out above and the functions below the operator and basis functions. Fig. 3.2. Block diagram for the recoupling transformation indicated by the guide graph of Fig. 3.1 with amplitudes drawn as dash-dot lines. The invariant matrix elements are represented as hatched boxes with three input lines, two for the basis functions and one for the operator.

The tilde on A*I+ is omitted in the invariant matrix element, i.e., on the left-hand side of Eq. (3.17), as implied by the notation of Eqs. (2.21) and (2.22), together with Eq. (2.15). Thus, the tilde tensor, here A*I+, must, of course, appear on the left. The invariant matrix element (see Fig. 3.3 or 3.4) is represented by a hatched box with three entering lines, corresponding to the tensors, the tilde tensors entering the graph symbol in its top position. [The invariant matrix element (3.17) reduces to an overlap integral when any of its MI, J, KN indices vanishes.] The sign of the Wigner—Eckart reduced matrix element (3.8), normalized according to Wigner [1], ch. 21, proves inconsistent with the Condon—Shortley phase convention, as indicated by its relation to Eq. (3.17), St*j+ED*L+E/*k+T"(!)(j`L~k)[t*j+DD*L+D/*k+] .

(3.18)

This phase, (!)(j`L~k), must be carried along in any calculation involving interference effects when using the reduced rather than the invariant matrix element. Returning to our present problem we should now evaluate the matrix element Eq. (3.14) in terms of symbols [tI *j+D*L+/*k+]*0+,[t*j+DD*L+D/*k+] .

(3.19)

The required factorization into contributions from specified components can be carried out as in Section 2, but the present situation is somewhat more involved. The guide graph, Fig. 3.1, shows again an input of original invariants and an output that consists here of three amplitude overlaps and of two invariant transition matrix elements. The relevant recoupling must perform for the two matrix elements of Eq. (3.14) three separate tasks: separating the amplitudes from the ‘‘dynamics,” i.e., from basis states and operators, and recoupling the latter for entering the invariant matrix

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Fig. 3.3. Recoupling graph for block A of Fig. 3.2.

Fig. 3.4. Recoupling graph for block B of Fig. 3.2.

elements (tasks A and B); and recoupling the amplitudes to be overlapped (task C). The block diagram of the recoupling graph required for implementing these three tasks (A,B,C) is shown in Fig. 3.2. The expression for the overall transformation will be found by multiplying the values of the individual blocks of this diagram. The blocks A or B represent in fact the ‘‘full basic building blocks” referred to in the introduction to this section. Figs. 3.3, 3.4 and 3.5, show the details of the individual recoupling blocks, respectively. Consider constructing the graph for task A, Fig. 3.3. We begin at the left of the graph writing down from top to bottom the invariant products in the same order as in Eqs. (3.11), (3.12) and (3.13). Each factor, such as lK , is displayed explicitly. The recoupling operations needed for separating the amplitudes from the dynamics, and for recoupling the latter into their standard invariant form, are carried out by means of two 9j recoupling boxes. We postpone assigning values to the ‘‘new” quantum numbers appearing at the ouput of the 9j-boxes until the end, i.e., until all blocks have been designed. The recoupling graph for block B, Fig. 3.4, is constructed analogously to that of block A. Block C ( Fig. 3.5) recouples the amplitudes, which enter the block as two triple products, [aJ (db)] and [(bI d)a], into three invariant overlaps, [aDa], [bDb], [dDd], taking care that tilde lines enter each box at the top. Fig. 3.5 resolves this operation into two recoupling boxes complemented by several

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Fig. 3.5. Recoupling graph for block C of Fig. 3.2.

crossing boxes. Each crossing box represents a sign factor. The first two of these reorder the lines into positions required for the recoupling; the third one brings a tilde line to the top, an operation essential only when dealing with half-integer spins. We supply labels to each graph by entering the values of ‘‘new” quantum numbers at the output of each recoupling box, beginning with the last one, i.e., on the right. Consider thus the output of box C: Each pair of tensors must couple to zero upon entering an overlap element, thus requiring the vector sum of all ‘‘new” quantum numbers to vanish. This leaves the coupling of the input triple products indetermined. Continuing with box B, the top two lines in Fig. 3.4, tI and D, that shall enter the lower invariant matrix element must couple to match the angular momentum l carried by the bottom line, /, whereby all three lines couple to zero. This requirement implies a prior coupling to l of the two bottom lines exiting from the first recoupling box as well as of the top two (amplitude) lines, as required to match the overall coupling to zero of the box input. Furthermore, the total angular momentum input to block B also vanishes, hence that must be the case also for the output. Accordingly all amplitude tensors at the block’s output must emerge coupled to zero to match the zero angular momentum of the invariant matrix element. Similar considerations hold for all the other invariant matrix elements and recoupling boxes. These conditions determine all ‘‘new” labels fully. In particular the three amplitudes at the output of blocks A and B emerge as invariant triple products. This completes for the time being the discussion of technical points. All labels thus being set we read off the algebraic value of each block, namely the product of the algebraic expressions for all symbols appearing on a graph. Fig. 3.3 shows three normalization factors, viz., lK , 1ª, lK @, two 9j-type recoupling boxes, the invariant matrix element [/DDDt], and the output tensor [aJ d b]. Similarly, Fig. 3.4 shows, with necessary changes, the factors for the conjugate matrix element. The resulting algebraic expressions thus are

C DC D 1 1 0

l

l

0

l

0 [/DDDt][aJ *l+d*1+b*l{+]*0+

SUDDDWT"lK 1ª lK @ l@

l@

0

l

l

l

0

0 0 0

"[/*l+DD*1+Dt*l{+][aJ *l+d*1+b*l{+]*0+,

(3.20)

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C DC D l@

0

l

l

0

SWDDDUT"lK @ 1ª lK 1 1 0

l

l

0 [tDDD/][bI *l{+d*1+a*l+]*0+

l

l@

l

0

181

0 0 0

"[t*l{+DD*1+D/*l+][bI *l{+d*1+a*l+]*0+

(3.21)

The input of Fig. 3.5 consists of the amplitude output tensors of Figs. 3.3 and 3.4. In addition to two 9j-type recoupling boxes Fig. 3.5 contains three crossing boxes; the graph terminates in amplitude overlaps. Thus we read

C D

C D

l

l

0

l@

1 l

[aJ *l+d*1+b*l{+]*0+[bI *l{+d*1+a*l+]*0+"(!)2l l

l

0 (!)1`l{~l l@

1 l

0 0 0

0 0 0

1 ](!)2l{[a*l+Da*l+][b*l{+Db*l{+][d*1+Dd*1+]" . [lK lK @ 1ª]2

(3.22)

The complete squared matrix element (Eq. (3.14)) consists according to the block graph Fig. 3.2 of the product of the three recoupling blocks A,B,C, augmented by the input normalization factors and by the diverse invariant matrix elements, already present in Figs. 3.3, 3.4 and 3.5. It results from inserting the expression Eq. (3.22) into the product of Eqs. (3.20) and (3.21). Eq. (3.14) thus amounts to DMD2"[A][B][C] "lK @2 lK 2 1ª2

A B

"

A B

1 3 [a*l+Da*l+] [b*l{+Db*l{+] [d*1+Dd*1+] [/*l+DD*1+Dt*l{+] [t*l{+DD*1+D/*l+] 1ª lK lK @

1 2 [/*l+DD*1+Dt*l{+] [t*l{+DD*1+D/*l+] . 1ª lK lK @

(3.23)

Note that alternative procedures lead to the same recoupling transformation; generally some of them are more efficient than others that require unnecessary summations. Examples of such procedures appear in Appendix A. Inserting now the values of the invariant overlaps in Eq. (3.23) reduces it to 1 DM D2" [½*l{+D½*1+D½*l+]2Sf (n@, r)DrDf (n, r)T2 . 1 l{ l lK @21ª2lK 2

(3.24)

The invariant matrix element containing three spherical harmonics arises quite frequently in angular momentum calculations. Hence we provide it with a compact designation, defining [4], ch. 4

A

B

lK kK jK l k j [lDkD j],[½*l+D½*k+D½*j+]"(!)12(l`k`j) . J4p 0 0 0

(3.25)

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This non-negative quantity vanishes upon violation of the triangular condition on (l, k, j). Eq. (3.25) includes the invariant overlap of two spherical harmonics as in the special case J4p[lD0Dl]"lK .

(3.26)

The factor J4p in Eq. (3.26) compensates the normalization of the l"0 spherical harmonic. The normalization of amplitudes, Eq. (2.17) or Eq. (2.15), implies that state representatives are normalized to unity in Eqs. (3.11) and (2.29). Summation (rather than averaging) over the finalstate m-components, as in Eq. (3.9), demands here inserting the multiplicity index lK @2. The expressions Eqs. (3.10) and (3.24) coincide thereby since the reduced matrix element differs from the invariant matrix element only by a sign. In our description of the Racah calculus the bulk concerns technical details. The Racah calculus itself consists of posing the problem in Eq. (3.14), of drawing the graphs, Figs. 3.1, 3.2, 3.3, 3.4 and 3.5, of reading off the expressions, (3.20)—(3.22), and of simplifying them to arrive at Eq. (3.24). No recoupling was actually needed in the example of this section and just one m-summation occurs in the m-scheme treatment, Eq. (3.9). The Racah calculus affords thus here no real advantage over the m-scheme; even here it is, however, more compact and less error-prone. Having gone through the example of this section we now summarize the steps involved in drawing recoupling graphs. The main recoupling graph, Fig. 3.2, will start by listing on its left, from top to bottom, the several symbols appearing in Eq. (1.12), here Eq. (3.14), namely, SUD,Us, Ds,D, DWT,W, and SWD,Ws, D, DUT,U. Each of these symbols, accompanied by a label ‘‘0” to indicate ‘‘invariant” (i.e., scalar), forks into two lines representing its geometric (higher) and dynamic (lower) components, dashed line for the former and full line for the latter, each of them labelled by the angular momentum number (l or j) of its irreducible tensorial transformations. Next begins the task of grouping all the geometric elements above all the dynamic ones. This task has been here broken up into blocks A and B. Each of the line crossings thus required corresponds to a recoupling transformation, i.e., to a 9j-type coefficient represented by a rectangular box, as shown in Figs. 3.3 and 3.4. Newly coupled pairs of lines emerging on the right-hand side of each box (i.e., after the crossing) will be labelled with ‘‘new” quantum numbers, depending on circumstances, entering then a next recoupling box — if and as needed — and eventually a ‘‘terminal box” when the sorting-out is completed. This labelling should be postponed until the whole graph has been completed so that all selection rules can be immediately taken into account. Having started the process with a product of invariants, the complete sorting-out also results generally in invariants. Each dynamic line terminates in a ‘‘matrix-” or ‘‘overlap-” box. Three lines terminate at each matrix box, two of them representing base-sets of wavefunctions and the third one representing an operator set; their respective (l or j) labels must obey a ‘‘triangular condition”10 to ensure a non-zero value of the matrix element. The matrix elements emerge immediately as invariant matrix elements [4], ch. 4, since our formulation includes the angular momentum couplings

10 k#l5j5Dk!lD for j,k,l in any order.

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automatically. The value of this invariant matrix element determines the numerical value of each ‘‘matrix box”. Pairs of lines representing base sets of wavefunctions with equal j labels may converge alternatively into ‘‘overlap boxes”. Lines representing geometric coefficients converge instead into subsets representing their relevant products. In this section, the geometric coefficients, i.e., the unobserved amplitudes a, b, d and their complex conjugates terminate in overlap boxes, as shown in Fig. 3.5. In general, they form the building blocks of correlation functions, as explained below, beginning with the next section. Comparing the ‘‘full building block” of this section, say, block A, Fig. 3.3, where one has to reshuffle three invariants, with that of Section 2, Fig. 2.3, we see that block A consists in essence of two building blocks in Section 2. We will recognize these building blocks in the recoupling graphs of the later sections. This feature of the reappearance of building blocks facilitates greatly the design of recoupling graphs.

4. Photon–particle correlation We treat here the simplest transition matrix element leading to a photoelectron’s angular distribution correlated with the direction of the incident photon beam. We shall do so directly in the Racah technique, which will be seen to require an almost trivial extension of the previous section with regard to angular momentum problems. On the other hand this section will present many of the paper’s new elements. Prominent in this regard are definitions pertaining to continuum states and to diverse polarization vectors. We seek thus the angular distribution of photoelectrons from a hydrogen-like atom, including a generic orientation of the initial target state, still ignoring the electron spin. The results will actually take the form of a correlation function, with a detailed structural analysis. We will first extract the correlation function directly from the wave function, adequate for obtaining the directional polarization vectors since the wave function is actually a function of the relevant emission angles, but would prove rather clumsy for evaluating other observables, e.g., the spin, or, in nuclear physics, the isospin, for which formulating the observable as the expectation value of a pertinent operator is more appropriate. Therefore in preparation for the later sections we shall describe this method already as an alternative to obtain the angular distributions. 4.1. Extracting angular distributions from the wave function The matrix element to be evaluated has the same structure as in Eq. (3.1): DMD2"SUDDsDWTSWDDDUT ,

(4.1)

differing only by replacing the bound state wave function Eqs. (3.4) and (3.5) with the continuum state functions DWT"lK @[b*l{+(kK )t*l{+(r)]*0+,lK @[b*l{+(kK )½*l{+(rL )]*0+4pf (k, r) , l{

(4.2)

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Fig. 4.1. Guide graph as in Fig. 3.1 complemented by forming polarization vectors.

and SWD"lK @[bI *l{+(kK )tI *l{+(r)]*0+,lK @[bI *l{+(kK )½*l{+(rL )]*0+4pf (k, r) , (4.3) l{ with k indicating the photoelectron’s momentum. As to be expected from its tensorial properties, and confirmed below in Eqs. (4.29), (4.30) and (4.32), the parameter b*l{+(kK ) is proportional to ½*l{+(kK ). Furthermore, we now specify the direction and polarization of the incoming photon beam, which govern the direction of the outgoing electron. The present example also produces interferences, as the final state l@ may take alternative values l#1 and l!1 that interfere in the differential cross section. Fig. 4.1 provides the relevant guide graph, showing how to extract and use probability amplitudes of the initial state, of the outgoing particle and of the photon polarization in generating the ‘‘Experimental Layout” factor of Eq. (1.13). These parameters take the form P*L+ "[d*1+dI *1+]*L+ P,M M from Eq. (3.13) for the photon polarization, and for the orientation parameters

(4.4)

P*La+ "[a*l+aJ *l+]*La+ a,M M of the initial state, and

(4.5)

P*Lb+ "[b*l{+(kK )bI *lA+(kK )]*Lb+/QL A (4.6) b,M M l{l of the emitted electron, with a renormalization coefficient QL A. The parameters P will be called l{l ‘‘polarization vectors.” One of the amplitudes in the Eqs. (4.4), (4.5) and (4.6) originates in the ket, the other in the bra, of the two matrix elements of Eq. (4.1). Hence only one of them bears a tilde, the other one being shifted to first position through the cyclic-permutation symmetry; this ordering is chosen to avoid the appearance of a phase factor [14]. The coupling order is not important for Eq. (4.4) itself, whose amplitudes d*1+ are self-conjugate, but should be preserved for later consistency. Eqs. (4.2) and (4.3) pertain to pure states, since impure states are not represented by wave functions but only by polarization vectors extracted from density. Hence the left-hand side expressions of these equations apply throughout.

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Eq. (4.6) has introduced the normalization QL A which emerges from the spherical harmonics of l{l the emitted electron in both the bra and the ket of each final state wave function. In the expression of angular distributions it is preferable to display angular dependences in terms of a single spherical harmonic, thus involving the expression [½*l{+(kK )½*l{+(kK )]*L+"QL A½*L+(kK ) , M l{l M

(4.7)

which in fact defines the factor Q, whose explicit expression derives from a general procedure helpful to our graphical approach, viewing the product of the two spherical harmonics on the left of Eq. (4.7) as the product of a generic operator A and a base wave function /"½*l{+. One then views m its expansion into a set of wave functions t as the combination of A/ with a unit operator i + Dt TSt D, resulting, in Dirac notation, in i i i AD/T"+ Dt TSt DAD/T . i i i

(4.8)

The matrix element’s construction will usually restrict the summation over i substantially. The ‘‘closure” operation represented by the symbol + Dt TSt D is represented in Fig. 4.2 by a ‘‘timei i i reversed” overlap box branching into a pair of lines, the top one labelled by a tilde. [Proper care is required to preserve correct phases in the presence of half-integer angular momenta.] Our present application to Eq. (4.7) simply identifies the set MDt TN as the harmonic’s set M½*L+N, i M thus recasting Eq. (4.8), with the recoupling represented by Fig. 4.2, into

C D ¸ ¸ 0

[A*K+/*I+]*L+"¸K (!)2L 0 ¸ ¸ [t*L+DA*K+D/*I+]t*L+"(1/¸K )[t*L+DA*K+D/*I+]t*L+ . M M M ¸ 0 ¸

(4.9)

[Identifying Dt T with ½*L+ has simplified here the procedure, but analogous opportunities will often i M emerge elsewhere.] The mock-zero line in Fig. 4.2 implements the reduction of a 9j- to a 6jrecoupling coefficient as in Fig. 2.4. In the present case the transformation (4.7) provides the factor [½*l{+(kK )½*lA+(kK )]*L+"(1/¸K )[¸Dl@DlA]½*L+(kK )"QL A½*L+(kK ) , M M l{l M

(4.10)

the last part repeating the definition, Eq. (4.7), with the symbol [¸Dl@DlA] defined in Eq. (3.25). The block diagram of the recoupling graph, corresponding to the guide graph Fig. 4.1, is shown in Fig. 4.3. Comparing Figs. 3.1 and 4.1 shows blocks A and B to be unchanged except for differences in the final state angular momentum wave functions, indicated here by their separate labels l@ and lA, respectively, serving to represent the directional distribution. Only block C requires modification into C@ to couple its amplitudes into polarization vectors, as shown in Fig. 4.4. We shall show the full recoupling graph in Appendix A after introducing a ‘‘compact graph convention”. The normalization of continuum states’ amplitudes Eqs. (4.31), (4.32) and (4.33) below, suggests graphing the parameters as (lK b*l+),½*l+(kK ) rather then as simple b*l+, with a compensating additional

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Fig. 4.2. Recasting the combination (operator A and basis function /) as a superposition of basis functions t with i matrix-element coefficients, Eq. (4.8). The unit operator + Dt TSt D is represented as a ‘‘time-reversed” invariant norm, i i i with its tilde line emerging in the top position; summation over the complete set of quantum numbers of the basis Dt T is i implied.

Fig. 4.3. Block diagram for the recoupling transformation of the guide graph of Fig. 4.1, with emerging polarization vectors P , P , P . A b P

Fig. 4.4. Graph for the recoupling of amplitudes into polarization vectors according to the guide graph 4.1.

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187

factor 1/lK , shown explicitly. Reading off Fig. 4.4 yields for the recoupling box C@

C

D

C

l

l

0

l@

1

l

[C@]"(!)2l l

l

0 (!)2l~La (!)1`l{~l lA

1

l

D

QLbAlK @lK A[P*La+P*Lb+P*LP+]*0+ . (4.11) a b P l{l

¸ ¸ 0 ¸ ¸ ¸ a a b P a Since the initial states are invariants the final product, representing the coupling of the three directional vectors, to be called ‘‘correlation invariant”, arrives as an invariant triple product. Multiplying all factors indicated in Fig. 4.3 leads to the final result

C

l@ 1 11 1 1 ¸Y DMD2"lK 2 1ª2 lK @ lK A [/DDDt][tDDD/](!)La a lA 1 lK @ lK A lK 1ª lK @ lK 1ª lK A ¸Y lK 2 b ¸ ¸ b P

C

l l ¸

a

D

QLbAlK @lK A[P*La+P*Lb+P*LP+]*0+ a b P l{l

D

l@ 1 l ¸Y l [¸ Dl@DlA][lD1Dl@][lAD1Dl] "(!)La a lA 1 b ¸Y lK 2 b ¸ ¸ ¸ b P a (4.12) ](4p)2 Sf (n, r)DrD f (p, r)TSf A(p, r)DrD f (n, r)T[P*La+P*Lb+P*LP+]*0+ . l a b P l l{ l Absence of information on the initial state polarization would cause ¸ to vanish, thus enforcing a ¸ "¸ . b P In Eq. (4.12) we have explicitly extracted the factor Q, whereby the polarization vector P*Lb+ reduces to a spherical harmonic, a convention applied henceforth. The presence of a Q factor b in the graph and in the algebraic expression will thus signify that the corresponding polarization vector reduces to a single spherical harmonic. The transition probability results (to within kinematic factors) by summing over unobserved quantum numbers: N a b P[P*La+P*Lb+P*LP+]*0+" + DM2(l@, lA, ¸ ¸ ¸ )D2 (4.13) LLL a b P a b P A l{,l including the explicit values of radial integrals in the factor N. The total probability, in the form of Eq. (1.13), yields the formal final result, the correlation function, ¼" + N a b P[P*La+P*Lb+P*LP+]*0+ . (4.14) LLL a b P LaLbLP In the terminology of Eq. (1.13), ¼ represents the ‘‘Measurement”, the coefficient N a b P the LLL ‘‘Dynamics” and the invariant triple product of the polarization vectors, which we call ‘‘correlation invariant”, the ‘‘Experimental Layout”. Factoring out the ‘‘dynamical” coefficients N — whose details may be less accurately known from theory — from the correlation function ¼ in Eq. (4.14), affords the opportunity of inverting the equation to determine N from experimental data and details of the correlation function. This aspect will be discussed further in Section 9.2.

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Eq. (4.14) represents the correlation function of the three polarization vectors associated with the initial and final states of the emitted particle and of the photon, respectively. We now spell out how to calculate these vectors [4], ch. 5, beginning with the photon polarization, whose description requires representing Cartesian vectors in polar coordinates, i.e., by ‘‘spherical vectors” defined with appropriate phase factors. A generic Cartesian vector V"» e #» e #» e , x x y y z z

(4.15)

appears in our polar coordinate notation as V"1ª[»*1+e*1+]*0+ ,

(4.16)

with ‘‘spherical unit vector” components i(e #ie ) y , e*1+" x 1 J2

(4.17)

e*1+"!ie , 0 z

(4.18)

!i(e !ie ) x y , e*1+ " ~1 J2

(4.19)

and ‘‘spherical vector” components i(» #i» ) y , »*1+" x 1 J2

(4.20)

»*1+"!i» , 0 z

(4.21)

!i(» !i» ) x y . »*1+ " ~1 J2

(4.22)

These definitions represent the scalar product of two vectors as A ) B"1ª[A*1+B*1+]*0+ ,

(4.23)

and their vector product C"A]B ,

(4.24)

C*1+"J2[A*1+B*1+]*1+ , M M

(4.25)

as

without any phase factor. This notation also interprets the factor J6 of Eq. (3.16) as the product 1ªJ2 of the normalization factors in Eqs. (4.23) and (4.25); inserting Eq. (4.25) in Eq. (4.16) yields

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now the invariant triple product A]B"1ªJ2[A*1+B*1+e*1+]*0+ .

(4.26)

Photon polarization along the x-axis is then represented by the dipole interaction components i , d*1+" 1 J2

d*1+"0 , 0

!i d*1+ " . ~1 J2

(4.27)

Eq. (2.6) [or Eq. (2.15)] verifies that dI *1+"d*1+, i.e., that d*1+ is self-conjugate. m m Evaluating now the (Liouville space) polarization vector P*La+ , Eq. (4.4), explicitly with the a,M appropriate vector coupling coefficients, yields its scalar and quadrupole components 1 , P*0+" P J3

1 P*2+ " , P,0 J6

1 P*2+ "! , P,2 2

1 P*2+ "! . P,~2 2

(4.28)

All other components of the photon polarization vector vanish. Polarization along the y-axis yields instead positive values of P*2+ and P*2+ . P,2 P,~2 The angular distribution of photoelectrons involves the large-distance behavior of their wave functions. The state of neutral particles is represented here by the multipole expansion of a plane wave: e*k > r"+ ilP (kK ) rL )4pj (kr)"+ illK [½*l+(kK )½*l+(rL )]*0+4pj (kr) . (4.29) l l l l l The expansion of a photoelectron’s Coulomb wave function differs from Eq. (4.29) by replacing the spherical Bessel functions with functions including a logarithmic Coulomb phase. We thus represent a ‘‘plane Coulomb wave” by PC(k ) r)"+ illK [½*l+(kK )½*l+(rL )]*0+4pf (kr) , (4.30) l l where f (kr) indicates a radial Coulomb function, normalized as the spherical Bessel j (kr), namely l l [4], ch. 10:

P

=

p d(k!k@) f (kr) f (k@r)r2dr" . l l 2 kk@

(4.31)

o Hereafter f (kr) will stand for either the spherical Bessel function (for a neutral target residue) or the l Coulomb function (for an ionic residue). The ‘‘plane wave phase” il in the expansions (4.29) and (4.30) plays an important role in the interference of different multipolarities. The directional polarization vector for a beam, of emitted particles in the present case, involves amplitudes analogous to those of Eqs. (2.17) and (2.22). In accordance with Fig. 4.4 we thus define (as in [4], ch. 9) 1 b*l+(kK )" ½*l+(kK ) m lK m

(4.32)

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which indeed fulfills

P

(4.33) + Db*l+(kK )D2d2kK "1 , m m thus providing the polarization vector for the electron’s direction (Fig. 4.2) through the parameters included in Eq. (4.12) [l@DlAD¸] P*L+ (kK )"[b*l{+(kK )bI *lA+(kK )]*L+" ½*L+(kK ) , b,M M M lK @lK A¸K

(4.34)

and concluding our general development. It may be instructive to work out this example in full detail for unpolarized photons incident along the z-direction. The angular distribution arises then as the incoherent average of the x- and y-polarizations, cancelling the contributions of P*2+ components with MO0. (For linear polarizaP,M tion P*2+ would indeed retain a /-dependence.) Spherical symmetry of the initial state implies P,M l"0; hence ¸ "0, and ¸ "¸ , to be denoted by ¸. This choice requires setting l@"lA"1 in the a b P final state reflecting the dipole character of the transition. Defining now the ¸-independent factor, K"[0D1D1]2(4p)2S f (n, r)DrD f (p, r)T2 , l l{ Eq. (4.12) reduces to

(4.35)

C D

1 1 ¸ (!)L ¼ "K 1 1 ¸ [¸D1D1][P*L+½*L+(kK )]*0+ , L P ¸K 0 0 0

(4.36)

with the definition (3.25)

[¸Dl@DlA]"(!)12(L`l{`lA)

A

B

¸K lK @lK A ¸ l@ lA J4p 0 0 0

and with P*L+ as in Eq. (4.34). Further, the particular values: b,M 1 1 0 1 " , 0 0 0 J3

A A

B B

(4.37)

(4.38)

J2 " , 0 0 0 J15

(4.39)

1 S1 1 1!1D0 0T" , J3

(4.40)

1 S1 1 1!1D2 0T" , J6

(4.41)

1 S2 0 2 0D00T" , J5

(4.42)

1 1 2

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191

1 ½*0+(h, u)" 2Jp

(4.43)

J5 ½*2+(h, u)"! [2!3sin2h] 0 4Jp

(4.44)

lead to

C

D

C

D

¼ ¼ #¼ 1 1 1 2ª 1 1 1 1 J2 1 2" " 0 ½*0+ # ½*2+ 0 K K 1ª1ª J4p J3 J3 2ª 1ª1ª J4pJ15 J6 J5 1 1 " sin2h . 12p J2

(4.45)

4.2. Angular distributions as expectation values of observable operators As mentioned in the introduction to this section, the directional polarization vector, P , b Eq. (4.34) was constructed directly from the final state’s kets and bras, DWT and SWD, represented as explicit functions of the electron’s direction kK , in Eqs. (4.2) and (4.3). This method, though adequate when dealing only with parameters explicitly contained in the wave function, does not lend itself easily to more general parameters preferably represented by an operator. We indicate now the appropriate formulation. The present observable is the direction of electron emission, specified by a detector operator D(kK ) (not to be confused with the Wigner rotation matrices DL , Eqs. (2.3) and (2.4)), whose expectaMM{ tion value SD(kK )T"SWDD(kK )DWT ,

(4.46)

with a bra SWD and ket DWT, is represented as a function of directions kK @ and kK A, respectively. The matrix element (4.46) would then be inserted between the two matrix elements of (4.1), a process viewed as averaging D(kK ) over final states excited by photoabsorption, DWT"DDDUT,DWTSWDDDUT ,

(4.47)

SWD"SUDDD,SUDDDWTSWD .

(4.48)

Here the operator D(kK ) takes the explicit form D(kK )"d(kK @!kK )d(kK A!kK ) .

(4.49)

Returning to Eq. (4.6) once again and inserting the expression (4.49) into Eq. (4.46) yields the explicit form of the ejected electron’s polarization vector, namely, P*Lb+ "[b*l{+(kK @)bI *lA+(kK A)]*Lb+DkK @"kK A"kK "[b*l{+(kK )bI *lA+(kK )]*Lb+ , (4.50) b,M M M indeed reproducing the results achieved with Eqs. (4.13), (4.14) and (4.34). As mentioned above, only the left-hand side of Eq. (4.50) would represent impure states; but only polarization vectors occur in the expressions for angular correlation functions, as apparent, for e.g., in Eq. (4.12).

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5. Inclusion of spin measurement This section extends last section’s treatment by taking into account the particle spin. Specifically, as described at the end of the last section, we introduce a measurement of the emitted particle’s spin as the expectation value of an operator, here the spin operator S. We thus augment the treatment of Section 4 by including the electron spin and constructing the multiple correlation of photon polarization, electron direction, and electron spin. (See Ref. [21] for an extensive treatment of experimental and theoretical aspects of electron spin—orbital correlations.) We note that, in contrast to the electric dipole operator, higher multipole operators, which may be present in the interaction Hamiltonian, act in general not only on the orbital but also on the spin elements of wave functions. As mentioned at the end of Section 4, we have to evaluate the expectation value of a detector operator M: SMT"SWDMDWT ,

(5.1)

where W represents the final state of the system. This state is reached from the ground state U, which now includes the electron spin (s"1), 2 DUT"jK [A*j+m*s+/*l+]*0+ , (5.2) by the action of the transition operator as in Eq. (4.47) DWT"DDDUT .

(5.3)

Note the order of spin and orbit basis functions in Eq. (5.2). The transition operator is now represented by D"jK [d*j+D*j+]*0+

(5.4)

with generic multipolarity D*j+,(E*j+, or M*j+) for electric or magnetic transitions. The apparatus includes now a spin analyzer, represented by S, in the direction kK previously embodied in the directional operator D of Eq. (4.49). Our extended detector operator M will thus take the form M"SD .

(5.5)

Accordingly, Eq. (5.1) reads SMT"SWDSDDWT .

(5.6)

The invariant spin operator S consists here of a unit vector element S*1+ representing the direction of spin selection and of the Pauli spin vector p*1+ S"1ª[S*1+p*1+]*0+ ;

(5.7)

its expectation value corresponds to spin polarization in the direction S*1+. Setting S*1+"k*1+ selects thus the helicity representation [22]. The two factors of the measuring operator, S and D, act on different elements being thus evaluated independently; their invariant form requires no operator recoupling in Eq. (5.6). The final state wave function may be represented symbolically as a product of its invariant spin and

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193

Fig. 5.1. Guide graph for photo-absorption, including spin bases m and a spin polarization vector S. Two pairs of arrowed brackets correspond to the ‘‘time-reversed” unit operator norm of the t bases in Fig. 4.2. i

Fig. 5.2. Step guide graphs for the right half of Fig. 5.1. Step (a): insertion of the unit operator; step (b): construction of invariant matrix element [tDDD/], of the [mDm] overlap with the residues DmT and [bI dA]. Fig. 5.3. Step guide graphs for the left half of Fig. 5.1: steps (c) and (d) correspond to (a) and (b) in Fig. 5.2.

orbital parts (thus postponing the appropriate recoupling) DWT"DW W T. (5.8) m O We can now evaluate the expectation value of the directional operator, D, immediately as in the previous section, by inserting the direction kK in the wave functions W , unchanged from the O previous section and given in Eqs. (4.2) and (4.3), being then left with calculating the matrix elements of P"SUDDsDW W TSW DSDW TSW W DDDUT . (5.9) m O m m m O In this equation the pairs DW TSW D represent unit operators in spinor space, in analogy of the m m discussion of Eq. (4.7). At this point the need for recoupling arises: the unit operators DW TSW D are m m invariants, with their factor SW D, representing the spin coupled with the orbital part SW D to yield m O the state’s angular momentum j. The guide graph for this evaluation, shown in Fig. 5.1, implies a rather complex procedure. Hence we break it up into step-graphs, i.e., guide graphs for each recoupling step; they are shown in Fig. 5.2 for the right matrix element of Fig. 5.1, and in Fig. 5.3 for the left element whose corresponding recoupling graphs appear in Figs. 5.4, 5.5, 5.6 and 5.7. As for the t in Fig. 5.4, one of i

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Fig. 5.4. Insertion of the spinors according to step (a) of Fig. 5.2; here and in the following s"1. The arrow on the right 2 indicates continuation of the recoupling in graph b, Fig. 5.5.

Fig. 5.5. Evaluation of the invariant matrix elements according to step (b) of Fig. 5.2, with jA"lA$1. Input a arrives 2 from Fig. 5.4, output goes to g, Fig. 5.9.

the spin basis functions emerging from the ‘‘time-reversed” overlap box combines with the corresponding orbital in a transition matrix element, while the other one forms the ket or the bra in the spin expectation values illustrated in Figs. 5.5 and 5.7. The result of the overall transformation is then given by the product of the subgraphs’ values, denoted below by gothic letters. We can now read off the algebraic expressions for the step sub-graphs, in Figs. 5.4 and 5.5, the sub-graphs (a) and (b) (s"1) yielding 2 s s 0

C D C DC D

[a]"lK AsL (!1)2s lA jA

[b]"1ªIK

lA 0 "!jK A jA 0

1

1

0

lA jA s

I

I

0

jA jA 0

jA jA 0

(5.10)

s

0

s

C D s

A

(!1)1`I~j s

lA

jA

l

I (!)l`1~lAsL [tADDD/] ,

0 1

(5.11)

1

respectively, with the output product [m*s+[bI *lA+[d*1+A*I+]*jA+]*s+]*0+,[m*s+KI @*s+]*0+ .

(5.12)

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195

Fig. 5.6. Insertion of a spinor pair according to step (c) of Fig. 5.3. Continuation in graph d, Fig. 5.7.

Fig. 5.7. Evaluation of the invariant matrix elements according to step (d) of Fig. 5.3; I has half-integer value. Continuation in graph G, Fig. 8.9.

Figs. 5.6 and 5.7 yield similarly for sub-graphs (c), (d)

C D C DC D l@

[c]"lK @sL (!)2s

I

l@ 0

0 s l@

j@ s

(!1)l{`s~j{"!jK @,

(5.13)

C D

0

j@ j@

0

s

l

I

[d]"IK 1ª 1 1 0

l@ j@

s (!1)1`j{~I s

l@

j@ (!1)l{`1~lsL [/DDDt@] .

j@

I

s

j@

and the output tensor

0

s

0 s

(5.14)

0 1 1

¹@"[[[AI *I+d*1+]*j{+b*l{+]*s+mI *s+]*0+,[K@*s+m*s+]*0+ .

(5.15)

Here the spin amplitudes KI @*s+, KA*s+, of Eqs. (5.12) and (5.15) emerge at the output of (b) and (d) as products of three amplitudes coupled to s"1, because the index I of A*I+ is half-integer I. 2 In graphs (b) and (d) the property Eq. (3.5) of invariant triple products is employed: In graph (d) after recoupling the tensors emerge in the coupling scheme [((mI /I )D)(mt)] but are immediately considered to be coupled as [(mI /I )(D(mt))].

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Fig. 5.8. Guide graph for evaluating the invariant spin matrix element and assembling polarization vectors.

Fig. 5.8 serves as a guide graph for the last step, which evaluates the spin operator and assembles the polarization vectors, to be recoupled as in Fig. 5.9, with the algebraic expression

C

DC D D C D 0

s

s

0

[G]"¸K (!)2s ¸ ¸ 0 S S S s s 0

s

s

0 [mDp*LS+Dm](!1)LS`s~s(!1)lA`jA~s

C

j@

] jA

l@

lA

K ¸

s

s

s s

0 0 0 I

1

j@

(!1)1`I~j I

1

jA (!1)2I~LAQLAb , l l{ K

A

(5.16)

¸ ¸ ¸ b S A P with ¸ "1 if the spin orientation is observed, and ¸ "0 otherwise; Q as defined in Eq. (4.10). S S The final result emerges as the product of the sub-graph contributions, in the simplified form: DMD2,[a][b][c][d][G]

C DC DC s

A

lA

jA

s

l

I

j@

l@

s

l

I

s

l@

j@

jA

lA

s

0 1

1

0 1 1

"(!)j{`j `I~s`LS`LA s

K ¸ b

¸ S

DC

I

1

j@

I

1

jA

¸ ¸ K A P

D

]QLAb [/DDDtA][t@DDD/][mDp*LS+Dm][[P*LA+P*LP+]*K+P*Lb+S*LS+]*0+ . (5.17) A P b l l{ In Eqs. (5.16) and (5.17) we encounter the invariant matrix element of the spin operator, [mDp*1+Dm]"iJ6 ,

(5.18)

according to [4], ch. 10. In the absence of spin measurement, the particle spin is taken into account by setting ¸ to 0 and replacing the spin operator p*1+ with the unit operator and its amplitude S S*1+ by S*0+, with S*0+"1, together with [mD1*0+Dm]"J2 .

(5.19)

All correlations and directional distributions are displayed in Eq. (5.17). The coupling scheme of the correlation invariant is given in Fig. 5.10.

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197

Fig. 5.9. Recoupling graph displaying the evaluation of matrix elements and of the directional distributions. Inputs from the outputs of graphs d,b. The null angular momenta on the left side imply the invariant coupling of observable outputs on the right, anticipated in Section 1. The symbol s stands for 1 (as previously), and ¸ for 1 or 0 depending on the 2 S observation of spin orientation or lack of it.

Fig. 5.10. Coupling scheme of the four-fold correlation function, Eq. (5.17) or (5.20).

We finally write the result in the form of Eq. (1.13): N A P b S [[P*LA+P*LP+]*K+P*Lb+S*LS+]*0+ . ¼" + (5.20) L ,L ,L ,L ,K A P b LA,LP,Lb,LS,K As always, a complete experiment determines the weights N of the correlation invariants in the reaction, the analog of Section 4’s result (4.16). The new vector S*LS+ emerging in Eq. (4.16) and the new quantum number K provide additional information on the photoabsorption process, when determining radial integrals by inverting Eq. (5.17), as anticipated in Section 4. Recalling that the invariant product in Eq. (5.20) depends on the several angles defining the mutual orientations of the polarization vectors in this product, the inversion of Eq. (5.20) amounts to a substantial problem in analytic geometry which will be addressed in Section 9. We display the full recoupling graph (the ‘‘overall blueprint”) in Appendix A, after having introduced the ‘‘compact graph convention.” For magnetic multipole interactions, the transition operator M*j+, Eq. (5.4), interacts not only with the electron orbital but also with its spin. The invariant spin overlaps in Figs. 5.5 and 5.7 must

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be replaced by spin matrix elements with the spin matrix originating in the interaction operator. We leave the problem as an exercise for the reader.

6. Addition of a spectator core Systems of interest often include an inert core that may carry angular momentum and other quantum numbers, in addition to other observed elements. In our progression towards more complicated problems this Section complements the previous examples by introducing such a ‘‘spectator” core, whose state remains unchanged in the transition, except for orientation changes. Spin consideration is omitted again for simplicity. The matrix element to be evaluated is again indicated by Eq. (4.1) to within extension of the state functions to include a core factor s*j+, DUT"IK [A*I+[s*j+/*l+]*I+]*0+ ,

(6.1)

DWT"jK [B*j+s*j+]*0+l@K [b*l{+f *l{+]*0+ ,

(6.2)

functions to be complemented by the operator D from Eq. (3.13), D"1ª[d*1+D*1+]*0+ ,

(6.3)

and by their conjugates according to Eq. (2.15). The spectator function s*j+ and its conjugate sJ *j+ are to be eliminated by directing them into overlap boxes, as indicated in the guide graph of Fig. 6.1, and in the recoupling graph shown by Fig. 6.2 in block form. Note in Figs. 6.3 and 6.4 how the invariant dipole operator D, having initially been placed in a ‘‘wrong” position, i.e., below the wave functions, rejoins in the recoupling A the relevant wave functions entering the invariant matrix boxes in the proper middle position, after they separate from the core in the recoupling box D. A corresponding feature occurs in the recouplings E and B. The overlap of the spectator functions in boxes D and E of the graph Fig. 6.2 is shown in detail in Figs. 6.3 and 6.4, with the simple result represented by

C DC D C DC D I I 0

j

l

I

[D]"IK JK j

j

0

l

l

0

IK 0 j [sDs]" , jK 0 l l

j

j

0

j

0 j

[E]"IK JK I I

0

j

l

0

0 l

l

l

j

IK I [sDs]"(!1)l`j~I . jK l

(6.4)

(6.5)

After performing these overlaps one thus reverts to the problem of Section 4, as seen by comparing the guide graphs 6.1 and 4.1, whose only difference lies in replacing the Section 4 amplitudes a and aJ with the coupled amplitudes BI A and AI B, respectively, requiring some additional recoupling and disentangling. This operation is carried out in box F, Fig. 6.5, with the

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199

Fig. 6.1. Guide graph showing the overlap of the spectator core in each matrix element and of its amplitude in the final state.

Fig. 6.2. Recoupling graph in block form. l@ may be different from lA.

recoiling residues remaining unobserved,

C

j

I

l

[F]"(!)j`I~l j

I

l

0 ¸

A

¸ A

D

[BDB] .

(6.6)

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Fig. 6.3. Recoupling graph for block D of Fig. 6.2. Fig. 6.4. Recoupling graph for block E of Fig. 6.2.

Fig. 6.5. Recoupling graph for block F of Fig. 6.2; overlap of the recoiling system’s orientations. Fig. 6.6. Inserts x — — x and y — — y to be added to Fig. 6.2 when de-coupling / from antisymmetric s/ is required.

Combining these results yields, [D] [E] DMD2" [F]DM D2 , 4 lK lK

(6.7)

denoting by DM D2 the matrix element DMD2 of Section 4, Eq. (4.12), and omitting further details. 4 For initial states with an open shell one must undo the antisymmetrization of the core particles with the particle undergoing the transition. The easiest way to accomplish this utilizing a ‘‘fractional parentage” expansion [3]a, ch. 13; [3]b, ch. 8.2; [4], ch. 6, U(n, a)*J+" + [s(n!1, b)*K+/*l+(n)]*J+(ln~1bK; lDNlnaJ) , M M b,K

(6.8)

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201

with the following notation: In the fully antisymmetrized state U(n, a)*J+ of the configuration (/*l+)n, n M represents the number of particles and a the set of quantum numbers required to specify the state fully in addition to the angular momentum quantum numbers; s(n!1, b)*K+ indicates an antisymmetric state of n!1 particles with additional quantum number b, coupled to the function of the nth particle, /*l+(n), to angular momentum J. The overall antisymmetry of the state U(n, a)*J+ results M from the summation over b, K with the weights provided by the ‘‘coefficients of fractional parentage”, (ln~1bK; lDNlnaJ), which cancels the contributions of the non-antisymmetric components in individual terms of the sum. A method for computing CFPs is given in [4], ch. 6. Fig. 6.6 represents this development including graph symbols for the fractional parentage coefficients drawn from [4], ch. 6. Fig. 6.6(a) and 6.6(b) should be inserted into Fig. 6.2 on the left of the dashed lines x — — x and y — — y, respectively, and in the dashed boxes labelled CF in Figs. 6.4 and 6.5. The new quantum numbers, K and b (not to be summed over) reflect the structure of the final bound-state spectrum through weight factors inserted in the angular distributions. Thus, e.g., the sum in Eq. (6.8) would collapse into a single term if a single possible ground state were available for the system. This open-shell treatment is represented by multiplying DMD2 in Eq. (6.7) by the square of the fractional parentage coefficient of Eq. (6.8) and summing over the new additional quantum numbers, unless restricted by the final state.

7. Two-body interaction The contribution of two-body potentials to Hamiltonians is represented by scalar functions that generally yield angular momentum transfers between interacting particles, arising from the potential’s radial dependence. The angular momentum aspects of these transfers are rather simple, and lead to the need of evaluating radial integrals, of no concern to our treatment. The resulting graph provides an ‘‘interaction building block”, which will appear in the evaluation of problems containing two-body interactions. In fact this represents the central problem in bound-state calculations. The scalar nature and the translation invariance of a two-body potential are implied by the notation »"»(r)

(7.1)

with r"Dr !r D . 1 2 Shell-model calculations expand Eq. (7.1) into multipole components

(7.2)

»"+ F (r , r )¸K [½*L+(rL )½*L+(rL )]*0+ (7.3) L 1 2 1 2 L to be used with shell-model wave functions /*j+ that are not translationally invariant; the required correction taking the center-of-mass motion into account is described in Ref. ([13], ch. 8). The form of F (r , r ) depends on the potential (7.1). Its expansion (7.3) includes an infinite number of terms, L 1 2 of which only a small number contributes to matrix elements, as we shall see presently.

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Fig. 7.1. Guide graph for a two-body interaction. (Subscripts of operators and states correspond to particle coordinates.)

Spin—orbit couplings and spin—spin interactions generate ‘‘tensor forces” whose expansion (Eq. (7.3)) replaces the factor F (r , r ) by L 1 2 G (r , r )[[½*1+(rL )½*1+(rL )]*2+[p*1+p*1+]*2+]*0+. (7.4) L 1 2 1 2 1 2 The potential form (7.3) affords factorizing the angular parts of its interaction matrix element M"+ M "+ [[t*j1+(1)t*j2+(2)]*I+DF (r , r )¸K [½*L+(rL )½*L+(rL )]*0+D[/*j3+(1)/*j4+(2)]*I+] (7.5) L L 1 2 1 2 L L into single-particle elements. Each term M of Eq. (7.5) is to be evaluated as in Section 4 with the L pertinent recoupling transformations. The relevant recoupling of wave functions and operators is indicated in the guide graph Fig. 7.1, with its implied recouplings displayed in Fig. 7.2. Its transformation structure corresponds to that of boxes A or B of Fig. 3.2, except for replacing their graph amplitudes with wave functions, terminated in invariant matrix elements. The usual factorization of wave functions /*j+(r)"f (r)½*j+(rL ) m j m in the absence of spins, is replaced in their presence by

(7.6)

/*j+ (r)"f (r)[m*s+½*l+(rL )]*j+ . (7.7) j,l m m l,m The graph Fig. 7.2 indicates how the interaction integrals M , Eq. (7.5), resolve into products of L single particle multipole moments, leading to the expression

C D

j j I 1 2 IK M "¸K ¸ ¸ 0 [t*j1+(1)D»*L+D/*j3+(1)] t*j2+(2)D»*L+D/*j4+(2) (7.8) L jK jK j j I 34 3 4 whose product of invariant matrix elements appears in Fig. 7.2 on the right of the dashed lines x — — x and y — — y. This expression is somewhat symbolic in that the radial integral does not necessarily factorize as detailed by Eq. (7.7). According to Eq. (7.8) the triangular conditions limit the values of ¸ to the ranges

C

D

j #j 5¸5D j !j D , 1 3 1 3 j #j 5¸5D j !j D . 2 4 2 4 In the absence of spin the final expression sets in Eq. (7.8),

(7.10)

[t*j1+D»*L+D/*j3+][t*j2+D»*L+D/*j4+]"R[ j D¸D j ][ j ¸D j ] , 1 2 3 4

(7.11)

(7.9)

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203

Fig. 7.2. Recoupling graph for factorizing a two-body interaction into one-body invariant matrix elements. [Indices as in Fig. 7.1 and Eq. (7.5).]

Fig. 7.3. Elimination of spinors from spin-independent interactions.

with the coefficients [ j D¸D j ] defined in Eq. (3.25). In the presence of spins, the recoupling k l transformation must terminate spin lines into overlaps. To this end the portion of Fig. 7.2 on the right of the dashed line x — — x is replaced by the graph in Fig. 7.3; similarly for the line y — — y. The crossing boxes in Fig. 7.3 serve to bring together the lines required to form an invariant triple product with the proper ordering of tensors. The combined invariant triple products yield, according to Fig. 7.3, [t*j1+D»*L+D/* j3+][t*j2+D»*L+D/*j4+]

C DC D

(7.12)

R" f 1(r ) f 3(r )F (r , r ) f 2(r ) f 4(r )dr dr . j 1 j 1 L 1 2 j 2 j 2 1 2

(7.13)

s

l 1 "R(!)j3~j1`j4~j2 s l 3 0 ¸ with

P

j 1 j 3 ¸

s

l 2 s l 4 0 ¸

j 2 j sL 2[l D¸Dl ][l D¸Dl ] 4 1 3 2 4 ¸

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This radial integral remains in general non-factorizable as a two-dimensional integral. The angular momentum treatment has thus resulted in replacing a six-dimensional integral by a small number of two-dimensional integrals with known numerical coefficients. Spin dependences of the potential are treated as discussed in Section 5. When dealing with many-body systems one must extract the two-body functions entering the matrix element (7.5), i.e. the functions [t*j1+(1)t*j2+(2)]*I+ and [/*j3+(1)/*j4+(2)]*I+, from the fully antisymmetrized wave functions. As in Section 6 this can be best performed using the fractional parentage expansion. In case the particles belong to the same shell the procedure (called ‘‘double CFP’’) extracts the particles sequentially. This process occurring frequently in bound-state calculations, is treated in Refs. [13]; [4], ch. 6.

8. Auger transitions following photo-ionization This section illustrates the treatment of a two-body force by analyzing an Auger process initiated by deep-shell photo-ionization followed by filling the vacancy with emission of a second electron. As an example consider a noble gas whose ground-state configuration includes two subshells (n s)2(n p)6. Photoabsorption by an s electron yields the intermediate structure (n s)(n p)6(el), s p s p with e indicating the photoelectron energy. (Processes in which (el) is replaced by a highly excited bound state (nl) are called ‘‘resonant Auger”.) The (n s) vacancy is then filled by transfer of s an (n p) electron, with the energy thus released being transferred by a two-body potential, ejecting p another electron (n pPe@l@), to reach a final configuration (n s)2(n p)4(el)(e@l@). The final state, with p s p the total angular momentum imparted by the photon, includes generally alternative partitions of this momentum among (n p)4, alternative (l, l@) pairs and electron spins; alternatives that superpose p coherently. The transfer of two electrons out of the (n p)6 configuration would involve averaging with p fractional parentage coefficients, as in Section 6. Here we eliminate this step by dealing with a fictitious example involving only three electrons, each initially in a single-occupancy sub-shell with spin-orbitals (/ , / , / ), labelled by j-quantum numbers, coupled to a spectator core, possibly a b c without spin. Photoabsorption transfers the / electron to t , while the Auger two-body interaca f tion transfers the pair (/ / ) to (t t ). b c e g Since polarization vectors describing polarizations and angular distributions are easily generated as in Section 4, we shall treat the general case directly, constructing all the relevant polarization vectors and thus obtaining a sevenfold correlation function. A complete experiment to determine this function fully is difficult, if not impossible, to achieve, of course. The relevant matrix element analogous to the already simplified form (5.9) reads (as discussed in the beginning of Section 5) DMD2"SUDD»DWTSm DSDm TSm DSDm TSWD»DDUT . (8.1) 1 1 2 2 Recouplings of states are required here only to factor out each matrix element since the operators D and » act here on different basis states. The initial and final states are coupled as DUT"IK [A*I+[/*j1+[/*j2+/*j3+]*j23+]*K+s*j+]*0+ , a b c DWT"JK [B*J+t*j@e+s*j+]*0+lK @ [b*l@f+f *l@f+]*0+lK @ [c*l@g+g*l@g+]*0+ e f g

(8.2) (8.3)

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Fig. 8.1. Overall guide graph. For compactness / ,/ ,/ ,t are denoted u, v, w, y, respectively. The dipole and two-body a b c e interactions act independently. The final state W is factored as in Eq. (8.3). Insertion of spin functions m proceeds as in Section 5; m indices have been omitted here.

with amplitudes A and B. The dipole operator D remains the same as in previous sections, » being the scalar operator of Eq. (7.1). In this expression, as in the previous sections, s represents the spectator wave functions; f and g the orbital parts of continuum functions t and t . Interference f g between final states will be indicated as in section Section 4 through amplitudes b, c, b@, c@, bA, cA. Continuum functions will display their spin—orbit parts explicitly, as in Section 5, t *j@f+"[m*s+f *l@f+]*j@f+ f

(8.4)

providing the directional and spin information required to generate polarization vectors. A full description of the reaction would include a propagator Green’s function, i.e., essentially an energy denominator, between the D and » operators separating their respective actions, and identifying the Feynman diagrams of the photoabsorption and Auger emission in alternative sequence. We ignore this Green’s function since, as a scalar, it does not affect angular distributions or the states’ structure. Note here the multiple couplings in the initial state wave function that provide all the information required for our purposes. We select the initial coupling to simplify following recouplings; alternative realistic structures would require additional specification. The relevant guide graph shown in Fig. 8.1, including all polarizations and correlations, resolves into sub-graphs pertaining to individual operation steps, as in Figs. 5.2 and 5.3, dealing first with the matrix elements and then assembling the polarization vectors. The sequence of steps required to evaluate the left-hand matrix element of Fig. 8.1 appears in Fig. 8.2: We first overlap spectator s away (g), dealing next with the two-body interaction », (h), and last the photoabsorption matrix element (i). The corresponding steps for the right matrix element are shown in Fig. 8.3. Fig. 8.4 presents the overall block diagram for evaluating of DM2D; the labels in the individual blocks correspond to those of Figs. 8.2 and 8.3. As can be seen, the inter-relations between different steps are rather involved, to be displayed in detailed recoupling graphs, which we now address. Consider the individual steps, beginning with Fig. 8.2. The separation of amplitudes from basis functions in Eqs. (8.2) and (8.3) is complicated here by the occurrence of multiple function products. An initial step (g) separates out the spectator factors s of W and U in the left-hand matrix element of

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Fig. 8.2. Step guide graphs for the right matrix element of Fig. 8.1. Fig. 8.3. Step guide graphs for the left matrix element of Fig. 8.1.

Fig. 8.4. Block graph for the recoupling operations. The labels of the boxes correspond to the step labels of Figs. 8.2 and 8.3.

(8.1) as indicated in Fig. 8.5, yielding a factor

C D j

j

0

[g]"IK JK @ j

j

0 [sDs] .

(8.5)

0 0 0

A second step (h), Figs. 8.2 and 8.6, extracts the two-particle state [/ / ] from U, leaving the rest b c coupled with angular momentum R@. The function [t t ] combines with the two-body interaction b c » and with the corresponding two-function state [/ / ] from W. (Actually, t consists of the e g g orbital g and spin m .) The remainder of sub-graph 8.6 is represented by g [[AI *I+ /I *j1+]*R{+B*J{+[c*l@g+ mI *s+]*j@g+]*0+ , (8.6) a

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207

Fig. 8.5. Recoupling graph for step (g) of Fig. 8.2.

Fig. 8.6. Recoupling graph for step (h) of Fig. 8.2.

contributing the factor

C

I

DC D

I

l@ l@ 0 g g [h]" j j K sL lK @ s s 0 (!1)l@g`s~j@g 1 23 g R@ j j j@ j@ 0 23 g g 0

C

J@

j@ j e j@ j@ 0 g g R@ j j 23

DC

R@ j j 23 R@ j j 23 0 0 0

D

][[/*j2+ /*j3+]*j23+D»D[t*j@e+ t*j@g+]*j23+] . b c e g The last step, Fig. 8.2, involving no new elements, is described in Fig. 8.7, yielding @

@ f

[i]"lK @ sL (!)l f`s~j f

C D C DC l@ l@ 0 1 1 0 f f s s 0 1ª j@ j@ 0 f f j@ j@ 0 j j 0 f f 1 1

I

j

1

R@

D

j 0 [/*j1+DD*1+Dt*j@f+] . 1 1 a f R@ 0 R@ j

(8.7)

(8.8)

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Fig. 8.7. Recoupling graph for step (i) of Fig. 8.2.

The combined output of these graphs and of Eqs. (8.5), (8.6), (8.7) and (8.8) forms the composite invariant triple product ¹@"[AI *I+[d*1+[mI *s+ b*l@f+]*j@f+]*j1+[B*J{+[c*l@g+ mI *s+]*j@g+]*R{+]*0+ .

(8.9)

Similarly recoupling graphs of individual steps, (d), (e), ( f ) in the guide graphs Fig. 8.3 lead to Figs. 8.8, 8.9 and 8.10, yielding factors

C D C DC D C D C DC D C D C DC D j

j

0

[j]"JK AIK j

j

0 [sDs] ,

(8.10)

0 0 0

I

[k]" j

0

I

j K 23 RA j j 23

JA

jA j e RA j j sL lK A(!)2s g 23 jA jA 0 g g

jA jA 0 jA g g g ](!)lAg`s~jAg 0 jA jA 0 e e jA j jA jA g 23 e g s s 0 jA jA f f [l]"sL lK A (!)2s lA lA 0 1ª 1 1 f f f jA jA 0 j j f f 1 1

lA lA g g s s

0

jA jA g g

0

j jA 23 g j j 23 23 0 jA g 0 j 1 0 I 0

0

[[t*jAe+ t*jAg+]*j23+D»D[/*j2+ /*j3+]*j23+] . e g b c

j

1

0

RA [t*jAf+DD*1+D/*j1+] . 1 f a RA 0 RA j

(8.11)

(8.12)

Again, the output of these graphs leads to the invariant triple product, ¹A"[[BI *JA+cJ *lAg+ [m*s+]*jAg+]*RA+[bI *lAf+ [m*s+]*jAf+d*1+]*j1+A*I+]*0+ . g f

(8.13)

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209

Fig. 8.8. Recoupling graph for step (j) of Fig. 8.3. Fig. 8.9. Recoupling graph for step (k) of Fig. 8.3.

Fig. 8.10. Recoupling graph for step (l) of Fig. 8.3.

The final operation extracts all polarization vectors — including the spin orientations of both electrons — from the product of invariants ¹@ and ¹A. Except for the spin functions, m*s+, which must be used to obtain the expectation values of the spin operators, this product contains the amplitudes which must be combined to obtain the polarization vectors as exemplified by Eq. (4.4)ff, making up the correlation invariant, i.e., the ‘‘Experimental Layout” of Eq. (1.13). All these operations are indicated in guide graph Fig. 8.11. The resulting recoupling graph, Fig. 8.12, yields the algebraic expression

C

J@

J@

0

[H]"(!1)R{`J{~j g(!1)j1`I~R (!1)2I~LA(!1)j `1~j1 JA

JA

0

@

C

I

j

] I

j

¸ A

1

1 N

A

R@ RA M

DC

A f

D

j@ j f 1 1 jA j (!1)s`lAf~jAf f 1 ¸ I N P f 1

C

DC

R@ RA

¸ ¸ 0 M B B l@ s j@ f f lA s jA QL@ fA ¸K f f f lfl f S ¸ ¸f I f S f

D

j@ J@ g jA JA g I ¸ g B

D

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Fig. 8.11. Guide graph for the remainder of the overall guide graph Fig. 8.1: evaluation of the spin matrix elements and construction of the polarization vectors.

C C

D D

C

D

¸ ¸f I l@ s j@ f S f g g jA QL@gg Ag¸K g ] ¸ f ¸ f 0 (!1)s`LSf~s[m Dp*LSf+Dm ] lA s S g S f f g ll S ¸ ¸g I I 0 I g S f f g ¸ ¸g I g S g ] ¸ g ¸ g 0 (!1)s`LSg~s[m Dp*LSg+Dm ] . (8.14) S S g g I 0 I g g This same graph (Fig. 8.12) shows the correlation invariant, namely the ‘‘Experimental Layout” factor of Eq. (1.13), to be fairly involved [K]"[P*LB+[P*LA+[P*LP+[P*Lf+S*LSf+]If+]*N+]*M+[P*Lb+ S*LSb+]*Ig+]*0+ . (8.15) B A P f f b b This formula represents the correlation invariant of the desired seven-fold correlation function containing: the orientation of the initial system, P , the polarization and direction of the incoming A photons, P , the orientation of the recoiling system, P , and the directions P , P , and spin P B f g orientations S , S of the emitted particles, respectively. All the corresponding quantum numbers, f g ¸ , ¸ , ¸ , ¸ , ¸ , ¸ , ¸ , M, N, I , I , are observable, by extracting the magnitudes of the A B P f Sf g Sg f g respective tensors from experimental data, much as in the simpler example of Section 4. Since the graph Fig. 8.12 is fairly complex we show the coupling scheme of our correlation function separately in Fig. 8.13. The complete expression of Eq. (8.1), i.e., the correlation function, equals the product DMD2"[g][h][i][j][k][l][H][K]" + (!)lf@ `lAf`lg@ `lAg~jf@ ~jAf~jg@ ~jAg`JA`RA`jAf`jgA`2I`j1`LA`LSf`LSg R{,RA

C

I IK 2 jK @ jK A 1ª2 ¸Y jK jK ¸Y B B j ] g g RY @ RY A jK 2 jK 2 ¸Y g¸Y f 1 S S R@ 23

C

R@

] RA M

j J@ g jA JA g I ¸ g B

DC

l@ g lA g ¸

s s g

¸

Sg

0

I

j

K

j

23

j

23

j@ g jA g I g

DC

I I ¸ A

DC

I

I

0

j j 1 23 RA j 23 j@ R@ 1 j RA 1 N M

K j

DC

DC

J@ j@ g R@

j@ j e j@ 0 g j j 23

D

j@ j f 1 1 jA j f 1 ¸ I N p f 1

DC

JA

jA j e jA jA 0 g g RA j j 23

D

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211

Fig. 8.12. Recoupling graph for the guide graph Fig. 8.11.

C

l@ s f ] lA s f ¸ ¸f f S

D

j@ f jA QL@ fA QL@ gA [m*s+Dp*LSf+Dm*s+][m*s+Dp*LSg+Dm*s+][[/*j2+ /*j3+]*j23+D» D[t*j@e+ t*j@g+]*j23+] f b c 12 e g lfl f lgl g I f

][[t*jAe+ t*jAg+]*j23+D» D[/*j2+ /*j3+]*j23+][/*j1+DD*1+Dt*j@f+] e g 12 b c a f ][t*jAf+DD*1+D/*j1+][P*LB+[P*LA+[P*LP+[P*Lf+S*LSf+]*If+]*N+]*M+[P*Lg+ S*LSg+]*Ig+]*0+ . f a B A P f f g g

(8.16)

The factors Q are defined in Eq. (4.10). Here and in the preceding formulae the matrix elements have not been resolved into their spin and orbital parts. Two dummy summations are required here, viz., over the indices R@ and RA, included in the recoupling steps (h) and (k), respectively, as implied in (H) and explicitly in Eq. (8.16), but do not occur in the correlation invariant [K], Eq. (8.15). These dummy summations might be by-passed by alternative recouplings, which we have been unable to find. On the other hand, four dummy summations would have been required by constructing the photon absorption matrix element before the two-body matrix element. As always, failure to observe one or more polarization vectors sets the respective angular momenta to zero. Failure to observe the spin polarization is detailed in Eq. (5.20) and in its discussion. The most complicated of the blocks of this section, b,c,e,f, correspond to the ‘‘full” building blocks of Section 3 and to the ‘‘interaction” building blocks of Section 7. Similarly, the other blocks are seen to contain the building blocks, together with some additions.

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Fig. 8.13. Coupling scheme of the correlation function, Eq. (8.15).

9. Processing of results Extracting tensorial or invariant parameters from experimental measurements requires two steps of inversion. First, raw data of experimental measurements must be recast into the values of the tensorial elements of the ‘‘Experimental Layout”. Next the known geometrical relations among tensorial parameters developed in the present paper must be recast into one or more equations whose solution would determine the unknown parameters of interest. The previous sections have shown how to construct correlation functions, specifically their factors called ‘‘Dynamics” and ‘‘Experimental Layout” in Eq. (1.13). The ‘‘Dynamics”, denoted by the symbol NM N, consisting of invariant matrix elements and recoupling coefficients, contains a information, unrelated to geometry, in the radial integral factors of Eqs. (3.10), (4.12), (4.35) and (7.13), and implicitly in Eqs. (7.8) and (8.16), which frequently are least known among the reaction parameters. The latter invariant factor, the ‘‘Experimental Layout”, called here a correlation invariant, and frequently denoted by ¹, contains its geometry, i.e., the direction and polarization of the incoming beam, the target orientation, the directions and polarizations of the emitted particles, etc. The present section constructs the invariant ¹ explicitly in terms of specific examples directly related to experiment, showing also how to obtain values of radial integrals from the experimental data. Throughout this paper we have expressed the resulting correlation functions in the form ¼"+ NM N¹*0+ . (9.1) a MaN M N a The functions ¼ themselves are not directly measurable quantities, being folded with the resolution function, say X , describing the angular windows, efficiencies, etc., of the diverse counters in c a specific setup, c, of the experimental system. The counting rate in that specific setup of the system, ¼ I , is accordingly replaced by the integral c

P

¼ I " d) X ¼ , (9.2) c c %9 c that now provides a discrete set of numbers directly equivalent to the discrete set of experimentally determined data. The first step of our analysis requires us thus to extract from the experimental data the ‘‘experimental” correlation function, ¼ , by inverting the relations (9.2), a task wellex known to experimentalists. We shall assume this task to have been accomplished, hence dropping the subscript ex.

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213

This section presents typical examples of explicit calculations of ¹; this invariant factor consists of irreducible tensors coupled to total angular momentum zero, but might prove more user-friendly if recast in terms of trigonometric functions. This section also presents procedures to extract radial integrals from the experimental data, concluding with some general remarks. 9.1. Correlation invariants Simple examples of correlation invariants encompassing the relevant layout have appeared in Sections 4 and 5, e.g., in Eq. (4.14), cast as the invariant triple product of polarization vectors: ¹"[P*La+P*Lb+P*LP+]*0+ . (9.3) a b P Two kinds of polarization vectors P occur here: (i) P*Lb+ arising directly as a spherical harmonic, b whereas (ii) P*La+ and P*LP+ consist of orientation amplitudes, transforming under coordinate a P rotations according to Eq. (2.4). Spherical harmonics arise as shown for example in Eq. (4.34): [l@DlAD¸ ] b ½*Lb+(kK ) , (9.4) P*Lb+ (kK )"[b*l{+(kK )bI *lA+(kK )]*Lb+" b,M M M lK @lK A¸Y b whereas a polarization vector of type (ii), previously discussed in Section 4, Eq. (4.4), is represented by P*L+ "[d*1+dI *1+]*L+ ; (9.5) P,M M the polarization vector P*La+ of Eq. (9.3) provides another example: a a a P*L + "[a*l+aJ *l+]*L + . (9.6) a,M M From these quantities the correlation invariant is evaluated by explicit calculation, as for example, the parameter ¹ of Eq. (9.3)

A

B

¸ ¸ ¸ a b P P*La+ P*Lb+ P*LP+ , (9.7) [P*La+P*Lb+P*LP+]*0+"(!1)La~Lb`LP + a b P a,ma b,mb P,mP m m m ma,mb,mP a b P including a 3j symbol. Constructing the geometric amplitudes frequently utilizes the ‘‘spherical unit vectors”, previously introduced in Eqs. (4.16), (4.17), (4.18) and (4.19): i(e #ie ) y , eL *1+" x 1 J2 eL *1+"!ie , 0 z !i(e !ie ) x y . eL *1+ " ~1 J2

(9.8) (9.9) (9.10)

Describing atomic or nuclear alignments by crystalline electric quadrupole fields also involves the spherical rank-2 unit tensors eL *2+"[eL *1+eL *1+]*2+ , M M

(9.11)

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occurring in the invariant quadrupole-field tensor Q"2ª[Q*2+eL *2+]*0+ ,

(9.12)

an analogue of the invariant magnetic-field form, Eq. (4.16), B"1ª[B*1+eL *1+]*0+ .

(9.13)

Other elements include the photon polarization vector, already discussed in Section 4, where Eq. (4.27) represents light polarized along the x-axis through the vector components i , d*1+" 1 J2

d*1+"0 , 0

!i d*1+ " , ~1 J2

(9.14)

from which Eq. (9.5) generates the polarization vectors, 1 P*0+ "+ d*1+d*1+ (1 m, 1 !mD00)" ; P,0 m ~m J3 m

(9.15)

P*2+ "+ d*1+d*1+ (1 m, 1 M!mD2 M) , P,M m M~m m leading to Eq. (4.28). Similarly the correlation invariant for a measurement including the spin polarization as in Section 5, Eq. (5.20), has the form ¹"[[P*LA+P*LP+]*K+P*Lb+S*LS+]*0+ , A P b reading explicitly as

(9.16)

A

B

K ¸ ¸ b S (¸ m ¸ m DK m )]P*LA+ P*LP+ P*Lb+ S*LS+ . (9.17) (!)K~Lb`LS ¹" + A A P P K A,mA P,mP b,mb mS m m m mK,mA,mP,mb,mS K b S The polarization vector of the spin operator S*LS+ occurs here as in Section 5, Eq. (5.7), with ¸ "1: S S"1ª[S*1+p*1+]*0+ . (9.18) When measuring the spin polarization along the propagation direction of an electron, i.e., along kK , the Cartesian components of the spin polarization vector reduce to S "cos h , z corresponding to

S "sin h cos u , x

S "sin h sin u , y

(9.19)

S*1+"!i cos h , 0 i sin h(cos u#i sin u) i sin h " e*r , S*1+" 1 J2 J2 !i sin h(cos u!i sin u) !i sin h " e~*r . S*1+ " ~1 J2 J2

(9.20)

The alternative measurement of a spin along an azimuthal direction yields instead of its components, S "0, z

S "!sin u, x

S "cos u ; y

(9.21)

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215

i.e., S*1+"0 , 0

!i sin u!cos u , S*1+" 1 J2

!i sin u#cos u S*1+ " . ~1 J2

(9.22)

Finally an alignment of atoms or nuclei arises whenever a rotational symmetry is perturbed by, say, a crystalline field, which requires diagonalizing the relevant Hamiltonian. Weak fields break only the M-degeneracy, where M is the magnetic quantum number of the system. The system’s wave function is represented then, as in Eq. (2.1), in terms of the standard amplitudes A(I), M W "+ A(I)W*I+ I M M M determined from the secular equation

(9.23)

+ SW*I+ DHDW*I+TA(I) "E A(I) . M{ M n,M{ n n,M M{ The conjugation relation, Eq. (2.6), provides the corresponding ket amplitude

(9.24)

A*I+ "(!1)I~MA(I) . (9.25) n,M n,~M Corresponding bra amplitudes, serving to construct the polarization vector, are according to Eqs. (2.15) and (2.7), (9.26) AI *I+ "(!)2IA(I)* , n,M n,M where the star denotes complex conjugation. With these parameters one immediately computes from Eq. (4.5) the observable polarization vector P*L+ " + A*I+ AI *I+ A(I M@ I MAD¸ M) . n,M{ n{,M n,n{,M M{MA Thermal equilibrium implies alternatively impure states of the system, namely

(9.27)

(9.28) P*L+ "d Z~1e~En@kTP*L+ , n,n{,M T§n,n{,M n,n{ with the normalization constant Z"+ e~En@kT. The special case of a magnetic field along the z-axis n yields simply A*I+ "d and E "CMd , with C a constant. Inserting these values in Eq. (9.28) n,M n,M n n,M yields the well-known result, P*L+ "Z~1e~CM@kT . (9.29) T§M We conclude this section by expressing the invariant tensors for the correlation function, Eq. (9.3), in terms of trigonometric functions analogous to those in Section 4, i.e., without any initial state polarization, ¸ "0, and with unpolarized photons incident along the z-direction, i.e. with A vanishing P*2+ ; note that Eq. (9.15) is not relevant, the unpolarized photons being impure states. P,M We also set l"0; l@"lA"1, and select the first of the above experimental arrangements, which measures the polarization along the propagation direction of the electron kK . The several triangular conditions associated with the Eqs. (9.3), (9.4), (9.5) and (9.6) lead then to the restriction ¸ "¸ "2. As previously, the contributions of the x- and y-polarizations lead to cancellations, P b

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and finally the values given in Section 4 need only complementing with the values

A

2 2

1

B A S

2

"!

0 1 !1

2

B

1

1 " , 0 !1 1 J10

(9.30)

m 15 ½*2+(h, u)" cos h sin h e*mr m 2 2p

(9.31)

with DmD"1. The correlation invariant Eq. (9.7) equals thus

CA BA S A BA S

1 [P*LP+P*Lb+S*LS+]*0+" P b J6 #

1 ! J10

1

J10

BA BA

1 15 cos h sin h e*r 2 2p

1 15 ! cos h sin h e~*r 2 2p

!i sin h J2

i sin h J2

B

e~*r

BD

e*r

"0 ,

(9.32)

which vanishes as implied by the vanishing of P*2+ . [Photon polarization would lead instead to P,M a non-zero value of Eq. (9.32).] 9.2. Radial Integrals Quantum theory connects a reaction’s input elements (o) to its output elements (R) by matrices (S): R,So. Explicit data on o and R may thus provide equation systems apt to determine unknown elements of the S matrices, e.g., matrix elements indicated by ‘‘hatched boxes” (first appearing in Fig. 3.2) that are implicitly accessible to numerical evaluation but may contain unknown or incompletely known aspects of the system dynamics. On the other hand, the numerical values of all other ingredients of the graphs, i.e., mainly recoupling coefficients, are fully known and accessible to numerical computation. One may want to employ experimental data to determine the values of the unknown elements. To this end one may read out of our graph the two equivalent sets of linear equations relating: each element of the ‘‘output” set of observable data R"So, as well as its reciprocal o"S~1R. Combination of the two sets provides a number of (non-independent) linear equations equal to the number of elements of the o and R sets. A survey of the graphs presented here may indicate whether the number of accessible data and of the relevant equations suffices to determine the numerical values of hatched boxes for a system of interest. Examples of such sets appear in Sections 4, 5 and 8; Eqs. (4.14), (5.20) and (8.15). The previous developments have led from the physics, i.e., from the invariant form of the dynamics, to coordinate-dependent expressions — required to compare with data determined by the specific geometry of the experiment. Inverting of this procedure to extract radial integrals from experimental results, presents the opposite problem of extracting invariants from the coordinatedependent data. In Eqs. (4.12), (5.20) and (8.16) the radial integrals are included in the invariants N. As described above in Section 9.1, the correlation invariant ¹, and hence the correlation function ¼, may be a function of several angles in the ‘‘laboratory system”. Let us take the simple case of an angle-independent factor S*LS+ (e.g., where a spin detector is oriented along the z-direction, while the outgoing particle lies in the x—y plane). Then in the example of Section 5, in

M. Danos, U. Fano / Physics Reports 304 (1998) 155–227

217

the correlation function ¼ of Eq. (5.20), N A P b S [[P*LA+P*LP+]*K+P*Lb+S*LS+]*0+ , ¼" + (9.33) L ,L ,L ,L ,K A P b LA,LP,Lb,LS,K only the polarization vector P*Lb+ is actually a function of the angles. In general, depending on the b experiment, the factor S*LS+ may include the angular functions ½*LS+, as in Eqs. (9.21) and (9.22). We M M rewrite Eq. (9.33) as ¼" + N A P b S [P*Lb+R*Lb+]*0+ L ,L ,L ,L ,K b A P b S L ,L ,L ,L ,K

(9.34)

with (9.35) R*Lb+"[S*LS+[P*LA+P*LP+]*K+]*Lb+ , M A P M and expand the experimental ¼ into multipole components F "¸K [C*L+½*L+]*0+, i.e., evaluate L N A P b S S¸K [C*L+½*L+]*0+ D [P*Lb+R*Lb+]*0+T , ¼ , S F D ¼ T" + (9.36) L ,L ,L ,L ,K L L b LA,LP,Lb,LS,K thus removing from ¼ its explicit angular dependences. At this point one must utilize the conditions of the actual experimental setup, reverting to the M-components of the tensor R*L+. Thus, to obtain the components of the amplitude vector C*L+, one must project ¼ onto its M multipole amplitude C*L+, M (9.37) C*L+"S½*L+(rL ) D ¼T" + N A P S R*L+(¸!M¸MD00) . L ,L ,L,L ,K M M M b LA,LP,LS,K In this equation all entries, except for the weights N A P S , have known values. For each value of L ,L ,L,L ,K ¸4¸ one thus obtains 2¸#1 equations. Achieving an adequate number of equations will L generally require obtaining data for several values of ¸ ,¸ ,S, i.e., data for alternative beam and A P target alignments, polarizations, etc., providing an algebraic set of linear equations for the weights N A P b S . L ,L ,L ,L ,K This set of equations, once solved, yields a set of bilinear equations for the radial integrals, as is evident for example from Eqs. (4.12), (5.20) and (8.16). Solving these equations then completes our evaluation. 9.3. Remarks Only recoupling coefficients provide the actual evaluation of the correlation function, which in this paper occurs as a sum over invariants (see Eqs. (4.14), (5.20) and (8.15)). VC or 3j coefficients appear only in the explicit evaluation of these functions, such as Eq. (9.32), for a given specific experimental arrangement. To that end one utilizes a coordinate system, e.g., the laboratory system, with the relevant orientations and positions of the experimental components — the beam direction, the placement of the detectors, etc. — which then permits to express the polarization vectors through their m-components. Summing over m-components with vector coupling coefficients yields finally expressions which can be written in terms of desired trigonometric functions.

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Failure to measure some parameter restricts the multipolarity, i.e. the ¸ value, of a polarization vector to its zero value corresponding to a detector with uniform sensitivity over the full sphere, ‘‘responding” thus only to the ¸"0 component of that polarization vector. When polarization vectors are included in an overall invariant, this restriction to ¸"0 delimits the range of multipolarities of the remaining polarization vectors, and, hence, the form of observable correlation functions. Analyzing experimental data represents them in a form analogous to Eq. (5.20). Extracting the values of the radial integrals, or analogous dynamical parameters can be achieved in principle using a straightforward inversion algorithm, as explained above in Section 9.2. However, the quality of the data frequently will not suffice to permit this approach. In that case iterative procedures are required, in which one assumes values for the unknown parameters and computes the resulting correlation functions, comparing them with the data, changing the assumed values, until reaching satisfactory agreement with the data.

Acknowledgements The work of one of us (UF) was supported by the NSF Grant PHI-9510316 which we gratefully acknowledged.

Appendix A. General remarks on recoupling graphs A desired recoupling transformation may often be cast in alternative ways. An example is provided in Figs. 8.6 and 8.9 where very similar recoupling transformations are performed in quite different ways. Frequently, but not always, some recoupling schemes prove more efficient than others by requiring fewer intermediate dummy summations, as in the example at the end of Section 8. These alternatives imply the presence of identities difficult to prove algebraically, one of the earliest being the Biedenharn—Elliot identity arising from the recoupling graphs Fig. A.1, which read: S(ab)f, (cd)g, e; 0Da, (bc)k, (de)p; 0T

(A.1)

C DC DC D C DC D a b f

"+ 0 g g h a h e c

d g

" 0 e c

e

p f

b 0 b

k d h

c

0 e

d g

k d h

e

(A.2)

k p a

a b f 0 c

c .

(A.3)

a k p

These equations, representing the Biedenharn—Elliot identity, eliminate the dummy summation over h. All 9j-type recoupling coefficients of Eqs. (A.2) and (A.3) contain one zero index, thus reducing through Eqs. (2.32) and (2.33) to expressions in terms of 6j coefficients, the form appearing in textbooks.

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219

Fig. A.1. Graph for the Biedenharn—Elliot identity. The implied sum over h in the upper diagram yields an identity that reduces it to the lower form.

Fig. A.2. Same as Fig. A.1 in the compact graph convention (2nd paragraph of Appendix A).

Before proceeding to other examples we introduce a symbol which permits drawing recoupling graphs in a more compact form, namely, a new ‘‘coupling bracket”, drawn as ‘‘ ) ” to bracket two L adjacent lines being coupled to angular momentum ¸, thus replacing the tapered confluence or divergence of lines. Re-drawing Fig. A.1 with this ‘‘compact graph convention” yields Fig. A.2. The resulting shortening of the graph becomes even more effective for more complicated graphs, as will become evident below in the context of Figs. A.7 and A.8. An example of recouplings with alternative efficiencies occurs in evaluating the invariant matrix element of the two-body potential for a complex state: »"[[[t*j+t*k+]*J+[t*j+t*k+]*K+]*L+D» D[[t*j+t*k+]*J+[t*j+t*k+]*K+]*L+] , (A.4) 1 3 2 4 1,2 1 3 2 4 whose guide graph, including the factorization of the interaction according to Section 7, is shown in Fig. A.3. Two recoupling orders present themselves, viz., those of Figs. A.4 and A.5. The first of these seems more natural in that it immediately assembles the interacting particles. Reading off these graphs we have

C DC DC DC DC D j

k

J

j

k

J

P Q ¸

j

j

P

k k Q

»"+ j k K P P Q ¸

j

k

K

P Q ¸

j

j

P

k k Q

0

0 0 0

0 0 0

P Q ¸

]jK kK [t*j+D»*0+Dt*j+][t*k+D»*0+Dt*k+]

0

0

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M. Danos, U. Fano / Physics Reports 304 (1998) 155–227

Fig. A.3. Guide graph for the two-body matrix element Eq. (A.4). [Particles (3,4) are spectators.] Fig. A.4. Recoupling graph for guide graph Fig. A.3. [Summation over P,Q required.]

Fig. A.5. Alternative recoupling graph for Fig. A.3. [No dummy summations.]

C DC DC D J K ¸

j

k

J

j

k J

" J K ¸

j

k

J

j

k J jK kK [t*j+D»*0+Dt*j+][t*k+D»*0+Dt*k+] .

0 0

0

0 0 0

0

0

0

(A.5)

As could be already seen by inspecting the graphs, the seemingly less natural recoupling order is by far more economical in requiring fewer recouplings and no dummy summations. In contrast, the

M. Danos, U. Fano / Physics Reports 304 (1998) 155–227

221

Fig. A.6. Recoupling identity for 8 tensors. (a) Recoupling with summation over (K,L,M,N). (b) Recoupling with summation over (P,Q).

evaluation of the seemingly more natural recoupling order in which one begins by separating the interacting particles from the spectator particles, involves a summation over the dummy index P. The recoupling identity underlying the Figs. A.4 and A.5 is shown in Fig. A.6; its algebraic expression is

C DC DC DC DC C DC DC DC D a

b

A

e

f

C

K

¸

E

a

c

K

b

d

¸

+ c d B KLMN K ¸ E

g

h

D

M N

F

e

g

M

f

h

N

¼ I

R

S »

A B

M N F

»

¹ º

E

a

b

A

c

d

B

R

¹

P

"+ C D F PQ P Q I

e

f

C

g

h

D

S

º

Q ,

R

¹ P

S º Q

¼

D (A.6)

» ¼ I

where four summations reduce to two. It is the most general identity involving the 9j-type, and hence also the 6j-type, recoupling transformations. All known identities, e.g., the Biedenharn—Elliot identity, arise from Eq. (A.6) by substituting zero for one or more of its quantum numbers. Compact graphs abbreviate the overall recoupling graph, denoted as ‘‘overall blueprint” in the Introduction, that allow a direct check of the results. As an example Fig. A.7 implements the guide graph Fig. 4.1, i.e., the complete recoupling graph for the transformation SUDDsDWTSWDDDUT"+ NM N[P*La+P*Lb+P*LP+]*0+ a a b P M N a

(A.7)

of Section 4 where MaN"M¸ ¸ ¸ N represents the quantum numbers specifying the states, which a b P combines the Figs. 4.2 (actually Figs. 3.3 and 3.4, modified for l@OlA) and 4.5. We also give the full graph for the evaluation of Eq. (5.9) of Section 5, i.e., the implementation of the guide graph Fig. 5.1, which combines Figs. 5.3, 5.4, 5.5, 5.6, 5.7, 5.8 and 5.9, as shown in Fig. A.8. This ‘‘overall

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Fig. A.7. Overall recoupling graph for the matrix element Eq. (3.14) of Section 4 combining Figs. 3.3, 3.4 and 3.5.

Fig. A.8. Overall recoupling graph for the matrix elements Eq. (5.9) of Section 5. The basic structure is the same as that of Fig. A.7, with the treatment of the spin added. Note the placing of spin detection on lower right-hand side.

blueprint” is the graph for the transformation SUDDsDWTSWDDDUT"+ NM N[[[P*LA+P*LP+]*K+]*LS+P*Lb+]*LS+S*LS+]*0+ , A P b b M N b

(A.8)

M. Danos, U. Fano / Physics Reports 304 (1998) 155–227

223

where MbN"M¸ ,¸ ,¸ ,¸ ,KN. The coupling scheme of the resulting correlation invariant is visible A P b S on the top right of Fig. A.8; coupling the tensors P and P to K, Eq. (A.8), is indicated in the graph A B by the coupling bracket labelled K, connecting the tensors P and P . Coupling this pair with P to A P b angular momentum ¸ is indicated by the coupling bracket labelled ¸ , finally coupling the S S resulting group of tensors with the tensor S to the overall angular momentum zero is indicated by the overall bracket’s label. This set of three coupling brackets is equivalent to the representation of the coupling scheme in Fig. 5.10 — which actually depicts a different grouping of the invariant triple product. Comparing the Figs. A.7 and A.8 one can recognize the persistence of the basic structure of the graph: the building blocks A and B, shown in Figs. 3.3 and 3.4, and recognizable in Fig. A.7, survive in Fig. A.8 with modifications to accommodate the spin. Similarly, the recoupling of the amplitudes to the polarization vectors of Fig. A.8 repeats that of Fig. A.7 except for the additions associated with the spin. In fact, the Fig. A.7 was drawn essentially by erasing in Fig. A.8 the parts associated with the spin. We give here only the algebraic expression corresponding to Fig. A.8 (in the notation of Section 5):

C DC D C C D C D C C D C C DC DC C D I

I

l@ l@ 0

0

P"IK 1ªlK @ 1 1 0 sL s j@

j@

s

j@

j@

0

0 (!1)l{`s~j{ j@

j@

0

j@ j@ 0

0

0 0 0

DC D j@ 0 j@

l@ s

j@

s

0

s

l

I

](!1)1`j{~I s

l@

j@ sL (!1)l{`1~l[/*l+DD*1+Dt*l{+]

s

0 1 1

lA lA 0

]lK A1ªIK sL (!1)2s s

A

jA

1

0 (!1)l `s~j I

s

jA jA

A

1

A

0

0 jA

s

s

jA (!1)1`I~j s

s

s

0

jA jA

lA jA

A

](!1)2j lA

I

D

D

C D jA jA

0

0 (!1)1`I~j jA jA

0

0

0

0

A

0

0

I sL (!1)l`1~lA[t*lA+DD*1+D/*l+]

l

0 1

1

D

s

s

0

j@

l@

s

I

1

j@

] s

s

0

jA

lA

s

I

1

jA QLbA(!1)2I~LA l{l K

¸ 0 K ¸ ¸ ¸ ¸ S b S A P 0 ¸ ¸ S S ]¸K ¸ ¸ 0 (!1)s`1~s[mDp*LS+Dm][[[P*LA+P*LP+]*K+P*Lb+]*LS+S*LS+]*0+. (A.9) S S S A P b ¸ 0 ¸ S S An overall graph also serves to elucidate the transfer of ‘‘information” on correlations from the wave function to polarization vectors. As an example, consider the accummulation of information pertaining to the spin polarization vector S*LS+ of Eq. (5.7). To that end we will trace it backwards, ¸

S

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ascertaining how the input data transfer the angular momentum information to this tensor. Thus begining at the right of Fig. A.8 with the polarization vector S, follow its line which goes through a recoupling box, to arrive at the injection of the spin operator S&[S p], where the line ‘‘turns around”, continuing as the spin matrix p which, having been coupled with S to angular momentum zero, continues to carry the complete orientational information. After again passing through some recoupling boxes, this line arrives at the invariant matrix element [mDpDm], where it receives information from mI and m. Follow now the mI , i.e., the top line, which after passing through some recoupling boxes combines with the amplitude b*l{+ to the angular momentum j@. At this point the tensor mI receives information of being in a jj-coupling state. Continuing its journey, mI soon turns around in the injection box, emerging as m, to be coupled with the orbital function t again to angular momentum j@. After some more recouplings its line turns around in the context of the dipole transition matrix element, to be coupled now with the function /I to j@, to finally arrive at the input as part as the input function Us"[mI /I ]. (In any turn-around, either in an overlap box, or in a time-reversed overlap box, a line continues to carry the full orientational information; it is preserved in turn-arounds.) A similar sequence of recouplings and turn-arounds occurs in the line m, originating in the spin matrix element [mDpDm], except for its coupling to angular momentum jA rather than j@. These paths where spin lines are coupled to orbital lines transmit the information that the system is in jj coupling to the overall recoupling transformation, and thus to the value of the overall transition matrix element and of the angular correlation function. In analogous manner one can follow the information transfer from the input to the output of the graph, to the individual polarization vectors, and hence to the overall correlation function. Another process requiring extensive angular momentum calculus occurs in the excitation of the photonuclear giant resonance. The elementary process of photon absorption would excite a single particle to a state to be transformed by the repeated action of two-body interactions into a final excited collective state. This process, best described in the particle—hole formalism with creation—annihilation operators [4], ch. 8, will not be discussed here beyond remarking that the recoupling techniques described in this article are also immediately applicable to it. The compact graph convention is applied extensively in [4], which provides further details in the use of graphs.

Appendix B. Definitions and notation Conjugation F(j)"(!1)j`mF*j+ , m ~m F*j+s"FI (j)"(!1)(j`m)FI *j+ , m m ~m ½*l+(rL )"(!i)l½ (rL ) , m l,m ½*l+*(rL )"(!1)l`m½*l+ (rL ) . m ~m Coupling [a(A)b(B)](L)" + (¸MDAm,Bm@)a(A)b(B), M m m{ m,m{

(B.1) (B.2) (B.3) (B.4)

(B.5)

M. Danos, U. Fano / Physics Reports 304 (1998) 155–227

[a*A+b*B+]*L+" + a*A+b*B+(Am,Bm@D¸M), M m m{ m,m{ jK "J2j#1 ,

225

(B.6) (B.7)

(!1)j`m d . (j!m, jm@D00)" m,m{ jK

(B.8)

States DWT"+ a(j)t*j+"W"jK [a*j+ t*j+]*0+ , m m m SWD"Ws"jK [aJ *j+ tI *j+]*0+ .

(B.10)

U"DUT"jK [a*j+ m*s+ /*l+]*0+ ,

(B.11)

Us"SUD"jK [aJ *j+ mI *s+ /I *l+]*0+.

(B.12)

(B.9)

s"1 2

Operators D"jK [d*j+ D*j+]*0+,

(B.13)

S"1ª[S*1+ p*1+]*0+.

(B.14)

Invariant overlaps [t*j+ D t*j+]" [tI *j+t*j+]*0+"jK ,

(B.15)

1 [a*j+Da*j+]"+[aJ *j+a*j+]*0+" . jK

(B.16)

Invariant matrix elements [A*I+DB*J+DC*K+]" [AI *I+B*J+C*K+]*0+ ,

(B.17)

A

B

1(l`k`j) lK kK jK l k j [½*l+D½*k+D½*j+],[lDkDj]"(!)2 , J4p 0 0 0

(B.18)

[m*s+Dp*1+Dm*s+]"iJ6 ,

(B.19)

[½*l{+(kK )½*lA+(kK )]*L+"½*L+(kK ) QL A , M M l{l QL A"(1/¸K )[¸Dl@DlA] , l{l P*La+ "[a*l+aJ *l+]*La+ . a,M M Reduced matrix elements SA*I+EB*J+EC*K+T"(!1)(I`J~K)[A*I+DB*J+DC*K+]

(B.20) (B.21) (B.22)

(B.23)

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Spherical unit vector components i(e #ie ) y , e*1+" x 1 J2

(B.24)

e*1+"!ie , 0 z !i(e !ie ) x y. e*1+ " ~1 J2

(B.25) (B.26)

Spherical vector components i(» #i» ) y , »*1+" x 1 J2

(B.27)

»*1+"!i» , 0 z !i(» !i» ) x y, »*1+ " ~1 J2

(B.28) (B.29)

V"» e #» e #» e "1ª[»*1+ e*1+]*0+, x x y y z z A ) B"1ª[A*1+B*1+]*0+,

(B.30)

A]B"1ªJ2[A*1+B*1+e*1+]*0+.

(B.32)

(B.31)

Fractional parentage expansion U(n,a)*J+" + [s(n!1,b)*K+/*l+(n)]*J+(ln~1bK;lDNln aJ) . M M b,K Neutral and Coulomb plane waves eik > r"+ ilP (kK ) rL )4p j (kr)"+ illK [½*l+(kK )½*l+(rL )]*0+4p j (kr) , l l l l l PC(k r)"+ illK [½*l+(kK )½*l+(rL )]*0+4p f (kr) , l l p d(k!k@) = = f (kr)f (k@r)r2dr" j (kr)j (k@r)r2 dr" l l l l 2 k k@ o o Recoupling

P

P

C

A B

M

D

[[a*A+b*B+]*M+[c*C+d*D+]*N+]*K+"+ C D N [[a*A+c*C+]*R+[b*B+d*D+]*S+]*K+, RS R S K

C

A B

M

D

G

A

B

M

H

C D N "M K NK RK SK C

D N ,

R S

S

K

R

K

(B.33)

(B.34) (B.35) (B.36)

(B.37)

(B.38)

M. Danos, U. Fano / Physics Reports 304 (1998) 155–227

C DC D A B

C

A B

C " B

0

0

0

227

A A 0

CK 0 " . AK BK C C 0 B

(B.39)

References [1] E.P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959. [2] B.R. Judd, Lie groups for atomic shells, Phys. Rep. 285 (1997) 1—76. [3] (a) U. Fano, G. Racah, Irreducible Tensorial Sets, Academic Press, New York, 1959. (b) U. Fano, A.R.P. Rau, Symmetries in Quantum Physics, Academic Press, New York, 1996. [4] M. Danos, V. Gillet, Angular Momentum Calculus in Quantum Physics, World Scientific, Singapore, 1990. Extensive introduction to recoupling graphs with emphasis on bound-state problems including Fock space; it contains only cursory treatment of reactions. [5] A.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, Princeton, NJ, 1957. [6] A.P. Yutsis, I.B. Levinson, V.V. Vanagas, Matematicheskii Apparat Teorii Momenta Kolichesta Dvizheniya, Vil’nyus, 1960 (in Russian). [7] E. El-Baz, B. Castel, Graphical Methods of Spin Algebras in Atomic and Nuclear and Particle Physics, Marcel Dekker New York, 1972. [8] P.G. Sandars, Adv. Chem. Phys. 14 (1969) 365. [9] J.S. Briggs, Rev. Mod. Phys. 43 (1971) 189—230. [10] I. Lindgren, J. Morrison, Atomic Many-Body Theory, Springer, Berlin, 1982. [11] M.E. Rose, Elementary Theory of Angular Momentum, Wiley, New York, 1957. [12] M. Danos, Ann. Phys. (NY) 63 (1971) 319—334. [13] M. Danos, V. Gillet, M. Cauvin, Methods in Relativistic Nuclear Physics, North-Holland, Amsterdam, 1984. [14] M. Danos, Irreducible Density Matrices, National Bureau of Standards publication NBSIR 85-3270, November 1985. [15] U. Fano, Liouville representation of quantum mechanics with application to relaxation processes, in: E.R. Caianello, (Ed.), Lectures on the Many-Body Problem, vol. II, Academic Press, New York, 1964, p. 217. [16] U. Fano, Rend. Fis. Acc. Lincei, Ser. 9 (6) (1995) 123—130. [17] U. Fano, Rev. Mod. Phys. 29 (1957) 74—93. [18] H.T. Williams, R.R. Silbar, J. Comput. Phys. 99 (1992) 299—309. [19] M. Rotenberg, et al., The 3!j and 6!j Symbols, Cambridge Technical Press, MIT, Cambridge, 1960. [20] L.I. Schiff, Quantum Mechanics, McGraw-Hill, New York, 1955. [21] J. Kessler, Electron Spin, Springer, Berlin, 1976. [22] M. Jacob, G.C. Wick, Ann. Phys. (NY) 7 (1959) 404—428.

Physics Reports 304 (1998) 229—354

Driven quantum tunneling Milena Grifoni, Peter Ha¨nggi* Institut f u( r Physik Universita( t Augsburg, Universita( tstra}e 1, D-86135 Augsburg, Germany Received December 1997; editor: J. Eichler Contents 1. Introduction 2. Floquet approach 2.1. Floquet theory 2.2. General properties, spectral representations 2.3. Time-evolution operators for Floquet Hamiltonians 2.4. Generalized Floquet methods for nonperiodic driving 2.5. The (t, t@) formalism 3. Driven two-level systems 3.1. Two-state approximation to driven tunneling 3.2. Linearly polarized radiation fields 3.3. Tunneling in a cavity 3.4. Circularly polarized radiation fields 3.5. Curve crossing tunneling 3.6. Pulse-shaping strategy for control of quantum dynamics 4. Driven tight-binding models 5. Driven quantum wells 5.1. Driven tunneling current within Tien—Gordon theory 5.2. Floquet treatment for a strongly driven quantum well 6. Tunneling in driven bistable systems 6.1. Limits of slow and fast driving

232 233 233 7. 235 238 240 241 243 243 245 251 251 253

8.

9.

255 256 260 10. 261 263 267 269

6.2. Driven tunneling near a resonance 6.3. Coherent destruction of tunneling in bistable systems Sundry topics 7.1. Pulse-shaped controlled tunneling 7.2. Chaos-assisted driven tunneling 7.3. Even harmonic generation in driven double wells 7.4. General spin systems driven by circularly polarized radiation fields Driven dissipative tunneling 8.1. The harmonic thermal reservoir 8.2. The reduced density matrix 8.3. The environmental spectral density Floquet—Markov approach for weak dissipation 9.1. Generalized master equation for the reduced density operator 9.2. Floquet representation of the generalized master equation 9.3. Rotating-wave approximation Real-time path integral approach to driven tunneling 10.1. The influence-functional method 10.2. Numerical techniques: The quasiadiabatic propagator method

* Corresponding author. 0370-1573/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 2 2 - 2

271 272 274 275 276 279 281 283 283 285 286 287 287 288 289 290 290

292

DRIVEN QUANTUM TUNNELING

Milena GRIFONI, Peter HA® NGGI Institut f u¨ r Physik, Universita¨ t Augsburg, Universita¨ tstra}e 1, D-86135 Augsburg, Germany

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

M. Grifoni, P. Ha( nggi / Physics Reports 304 (1998) 229—354 11. The driven dissipative two-state system (general theory) 11.1. The driven spin—boson model 11.2. The reduced density matrix (RDM) of the driven spin—boson system 11.3. The case a"1/2 of Ohmic dissipation 11.4. Approximate treatments 11.5. Adiabatic perturbations and weak dissipation 12. The driven dissipative two-state system (applications) 12.1. Tunneling under ac-modulation of the bias energy 12.2. Pulse-shaped periodic driving 12.3. Dynamics under ac-modulation of the coupling energy

294 294 296 301 303 308 310 310 327

12.4. Dichotomous driving 12.5. The dissipative Landau—Zener— Stu¨ckelberg problem 13. The driven dissipative periodic tight-binding system 14. Dissipative tunneling in a driven double-well potential 14.1. The driven double—doublet system 14.2. Coherent tunneling and dissipation 14.3. Dynamical hysteresis and quantum stochastic resonance 15. Conclusions Note added in proof Acknowledgements References

231 331 333 335 340 340 341 343 345 346 347 347

329

Abstract A contemporary review on the behavior of driven tunneling in quantum systems is presented. Diverse phenomena, such as control of tunneling, higher harmonic generation, manipulation of the population dynamics and the interplay between the driven tunneling dynamics and dissipative effects are discussed. In the presence of strong driving fields or ultrafast processes, well-established approximations such as perturbation theory or the rotating wave approximation have to be abandoned. A variety of tools suitable for tackling the quantum dynamics of explicitly time-dependent Schro¨dinger equations are introduced. On the other hand, a real-time path integral approach to the dynamics of a tunneling particle embedded in a thermal environment turns out to be a powerful method to treat in a rigorous and systematic way the combined effects of dissipation and driving. A selection of applications taken from the fields of chemistry and physics are discussed, that relate to the control of chemical dynamics and quantum transport processes, and which all involve driven tunneling events. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 03.65.!w; 05.30.!d; 33.80.Be

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1. Introduction During the last few decades we could bear witness to an immense research activity, both in experimental and theoretical physics, as well as in chemistry, aimed at understanding the detailed dynamics of quantum systems that are exposed to strong time-dependent external fields. The quantum mechanics of explicitly time-dependent Hamiltonians generates a variety of novel phenomena that are not accessible within ordinary stationary quantum mechanics. In particular, the development of laser and maser systems opened the doorway for creation of novel effects in nonlinear quantum systems which interact with strong electromagnetic fields [1—7]. For example, an atom exposed indefinitely to an oscillating field eventually ionizes, whatever the values of the (angular) frequency and the intensity of the field. The rate at which the atom ionizes depends on both, the driving frequency and the intensity. Interestingly enough, in a pioneering paper by H. R. Reiss in 1970 [8], the seemingly paradoxical result was established that extremely strong field intensities lead to smaller transition probabilities than more modest intensities, i.e., one observes a declining yield with increasing intensity. This phenomenon of stabilization that is typical for above threshold ionization (ATI) is still actively discussed, both in experimental and theoretical groups [9,10]. Other activities that are in the limelight of current topical research relate to the active control of quantum processes; e.g. the selective control of reaction yields of products in chemical reactions by use of a sequence of properly designed coherent light pulses [11—13]. Our prime concern here will focus on the tunneling dynamics of time-dependently driven nonlinear quantum systems. Such systems exhibit an interplay of three characteristic components, (i) nonlinearity, (ii) nonequilibrium behavior (as a result of the time-dependent driving), and (iii) quantum tunneling, with the latter providing a paradigm for quantum coherence phenomena. By now, the physics of driven quantum tunneling has generated widespread interest in many scientific communities [1—7] and, moreover, gave rise to a variety of novel phenomena and effects. As such, the field of driven tunneling has nucleated into a whole new discipline. Historically, first precursors of driven barrier tunneling date back to the experimentally observed photon-assisted-tunneling (PAT) events in 1962 in the Al—Al O —In superconductor—insula2 3 tor—superconductor hybrid structure by Dayem and Martin [14]. A clear-cut, simple theoretical explanation for the step-like structure in the averaged voltage—tunneling current characteristics was put forward soon afterwards by Tien and Gordon [15] in 1963, who introduced the physics of driving-induced sidechannels for tunneling across a uniformly, periodically modulated barrier. The phenomenon that the quantum transmission can be quenched in arrays of periodically arranged barriers, e.g. semiconductor superlattices, leading to such effects as dynamic localization or absolute negative conductance have theoretically been described over twenty years ago [16—18]; but these have been verified experimentally only recently [19—22]. The role of time-dependent driving on the coherent tunneling between two locally stable wells [23] has only recently been elaborated [24]. As an intriguing result one finds that an appropriately designed coherent cw-drive can bring coherent tunneling to an almost complete standstill, now known as coherent destruction of tunneling (CDT) [24—26]. This driving induced phenomenon in turn yields several other new quantum effects such as low frequency radiation and/or intense, nonperturbative even harmonic generation in symmetric metastable systems that possess an inversion symmetry [27—29].

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233

We shall approach this complexity of driven quantum tunneling with a sequence of sections. In the first half of seven sections we elucidate the physics of various novel tunneling phenomena in quantum tunneling systems that are exposed to strong time-varying fields. These systems are described by an explicitly time-dependent Hamiltonian. Thus, solving the time-dependent Schro¨dinger equation necessitates the development of novel analytic and computational schemes which account for the breaking of time translation invariance of the quantum dynamics in a nonperturbative manner. Beginning with Section 8 we elaborate on the effect of weak, or even strong dissipation, on the coherent tunneling dynamics of driven systems. This extension of quantum dissipation [30—35] to driven quantum systems constitutes a nontrivial task: Now, the bath modes couple resonantly to differences of quasienergies rather than to unperturbed energy differences. The influence of quantum dissipation to driven tunneling is developed theoretically in Sections 8—11, and applied to various phenomena in the remaining Sections 12—14. The authors made an attempt to comprise in this review many, although necessarily not all important developments and applications of driven tunneling. In doing so, this review became rather comprehensive. As an inevitable consequence, the authors realize that not all readers will wish to digest the present review in its entirety. We trust, however, that a reader is able to choose from the many methods and applications covered in the numerous sections which he is interested in. There is the consistent underlying theme of driven quantum tunneling that runs through all sections, but nevertheless, each section can be considered to some extent as self-contained. In this spirit, we hope that the readers will be able to enjoy reading from the selected fascinating developments that characterize driven tunneling, and moreover will become invigorated doing own research in this field.

2. Floquet approach 2.1. Floquet theory In presence of intense fields interacting with the system it is well known [37—39] that the semiclassical theory (i.e., treating the field as a classical field) provides results that are equivalent to those obtained from a fully quantized theory, whenever fluctuations in the photon number (which, for example, are of importance for spontaneous radiation processes) can safely be neglected. We shall be interested primarily in the investigation of quantum systems with their Hamiltonian being a periodic function in time, i.e., H(t)"H(t#T) ,

(1)

with T being the period of the perturbation. The symmetry of the Hamiltonian under discrete time translations, tPt#T, enables the use of the Floquet formalism [36]. This formalism is the appropriate vehicle to study strongly driven periodic quantum systems: Not only does it respect the periodicity of the perturbation at all levels of approximation, but its use intrinsically avoids also the occurrence of so-called secular terms (i.e., terms that are linear or not periodic in the time variable). The latter characteristically occur in the application of conventional Rayleigh—Schro¨dinger timedependent perturbation theory. The Schro¨dinger equation for the quantum system with coordinate

M. Grifoni, P. Ha( nggi / Physics Reports 304 (1998) 229—354

234

q may be written as (H(q, t)!i+ /t)W(q, t)"0 .

(2)

For the sake of simplicity only, we restrict ourselves here to the one-dimensional case. With H(q, t)"H (q)#H (q, t), H (q, t)"H (q, t#T) , (3) 0 %95 %95 %95 the unperturbed Hamiltonian H (q) is assumed to possess a complete orthonormal set of eigen0 functions Mu (q)N with corresponding eigenvalues ME N. According to the Floquet theorem, there n n exist solutions to Eq. (2) that have the form (so-called Floquet-state solution) [36] W (q, t)"exp(!ie t/+)U (q, t) , (4) a a a where U (q, t) is periodic in time, i.e., it is a Floquet mode obeying a U (q, t)"U (q, t#T) . (5) a a Here, e is a real-valued energy function, being unique up to multiples of +X, X"2p/T. It is a termed the Floquet characteristic exponent, or the quasienergy [37—39]. The term quasienergy reflects the formal analogy with the quasimomentum k, characterizing the Bloch eigenstates in a periodic solid. Upon substituting Eq. (4) into Eq. (2) one obtains the eigenvalue equation for the quasienergy e , i.e., with the Hermitian operator a H(q, t),H(q, t)!i+ /t , (6) one finds that H(q, t)U (q, t)"e U (q, t) . a a a We immediately notice that the Floquet modes

(7)

U (q, t)"U (q, t) exp(inXt),U (q, t) (8) a{ a an with n being an integer number n"0,$1,$2,2 yield the identical solution to that in Eq. (4), but with the shifted quasienergy e Pe "e #n+X,e . (9) a a{ a an Hence, the index a corresponds to a whole class of solutions indexed by a@"(a, n), n"0,$1,$2,2 The eigenvalues Me N therefore can be mapped into a first Brillouin a zone, obeying !+X/24e(+X/2. For the Hermitian operator H(q, t) it is convenient to introduce the composite Hilbert space R?T made up of the Hilbert space R of square integrable functions on configuration space and the space T of functions which are periodic in t with period T"2p/X [40]. For the spatial part the inner product for two square integrable functions f (q) and g(q) is defined by

P

S f DgT " : dq f *(q)g(q) ,

(10)

M. Grifoni, P. Ha( nggi / Physics Reports 304 (1998) 229—354

yielding with f(q)"u (q) and g(q)"u (q) n m Su Du T"d . n m n,m The temporal part is spanned by the orthonormal set of StDnT"exp(inXt), n"0,$1,$2,2, and the inner product in T reads

P

1 (m, n) " : T

T

dt(e*mXt)*e*nXt"d . n,m

235

(11) Fourier

vectors

(12)

0 Thus, the eigenvectors of H obey the orthonormality condition in the composite Hilbert space R?T, i.e.,

P P

1 |U DU } " : a{ b{ T

T

dt

=

dq U* (q, t)U (q, t)"d "d d , a{ b{ a{,b{ a,b n,m

(13)

~= 0 and form a complete set in R?T,

(14) + + U* (q, t)U (q@, t@)"d(q!q@)d(t!t@) . an an a n Note that in Eq. (14) we must extend the sum over all Brillouin zones, i.e., over all the representatives n in a class, cf. Eq. (9). For fixed equal time t"t@, the Floquet modes of the first Brillouin zone U (q, t) form a complete set in R, i.e., a0 + U* (q, t)U (q@, t)"d(q!q@) . (15) a a a Clearly, with t@Ot#mT"t (mod T), the functions MU* (q, t), U (q@, t@)N do not form an orthonora a mal set in R. 2.2. General properties, spectral representations With a monochromatic perturbation H (q, t)"!Sq sin(Xt#/) , (16) %95 the quasienergy e is a function of the parameters S and X, but does not depend on the arbitrary, a but fixed phase /. This is so because a shift of the time origin t "0Pt "!//X will lift 0 0 a dependence of e on / in the quasienergy eigenvalue equation in Eq. (7). In contrast, the a time-dependent Floquet function W (q, t) depends, at fixed time t, on the phase. The quasienergy a eigenvalue equation in Eq. (7) has the form of the time-independent Schro¨dinger equation in the composite Hilbert space R?T. This feature reveals the great advantage of the Floquet formalism: It is now straightforward to use all theorems characteristic for time-independent Schro¨dinger theory for the periodically driven quantum dynamics, such as the Rayleigh—Ritz variation principle for stationary perturbation theory, the von-Neumann-Wigner degeneracy theorem, or the Hellmann— Feynman theorem, etc.

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236

With H(t) being a time-dependent function, the energy E is no longer conserved. Instead, let us consider the averaged energy in a Floquet state W (q, t). This quantity reads a T 1 HM , . (17) dtSW (t)DH(t)DW (t)T"e # U i+ U a T a t a a a a 0 If we invoke a Fourier expansion of the time-periodic Floquet function U (q, t)"+ c (q) exp(!ikXt), + :dqDc (q)D2"1"+ Sc Dc T, Eq. (17) can be recast as a sum over a k k k k k k k k, i.e.,

P

TT K K UU

= = HM "e # + +kXSc Dc T" + (e #+kX)Sc Dc T . (18) a a k k a k k k/~= k/~= Hence, HM can be looked upon as the energy accumulated in each harmonic mode of a W (q, t)"exp(!ie t/+)U (q, t), and averaged with respect to the weight of each of these harmonics. a a a Moreover, one can apply the Hellmann—Feynman theorem, i.e.,

TT K

de (X) a " dX

U (X) a

K UU

H(X) U (X) a X

.

(19)

Setting q"Xt and H(q, q)"H(q, q)!i+X/q, one finds

A B

(20)

e (S, X) HM "e (S, X)!X a . a a X

(21)

1 H "!i+ "!i+ , q X t X q and consequently for Eq. (17) [42]

This connection between the averaged energy HM and the quasienergy e in addition provides a a a relationship between the dynamical phase sa of a Floquet state W , i.e., [41] D a 1 T 1 sa "! dtSW DH(t)DW T"! THM , (22) D a a a + + 0 and a nonadiabatic (i.e. generalized) Berry phase sa , where B s"sa #sa "!e T/+ ; (23) D B a thus yielding

P

TT K K UU

sa "iT B

W W a t a

2p e (S, X) a "! . + X

(24)

The part sa of the overall phase s describes an intrinsic property of a cyclic change of parameters in B the periodic Hamiltonian H(t)"H(t#T) that explicitly does not depend on the dynamical time interval of cyclic propagation T.

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237

Next we discuss general features of quasienergies and Floquet modes with respect to their frequency and field dependence. As mentioned before, if e "e possesses the Floquet mode a a0 U (q, t), the modes a0 U PU "U (q, t) exp(iXkt), k"0,$1, 2,2 , a0 ak a0

(25)

are also solutions with quasienergies e "e #+kX , ak a0

(26)

yielding identical physical states, W (q, t)"exp(!ie t/+)U (q, t)"W (q, t) . a0 a0 a0 ak

(27)

For an interaction SP0 that is switched off adiabatically, the Floquet modes and the quasienergies obey U (q, t) S?0 P U0ak(q, t)"ua(q) exp(iXkt) ak

(28)

e (S, X) S?0 P e0ak"Ea#k+X , ak

(29)

and

with Mu , E N denoting the eigenfunctions and eigenvalues of the time-independent part H of the a a 0 Hamiltonian in Eq. (3). Thus, when S"0, the quasienergies depend linearly on frequency so that at some frequency values different levels e0 intersect. When SO0, the interaction operator mixes ak these levels, depending on the symmetry properties of the Hamiltonian. Given a symmetry for H(q, t), the Floquet eigenvalues e can be separated into symmetry classes: Levels in each class mix ak with each other, but do not interact with levels of other classes. Let us consider levels e0 and e0 of an bk the same group at resonances, i.e., E #n+X "E #k+X , a 3%4 b 3%4

(30)

with X being the frequency of an (unperturbed) resonance. According to the von3%4 Neumann—Wigner theorem [43], these levels of the same class will no longer intersect for finite SO0. In other words, these levels develop into avoided crossings, cf. Fig. 1a. If the levels obeying Eq. (30) belong to a different class, for example to different generalized parity states, the quasienergies at finite SO0 exhibit exact crossings; cf. Fig. 1b. These considerations, conducted without any approximation, leading to avoided vs. exact crossings, determine many interesting and novel features of driven quantum systems. Some interesting consequences follow immediately from the structure in Fig. 1: Starting out from a stationary state W(q, t)"u (q) exp(!iE t/+) the smooth adiabatic switch-on of the interaction 1 1 with X(X (X'X ) will transfer the system into a quasienergy state W [44—48]. Upon 3%4 3%4 10 increasing (decreasing) adiabatically the frequency to a value X'X (X(X ) and again 3%4 3%4 smoothly switching off the perturbation, the system generally jumps to a different state

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Fig. 1. Quasienergy dependence on the frequency X of a monochromatic electric-dipole perturbation near the unperturbed resonance X between two levels. The dashed lines correspond to quasienergies for SP0. In panel (a), we depict an 3%4 avoided crossing for two levels belonging to the same symmetry related class. Note that with finite S the dotted parts belong to the Floquet mode U , while the solid parts belong to the state U . In panel (b), we depict an exact crossing 2m 1n between two members of quasienergies belonging to different symmetry-related classes. With SO0, the location of the resonance generally undergoes a shift d"X (SO0)!X (S"0) (so-called Bloch—Siegert shift [66]) that depends on 3%4 3%4 the intensity S. Only for SP0 does the resonance frequency coincide with the unperturbed resonance X . 3%4

W(q, t)"u (q) exp(!iE t/+). For example, this phenomenon is known in NMR as spin exchange; 2 2 it relates to a rapid (as compared to relaxation processes) adiabatic crossing of the resonance. Moreover, as seen in Fig. 1a, the quasienergy e and Floquet mode U as a function of frequency 2k 2k exhibit jump discontinuities at the frequencies of the unperturbed resonance, i.e., the change of energy between the two parts of the solid lines (or dashed lines, respectively). 2.3. Time-evolution operators for Floquet Hamiltonians The time propagator º(t, t ), defined by 0 (31) DW(t)T"º(t, t )DW(t )T, º(t , t )"1 , 0 0 0 0 assumes special properties when H(t)"H(t#T) is periodic. In particular, the propagator over a full period º(T, 0) can be used to construct a discrete quantum map, propagating an initial state over long multiples of the fundamental period by observing that º(nT, 0)"[º(T, 0)]n .

(32)

This important relation follows readily from the periodicity of H(t) and its definition. Namely, we find with t "0 (Á denotes time ordering of operators) 0 i nT i n kT U(nT,0)"Á exp ! dt H(t) "Á exp ! + dt H(t) , + + 0 k/1 (k~1)T which with H(t)"H(t#T) simplifies to

C P C

D

P

C

D

P

D

C P

D

T i n n i T dt H(t) "Á < exp ! dt H(t) . º(nT, 0)"Á exp ! + + + 0 k/1 k/1 0

(33)

M. Grifoni, P. Ha( nggi / Physics Reports 304 (1998) 229—354

239

Because the terms over a full period are equal, they do commute. Hence, the time-ordering operator can be moved in front of a single term, yielding

C P

D

i T n dt H(t) "[º(T, 0)]n . º(nT, 0)" < Á exp ! + 0 k/1 Likewise, one can show that with t "0 the following relation holds: 0 º(t#T, T)"º(t, 0) ,

(34)

(35)

which implies that º(t#T, 0)"º(t, 0)º(T, 0) .

(36)

Note that º(t, 0) does not commute with º(T, 0), except at times t"nT, so that Eq. (36) with the right-hand-side product reversed does not hold. A most important feature of Eqs. (30)—(36) is that the knowledge of the propagator over a fundamental period T"2p/X provides all the information needed to study the long-time dynamics of periodically driven quantum systems. That is, upon a diagonalization with an unitary transformation S, Ssº(T, 0)S"exp(!iD) , with D a diagonal matrix, composed of the eigenphases Me TN, one obtains a º(nT, 0)"[º(T, 0)]n"S exp(!inD)Ss .

(37)

(38)

This relation can be used to propagate any initial state DW(0)T"+ c DU (0)T, c "SU (0)DW(0)T , (39) a a a a a in a stroboscopic manner. This procedure generates a discrete quantum map. With W (q, t"0)"U (q, t"0), its time evolution follows from Eq. (4) as a a W(q, t)"+ c exp(!ie t/+)U (q, t) . a a a a With W(q, t)"SqDº(t, 0)DW(0)T, a spectral representation for the propagator, i.e.,

(40)

K(q, t; q , 0) " : SqDº(t, 0)Dq T , 0 0 follows from Eq. (41) with W(q, 0)"d(q!q ) as 0

(41)

K(q, t; q , 0)"+ exp(!ie t/+)U (q, t)U* (q , 0) . 0 a a a 0 a This relation is readily generalized to arbitrary propagation times t't@, yielding

(42)

K(q, t; q@, t@)"+ exp(!ie (t!t@)/+)U (q, t)U* (q@, t@) . a a a a

(43)

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240

This intriguing result generalizes the familiar form of time-independent propagators to timeperiodic ones. Note again, however, that the role of the stationary eigenfunction u (q) is taken over a by the Floquet mode U (q, t), being orthonormal only at equal times t"t@ mod T. a 2.4. Generalized Floquet methods for nonperiodic driving In the previous subsections we have restricted ourselves to pure harmonic interactions. In many physical applications, e.g. see in [7,11—13], however, the time-dependent perturbation has an arbitrary, for example, pulse-like form that acts over a limited time regime only. Clearly, in these cases the Floquet theorem cannot readily be applied. This feature forces one to look for a generalization of the quasienergy concept. Before we start doing so, we note that the Floquet eigenvalues e in Eq. (9) can also be obtained as the ordinary Schro¨dinger eigenvalues within a twoan dimensional formulation of the time-periodic Hamiltonian in Eq. (3). Setting Xt"h, Eq. (3) is recast as H(t)"H (q, p)#H (q, h(t)) . (44) 0 %95 With hQ "X, one constructs the enlarged Hamiltonian HI (q, p; h, p )"H (q, p)#H (q, h)#Xp , h 0 %95 h where p is the canonically conjugate momentum, obeying h hQ "HI /p "X . (45) h The quantum mechanics of HI acts on the Hilbert space of square-integrable functions on the extended space of the q-variable and the square-integrable periodic functions on the compact space of the unit circle h"h #Xt (periodic boundary conditions for h). With H (q, t) given by Eq. (16), 0 %95 the Floquet modes U (q, h) and the quasienergies e are the eigenfunctions and eigenvalues of the ak ak two-dimensional stationary Schro¨dinger equation, i.e., with [h, p ]"i+, h !+2 2 #» (q)!Sq sin(h#/)!i+X U (q, h)"e U (q, h) . (46) 0 ak ak 2m q2 h ak

G

H

This procedure opens a door to treat more general, polychromatic perturbations composed of generally incommensurate frequencies. For example, a quasiperiodic perturbation with two incommensurate frequencies X and X , e.g. 1 2 H (q, t)"!qS sin(X t)!qF sin(X t) , (47) %95 1 2 can be enlarged into a six-dimensional phase space (q, p ; h , p 1; h , p 2), with Mh "X t; h "X tN 1 1 2 2 q 1 h 2 h defined on a torus. The quantization of the corresponding momentum terms yield a stationary Schro¨dinger equation in the three variables (x, h , h ) with a corresponding Hamiltonian operator 1 2 HI given by HI "H(q, h , h )!i+X (/h )!i+X (/h ) (48) 1 2 1 1 2 2 with eigenvalues Me 1 2N and generalized stationary wavefunctions given by the generalized a,k ,k Floquet modes U 1 2(q, h , h )"U 1 2(q; h #2p; h #2p). An application of this extended twoa,k ,k 1 2 a,k ,k 1 2 frequency Floquet theory could be invoked to study the problem of bichromatic field control of tunneling in a double well [49].

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There are important qualitative differences between the periodic and the quasi-periodic forcing cases. Let us consider a Hilbert space of finite dimension, such as e.g., a spin 1 system or a N-level 2 system in a radiation field. For a periodic time-dependent driving force the corresponding Floquet operator is a finite-dimensional unitary matrix. Thus, the corresponding quasienergy spectrum is always pure point. This, however, is no longer the case for quasi-periodic forcing [50]. We note that a point spectrum implies stable quasiperiodic dynamics (i.e., almost-periodic evolution); in contrast, a continuous spectrum signals an instability (leading to an unstable chaotic behavior) with typically decaying asymptotic correlations. With two incommensurate driving frequencies X and 1 X the spectrum of the generalized Floquet operator in Eq. (48) can be pure point, absolutely 2 continuous or also purely singular continuous [51—53]. A general perturbation, such as a time-dependent laser-pulse interaction, consists (via Fourierintegral representation) of an infinite number of frequencies, so that the above embedding ceases to be of practical use. The general time-dependent Schro¨dinger equation i+(/t)W(q, t)"H(q, t)W(q, t) ,

(49)

with the initial state given by W(q, t )"W (q), can be solved by numerical means, by a great variety 0 0 of methods [25,54—56]. All these methods must involve efficient numerical algorithms to calculate the time-ordered propagation operator º(t, t@). Generalizing the idea of Shirley [37] and Sambe [40] for time-periodic Hamiltonians, it is possible to introduce a Hilbert space for general time-dependent Hamiltonians in which the Schro¨dinger equation becomes time independent. Following the reasoning by Howland [57], we introduce the reader to the so-called (t, t@)-formalism [58]. 2.5. The (t, t@)-formalism The time-dependent solution W(q, t)"º(t, t )W(q, t ) 0 0 for the explicitly time-dependent Schro¨dinger equation in Eq. (49) can be obtained as

(50)

W(q, t)"W(q, t@, t)D , t{/t where

(51)

W(q, t@, t)"exp[(!i/+)H(q, t@)(t!t )]W(q, t@, t ) . 0 0 H(q, t@) is the generalized Floquet operator

(52)

H(q, t@)"H(q, t@)!i+ /t@ .

(53)

The time t@ acts as a time coordinate in the generalized Hilbert space of square-integrable functions of q and t@, where a box normalization of length T is used for t@ (0(t@(T). For two functions / (q, t), / (q, t) the inner, or scalar product reads a b 1 T = |/ D/ }" dt@ dq /* (q, t@)/ (q, t@) . (54) a b a b T 0 ~=

P P

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The proof for Eq. (51) can readily be given as follows [58]: Note that from Eq. (52) i+ W(q, t@, t)"H(q, t@) exp[!iH(q, t@)(t!t )/+]W(q, t@, t ) 0 0 t "!i+

W(q, t@, t)#H(q, t@)W(q, t@, t) . t@

(55)

Hence,

A

i+

B

# W(q, t@, t)"H(q, t@)W(q, t@, t) . t t@

(56)

Since we are interested in t@ only on the contour t@"t, where t@/t"1, one therefore finds that

K

W(q, t@, t) t@

K

W(q, t@, t) W(q, t) , (57) # " t t t{/t t{/t which with Eq. (56) for t"t@ consequently proves the assertion in Eq. (51). Note that a long time propagation now requires the use of a large box, i.e., T must be chosen sufficiently large. If we are not interested in the very-long-time propagation, the perturbation of finite duration can be embedded into a box of finite length T, and periodically continued. This so constructed perturbation now implies a time-periodic Hamiltonian, so that we require timeperiodic boundary conditions W(q, t@, t)"W(q, t@#T, t) ,

(58)

with 04t@4T, and the physical solution is obtained when t@"t mod T .

(59)

Stationary solutions of Eq. (55) thus reduce to the Floquet states, as found before, namely W (q, t@, t)"exp(!ie t/+)U (q, t@) , (60) a a a with U (q, t@)"U (q, t@#T), and t@"t mod T. We remark that although W(q, t@, t), W (q, t@, t) are a a a periodic in t@, the solution W(q, t)"W(q, t@"t, t) is generally not time periodic. The (t, t@)-method hence avoids the need to introduce the generally nasty time-ordering procedure. Expressed differently, the step-by-step integration that characterizes the time-dependent approaches is not necessary when formulated in the above generalized Hilbert space where H(q, t@) effectively becomes time-independent, with t@ acting as coordinate. Formally, the result in Eq. (55) can be looked upon as quantizing the new Hamiltonian HK , defined by HK (q, p; E, t@)"H(q, p, t@)!E ,

(61)

using for the operator EPEK the canonical quantization rule EK "i+ /t; with [EK , tK ]"i+ and tK /(t)"t/(t). This formulation of the time-dependent problem in Eq. (49) within the auxiliary t@ coordinate is particularly useful for evaluating the state-to-state transition probabilities in pulsesequence-driven quantum systems [11,13,58].

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We conclude this section by commenting on exactly solvable driven quantum systems. In contrast to time-independent quantum theory, such exactly solvable quantum systems with time-dependent potentials are extremely rare. One such class of systems are (multidimensional) systems with at most quadratic interactions between momentum and coordinate operators, e.g. the driven free particle [59,60], the parametrically driven harmonic oscillator [61,62], including generalizations that account for quantum dissipation via bilinear coupling to a harmonic bath [63]. Another class entails driven systems operating in a finite or countable Hilbert space, such as some spin system, cf. Sections 3.4 and 7.4, or some tight-binding models, cf. Section 4.

3. Driven two-level systems 3.1. Two-state approximation to driven tunneling In this section we shall investigate the dynamics of driven two-level-systems (TLS), i.e., of quantum systems whose Hilbert space can be effectively restricted to a two-dimensional space. The most natural example is that of a particle of total angular momentum J"+/2, as for example a silver atom in the ground state. The magnetic moment of the particle is l"1+cr, where 2 c is the gyromagnetic ratio and r"(p , p , p ) are the Pauli spin matrices.1 When the particle is x y z placed in a time-dependent magnetic field B(t) the time-dependent magnetic Hamiltonian H thus M reads H (t)"!l ) B"!1+c(p B (t)#p B (t)#p B (t)) . (62) M 2 z z x x y z For a generic quantum system one considers the case in which only a finite number of quantum levels strongly interact under the influence of the time-dependent interaction. This means that a truncation to a multi-level quantum system in which only a finite number of quantum states strongly interact is adequate. In particular, the truncation to two relevant levels only is of enormous practical importance in nuclear magnetic resonance, quantum optics, or in low temperature glassy systems, to name only a few. Setting +D "E !E , this truncation in the energy 0 2 1 representation of the ground state D1T and excited state D2T is in terms of the Pauli spin matrices p , p and p given by z x y H "!1+D p , (63) E 2 0 z where the subscript “E” indicates that this is the energy representation. Very frequently in the chemical literature the effects of a static electric field E coupling to the 0 transition dipole moment k between the two levels of a molecule are investigated. Setting

1 We use for the Pauli matrices the representation

A B

A

B

A

B

0 1 0 !i 1 0 p" , p" , p" . x y z 1 0 i 0 0 !1 That is, p is chosen to be diagonal. z

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!+e /2"E k, we end up with the TLS Hamiltonian 0 0 H "!1+(D p !e p ) , (64) 015 2 0 z 0 x which is widely used to investigate optical properties. Diverse physical or chemical systems can be described by the Hamiltonian (63). In particular, a very important class concerns those systems moving in an effective double well potential in the case in which only the lowest energy doublet is occupied. These systems are the simplest one exhibiting quantum interference effects: a system which is initially localized in one well will oscillate clockwise between the eigenstates in the left and right well due to quantum mechanical tunneling. A well-known example is the ammonia molecule NH : The nitrogen atom, much heavier than its 3 partners is motionless; the hydrogen atoms form a rigid equilateral triangle whose axis passes through the nitrogen atom. The potential energy is thus only a function of the distance q between the nitrogen atom and the plane defined by the hydrogen atoms. Hence, the potential energy of the system has two equivalent configurations of stable equilibrium, which are symmetric respect to the q"0 plane. Quantum mechanically the hydrogens can tunnel back and forth between the two potential minima [23]. To be more explicit, we characterize the double well by the barrier height E , by the separation B +u ;E of the first excited state from the ground state in either well, and by an intrinsic 0 B asymmetry energy +e ;+u between the ground states in the two wells. At zero temperature, the 0 0 system will be effectively restricted to the two-dimensional Hilbert space spanned by the two ground states. Taking into account the possibility of tunneling between the two wells, the TLS is then governed by the pseudospin Hamiltonian (tunneling or localized representation) H "!1+(D p #e p ) . (65) TLS 2 0 x 0 z This is the representation which is convenient to investigate tunneling problems. Here the basis is chosen such that the states DRT (right) and D¸T (left) are eigenstates of p with eigenvalues 1 and !1, z respectively. Hence, p is related to the position operator q in this discretized representation by z q"p d/2, with d being the spatial distance between the two localized states. The interaction energy z +D is the energy splitting of a symmetric (i.e., e "0) TLS due to tunneling. 0 0 We observe now that the unitary transformation R(h)"exp(!ihp /2), !tan h"D /e , (66) y 0 0 tranforms H as TLS RH R~1"!1Ep , E"+JD2#e2 , (67) TLS 2 z 0 0 which is of the same form of Eq. (63) with energy splitting E. Note that if the static asymmetry e originates from an externally applied dc-field it may be convenient to apply the transformation 0 RI "exp(ipp /4) to the Hamiltonian H . Such a procedure yields y TLS RI H RI ~1"!1+(D p !e p )"H . (68) TLS 2 0 z 0 x 015 Hence, the two transformations R and RI coincide when e "0. 0 At this point it is indeed interesting to recall that for the two-level problem, the corresponding density matrix o is a 2]2 matrix with matrix elements o (where p, p@"$1 are the eigenvalues p,p{

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of p ), whose knowledge enables the evaluation of the expectation value of any observable related z to the TLS. In particular, because any 2]2 matrix can be written as a linear combination of the Pauli matrices and of the unit matrix I, one has o(t)"I/2#+ Sp T p /2, where i/x,y,z i t i Sp T " : TrMo(t)p N (with Tr denoting the trace). This yields i t i Sp T "o (t)!o (t), Sp T "o (t)#o (t), z t 1,1 ~1,~1 x t 1,~1 ~1,1 Sp T "i[o (t)!o (t)] , (69) y t 1,~1 ~1,1 where we denoted by o , and o the diagonal matrix elements of the density matrix, and by ~1,~1 1,1 o ,o the off-diagonal elements (so-called coherences). We have then for the expectation ~1,1 1,~1 values P(t) of the position operator (70) P(t) " : TrMo (t)p N" cos h TrMo (t)p N# sin h TrMo (t)p N , TLS z E z E x where o2 is the density matrix in the corresponding representation. Hence, in the localized basis Sp T describes the population difference of the localized states. It provides direct informations on z t the dynamics of the TLS. In the same way, one obtains for the difference N(t) in the population of the energy levels N(t) " : TrMo (t)p N" cos h TrMo (t)p N! sin h TrMo (t)p N . (71) E z TLS z TLS x In particular, N(t) coincides with TrMo (t)p N for the case of a symmetric (e "0), undriven TLS. TLS x 0 In the absence of driving, the density matrix can be evaluated straightforwardly. A TLS prepared in a localized state at an initial time t"t will exhibit an oscillatory, undamped, behavior due to 0 quantum interference effects. One finds in the localized basis +2e2 +2D2 0 cos [E(t!t )/+] , Sp T " 0# z t 0 E2 E2

(72)

e D Sp T "+2 0 0M1! cos [E(t!t )/+]N , x t 0 E2

(73)

+D Sp T " 0 sin [E(t!t )/+] . 0 y t E

(74)

Hence, a TLS prepared in a localized state at an initial time t will undergo quantum coherent 0 oscillations between its localized states with frequency E"+(D2#e2)1@2, corresponding to the 0 0 energy splitting of the isolated TLS. Correspondingly, as follows from Eq. (71), the difference population of the energy levels is constant for an initially localized state, and given by N(t)"+e /E. 0 In the following, we shall investigate the evolution of some observables related to the driven TLS in the presence of different kinds of driving forces. 3.2. Linearly polarized radiation fields Let us next consider, for example, the interaction of the TLS with a classical electric field E(t). We end up with the time-dependent Hamiltonian H (t)"!1+D p !kE sin (Xt#/)p , E 2 0 z x

(75)

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with k,S2DqD1T being the transition dipole moment, and D1T, D2T the energy eigenstates. Here we have used a scalar approximation of the electric field E along the q-direction. The linearly polarized field in Eq. (75) can, with +eL /2,kE, be regarded as a superposition of left- and right-circularly polarized radiation. Namely, setting /"p/2, we have eL cos Xt"eL [exp(!iXt)#exp(iXt)]/2 .

(76)

The effects of the two contributions exp($iXt) can be better understood if one considers the quantized version, cf. Eq. (103) below, of the time-dependent Hamiltonian (75). Let us next introduce the product states Di, nT " : DiTDnT, with (i"1, 2) and with DnT being a Fock state of the quantized electric field with occupation number n. For the absorption process D1, nTPD2, n!1T, the term eL exp(!iXt) supplies the energy +X to the system. It corresponds to the rotating-wave (RW) term, while the term eL exp(iXt) is called the anti-rotating-wave term. This anti-RW term removes the energy +X from the system, i.e., D1, nTPD2, n#1T, and is thus energy nonconserving. Likewise, the term exp (iXt) governs the process of emission D2, nTPD1, n#1T, i.e., it is a RW term, while the second-order process D2, nTPD1, n!1T is again an energy-nonconserving anti-RW term. Another example of a driven TLS can be obtained in the context of electron-spin resonance (ESR), nuclear magnetic-spin resonance (NMR) or atomic-beam spectroscopy. In this case the TLS is a particle of total angular momentum J"+/2 placed in both a static magnetic field B in the 0 z-direction, and a time-dependent oscillating magnetic field 2B cos (Xt) in x-direction. The mag1 netic moment of the particle is k"cJ, where c is the gyromagnetic ratio. The Hamiltonian H for the particle in the time-dependent magnetic field thus reads, cf. Eq. (62), NMR H (t)"!l ) B"!1+cp B !+cp B cos (Xt) . NMR 2 z 0 x 1

(77)

D "cB , 0 0

(78)

With kE"+eL /2"+cB , 1

this Hamiltonian coincides with the laser-driven TLS in Eq. (75) with /"n/2. Finally, if one considers the effects of an ac electric field which is modulating the asymmetry energy of the underlying bistable potential, one ends up, in the tunneling representation, with H (t)"!1+(e #eL cos Xt)p !1+D p . TLS 2 0 z 2 0 x

(79)

Note that the same form of the Hamiltonian (79) is obtained if one considers the combined effect of a dc—ac electric field on a symmetric TLS. 3.2.1. Time evolution of the population of the energy levels The driven two-level system has a long history, with recent reviews being available [64]. A pioneering piece of work must be attributed to Rabi [65], who considered the two-level system in a circularly polarized magnetic field, a problem that he could solve exactly, see below in Section 3.4. He thereby elucidated how to measure simultaneously both the sign as well as the magnitude of magnetic moments. However, as Bloch and Siegert experienced soon after [66], this problem is no longer exactly solvable in analytical closed form when the field is linearly polarized,

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rather than circularly. Let us start with the linearly polarized case as given by the driven Hamiltonian (75), (76). We set for the wave function in the energy basis D1T, D2T,

AB

AB

1 0 #c (t) exp(!iD t/2) , W(t)"c (t) exp(iD t/2) 2 0 1 0 0 1

(80)

where Dc (t)D2#Dc (t)D2"1. The Schro¨dinger equation thus has the form 1 2 eL cos Xt c (t) exp(iD t/2) d c1(t) exp(iD0t/2) + D0 1 0 i+ "! , dt c (t) exp(!iD t/2) 2 eL cos Xt !D c (t) exp(!iD t/2) 2 0 0 2 0 and provides two coupled first-order equations for the amplitudes,

A

B

A

BA

B

(81)

dc /dt"1ieL [exp[i(X!D )t]#exp[!i(X#D )t]]c , 0 0 2 1 4 (82) dc /dt"1ieL [exp[!i(X!D )t]#exp[i(X#D )t]]c . 2 4 0 0 1 With an additional differentiation with respect to time, and substituting cR from the second 2 equation, we readily find that c (t) obeys a linear second order ordinary differential equation with 1 time periodic (T"2p/X) coefficients (so termed Hill equation). Clearly, such equations are generally not solvable in analytical closed form. Hence, although the problem is simple, the job of finding an analytical solution presents a hard task! To make progress, one usually invokes, at this stage, the so-called rotating-wave approximation (RWA), assuming that X is close to D (near 0 resonance), and eL not very large. Then the anti-rotating-wave term exp[i(X#D )t] is rapidly 0 varying, as compared to the slowly varying rotating-wave term exp [!i(X!D )t]. It thus cannot 0 transfer much population from state D1T to state D2T. Neglecting this anti-rotating contribution one finds in terms of the detuning parameter d,X!D (rotating-wave approximation), 0 dc eL dc eL 1"i exp(idt)c , 2"i exp(!idt)c . (83) 2 1 dt 4 dt 4 From Eq. (83) one finds for c (t) a linear second-order differential equation with constant coeffi1 cients — which can be solved readily for arbitrary initial conditions. For example, setting c (0)"1, c (0)"0, we arrive at 1 2 c (t)"exp(idt)[cos (1X t)!i(d/X ) sin (1X t)] , R 2 R 1 2 R (84) c (t)"exp(!idt)(ieL /2X ) sin (1X t) , 2 R 2 R where X denotes the celebrated Rabi frequency R X "(d2#eL 2/4)1@2 . (85) R The populations as a function of time are then given by Dc (t)D2"(d/X )2#(eL /2X )2 cos2(1X t) , 1 R R 2 R Dc (t)D2"(eL /2X )2 sin2(1X t) . 2 R 2 R

(86) (87)

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Fig. 2. The population probability of the upper state Dc (t)D2 as a function of time t at resonance, i.e., d"0 (solid line), 2 versus the non-resonant excitation dynamics (dashed line) at d"eL /2O0.

Note that at short times t, the excitation in the upper state is independent of the detuning, i.e., ; Rt 1eL 2t2/16. This behavior is in accordance with perturbation theory, valid at small times. Dc (t)D2XP 2 Moreover, the population at resonance X"D completely cycles the population between the two 0 states, while with dO0, the lower state is never completely depopulated, see Fig. 2. Up to now, we have discussed approximate RWA solutions. Explicit results for the time-periodic Schro¨dinger equation require numerical methods, i.e., one must solve for the quasienergies e and an the Floquet modes W (q, t). Without proof we state here some results that are of importance in an discussing driven tunneling in the deep quantum regime. For example, Shirley [37] already showed that in the high-frequency regime D ;max[X, (eL X)1@2] the quasienergies obey the difference 0 e !e "+D J (eL /X) , (88) 2,~1 1,1 0 0 where J denotes the zeroth-order Bessel function of the first kind. The sum of the two quasiener0 gies obeys the rigorous relation [37] e #e "E #E "0 (mod +X) . (89) 2n 1k 1 2 For weak fields, one can evaluate the quasienergies by use of the stationary perturbation theory in the composite Hilbert space R?T. In this way one finds: 1. Exact crossings at the parity-forbidden transitions where D "2nX, n"1, 2,2 , so that 0 e "0 (mod +X) . (90) 2,1 2. At resonance X"D : 0 1 e "$ +D (1#(eL /2D )J1!eL 2/64D2) (mod +X) . (91) 2,1 0 0 2 0 3. Near resonance, one finds from the RWA approximation e "$1+X(1#X /X) 2,1 2 R

(mod +X) ,

(92)

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where X denotes the Rabi frequency in Eq. (85). Correcting this result for counterrotating terms, R an improved result, up to order O(eL 6), reads [64] e "$1+X(1#XM /X) (mod +X) , 2,1 2

(93)

with the effective Rabi frequency XM given by D eL 4 D eL 2 0 0 ! . XM 2"d2# 2(X#D ) 32(X#D )3 0 0

(94)

Notice that the maximum of the time-averaged transition probability in Eq. (87) occurs within RWA precisely at X"D . This result no longer holds with Eq. (94) where the maximum with 0 X PXM in Eq. (86) undergoes a shift, termed the Bloch—Siegert shift, i.e., X OD [64,66]. From R R 3%4 0 (XM 2/D ) L "0 this shift is evaluated as [64,66] 0e (eL /4)4 eL 2 # . X "D # 3%4 0 16D 4D3 0 0

(95)

This Bloch—Siegert shift presents a characteristic measure of the deviation beyond the RWAapproximation, as a result of the nonzero anti-rotating-wave terms in Eq. (82). 3.2.2. Coherent destruction of tunneling (CD¹) Let us now consider the influence of a monochromatic perturbation on the quantum coherent motion of a tunneling particle between the localized states D¸T and DRT. This problem is conveniently addressed within the localized basis for the TLS Hamiltonian, cf. Eq. (79). We consider for simplicity only the case of a symmetric TLS, i.e., e "0 in Eq. (79). In this case we have 0 D¸T"2~1@2(D1T!D2T), and DRT"2~1@2(D1T#D2T), with D1T and D2T being the unperturbed energy eigenstates. For the state vector in this localized basis D¸T and DRT, we set DW(t)T"c (t)exp[!i(eL /2X) sin (Xt)]D¸T#c (t)exp[#i(eL /2X) sin (Xt)]DRT . L R

(96)

Given the cos(Xt) perturbation (i.e., /"p/2), we consequently obtain from the Schro¨dinger equation for the amplitudes Mc (t)N the equation R,L i(d/dt)c (t)"!1D exp[$i(eL /X) sin (Xt)]c (t) . L,R 2 0 R,L

(97)

For large frequencies X
(98)

where J (x)"(X/2p):T ds exp[ix sin (Xs)], is the zeroth-order Bessel function of the first kind. This 0 0 yields equations of motion that mimick those of a static TLS — which are different from the RWA in Eq. (83) — with a frequency-renormalized splitting, cf. also Eq. (88), i.e., D PJ (eL /X)D " : D . 0 0 0 %&&

(99)

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The static TLS is easily solved to give with the initial preparation c (t"0)"1 for the return L probability P (t) the approximate result L P (t)"Dc (t)D2" cos2(J (eL /X)D t/2) . (100) L L 0 0 Thus, at the first zero j of J (x), i.e. when eL /X"2.404822, P (t) in Eq. (100) precisely equals 0,1 0 L unity, i.e., the effective tunnel splitting vanishes, in agreement with the result in Eq. (88). Thus, one finds a complete coherent destruction of tunneling (CDT) [24,25,67—72]. This high-frequency TLS approximation for the localization manifold M1 (eL , X), as determined -0# by the first root of J (eL /X), is depicted in Fig. 10a in Section 6.3 by a solid line, and is compared 0 with the result for CDT in a continuous bistable system, cf. Section 6. Higher roots j yield an 0,n approximation for the localization manifold Mn (eL , X) with n'1. Moreover, we note that -0# J (x)&x~1@2 as xPR. This implies, within the TLS-approximation to driven tunneling, that 0 tunneling is always suppressed for X'D for appropriate values of the ratio eL /X. An improved 0 formula for P (t) in Eq. (100) that contains also higher odd harmonics of the fundamental driving L frequency X has recently been given in [71]. This driven TLS is closely connected with the problem of periodic, nonadiabatic level crossing. In the diabatic limit d,D2/(eL X)P0, corresponding to a large amplitude driving, the return probabil0 ity has been evaluated by Kayanuma [72]. In our notation and with eL 'max(X, D ) this result 0 reads

G

C A

p eL P (t)& cos2 (2X/eL p)1@2 sin # L X 4

BD H

D t 0 . 2

(101)

With J (x)&J2/px sin (x#p/4), for x<1, Eq. (101) reduces for eL /X<1 to Eq. (100). Here, the 0 phase factor of p/4 corresponds to the Stokes phase known from diabatic level crossing [72,73], and eL /X is the phase acquired during a single crossing of duration T/2. The mechanism of coherent destruction of tunneling in this limit eL 'max(X, D ) is related, therefore, to a destructive interfer0 ence between transition paths with eL /X"np#3p/4. Additional insight into CDT has been provided by Wang and Shao [69]: They discussed the phenomenon by mapping the driven quantum dynamics to the classical movement of a charged particle moving in a two-dimensional harmonic oscillator potential under a periodically acting, perpendicular magnetic field. The motion of the particle is constrained to the interior of a unit circle in the plane. The behavior of tunneling and localization is described by the radial trajectory of the classical particle. In doing so, the authors have demonstrated that for a particle that starts at the edge of the unit circle localization happens only if the radial trajectory is periodic and essentially runs in a narrow concentric ring within the unit circle. Finally, for the case of an asymmetric TLS (e O0), one finds for the resonant case, i.e., 0 e /X"n (n"1, 2,2), that tunneling is still coherently suppressed at high driving frequencies 0 X
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3.3. Tunneling in a cavity The phenomenon of coherent destruction of tunneling also persists if we use a full quantum treatment for the semiclassical description of the field: For an electric field operator defined as EK "(+g/k)(as#a), where k is the dipole matrix element between the TLS eigenstates D1T and D2T, the quantized version of Eq. (75) reads, see e.g. in [76], (103) H"!1+D p #+Xasa!+g(as#a)p , x 2 0 z where a, as are the bosonic creation and annihilation operators for the field mode with frequency X. With SnT"SasaT being the average value of the occupation number n for a coherent state DnT of the field, the coupling constant g is related to the semiclassical field strength eL by 4gJSnT"eL .

(104)

The vanishing of the quasienergy difference D is then controlled by the roots of the Laguerre %&& polynomial ¸ of the order of the photon number n [77—79], i.e., n D "D exp(!2a)¸ (4a) , (105) %&& 0 n where a"(g/X)2. Note that the semiclassical limit is approached with gP0, nKSnTPR, such that 4gJn"eL "const, with ¸ converging to the zeroth order Bessel function. Hence, we recover n the effective tunneling splitting inherent in Eq. (99), i.e., D "D J (eL /X). %&& 0 0 The mechanism of CDT has been investigated in [79] from the viewpoint of quantum optics by studying the photon statistics of the single-mode cavity field in terms of the product states DiTDnT " : Di, nT, with i"¸, R. Given the initial preparation DW(t"0)T"D¸TDnT, the transition probability P (t) to find the system in the product state Di, mT for any i"¸,R after the interaction n?m time t is given by P (t)" + DSi, mD exp(!iHt/+)D¸, nTD2 . (106) n?m i/L,R For fixed t this quantity yields the photon number distribution with P (t"0)"d . For fixed n?m n,m m, this transition probability describes the cavity mode-TLS interaction. The tunneling dynamics is then characterized by an amplitude modulation of the (fast) cavity oscillations. The authors in [79] have demonstrated that CDT manifests itself in the limit of large n in a characteristic slowing down of the amplitude modulation of the quantity P (t) monitored versus interaction time t. n?n The dynamics of the Hamiltonian in Eq. (103) is very rich: As a function of its parameters is able to depict quantum revivals, regular and irregular behavior of occupation probabilities of the TLS, non-integrable semiclassical dynamics, etc. [80]. As such, this simple model constitutes an archetype for studying different aspects of quantum chaos. 3.4. Circularly polarized radiation fields Here, we explicitly consider the case pioneered by Rabi [65], i.e., a TLS driven in a spatially homogeneous, circularly polarized external radiation field. This represents, for example, the case of

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a spin 1 subject to a static magnetic field B along the z direction, and to a time-dependent magnetic 2 0 field in the x and y directions, i.e., cf. Eq. (77), H(t)"!l ) B"!1+cp B !+cB [p cos (Xt)!p sin (Xt)] . 2 z 0 1 x y

(107)

D "cB , 0 0

(108)

With eL /2"cB , 1

this leads, in the energy representation, to the Hamiltonian 1 1 H(t)"! +D p ! +eL [p cos (Xt)!p sin (Xt)] x y 2 0 z 2

A

1 D "! + 0 2 eL exp(!iXt)

B

eL exp(iXt) !D

0

.

(109)

Absorbing the phase exp($iD t) into the time dependence of the coefficients, i.e., setting 0 a (t)"c (t) exp($iD t), we rotate the states around the z-axis by the amount Xt. With 1,2 1,2 0 S "1+p , one has z 2 z

A B

A

A

B

BA B A

B

b (t) a (t) a (t) exp(!iXt/2) 1 "exp ! i S Xt 1 " 1 . (110) + z b (t) a (t) a (t) exp(#iXt/2) 2 2 2 Upon a substitution of Eqs. (109) and (110) into the time-dependent Schro¨dinger equation, and collecting all the terms, results in a time-independent Schro¨dinger equation for the states (b (t), b (t)), which reads 1 2 !ibQ "!1(X!D )b #1eL b , 1 2 0 1 4 2 (111) !ibQ "1eL b #1(X!D )b . 0 2 2 4 1 2 Thus, one obtains a harmonic oscillator equation for b (t) (and similarly for b (t)), i.e., 1 2 eL 2 d2 # b "0 . (112) b® # 1 16 4 1 It describes an oscillation with frequency X "JeL 2/4#d2, where X coincides precisely with the R R previously found Rabi frequency in Eq. (85). With c (0)"a (0)"b (0)"1, and 1 1 1 c (0)"a (0)"b (0)"0 the populations are given by the corresponding relations in Eq. (86), 2 2 2 which in this case are exact. In particular, the transition probability ¼ (t)" 1?2 DS2DW(t)TD2"Da (t)D2"Db (t)D2"Dc (t)D2 obeys 2 2 2 (113) ¼ (t)"(eL 2/4X2) sin2(1X t) . 1?2 R 2 R At resonance, d"0, X"D , it assumes with X2"eL 2/4 its maximal value ¼ (tK )"1 at times 0 R 1?2 tK obeying eL tK /2"p (mod 2p).

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We point here to the importance of initial preparation: The coherence effects of a circularly polarized radiation field for an initially localized TLS has been investigated in [81]: Interestingly enough, one again finds the phenomenon of CDT if, for sufficiently strong driving strength eL , the driving frequency X matches precisely the Rabi frequency X . R 3.5. Curve crossing tunneling In this subsection we shall investigate the problem proposed by Landau [82], Zener [83] and Stu¨ckelberg [84], known as the Landau—Zener—Stu¨ckelberg problem of nonadiabatic crossing of energy levels. Put differently, we consider the explicit time-dependent situation in which the (parametric) energy levels of a quantum mechanical Hamiltonian are brought together in the course of time by means of an external perturbation. The original work by Zener concentrates on the electronic states of a bi-atomic molecule, while in that by Landau and Stu¨ckelberg one considers two atoms which undergo a scattering process. Because Zener, Landau and Stu¨ckelberg restrict themselves to the Hilbert space generated by two electronic functions only, we have to deal with a two-level system. Quantitatively, for a two-level crossing, the problem is described by the time-dependent Hamiltonian H (t)"!1+(vtp #D p ) , LZS 2 z 0 x

(114)

where D is the coupling between the two levels, and +v is the rate of change of the energy of the 0 uncoupled levels due to an external driving bias. In this representation the eigenstates of p are the z diabatic states D¸T, DRT which satisfy p D¸T"!D¸T (p DRT"DRT), while the eigenstates of H (t), z z LZS at fixed time t are the adiabatic states D$T with adiabatic energies E (t)"$+(D2#v2t2)1@2/2, cf. B 0 Fig. 3. Note that in the infinite past and future the diabatic and adiabatic representations coincide. We assume that at an initial time far away in the past the system is in the lower state. We then ask for the transition probability ¼ (tPR)"DS#Dº(R,!R)D!TD2"DS¸Dº(R,!R)D¸TD2 " : P ~?` LZS

(115)

of finding the system in the excited level in the far future, cf. Fig. 3. º(t , t )" 2 1 Á expM(i/+):t21H (t) dtN denotes the time evolution operator. To obtain this transition probability, t LZS we follow here closely the approach by Kayanuma [85]. To start, it is convenient to split the Hamiltonian (114) as H (t)"H (t)#HD, where H (t) is the time-dependent diagonal part. LZS $ $ Hence, the propagator can be expressed by use of Feynman’s disentangling theorem [86] as º(R,!R)"º (R,!R)ºD(R,!R) $

A P

B A P

B

i = i = H (t) dt Á exp ! HI D(t) dt , "exp ! $ + + ~= ~=

(116)

where

AP

HI D(t)"exp

B

A P

B

i t i t H (t@) dt@ HD exp ! H (t@) dt@ . $ $ + + ~= ~=

(117)

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Fig. 3. Crossing of the diabatic energy levels (solid lines) in the Landau—Zener—Stu¨ckelberg problem in the course of time. D is the coupling between the levels, while the time q characterizes the time spent in the crossover region. The 0 LZS dashed lines represent the adiabatic energy levels E (t). B

Expanding the time-ordered exponent in Eq. (116) one obtains

A B

= i n S¸Dº (R,!R)ºD(R,!R)D¸T" + ! $ + n/0 `= tn t2 dt ] dt 2 dt S¸Dº (R,!R)HI D(t )HI D(t )2HI D(t )D¸T . n n n~1 1 n~1 1 $ ~= ~= ~= We observe that the operator º is diagonal, i.e., $ iv Sº (t , t )D¸T"exp ! (t2!t2 ) D¸T , j~1 $ j j~1 4 j

P

P

P

A A

B B

iv Sº (t , t )DRT"exp # (t2!t2 ) DRT , $ j j~1 j~1 4 j

(118)

(119)

while the operator HD is non-diagonal and generates spin flips, i.e., HDD¸T" (+D /2)DRT, HDDRT"(+D /2)D¸T. Thus, only an even number of spin flips contribute to the matrix 0 0 element (118). One finds

A B C

= D2 n S¸Dº (R,!R)ºD(R,!R)D¸T" + ! 0 $ 4 n/0 iv 2n `= t2n t2 dt dt dt exp ! + (!1)jt2 . ] 2 2n 2n~1 1 j 2 ~= ~= ~= j/1 By making the variable transformation

P

P

P

p~1 x "t , x "t # + (t !t ), y "t !t , 1 1 p 1 2j`1 2j p 2p 2p~1 j/1

D

(120)

(121)

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and upon noting that the resulting integrand is symmetric under exchange of the integration variables x , one can perform the integrations to obtain p = 1 pD2 n 0 . S¸Dº (R,!R)ºD(R,!R)D¸T" + (122) $ n! 2DvD n/0 This result is recast as

A B

P "exp(!2pc) , (123) LZS where c"D2/(2DvD). Note that this result is exact. The transition probability P depends only on 0 LZS the ratio between the time q "D /DvD spent in the crossover region and the coherent tunneling LZS 0 time q "2D~1, i.e., c"q /q . #0) 0 LZS #0) 3.6. Pulse-shaping strategy for control of quantum dynamics The phenomenon of coherent destruction of tunneling discussed previously in Section 3.2.2 is an example of a dynamical quantum interference effect by which the quantum dynamics can be manipulated by an observer. More generally, the dependence of quasienergies on field strength and frequency can be used to control the emission spectrum by either generating or by selectively eliminating specific spectral lines. For example, the near crossing of quasienergies in a symmetric double well generates anomalous low-frequency lines and — at exact crossing — doublets of intense even-harmonic generation (EHG) [27], see Section 7.3. This control by a periodic continuous-wave driving can be generalized by recourse to more complex perturbations. The goal by which a pre-assigned task for the output of a quantum dynamics is imposed from the outside by applying a sequence of properly designed (in phase and/or shape) pulse perturbations is known as quantum control [11,13]. For example, a primary goal in chemical physics is to produce desired product yields or to manipulate the atomic and molecular properties of matter [11,13,87,88]. As an archetype situation, we present the control of the quantum dynamics of two coupled electronic surfaces — assuming throughout validity within a rotating wave approximation — i.e.,

A B C

DA B

H (R) !k(R)E(t) W W1 1 1 , i+ " (124) * t W !k(R)E (t) W H (R) 2 2 2 where R denotes the nuclear coordinates and H , H are the Born—Oppenheimer Hamiltonians for 1 2 the ground and excited field free surfaces, respectively. The surfaces are coupled within the dipole approximation by the transition dipole operator k(R) and the radiation field E(t). Notice that the structure in Eq. (124) is identical to that obtained in the driven TLS. Following Kosloff et al. [88], the rate of change to the ground state population n (t)"SW (t)DW (t)T is readily evaluated to read 1 1 1 dn /dt"2ReSW DWQ T"!(2/+)Im(SW (t)Dk(R)DW (t)TE(t)) . (125) 1 1 1 1 2 If we now choose E(t) as a control, i.e., E(t)PE (t)"SW (t)Dk(R)DW (t)TC(t) , # 2 1

(126)

256

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with C(t) a real-valued function, we can freeze the population transfer (null-population transfer), i.e., with Eq. (126), dn /dt"0 , (127) 1 for all times t. In other words, the population in the ground electronic surface, and necessarily also the population of the excited surface, remains fixed. If we were to choose E(t)PiE (t), it would # cause population to be transferred to the upper state, while E(t)P!iE (t) would dump popula# tion down to the ground state. Hence, by controlling the phase of a laser, we can control the population transfer at will. This phenomenon applies equally well to driven tunneling in a TLS. What can we achieve if we manipulate the amplitude C(t)? The change in the energy of the ground state surface, which can be varied by exciting specific ground-state vibrational modes, is obtained as dE d SW (t)DH DW (t)T 1" 1 1 1 . dt dt SW (t)DW (t)T 1 1 Under the null-population-transfer condition in Eq. (126), this simplifies to [88]

(128)

dE 2C(t) 1"! ImSW (t)DH k(R)DW (t)TSW (t)Dk(R)DW (t)T . (129) 1 1 2 2 2 dt +n 1 It follows that the sign of C(t) can be used to “heat” or “cool” the ground-state wavepacket; the magnitude of C(t) in turn controls the rate of heating (or cooling). With this scheme of phase and amplitude control of a laser pulse it is possible to excite vibrationally the lower state surface while minimizing radiation damage either by ionizing or by dissociating the corresponding quantum system.

4. Driven tight-binding models In the previous section we have considered driven two-state systems. Here we shall consider the driven, infinite tight-binding (TB) system. Similar to the case of a two-state system, the infinite TB system can result from the reduction of the tunneling dynamics of a particle moving in an infinite periodic potential » (q)"» (q#d) within its lowest energy band. To be definite, if one fixes the 0 0 potential depth such that the Wannier states DnT building up the lowest band have appreciable overlap only with their nearest neighbors Dn$1T, one obtains the Hamiltonian H of the bare TB multistate system, i.e., +D = H "! 0 + (DnTSn#1D#Dn#1TSnD) , (130) TB 2 n/~= where 2+D is the width of the lowest energy band. The influence of the driving force is described by 0 the perturbation H (t)"!q+[e #eL cos (Xt)]/d, q"d+ nDnTSnD , %95 0 n

(131)

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Fig. 4. A periodic potential with lattice constant d and potential drop +e per period. Within the miniband, tight-binding 0 approximation, the Wannier state DnT localized at the site nd has appreciable overlap only with its nearest neighbors Dn$1T.

where the operator q measures the position of the particle on the lattice, while +e(t)/d"+(e #eL cos Xt) measures the potential drop per lattice period d due to externally applied 0 dc- and ac-fields, cf. Fig. 4. In order for the single band approximation to be still consistent, the driving frequency X, as well as the ac and dc-bias strengths e and eL , respectively should be chosen 0 much smaller than the gap between the lowest two bands. The problem described by H(t)"H #H (t) has a great variety of applications as, for example, in the investigation of the TB %95 dynamics of ultracold atoms in standing light waves [89,90], where lattice defects are absent, in high-quality superlattices [16,19—22,91—98], or in thin crystals [99]. Suppose now that the particle has been prepared at time t "0 at the origin Dn"0T. Then, the 0 dynamical quantity of interest is the probability P (t) for finding the particle at site n at time t'0. In n turn, the knowledge of P (t) enables the evaluation of all the dynamical quantities of interest in the n problem as, for example, the position’s expectation value P(t), as well as the diffusive variance S(t), i.e., = P(t) " : SqT "d + nP (t) , t n n/~= (132) = S(t) " : Sq2T !SqT2"d2 + n2P (t)!P2(t) . t t n n/~= The expectation value P(t) represents the generalization of the quantity P(t)"Sp T , cf. Eq. (70), z t introduced in the study of the dynamics of a two-state system. If there is no external ac—dc field, then the Bloch waves [100] s "+ exp(!iknd)DnT (133) k n solve the stationary Schro¨dinger equation H s "E(k)s with E(k) an absolutely continuous TB k k spectrum, i.e., E(k)"!+D cos (kd). Thus, 2+D is the width of the resulting energy band. For this 0 0 problem the propagator º0 (t, t@) from state DlT to state Dl@T is explicitly evaluated to read l,l{ º0 (t, t@)"J (D (t!t@))e(*p@2)(l~l{) , (134) l,l{ l~l{ 0 where J (x) is the lth-order Bessel function. Given the initial condition P (t"0)"d , one then l n n,0 finds P(t)"0 and S(t)"d2D2t2/2 , (135) 0 i.e., the variance of an initially localized wave packet spreads ballistically in time (superdiffusion).

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In presence of a dc-field +e /d, the Wannier—Stark states [101,102] 0 / "+ J (D /e )DnT (136) m n~m 0 0 n constitute a complete set of energy eigenstates with corresponding — now discrete — eigenvalues E "m+e forming the Wannier—Stark ladder [101—103]. A particular Wannier—Stark state / is m 0 m localized around the mth site with a localization length of the order of 2D /e lattice periods. In this 0 0 case the propagator takes the form

A

A

BB G

C

D

H

2D e (t!t@) p!e (t!t@) 0sin 0 0 exp i(l!l@) !il@e (t!t@) , ºDC(t@t@)"J 0 l,l{ l~l{ e 2 2 0 which, with the initial condition P (t"0)"d , leads to n n,0 P (t)"J2((2D /e )sin (e t/2)) . n n 0 0 0 Because Eq. (138) is symmetric in n, one obtains P(t)"0 and

(137)

(138)

S(t)"2d2(D /e )2sin2(e t/2) . (139) 0 0 0 Hence, a wavepacket which is initially localized on one site spreads in space and returns back to its initial shape after the Bloch period T "2p/e . The center of the wavepacket, though, remains B 0 fixed. On the contrary, if one considers a wavepacket which initially is not sharply localized, one finds that the center of the wavepacket oscillates in space with an amplitude which is inversely proportional to the dc-field strength e , and with temporal period T . The shape of the wavepacket 0 B remains almost invariant. This is the famous Bloch oscillation that was first discussed by Zener [104]. It has been observed only recently in semiconductor superlattices through the emission of THz radiation by the electrons [91], as well as in optical lattices [89,90]. If only an ac-field is applied, the Floquet states W (t) solve the time-dependent Schro¨dinger k equation. They read

A

P

B

t W (t)"+ DnT exp !iq (t)nd!(i/+) dqE(q (q)) , (140) k k k 0 n with q (t)"k#(eL /X)sin (Xt)/d. Setting W (t)"U (t) exp(!ie(k)t/+) with U (t#T)"U (t), the k k k k k quasienergies form again an absolutely continuous spectrum which reads e(k)"J (eL /X)E(k), mod +X . (141) 0 Given the initial condition P (t"0)"d , one finds for the two transport quantities in n n,0 Eq. (132) [105] P(t)"0, where

P P

u(t)" v(t)"

t

0 t

0

S(t)"d2(D2/2)[u2(t)#v2(t)] , 0

(142)

dt@sin [eL /X sin (Xt@)] , (143) dt@cos [eL /X sin (Xt@)] .

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The above equations, Eqs. (142) and (143), can be simplified at long times t<2p/X to yield S(t)"d2D2 t2/2, D "J (eL /X)D , (144) %&& %&& 0 0 i.e., the problem of diffusion reduces effectively to that of an undriven particle, cf. Eq. (135), with a renormalized energy bandwidth 2+D 42+D . In particular, when eL /X coincides with a kth zero %&& 0 j of the Bessel function J , the wavepacket reassembles itself after a period, as it can be verified 0,k 0 from the exact result (142). Hence, on a long time average, it neither moves nor spreads. This phenomenon, termed dynamic localization (DL) [16,92,105,109,110], resembles the phenomenon of coherent destruction of tunneling (CDT) discussed for the two-state system in Section 3.2.2. Note, however, that the two phenomena are very distinct: The DL is based on a collapse of a band, cf. Eq. (141), while CDT necessitates the crossing of two tunneling related discrete quasienergies. More importantly, the probability of an initially localized preparation P (t"0)"d does not n n,0 stay localized. To be definite, with the probability to stay at site n"0 given by [105] P (t)"J2(D Ju2(t)#v2(t)) , (145) 0 0 0 P (t) at band collapse becomes an oscillating function with period 2p/X varying between zero and 0 one. It exhibits stroboscopic localization only at multiples of the driving period, i.e., at t "n2p/X, n"0, 1, . . ; cf. also Fig. 3a in Ref. [105]. Further, we remark that, while for the n two-state system the renormalization (89) of the intersite coupling leading to CDT is a perturbative result in the ratio (D /X) between the intersite coupling and the external frequency, the correspond0 ing equation (141) for the quasienergy band is an exact result. This is due to exact translational symmetry of an infinite tight-binding lattice with nearest-neighbor intersite interactions. Finally, in the presence of both ac- and dc-fields the Houston states [106] solve the timedependent Schro¨dinger equation. They have the same form as the Floquet states in Eq. (140) solving the Schro¨dinger equation in the absence of a dc-field, but with q (t) in Eq. (140) substituted k by q (t)"k#[e t#(eL /X)sin (Xt)]/d . (146) k 0 However, it must be recognized that the problem now contains two frequencies, the ac-frequency X and the Bloch frequency X "e . As a consequence, it turns out that the Houston states coincide B 0 with the Floquet states of H #H (t) only at resonance, when multiples of the driving frequency TB %95 match the rungs of the Wannier—Stark ladder, i.e., e "nX [96,107], see Fig. 5. In this case 0 W (t)"U (t) exp(!ie(k)t/+) assumes the Floquet form with U (t#T)"U (t) and corresponding k k k k quasienergies given by

P

1 e(k)" T

T

AB

eL dtE(q (t))"(!1)nJ E(k), mod +X . k n X

(147) 0 The quasimomentum k then remains a good quantum number for the eigenfunctions U (t) despite k the presence of the dc-field. Whereas the Wannier—Stark states are localized, an additional resonant ac-field leads again to extended quasienergies states that form quasienergy bands of finite width. Therefore, a resonant ac-field can destroy Wannier—Stark localization even if its amplitude is infinitesimally small [111].

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Fig. 5. A tight-binding lattice driven by dc- and ac-fields exhibits quasi-energy bands if the energy n+X of n photons precisely matches the energy difference +e between adjacent rungs of the Wannier—Stark ladder. The emergence of the 0 quasi-energy band occurs because, as depicted schematically in this figure for n"1, at resonance, the energies of the atomic states “dressed” by the ac-field are exactly aligned with the rungs of the Wannier—Stark ladder.

In presence of random static disorder these delocalized states tend to localize again [95,96,108], and it is even possible to control the degree of localization, i.e., the localization length, by appropriately varying the ac-field amplitude [95—97]. An arbitrary, initially localized wavepacket W(t"0) can be represented as a superposition of Floquet states with a certain momentum distribution fK (k, 0), and the time evolution of each component is given by fK (k, T)"fK (k, 0) exp(!ie(k)T/+). Assuming that the initial momentum distribution is sharply peaked around k , the time-averaged group velocity of the wavepacket is 0 D d de(k) eL vN " "(!1)n 0 J sin (k d) . (148) 0 2 n X dk 0 k Hence, the wavepacket generally moves along the whole lattice, and simultaneously is spreading. When eL /X coincides with a kth zero j of the Bessel function J the wavepacket reassembles itself n,k n after a period. Hence, on average, it neither moves nor spreads and the dynamic localization phenomenon occurs. We observe that interesting new features occur when coupling to higher bands start to play a role. As pointed out in [112], in the presence of a static electric field the spectrum consists of two interpenetrating Wannier—Stark ladders. In the presence of ac-fields, quasienergy bands occur [103,113,114] whose bandwidth can be controlled by appropriately tuning of the ac-field parameters. Finally, a nonperturbative investigation of the influence of a strong laser field on the band structure of a one-dimensional Kronig—Penney periodic delta potential has been investigated in [115].

K K

AB

5. Driven quantum wells With the recent advent of powerful radiation sources, which are able to deliver extremely strong and coherent electric driving fields, the quantum transport in solid state devices, such as in

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superconductors and semiconductor heterojunctions, has been revolutionalized. The latter, by now, can even be nanostructured. Of particular interest is the role of photon-assisted tunneling (PAT) in single, double-barrier and periodic superlattice-like structures. The mechanism of PAT has originally been observed in the experiments by Dayem and Martin [14] in the stationary dc-current—voltage characteristics of a superconductor (Al)—insulator (Al O )—superconductor (In) 2 3 (SIS)—structure. Motivated by this very experiment, Tien and Gordon [15] put forward a simple theory, which continues to enjoy great popularity up to these days. 5.1. Driven tunneling current within Tien—Gordon theory Following Tien and Gordon [15] one considers two (superconducting) films A and B with a time-dependent electric field acting normal to their surface. In doing so, one neglects the intermediate insulating regime where the wave functions drop off sharply. Neglecting electron—field interactions in this region one can use the potential of film A as a reference, and the main effect of the electromagnetic field is accounted for by adding a time-dependent, but spatially constant, potential that is being built up across the insulator in regions with coordinates q in film B. Hence, neglecting the interface, the total Hamiltonian reads, see Fig. 6,

G

H(t)"

H (q)#»K cos Xt, 0 H (q), 0

q in B ,

(149)

q in A .

Note that this Hamiltonian describes driven quantum transport that does not involve the spatial dependence. Hence, within the region of the film B, the wave function simply assumes an overall phase factor given by

G

P

expM!ig(t)/+N"exp (!i/+)

t

0

H

ds »K cos (Xs) ,

(150)

meaning that the wave function with coordinates q in part of film B is given exactly by W(q, t)"u(q) expM!iEt/+N expM!ig(t)/+N .

(151)

Put differently, the interaction in Eq. (149) does clearly not change the spatial dependence of the unperturbed Hamiltonian H , whose wave function is given by the first two contributions in 0 Eq. (151). Note however, that the solution in Eq. (151) is not an exact wave function W(q, t) in all space of film A (where H(t) equals H ) and film B, because no wave function matching at the 0 interface has been accounted for. The phase can be readily written as expM!ig(t)/+N" expM!i(»K /+X)sin (Xt)N, which, by use of the generating function of ordinary Bessel functions, is recast as = expM!ig(t)/+N" + J (»K /+X) exp(!inXt) . (152) n n/~= This form allows for a simple interpretation for PAT: In this theory, photon absorption (n'0) or emission (n(0) is viewed as creating a new set of (electron) eigenstates of energy E#n+X, where

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Fig. 6. The Tien—Gordon model: a time-dependent uniform electric field acting normal to the surfaces A and B of two superconducting films builds up a potential modulation across the tunneling barrier separating the two films. The potential of film A is taken as a reference. The electric field then sets up a potential difference »K cos(Xt) in film B.

E denotes the eigenenergy of the original state with H(t)"H throughout A and B. The probability 0 of absorbing (or emitting) n photons equals each other, and is given by the weight J2(a), with a being n the dimensionless radiation quantity a" : »K /+X .

(153)

We remark that this argument is inversely proportional to the first power in X. The theory allows for a straightforward evaluation of the average dc-current I (» ,»K , X) in presence of an additional 0 0 applied dc-bias » . In terms of the dc-tunneling-current, i.e., 0

P

I$# (» )"C AB 0

=

~=

dE[ f (E!» )!f (E)]. (E!» ). (E) , 0 A 0 B

(154)

where C is a proportionality constant, the f ’s are the Fermi factors, and . are the densities of A,B states in A and B, respectively, the effect of an ac-modulating voltage drop in film B results in a nonzero, average tunneling current. It explicitly reads [15] = I (» ,»K , X)" + J2(»K /+X)I$# (» #n+X) . (155) 0 0 n AB 0 n/~= We note that such SIS-structures are commonly used for high-frequency detection [116]. The Tien—Gordon theory — for additional insight into this approach see also [116,117] — has recently been successfully applied in explaining qualitatively the experimental findings of PAT through quantum dots [118—120], in quantum point contacts [121], for stimulated emission and absorption in THz-irradiated resonant tunneling diodes [122]. Interestingly enough, the simple theory in Eq. (154) for the average dc-current also qualitatively describes the experimentally observed PAT-phenomena in superlattices, particularly the emergence of “absolute negative conductance” near zero bias voltage » . In other words, it might occur that I (» ,»K , X)(0 for 0 0 0 small but positive applied dc-bias » [16—22,98]; see also in Section 13. In recent years, more 0 realistic theoretical descriptions beyond the Tien—Gordon level have been developed by various groups [123—138].

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Fig. 7. Driven quantum wells: An infinite high square well sandwiched between two walls at q"$¸/2, and driven by (a) an external field !qS cos(Xt), or (b) driven by a uniform in space, time-dependent potential »K (t)"»K cos(Xt). The situation in (b) is generally referred to as “Tien—Gordon” case. The location of the lowest quasienergy e is indicated by 0 the dashed line. For case (a) one has e "E@ #S2/(4mX2), and e (S, X)S?0 P E "+2p2/2m¸2, being the ground state 0 0 0 0 energy of an undriven square well. For case (b) one has e "E . 0 0

In principle, the exact description of strong ac-field interactions necessitates a full treatment in terms of Floquet theory for the total Schro¨dinger equation of system — leads — ac-field interactions, which generally constitutes a difficult task. In the following we address in some detail such analysis for a strongly driven quantum well. 5.2. Floquet treatment for a strongly driven quantum well In the previous subsection we considered the effects of a — spatially constant — time-dependent modulation across a tunneling barrier. In cases where the coherence length of the wave function is smaller than the region of uniform modulation, boundary conditions at the interfaces must be taken into account. The case of a quantum well of width ¸ and finite height » , where only the 0 quantum well but not the neighboring regions is subject to a uniform bias modulation has been considered in [127,128,133,134], see Fig. 7b for the case of a quantum well with infinte high walls. This situation is of interest, for example, to investigate transmission probabilities through a double-barrier resonant tunneling diode [122], or driven tunneling through a quantum dot [118—120]. The Hamiltonian inside the quantum well reads +2 2 H(t)"! #»K cos(Xt), 2m q2

¸ ¸ ! 4q4 , 2 2

(156)

with m denoting the effective mass. At the interfaces, q"$¸/2, the wave function and its flux must be continuous. To satisfy these requirements, one needs to superimpose many wave functions of the general form (151) inside the quantum well. This yields for the wave function inside the

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quantum well the solution = W (q, t)" + exp[!i(E#n+X)t/+][A exp(ik q)#B exp(!ik q)] E n n n n n/~=

A

B

»K ] exp !i sin(Xt) , +X

(157)

where E denotes a bound state energy in the undriven quantum well. Hence, the wave function W (q, t) is of the Floquet form E W (q, t)"exp(!iet/+)U (q, t) , E E

(158)

with U (q, t) periodic in T"2p/X, and with quasienergy e"E mod +X. In the form given by E Eq. (157) the full wave function is built up by an infinite sum over transport sidebands, where the nth transport sideband is characterized by the quasiwavevector k , defined as n +k "J2m(E#n+X) . n

(159)

The coefficients A and B are determined by the matching conditions for the wave function and its n n flux at the interfaces q"$¸/2. Hence, when time-dependent transport through a barrier is investigated, transport sidebands with a definite quasiwavevector k must be considered. It is n important to notice that because of the residual periodic time dependence carried by each transport sidebands, these transport sidebands are not obtained by a Fourier decomposition of the periodic Floquet mode, in contrast to the Tien—Gordon case discussed in the preceding subsection. For this problem the probability S to find the particle in a particular transport sideband n has been n evaluated in [133,134] under the assumption that n+X;E and n+X;» !E, for all sufficiently 0 populated sidebands. To lowest order in the quantities n+X/E and n+X/(» !E), one obtains 0 S "J2(c»K /+X) , n n

(160)

where the coefficient c41 depends on the characteristics of the quantum well. In the limit ¸PR the Tien—Gordon problem discussed in the previous section is recovered. Correspondingly, cP1. One can directly apply these results to the problem of evaluating the transmission probability of an electron traversing a double-barrier structure with a central oscillating quantum well. Assuming that the incoming electron enters the quantum well in the channel n and exits via the channel n , */ 065 the total transmission probability ¹ */ 065 from channel n to channel n is n ,n */ 065 ¹ */ 065"¹ S */S 065 , 0 n n n ,n

(161)

where ¹ (E) denotes the transmission probability n "0Pn "0 for a static double barrier 0 */ 065 structure at a corresponding energy E. With incoming energy E of the scattering particle taken at resonance, the transmission probability ¹ (E) in a symmetric double barrier equals one. Hence, for 0 an incoming electron in the sideband n quenching of the transmission probability ¹ */ 065 occurs at */ nn the zeros of J */(c»K /+X) and/or J 065(c»K /+X). n n

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Qualitative new results are obtained when the assumption of a uniform spatial modulation of the quantum well is abandoned. In the following we consider the problem of a strongly driven quantum well of width ¸, sandwiched between two infinitely high walls at q"$¸/2, as depicted in Fig. 7a, being harmonically driven by an electric field !Sq cos(Xt), i.e., its Hamiltonian reads +2 2 H(t)"! #Sq cos(Xt), 2m q2

¸ ¸ ! 4q4 . 2 2

(162)

The Hamiltonian in Eq. (162) appears very simple. Nevertheless, in contrast to the familiar time-independent case with S"0, no exact solutions are known when SO0 [139—142]. Usually, only perturbative solutions valid to first order in the driving strength S, i.e., a linear response treatment, is deduced [144,145]. As mentioned earlier, with continuous coordinates q and corresponding continuous momenta p, the total time-dependent Hamiltonian must contain interactions that are constant, linear, or at most quadratic in q and/or p for it to be exactly solvable, see Refs. [59—63]. The Hamiltonian in Eq. (162) possesses a generalized parity symmetry P, where P : qP!q; tPt#p/X. The lowest discrete quasienergy e associated to the Floquet state 0 W (q, t)"exp(!ie t/+)U (q, t), with U (q, t) periodic in T"2p/X, can be expressed in the form 0 0 0 0 [142] e "E@ (S, X)#S2/4mX2, mod +X , (163) 0 0 where E@ (SP0, X) approaches the lowest unperturbed bound state E "+2p2/2m¸2. The periodic 0 0 Floquet mode U of even parity reads 0 = U (q, t)" + A Mexp[!ik (q!S cos(Xt)/mX2)] 0 n n n/~= #(!1)n exp[!ik (q!S cos(Xt)/mX2)]N exp(!inXt) n ] exp(!iSq sin(Xt)/+X) exp(iS2 sin(2Xt)/8+mX3) , (164) where +2k2(S, X)"2m[E@ (S, X)#n+X] . (165) n 0 In the form given by Eq. (164) the wave function is built up by an infinite sum over sidebands which, due to the residual q-dependence and periodic t-dependence of the phase, cf. last line in Eq. (165), can neither be interpreted as energy sidebands as in the Tien—Gordon case, nor as transport sidebands as for the spatially uniform driven quantum well. The set of coefficients MA (S, X)N and the energy E@ (S, X) must be obtained from the boundary condition that a Floquet n 0 mode of even parity (odd parity) U (q, t)(U (q, t)) obeys U ($¸/2, t)"U (G¸/2, t# %7%/ 0$$ %7%/ %7%/ p/X)"0, (U ($¸/2, t)"!U (G¸/2, t#p/X)"0), at all times t. This set MA (S, X)N is conve0$$ 0$$ n niently obtained by using again the generating function of Bessel functions J occurring implicitly n in Eq. (164), cf. Eq. (152); however, the resulting system of equations is far too complicated to be solved exactly in analytical terms, see Ref. [142] where an explicit approximation valid to order v2 of the relative spacing v"+X/E@ is presented. In terms of the quantities 0 v"+X/E@ (S, X), u"k (S, X)S/mX2 , (166) 0 0

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and observing that k "k (1#nv)1@2, see Eq. (165), one finds from this approximative solution to n 0 leading order

A

B

u2v2 E@ (S, X)"E 1! , 0 0 8

A

B

p u2v2 k (S, X)" 1! . 0 ¸ 16

(167)

In the same way, the coefficients A can be evaluated to second order in v [142]. Finally, much n information on the transmission probability can be gained from an investigation of the side-bands weights S , where S is the probability of the nth sideband, i.e., n n S "w DA D2/+ w DA D2 , (168) n n n n n n where w "D1#(!1)nJ (2k S/mX2) sin(k ¸)/k ¸D. With v;1, the weights S go to zero at the n 0 n n n n roots of J (k S/mX2). The weights S of these sidebands are depicted in Fig. 8. These weights n 0 n strongly vary as a function of the driving strength S. In particular, we observe that channels with n(0 are generally stronger than those with n'0. This is in contrast to the case of a spatially uniform driven quantum well. The latter predicts from Eq. (160) equal weights for the sidechannels. Most importantly, the transport side-band amplitudes are a nonmonotonic function of the driving strength S, see Fig. 8, and these sidebands scale with Sk /mX2JX~2 rather than with 0 a"»K /+XJX~1, being typical for a situation which is ruled by a uniform potential modulation as in the Tien—Gordon theory, see in this context also the corollary in Ref. [131]. This reasoning can be extended to include Floquet states originating from higher levels. In this case the original spectrum of bound states E "+2(n#1)2p2/2m¸2 evolves into a discrete n quasienergy spectrum e (S, X)"E@ (S, X)#S2/4mX2, mod +X , n n

(169)

with E@ (S, X) " : (+k(n)(S, X))2/2m, i.e., with vanishing field e (S, X) P S?0 E and k(n)(S, X)¸ P S?0(n#1)p, n 0 n n 0 and corresponding Floquet modes of even and odd generalized parity symmetry. For a numerical study of E@ (S, X) vs. S see Ref. [141]. Next, we note that in the case of a uniform potential n modulation, and infinite height walls at q"$¸/2, cf. part b in Fig. 7, the Tien—Gordon theory becomes exact, yielding the quasienergies e (S, X)"E , mod +X , n n

(170)

with corresponding Floquet modes, which simply acquire a time-dependent phase, i.e., U (q, t)"u (q) expM!i»K sin(Xt)/+XN . n n

(171)

Here, u (q) are again the unperturbed wave functions of an infinitely high square well. Clearly, due n to the q-independent, uniform interaction »K cos(Xt) the spatial dependence of the Floquet modes U (q, t) cannot be affected, nor is the value of the quasienergy, i.e., e "E . n n n We end this section by addressing the relevance of these results for PAT in realistic situations with a single quantum well of finite barrier height, as it occurs with a double-barrier structure. As shown in Ref. [142], the analytical prediction for quenching the PAT-assisted tunneling current,

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Fig. 8. Weights of the sidebands, see Eq. (168), of a driven single electron square quantum well. The weights S are n evaluated for a well of width ¸"16 nm (yielding for the static ground state energy E "21 923 meV) vs. driving laser 0 field strength F " : S/e and with photon energy +X"5.38 meV. The top panel depicts the numerical exact solution; the lower panel shows the analytic prediction valid up to fourth order in v"+X/E@ &1. Note that the weights are different 0 4 for $n.

implying that S P0, are in good agreement whenever the photon energy +X exceeds the tunneling n linewidth D of a static resonance. Note that the case with an infinitely high quantum well 0 implicitly assumes that D "0. The theory outlined in this section also applies to multiple 0 quantum wells as they occur in PAT in superlattices: There, quantum ac-transport predominantly proceeds via sequential resonant tunneling. Hence, here too, it is sufficient to consider the physics in a single driven quantum well to describe the main features of PAT.

6. Tunneling in driven bistable systems In this section we address the physics of coherent transport in bistable systems. These systems are abundant in the chemical and physical sciences. On a quantum mechanical level of description bistable, or double-well potentials, are associated with a paradigmatic coherence effect, namely quantum tunneling. Here we shall investigate the influence of a spatially homogeneous monochromatic driving on the quantal dynamics in a symmetric, quartic double well. This archetype system is particularly promising for studying the interplay between classical nonlinearity — its classical dynamics exhibits chaotic solutions — and quantum coherence. Its Hamiltonian reads [24,25] H(q, p; t)"(p2/2m)#» (q)#qS sin(Xt#/) , 0

(172)

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with the quartic double-well potential » (q)"!(mu2/4)q2#(m2u4/64E )q4 . (173) 0 0 0 B Here, m denotes the mass of the particle, u is the classical frequency at the bottom of each well, 0 E the barrier height, and S, X and / are the amplitude, angular frequency and phase of the driving. B The number of doublets with energies below the barrier top is approximately given by D"E /+u . B 0 The classical limit hence amounts to DPR. For ease of notation, we introduce the dimensionless variables Jmu + 0 , qN " q and

p pN " Jmu + 0

(174)

X S uN " , SM " , (175) u Jmu3 + 0 0 where the overbar is omitted in the following. This is equivalent to setting in Eq. (173) formally m"+"u "1. 0 As discussed in Section 2, the symmetry of H(t) reflects a discrete translation symmetry in multiples of the external driving period T"2p/X, i.e., tPt#nT. Hence, the Floquet operator describes the stroboscopic quantum propagation tM "u t, 0

º(nT, 0)"[º(T, 0)]n .

(176)

Besides the invariance under discrete time translations, this periodically driven symmetric system exhibits a generalized parity symmetry P, i.e., P : qP!q;

tPt#T/2 .

(177)

This generalized parity is a symmetry in the composite Hilbert space R?T. Denoting the reflection at the origin, i.e., qP!q by the ordinary operator P, Eq. (177) implies H(t#T/2)"PH(t)P .

(178)

From this identity it follows that the second half of a full period T can be expressed by the propagator over the first half, i.e., º(T, T/2)"Pº(T/2, 0)P, yielding º(T, 0)"º(T, T/2)º(T/2, 0) "(Pº(T/2, 0)P)º(T/2, 0)"(Pº(T/2, 0))2 .

(179)

Just as in the unperturbed case with S"0, this allows the classification of the corresponding quasienergies e into an even and an odd subset. For very weak fields SP0, the quasienergies an e follow from Eq. (29) as ak e0 (S, X)"E #kX, k"0,$1,$2,2 , (180) ak a with ME N being the unperturbed eigenvalues in the symmetric double well. As pointed out in a Eq. (9), this infinite multiplicity is a consequence of the fact that there are infinitely many possibilities to construct equivalent Floquet modes, cf. Eq. (8): The multiplicity is lifted if we

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consider the cyclic quasienergies mod X. Given a pair of quasienergies e , e , aOa@, a physical a,k a{,k{ significance can be attributed to the difference Dk"k@!k. For example, a crossing e "e a,k a{,k`Dk can be interpreted as a (Dk)-photon transition. With S'0, the equality in Eq. (180) no longer provides a satisfactory approximation. Nevertheless, the driving field is still most strongly felt near the resonances e +e . The physics of periodically driven tunneling can be qualified by the a,k a{,k{ following two properties: (i) First we observe, by an argument going back to von Neumann and Wigner [43], that, in absence of any asymmetry, two parameters must be varied independently to locate an accidental degeneracy. In other words, exact quasienergy crossings are found at most at isolated points in the parameter plane (S, X), i.e., the quasienergies exhibit typically avoided crossings. In presence of the generalized parity symmetry in Eq. (177) in the extended space R?T, however, this is true only among states belonging to the same parity class, or for cases of driven tunneling in presence of an asymmetry (then Eq. (177) no longer holds). With the symmetry in Eq. (177) present, however, quasienergies associated with eigenstates of opposite parity do exhibit exact crossings and form a one-dimensional manifold in the (S, X)-plane, i.e., Me(S, X)N exhibit an exact crossing along lines. With SP!S, implying e(S, X)"e(!S, X), these lines are symmetric around the X-axis. (ii) Second, the effective coupling due to the finite driving between two unperturbed levels at the crossing E "E !DkX, as reflected in the degree of splitting of that crossing at SO0, rapidly a a{ decreases with increasing Dk, proportional to the power law SDk. This suggests the interpretation as a (Dk)-photon transition. Indeed, this fact can readily be substantiated by applying the usual (Dk)th-order perturbation theory. As a consequence, for small driving S only transitions with Dk a small whole number do exhibit a significant splitting. 6.1. Limits of slow and fast driving In the limits of both slow (adiabatic) and fast driving, we have a clearcut separation of time scales between the inherent tunneling dynamics and the external periodic driving. Hence, the two processes effectively uncouple. As a consequence, driven tunneling results in a mere renormalization of the bare, lowest order, tunnel splitting D , and likewise for higher-order tunnel splittings 0 D "E !E . This result can be substantiated by explicit analytical calculations [25]. Let us n n`2 n`1 briefly address the adiabatic limit, i.e., the driving frequency X satisfies X;D . Setting 0 /I ,(Xt#/), the tunneling proceeds in the adiabatic potential »(q, /I )"» (q)#qS sin /I . (181) 0 The use of the quantum adiabatic theorem predicts that W(q, t) will cling to the same instantaneous eigenstates. Thus, the evaluation of the periodic-driving renormalized tunnel splitting follows the reasoning used for studying the bare tunnel splitting in presence of an asymmetry e , 0 e "»(q , /I )!»(q , /I ) , (182) 0 ~ ` with q denoting the two minima of the symmetric unperturbed metastable states. With B the instantaneous splitting determined by D(S, /)"(D2#e2)1@2, the averaging over the phase / 0 0 between [0, 2p] yields for the renormalized tunnel splitting D (S), the result [25] !$ D (S)"(2D /p)(1#u)1@2E[Ju/(1#u)]5D , (183) !$ 0 0

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with u"32S2D/D2, and E[x] denoting the complete elliptical integral. This shows that D in!$ creases proportional to S2 as u;1, and is increasing proportional to S for u<1. Hence, a particle localized in one of the two metastable states will not stay localized there (this would be the prediction based on the classical adiabatic theorem) but rather will tunnel forth and back with an increased tunneling frequency D 'D . Obviously, with the slowly changing quantum system !$ 0 passing a near degeneracy (tunnel splitting) the limits +P0, X fixed and small (classical adiabatic theorem) and XP0, + fixed (quantum adiabatic theorem) are not equivalent. The limit of high-frequency driving can be treated analytically as well. The unitary transformation W(q, t)"exp(!i(S/X) cos(Xt#/)q)g(q, t)

(184)

describes the quantum dynamics within the familiar momentum coupling in terms of an electromagnetic potential A(t)"!(S/X) cos(Xt#/), i.e., the transformed Hamiltonian reads HI (q, t)"H (q, p)!A(t)p , 0

(185)

with H the Hamiltonian in the absence of the driving field. We drop all time-dependent 0 contributions that do not depend on q and p. Next we remove this A(t)p-term by a Kramers—Henneberger transformation [143], i.e.,

A P

g(q, t)"exp !i

t

B

dt@A(t@)p f (q, t)

(186)

to yield HK (q, t)"1p2#» (q!(S/X2) sin(Xt#/)) , 2 0

(187)

resulting in a removal of the A(t)p-term, and the time dependence shifted into the potential »(q, t). After averaging over a cycle of the periodic perturbation we obtain an effective Hamiltonian

C

A BD

1 3 S 2 1 1 # q4 , H " p2! q2 1! H 2 4 16D X2 64D

(188)

with a frequency-dependent curvature. This large-frequency approximation results in a highfrequency renormalized tunnel splitting [25],

A

A BB

3 S 2 exp(2S2/X4)5D . D "D 1! 0 H 0 16D X2

(189)

Hence, fast driving results in an effective reduction of barrier height, thereby increasing the net tunneling rate. Note here the different behavior of the effective tunnel splitting on the angular frequency of periodic driving: In the two-state model (99), the argument in the function yielding the driving induced, effective tunnel splitting for fast fields X'D exhibits a scaling proportional to 0 X~1; in contrast, here, with a spatially extended coordinate q for a bistable potential, the scaling becomes proportional to X~2 in the regime X
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In conclusion, the regime of adiabatic slow driving and very-high-frequency driving (away from high-order resonances) can be modeled via a driving-induced enhancement of the tunnel splitting. 6.2. Driven tunneling near a resonance Qualitative changes of the tunneling behavior are expected as soon as the driving frequency becomes comparable to internal resonance frequencies of the unperturbed double well with energy eigenstates E , E ,2 with corresponding eigenfunctions u (x), u (x),2 Thus, such resonances 1 2 1 2 occur at X"E !E , E !E , E !E ,2 etc. A spectral decomposition of the dynamics re3 2 4 1 5 2 solves the temporal complexity which is related to the landscape of quasienergies planes e (S, X) in a,k parameter space. Most important are the features near close encounters among the quasienergies. In particular, two quasienergies can cross one another if they belong to different parity classes, or otherwise, they form an avoided crossing. The situation for a single-photon transition-induced tunneling is depicted in Fig. 9 at the fundamental resonance X"E !E . For S'0, the corre3 2 sponding quasienergies e and e form avoided crossings, because they possess equal parity 2,k 3,k~1 quantum numbers. Starting from a state localized in the left well, we depict in Fig. 9a the probability to return P (t )"DSW(0)DW(t )TD2, t "nT. Instead of a monochromatic oscillation, L n n n which characterizes the unperturbed tunneling we observe in the driven case a complex beat pattern. Its Fourier transform reveals that it is mainly composed of two groups of three frequencies each (Fig. 9b). These beat frequencies can be associated with transitions among Floquet states at the avoided crossing pertaining to the two lowest doublets. The lower triplet is made up of the differences of the quasienergies e , e , and e ; the higher triplet is composed of the differences 3,~1 2,0 1,0 e !e ,e !e , e !e [25]. An analytical, weak-field and weak-coupling 4,~1 3,~1 4,~1 2,0 4,~1 1,0 treatment of a resonantly driven two-doublet system has been presented with Refs. [146,147]. The use of a nonlinear resonance for controlling tunneling has been studied for the case of a driven double-square well system in [148].

Fig. 9. Tunneling in a double well driven at the fundamental resonance X"E !E . (a) Time evolution of the 3 2 return probability P (t ), t "nT for a state initially localized in the left well over the first 105 time steps; L n n (b) corresponding local spectral two-point correlations PI (u) [25]. The dimensionless parameter values are D"2, S"2]10~3, and X"0.876.

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6.3. Coherent destruction of tunneling in bistable systems A particularly interesting phenomenon occurs if we focus on nearly degenerate states that are tunnel splitted. For example, let us consider in the deep quantum regime the two quasienergies e (S, X) and e (S, X). The subsets Me (S, X)N and Me (S, X)N belong to different parity classes so that 1,k 2,k 1 2 they can form exact crossings on one-dimensional manifolds, see below Eq. (180); put differently, at the crossing the corresponding two-photon transition that bridges the unperturbed tunnel splitting D is parity-forbidden. To give an impression of driven tunneling in the deep quantum regime, we 0 study how a state, prepared as a localized state centered in the left well, evolves in time under the external force. Since this state is approximately given by a superposition of the two lowest unperturbed eigenstates, DW(0)T+(DW T#DW T)/J2, its time evolution is dominated by the Floquet-state 1 2 doublet originating from DW T and DW T, and the splitting e !e of its quasienergies. Then 1 2 2 1 a vanishing of the difference e !e might have an intriguing consequence: For an initial state 2,~1 1,1 prepared exactly as a superposition of the corresponding two Floquet states W (q, t) and W (q, t), cf. 1 2 Eqs. (4) and (8), the probability to return P (t ), probed at multiples of the fundamental driving period L n T"2p/X, becomes time independent. This gives us the possibility that tunneling can be brought to a complete standstill [24,25]. For this to happen, it is necessary that the particle neither spreads and/nor tunnels back and forth during a full cycle of the external period T after which the two Floquet modes assemble again [67]. Hence, this condition [24,67], together with the necessary condition of exact crossing between the tunneling related quasienergies e "e (n is 2n,k~1 2n~1,k`1 number of tunnel-splitted doublet) guarantees that tunneling can be brought to a complete standstill in a dynamically coherent manner. In Fig. 10a we depict the corresponding one-dimensional manifold of the jth crossing between the quasienergies that relate to the lowest tunnel doublet, i.e., Mj/1(S, X), which is a closed curve that is reflection symmetric with respect to the line S"0. There -0# a localization of the wave function W(q, t) can occur. A typical time evolution of P (t ) for a point on L n the linear part of that manifold is depicted in Fig. 10b.

Fig. 10. Suppression of tunneling in a driven double well at an exact quasienergy crossing, e "e . (a) One of the 2,~1 1,1 manifolds in the (S, X)-plane where this crossing occurs (data obtained by diagonalization of the full Floquet operator for the driven double well are indicated by crosses, the full line has been derived from a two-state approximation, the arrow indicates the parameter pair for which part (b) of this figure has been obtained); (b) time evolution of the return probability P (t ) over the first 1000 time steps starting from an initial state prepared as a coherent state in the left well. L n Dimensionless parameters are D"2 (i.e., D "1.895210~4), S"3.171, X"0.01. 0

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Moreover, a time-resolved study over a full cycle (not depicted) does indeed show that the particle stays localized also at times tOt . Almost complete destruction of tunneling is found to n occur on M1 for D (X(E !E . For XPE !E , the strong participation of a third -0# 0 3 2 3 2 quasienergy mixes nonzero frequencies into the time dependence so that coherent destruction of tunneling at all times ceases to exist. For small frequencies, D /24X4D , and corresponding 0 0 small driving strengths S, as implied by M1 (S, X), the driven quantum mechanics approaches the -0# unperturbed quantum dynamics. In particular, it follows from (28) and (29) for XPD /2 and SP0, 0 i.e., U (q, t)"u (q) exp (iXt), U (q, t)"u (q) exp (!iXt), that 1,1 1 2,~1 2 (190) P (t)"DSW(0)DW(t)TD2" cos2(D t/2), X"D /2 . L 0 0 For X+D , e and e exhibit an exact crossing. With corresponding Floquet modes deter0 1,1 2,~1 mined from perturbation theory as 1 U (q, t)+ [u (q) exp(iXt)#iu (q)] , 1,1 2 J2 1 1 U (q, t)+ [u (q) exp(!iXt)#iu (q)] , 1 2,~1 J2 2

(191)

the result for P (t), with W(q, 0)"[u (q)#u (q)]/J2, localized in the left well, becomes L 1 2 (192) P (t)+1[3# cos (2D t)], X+D . 0 0 L 4 For larger frequencies obeying D (X(E !E , the Floquet modes can be approximated by 0 3 2 [67] U (q, t)+u (q)D sin(Xt)D!iu (q) cos(Xt) , 1,1 2 1 U (q, t)+u (q)D sin(Xt)D!iu (q) cos(Xt) . 2,~1 1 2 This results in a complete localization, i.e.,

(193)

P (t)"1, D (X(E !E . (194) L 0 3 2 Throughout Eqs. (190)—(194), we set the initial phase in Eq. (172) equal to zero. Starting from a coherent state localized in the left well, taken as the ground state of the harmonic approximation, we depict in Fig. 11 the spatially resolved tunneling dynamics for DW(q, t)D2 at time t"0 and at time t"458 T for X"0.01"52.77D , and S"3.17]10~3, yielding an exact 0 crossing between e and e . For this value of n"458, the deviation which originates from 1,1 2,~1 small admixtures of higher-lying quasienergy states to the initial coherent state, is exceptionally large. For other times the localization is even better. It is hence truly remarkable that the coherent destruction of tunneling on M1 (S, X), with D (X(E !E , is essentially not affected by the -0# 0 3 2 intrinsic time dependence of the corresponding Floquet modes, nor by the presence of other quasienergy states e , a"3,4,2 a,k Hence, as discussed in Section 3.2.2, additional insight into the mechanism of coherent destruction of tunneling can be obtained if one simplifies the situation by neglecting all of the spatial

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Fig. 11. Coherent destruction of tunneling in a double well. The probability DW(q, t)D2 at t"458 T (full line) is compared with the initial state (dashed line, the dotted line depicts the unperturbed symmetric bistable potential). The parameters are D"2, S"3.171]10~3 and X"0.01, i.e., X equals 52.77 times the unperturbed tunneling splitting.

information contained in the Floquet modes U (q, t) and restricting the influence of all quasienera gies to the lowest doublet only [67—72]. Such a two-state approximation cannot reproduce those sections of the localization manifolds that are affected by resonances, e.g. the part in Fig. 10a that bends back to S"0 for X(E !E , nor can it reproduce the correct dynamics for X'E !E . 3 2 3 2 Nevertheless, it provides a simple explanation of the CDT phenomenon for frequencies E !E 'X'D . Then, the average over a complete cycle of the external perturbation yields 3 2 0 a static approximation with a frequency-renormalized splitting D PD , where 0 %&& (195) D "J (eL /X)D , %&& 0 0 with SSu DqDu T"eL /2, and J the zeroth-order Bessel function of the first kind. This gives for the 1 2 0 return probability P (t) the approximate result of Eq. (100), i.e., P (t)"cos2(J (eL /X)D t/2). On the L L 0 0 localization manifold M1 (eL , X), we find localization at the first zero j of J (eL /X), i.e., -0# 0,1 0 eL /X"2.404822, in agreement with the result in Eq. (88). On this manifold, P (t) precisely equals L unity, i.e., the effective tunnel splitting vanishes. Thus one finds a complete coherent destruction of tunneling. This high-frequency TLS approximation, as determined by the first root of J (eL /X), is 0 depicted in Fig. 10a by a solid line. Note that, within the TLS-approximation to driven tunneling, tunneling is always suppressed when X'D . With X'E !E , this two-level prediction thus 0 2 1 fails to describe the numerically established enhancement of tunneling in the full double well away from resonant transitions [24,25].

7. Sundry topics The physics of driven tunneling, as demonstrated with the foregoing sections, exhibits a rich structure and carries a potential for various prominent applications. In the following, we shall report on some special applications that provide the seed for interesting new physics. Our selection is by no means complete but has been determined mainly by our knowledge and prejudices only. For example, we do not discuss here in detail the role of nonadiabatic CDT that controls the

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experimentally observed anomalous enhancement of ionization rates of laser-driven diatomic molecules [149], or the vibrational mode-specific CDT-induced quenching of proton tunneling in jet-cooled tropolene molecules [150]. Yet another application of CDT in nuclear physics is the tunneling control in the periodically driven Lipkin N-body fermion system with SU(2) symmetry: At CDT conditions a coherent suppression of tunneling between Hartree—Fock minima in the N-fermion quantum system occurs [151]. 7.1. Pulse-shaped controlled tunneling Some important aspects involve the shaping of the driving field that itself can influence the driven tunneling dynamics. For example, Bavli and Metiu [152] have demonstrated that a semi-infinite laser pulse of appropriate shape can be used to conveniently localize a particle in a double-well potential, hence creating a localized initial state as a superposition of energy eigenstates. This very initial condition plays a key role for most of our discussion on driven tunneling throughout this report. Pulse-shaping can be used also to directly influence the dynamics of the tunneling process. Following the reasoning of Holthaus [153] we show that an appropriate shaping of the laser pulse opens a pathway of controlling in situ the tunneling process. As a result, one finds that population transfer in a bistable potential can be achieved on time scales which can be orders of magnitude shorter than the bare tunneling time scale of the undriven system. Hence, this process is opposite to the localization that characterizes CDT; it rather describes an acceleration effect for the tunneling process. Within a pragmatic approach we use the bistable model discussed in Section 6, i.e., a driven tunneling dynamics expressed in the dimensionless units, see Eqs. (174) and (175) by the Hamiltonian p2 q2 q4 H(q, p; t)" ! # #qj(t) sin(Xt#/) , 2 4 64D

(196)

where j(t) describes the pulse shape. If E and E 'E denote the lowest eigenenergies of the 1 2 1 unperturbed double well with corresponding wave function u (q) and u (q), the tunneling dynam1 2 ics of the undriven system evolves in time as W(q, t)"2~1@2e~*E1t[u (q)#u (q)e~*(E2~E1)t] , 1 2

(197)

for which, after the “tunneling time” q "p/(E !E )"p/D , 0 2 1 0

(198)

the relative phase of the two wave functions has acquired the value of p. This implies that an initially localized particle in one of the wells has tunneled after a time q into the neighboring well. 0 For a fixed strength j(t)"j, the Floquet modes U (q, t; j)"U (q, t#T; j), see Eq. (7), with a a T"2p/X, are solutions of a generalized Schro¨dinger equation with the eigenvalues becoming the quasienergies ej. Let us next invoke the adiabatic principle for Floquet states: If the system is a prepared in a Floquet state and if the parameter j(t) is varied sufficiently slowly, the system remains in the Floquet state connected to the initial one. This adiabatic Floquet principle [44—48] is not obvious: Note that the quasienergies are dense in the first Brillouin zone !X/24e(X/2, and

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that the usual gap condition of the standard adiabatic theorem must be replaced by a condition on the ineffectiveness of resonances [47,48]. Let the pulse j(t) now be switched on slowly from zero, i.e., the pulse duration T is much longer than the driving period T. The Floquet state W(q, t), 1 starting from W(q, t"0) " : u , L see Eq. (197), evolves then adiabatically into

G

C P

W(q, t)"2~1@2 W (q, t; j(t)) exp !i 1

t

(199)

D

C P

t

DH

ds ej(s) 2 0 0 with the relative phase acquiring after a complete duration T of a pulse the value 1

P

D (T )" T 1

T 1

ds(ej(s)!ej(s)) . 2 1 0 If the conditions are such that

ds ej(s) #W (q, t; j(t)) exp !i 1 2

,

(200)

(201)

D (T )"(2n#1)p, n"0,$1,$2,2 (202) T 1 the initial localized wave function has, up to an irrelevant overall phase factor, evolved into a localized wave function in the neighboring well. Thus, the pulse has forced the particle to tunnel in time T through the barrier. The advantageous feature of strong laser-pulse driving is now that 1 the quasienergy difference ej !ej may become much larger than the original tunnel splitting 2 1 D "E !E ; thus, the actual tunneling time q , which now is defined by p"D (q ), can assume 0 2 1 T T T values much shorter than the bare tunneling time q in Eq. (198). The viability of this mechanism 0 has been demonstrated with Ref. [153], wherein the time-dependent Schro¨dinger equation corresponding to Eq. (196) was solved numerically with j(t) conveniently chosen as j(t)"j sin2(pt/T ), 04t4T . (203) .!9 1 1 The results are depicted in Fig. 12. The manipulation of the relative phase between the two adiabatically moving components of the wave function results in speeding up the tunneling time q by orders of magnitude, e.g. q K10~2q in the calculation in Fig. 12. Intuitively, this speed-up T T 0 effect is rooted in the large differences of quasienergies with the corresponding Floquet states acquiring admixtures from excited states that are located closer to the barrier top and, therefore, these tunnel faster. It surely would be of interest to measure this tunneling acceleration with suitably engineered semiconductor double-quantum well structures, or in laser-assisted tunneling of proton transfer in molecules. 7.2. Chaos-assisted driven tunneling A periodically driven nonlinear classical dynamics generally exhibits chaos [154,155]. Clearly, the archetype system of a periodically driven, symmetric double well in Eqs. (172)—(175) is an ideal candidate to study the interplay between nonlinear tunneling quantum dynamics and classical Hamiltonian chaos. The parameter that controls the semiclassical limit is the scaled barrier height

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Fig. 12. Pulse-shaped controlled tunneling in the symmetric double well, Eq. (196), with dimensionless parameters D"2.5, q "49766 T, T"2p/X, and X"1.5, and with q "p/(E !E ) the tunneling time. The pulse function 0 0 2 1 j(t)"j sin2(pt/T ) with T "500T is used. The upper-half depicts the tunneling phase functional D (T )/p vs. the .!9 1 1 T 1 maximal amplitude j . For the maximal amplitudes j satisfying the condition in Eq. (202), note the vertical lines in .!9 .!9 the upper panel, a complete transfer of localization from “left” to “right” occurs, i.e., W(q, t"0)"u PW(q, t"T )" L 1 u . The lower half depicts the tunneling probability, DSu DW(T )TD2, vs. j , as calculated numerically from the R R 1 .!9 time-dependent Schro¨dinger equation for the Hamiltonian (196). The agreement with the prediction from the adiabatic Floquet principle is remarkable.

D"E /+u . While the phenomenon of coherent destruction of tunneling in a double well is B 0 controlled by the dynamics in the deep quantum regime with D small, the semiclassical limit amounts to letting DPR. In the latter case the number of doublets below the barrier becomes very large. The idea that chaotic layers separating symmetry-related regular regions can give rise to quantal tunneling is known as dynamical tunneling [156]. The fingerprints of dynamical tunneling in the spectrum has been discussed by Bohigas et al. [157—160]. These works on chaos-assisted tunneling were based on time-independent two-dimensional nonlinear oscillator systems. Moreover, tunneling through dynamical barriers has been studied with forced nonlinear oscillator systems [161—163]. The effect of a chaos-assisted increase of the tunneling rate in a driven double well was first pointed out by Lin and Ballentine [164,165], and by Plata and Gomez-Llorente [166]. By monitoring the Husimi distributions [167,168] of the wave functions, these authors showed that for certain values of the driving parameters for the Hamiltonian in Eq. (172) the rate of tunneling can be enhanced by orders of magnitudes. This enhancement phenomenon should not be confused with

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Fig. 13. Chaos-assisted driven tunneling: (a) Tunneling splittings D "De !e D of associated (even and odd) pairs of n n,% n,0 quasienergies in a driven, symmetric double well, see Eqs. (172)—(174) vs. driving strength S of the periodic driving !qS cos(Xt). (b) Corresponding overlaps C "1 :T dt:K dp dq h (q, p, t) of the corresponding Husimi representation n T0 n,% h for the Floquet state W with the chaotic layer K around the separatrix vs. driving strength S. The dimensionless n,% n,% parameter values are D"8 and X"0.95.

the pulse-shaped enhancement discussed in Section 7.1; the latter is controlled by the (adiabatic) lowest two quasienergy states in the deep quantum regime, the former is related to the interplay between different topologies of phase space in the strongly driven system near the classical limit with D<1. The interplay between regular and chaotic regions in driven tunneling has been investigated in terms of the Floquet spectrum by Utermann et al. [169,170]. The underlying scenario for chaos-assisted driven tunneling is as follows. For not too strong driving, the classical phase space of this system is characterized by a pair of symmetry-related regular regions near the minima of the unperturbed potential, embedded in a chaotic layer along the unperturbed separatrix. Quantal quasienergy states localized in the regular regions form tunnel doublets with exponentially small splittings. These doublets do break up, and their splittings will reach values of the order of the mean level spacing, as soon as the associated pair of quantizing tori dissolves in the chaotic layer. Therefore, the tunnel splittings depend strongly on the nature of regular vs. chaotic phase space regions to which the corresponding pair of Floquet states predominantly belongs. These ideas have been tested by studying the break-up of tunnel splittings D versus the driving amplitude S, and comparing the n behavior with the measure of overlap C of the Husimi representation of associated (even) Floquet n,% state with the chaotic region in the vicinity of the separatrix; for further details we refer to [169,170]. Fig. 13, which depicts these measures as a function of driving strength S, shows a striking qualitative agreement between the functional dependences of the splittings D and the correspondn ing overlaps C of its corresponding Husimi representation with the chaotic layer. Both quantities n exhibit a transition to steep exponential growth as the chaotic sea (not shown) along the separatrix spreads at the expense of regular regions around the potential minima.

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7.3. Even harmonic generation in driven double wells As Bavli et al. [27—29] have shown, the phenomenon of coherent destruction of tunneling for electrons in a quantum double well implies striking new features when related spectroscopic quantities are investigated. By studying the dipole moment k(t) that is dynamically induced by the action of the laser driving field, one encounters new phenomena such as anomalous evenharmonic-generation (EHG), and a novel low-frequency radiation generation (LFG). These phenomena are a priori rather surprising: The double well of the type in Eq. (172) in which the electron is moving is symmetric, i.e., the undriven Hamiltonian is invariant with respect to a parity transformation of the electron position q. Perturbation theory to all orders consequently implies that such a system cannot produce even harmonics. We note, however, that the violation of this selection rule takes place precisely at the same points in the parameter space MS, XN where the necessary (but not sufficient) condition for CDT occurs, i.e., exact crossings between associated pairs of quasienergies of opposite generalized parity symmetry P, cf. Eq. (177). The new phenomena can be best understood in terms of the spectral properties of quasienergies. In terms of the time-dependent wave function W(q, t) of the time-dependent Schro¨dinger equation, i.e.,2 W(q, t)"+ A exp(!ie t/+)U (q, t) , a a a a and expanding the periodic Floquet mode U (q, t)"U (q, t#T) into Fourier series, i.e., a a = W(q, t)"+ A exp (!ie t/+) + exp(!inXt)c (q; a) , a a n a n/~= we find for the time-dependent dipole moment k(t)"SW(t)DqDW(t)T

(204)

(205)

(206)

the spectral Floquet representation

P

= = = k(t)"+ + A*A exp [i(e !e )t/+] + dq c*(q; a)qc (q; b) . + exp[i(n!m)Xt] a b a b n m ~= a b n/~= m/~=

(207)

The generalized parity P : qP!q; tPt#T/2, cf. Eq. (177), implies even or odd generalized parity P(a)"$1 for the Floquet modes U (q,t#T/2)"$U (q, t). In particular, one finds that a a the integral in Eq. (207) vanishes, i.e.,

P

=

dq c* (q; a)qc (q; b)"0 , n m

(208)

~= if P(a)"P(b) and n!m is even, or P(a)OP(b) and n!m is odd.

2 If the drive has a constant amplitude, i.e., a pure periodic driving field, the coefficients A are independent of time. If a the system is driven by a semi-infinite pulse, the coefficients A become dependent on time, i.e., A PA (t), but become a a a again independent of time when the transients excited by turning on the pulse quiet down.

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These selection rules describe all of the novel spectroscopic features discussed in Ref. [27]. There, Bavli and Metiu have shown that EHG appears and LFG disappears when two quasienergies become degenerate. Moreover, the selection rules explain why one cannot detect pure even harmonics when the quasienergies do not exhibit exact crossings, and that one observes for an initially localized state with P(a)OP(b) no shifted odd harmonics regardless of angular driving frequency X and driving strength S. In somewhat greater detail we observe that the frequencies u(a, b; n!m) occurring in the spectrum of k(t) have the form +u(a, b; n!m)"e (X, S)!e (X, S)#(n!m)+X . a b

(209)

Therefore, we can characterize the spectrum according to four types: 1. The static dipole component corresponds to a"b and n"m, and is independent of time. For a symmetric system this term is always zero, except when n"m and P(a)OP(b) with e (X, s)"e (X, S), i.e., at the exact crossing points that rule CDT. The dynamic localization a b yielding CDT thus generates also a nontrivial dynamically induced static dipole moment! 2. Purely harmonic terms correspond to a"b and nOm. For a symmetric double well no terms with (n!m) even appear, hence — away from points of accidental degeneracy — no even harmonics are present in the spectrum. 3. Shifted harmonic terms correspond to aOb and nOm, and a progression of doublets occur at +u(a, b; n!m)"(n!m)+X$[e (X, S)!e (X, S)]. If P(a)"P(b), then no shifted even hara b monics occur; if P(a)OP(b), then no shifted odd harmonics occur. 4. Finally, the fourth type occurs for aOb and n"m. They oscillate with frequency u(a, b; 0)"[e (X, S)!e (X, S)]/+ and are termed “bandhead” terms by Bavli and Metiu. a b As already mentioned, the spectrum changes as two Floquet states a and b possessing a different generalized parity P(a)OP(b) approach each other, and become degenerate. Then, the frequency of a “bandhead” term is becoming smaller, and eventually approaches zero. This is the effect of LFG, which in turn provides the generation of a nonzero static dipole. Moreover, the two shifted harmonics having the frequencies u(a, b; 2n)"2nX$[e (X, S)!e (X, S)]/+ collapse at the point of a b degeneracy e "e with P(a)OP(b). In this limit, one observes a progression of (doubly degenerate) a b even harmonics of frequency 2nX; hence, EHG. The dependence of the spectral weights on the choice of initial condition, i.e., optical k(t"0)"0, versus tunneling initial conditions, i.e., k(t"0)"1, have been investigated in Ref. [28]. As mentioned at the beginning of this section, tunneling initial conditions can be realized with a semi-infinite laser pulse [27,152]. Compared to optical initial conditions, i.e., preparation in the ground state, localized initial conditions are more difficult to realize, but preparation of such coherent states has been recently achieved experimentally [171,172]. As demonstrated by Dakhnovskii and Bavli [28], a considerable amount of LFG, but only weak EHG, can be obtained for excitation from the ground state. For a discussion of the possibility to observe these effects experimentally we refer to the literature [27]. They are not easy to detect, due to various dephasing effects. Good candidates to observe these phenomena of EHG and LFG are quantum double-well heterostructures and optical systems. Alternatively, generation of strong higher-order harmonics has been predicted for short laser pulse driven diatomic molecular ions [173,174].

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7.4. General spin systems driven by circularly polarized radiation fields The fact that the time evolution of a TLS in a circularly polarized field can be factorized in terms of a time-independent Hamiltonian in Eq. (111) is surprising. We note that this factorization involves a rotation around the z-axis, i.e., Da(t)TPDb(t)T" exp(!iS Xt/+)Da(t)T. This feature can be z generalized to any higher-dimensional system, such as a magnetic system or a general quantum system that can be brought into the structure which, in a representation where J is diagonal, is of z the form H(t)"H (J2)#H (J )!eL [J cos (Xt)!J sin (Xt)] . (210) 0 1 z x y Here, H (J2) contains all interactions that are rotationally invariant (Coulomb interactions, 0 spin—spin and spin—orbit interactions). If we now define R(t) " : exp(!iJ Xt/+) and observe that z R(t)J R(t)~1"J cos Xt#J sin Xt , x x y R(t)J R(t)~1"!J sin Xt#J cos Xt , (211) y x y R(t)J R(t)~1"J , z z we find upon substituting Eq. (211) into Eq. (210)3 HI (t),R(t)H(t)R~1(t)"H (J2)#H (J )!eL J . (212) 0 1 z x Hence, the transformed Hamiltonian becomes independent of time. With the propagator obeying i i º(t, t )"! H(t)º(t, t )"! R~1(t)HI R(t)º(t, t ) , 0 0 0 t + + we find from i [R(t)º(t, t )R~1(t )]"! HK [R(t)º(t, t )R~1(t )] , 0 0 0 0 + t

(213)

where HK "HI #XJ "H #H (J )!eL J #XJ , z 0 1 z x z that the propagator factorizes into the form

A B A

B A

(214)

B

i i i º(t, t )"exp J Xt exp ! HK (t!t ) exp ! J Xt . 0 0 + z + + z 0

(215)

3 Such a factorization method equally well applies to rotationally invariant Hamiltonians H which are subject to 0 externally applied, monochromatic, spatially homogeneous, circularly polarized electric fields E(t)"!(d/dt)A(t), with A(t)"X~1(E cos (Xt),E sin (Xt),0). Then the interaction with the field transforms into R(t)[p!QA(t)]2R~1(t)" 0 0 [p!QAI ]2, with AI "X~1(E ,0,0), and Q being the charge. 0

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Because exp(iJ Xt/+) at times t"2p/X equals 1 for integer values of the angular momentum and z !1, for half-integer values, respectively, the propagator in Eq. (215) can be recast into the Floquet form in Eq. (36), i.e., º(t#nT, 0)"º(t, 0)[º(T, 0)]n .

(216)

With J "$1,$2,2 , the Floquet form, cf. Eq. (35), is achieved already with Eq. (215). For z half-integer spin the corresponding Floquet form is obtained by setting for the propagator

AA

B B A A

B

B A A

B B

+ i + i + i º(t, t )"exp J # Xt exp ! HK # X (t!t ) exp ! J # Xt , 0 0 0 + 2 + z 2 + z 2

(217)

since the first and third contribution are now periodic with period T. Given the eigenvalues MeL N of a HK , the exact quasienergies are given by the relation e "(eL #+X/2) mod +X . (218) a a The general results derived here carry a great potential for applications involving time-dependent tunneling of spin in magnetic systems with anisotropy, and strongly driven molecular and quantum optical systems as well. 7.4.1. Tunneling of giant spins in ac-fields In recent years, macroscopic tunneling of magnetization has been the subject of intense research [175,176]. More recently, the problem of tunneling of a giant spin with spin quantum number J<1 under periodic magnetic field perturbation has attracted interest in connection with the behavior of macroscopic magnetic moments in an anisotropic field [177,178]. Introducing a WKB-analysis for semiclassical-driven quantum systems, these authors assess that, as a rule, tunneling of spin is generally hampered rather than accelerated. One particular driven anisotropic spin system is given by the Hamiltonian H"!cJ2!eL [J cos (Xt)#J sin (Xt)] , (219) z x y which is of the circularly polarized type considered in Eq. (210). The anisotropy, which produces the “energy barrier”, is given by the term !cJ2. For X"0, the ground-state splitting results as z [179]

A B

eL 2J~1 +D "eL , 0 +cJ

J<1 .

(220)

Tunneling of giant spin is shown to occur at frequencies X"+c, 2+c,2, J+c, with blocking of tunneling occurring in between. In particular, for X obeying [178] (2eL /e+cJ)4J;2X/+c;1,

J<1 ,

(221)

where e denotes the basis of the natural logarithm, coherent tunneling is hampered. We note that, by expressing (221) in terms of the energy splitting (220), this condition implies high-frequency driving X
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8. Driven dissipative tunneling Up to this point in our review we have investigated the effects of driving on a quantum system with few degrees of freedom, under the hypothesis that this system could be considered as isolated from its surroundings. This idealization often fails to describe thermal and dynamical properties of real physical or chemical systems when the quantum system is in mutual contact with a thermal reservoir. In this case, the system has to be considered as an open system. The coupling with a heat bath causes damping of the quantum system and no longer is the dynamics predicted by the Schro¨dinger equation [30—35]. In the following, we shall be interested in the investigation of the dynamical behavior of a “small” quantum system in contact with a heat bath, and that additionally is subject to an external, generally time-dependent force. We start the discussion of dissipative-driven tunneling by presenting first the basic methodology used to describe quantum dissipation. In the following section we discuss a Floquet—Markov—Born approximation to the dynamics which can be applied to highly nonlinear systems under the assumption of weak coupling between the system and the reservoir, and whenever the external driving field is periodic. An application of this method is considered in Section 14.2, where driven tunneling in a dissipative double-well potential is discussed. In Section 10, we shall present the real-time path integral technique, which allows to treat the dissipative influence systematically. In particular, for a harmonic thermal bath, the bath degrees of freedom can be eliminated exactly. We shall apply this method to investigate driven tunneling in dissipative tight-binding systems in Sections 11—13, and the extended double-well system in Section 14.3. A comparison between the path integral results and those of other methods will also be discussed. 8.1. The harmonic thermal reservoir A great number of methods and techniques have been developed to investigate the thermodynamics and dynamics of open quantum systems in the absence of driving. A basic approach is to regard the system and reservoir as constituting a conservative many-body system. In this picture, dissipation comes into play because of energy transfer between the small system and the large environment. Once the energy is deposited into the environment, it is not returning within any finite time scale of physical relevance (infinite Poincare´ recurrence time). If the equations of motion for the closed system are linear in the bath coordinates, the latter can be eliminated exactly and one obtains closed equations for the damped system alone. This restriction on linearity assumes that any bath degree of freedom is sufficiently weakly perturbed by the coupling with the small system [30—35]. This is a physically reasonable assumption when the reservoir is macroscopic. It is important to notice that this does not necessarily imply that the dissipative influence of the environment on the small system is weak, because the number of bath degrees of freedom which couple to the small system is very large. For a more detailed and complete description we refer to books, reviews and articles such as, for example, Refs. [32—35]. Here, we generalize these concepts to include the effects of time-dependent driving. A general form of the Hamiltonian of the driven system-plus-reservoir complex which satisfies the requirement of a linear coupling of the bath degrees of freedom to the system degree of freedom is given by H(t)"H #H #H #H (t) , 0 B SB %95

(222)

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where H is the Hamiltonian of the isolated, generally nonlinear system, 0 H "p2/2m#» (q) . 0 0

(223)

Here we restrict ourselves to a one-dimensional system coordinate q. A generalization of the formalism to higher-dimensional cases is straightforward. The terms H #H describe the B SB reservoir as a set of N harmonic oscillators linearly coupled in the bath coordinates x to the system i via the interaction term H . That is, SB

C

A

BD

1 N p2 c 2 i #m u2 x ! i F(q) H #H " + . B SB 2 i i i m u2 m i i i i/1

(224)

Here, the term F2(q)+ c2/2m u2 has been introduced to avoid renormalization effects of the i i i i potential » (q) induced by coupling to the reservoir. The generally nonlinear function F(q) 0 describes a nonlinear system-bath coupling. Finally, H (t)"!S(q)E(t) %95

(225)

describes the interaction of the quantum system with an external force. For the sake of simplicity, we shall further restrict our discussion to the case that the dissipation mechanism is strictly linear in q, that is, F(q)"q, which describes a bi-linear system-bath coupling. Defining next » (q, t)"» (q)!S(q)E(t), we find 1 0

C

A

BD

p2 1 N p2 c 2 i #m u2 x ! i q H(t)" #» (q, t)# + . 1 i i i 2m 2 m u2 m i i i i/1

(226)

Without driving, this Hamiltonian has been used to investigate quantum dissipation since the late 1950s [30,33—35,180—184]. In particular, Magalinskij [180] pioneered (see also [182,184]), the so-termed “quantum Langevin equation” for the coordinate operator q, by starting from the full system-plus-bath Hamiltonian. Generalized to our time-dependent Hamiltonian this Heisenberg operator equation for the coordinate operator q takes the form

P

d2q(t) dq(t@) » (q, t) t m #m dt@c(t!t@) # 1 "f(t)!mc(t!t )q(t ) , 0 0 dt2 dt@ q t0

(227)

where t denotes the time of initial preparation. Note that in the literature (but not here) the initial 0 c(t!t )"0. term mc(t!t ) is frequently dropped4 assuming that with time t P!R, lim 0 0 0 0 t ?~= By doing so, however, one neglects transient effects. Here,

C

c f(t)"+ c x (t ) cos [u (t!t )]# i p (t ) sin [u (t!t )] i i 0 i 0 i 0 mu i 0 i i i

D

(228)

4 For a detailed discussion of the use and abuse of (quantum) Langevin equations using the stochastic force m(t)"f(t)!mc(t!t )q(t ) we refer to [185]. 0 0

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is a random force operator obeying Gaussian statistics, with zero mean Sf(t)T "0, and force—force b correlation

P

+m = K(t)"Sf(t)f(0)T " du ucJ @(u)[coth(+bu/2) cos (ut)!i sin (ut)] . b p 0

(229)

Here, b"1/k ¹ is the inverse temperature. The bracket S T indicates the statistical average with B b respect to the bath Hamiltonian H . Moreover, cJ @(u) is the real part of the Fourier transform cJ (u) of B the time-dependent damping coefficient c(t) given by 1 N c2 i cos[u (t!t@)] . c(t!t@)" + i m m u2 i/1 i i

(230)

It is interesting to observe that the result (229) for the force—force correlation function derived for the oscillator model is a general result of the fluctuation-dissipation theorem, and is therefore independent of the microscopic details of the harmonic bath [180]. In the classical limit +P0 the quantum Langevin equation reduces to the well-known phenomenological Langevin equation with memory-dependent damping, and where the force—force correlator of the classical force f (t) #satisfies Sf (t)f (t@)T "2mk ¹c(Dt!t@D). #- #b B 8.2. The reduced density matrix Starting from the full system-plus-bath Hamiltonian (222), the aim of the following sections will be to evaluate the reduced dynamics of the system described by the reduced density operator o(t)"Tr M¼(t)N, where Tr denotes the trace over the bath degrees of freedom of the Hamiltonian B B (226), and with ¼(t) being the density operator of the system-plus-reservoir complex. In terms of the density operator at initial time ¼(t ) and of the time evolution operator º(t, t )" 0 0 Á expM!i:t0 dt@ H(t@)/+N, where Á denotes the time-ordering operator, it reads t ¼(t)"ºs(t, t )¼(t )º(t, t ) . 0 0 0

(231)

In the following it will be convenient to consider an intial condition on ¼(t) of the Feynman— Vernon type [186], that is, to assume that at the initial time t"t the bath is in thermal 0 equilibrium and uncorrelated to the system. Hence, ¼(t ) has the form 0 ¼(t )"o (t )¼ , 0 S 0 B

(232)

where ¼ "exp(!bH )/Tr Mexp(!bH )N is the canonical density operator of the heat bath, and B B B B o that of the system described by the Hamiltonian H (t)"H #H (t). The inclusion of more S S 0 %95 realistic initial preparations goes beyond the scope of this review. We refer to Refs. [34,187] where this generalization is implemented for open systems without driving. Nevertheless, we observe that while the effects of the initial preparation are crucial in the investigation of the transient dynamics of the system, if the total system is ergodic, initial preparation effects will have decayed when the asymptotic dynamics is considered.

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8.3. The environmental spectral density It is convenient at this point to introduce the spectral function J(u) [32] of the environmental coupling, i.e., p N c2 i d(u!u ) . J(u)" + (233) i 2 m u2 i/1 i i Because we aim at the environment to constitute a proper heat bath, we shall consider J(u) to be continuous. Hence, the sum in Eqs. (230) and (233) may be replaced by an integral. In doing so one obtains J(u)"umcJ @(u) .

(234)

The model described by the Hamiltonian (226) is thus completely fixed by the mass m, the time-dependent potential » (q, t) and by the spectral density J(u).5 Moreover, the latter is 1 completely determined by the frequency-dependent damping coefficient cJ @(u) which already appears in the classical equations of motion. Hence, whenever J(u) cannot be extracted from microscopical considerations on the composite system, a conventional molecular dynamics simulation may be used to compute cJ @(u), and in turn J(u). When the damping coefficient is frequencyindependent, i.e., cJ (u)"c, the dynamics is memoryless, and the spectral density assumes the Ohmic form J(u)"mcu .

(235)

This corresponds in the classical case to a Langevin equation with delta-correlated Gaussian random forces (white noise). In reality, any physical spectral density falls off in the limit uPR. Hence, a cutoff frequency u on the spectral density has to be introduced, which sets the time scale # q "1/u for inertia effects in the reservoir. For example, for the Drude-regularized kernel # # c(t)"cu exp(!u t) the spectral density reads # # mcu J(u)" . (236) 1#u2/u2 # This is of the Ohmic form when u;u . Very often in the literature the case of Ohmic friction with # an exponential cutoff is considered. In this case J(u)"mcue~u@u# .

(237)

This reasoning can be extended to consider the case of general frequency-dependent damping. That is, it is assumed that the spectral density has a power-law dependence on the frequency for frequencies less than a characteristic cutoff frequency u . Introducing a cutoff function f (u, u ), the # # spectral density then has the form J(u)Jc usf (u, u ), f (u, u )P0 if u
(238)

5 This fact, for example, no longer holds true if one considers a different system-bath coupling, such as a coupling to a bath composed of two-level systems.

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where s'0. As already discussed, the case s"1 describes Ohmic damping. The cases 0(s(1, and s'1, are known as sub-Ohmic and super-Ohmic, respectively. The case of Ohmic friction is important, e.g., for charged interstitials in metals or in Josephson systems. On the other hand, super-Ohmic dissipation with s"3 or s"5 (depending on the properties of the strain field) describes the low-temperature dynamics of a phonon bath in the bulk [34]. It is clear that the assumption of Eq. (238) may oversimplify the problem for frequencies u of the order of, or greater than u . Nevertheless, whenever J(u) falls off at least as some negative power of # u when uPR, the effects of the environmental high-frequency modes can be absorbed into a renormalization of the parameters appearing in the Hamiltonian H of the system; see [188] for S the case of a double-well system. 9. Floquet—Markov approach for weak dissipation In this section we shall describe the Floquet—Markov—Born approximation to the dynamics, which can be applied to strongly nonlinear systems under the assumption of weak coupling between the system and the reservoir, and whenever the external driving field E(t) is periodic, i.e., if E(t#T)"E(t), T"2p/X .

(239)

This method combines the Markov—Born approach to quantum dissipation (leading to a master equation for the reduced density operator) with the Floquet formalism, cf. Section 2, that allows to treat time-periodic forces of arbitrary frequency and strength. While the Floquet formalism amounts to using an optimal representation and is exact, the simplification brought by the Markov—Born approximation is achieved at the expense of the regime of validity of the method. The Markov—Born approximation, in fact, restricts the application of the master equation for the reduced density operator to weakly damped systems for which the dissipative relaxation times are large compared to the relevant time scales of the isolated system, and also to the thermal time scale +b. For example, for Ohmic damping, the Markov—Born approximation requires that the conditions +c;k ¹, and c;D are satisfied, where D are the transition frequencies of the central B a,l a,l system, see Eq. (247) below. A subtle point is that the results so obtained depend on whether the Markovian approximation is made with respect to the eigenenergy spectrum of the small quantum system without driving, or with respect to the quasienergy Floquet spectrum of the small quantum system in the presence of driving [189], that is, the two procedures do not commute. Here we shall report on the results of a Markovian description of the dynamics based on a quasienergy spectrum. This method has recently been applied to the parametrically driven harmonic oscillator [190], and to the dissipative tunneling of a particle in a driven bistable potential [191,192]. The latter case is discussed in Section 14.2. 9.1. Generalized master equation for the reduced density operator The starting point of this method is the equation of motion for the reduced density operator o(t)"Tr M¼(t)N (where Tr denotes the trace over the bath degrees of freedom of the Hamiltonian B B (222)), as it can be derived from the Liouville—von Neumann equation ¼ Q (t)"!(i/+)[H(t), ¼(t)]

(240)

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288

for the full density operator ¼(t) of system-plus-reservoir. For simplicity, we assume that at the initial time t"t the bath is in thermal equilibrium and is uncorrelated with the system, cf. (232). 0 Under the hypothesis of a weak coupling to the reservoir, a transformation into the interaction picture allows a weak coupling expansion with respect to the interaction Hamiltonian H . Tracing SB out the bath degrees of freedom, and by making a Markovian approximation by assuming that the autocorrelations of the bath decay on a time scale +/k ¹ much shorter than the time scale 1/c of the B system correlations, one arrives at equations of motion for the reduced density matrix. They have the form [193—195]

P

i 1 = oR (t)"! [H (t), o(t)]! dqMK@(q)[q, [qJ (t!q, t), o(t)]]!iKA(q)[q, [qJ (t!q, t), o(t)] ]N , ` + S +2 0 (241) where K@(q) and KA(q) are the real and imaginary parts, respectively, of the force—force correlation function K(q) in Eq. (229). Here, [A, B] "AB#BA is the anticommutator, while the tilde ` denotes the interaction picture defined by AI (t, t@)"ºs (t, t@)Aº (t, t@) , 0 0

A P

(242)

B

i t º (t, t@)"Á exp ! dtA (H (tA)#H ) , 0 S B + t{

(243)

with Á the time-ordering operator. 9.2. Floquet representation of the generalized master equation In Section 2 we saw that for a time-periodic Hamiltonian H (t), the Floquet theorem asserts that S there exists a complete set of solutions of the form DW (t)T"e~*eat@+DU (t)T, DU (t#T)T"DU (t)T . a a a a

(244)

The quasienergy e plays the role of a phase and therefore is only defined mod +X. We shall apply a the Floquet theorem to the Hamiltonian H (t)"H #H (t), cf. Eq. (222), and use the basis S 0 %95 MDW (t)TN of the undamped system as an optimal representation to decompose states and operators. a In particular, the matrix elements X (t) of the position operator q with the representation of states ab DW (t)T read a X (t)"e*(ea~eb)t@+SU (t)DqDU (t)T"+ e*DabktX , ab a b abk k

(245)

where

P

1 X " abk T

T

0

dt e~*kXtSU (t)DqDU (t)T , a b

(246)

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289

and where the transition energies are given by +D "e !e #k+X . (247) abk a b The fact that the Floquet states DW (t)T solve the Schro¨dinger equation allows for a formal a simplification of the master equation (241). Representing the density operator in this basis, i.e., o (t)"SW (t)Do(t)DW (t)T , ab a b the master equation takes the form

P

P

(248)

1 = = oR (t)" du J(u)n (u) dq e*uqR (t!q, t) ab 5) ab p+ 0 0 1 = = # du J(u)[1#n (u)] dq e~*uqR (t!q, t)#h.c. , (249) 5) ab p+ 0 0 where n (u)"1/[exp(+u/k ¹)!1] gives the thermal occupation of the bath oscillator with 5) B frequency u, and

P

P

R (t!q, t)" + (X (t!q)o (t)X* (t)!X* (t)X (t!q)o (t)) . (250) ab aa{ a{b{ bb{ a{a a{b{ b{b a{b{ Inserting next the identity := dq e*uq"pd(u)#P(i/u), with P denoting the principal value, and the 0 Fourier expansion (246) of the matrix elements X (t), the equation of motion (249) for oR can be ab ab expressed in terms of the time-independent matrix elements X X* , multiplied by phase factors abk a{b{k{ exp[i(D !D )t]. abk a{b{k{ 9.3. Rotating-wave approximation The resulting equation is still rather complicated so that, frequently, the rotating-wave approximation (RWA) is introduced [189]; cf. also Section 3.2.1. In the RWA it is usually assumed that phase factors exp[i(D !D )t] oscillate faster than all other time dependences, and hence can abk a{b{k{ be neglected. This argument applies, however, only to quasienergy spectra that do not possess systematic degeneracies or quasidegeneracies. For the case of the parametric harmonic oscillator, for example, which has the peculiarity of equidistant quasienergy levels the condition (a!b, k)"(a@!b@, k@) is sufficient to ensure D "D ; thus, these terms have to be kept in the abk a{b{k{ RWA [190]. For the driven double well, in the vicinity of the manifold where relevant quasienergy crossings occur, cf. Section 6, the conservative time evolution contains very small energy scales, and correspondingly large time scales. In order to obtain an adequate description for the reduced dynamics one can only drop terms with kOk@ [191,192]. Within the simplest RWA one finds [189] oR (t)"+ [A o (t)!A o (t)] , aa al ll la aa l 1 oR (t)"! + (A #A )o (t), aOb , ab la lb ab 2 l

(251)

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290

where = A " + [c #n (DD D)(c #c )] , ab abk 5) abk abk ba~k k/~= c "2ph(D )J(D )DX D2 . abk abk abk abk

(252)

Hence, within the simplest RWA two decoupled subsets of equations are obtained. One subset describes the approach of the diagonal elements towards a steady state (dissipation), while the other is concerned with the decay of the off-diagonal elements (effect of decoherence). The time-independent coefficients A contain the specifics of the potential and driving through the ab quasienergy differences D , and Floquet modes via X . The equation for the nondiagonal abk abk elements is easily integrated, and predicts exponentially vanishing off-diagonal elements at long times. Hence, the reduced density matrix becomes diagonal in the Floquet basis at long times. These features are a consequence of the RWA, they do not hold true if such an approximation is not performed. We shall discuss the predictions of Eq. (241) for the case of a particle in a driven double-well potential in greater detail in Section 14.2 below.

10. Real-time path integral approach to driven tunneling 10.1. The influence-functional method A very useful tool to describe non-equilibrium time-dependent dissipative phenomena is the real-time path integral approach. Within the simple phenomenological model described by the Hamiltonian in Eqs. (224) and (226), it enables to evaluate the trace operation involved in the definition of the reduced density matrix o(t)"Tr M¼(t)N of the relevant system exactly. Here, B we shall assume the factorized intial condition (232) for the total density matrix ¼(t), and refer to [34,187] for the case of more general initial conditions. As proposed by Feynman and Vernon [186], to carry out the reduction it is convenient to express the reduced density matrix (RDM) in the coordinate representation of the full time evolution operator º(t, t ). Introducing the N-component vector x(t ) " : x "(x ,2,x ) for the 0 k k k,1 k,N bath coordinates at time t , and setting x(t)"x , the matrix elements of the RDM at time t read k &

P

o(q , q@ ; t) " : dx Sq x D¼(t)Dq@ x T . & & & & & & &

(253)

Using next the path-integral representation of the time evolution operator in the coordinate representation, o(q , q@ ; t) may be expressed in terms of the initial density matrix o(q , q@; t ) at the & & * * 0 time t as [34,187] 0

P P

o(q , q@ ; t)" dq dq@J(q , q@ , tDq , q@, t )o(q , q@; t ) . & & * * & & * * 0 * * 0

(254)

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291

Here J is the propagating function of the reduced density which can be expressed as the double path-integral

P P

J(q , q@ , tDq , q@, t )" Dq Dq@A[q]B[q]A*[q@]B*[q@]F[q, q@] , & & * * 0

(255)

where the sum is over all real-time paths q(t@), q@(t@) obeying the constraints q(t )"q , q@(t )"q@, 0 * 0 * and q(t)"q , q@(t)"q@ . The quantity & &

GP C

A[q]"exp

DH

i t 1 dt@ mqR 2!» (q) 0 + 0 2 t

(256)

is the probability amplitude of the quantum system to follow the path q(t@) in the absence of external driving and fluctuating forces f(t). The factor

GP

B[q]"exp

H

i t dt@E(t@)S(q(t@)) + 0 t

(257)

incorporates the effect of the external driving force, cf. (225), and F[q, q@] is the Feynman—Vernon influence functional which describes the influences of the fluctuating force. For a harmonic bath (Gaussian statistics), it takes the form F[q, q@]"expM!U[q, q@]/+N, with

P P P

U[q, q@]"i(mc(t )/2) 0

t

t

dt@ (q2(t@)!q@2(t@))

0

1 t t{ dt@ dtA(q(t@)!q@(t@))(K(t@!tA)q(tA)!K*(t@!tA)q@(tA)) , # + 0 t t0

(258)

where K(t) is the force autocorrelation function given in Eq. (229). Finally, :Dq symbolically means summation in function space over all paths with the end-points (q , q@ ) held fixed. Hence, the price & & to be paid for the exact evaluation of the trace over the bath degrees of freedom is the appearance of a term that is nonlocal in time in the influence functional F[q, q@]. The latter couples the same path q or q@, as well as the two different paths q, q@, at different times. To better illustrate the effect of the influence functional F[q, q@], it is useful to introduce center of mass and relative coordinates, i.e., g(t)"q(t)#q@(t),

m(t)"q(t)!q@(t) .

(259)

Here g(t) describes propagation along the diagonal of the density matrix, while the path m(t) is a measure of how far the system gets off-diagonal while propagating. In terms of these new paths one obtains for UI [g, m]"U[q, q@] [34,187],

P P

1 t t{ UI [g, m]" dt@ dtAm(t@)K@(t@!tA)m(tA) + 0 t t0

P P

# i(m/2)

t

t

dt@

0

P

t{ t dtAm(t@)c(t@!tA)gR (tA)# i(m/2)g dt@ c(t@)m(t@) , 0 t0 t0

(260)

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292

with g "g(t ), and where K@(t) is the real part of the kernel K(t), cf. (229). Equivalently, by 0 0 performing two partial integrations, the influence functional can be recast in the form

P P P

1 t t{ dt@ dtA[mQ (t@)¸@(t@!tA)mQ (tA)#imQ (t@)¸A(t@!tA)gR (tA)] UI [g, m]"! + 0 t t0 t t # m(t) dtA[¸@(t!tA)mQ (tA)#i¸A(t!tA)gR (tA)]# ig m(t)¸A(t)! dt@m(t@)¸A(t@) , 0 t0 t0 (261)

C

P

D

with ¸@(t) and ¸A(t) being the real and imaginary parts of

P

J(u) cosh(+ub/2)!cosh[u(+b/2!it)] 1 = du , (262) ¸(t)" u2 +p sinh(+ub/2) 0 respectively. Using this method an exact analytical solution has been discussed for the driven, dissipative parametric harmonic oscillator [63]. In Sections 11—13 we shall describe an application to the study of tunneling in driven tight-binding systems. In the remainder of this section, instead, we shall briefly discuss a numerical methodology to discretize the full Feynman propagator. Using this method some results on the driven two-level-system and on the driven double-well system are discussed below in Sections 12 and 14.3, respectively. 10.2. Numerical techniques: ¹he quasiadiabatic propagator method The quasiadiabatic propagator path-integral method (QUAPI) [196—198] allows efficient realtime path-integral calculations that may even span long time intervals. In the QUAPI, an appropriate partitioning of the propagator leads to a modified path-integral expression which contains the one-dimensional propagator describing the evolution of the uncoupled system along bare paths, and a nonlocal influence functional that includes multidimensional corrections to the bare paths. To achieve this, one divides the total time into M small time intervals Dt"(t!t )/M. 0 Denoting now H #H (t)"H (t), and the bath contribution H #H of the N harmonic 0 %95 S B SB oscillators in Eq. (226) as N H #H "H " : + H(j)(q) , B SB R R j/1 the propagator between time t "kDt and t "t #Dt is split symmetrically as k k`1 k !i/+: t dt@H(t@) !i/+[H Dt#: t dt@H (t@)] R S º(t , t )"Áe "Áe t t k`1 k +e~*HRDt@(2+)º (t ,t )e~*HRDt@(2+) , S k`1 k where k`1 k

tk`1 dt@H (t@) S k

º (t , t )"Áe!i/+: t S k`1 k

(263)

k`1 k

.

(264)

(265)

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293

Because the operators H and H do not commute, the splitting is not exact, and the error is of the S R order of [H , [H , H ]]Dt3. The short-time propagator in the position representation between the R S R time t and the time t is then given by k k`1 Sq x Dº(t ,t )Dq x T+Sq Dº (t ,t )Dq T k`1 k`1 k`1 k k k k`1 S k`1 k k N ] < Sx De~*H(j)R(qk`1)Dt@2+e~*H(j)R(qk)Dt@2+Dx T , k`1,j k,j j/1 and is usually termed the quasiadiabatic propagator. It describes the exact dynamics of the relevant system in the limit of vanishing coupling to the bath. Introducing next the discretized (with respect to time) paths q(t) and q@(t), i.e., the evolution of the paths q, q@ from the time t to the time t"MDt 0 is described by the finite sequences Mq , q ,2, q N, Mq@ ,2, q@ N, and performing the trace over the 0 1 M 0 M bath degrees of freedom, the time evolution of the RDM is described by the multidimensional path integral

P

P

P

P

o(q , q@ , t)" dq 2 dq dq@ 2 dq@ & & 0 M~1 0 M~1 ]Sq Dº (t, t )Dq T Sq Dº (t , t )Dq T & S M~1 M~1 2 1 S 1 0 0 ]Sq Do(t )Dq@ TSq@ Dºs(t , t )Dq@ T2Sq@ Dºs(t, t )Dq@ T 0 0 0 0 S 1 0 1 M~1 S M~1 & ]F[Mq , q ,2, q N,Mq@ , q@ ,2, q@ N] . 0 1 & 0 1 &

(266)

Here, F[Mq , q ,2, q N,Mq@ , q@ ,2, q@ N] is the Feynman—Vernon influence functional evaluated on 0 1 & 0 1 & the discretized paths q, q@ where we have set q "q , q@ "q@ . In the limit MPR the expression & M & M (258) for the influence functional is recovered. To calculate the multidimensional path-integral numerically, one needs to discretize the paths also in space. A conventional trapezoidal calculation would require a fine grid which would lead to very high-dimensional sums. A more convenient method is to use the minimal number of basis functions for which evaluating the matrix elements in Eq. (266) is sufficient to describe the system dynamics. A useful observation is that, in the absence of driving and of coupling to the bath, the propagator for the uncoupled system is diagonal in the basis MDu TN of the energy eigenstates of the time-independent Hamiltonian H . On the other hand, * 0 the influence functional is conveniently expressed in the basis MDu TN in which the operator q is * diagonal. Hence, in the absence of driving, a useful choice for the basis functions is the discrete variable representation (DVR) [199]. This amounts to choose the first n energy eigenstates Du T of j the system Hamiltonian H , and to perform the unitary transformation 0 n Du T" + ¸ Du T , (267) * i,j j j/i that selects n position eigenstates Du T by means of the orthogonal transformation matrix ¸ . This * i,j diagonalizes the coupling and gives an appropriate approximation for the requirement of completeness of the discrete basis. The number n of energy eigenstates chosen depends on the problem, and it is usually between 2 and 20. The path integral is then decomposed into sums over matrix

294

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elements in the DVR basis multiplied with the influence functional evaluated at the eigenvalues of the position operator. In the presence of driving, the time-dependent Hamiltonian H (t) is not S anymore diagonal in the basis of H , though the choice of the discrete set MDu TN continues to be 0 * convenient for the numerical calculations. Recently, a novel basis set constructed in terms of appropriate moving basis sets has also been proposed [196]. The practical applicability of the QUAPI method depends critically on the dimension of the multidimensional path-integral. The numerical effort for global evaluation of the path integral grows exponentially with the number of time slices. If one is interested in the long-time dynamics, iterative schemes are therefore necessary. For this to work, the fact that the range of nonlocal interactions in the influence functional described by the correlation function K(t) in Eq. (229) is finite in time, with a maximum at t"0, is crucial. Then, the path integration can be truncated, and the time evolution equation leads to an iterative tensor multiplication scheme [197,198]. The number of time steps after which the path integration is truncated determines the rank of the tensor in the multiplication scheme. Despite the fact that the range of nonlocal interactions is finite in time, it decays only algebraically for Ohmic damping at temperature ¹"0, and exponentially for ¹'0 [34]. This has to be taken into account at very low temperatures, when the length of the convergence factor, as given by the bath correlations, diverges. Finally, it is worth remarking that the discrete variable representation of the QUAPI offers a direct way of reducing an extended double-well dynamics to a reduced, finitelevel-system dynamics continuously, by decreasing the number of DVR states. More generally, the DVR method should provide an unified approach to investigate the dynamics of continuous or tight-binding models. The QUAPI method has been sucessfully applied to investigate the driven spin—boson system [200—202], and part of the results found will be discussed in Section 12. In Section 14.3 we discuss the results of the QUAPI on dynamical hysteresis in the driven and dissipative double-well potential [203,204].

11. The driven dissipative two-state system (general theory) 11.1. The driven spin-boson model In this section we shall investigate the driven dynamics of a dissipative two-level system (TLS). As discussed in Section 3, the driven TLS can model the physics of intrinsic two-level systems, such as spin 1 particles in external magnetic fields or, more generally, the motion of a quantum particle 2 at low temperatures in an effective double-well potential in the case that only the lowest energy doublet is occupied. In this latter case, the dissipative TLS can describe, for example, hydrogen tunneling in condensed media [205], tunneling of atoms between an atomic-force microscope tip and a surface [206],6 or of the magnetic flux in a superconducting quantum interference device (SQUID) [208]. Recent experiments on submicrometer Bi wires have measured transition rates of two-level systems coupled to conduction electrons [209,210]. Moreover, since the seminal works

6 Here, the quantum dissipation for Xe-atoms tunneling between an atomic-force-microscope tip and a Ni(100) surface is Ohmic-like [207].

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by Marcus [211] and Levich [212] this same model has been applied to describe nonadiabatic chemical reactions in the condensed phase, such as electron transfer (ET) [35,213—217] or proton transfer reactions [218,219]. As illustrated in Section 3, to investigate tunneling problems, the two-level system is conveniently described by the pseudospin Hamiltonian (65) + : H "! (D p #e p ) . H " 0 TLS 0 z 2 0 x

(268)

The basis is chosen such that the states DRT (right) and D¸T (left) are eigenstates of p with z eigenvalues #1 and !1, respectively. The interaction energy +D is the energy splitting of 0 a symmetric TLS due to tunneling. Because of its fundamental role in the understanding of the interplay among quantum coherence, dissipation and driving in nonlinear quantum systems, we shall dedicate all of this section to the driven and dissipative two-level system. Moreover, this problem can — to a large extent — be treated in analytical terms. Quite generally, the stochastic influence of the bath results in a reduction of the coherent tunneling motion by incoherent processes [32—35], and may even lead to a transition to localization at zero temperature [220]. An important question is to which degree the dissipative TLS dynamics is influenced by externally applied time-dependent fields [67,68,191,192, 200—202,221—250]. In particular, as discussed in Section 3.2.2, in the absence of interaction with a bath, a complete destruction of tunneling can be induced by a coherent driving field of appropriate frequency and strength [24,67,68]. This effect persists in the presence of dissipation [191,192,225,226,229,238]. The transition temperature above which quantum coherence is destroyed by a stochastic environment is modified by a driving field [225,226,238]. The possibility to control a priori the proton transfer by an electric field has been first addressed in [223] in the classical limit. This same issue applied to the case of electron transfer rate in a polar solvent was subsequently discussed in [227,228]. Long-time coherent oscillations may arise due to driving induced correlations between tunneling transitions [191,192,200—222,229,246]. Other interesting examples of control of the TLS dynamics are the enhancement of the TLS response to a weak coherent signal [232,233] for optimal values of the temperature, i.e., quantum stochastic resonance (QSR), see Section 12.1.4, or of the periodic asymptotic tunneling amplitude vs. friction strength [200]. The possibility of inversion of the population between the donor and the acceptor in an ET process has been recently investigated in [225,228,236], while a stabilization of the transfer rate by a periodic field has been discussed in [201,202]. Novel effects, such as an exponential enhancement of the transfer rate by an electric field modulating the coupling between the localized states has been addressed recently in [238]. Finally, interesting phenomena arise when a stochastic (e.g. of the dichotomous type) modulation of the intersite coupling [239,240] or of the asymmetry energy between the two-state-system [241] is applied. We shall apply here the path-integral formalism described in the previous section to investigate the reduced dynamics of the driven and dissipative TLS. As a first step to characterize the driven dissipative TLS, we have to specifiy the Hamiltonian term H (t) introduced in Eq. (222) %95 describing the interaction with external time-dependent fields. Usually, only the effects of a driving field modulating the asymmetry energy between the two wells are considered [200—202,221—236,244,246—250]. Here, following [238,243], we generalize the model to include the

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possibility of external (multiplicative or additive) modulation of the coupling energy between the localized states. This time dependence of the coupling parameter could arise, for example, from an ac-modulation of the barrier height or width of the underlying double-well potential, and could be realized in a superconducting loop with two Josephson junctions [208]. We observe that (see the detailed discussion in Section 12 below), while the effect of asymmetry modulation is additive, a barrier modulation results in a multiplicative (exponential) modification of the coupling parameter. Another example of time-dependent tunneling-coupling may arise in long-range bridgeassisted electron-transfer (ET) [239,240]. In this case, the direct coupling between donor and acceptor is negligibly small to cause ET. However, the intervening medium can provide states which serve as a “bridge” in the long-range ET, and hence induce a stochastically fluctuating effective electronic coupling between donor and acceptor states. A general Hamiltonian for the driven TLS reads H #H (t)"!1+[D(t)p #e(t)p ] . (269) TLS %95 2 x z Here, D(t) and e(t) are explicitly time-dependent, but generally are not periodic. The specific form of this time dependence depends on the characteristics of the driving fields. Finally, like in Eq. (224), we characterize the thermal bath as an ensemble of harmonic oscillators described by the Hamiltonian H , and we consider bi-linear couplings to the heat bath B that are sensitive to the value of the position operator p . For example, a dipole-local-field coupling z provides a simple physical model for this type of coupling. In this discretized representation the interaction Hamiltonian H assumes the form H "!(p d/2)+ c x " : !p X/2, where x is SB SB z i i i z i a bath coordinate and d is the tunneling distance. We end up with the time-dependent spin—boson Hamiltonian7 H(t)"!1+[D(t)p #e(t)p ]!1p X#H , 2 x z 2 z B

(270)

with

A

B

1 p2 H " + i #m u2x2 . (271) B 2 i i i m i i In the following we derive formally exact solutions for the reduced density matrix and discuss analytical and numerical approximations. Moreover, we illustrate some important examples of control of tunneling by external monochromatic or polychromatic fields and discuss the phenomenon of quantum stochastic resonance. 11.2. The reduced density matrix (RDM) of the driven spin—boson system We wish to compute the reduced density matrix (RDM) o(t)"Tr M¼(t)N of the driven B spin—boson system. We consider a factorized initial state ¼(t )"o (t )¼ for the density matrix 0 S 0 B ¼(t) of system-plus-reservoir.

7 Note that throughout this and the next section the “bare” tunneling splitting D in Eq. (268) denotes the tunneling 0 splitting that is already renormalized by high-frequency mode contributions of the bath with frequency u'u , with # u appearing in the spectral density J(u) in Eq. (238) [188]. #

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In particular, we suppose that at times t(t the particle is held at the site p "1 with the bath 0 z being in thermal equilibrium. For this problem the RDM is a 2]2 matrix with matrix elements o " : o(p, p@; t), where p, p@"$1, whose knowledge enables the evaluation of the expectation p,p{ value of any observable related to the TLS. In particular, as discussed in Section 3, the knowledge of the RDM is equivalent to knowing the expectation values Sp T " : TrMo(t)p N, cf. Eq. (69). In the i t i localized representation we prefer to use, Sp T describes the population difference of the two z t localized states. It provides therefore direct information on the dynamics of the TLS. In order to evaluate other observables related to the TLS, such as e.g. the coherences, one must evaluate the expectations values Sp T and Sp T . x t y t In the presence of driving and dissipation the quantum coherent motion described earlier by Eqs. (72)—(74) will be modified. Let us next investigate these modifications. 11.2.1. Exact path-integral solution for the RDM For a harmonic bath, the trace over the bath degrees of freedom prescribed in the definition of o(t) can be performed exactly by using the Feynman—Vernon path-integral method [186] discussed in Section 10. In particular, the matrix elements o of the reduced density matrix are expressed in p,p{ terms of the double path integral in Eq. (255) as

P P

o (t)" Dq Dq@A[q]B[q]A*[q@]B*[q@]F[q, q@] , p,p{

(272)

in which the sum is over all real-time paths q(t@), q@(t@) with the imposed constraints q(0)"q@(0)"d/2, q(t)"pd/2 and q@(t)"p@d/2 (p"$1, p@"$1).8 The quantity A[q] is the probability amplitude of the TLS to follow the path q(t@) in the absence of driving and fluctuating forces. The factor B[q] incorporates the effect of the driving force, and F[q, q@] is the Feynman— Vernon influence functional, cf. Eq. (258). For the driven TLS it is convenient to slightly modify the definition of the center of mass and relative coordinates in Eq. (259) in order to obtain dimensionless coordinates. We introduce m

(t)"[q(t@)!q@(t@)]/d, TLS

g (t@)"[q(t@)#q@(t@)]/d . TLS

(273)

Now, a two-state system starting out from a diagonal state of the density matrix is again in a diagonal state after any even number of transitions, and in an off-diagonal state after any odd number of transitions. Hence, m (t)"0 and g (t)"1 or !1 depending on whether one is in the TLS TLS state o " : RR or o " : LL of the RDM, respectively. On the other hand, g (t)"0 and 1,1 ~1,~1 TLS m (t)"1, or !1, if one is in the state o " : RL, or o " : LR. Suppose that we want to TLS 1,~1 ~1,1 evaluate Sp T . A general path will have 2n transitions at flip times t , j"1, 2,2, 2n, within the z t j

8 Note that this implicitly means that we express the double path integral in the discrete variable representation (DVR), see Section 10.2 and Eq. (267), in which the operator q is diagonal. For the case of a TLS the DVR basis and the localized basis formed by the vectors D¸T, DRT coincide. This does not hold anymore true if, for example, the reduction of the double-well potential to the first two doublets is considered, see Section 14.1.

298

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Fig. 14. Generic path with 2n transitions at flip times t , t , 2,t contributing to the population difference Sp T 1 2 2n z t between the localized states of a driven and dissipative two-level-system. The time intervals spent in a diagonal state RR or LL of the reduced density matrix are called sojourns, and the time intervals spent in an off-diagonal state RL or LR are termed blips. The initial and final sojourns are fixed by the condition o (t )"o (t),RR on the reduced density p,p{ 0 p,p{ matrix.

interval t (t@(t. Such a path can be parametrized by 0 n m(2n)(t@)" + m [h(t@!t )!h(t@!t )] , j 2j~1 2j j/1 (274) n g(2n)(t@)" + g [h(t@!t )!h(t@!t )] , j 2j 2j`1 j/0 where t ,t, h(t) is the unit step function, and where the labels m "$1 and g "$1 mark 2n`1 j j the two off-diagonal LR or RL, and diagonal states RR or LL of the RDM, respectively, cf. Fig. 14. On the other hand, if we want to evaluate Sp T or Sp T a general path will have 2n#1 transitions x t y t at corresponding flip times t , j"1, 2,2, 2n#1 within the interval t (t@(t. The periods j 0 t (t@(t in which the system is in a diagonal state RR or LL are usually referred to as 2j 2j`1 sojourns. Likewise, the periods t (t@(t in which the system stays in off-diagonal states RL 2j~1 2j or LR are termed blips [32,34], see Fig. 14. To evaluate now the dissipative influence, it is convenient to consider the influence functional in the form given by Eq. (261), where the time derivatives gR (t) and mQ (t) of the sequences of sojourns, and of blips, respectively, occur. This TLS TLS yields n mQ (2n)(t@)" + m [d(t@!t )!d(t@!t )] , j 2j~1 2j j/1 (275) n gR (2n)(t@)" + [g d(t@!t )!g d(t@!t )] . j 2j j~1 2j~1 j/1 For later convenience we note that a blip transition acts as an effective neutral dipole, where to the jth blip with spin flips at times t and t are associated the charges / " : m and / " : !m , 2j~1 2j 2j~1 j 2j j respectively. This does not hold true for the sojourn transitions. Let us next introduce the functions Q " : Q(t !t ), and j,k j k K "Q@ #Q@ !Q@ !Q@ , j,k 2j,2k~1 2j~1,2k 2j,2k 2j~1,2k~1 (276) X "Q@@ #Q@@ !Q@@ !Q@@ , j,k 2j,2k`1 2j~1,2k 2j,2k 2j~1,2k`1

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with Q@(t) and QA(t) being the real and imaginary parts of the bath correlation function Q(t) " : d2¸(t), respectively, with ¸(t) defined in Eq. (262), i.e.,

P

J(u) cosh (+ub/2)!cosh [u(+b/2!it)] d2 = du Q(t)" . u2 +p sinh(+ub/2) 0

(277)

The function K describes the interblip correlations of the dipole pair Mj, kN, while the function j,k X describes the correlations of the blip j with a preceding sojourn k. The influence function UI (n), j,k cf. Eq. (261), for a path with n transitions in the off-diagonal states of the RDM then becomes n j~1 n j~1 n UI (n)" + Q@ # + + m m K !i + + m g X . (278) 2j,2j~1 j k jk j k jk j/1 j/2 k/1 j/1 k/0 Finally, we point out that within this parametrization the sum over histories of paths is represented (i) by the sum over any number of flip times, (ii) by the time-ordered integrations over the flip times Mt N, and (iii) by the sum over all possible arrangements Mm "$1N and Mg $1N of the blip and j j j sojourn intervals. Introducing for the time integrals the compact notation

P

t

t0

P P

D Mt N2 " : n j

t

t0

dt n

tn

t0

P

dt 2 n~1

t2

t0

dt 2 , 1

(279)

the path summation takes the form

P P

P

t = Dq Dq@2 N + D Mt N + + 2 . (280) n j M N M N mj gj n/0 t0 The double path-integral can now be performed, and the formal solution for Sp T and Sp T reads x t y t [243]

A BP A BP

1 = 1 n t Sp T " + ! D Mt Nd Mt N + m (F(`) C~ #F(~) C` ) , x t 2 2n`1 j 2n`1 j M n`1 n`1 n`1 n`1 n`1 2 t0 mj/B1N n/0 1 n t 1 = D Mt Nd Mt N + (F(`) C` !F(~) C~ ) , Sp T " + ! 2n`1 j 2n`1 j M n`1 n`1 n`1 n`1 y t 2 2 t0 mj/B1N n/0 where t "t. For Sp T one obtains 2n`2 z t

(281)

(282)

A BP

= 1 n t Sp T "1# + ! D Mt Nd Mt N + (F(`)C`!F(~)C~) . (283) z t 2n j 2n j M n n n n 2 t0 n/1 mj/B1N Here, the set Mm N labels the two off-diagonal states of the reduced density matrix, while the sum j over the diagonal states visited at intermediate times has already been performed. The influence of the driving is contained in the factors n d Mt N" < D(t ) , n j j j/1

(284)

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and C`"cos / , n n where

C~"sin / , n n

(285)

P

n t (286) / " + m f(t , t ), f(t, t@)" dt@@e(t@@) . n j 2j 2j~1 t{ j/1 All the dissipative influences are in the functions F(B). Combining all intrablip and interblip n correlations in the expression

A

B

n n j~1 G "exp ! + Q@ !+ + mmK , (287) n 2j,2j~1 j k j,k j/1 j/2 k/1 and introducing the phases s "+n m X which describe the correlations between the kth n,k j/k`1 j j,k sojourn and the n!k succeeding blips, the functions F(B) take the form n n~1 F(`)"G < cos s , F(~)"F(`)tan s . (288) n n n,k n n n,0 k/0 Up to here the results are exact, though they are far too complex to be handled analytically. Hence, one has to resort to suitable approximation schemes and subsequent numerical computations. Before proceeding along these lines, it is advantageous to cast the expression for Sp T into the form z t of a master equation [222], and the expression for Sp T into an integral expression [243]. x t Interestingly, this can be performed exactly.

11.2.2. Exact master equations and integral expressions for the RDM To obtain the above-mentioned relations for the expectations Sp T , we define modified influence i t functions FI (B) involving suitable subtractions of products of lower-order influence functions, i.e., n n FI (B)"F(B)! + (!1)j + F(`) , F(`)2F(B) d n n m1 m2 mj m1`2`mj,n 2 j/2 m1, ,mj where the inner sum is over positive integers m . By definition, each subtraction involves again time j ordering for the flip times. In the subtracted terms, the bath correlations are only inside of each of the individual factors F(B) , i.e., there are no correlations between these factors. We next introduce mj the kernels ½(B)(t, t@) defined by the series expression

P

P

t t3 = 2 dt d Mt N2~(n`1) + m FI (Y) C(B) . ½(B)(t, t@)" + (!1)n dt 2n`1 2 2n`1 j n`1 n`1 n`1 M t{ t{ mj/B1N n/0 In the same way the kernels K(B)(t, t@) are defined by the series

P

P

= t t3 K(B)(t, t@)"D(t) + (!1)n dt Mt N2~(n`1) + FI (B) C(B) , 2 dt d n`1 n`1 2 2n`1 j 2n`1 M t{ t{ mj/B1N n/0

(289)

(290)

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where the label # (or !) indicates that the respective kernel is an even (or odd) function of the bias energy e(t). The product functions FI (Y)C(B), FI (B)C(B) depend on flip times, where the first and n n n n the last one are identified with t@ and t, respectively. All intermediate flip times are integrated over as prescribed by Eqs. (289) and (290). Taking the time derivative of Eqs. (281)—(283), it is found that the formal solution for Sp T can be recast in the form of an exact generalized master equation z t (GME) [222]

P

d t Sp T " dt@[K(~)(t, t@)!K(`)(t, t@)Sp T ] , z t z t{ dt t0

(291)

as follows from an iterative solution of Eq. (291) with (290). In contrast, there is no closed master equation for Sp T or Sp T . Instead, the following exact integral relations hold [243]: x t y t

P

Sp T " x t

t

t0

dt@[½(`)(t, t@)#½(~)(t, t@)Sp T ] z t{

(292)

1 d Sp T "! Sp T . y t D(t) dt z t

(293)

Here, the initial condition Sp T 0"1 is assumed. These exact results show that, for a localized z t initial condition, no integro-differential equation exists for the expectation values of Sp T and Sp T. x y This is indeed remarkable. 11.3. The case a"1/2 of Ohmic dissipation Let us consider the case in which the bath has an Ohmic spectrum with an exponential cutoff as in Eq. (237), i.e., J(u)"(2p+/d2)aue~u@u# ,

(294)

where the dimensionless coupling constant a"mcd2/2p+ has been introduced. For this particular form, the bath correlation functions Q@(q) and Q@@(q), see Eq. (277), can be evaluated exactly [251,252], yielding

K

K

C(1#1/+bu ) # Q@(q)"a ln(1#u2q2)#4a ln , # C(1#1/+bu #iq/+b) # Q@@(q)"2a atan(u q) , #

(295)

where C(z) denotes the gamma function. In the following we shall discuss a closed-form solution for the special value a"1 of the Ohmic 2 strength in the limit of a large cutoff u . In fact, in this limit, the case a"1 represents a sort of # 2 “critical point” (in the absence of driving) in the investigation of the destruction of quantum coherence by bath-induced incoherent tunneling transitions. For the case of a symmetric system, a"1 represents the “critical” dissipative value for which coherent tunneling is destroyed by the 2

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302

Fig. 15. Fundamental spectral amplitude g vs. driving frequency X for different temperatures for the case a"1 of 1 2 Ohmic dissipation. At high temperatures only incoherent relaxation occurs (dashed lines). As the temperature is decreased, resonances appear at submultiples X"e /n (n"1, 2,2) of the static bias e (dot-dashed and full lines). These 0 0 denote the occurrence of driving-induced coherence. Frequencies are in units of C "pD2/u , temperatures in units of 0 0 # +C /k . 0 B

dissipative environment for any temperature [32,34]. Hence, for a51, the tunneling dynamics is 2 always incoherent down to ¹"0. Note also that the critical value of a for the destruction of quantum coherence depends on the specific transport quantity which is investigated [253,254]. In the presence of driving this situation is modified due to driving-induced coherent processes that render the low-temperature dynamics non-Markovian, see Fig. 15. To be definite, we shall term the “scaling limit” the limit u q<1 and +u
P

P(!)(t)"

t

0

t

P

dt e!: t dsC(s) 2 0

t

P(4)(t)"e!: 0 dsC(s) .

t2

0

t2

dt D(t )D(t )e~Q{(t2~t1)e!: t dsC(s)/2 sin[f(t , t )] , 2 1 1 2 1 1

(298)

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The function f(t, t@) is defined in Eq. (286), and additionally we have introduced the relaxation rate C(t)"pD2(t)/2u . Here, P(!)(t) represents the contribution to P(t) that is antisymmetric with respect # to an inversion of the bias eP!e. It determines the long-time dynamics. The investigation of the symmetric contribution P(4)(t) is, on the other hand, of interest to obtain information on the transient dynamics. It decays to zero as tPR. As expected, Eq. (298) predicts that in the absence of driving a symmetric TLS undergoes, for any temperature, incoherent relaxation with rate C " : pD2/2u . Note that for this special case a"1 the relaxation rate coincides with the tunneling 0 0 # 2 splitting D , see Eq. (310) below, renormalized by bath modes with u(u . Further, P(4)(t) is not % # sensitive to asymmetry modulation. Along the same line is Sp T ,0. To conclude, we observe that, x t as argued in Ref. [32], the error introduced in P(t) by approximating the bath correlation functions Q@(q) and Q@@(q) as in Eq. (296) can in the worst case become comparable to the value assumed by P(t) itself only at long times when the condition t'ln(u /D )/C is satisfied. # 0 0 For general forms of dissipation, the exact master equation (291) and the related equations for Sp T , Sp T cannot be solved exactly. In the following, we shall discuss approximations that render x t y t the analytical analysis more tractable. 11.4. Approximate treatments 11.4.1. The noninteracting-blip approximation (NIBA) The noninteracting-blip approximation (NIBA) has been reviewed in [32], and since, successfully applied to investigate the tunneling dynamics in different parameter regimes. An extension of the NIBA to the driven case has been discussed in [221,225,226,229,236,243]. The NIBA is a rather good approximation for sufficiently large temperatures and/or dissipative strength, and/or for strong bias. Moreover, in the absence of driving, it provides a systematic weak damping approximation down to the lowest temperatures for the case of a symmetric system, i.e., e "0. This has 0 been confirmed via a direct comparison between the NIBA prediction and its next order correction given by the interacting-blip-chain-approximation (IBCA), cf. Section 11.4.3, real-time MonteCarlo calculations [252,256], and by calculations via the numerical QUAPI method discussed in Section 10.2 [201]. In the NIBA one formally neglects the blip—blip interactions K . Furthermore, the sojourn—blip j,k interactions X are disregarded except neighboring ones with k"j!1, and they are approxij,k mated by X "QA(q ). Hence, in the NIBA, the influence functions UI (n) in Eq. (278) depend on j,j~1 j the blip length alone, i.e., n UI (n) " + [Q@(q )!im g QA(q )] , (299) NIBA j j j~1 j j/1 with q "t !t the length of the jth blip interval. Because interblip correlations are disj 2j 2j~1 regarded in the NIBA, the modified influence functions FI (B) are zero for all nO1, and therefore the n series expansions (289) and (290) reduce to the terms of lowest order in D(t). Under these approximations, defining P(t) " : Sp T , we obtain from Eqs. (291)—(293) the explicit form z t

P

d t P(t)" dt@[K(~)(t, t@)!K(`)(t, t@)P(t@)] dt t0

(300)

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304

and

P

Sp T " x t

t

dt@[½(`)(t, t@)#½(~)(t, t@)P(t@)]

(301)

t0

1 d P(t) , Sp T "! y t D(t) dt

(302)

with the NIBA kernels K(`)(t, t@)"D(t)D(t@)e~Q{(t~t{)cos [Q@@(t!t@)]cos [f(t, t@)] , K(~)(t, t@)"D(t)D(t@)e~Q{(t~t{)sin[Q@@(t!t@)]sin[f(t, t@)]

(303)

and ½(`)(t, t@)"D(t@)e~Q{(t~t{)sin[Q@@(t!t@)]cos [f(t, t@)] , ½(~)(t, t@)"D(t@)e~Q{(t~t{)cos [Q@@(t!t@)]sin[f(t, t@)] .

(304)

Evaluation of the GME (300) and of Eqs. (301) and (302) gives the dynamics of the spin—boson model within the NIBA. We observe that, as shown in [257,258] for the undriven case, and generalized to the driven case in [225,226,236], the polaron transformation approach used in the small polaron model [259] leads, to lowest-order perturbation theory in the bath renormalized tunneling splitting, to a master equation analogous to Eq. (300), and possessing identical kernels (303). Thus far, the NIBA has been discussed on a purely formal level only. The key idea that underpins the NIBA is that a blip transition acts like an effective dipole, cf. Eq. (275), which interacts only weakly with a neighboring dipole. Let us consider a spectral density of the form, cf. Eq. (238), J(u)"(2p+/d2)a uJ 1~sus exp(!u/u ) , (305) s # where a is the dimensionless coupling constant, and where a " : a is the Ohmic coupling constant s 1 already introduced in Section 11.3; cf. Eq. (294). Let us first consider the undriven case. Then, we can distinguish between three different regimes in which the NIBA is justified: (i) The first regime is encountered when the average blip length is so small compared to the average sojourn length, that the blip—blip interactions K and the sojourn—blip interactions may j,k be neglected. The suppression of the blip length against the sojourn length is favoured by the particular form Jexp(!Q@(t)) of the intrablip interactions appearing in the interaction function G in Eq. (287), which tend to suppress the blip length. On the contrary, the average blip-plusn sojourn length is constant and of the order of t/n for a path with 2n transitions up to time t. We observe then that the function Q@(t) behaves at long times as t1~s at zero temperature, and as t2~s at finite temperatures. Hence, long blips q are exponentially suppressed by the intrablip interactions Jexp(!Q@(q)) for any temperature in the sub-Ohmic case 0(s(1, and in the case 14s(2 at finite temperatures. In turn, because it is the difference of the Q@ that enters the interblip correlation factor Jexp(!K ), cf. Eq. (276), this factor can be set equal to unity with negligible error. In j,k particular, if the function Q@ increases linearly with t, then the interblip interactions K cancel out j,k exactly.

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305

It then turns out that the NIBA is justified always in the sub-Ohmic case s(1, in the Ohmic case s"1 for high enough temperatures and/or dissipative strength, and in the case 1(s(2 for high enough temperatures. To be precise, for Ohmic dissipation it is always justified when a'1, or when k ¹5+D , with a arbitrary. B 0 (ii) For super-Ohmic dissipation with s'2 the intrablip interactions are not effective in suppressing the blip lengths. Though, the NIBA can be justified for arbitrary dissipation as long as the system exhibits overdamped dynamics, as it is, for example, for a strong static bias e 'D . 0 0 (iii) In addition, the NIBA is a systematic weak damping approximation down to zero temperature for the case of a symmetric system (e "0). In fact, the effect of the interblip correlations K is 0 j,k of the order of the square of the coupling constant a , whereas that of the intrablip correlations s considered in the NIBA is of linear order. In the presence of driving, the parameter regime in which conditions (i)—(iii) are valid become modified. Let us, for example, consider the case of a driving field which modulates the asymmetry energy of the TLS (i.e., D(t)"D in Eq. (269)). For a very slow driving field the overall effect is an adiabatic 0 modulation of the asymmetry energy of the TLS [221]. In general, condition (i) still holds true. Condition (ii) gets modified because the system will show an overdamped dynamics when De(t)D'D . On the contrary, a fast oscillating driving field tends to suppress long blips [229]. Hence, 0 NIBA should be justified even better in the presence of a fast oscillating field, if it is already justified in the undriven case. For example, denoting by D 4D the tunneling splitting renormalized by %&& 0 a fast ac-field, cf. Eq. (359) below, the NIBA condition in (i) for the case of an Ohmic bath becomes a'1 or k ¹'+D . B %&& We observe, on the contrary, that a fast ac-field which is modulating the coupling energy of the TLS, cf. Eq. (383) below, tends to enhance quantum coherence. In this case the validity range of NIBA is diminished. For example, denoting by D 5D the field renormalized tunneling %&&,d 0 splitting, cf. Eq. (386) below, the NIBA condition in (i) for the case of an Ohmic bath becomes a'1, or k ¹'+D . B %&&,d In the absence of driving, Eq. (326) below is conveniently solved by Laplace transformation [32,34]. The study of the resulting pole equation for the Laplace transform of P(t) " : Sp T gives z t information on the transient dynamics. In particular, for Ohmic dissipation and weak Ohmic coupling a;1, cf. Eqs. (238) and (294), the NIBA predicts a destruction of the damped coherent motion described by Eq. (72) above a crossover temperature ¹*(a)K+D /k a [32,34]. For a51 0 B 2 the dynamics is incoherent down to ¹"0, cf. also [254]. For super-Ohmic dissipation with s'2, the NIBA predicts underdamped motion up to high temperatures ¹*. For the case s"3 we have s ¹* K(+uJ /k Ja ) [260]. These predictions have been confirmed by Monte-Carlo calculations 3 B 3 [252,256]. Finally, the NIBA predicts that the TLS dynamics approaches incoherently a stationary equilibrium value P "Sp T "tanh(+e /2k ¹) with relaxation rate C given by 45 z = 0 B 0 = dq h(`)(q) cos(e q) , (306) C "D2 0 0 0 0 and with P "o /C , where 45 0 0 = o "D2 dq h(~)(q) sin(e q) . (307) 0 0 0 0

P

P

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306

For later convenience we have introduced here the kernels h(`)(q)"e~Q{(q) cos[QA(q)] , h(~)(q)"e~Q{(q) sin[QA(q)] .

(308)

Inserting the expressions (296) for the Ohmic kernels Q@(q) and QA(q) valid in the scaling approximation u q<1, +u
A B

D2 +bu 1~2aDC(a#i+be /2p)D2 # 0 C " 0 cosh(+be /2) , 0 2u 2p 0 C(2a) #

(309)

where C(z) denotes the Gamma function. The above expression (309) is defined for any value of the coupling constant a. To investigate the tunneling dynamics for weak coupling a;1 it is convenient to introduce D "D (D /u )a@(1~a)[cos(pa)C(1!2a)]1@(2~2a) , % 0 0 #

(310)

which is the bath-renormalized tunneling splitting when a(1. In terms of D the static rate, when % a(1, has then the form

A B

D +bD 1~2a % C "sin(pa) % DC(a#i+be /2p)D2 cosh(+be /2) . 0 0 0 2p 2p

(311)

On the contrary, for nonzero bias e O0, NIBA breaks down for weak damping and low 0 temperatures. In fact, because DP DP1 when ¹P0, the NIBA predicts an incorrect symmetry 45 breaking at low temperatures, weak coupling and arbitrary small bias energies e . In the same way, 0 the NIBA predicts for Sp T a stationary equilibrium value Sp T "(D /e ) tanh(+e /2k ¹). That x t x = 0 0 0 B is, it occurs an unphysical divergence Sp T PR when ¹P0 and for arbitrary small bias. x = For a discussion of the predictions of the NIBA resulting from the above equations in the presence of driving we refer the reader to the next section. In the following subsections, by taking into account subsequent terms of the series expansions (289) and (290), which thereby account for interblip correlations, we calculate corrections to the NIBA systematically. 11.4.2. Systematic weak damping approximation For weak damping, nonzero bias and low temperatures the NIBA breaks down because the bath correlations K and X contribute to effects which depend linearly on J(u). The kernels K(B)(t, t@) j,k j,k in Eq. (290) have been discussed for weak-damping in Ref. [222]. They have the form K(`)(t, t@)"D(t)D(t@) cos[f(t, t@)][1!Q@(t!t@)]

P P

#

t

t{

dt 2

t2

t{

dt d Mt N sin[f(t, t )]P (t , t ) sin[f(t , t@)] 1 4 j 2 0 2 1 1

][Q@(t!t@)#Q@(t !t )!Q@(t !t@)!Q@(t!t )] 2 1 2 1

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307

and K(~)(t, t@)"D(t)D(t@) sin[f(t, t@)]QA(t!t@)

P P

!

t

t{

dt 2

t2

dt d Mt N sin[f(t, t )]P (t , t ) cos[f(t , t@)][QA(t!t@)!QA(t !t@)] . 1 4 j 2 0 2 1 1 2 t{ (312)

In these weak-coupling expressions, the first part represents the NIBA. The residual contributions have been written in such a way that the term P (t , t ) is sandwiched between two blip-like 0 2 1 contributions occurring in the time intervals t !t@ and t!t . The term P (t , t ) takes into 1 2 0 2 1 account all the tunneling transitions of the undamped system during the interblip interval t !t . It 2 1 reads

P

= n t P (t, t )"1# + (!1)n D Mt Nd Mt N < cos[f(t , t )] , (313) 0 0 2n j 2n j 2j 2j~1 0 t n/1 j/1 and it reduces to Eq. (72) in the absence of driving. Note that, accordingly, P (t, t ) satisfies 0 0 a master equation as in Eq. (291) with kernels K(`)(t, t@)"D(t)D(t@) cos[f(t, t@)] and K(~)(t, t@)"D(t)D(t@) sin[f(t, t@)]. The weak-damping form of Sp T is obtained from the solution of the GME (291) with the z t kernels (312) substituted. Again, Sp T results by differentiation according to Eq. (293). On the y t other hand, the weak-damping form of Sp T results from the integral relation x t t Sp T " dt@ D(t@) sin [f(t, t@)]Sp T . (314) x t z t{ t0 The coupled set of Eq. (291) can be investigated by Laplace transformation methodology thereby generalizing the method developed in Refs. [34,32,261] for the static case, and in [222,229] for the driven case. The case of ac-monochromatic driving modulating the asymmetry energy will be discussed in Section 12.

P

11.4.3. The interacting-blip chain approximation In this subsection we briefly describe an approximation beyond the NIBA, which has been introduced only recently [244]. We set t "0. In addition to the NIBA-interactions in Eq. (299), 0 this improved approximation accounts also for the interactions of all nearest-neighbor blip pairs K and the full interactions of the nearest-neighbor sojourn—blip pairs X as well. Diaj,j~1 j,j~1 gramatically, we then have a chain of blips with nearest-neighbor interactions. Therefore, this approximation has been called the interacting-blip chain approximation (IBCA) [244]. A pictorial description illustrating the chain of interacting blips as considered in the IBCA is sketched in Fig. 16 for the three-blip contribution to Sp T . In the IBCA, the influence function UI (n), cf. z t Eq. (278), is given by, n n n UI (n) " + Q@ #+ mm K !i + m g X IBCA 2j,2j~1 j j~1 j,j~1 j j~1 j,j~1 j/1 j/2 j/1

(315)

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Fig. 16. Three-blip contribution in the IBCA. The solid line represents a sojourn, and the curly line a blip interval. The blip—blip correlations K are sketched by a dashed curve, and the sojourn—blip correlations X by a dotted curve. j,j~1 j,j~1 The intrablip interactions as well as the individual contributions in K are not depicted. j,j~1

with the full nearest-neighbor sojourn—blip interaction reading X "QA #QA !QA . (316) j,j~1 2j,2j~1 2j~1,2j~2 2j,2j~2 To investigate the dynamics of Sp T in the IBCA, one introduces the conditional probabilities i t R (t; q) and R (t; q) for the particle with initial diagonal state g "#1 to hop at time t!q into ` ~ 0 the final off-diagonal state m "#1 and m "!1, respectively, and afterwards staying there until & & time t. After that, the period q left until time t is spent in this off-diagonal state. Upon integrating the conditional probabilities R (t; q) over the period q, one obtains the off-diagonal elements of the B reduced density matrix at time t as

P P

t

dq[R (t; q)#R (t; q)] , (317) ` ~ 0 t Sp T "i dq[R (t; q)!R (t; q)] . (318) y t ` ~ 0 The quantity Sp T is obtained by integrating Sp T , i.e., from Eq. (293) z t y t t (319) Sp T "1! dt@ D(t@)Sp T . y t{ z t 0 According to Eqs. (317)—(319), the dynamical problem is solved up to quadratures once the quantities R (t@; q) are known in the interval 04t@4t. As discussed in [244], the quantities B R (t@; q) obey inhomogenous coupled integral equations which can be solved numerically by an B iteration scheme. The predictions of the IBCA for both Sp T and Sp T are compared with those of the NIBA in z t x t Fig. 17. Note that, even though low temperatures and moderate damping are assumed, the NIBA provides a fairly good approximation for Sp T . This holds no longer necessarily true for Sp T . z t x t Sp T " x t

P

11.5. Adiabatic perturbations and weak dissipation In this section we discuss how to tackle the time-dependent problem for the case of a perturbation which is varying slowly as compared to the time scales of the unperturbed system. Let us consider the Hamiltonian (269) of the form H (t)"!1+[D p #e(t)p ] , TLS 2 0 x z

(320)

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Fig. 17. The population difference Sp T between the localized states of a two-level-system and the coherence Sp T are z t x t depicted for the case of a symmetric TLS (e "0) driven by an ac-field eL cos(Xt) and Ohmic dissipation. The field 0 parameters are eL "2.5D , X"5D , and the dissipation parameters are a"0.25, k ¹"0.05+D , i.e., moderate damping 0 0 B 0 and low temperatures. The full-line curve depicts the IBCA prediction and the dash—dotted the NIBA. Minor deviations of the IBCA from the NIBA curve occur for Sp T while the differences are drastic for Sp T . At long times only odd z t x t harmonics contribute to Sp T , and only even ones to Sp T . z t x t

where Dde(t)/dtD;D2 . 0 In this case it is convenient to perform a time-dependent unitary transformation R(t)"expM!ih(t)p /2N, h(t)"!arccot[e(t)/D ] , y 0

(321)

(322)

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310

which, at any instant, diagonalizes the time-dependent TLS Hamiltonian in Eq. (320). Taking also into account the TLS interaction term with the bath, the Hamiltonian H(t) in Eq. (270) is transformed into 1 +D +e(t) H (t)"R(t)H(t)R~1(t)"! E(t)p # 0Xp # Xp #H , !$ z E(t) z E(t) x B 2

(323)

where E(t)"+(D2#e2(t))1@2 is the adiabatic energy splitting. Due to the explicit time dependence 0 of the transformation (322), however, the transformed Hamiltonian H (t) does not commute with !$ the transformed time-evolution operator º (t ,t )"Á expM!i/+:t21 dt H (t)N. Here, !$ 2 1 %&& t H (t)"H (t)!i+R(t) (dR~1/dt) . (324) %&& !$ By evaluating !i+R(t)(dR~1/dt)"(+3D /E3)(de(t)/dt), one readily observes that the second term 0 on the r.h.s. of Eq. (324) can be treated as a perturbation only when inequality (321) holds, i.e., for adiabatic driving. Moreover, because of the rotation (322), the bath couples both to p and to p . z x Thus, the transformation turns out to be useful only if, in addition to adiabaticity, the bath is weakly coupled to the TLS, hence allowing a perturbative expansion in the bath—TLS interaction. This method will be applied in Section 12.1.2 to evaluate the TLS dynamics for weak dissipation and in the presence of a slow monochromatic field, and in Section 12.5 in the context of the dissipative Landau—Zener—Stu¨ckelberg problem.

12. The driven dissipative two-state system (applications) 12.1. Tunneling under ac-modulation of the bias energy To make quantitative predictions the detailed form of the driving field has to be specified. In this section we shall study the effects of a driving field which periodically modulates the bias energy of the undriven TLS. To be definite, we consider a time-dependent TLS of the form H (t)"!1+MD p #[e #eL cos(Xt)]p N . (325) TLS 2 0 x 0 z In the following, we shall focus our attention to the investigation of the position expectation value P(t) " : Sp T which is the relevant quantity in the context of control of tunneling. The same z t line of reasoning applies to the quantities Sp T and Sp T . x t y t Before we restrict ourselves to parameter regimes for which the approximations on the bath correlation functions discussed in the previous section apply, we outline some general characteristics of the driven dynamics. We observe that the generalized master equation (291) is conveniently solved by Laplace transformation techniques. Introducing the Laplace transform PK (j)" := dt e~jtP(t) of P(t), one obtains 0 = jPK (j)"1# dt e~jt[KK (~)(t)!KK (`)(t)P(t)] , (326) j j 0 where KK (B)(t)":= dt@ e~jt{K(B)(t#t@, t). In the absence of driving, the kernels KK (B) do not depend j 0 j on time, and Eq. (326) reproduces all the known results in the literature [32,34].

P

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For periodic driving the kernels KK (B)(t) have the periodicity of the external field, and thus can be j expanded in a Fourier series, i.e., = KK (B)(t)" + kB(j)e~*mXt . j m m/~=

(327)

This allows for a recursive solution of Eq. (326) [229]. In particular, the asymptotic dynamics is determined by the poles of the recursive solution at j"$imX, where m is an integer number. Thus, the asymptotic dynamics is periodic in time with the periodicity T"2p/X of the driving force, i.e., = lim P(t)"P(!4)(t)"P(!4)(t#T)" + p e~*mXt , m t?= m/~=

(328)

with [229] k~(0) k`(0) p " 0 !+ m p , 0 k`(0) k`(0) m 0 mE0 0

(329)

and for mO0

A

B

i p " k~(!imX)!+ k` (!imX)p . m mX m m~n n n

(330)

From Eqs. (329) and (330) it is found that the asymptotic expectation obeys the integrodifferential equation

P

X 2p@X dt@ P(!4)(t@)L(t@, t!t@) , PQ (!4)(t)"F(t)! 2p 0

(331)

which describes the time evolution within a period T. Here, F(t)"+ e~*nXtk~(!inX) , n n L(t, t@)"+ e~*mXte~*nXt{k`(!inX) . m m,n

(332)

This result is still exact. It explicitly shows that also the asymptotic-driven dynamics is intrinsically non-Markovian, and is not invariant under continuous-time translations. An analysis of the symmetry properties of the kernels KK (B) upon inverting the bias eP!e, j depicts selection rules for the harmonics p . From Eq. (330) it follows that for a symmetric TLS, i.e., m if e "0, in the presence of monochromatic driving only the odd harmonics of Sp T are different 0 z t from zero [229]. This is known to hold true also in classical systems [262]. In the same way, Sp T x t and Sp T obey selection rules. Namely, only the even harmonics of Sp T and the odd harmonics of y t x t Sp T are different from zero if e "0. y t 0

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312

12.1.1. Control of tunneling within the NIBA Let us next consider the dynamics of P(t) within the NIBA. For the case of monochromatic driving the functions kB can be explicitly evaluated to yield [229] m = dq e~jqh(B)(q)AB(q) , (333) kB(j)"D2 m m 0 0 where the functions h(B)(t) have been introduced earlier in Eq. (308), and where

P

A

B

Xq eL A~ (q)"(!1)me~*mXq sin(e q)J sin , 2m 0 2m X 2

A

B

Xq 2eL A~ (q)"(!1)me~*(m`1@2)Xq cos(e q)J sin . 2m`1 0 2m`1 X 2

(334)

Here J (z) denotes the ordinary Bessel function of order m, and m Xq 2eL A` (q)"(!1)me~*mXq cos(e q)J sin , 2m 0 2m X 2

A

B

A

B

Xq 2eL A` (q)"(!1)m`1e~*(m`1@2)Xq sin(e q)J sin . 2m`1 0 2m`1 X 2

(335)

Hence, knowledge of the functions kB allows a spectral investigation of the asymptotic dynamics; m cf. the study of nonlinear Quantum Stochastic Resonance as in Section 12.1.4, or the phenomenon of dynamical hysteresis as in Section 14.3. Some characteristics of the driven dynamics obtained by a numerical integration of the NIBA master equation are shown in Figs. 18 and 19. The bath is assumed to have an Ohmic spectrum with an exponential cutoff as in Eq. (294), i.e., J(u)"(2p+/d2)aue~u@u# .

(336)

In Fig. 18, some general effects of dissipation (dashed curve), and of dissipation-plus-driving (dot-dashed curve) on the undriven and undamped coherent tunneling dynamics (full-line curve) are shown. Dissipation tends to destroy quantum coherence and, at long times, in the absence of driving, the stationary equilibrium value P is reached. When an additional driving field is acting, 45 the driven and dissipative motion becomes periodic at long time, the resulting dynamics being governed by Eq. (331). In Fig. 19 is depicted that driving-induced correlations may lead, for example, to large amplitude oscillations in the asymptotic dynamics of a strongly biased, e
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313

Fig. 18. Influence of dissipation and driving on the tunneling dynamics of a symmetric, e "0, two-level-system. 0 Depicted is the population difference P(t) " : Sp T between the localized states. For comparison, are depicted also the z t nondissipative, undriven coherent tunneling case (solide line), and the Ohmic damped, undriven case (dashed curve). The influence of dissipation results in a decay of quantum coherence towards equal populations of the localized states. The effect of an additional ac-field eL cos(Xt) results in an oscillatory relaxation towards an asymptotic, periodic long-time dynamics (dot-dashed curve). The chosen driving parameters are X"2 and eL "6. Frequencies are given in units of D , 0 temperatures in units of +D /k . 0 B

As shown by Eq. (300) or Eqs. (326) and (331) the transient dynamics, as well as the long-time dynamics, depend on an intriguing interplay between the stochastic and the external driving forces. In the following we shall first briefly discuss the linear response limit of Eqs. (329) and (330), being valid for any angular driving frequency X. Subsequently, we release the assumption of weak external fields and we study the nonlinear dynamics in different parameter regimes. 12.1.1.1. Linear response approximation. On the assumption that the driving force is weak, Eqs. (329) and (330) can be evaluated by linearizing them in the amplitude eL of the time-dependent force. To linear order in eL , only the coefficients kB with m"0,$1 contribute. Further, the m linearized coefficients kB(0) are of order zero in eL , while the coefficients kB(1) are of order eL . In terms 0 1 of these linearized coefficients, and of the zeroth-order function

P

v(X, e )"D2 0 0

=

dq e*Xqh(`)(q) cos(e q) , 0

0 one finds for the linear susceptibility sJ (X) to Sp T , i.e., for z t i = sJ (X)" dq h(q)S[p (q), p (0)]T , z z b 4+ ~=

P

(337)

(338)

314

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Fig. 19. Driven and damped dynamics of a biased, e O0, two-level-system. Large amplitude oscillations in the 0 long-time dynamics of the population difference P(t) " : Sp T are induced by a monochromatic field of intensity eL and z t intermediate frequency X. Ohmic dissipation is considered. For comparison, the dissipative behavior in the absence of driving (static case) is also shown. The fine resonance structure can be interpreted as the result of multiphoton absorption or emission at the proper frequency of the TLS: the number n of resonances observed satisfies the relation n"e /X, where 0 +e is the static asymmetry energy of the TLS. Frequencies are in units of d"D /2, temperatures in units of +d/k . 0 0 B

the result [229]

A

B

k~(0)(0) 1 k~(1)(!iX)!k`(1)(!iX) 0 . (339) sJ (X)" 1 k`(0)(0) +eL [!iX#v(X, e )] 1 0 0 In Eq. (338), h(q) is the Heaviside function, [ , ] denotes the commutator and S T the thermal b average of the full system in the absence of the external periodic force (eL "0). This susceptibility leads for the asymptotic expectation to the linear response result (340) P(!4)(t)"P #+eL [sJ (X)e~*Xt#sJ (!X)e*Xt] , 45 where P " tanh(+e /2k ¹)"k~(0)(0)/k`(0)(0), with k`(0)(0)"C , cf. Eq. (306). This result, within 45 0 B 0 0 0 0 NIBA, holds for any driving frequency X. It is now interesting to observe that for low enough frequencies XP0, one can linearize (in X) the integrals defining v(X, e ) and the linearized coefficients kB(1)(!iX). We obtain 0 1 v(X, e )Kv(0, e )"k`(0)(0) together with 0 0 0 eL d k~(1)(!iX)Kk~(1)(0)" k~(0)(0) , 1 1 2 de 0 0 (341) eL d k`(0)(0) . k`(1)(!iX)Kk`(1)(0)" 1 1 2 de 0 0

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Hence, within this adiabatic limit the linear susceptibility is readily found to assume a ¸orentzian form, i.e., 1 1 1 sJ (X)KsJ (X)" . !$ 4k ¹ cosh2(+e /2k ¹) 1!iXC~1 B 0 B 0

(342)

If the specific form, Eq. (336), of the Ohmic interaction is assumed, in the temperature regime +u
(343)

with the time-dependent rate given by

P

C (t)"D2 M 0

=

dq h(`)(q) cos[f(t, t!q)] ,

(344)

0

with f(t, t@)":t dtA e(tA). The time-dependent, nonstationary quantity P (t) is given by t{ M,/4 P (t)"o (t)/C (t) , M,/4 M M

(345)

where o (t) reads M

P

o (t)"D2 M 0

=

dq h(~)(q) sin[f(t, t!q)] .

(346)

0

In the limit eL P0 the rate C (t) reduces to the static rate C in Eq. (306). Eq. (343) is readily solved M 0 in terms of quadratures [221,229]. Note that, in general, C (t)Olim KK (`)(t), and M j?0 j o (t)Olim KK (~)(t). This holds true only in the adiabatic limit (Xq )P0, for which M j?0 j K f(t, t!q)Pqe(t), P

(t)Ptanh[+be(t)/2] " : P . !$,/4

M,/4

(347)

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Hence, in this adiabatic limit e(t) behaves like a time-dependent asymmetry, and an adiabatic detailed balance condition is fulfilled in analogy with the static case. In general, however, the detailed balance condition does not hold in the presence of driving; see also point (v) in the next section for the case of a high-frequency field. In Fig. 20 the predictions of the Markovian approximation given by Eq. (343) are compared with those of the exact NIBA equation (300) for the case of moderately small Ohmic friction, see Eq. (294). The predictions of the high-frequency non-Markovian approximation (348) discussed in the next subsection are also depicted. In Fig. 20a, a moderate driving frequency X"D "e /4 is 0 0 considered. For these parameters the Markovian approximation (343) is very good. The frequency is increased to X"15D in Fig. 20b. In this latter parameter regime the Markovian approximation 0 breaks down but the high-frequency approximation (350) discussed in the next section describes well the non-Markovian dynamics. 12.1.1.3. High-frequency regime. In the high-frequency regime X<MC , e , D N, the driving field H 0 0 oscillates too fast to account for the details of the dynamics within one period. Hence, a first approximation to the true dynamics described by Eq. (300) amounts to approximate the kernels KB(t, t@) in Eq. (300) with their average SK(B)(t, t@)TT " : K(B)(t!t@) over a period [or KK (B)(t) in 0 j Eq. (326) with their corresponding averages kB(j) given by the term with m"0 in Eq. (333)]. 0 : P (t) [222,227—229,236]. Time Hence, the essential dynamics of P(t) is described by SP(t)TT " H translation invariance is recovered by such an averaging procedure, and the resulting equation for P (t) obtained from Eq. (291) is of convolutive form. It reads H t t PQ (t)"! dt@K(`)(t!t@)P (t@)# dt@K(~)(t!t@) . (348) H 0 H 0 0 0 From Eq. (326) one obtains (for convenience we explicitly indicate the field dependence of the kernels kB(j)), 0 1#k~(j, eL )/j 0 PK (j)" . (349) H j#k`(j, eL ) 0 From Eqs. (348) and (349) some useful and insightful features of the high-frequency driving dynamics can be discussed. (i) A high-frequency field suppresses the periodic long-time oscillations, and [as seen by approximating kB(j)KkB(0)] the driven TLS satisfies at long times the rate equation 0 0 PQ (t)"!C [P (t)!P ] , (350) H H H H,/4 where P "k~(0)/k`(0) is the averaged nonstationary equilibrium value at high frequencies. The H,/4 0 0 high-frequency relaxation rate k`(0) " : C is given by 0 H Xq = 2eL C "D2 sin . (351) dq h(`)(q) cos (e q)J H 0 0 0 X 2 0 Note that C "SC (t)TT, with C (t) the Markovian rate of Eq. (343). From Eq. (350) we may H M M conclude that for high-frequency monochromatic driving the long-time behavior is governed within a good approximation by a Markovian dynamics.

P

P

P

A

B

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Fig. 20. Comparison between the predictions of the NIBA equations (291) and (303) for the population difference P(t) " : Sp T and those of the Markovian and high-frequency approximations (343) and (350), respectively. Ohmic z t dissipation is considered. Frequencies are in units of d"D /2, temperatures in units of +d/k . Panel (a) A monochromatic 0 B field of intermediate frequency X"D "e /4 is used. Panel (b) The frequency is increased to be X"15D . In this latter 0 0 0 case the Markovian approximation breaks down, but the non-Markovian high-frequency approximation (350) describes the dynamics well.

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(ii) Note, however, that while Eq. (348) constitutes a valid approximation to investigate the high-frequency, transient dynamics, it misses the periodic oscillations around the asymptotic mean value P . Going beyond the simple approximation (348), one obtains to leading order in H,/4 Dk`(inX)D/X;1 the result [229] m P(!4)(t)"P

# + p (X) e~*mXt , H,/4 m mE0

i p (X)" [k~(!imX)!k`(!imX)P ] . m H,/4 m mX m

(352)

(iii) Eq. (350) may be used to investigate the modification of the transfer rate C as compared to H the static one C . This issue is prominent in the context of control of chemical reaction. In the case 0 of long-range electron transfer (ET) reactions [227,228,236], for example, C determines the H transfer rate between the donor and acceptor states. Let us consider the case of Ohmic dissipation in Eq. (294). As a first feature, because DJ (z)D41, it 0 is apparent that for a symmetric TLS the effect of a fast asymmetry modulation results in an overall reduction of the incoherent tunneling rate C as compared to the Ohmic static one C , cf. Eq. (309), H 0 whenever a(1. 2 Secondly, the relaxation rate can be controlled by appropriately choosing the external field parameters. To see this, it is convenient to expand the high-frequency rate into Fourier series as = C " + J2(eL /X)C (e #nX) , (353) H n 0 0 n/~= where C is the static rate (306). Hence, via photon absorption or emission the ac-field opens new 0 tunneling channels, each weighted by the Bessel function J2(eL /X). As shown in Fig. 21, the rate n might be enhanced as compared to the static case when the resonance condition e "nX is fulfilled, 0 and when the corresponding channel has a nonzero weight J2(eL /X). Vice versa, as shown in the inset n of Fig. 21, C can become strongly suppressed. H (iv) As first discovered in [201,202], a fast monochromatic field can stabilize localized states without much fine-tuning of external conditions. Some numerical results obtained in [201,202] by using the tensor propagator method discussed in Section 10.2 are shown in Fig. 22, where the decay rate C is depicted as a function of the Ohmic coupling parameter a. In contrast to the undriven case (dashed curve), it is shown that the decay rate C is largely independent on the dimensionless Ohmic coupling parameter a and on the driving frequency X, depending mainly on the driving strength eL . A similar behavior is also demonstrated in [201] for the case of a superOhmic bath. Finally note that, as confirmed in [201,202], the high-frequency NIBA rate C represH ents an extremely good approximation to the exact numerical rate C in the parameter regime considered for the numerical calculations. (v) Eq. (350) also leads to the possibility of inversion of asymptotic populations of the localized states of the TLS by a fast ac-field. This feature again entails important applications as, for example, the possible inversion of the direction of long-range electron-transfer chemical reactions [228,236]. This fact relies on the observation that, in the presence of a fast ac-field, the detailedbalance condition between the forward C "(k`(0)#k~(0))/2, and backward C " & 0 0 "

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Fig. 21. Dependence of the high-frequency rate C of a driven and Ohmically damped TLS vs. the static asymmetry e . H 0 The peaks at e "nX describe photon absorption or emission processes. In the inset the amplitude eL of the ac-field is 0 chosen such that the Bessel function J (eL /X) vanishes; hence, the first resonance peak disappears. Frequencies are in units 1 of d"D /2, temperatures in units of +d/k . 0 B Fig. 22. Numerical results obtained by using the QUAPI method for the decay rate C of a dissipative TLS driven by a high frequency ac-field vs. the Ohmic coupling parameter a. In contrast to the undriven case (dotted line), the high frequency rate is largely independent of the parameters of the environment and of the driving frequency, depending only on the driving strength. The dissipative parameters are k ¹"5+D , and u "10D . Moreover, the solid squares indicate B 0 # 0 results for a generic ac-field with eL "30D , X"7.5D while the hollow circles refer to parameters such that CDT occurs 0 0 in the absence of dissipation, i.e., eL "30D , X"5.44D . In this parameter regime the decay rate C is analytically well 0 0 approximated by the high-frequency NIBA rate C (not shown). H

(k`(0)!k~(0))/2 rates does not hold anymore true. The destruction of detailed balance by a strong 0 0 periodic field may be characterized by an effective energy bias, i.e., +e "k ¹ ln(C /C ) . %&& B & "

(354)

In the limit of vanishing field eL P0, this apparent bias e coincides with the static bias e . %&& 0 (vi) Eq. (349) yields information on both the transient and the long-time dynamics of the TLS. Hence, it can be used to investigate the modification of the quantum coherent motion of the isolated TLS by thermal and external driving forces [225,226,238]. Here we restrict ourselves to the case of a symmetric TLS, i.e., e "0, and of Ohmic dissipation (294), and study the poles of (349) 0 resulting from the equation j#k`(j; eL )"0 in the high-frequency regime X<2pak ¹/+, D . We 0 B 0 refer to [226] for a discussion of the influence of a phononic bath where J(u)Ju3. For our purposes it is convenient to express the kernels k`(j; eL ) in terms of the static one 0

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K(j) " : lim L k`(j; eL ) as e?0 0 = k`(j; eL )" + J2(eL /X)K(j#inX) , 0 n n/~= where, when a(1 [34,32],

(355)

A B

D +bD 1~2a R(j) % K(j)" % , p 2p a#+bj/2p

(356)

R(j)"C(1#a#+bj/2p)/C(1!a#+bj/2p) .

(357)

In Eq. (356) above, C(z) denotes the Gamma function, and D is the bath-renormalized tunneling % splitting when a(1, cf. Eq. (310). Here we made the high cutoff approximation u q<1, +u
C

A

B

D

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[24,67—72,191,192]. The condition J (eL /X)"0 was found to be the necessary, but not sufficient 0 [67], criterion for CDT in the case of a driven TLS in the absence of dissipation [24,67—72]. In the presence of (weak) Ohmic dissipation, it can be deduced from the discussion in (iv) that, because C (C ;X, suppression of tunneling may be stabilized over several periods of the driving force H 0 in accordance with previous findings [191,192]. The same phenomenon is found to happen for a super-Ohmic phonon bath [226]. 12.1.2. Quantum coherence for weak dissipation and low temperatures At low temperatures, weak dissipation and nonzero static bias, the NIBA breaks down because inter-blip correlations contribute to linear order in the spectral density J(u). Hence, in this regime the tunneling dynamics needs to be investigated using approximation schemes different from the NIBA. 12.1.2.1. High-frequency regime. Using the weak coupling expressions (312), the transient driven dynamics in the high-frequency regime X
(360)

(361)

P(0)(t)"p #+ p cos(l t) (362) H 0 j j j with the amplitudes p , p , p defined in a cyclic way. One obtains, 1 2 3 (l2!e2)(l2!e2)(l2!e2) 3 e2 1 1 2 1 3 , p " 1 p "< j , (363) 1 0 l2(l2!l2)(l2!l2) l2 1 1 2 1 3 j/1 j with the sum rule +3 p "1 being obeyed. j/0 j On the other hand, as a result of the bath influence, the poles of the Laplace transform of P(0)(t) H are shifted from the imaginary axis into the negative-real halfplane, and there appears an additional

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pole at the origin. The residuum of this pole (denoted by P ) represents the equilibrium H,/4 population reached at long times. The dynamics is described by the expression9 [222] 3 P (t)" + p cos (l t) e~Cjt#(p !P ) e~CHt#P , (364) H j j 0 H,/4 H,/4 j/1 where the four damping rates C , j"1, 2, 3, and C depend explicitly on the characteristics of the j H bath, and on the driving parameters X and eL [222]. Hence, in this parameter regime, P (t) still H displays damped coherent oscillations as in the undriven case. Though, due to the driving, three main oscillation frequencies l as well as corresponding damping rates C come into play. j j (ii) When eL 5X the parameter regime is met where CDT may occur. This effect has been investigated in Ref. [245] using a generalized Bloch equation formalism applied to a super-Ohmic bath where J(u)Ju3, cf. Eq. (238). There, it is argued that, at high temperatures, the effect of the driving field is simply to renormalize the bare tunneling frequency D as D PD J (eL /X). Hence, the 0 0 0 0 localization transition occurs again near the zeros of the zeroth-order Bessel function J (eL /X). As in 0 the NIBA case, because the transition rate vanishes faster than the tunneling frequency, this implies that in the symmetric case, e "0, there will be a slow coherent oscillation near the localization 0 transition, in agreement with [226]. On the other hand, for the biased case one expects pure exponential decay in the vicinity of the transition. 12.1.2.2. Adiabatic frequency regime. For weak dissipation, low temperatures and in the adiabatic regime +X;E(t), D+hQ (t)D;E(t), where h(t)"!arccot(e(t)/D ), the tunneling dynamics can be 0 investigated with the help of the adiabatic approach discussed in Section 11.5 [263,264]. Here, E(t)"+(D2#e2(t))1@2 is the adiabatic level splitting of the TLS Hamiltonian H (t) in Eq. (320). In 0 TLS this case, in contrast to the adiabatic regime discussed within the NIBA, cf. Eq. (347), the relevant dynamical variable is the difference N(t) between occupation numbers in the lower and upper adiabatic states of H (t), cf. Eq. (71) for the zero driving case, and not the tunneling quantity P(t). TLS Neglecting the adiabatic term !i+R(t)(dR~1/dt) in Eq. (324), one then finds that, at long times, the dynamics of N(t) is governed by a rate equation which is formally similar to Eq. (343) with Eq. (347), i.e., NQ (t)"!C(t)[N(t)!N (t)] , (365) !$,/4 where the adiabatic equilibrium distribution of the TLS is given by N (t)"tanh(E(t)/2k ¹), and !$,/4 B where second-order perturbation theory in the bath-TLS coupling leads to the rate expression C(t)"(+D /E(t))J(E(t)/+) coth (E(t)/2k ¹) , (366) 0 B with J(u) being the environmental spectral density, cf. Eq. (238). A formal solution for N(t) is obtained in terms of quadratures as

P

N(t)"

t

~=

A P

dt@ exp !

t

t{

B

dtAC(tA) C(t@)N (t@) . !$,/4

(367)

9 We disregard real-valued shifts of the oscillation frequencies l , as well as shifts of the amplitudes p caused by j j coupling to the environment, as they are of secondary interest here.

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The position expectation value P(t) can now be evaluated from the relation, cf. Eq. (70) for the static case, P(t)" cos[h(t)]N(t)# sin[h(t)]M(t) ,

(368)

with M(t) " : TrMo (t)p N, N(t) " : TrMo (t)p N, and o (t) being the reduced density matrix evolving in E x E x E the adiabatic basis. Observing that asymptotically M(tPR)+0, the asymptotic behaviour of P(t) may be evaluated from Eq. (367) together with the relation P(t)"(+e(t)/E(t))N(t). We refer to [263,264] for an application of these results to the investigation of the nonlinear acoustic response in vibrating-reed experiments on dielectric glasses at low temperatures [265]. In this case, in fact, the dissipative mechanism mainly originates from the interaction between thermal phonons and TLS-like low-energy excitations arising from the disordered glassy structure [266,267]. A weak coupling approach in the TLS—phonon interaction is generally correct to treat the low-temperature dynamics. Thus, Eq. (367) with Eq. (366) given in terms of the phononic spectral density J(u)Ju3 yields the formal solution of the dynamics for insulating glasses driven by low-frequency strain fields, being the case for the vibrating-reed experiments [265]. In contrast, when vibrating-reed experiments in metallic glasses are considered [265], the interaction of the TLS with electrons cannot be described within a perturbative treatment. In this case the NIBA equation (343) within the adiabatic limit (347) properly describes the dynamics [268]. 12.1.3. Driving-induced coherence: ¹he case a"1/2 In the limit of a large cutoff u , where the approximation (296) for the Ohmic kernels holds, the # equation for P(t) for a"1 is given by Eq. (298). Let us consider only asymmetry driving as in the 2 Hamiltonian of Eq. (325). From Eq. (298), the harmonics p (X, eL ) of P(t) are given by the expression m [221] C 2u 0 #h (!imX, C ) (369) p (X, eL )" 0 m C !imX p m 0 with = h (j, C )" dq e~jq~C0q@2~Q{(q)A~ (q) , 2k 2k 0 0 (370) = h (j, C )" dq e~jq~C0q@2~Q{(q) A~ (q) , 2k`1 0 2k`1 0 where the coefficients A~(q) have been defined in Eq. (334). Here, C "pD2/2u . In Fig. 15 the k 0 0 # fundamental spectral amplitude g (X, eL )"4pDp (X, eL )/eL D2 is plotted vs. frequency for different 1 1 temperatures. As the temperature is decreased, resonances are found at submultiples X"e /n (n"1, 2,2) of the static bias. Correspondingly, the dynamics is intrinsically non0 Markovian. As the temperature is increased, the coherence is increasingly lost, note the behavior of the dot-dashed and dashed lines in Fig. 15.

P

P

12.1.4. Quantum stochastic resonance The phenomenon of stochastic resonance constitutes a cooperative effect of noise and periodic driving in bistable systems, resulting in an increase of the response to the applied periodic signal for

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some optimal value of the noise strength. Since its discovery in 1981 [269], this intriguing phenomenon has been the object of many investigations in classical contexts [270—272]. Classically, the maximal enhancement in the output signal is assumed when the thermal hopping frequency is near the frequency of the modulation. Hence, the term resonance. Here, we shall focus on the deep quantum regime, where tunneling is the only channel for barrier crossing and one can treat the bistable system within the two-level system approximation discussed in this section. As we shall see, qualitative new features occur as compared to the classical case [232—234]. As the temperature is increased, the investigation of quantum stochastic resonance (QSR) requires to go beyond the two-level system approximation discussed in this section. QSR in the regime where both quantum tunneling and thermal hopping over the barrier contribute to the escape mechanism has been investigated in [235] by use of a semiclassical analysis. Recently, this regime has been investigated numerically in [203] via the QUAPI method discussed in Section 10.2. The relevant theoretical quantity describing the dissipative dynamics under the external perturbation is the expectation value P(t)"Sp T . On the other hand, the quantity of experimental z t interest for Quantum Stochastic Resonance (QSR) is the time-averaged power spectrum

P

SM (u)"

`=

~=

dqe*uqCM (q) ,

(371)

defined as the Fourier transform of the time-averaged, asymptotic (i.e., t P!R) symmetrized 0 correlation function CM (q) where

P

X 2p@X 1 CM (q)" dt Sp (t#q)p (t)#p (t)p (t#q)T. z z z 2p 2 z 0

(372)

The combined influence of dissipative and driving forces render extremely difficult an evaluation of the correlation function CM (q) (and hence of the power spectrum) at short times. Matters simplify for times t, q large compared to the time scale of the transient dynamics, when P(t) acquires the periodicity of the external perturbation, cf. Eq. (328). Correspondingly, CM (q) and its related power spectrum in the asymptotic regime read = lim CM (q)"C(!4)(q)" + Dp (X, eL )D2e~*mXq m q?= m/~=

(373)

= S(!4)(u)"2p + Dp (X, eL )D2d(u!mX) . m m/~=

(374)

and

Hence, the amplitudes Dp D, as given in Eq. (330), of the super-harmonics of P(!4)(t) determine the m weights of the d-spikes of the power spectrum in the asymptotic state. In order to investigate QSR [232—234] we shall examine the scaled power amplitude g in the mth frequency component of m S (u) [273] !4 g (X, eL )"4pDp (X, eL )/+eL D2 . m m

(375)

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The quantitative study of QSR requires thus to solve the asymptotic dynamics of the nonlinearly driven dissipative bistable system. Here, we shall discuss some characteristics of QSR as they emerge from the study of the dynamics of a TLS in an Ohmic bath, cf. Eq. (294), and under ac-modulation of the bias energy as in Eq. (325). We shall focus on the low temperature regime and high cutoff limit +u
Fig. 23. Quantum stochastic resonance (QSR) in the deep quantum regime. The power amplitude g (X, eL ) is plotted as 1 a function of the temperature for different driving strengths eL for the case a"1 of Ohmic dissipation. Frequencies are in 2 units of C "pD2/u , temperatures in units of +C /k . For highly nonlinear driving fields eL 'e , the power amplitude 0 0 # 0 B 0 decreases monotonically as the temperature increases (uppermost curve). As the driving strength eL is decreased, a shallow minimum followed by a maximum appears when the static asymmetry e equals, or slightly overcomes, both the external 0 frequency X and strength eL (intermediate curves). For even smaller external amplitudes, the nonlinear QSR can be studied within the linear response theory (dashed curve).

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temperatures in units of +C /k ). For highly nonlinear driving fields eL 'e the power amplitude 0 B 0 decreases monotonically as the temperature increases (uppermost curve). As the driving strength eL is decreased, a shallow minimum followed by a maximum appears when the static asymmetry e equals or slightly overcomes both the external frequency X and strength eL (intermediate curves). 0 For even smaller external amplitudes, the nonlinear QSR can be studied within the linear response theory (dashed curve). In the linear region the shallow minimum is washed out, and only the principal maximum survives. It is now interesting to observe that, because for the undriven case the TLS dynamics for a"1 is always incoherent down to ¹"0, the principal maximum arises at 2 the temperature ¹ at which the relaxation process towards thermal equilibrium is maximal. On the other hand, the minimum in g appears in the temperature region where driving-induced coherent 1 processes are of importance, cf. Fig. 15. This means that the power amplitude g plotted versus 1 frequency shows resonances when X+e /n (n"1, 2,2). As the temperature is increased, this 0 coherence is increasingly lost. (iii) QSR can be well understood within a linear response analysis in an intermediate-to-lowfrequency regime. Within linear response, only the harmonics 0,$1 of P(!4)(t) in Eq. (373) are different from zero, p "P being just the thermal equilibrium value in the absence of driving, and p "eL sJ ($X) 0 45 B1 being related to the linear susceptibility sJ (X) by Kubo’s formula. With increasing strength eL linear response theory fails, and higher harmonics become important. Let us consider the regime +X;2pak ¹ and k ¹'+D , and/or a'1, and/or e
A B

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Fig. 24. Noise-induced suppression (NIS) of higher harmonics. The third power amplitude g when plotted versus 3 temperature reveals a striking characteristic: As the driving frequency is decreased, a noise-induced suppression (NIS) occurs in correspondence of the QSR maximum in the fundamental harmonic. Frequencies are in units of the bath-renormalized tunneling splitting D , cf. Eq. (310), temperatures in units of +D /k . % % B

Similar qualitative results are obtained also in the parameter region of low temperatures k ¹4+D and weak coupling a;1 where underdamped quantum coherence occurs and NIBA B 0 breaks down [233,234]. (iv) Even if QSR can be well understood within a linear response theory, novel interesting effects arise in the regime of nonlinear QSR, when higher order harmonics become resolvable. In particular, quantum noise can substantially enhance, but also suppress, the nonlinear response. Fig. 24 depicts the behavior of the third power amplitude g versus temperature. It reveals 3 a striking effect: As the driving frequency is decreased, a noise-induced suppression (NIS) of higher harmonics occurs in correspondence of the stochastic maximum in the fundamental harmonic g . 1 To understand the results shown in Fig. 24 it is convenient to rewrite p (X) as the sum of m a quasistatic (frequency-independent) contribution

P

1 2p p(24)" dx tanh [+b(e #eL cos x)/2]cos(mx) , (378) m 0 2p 0 and a retarded one p(3%5)(X). The latter becomes negligible when XP0. For the parameters in m Fig. 24 this amounts to X410~4D . A numerical evaluation shows that the NIS is most 0 pronounced when the retarded contribution becomes negligible, so that p (X)Kp(24). m m The same reasoning holds true in the parameter regime of low-driving frequencies, low temperatures and weak coupling to the bath where adiabatic quantum coherence occurs [233,234]. Hence, the discussion of this subsection makes clear that quantum noise does not necessarily represent a nuisance but rather can be a useful tool when used within the interplay of nonlinearity, driving and (quantum) noise. 12.2. Pulse-shaped periodic driving Given the results of the previous subsections, we have all the necessary tools to discuss the case of pulse-shaped periodic driving of the form e(t)"e #eL g(t) cos(Xt), 0

g(t#T )"g(t) , 1

(379)

328

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Fig. 25. Different periodic driving signals can be used to control tunneling: (a) monochromatic driving; (b) and (c) pulse-shaped polychromatic driving of period T . The latter are obtained by modulating a monochromatic signal by 1 a pulse function of pulse duration T . 0

and D(t)"D . The quantity g(t) is a “pulse” function of period T "2p/X 5T"2p/X 0 1 1 that modulates the monochromatic part, cos(Xt), of the signal. Depending on the particular shape of the ‘pulse’ function g(t), different experimental realizations may be mimicked, as shown in Fig. 25. Because Eqs. (343) and (350) are restricted only by the assumption of a separation of time scales, in the parameter regime q~1<X, X , C , with C the appropriate time-averaged relaxation rate, K 1 H,g H,g the resulting dynamics is Markovian. The dynamics can then be approximated by Eq. (343). An additional approximation to Eq. (343) can be discussed in the parameter regime X< C <X . H,g 1 That is, we assume that the pulse-shape function g(t) is a slowly varying function on the time scale set by the transient dynamics, while, on the contrary, the monochromatic part cos(Xt) is fastly changing. Although the oscillatory long-time dynamics will assume the periodicity of the slow pulse shape function g(t), see Eq. (328), the dynamics within a pulse period T is also determined by 1 the fast monochromatic signal. Hence, due to the assumption X
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The average over the period T"2p/X yields [231,246]

A A

Ssin [f(t, t@)]TT"J

A A

BB BB

2eL g(t) X(t!t@) sin X 2

cos[e (t!t@)] , 0

2eL g(t) X(t!t@) sin 0 X 2

sin [e (t!t@)] . 0

Scos [f(t, t@)]TT"J 0

(380)

The smooth, slowly oscillating function SP (t)TT therefore satisfies the rate equation (343) with M relaxation rate given by

P

C (t)" M,g

=

0

A

A BB

2eL g(t) Xq sin X 2

dq h(`)(q)J 0

cos(e q) . 0

(381)

The nonstationary asymptotic quantity is P (t)"o (t)/C (t), where /4,g M,g M,g

P

o (t)" M,g

=

0

dq h(~)(q)J 0

A

A BB

2eL g(t) Xq sin X 2

sin (e q) , 0

(382)

with h(B) defined in Eq. (308). Note that when g(t)"1 the high-frequency approximation (350) is recovered. This is in agreement with the physical intuition saying that a monochromatic signal can be thought to be a pulse-shaped signal with an infinite pulse period T . The asymptotic oscillatory 1 dynamics for the case of Ohmic friction and pulse-shaped driving of the type (b) in Fig. 25 is shown in Fig. 26. For comparison, the prediction of the rate equation (343) with Eqs. (344) and (346) is also depicted. In addition, we show the prediction of Eq. (343) where the approximated equations (381) and (382), obtained in the Markovian limit of low X and high X, are used. This latter Markov 1 —#— high X approximation describes the TLS oscillations between the two bounding values P " : P and P " : P . These bounds are indicated by the arrows in Fig. 26. The low 45 /4,g/0 H,/4 /4,g/1 X -high X approximation scheme clearly misses the fine structure due to the monochromatic 1 driving. The possibility to use this effect to control long-range electron-transfer reactions have been proposed in [230]. 12.3. Dynamics under ac-modulation of the coupling energy In this section our focus is on the effect of ac-modulation of the coupling energy of the TLS. We shall assume for the driven TLS the form H (t)"!1+[D exp(d cos X t)p #e p ] . TLS 2 0 d x 0 z

(383)

This time dependence of the coupling parameter could arise for example from an ac-modulation of the barrier height or width of the underlying double-well potential. For example, for a particle of mass m moving in a double-well potential with barrier height E , vibrational frequency u at the B 0 bottom of each well, and tunneling distance d, the bare tunneling splitting D in WKB approxima0 tion is given by D "u exp[!d(2mE /+)1@2]. This kind of driving could be realized in a supercon0 0 B ducting loop with two Josephson junctions [208]. It is important to observe that, while the effect of

330

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Fig. 26. Dynamics of the population difference P(t) induced by a periodic pulse of the form in Fig. 25b. A low-frequency periodic square pulse of period T "2p/X and pulse duration T "T /2 is multiplied by a monochromatic signal of 1 1 0 1 intermediate frequency X. The oscillatory dynamics within the exact NIBA (solide line) is depicted in the inset where P(t) is plotted for longer times. The predictions of the exact NIBA are compared with two Markovian approximations, namely the Markovian approximation in Eq. (343) with Eqs. (344) and (346) (dashed curve), or with Eqs. (381) and (382) (dot—dashed curve) being valid for slow pulses with internal high-frequency oscillations. The latter Markov —#— high X approximation describes the population oscillations between the two bounds P and P , as indicated by the arrows. 45 H,/4 Frequencies are in units of d"D /2, temperatures in units of +d/k . 0 B

asymmetry modulation is additive, a barrier modulation results in a multiplicative (exponential) modification of the coupling parameter. In the following, we shall refer to the situation of pure asymmetry modulation as “case I”, to that of pure barrier modulation as “case II”, respectively. Due to the generality of the master equation within the NIBA (300), and following the lines of the analysis of the previous sections, it is straightforward to investigate the dynamics under the driving in Eq. (383). The obtained results are as follows [238]: (i) As for the case I of bias modulation, the asymptotic dynamics is periodic and an equation for the harmonics of P(!4)(t) is given by Eq. (330), where for the driving of Eq. (383) the functions kB are m explicitly given by

P P

k~(j, d)"(!1)mD2 m 0

k`(j, d)"(!1)mD2 m 0

=

0 =

0

A A

B B

Xq dq e~jq~Q{(q)e~*mXd q@2sin [QA(q)]sin(e q)I 2d cos d , 0 m 2

Xq dq e~jq~Q{(q)e~*mXd q@2cos [QA(q)]cos (e q)I 2d cos d , 0 m 2

where I (z) denotes the modified Bessel function of order m. m

(384)

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331

(ii) As shown in Section 12.1 above, for a symmetric TLS, spatial symmetry properties of the kernels kB(j) imply that all the Fourier components of P(!4)(t) with even index vanish for case I, m cf. Eq. (333) and the discussion below. In contrast, here we find that all super-harmonics vanish identically for ac-modulation of the coupling energy. (iii) Some interesting new features of case II driving can be found in the high frequency regime X
C

A

B

D

12.4. Dichotomous driving Up to now we considered the effects of a deterministic external driving on the TLS dynamics. Here, we shall consider the case where the driving constitutes an external discrete two-state stochastic process (dichotomous process). In contrast to equilibrium noise produced by the harmonic (Gaussian) bath in the spin—boson model, such a noise must be considered as a source of nonequilibrium noise. This situation arises, for example, in proteins, where the transfer of electrons between different sites may be influenced not only by thermal phonons, but also by large-amplitude local excitations [239—241]. If one considers the simplest case of a dichotomous Markovian process (DMP), this can simulate, for example, the hopping of a relevant molecular group or ion between two equivalent positions. Let us thus consider the noisy TLS H (t)"!1+[D p #(e #eL g(t))p ] , (387) TLS 2 0 x 0 z where g(t)"$1 is a stationary DMP with zero mean Sg(t)T "0 and exponentially decaying g autocorrelation function Sg(t)g(t@)T "exp[!lDt!t@D] . g

(388)

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Here, S T denotes the average over the stochastic process. It should be distinguished from the g thermal statistical average denoted by S T in the previous sections. We shall derive the NIBA equations for the TLS dynamics averaged over the discrete two-state stochastic process. Generally, this is a nontrivial problem. It can be done exactly, however, for the case of a DMP. For a more detailed discussion we refer to Refs. [239,240]. The case of a dichotomously fluctuating tunneling matrix element has instead been treated in Refs. [241,242]. Within the NIBA, the dynamics of the driven and dissipative TLS is described by the equations of motion (300) for the quantum expectation value P(t)"Sp T of the position. To average z t Eq. (300) one has to evaluate the correlator SS(t, t@)P(t@)T , where S(t, t@)"exp[!ieL :t dtAg(tA)]. As g t{ follows from a theorem due to Bourret et al. [274] one finds that SS(t, t@)P(t@)T "S (t!t@)SP(t@)T #S (t!t@)Sg(t@)P(t@)T , g 0 g 1 g

(389)

where S (t!t@)"SS(t, t@)T , and S (t!t@)"Sg(t@)S(t, t@)T . In the same way one has 0 g 1 g Sg(t)S(t, t@)P(t@)T "S (t!t@)SP(t@)T #S (t!t@)Sg(t@)P(t@)T , g 1 g 2 g

(390)

where S (t!t@)"Sg(t)g(t@)S(t, t@)T . The next step consists in the evaluation of the averaged 2 g functions S , S , S . This can be done with the help of a theorem due to Shapiro and Loginov 0 1 2 [275], namely,

T

U

dR(t) d Sg(t)R(t)T "!lSg(t)R(t)T # g(t) g g dt dt

(391)

g

holds for any retarded functional R(t). Applying this theorem to the correlator Sg(t)S(t, t@)T , one g obtains a closed set of differential equations for S (t!t@) and S (t!t@). The solution of these 0 1 equations yields

A BC A B

lq S (q)"exp ! 0 2

cosh

A BD

l6 q l6 q l # sinh 2 2 l6

,

(392)

where l6 "Jl2!4eL 2. In the same way one obtains

A B AB A BC A B A BD

i2eL lq lN q S (q)"! exp ! sinh , 1 l6 2 2 lq S (q)"exp ! 2 2

cosh

l6 q lN q l ! sinh 2 2 l6

(393) .

Finally, by applying again the theorem (391) to the correlator Sg(t)P(t)T one obtains an exact set of g coupled equations for SP(t)T and Sg(t)P(t)T , i.e., g g

P

d t SP(t)T "! dq[S (q)h(`)(q) cos(e q)SP(t!q)T 0 0 g g dt 0 !iS (q)h(`)(q) sin(e q)Sg(t!q)P(t!q)T #S (q)h(~)(q) sin(e q)] 1 0 g 0 0

(394)

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333

and

P

t d Sg(t)P(t)T "!lSg(t)P(t)T ! dq[S (q)h(`)(q) cos (e q)Sg(t!q)P(t!q)T 2 0 g g g dt 0 !iS (q)h(`)(q) sin(e q)SP(t!q)T #iS (q)h(~)(q)cos (e q)] . (395) 1 0 g 1 0 The initial conditions are SP(0)T "1, and Sg(0)P(0)T "0. These equations should be considered g g as the generalization of the NIBA to the case of dichotomous driving. They account for the effects of the discrete two-state stochastic process exactly. The above equations are still rather complicated. In the case of a TLS with zero bias, e "0, they simplify because the two coupled equations 0 separate into two independent equations. In particular, the one for the expectation value SP(t)T g reads

P

d t (396) SP(t)T "! dq S (q)h(`)(q)SP(t!q)T . 0 g g dt 0 Eq. (396) can be solved by Laplace transformation. The resulting dynamics can be coherent or incoherent, depending on the interplay between quantum interference effects, dissipation and driving. In particular, in the incoherent regime the resulting decaying rate C has the form DMP 1 l l l#lN l!lN C " 1! hK (`) # 1# hK (`) , (397) DMP 2 lN lN 2 2

CA B A B A B A BD

where hK (`)(j) is the Laplace transform of the function h(`)(t). The predictions of Eq. (396) with Eq. (397) for the case of long-range electron transfer reactions have been worked out in Ref. [241]. Finally, the combined effects of a symmetric DMP and of an additional, externally applied dc—ac field have been investigated recently in [276]. 12.5. The dissipative Landau—Zener—Stu¨ ckelberg problem In this section we investigate the influence of dissipation on the Landau—Zener—Stu¨ckelberg problem discussed in Section 3.5 [247—250,277]. In this context, one considers the time-dependent variation of the energy levels of a dissipative quantum system. This situation is encountered frequently ranging from the case of the solar neutrino puzzle [278], to the physics of nuclear magnetic resonance [279]. It turns out to be of relevance also in the context of macroscopic quantum tunneling in a rf SQUID [248], when the external flux is changed at a constant rate. Quantitatively, the problem is described by the time-dependent spin—boson Hamiltonian (270), with TLS Hamiltonian H (t)"!1+(D p #vtp ) , (398) LZS 2 0 x z where +D is the energy gap between the two levels, cf. Eq. (114). We then ask for the probability 0 P " : (1!Sp T )/2 of finding the system in the excited state in the far future, cf. Fig. 3. LZS z t?= In the absence of dissipation the exact result P "e~2pc in (123) holds, where c is the LZS dimensionless parameter c"D2/(2DvD) and, as shown in Fig. 3, q "D /DvD is interpreted as the 0 LZS 0

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334

time spent in the crossover region. With coupling to the environment, two other time scales q "1/u and q "u /DvD come into play. Because u is the highest frequency of the environment, # # v # # cf. Eq. (238), q defines the reaction-time of the bath to an external perturbation. # In the fast-passage limit c;1, and with u ;JDvD, the system remains only a small time in the # cross-over region as compared to both the tunneling period and the bath reaction time. That is, the undamped result is not modified by dissipation [248,249]. One obtains to first order in c the result P "1!2pc . (399) LZS In the slow passage limit c<1 the situation becomes more complex. Here, we consider the case of very weak dissipation. To tackle the problem, it is convenient to perform the adiabatic transformation (322) which leads to the effective Hamiltonian appearing in the time-evolution operator, cf. Eq. (324), H "!1E(t)p #H !(+vt/E(t))Xp #(+D /E(t))Xp #(+3vD /E2(t))p , %&& 2 z B z 0 x 0 y

(400)

where $E(t)"$+Jv2t2#D2 define the corresponding adiabatic energies, cf. Fig. 3. In the limit 0 cPR, i.e., DvDP0, the adiabatic term (+3vD /E2(t))p vanishes. Note that, of course, the bath 0 y contribution Jp remains finite. In fact, the problem becomes time-independent and the rotation x (322) simply transforms the original Hamiltonian as p Pp and p P!p . This transformation x z z x has been first applied to the Landau—Zener—Stu¨ckelberg problem in Refs. [248,249]. Though, the authors there identify both of the off-diagonal terms in the Hamiltonian (400) as adiabatic contributions, and in turn make a first-order expansion in both the true adiabatic contribution Jp , and the bath-induced contribution Jp . The bath-induced contribution Jp , however, has y x z been considered to all orders. This neither constitutes an adiabatic approximation nor a weak coupling expansion in the interaction with the bath. Thus, this procedure seems awkward to us; see also in [277], where in the adiabatic passage limit c<1 and in the limit of weak coupling with the bath, a first-order expansion both in the adiabatic term and in the bath interaction is done. One then obtains P "P #DP, where P is the Landau—Zener—Stu¨ckelberg probability in the LZS 0 0 absence of dissipation, while DP gives the first-order correction due to the environment. It reads [277]

P

P

1 `= +2D2 `= 0 , DP" (401) dt dt e*g(t2,t1)@+K(t !t ) 2 1 E(t )E(t ) 1 2 +2 2 1 ~= ~= where K(t) is the force—force correlation of the bath in Eq. (229), and g(t , t )":t21 dt E(t). This t 2 1 equation, though, is still too complicated to be handled without approximations. In the lowtemperature regime +bD <1 one obtains 0 e~+bD0@2 , (402) DP+2a pe~4pc#c +bD /2 0 while in the high-temperature regime +bD ;1, but still +bu <1, one finds 0 # 1 . (403) DP+2a pe~4pc#c +bD /2 0

A

B

A

B

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335

Though approximate, the results (402) and (403) show that the interaction with the bath modifies the non-dissipative result even at ¹"0. Moreover, the transition probability increases monotonically as the temperature grows. This is in contrast with the results in [248,249] where no influence of the bath was found at ¹"0, and with those in [250] where even a reduction of the transition probability was predicted.

13. The driven dissipative periodic tight-binding system In this section we consider the effect of dissipation on the driven infinite tight-binding (TB) model introduced in Section 4. The dissipative TB system can serve as an idealized model for the diffusion of a quantum particle on a surface of a single-crystal, or among interstitials inside a crystal [280]. It can also be invoked to investigate quantum effects in the current—voltage characteristic of a small Josephson junction [281] or of superlattices driven by strong dc- and ac-fields [18—22,92]. Finally, this multistate system can be related to the Luttinger liquid model [282,283] for the conductance between two one-dimensional quantum wires connected by a weak link via a formally exact mapping [284]. The dissipative multistate system in the absence of time-dependent driving has been the object of intense research in the past years [285—290]. In contrast, in Section 4 we investigated localization effects in dc—ac-driven TB lattices in the absence of thermal noise. Here, we extend the analysis by investigating the combined effects of dissipation and driving on the tunneling dynamics. As a starting model we consider the one-dimensional infinite tight-binding lattice described by the Hamiltonian H(t)"H #H (t)#H #H , cf. Eq. (222). The first term H is the Hamiltonian TB %95 SB B TB of the bare multistate system, cf. Eq. (130), i.e., +D H "! 0+ (DnTSn#1D#Dn#1TSnD) , (404) TB 2 n where DnT denotes a (Wannier) state localized at the nth site, while 2+D is the width of the single 0 energy band. The influence of the driving force is described by Eq. (131), i.e., H (t)"!q+[e #eL cos(Xt)]/d, q"d+ nDnTSnD , (405) %95 0 n where the operator q measures the position of the particle on the lattice. Finally, the Hamiltonian term H #H describes the influence of an environmental bath of harmonic oscillators as given by SB B Eq. (224). Suppose that the particle has been prepared at time t "0 at the origin, with the bath having 0 a thermal distribution at temperature ¹. Then, the dynamical quantity of interest is the probability P (t) for finding the particle at site n at time t'0. In turn, the knowledge of P (t) enables the n n evaluation of all the transport quantities of interest in the problem. For example, the nonlinear mobility k(e , eL ) reads 0 d X t`2p@X k(e , eL )"lim dt@ PQ (t@) , (406) 0 +e 2p t t?= 0

P

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336

and the nonlinear diffusion coefficient D(e , eL ) is given by 0

P

X t`2p@X D(e ,eL )"lim dt@ SQ (t@) , 0 4p t t?=

(407)

where the position’s expectation value P(t) " : SqT , as well as the variance S(t) " : Sq2T !SqT2, are t t t defined in Eq. (132). The path-integral evaluation of the velocity v(t)"PQ (t) in the presence of ac—dc fields has been addressed in Ref. [291], to the lowest perturbative order D2 in the tunneling matrix element, under 0 the assumption that the particle tunnels incoherently from site to site. As demonstrated in Ref. [292] for the case of a dc-field, and in [293] in the presence of a time-dependent field, in this case the resulting dynamics is identical to that of a nearest-neighbor hopping model in which the occupation probabilities obey rate equations. These equations can be thought as the generalization of the NIBA Markovian equation (343) to the case of a tight-binding lattice. More generally, by abandoning the Markovian assumption, the NIBA set of coupled equations for a driven tightbinding lattice reads [294]

P

PQ (t)" n

t

0

dt@[¼ (t, t@)P (t@)#¼ (t, t@)P (t@)] & n~1 " n`1

P

!

t

0

dt@(¼ (t, t@)#¼ (t, t@))P (t@) , & " n

(408)

where the backward ¼ (t, t@) and forward ¼ (t, t@) kernels can be expressed in terms of the NIBA " & kernels K(B)(t, t@) in Eq. (303) by using the relations K(`)(t, t@)"¼ (t, t@)#¼ (t, t@) and & " K(~)(t, t@)"¼ (t, t@)!¼ (t, t@). In the Markovian limit, one obtains from Eq. (408) the coupled set & " of equations [293] PQ (t)"C (t)P (t)#C (t)P (t)!(C (t)#C (t))P (t) , n & n~1 " n`1 & " n

(409)

where the backward C and forward C tunneling rates can be expressed in terms of the NIBA " & Markovian kernels C (t) and o (t) in Eqs. (344) and (346) using the relations C (t)"C (t)#C (t) M M M & " and o (t)"C (t)!C (t). M & " The general solution of Eq. (409) can be obtained in the form P (t)"exp[!(F(t)#B(t))](F(t)/B(t))~n@2I (2JF(t)B(t)) , n n

(410)

with

P

B(t)"

t

0

dt@ C (t@), "

P

F(t)"

t

0

dt@ C (t@) , &

(411)

and I (z) is the modified Bessel function of order n. This in turn leads to the result n P(t)"d[F(t)!B(t)] ,

S(t)"d2[F(t)#B(t)] .

(412)

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In the dc-limit eL ,0 one recovers the known relations P(t)"do t and S(t)"d2c t, where 0 0 C "C D L and o "o D L , cf. Eqs. (306) and (307). Hence, the particle diffuses linearly in time 0 M e/0 0 M e/0 with nonlinear dc-mobility k (e )"d2o (e )/+e and dc-diffusion coefficient D (e )"d2C (e )/2. 0 0 0 0 0 0 0 0 0 Turning next to the explicit expression for the nonlinear mobility k(e , eL ) and the nonlinear 0 diffusion coefficient D(e , eL ), one finds [293] 0

P P

A A BB A A BB

d2D2 = eL Xq 0 k(e , eL )" sin dq J 0 0 +e X 2 0 0

h(~)(q)sin(e q) , 0

(413)

d2 = eL Xq dq J D(e , eL )" D2 sin 0 0 0 2 X 2 0

h(`)(q)cos(e q) , 0

(414)

with h(B)(q) defined in Eq. (308). Hence, the nonlinear diffusion has the same form as the incoherent tunneling rate of a two-state system driven by a fast ac-field, cf. Eq. (351). Thus far, our NIBA results have been general. For quantitative calculations we restrict ourselves to the important case of Ohmic dissipation (294). In our reported calculations we have used a high-frequency cutoff u "20D . # 0 The mean square deviation S(t) vs. time is depicted by the curves in Fig. 27. The average slope of the curves in Fig. 27 is well approximated by twice the diffusion coefficient D(e , eL ) in Eq. (414). For 0 generic values of e , eL and X, the ac-controlled diffusion is reduced as compared to the undriven 0 case (e "eL "0). Though, near resonances e "nX, the diffusion can be enhanced. For values of 0 0 eL and X such that J (eL /X)K0, i.e., eL /XK2.4, see the dot—dashed curves in Fig. 27, the diffusion 0 becomes very slow, especially for zero dc-bias e and high driving frequencies X. This choice for eL /X 0 corresponds to the condition for dynamic localization (DL) found for nondissipative TB systems, cf. Section 4. This behavior can also be inferred by inspecting Fig. 21 where the high-frequency rate C "2D(e , eL )/d2 is plotted versus the dc-bias e . H 0 0 In Fig. 28a the dimensionless current I(e , eL ; a)/edD , where I"e(+e /d)k, with e denoting the 0 0 0 elementary electron charge, is plotted vs. the dissipative strength a for different driving frequencies X. Two striking effects occur for weak damping and strong damping, respectively: (i) A negative current at weak dissipation is observed. Moreover, for fixed eL and X the amplitude of the negative current is maximal for a characteristic value a"a* of the Ohmic strength. (ii) For stronger dissipation the current becomes independent of the ac-frequency and of dissipation on a wide range. As the frequency is increased further, the linear response regime eL /X;1 is approached and the current moves towards its dc-limit, full-line in Fig. 28a. In Fig. 28b, I(e , eL ; a)/edD is plotted vs. 0 0 the applied dc-voltage e for different values of the applied ac-voltage eL . A negative current vs. static 0 bias e , i.e., a “negative conductance” is observed for small dc-bias. 0 To explain the results of Figs. 27 and 28 it is convenient to expand J ((eL /X)sin(Xq/2)) in Fourier 0 series, yielding

AB

AB

= eL eL R(e , eL ; a)"J2 R (e ; a)# + J2 R (e #nX; a) , 0 0 X 0 0 n X 0 0 n/~=,nE0

(415)

where R (e ; a) " : R(e , eL "0; a), and R"I or D. Thus, the ac-voltage produces new channels for 0 0 0 dc-current flow or dc-diffusion due to photon emission (n(0) and absorption (n'0), each

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Fig. 27. Diffusion in the driven and dissipative tight-binding system. The dimensionless variance S(t)/d2"(Sq2T !SqT2)/d2 vs. time (full straight line and dot—dashed curves) can be reduced or enhanced as compared to t t the force-free case. It assumes its minimal values for zero dc-bias, high ac-frequencies X and when eL K2.4X. The dashed straight lines depict the curve 2D(e ,eL )t, where D is the diffusion constant in Eq. (414). Ohmic dissipation is considered. 0

Fig. 28. Dimensionless average current I/edD in the driven and dissipative tight-binding system. I/edD is plotted vs. the 0 0 dissipative Ohmic coupling strength a (panel (a)) and vs. the dc-bias e (panel (b)). For weak dissipation a negative current 0 occurs which is maximal for a nonzero value of a. For stronger dissipation the current becomes independent of both the ac-frequency X and of the dissipative strength a on a wide range.

weighted by the factor J2(eL /X). With R being the current I, we note that Eq. (415) formally n coincides with the driven tunneling current of Tien—Gordon theory [15] in Eq. (155): The result for the corresponding dc-tunneling current is here substituted by the dissipation renormalized static current, resulting from Eq. (413) with eL "0.

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Let us discuss the time-averaged current further. The current inversion in Fig. 28 is due to the competition between the channels with n50, each of which always gives a positive contribution to the total current, and higher-order channels with n(0 which may give a negative contribution. In fact, with the help of the blocking effect, i.e., J (eL /X)K0 of the n"0 channel, it can be more 0 favorable in current gain (in absolute value) to emit n photons than to absorb n of them, even if the weight J2(eL /X) of the sidebands is the same. For example, for the dot—dashed curve in Fig. 28a the n main contribution to the current is found to come from the first negative channel (n"!1) (not shown). Correspondingly, the maximal negative inversion occurs for a"a*O0 such that DI (e !X; a)D is maximal as a function of a, cf. Fig. 28a. For fixed eL , I(e , eL ; a) vs. e may have several 0 0 0 0 minima e(n) determined by the condition .*/ e(n) #e*"nX , .*/

(416)

where e* is the maximum of the dc-current I vs. the dc-bias, see Fig. 28b. Note that the positions 0 of these minima do not depend on the strength eL of the ac-field. The ac-field strength determines instead the weight of the different channels. For the parameters chosen in Fig. 28b, the maximal negative current occurs when eL is chosen such that the first channel dominates, i.e., J2(eL /X)K0, nO1. A similar reasoning applies to explain the behaviour of the variance in n Fig. 27. Finally, we observe that the structure in Eq. (415) is universal, i.e., it does not depend on the specifics of the thermal bath. Dissipation determines instead the shape of the dc-current I , and of 0 the dc-diffusion D . Hence, as long as the dc-current vs. the dc-bias presents the physical 0 characteristics of being antisymmetric, i.e., positive (negative) for e '0 (e (0) with a maximum 0 0 at e "e*, a current reversal is possible even for a dissipative mechanism which is different from the 0 Ohmic one. In the same spirit, in the investigated temperature regime k ¹;+u , the nonlinear B # diffusion will be strongly reduced in the parameter regime of dynamic localization. Indeed, a similar behavior for the current vs. the dc-bias in semiconductor superlattices at room temperature has been theoretically described in [16—18,92] within a phenomenological approach based on a classical Boltzman equation with a single collision time ansatz; and in [98] within the framework of a bias modulated Bardeen Hamiltonian for N quantum wells. Recently, such time-averaged negative currents vs. the dc-bias have been experimentally observed in semiconductor superlattices [20,21]. Note, however, the fact that the dc-current I P0 as the dc-strength e PR is solely a peculiar0 0 ity of the single band approximation used for the Hamiltonian H in Eq. (404). Such an TB approximation in fact breaks down whenever the externally applied dc- and ac-field strength e and 0 eL , as well as the driving frequency X, become comparable with the interband level spacing. The effect of higher bands on the dc-voltage characteristics in dc-ac driven superlattices has recently been discussed in [295] within the framework of the Bardeen Hamiltonian. There, the characteristic saw-tooth profile of the average current vs. dc-bias due to the formation of high-field domains is also discussed. Finally, we observe that an absolute negative conductance has recently been predicted also for a dissipative TB particle under the combined effect of a dc- and of a dichotomous field [294]. Again, this effect is found to be largely independent of the specifics of the shape of the dc-current I vs. dc-bias. 0

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14. Dissipative tunneling in a driven double-well potential In Section 6 the nondissipative tunneling motion of a particle moving in a driven double well, cf. Eqs. (172) and (173), has been discussed, while in Sections 11 and 12 we investigated the driven and dissipative dynamics restricting our attention only to the lowest doublet of the potential, i.e., in the two-level-system (TLS) approximation. Here we shall investigate the dissipative motion within the full double-well dynamics. 14.1. The driven double—doublet system The first new feature arising in the full double-well potential (DW), as compared to the TLS, is the possibility of vibrational relaxation (i.e., intradoublet transitions) in addition to left-to-right tunneling (i.e., interdoublet transition). The competition between intra- and interwell relaxation rates for the undriven DW has been discussed analytically in [296,297] and [298]. When an external ac-field is applied, the possibility of controlling tunneling by pumping population from the lower to the upper tunneling pairs occurs. Very recently, a calculation in this direction has been initiated in [299,300], where the authors restrict themselves to the case of a DW when only the two lowest energy doublets are thermally accessible. In [299,300], the polaron transformation [259] usually applied to investigate the two-level system [257,158] is extended to the double—doublet system (DDS) problem. A second-order perturbative calculation in the Franck—Condon renormalized tunneling doublets is then carried out. As suggested in [299,300], because the upper doublet is at least one order of magnitude larger than the lower one, even a small increase of the upper doublet’s population can lead to a significative increase in population transfer between the localized states of the underlying double-well potential. Some of the most intriguing results for an external field which is resonant with the interdoublet spacing can be summarized as follows: (i) The asymptotic upper doublet population p (R) is naturally increased by an external 61 resonant field as compared to the nondriven case. (ii) As long as the vibrational relaxation rate is larger than the interdoublet tunneling rates, the upper and lower doublet populations will reach their asymptotic values on a shorter time scale as compared to the left—right scale. The overall left—right rate constant is then the sum of the individual doublet left—right rate constants, each weighted by the corresponding steady-state doublet population. Because the asymptotic upper doublet population p (R) is increased by an 61 external field as compared to the nondriven case, the external field serves to pump population in the upper doublet, from where it can be more effectively transferred to the right well by left—right tunneling transitions. (iii) When, on the contrary, vibrational relaxation occurs on a much longer time scale as compared to interdoublet tunneling, the two doublets are effectively decoupled in the absence of an external field. In the presence of an external resonant field, the up—down population would oscillate indefinitely at the gap frequency if the left—right rate processes were absent. Dissipation qualitatively changes the characteristics of the left (right) population, which reaches asymptotically the steady value 1. The evolution of the left (right) population towards the steady-state value 1 may be underdamped 2 2 or extremely overdamped depending on the choice of the parameters.

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14.2. Coherent tunneling and dissipation Here, we shall apply the Floquet—Markov—Born approach of Section 9 to investigate numerically the effects of dissipation on driven tunneling in the DW. For the numerical calculations the dimensionless variables and parameters are given by D"E /+u , ¹M "k ¹/+u , c6 "c/u , B 0 B 0 0 X1 "X/u , SM "Sx /+u , where x "J+/mu , see also Eqs. (174) and (175), and c is the damping 0 0 0 0 0 coefficient which enters in the Ohmic spectral density Eq. (235). In the following we omit the overbars. The driven double-well potential » (q, t) introduced in Eq. (226) then reads 1 q2 q4 V (q, t)"! # #Sq cos(Xt) , 1 4 64D

(417)

where D is the dimensionless barrier height. We report on the predictions on the tunneling dynamics of the driven double-well resulting from the simple RWA in Eq. (251), and from its improved version, see discussion above Eq. (251). The latter form will have to be used when investigating the coherent destruction of tunneling induced by the monochromatic field, where quasidegeneracies in the quasienergy spectrum occur. A thermal bath with a spectral density J(u) of the Ohmic—Drude form as in Eq. (236) has been chosen. Using the results of Refs. [191,192], the slow dynamics generated by the simplest RWA equation (251) is depicted in Fig. 29 in a stroboscopic manner. In Fig. 29a we show the time evolution of the correlation function P(t )"TrMo(t )o(t )N/TrMo(t )2N, where o(t ) is the reduced density matrix at n n 0 0 0 the initial time, and t "nT. A pure coherent state centered at one of the classical attractors in the n right well is chosen as the initial state. The chosen parameters are D"6, c"10~5, ¹"0, S" 0.08485, and X"0.9, so that the driving frequency X is close to the fundamental first resonance. Note that for these values of the damping and barrier height the condition D
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Fig. 29. Tunneling in the ac-driven double well with Ohmic dissipation. The scaled parameters are D"6, X"0.9, S "0.08485, c"10~5, and ¹"0. (a) Time evolution of the autocorrelation function P(t ), t "2pn/X, over the first 104 n n time steps, starting from a state centered in one of the wells; (b) a section of the Fourier transform of (a) (dashed: the same function for the undamped case); (c) spatially resolved states at times t (graph 1) n"20, 2) n"40, 3) n"8910, 4) n n"5]104 and stationary state (thick curve); (d) Husimi distribution of the stationary state, compared with the corresponding classical distribution (sharp peaks).

equation in Eq. (251). Fig. 30a depicts the time evolution of the correlation function P(t ) for n parameters of the double well slightly offset from the localization manifold for c"10~6, D"2 and various values of the temperature ¹. P(t ) is evaluated in the improved master equation beyond n RWA, where only terms with kOk@ have been dropped. At low temperatures, P(t ) exhibits n a slowly decaying coherent oscillation with a very long period, due to the small detuning from the quasi-energy crossing. The fast decay of superimposed oscillations reflecting the admixture of higher order quasienergy states is shown in Fig. 30b. Asymptotically, the distribution among the wells (in this stroboscopic picture) is completely thermalized. Upon increasing temperature, the decay time of the coherent oscillations first decreases until this oscillation is suppressed from the beginning (not shown). After going through a minimum, however, the decay time q of P(t ), defined $ n by P(t )Kexp(!n/q ), increases again, until it reaches a resonance-like maximum at an optimal n $ temperature ¹*, see Fig. 30b. This maximum occurs at a temperature ¹* such that the time scale of the incoherent processes induced by the reservoir stabilizes the localization of the wave packet in one of the two wells.

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Fig. 30. Coherent suppression of tunneling in the presence of Ohmic dissipation. (a) Time evolution of the autocorrelation function P(t ) over the first 107 time steps. The scaled parameters being D"2, X"0.01, S"3.171]10~3 and n c"10~6. Inset: The first 2]104 time steps on an enlarged time scale; temperature dependence of the decay time q (defined by P(t )Kexp(!n/q ) for three values of the detuning DX"X!X (S) from the localization manifold. $ n $ -0# Graph (1) DX"!1.4]10~7, (2) DX"5]10~7 at S"3.1712]10~3, (3) DX"1.4]10~6 at S"3.1715]10~3. The other parameters are as in (a).

14.3. Dynamical hysteresis and quantum stochastic resonance The QUAPI method discussed in Section 10 is applied to investigate the dynamics in a driven double-well potential q4 q2 !Sq cos (Xt) , » (q, t)"! # 1 64D 4

(418)

in the presence of Ohmic dissipation, cf. Eq. (237) [203,204]. In parametric form, i.e., in a panel of SqT vs. F(t)"!S cos(Xt), one then observes at asymptotic times a hysteresis loop, cf. Fig. 31, t where the area A comprised by the hysteresis curve is conveniently characterized by

Q

P

A" SqT dF(t)"SX t

2p@X

dtSqT sin(Xt) . (419) t 0 The area A of the hysteresis curve is investigated in Fig. 32. The driving strength is assumed to be so strong that after the time interval q one of the two potential minima disappears and the "i potential exhibits only one global minimum. This driven bistable device can then be thought as a quantum switch. Moreover, in the investigated temperature regime both quantum and classical fluctuations contribute to the particle’s dynamics. In Fig. 32 the area A is plotted vs. driving frequency for different temperatures. For low driving frequencies the hysteresis area is very small, because the particle has enough time to tunnel from one metastable minimum to the global minimum induced by the driving force. Enhancing the frequency the area becomes larger. For even higher frequencies, though, the particle cannot follow anymore the details of the potential modulation and it feels an effective static barrier. In this latter case the area is reduced again, and the particle is on average situated in the potential barrier region. From this discussion it is clear that the maximal area occurs at some intermediate driving frequency. Similar as for the case of

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Fig. 31. Quantum hysteresis. The position expectation value SqT computed parametrically as a function of an external t force F(t)"!S cos(Xt) exhibits a hysteresis loop. As a signature of quantum dynamics, steps in the quantum hysteresis loop, as indicated by the arrows, occur. Inset: enlarged part of the hysteresis loop where steps can be observed. Parameters in scaled units are D"1.5, S"1.2, X"10~4, c"0.005, ¹"0.1. Fig. 32. The area A of the hysteresis loop for a driven and dissipative double well. The area A is shown vs. driving frequency X for dimensionless parameters c"1.0, D"1 and S"0.8 at different temperatures. A maximum occurs at finite frequencies which, at low temperatures, is independent of temperature.

classical hysteresis, it can be argued that the maximum occurs for the optimal frequency X such .!9 that half the time interval q J1/X during which the system is bistable balances the decay time "i .!9 q to cross the barrier due to incoherent tunneling [203]. This matching condition is confirmed by $ Fig. 32 which shows that at low temperatures the area is only weakly temperature dependent, in agreement with semiclassical results on the decay time q [33]. For higher temperatures, the $ optimal matching frequency X is shifted to lower values. This is surprising since it implies .!9 a decrease of the relaxation rate with increasing temperatures. An intriguing feature is shown in Fig. 31, where the quantum hysteresis loop exhibits steps [204]. The same feature has recently been observed experimentally in high-spin molecular magnets [306]. However, it should be remembered that the two physical systems are rather different. To understand better the origin of these steps, the transient time-dependent response SqT t is plotted vs. time in Fig. 33. The system has been prepared in a steady state with a strong tilt to the left, and subsequently has been let slowly evolve towards a positive tilt. Steps similar to those in the hysteresis loop occur. In correspondence to these steps, the energy spectrum computed parametrically as a function of a static control force (i.e., X"0) exhibits avoided level crossing. There, tunneling is enhanced, and a step occurs. The first step for negative tilt is characteristic for thermally assisted resonant tunneling. In agreement with such avoided level crossing features, a multiresonance structure of the relaxation rate plotted vs. a static force is observed [204].

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Fig. 33. Avoided level crossing dynamics for a quantum switch. Upper panel: Position expectation value SqT vs. t F(t)"!S cos(Xt). Characteristic steps are observed. The chosen parameters in scaled units are D"2.5, S"0.7, X"10~4, c"0.001, ¹"0.1. Lower panel: The first adiabatic energy levels (alternatingly dashed and solid lines) computed parametrically as a function of a static control force. The steps in SqT occur in correspondence with t avoided level crossings. Fig. 34. Quantum stochastic resonance (QSR) for a strongly driven double well. The spectral amplification g is plotted 1 vs. temperature at X"0.015, S"0.4 and D"1, for various Ohmic friction strengths c. The occurrence of the maximum at finite temperatures indicates QSR. As the damping strength c is decreased the maximum disappears, i.e., QSR vanishes when quantum coherent processes start to dominate over incoherent ones.

"+ p (X, S)e~*mXt, the spectral amplification t?= m m

Finally, upon noting that SqT

g "4pDp (S, X)/SD2 , (420) 1 1 describes quantum stochastic resonance (QSR), cf. Section 12.1.4. The temperature dependence of g is shown in Fig. 34 for different damping strengths. A maximum at finite temperatures indicates 1 QSR. It should be observed that for the parameters chosen in the figure the unperturbed double well potential has only one tunneling doublet below the barrier. Nevertheless, due to the very high driving strength considered, and the relatively high-temperature regime at which the maximum occurs, the two-level system approximation discussed in Section 11 is now no longer valid.

15. Conclusions With this long review we have attempted to present a “tour of horizon” of the physics that is governed by driven quantum tunneling events. With the present day availability of strong radiation sources such as e.g. free-electron laser systems yielding strong and coherent electric driving fields, together with the advances in micro- and nano-technology, the challenges involving driven

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quantum transport have moved into the limelight of present day scientific activities. In doing so, several novel interpretative concepts as well as new analytic and computational tools have been put forward. These tools, such as the Floquet approach, the (t, t@)-formalism or the computational quasiadiabatic propagator method, to name only a few, are intrinsically nonperturbative in nature. Thus, they are ideally suited to describe non-linear novel driven tunneling phenomena beyond conventional perturbation schemes. In the foregoing sections we hopefully have conveyed several novel features that characterize quantum tunneling as it arises as the result from an interplay involving nonlinearity, time-dependent forcing and dissipation. As shown repeatedly, the application of suitably tailored strong driving forces allows one to selectively control, a priori, quantum tunneling-dominated processes such as population transfer, energy transfer, tunneling probabilities, reaction rates, diffusion coefficients, or current—voltage characteristics. The material in the first seven sections solely deals with driven tunneling in absence of dissipative, environmental influences. The remaining parts account for the modifications that are acquired when frictional quantum effects additionally perturb the driven tunneling dynamics. Although the main scope of this review has been on the theoretical development we have interfaced the presentation with experimental corroboration and/or applications to the best of our knowledge. In doing so, the authors share the hope that both theory and even more experiment will become invigorated by readers of this present review. Certain topics involving driven tunneling have remained untouched. To a large extent the authors focused — although not exclusively — on tunneling problems for which the underlying quantum states are localized, implying bounded transport quantities. In the case of a pure ac-drive, these quantum states can be described in terms of a pure point spectrum for corresponding quasienergies and corresponding square integrable Floquet modes, respectively. In contrast, driven tunneling in periodic structures, cf. Sections 4 and 13, involve extended (scattering) states that belong to an absolute continuous spectrum, which in turn rule unbounded transport quantities. A field-induced barrier transparency in a periodically driven potential barrier has been recently discussed in [301]. In situations where extended quantum states determine the physics, such as in ionization, dissociation, decay of resonances, ac-driven quantum decay, etc., it is often necessary to rotate the coordinates of the Hamiltonian into the complex plane (complex scaling) [302,303]. Characteristic applications are ac-induced dissociation from a nonlinear quantum well [304] or ac-driven quantum decay out of a metastable well [305]. In both of these cases the quantum dynamics becomes governed by complex-valued quasienergies. The field of driven tunneling is presently enjoying widespread interest among the communities of both physics and chemistry. Without doubt, we shall experience still many novel effects that are predominantly ruled by driven quantum tunneling; nevertheless, we share the confident belief that this survey, although not complete, captures a good snapshot of the present state of the art of a very dynamic research area.

Note added in proof After submission of this review several contributions to a timely topic, namely the spontaneous dc-current generation in the presence of external, unbiased time-dependent fields have appeared.

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For example, this effect has been predicted in ac-driven semiconductor superlattices when a symmetry breaking mechanism due to repulsive electron—electron interaction occurs [307]. Likewise, a new class of discrete Brownian rectifiers occurs when considering the dissipative motion of a tightbinding particle in the presence of unbiased, asymmetric forcing, such as e.g. in the presence of harmonic mixing signals of the form eˆ cos()t)#eˆ cos ()t#u) [308], or in the presence of an 1 2 asymmetric dichotomic Markov field [309]. Directed, quantum noise induced current is generated also in quantum ratchets, i.e., in periodic structures that lack reflection symmetry. In this context, the induced current in a continuous ratchet potential has been investigated in [310], and for a discrete model in [311].

Acknowledgements M. Grifoni and P. Ha¨nggi greatly acknowledge the support of the Deutsche Forschungsgemeinschaft via the Schwerpunktprogramm “Zeitabha¨ngige Pha¨nomene und Methoden in Quantensystemen der Physik und Chemie” (HA 1517/14-2). We also thank L. Hartmann for preparing some figures for this review. Moreover, we thank L. Hartmann, M. Holthaus, N. Makri, M. Thorwart, R. Utermann, M. Wagner and M. Winterstetter for providing original figures relating to their works. In the course of writing this review we enjoyed numerous insightful discussions with friends and colleagues around the world, in particular, among others, with R. Aguado, R. Coalson, Y. Dakhnovskii, T. Dittrich, M. Holthaus, I. Goychuk, F. Grossmann, J. M. Gomez-Llorente, L. Hartmann, G. Ingold, P. Jung, H.-J. Korsch, S. Kohler, N. Makri, N. Moiseyev, P. Neu, P. Pechukas, U. Peskin, G. Platero, P. Reimann, M. Sassetti, J. Shao, J. Stockburger, M. Thorwart, D. Tannor, R. Utermann, M. Wagner, M. Winterstetter and U. Weiss. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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