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Quasiparticle Spectrum for the Dilute Bose-gas in an Inhomogeneous External Field

Alexander V. Zhukov

Published Jan 27, 2001• 0 citations
OpenAlex

Abstract

A quasiparticle description of various condensed media is a very popular tool in study of their transport and thermodynamic properties. I present here a microscopic theory for the description of diffusion processes in two-component gas of quasiparticles with arbitrary dispersion law and statistics. Particularly, I analyze the role of interaction within each subsystem (i.e. between identical quasiparticles) in relaxation of the whole system. The approach for solution of such kinetic problem allows to study the most important limiting cases and to clarify their physical sense. Classical results for diffusion coefficient of light particles in a massive gas (Lorentz model) and of massive particles in a light gas (Rayleigh model) are obtained directly from the general solution without using artificial approaches, as it was done earlier. This particularly provide a possibility to generalize these popular models on quasiparticle systems.

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A quasiparticle description of various condensed media is a very popular tool in study of their transport and thermodynamic properties. I present here a microscopic theory for the description of diffusion processes in two-component gas of quasiparticles with arbitrary dispersion law and statistics. Particularly, I analyze the role of interaction within each subsystem (i.e. between identical quasiparticles) in relaxation of the whole system. The approach for solution of such kinetic problem allows to study the most important limiting cases and to clarify their physical sense. Classical results for diffusion coefficient of light particles in a massive gas (Lorentz model) and of massive particles in a light gas (Rayleigh model) are obtained directly from the general solution without using artificial approaches, as it was done earlier. This particularly provide a possibility to generalize these popular models on quasiparticle systems.
The paper deals with the properties of Bose-gases in inhomogeneous external fields at very low temperatures, when a Bose–Einstein condensation occurs. This particularly refers to trapped alkali gases, which are of great experimental and theoretical interest. The system is considered here in the frame of the developed approach close to the Bogoliubov one with necessary modifications. In particular the generalization of Bogoliubov transformations made it possible to find the spectrum of quasiparticles (low-energy thermal excitations) with account of purely quantum parts of ψ-operators in first order.
The central problem of this chapter is temporal coherence of a three-dimensional spatially homogeneous Bose-condensed gas, initially prepared at finite temperature and then evolving as an isolated interacting system. A first theoretical tool is a number-conserving Bogoliubov approach that allows to describe the system as a weakly interacting gas of quasi-particles. This approach naturally introduces the phase operator of the condensate: a central actor since loss of temporal coherence is governed by the spreading of the condensate phase-change. A second tool is the set of kinetic equations describing the Beliaev-Landau processes for the quasi-particles. We find that in general the variance of the condensate phase-change at long times ttt is the sum of a ballistic term ∝t2\propto t^2∝t2 and a diffusive term ∝t\propto t∝t with temperature and interaction dependent coefficients. In the thermodynamic limit, the diffusion coefficient scales as the inverse of the system volume. The coefficient of t2t^2t2 scales as the inverse volume squared times the variance of the energy of the system in the initial state and can also be obtained by a quantum ergodic theory (the so-called eigenstate thermalisation hypothesis).
We derive a non-Markovian generalization to the quantum kinetic theory described by Walser et al. [Phys. Rev. A 59, 3878 (1999)] in the absence of a condensed fraction for temperatures above the Bose-Einstein condensation temperature i.e., T>Tc.T>{T}_{c}.T>Tc​. Within this framework, quasiparticle damping arises naturally due the finite duration of a binary collision and it leads to a systematic Markov approximation from the non-Markovian Born theory. Such a self-consistent theory conserves the total energy to second order in the interaction strength. By introducing an improved damping function, we demonstrate global energy conservation at the order of the perturbation theory. Finally, we apply this kinetic theory to a simple model of an inhomogeneous Bose gas that is confined in a spherical box. By studying numerically the real-time quantum evolution towards equilibrium, we obtain damping rates and frequencies of the collective modes and illustrate the emergence of differing time scales for correlation and relaxation.
We consider equilibrium canonical-ensemble fluctuations of the Bose-Einstein condensate (BEC) in a trap using the particle-number conserving operator formalism of Girardeau and Arnowitt [Phys. Rev. 113, 755 (1959)] and the concept of the canonical-ensemble quasiparticles. We find explicit expressions for the characteristic function and all cumulants of the ground-state occupation statistics both for the dilute weakly interacting and ideal Bose gases. We present analytical formulas for the moments of the ground-state occupation fluctuations in the weakly interacting and ideal Bose gases in a box (``homogeneous gas'') as well as in the ideal Bose gas in an arbitrary power-law trap, in particular, in an arbitrary harmonic trap. We calculate the effect of Bogoliubov coupling between quasiparticles on suppression of the ground-state occupation fluctuations at moderate temperatures and their enhancement at very low temperatures. We clarify a crossover between ideal-gas and weakly interacting gas statistics, which is governed by a pair-correlation, squeezing mechanism. The important conclusion is that in most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit.
A self-consistent field (SCF) model is constructed for spatially inhomogeneous Bose systems with broken symmetry. A system of self-consistent equations is obtained for the wave functions of the quasiparticles and the wave function of the condensate particles and another system of equations for the normal and anomalous single-particle density matrices. The many-particle wave function is found. The thermodynamics of multiparticle Bose systems is constructed on the basis of a microscopic treatment in the SCF approximation. It is stressed that Bose condensation in the ideal gas model is substantially different from Bose condensation in a system of interacting Bose particles because a pair condensate is necessarily present as well as the single-particle condensate, even for an arbitrarily weak interaction. The role of single-particle and collective excitations in Bose systems is discussed.