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arXiv:cond-mat/9703199 · Physical Review A

Microscopic treatment of binary interactions in the nonequilibrium dynamics of partially Bose-condensed trapped gases

Nick P. Proukakis, Keith R. Burnett, Henk T. C. Stoof

Published Feb 1, 1998• 112 citations
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Abstract

In this paper we use microscopic arguments to derive a nonlinear Schr\"odinger equation for trapped Bose-condensed gases. This is made possible by considering the equations of motion of various anomalous averages. The resulting equation explicitly includes the effect of repeated binary interactions (in particular ladders) between the atoms. Moreover, under the conditions where dressing of the intermediate states of a collision can be ignored, this equation is shown to reduce to the conventional Gross-Pitaevskii equation in the pseudopotential limit. Extending the treatment, we show first how the occupation of excited (bare particle) states affects the collisions, and thus obtain the many-body $T$-matrix approximation in a trap. In addition, we discuss how the bare particle many-body $T$ matrix gets dressed by mean fields due to condensed and excited atoms. We conclude that the most commonly used version of the Gross-Pitaevskii equation can only be put on a microscopic basis for a restrictive range of conditions. For partial condensation, we need to take account of interactions between condensed and excited atoms, which, in a consistent formulation, should also be expressed in terms of the many-body $T$ matrix. This can be achieved by considering fluctuations around the condensate mean field beyond those included in the conventional finite temperature mean field, i.e., Hartree-Fock-Bogoliubov, theory.

DOI
10.1103/physreva.57.1230

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In this paper we use microscopic arguments to derive a nonlinear Schr"odinger equation for trapped Bose-condensed gases. This is made possible by considering the equations of motion of various anomalous averages. The resulting equation explicitly includes the effect of repeated binary interactions (in particular ladders) between the atoms. Moreover, under the conditions where dressing of the intermediate states of a collision can be ignored, this equation is shown to reduce to the conventional Gross-Pitaevskii equation in the pseudopotential limit. Extending the treatment, we show first how the occupation of excited (bare particle) states affects the collisions, and thus obtain the many-body TTT-matrix approximation in a trap. In addition, we discuss how the bare particle many-body TTT matrix gets dressed by mean fields due to condensed and excited atoms. We conclude that the most commonly used version of the Gross-Pitaevskii equation can only be put on a microscopic basis for a restrictive range of conditions. For partial condensation, we need to take account of interactions between condensed and excited atoms, which, in a consistent formulation, should also be expressed in terms of the many-body TTT matrix. This can be achieved by considering fluctuations around the condensate mean field beyond those included in the conventional finite temperature mean field, i.e., Hartree-Fock-Bogoliubov, theory.
In this paper we use microscopic arguments to derive a nonlinear Schrödinger equation for trapped Bose-condensed gases. This is made possible by considering the equations of motion of various anomalous averages. The resulting equation explicitly includes the effect of repeated binary interactions (in particular ladders) between the atoms. Moreover, under the conditions where dressing of the intermediate states of a collision can be ignored, this equation is shown to reduce to the conventional Gross-Pitaevskii equation in the pseudopotential limit. Extending the treatment, we show first how the occupation of excited (bare particle) states affects the collisions, and thus obtain the many-body T-matrix approximation in a trap. In addition, we discuss how the bare particle many-body T matrix gets dressed by mean fields due to condensed and excited atoms. We conclude that the most commonly used version of the Gross-Pitaevskii equation can only be put on a microscopic basis for a restrictive range of conditions. For partial condensation, we need to take account of interactions between condensed and excited atoms, which, in a consistent formulation, should also be expressed in terms of the many-body T matrix. This can be achieved by considering fluctuations around the condensate mean field beyond those included in the conventional finite temperature mean field, i.e., Hartree-Fock-Bogoliubov, theory.
Using the Kadanoff-Baym non-equilibrium Green's function formalism, we derive kinetic equations for the non-condensate atoms at finite temperatures which include the effect of binary collisions between atoms. The effect of collisions is included using the second-order self-energy given by the Beliaev (gapless) approximation. We limit our discussion to finite temperatures where we can use the single-particle Hartree-Fock spectrum for the excited atoms. In this limit, we can neglect the off-diagonal propagators (g~12\tilde{g}_{12}g~​12​ and g~21\tilde{g}_{21}g~​21​). As expected, this leads to the kinetic equations and collision integrals used in recent work by Zaremba, Nikuni, and Griffin (ZNG) [1]. We also derive a consistent equation of motion for the condensate wavefunction, involving a finite-temperature generalization of the well-known Gross-Pitaevskii equation which includes a dissipative term, as well as the mean field of the non-condensate.
Recent experimental observations of collective excitations of Bose condensed atomic vapours have stimulated interest in the microscopic description of the dynamics of a Bose-Einstein condensate confined in an external potential. We present a finite temperature field theory for collective excitations of trapped Bose-Einstein condensates and use a finite-temperature linear response formalism, which goes beyond the simple mean-field approximation of the Gross-Pitaevskii equation. The effect of the non-condensed thermal atoms we include using perturbation theory in a quasiparticle basis. This presents a simple scheme to understand the interaction between condensate and non-condensed atoms and enables us to include the effect the condensate has on collision dynamics. At first we limit our treatment to the case of a spatially homogeneous Bose gas. We include the effect of pair and triplet anomalous averages and thus obtain a gapless theory for the excitations of a weakly interacting system, which we can link to well known results for Landau and Beliaev damping rates. A gapless theory for trapped systems with a static thermal component follows straightforwardly. We then investigate finite temperature excitations of a condensate in a spherically symmetric harmonic trap. We avoid approximations to the density of states and thus emphasise finite size aspects of the problem. We show that excitations couple strongly to a restricted number of modes, giving rise to resonance structure in their frequency spectra. Where possible we derive energy shifts and lifetimes of excitations. For one particular mode, the breathing mode, the effects of the discreteness of the system are sufficiently pronounced that the simple picture of an energy shift and width fails. Experiments in spherical traps have recently become feasible and should be able to test our detailed quantitative predictions.
This paper extends an earlier quantum kinetics treatment for dilute, weakly interacting, partially Bose-Einstein condensed gases, presented by the author elsewhere (Proukakis N P and Burnett K 1996 J. Res. Natl Inst. Stand. Technol. 101 457), by consistently treating the dynamics of the uncondensed atoms to the same level of approximation as the condensed atoms. Our method is based on a hierarchy of coupled equations of motion for the condensate mean field and fluctuations around this mean field, truncated to second order in the (effective two-body) interatomic potential, and with suitable decoupling approximations for higher-order correlations. By applying perturbation theory in the Hartree-Fock-Bogoliubov basis, we re-derive the quantum kinetic theory of Walser et al (Walser R, Williams J, Cooper J and Holland M 1999 Phys. Rev. A 59 3878), which further indicates the consistency of our treatment with the Kadanoff-Baym non-equilibrium Green function formalism for trapped gases.
This paper extends an earlier quantum kinetics treatment for dilute, weakly interacting, partially Bose-Einstein condensed gases, presented by the author elsewhere (Proukakis N P and Burnett K 1996 J. Res. Natl Inst. Stand. Technol. 101 457), by consistently treating the dynamics of the uncondensed atoms to the same level of approximation as the condensed atoms. Our method is based on a hierarchy of coupled equations of motion for the condensate mean field and fluctuations around this mean field, truncated to second order in the (effective two-body) interatomic potential, and with suitable decoupling approximations for higher-order correlations. By applying perturbation theory in the Hartree-Fock-Bogoliubov basis, we re-derive the quantum kinetic theory of Walser et al (Walser R, Williams J, Cooper J and Holland M 1999 Phys. Rev. A 59 3878), which further indicates the consistency of our treatment with the Kadanoff-Baym non-equilibrium Green function formalism for trapped gases.
We describe a method for studying the dynamics of a condensate in a weakly interacting, dilute Bose gas which is based on a finite-temperature extension of the Gross-Pitaevskii equation. We present a general finite-temperature linear response formalism for nonuniform systems and give explicit results for homogeneous ones, where we can link our treatment to well-known results. This treatment leads to a consistent V2{V}^{2}V2 calculation of the excitations for a weakly interacting Bose gas, including Landau and Beliaev damping as well as collisional lifetimes. We do not make use of the Popov approximation, but include pair and triplet anomalous averages to second order in the interaction potential and thus obtain a gapless theory for the excitations of a weakly interacting homogeneous gas. We also show how the formalism can be used to obtain self-consistent theories. We explicitly derive gapless equations for a weakly coupled finite-temperature condensate with a static thermal component and give a clear criterion of their validity in trapped systems.