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arXiv:cond-mat/9901172

Theory of excitations of the condensate and non-condensate at finite temperatures

Allan Griffin

Published Jan 18, 1999• 2 citations
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Abstract

We give an overview of the current theory of collective modes in trapped atomic gases at finite temperatures, when the dynamics of the condensate and non-condensate must both be considered. A simple introduction is given to the quantum field formulation of the dynamics of an interacting Bose-condensed system, based on equations of motion for the condensate wavefunction and single-particle Green's functions for the non-condensate atoms. We discuss the nature of excitations in the mean-field collisionless region, including the Beliaev second-order approximation for the self-energies. We also sketch the derivation of coupled two-fluid hydrodynamic equations using a simple kinetic equation which includes collisions between condensate and non-condensate atoms.

DOI
10.48550/arxiv.cond-mat/9901172

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Cambridge University Press eBooks•Feb 19, 2009•0 citations
We give an overview of the current theory of collective modes in trapped atomic gases at finite temperatures, when the dynamics of the condensate and non-condensate must both be considered. A simple introduction is given to the quantum field formulation of the dynamics of an interacting Bose-condensed system, based on equations of motion for the condensate wavefunction and single-particle Green's functions for the non-condensate atoms. We discuss the nature of excitations in the mean-field collisionless region, including the Beliaev second-order approximation for the self-energies. We also sketch the derivation of coupled two-fluid hydrodynamic equations using a simple kinetic equation which includes collisions between condensate and non-condensate atoms.
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.
This is based on 4 lectures given at the 13th Australian Physics Summer School, Australia National University, Canberra, Jan 17-28, 2000. The main topic is the theory of collective modes in a trapped Bose gas at finite temperatures. A generalized Gross-Pitaevskii equation is derived at finite temperatures, which is used to discuss a new mechanism for damping in the collisionless region arising from interactions with a static thermal cloud of non-condensate atoms. Next, introducing a kinetic equation for the thermal cloud, we derive two-fluid equations of motion for the condensate and non-condensate components in the collision-dominated hydrodynamic region. We show that these are precisely the equivalent of the Landau two-fluid equations in the limit that the two components are in diffusive local equilibrium. However, our equations also predict the existence of a new zero frequency relaxational mode, in addition to the usual Landau hydrodynamic modes (such as first and second sound). The special importance and simplicity of two-fluid hydrodynamics is stressed.
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.
The discovery of Bose Einstein condensation (BEC) in trapped ultracold atomic gases in 1995 has led to an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first treatment of BEC at finite temperatures, this book presents a thorough account of the theory of two-component dynamics and nonequilibrium behaviour in superfluid Bose gases. It uses a simplified microscopic model to give a clear, explicit account of collective modes in both the collisionless and collision-dominated regions. Major topics such as kinetic equations, local equilibrium and two-fluid hydrodynamics are introduced at an elementary level. Explicit predictions are worked out and linked to experiments. Providing a platform for future experimental and theoretical studies on the finite temperature dynamics of trapped Bose gases, this book is ideal for researchers and graduate students in ultracold atom physics, atomic, molecular and optical physics and condensed matter physics.
We study the low-energy collective oscillations of a dilute Bose gas at finite temperature in the collisionless regime. By using a time-dependent mean-field scheme we derive for the dynamics of the condensate and noncondensate components a set of coupled equations, which we solve perturbatively to second order in the interaction coupling constant. This approach is equivalent to the finite-temperature extension of the Beliaev approximation and includes corrections to the Gross-Pitaevskii theory due to both quantum and thermal fluctuations. For a homogeneous system we explicitly calculate the temperature dependence of the velocity of propagation and damping rate of zero sound. In the case of harmonically trapped systems in the thermodynamic limit, we calculate, as a function of temperature, the frequency shift of the low-energy compressional and surface modes.
In contrast with Chapter 2, in this chapter we include the dynamics of the thermal cloud. As noted in Chapter 1, we treat the noncondensate atoms using the simplest microscopic model approximation that captures the important physics. In particular, we consider only temperatures high enough ( T ≥ 0.4 T BEC ) that the noncondensate atoms can be described by a particle-like Hartree–Fock (HF) spectrum. To extend the analysis to very low temperatures is in principle straightforward (see Chapter 7). However, the details are more complicated since the excitations of the thermal cloud take on a collective aspect (i.e. a Bogoliubov-type quasiparticle spectrum must be used). In trapped Bose gases, the HF single-particle spectrum gives a good approximation down to much lower temperatures than in the case of uniform Bose gases, as first emphasized by Giorgini et al . (1997). In Section 3.1, we derive a generalized form of the Gross–Pitaevskii equation for the Bose order parameter Φ( r , t ) that is valid at finite temperatures. It involves terms that are coupled to the noncondensate component (the thermal cloud) and thus its solution in general requires one to know the equations of motion for the dynamics of the noncondensate atoms. In Section 3.2 we restrict ourselves to finite temperatures high enough that the noncondensate atoms can be described by a quantum kinetic equation for the single-particle distribution function f ( p, r , t ). A detailed microscopic derivation of this kind of kinetic equation is given in Chapters 6 and 7 using the Kadanoff–Baym Green's function formalism. A characteristic feature of a Bose-condensed gas is that the kinetic equation governing f ( p, r , t ) involves a collision integral C 12 [ f , Φ] describing collisions between condensate and noncondensate atoms.