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arXiv:1112.3557 · Physical Review A

Quantum stochastic description of collisions in a canonical Bose gas

Patrick Navez, Achilleas Lazarides

Published Dec 12, 2012• 0 citations
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Abstract

We derive a stochastic process that describes the kinetics of a one-dimensional Bose gas in a regime where three-body collisions are important. In this situation the system becomes nonintegrable, offering the possibility of investigating dissipative phenomena more simply than in higher-dimensional gases. Unlike the quantum Boltzmann equation describing the average momentum distribution, the stochastic approach allows a description of higher-order correlation functions in a canonical ensemble. As will be shown, this ensemble differs drastically from the grand-canonical one. We illustrate the use of this method by determining the time evolution of the momentum-mode particle-number distribution and the static structure factor during the evaporative cooling process.

DOI
10.1103/physreva.86.063612

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We derive a stochastic process that describes the kinetics of a one-dimensional Bose gas in a regime where three-body collisions are important. In this situation the system becomes nonintegrable, offering the possibility of investigating dissipative phenomena more simply than in higher-dimensional gases. Unlike the quantum Boltzmann equation describing the average momentum distribution, the stochastic approach allows a description of higher-order correlation functions in a canonical ensemble. As will be shown, this ensemble differs drastically from the grand-canonical one. We illustrate the use of this method by determining the time evolution of the momentum-mode particle-number distribution and the static structure factor during the evaporative cooling process.
A technique to simulate the grand canonical ensembles of interacting Bose gases is presented. Results are generated for many temperatures by averaging over energy-weighted stochastic paths, each corresponding to a solution of coupled Gross-Pitaevskii equations with phase noise. The stochastic gauge method used relies on an off-diagonal coherent-state expansion, thus taking into account all quantum correlations. As an example, the second-order spatial correlation function and momentum distribution for an interacting 1D Bose gas are calculated.
Many-mode interacting Bose gases (1D,2D,3D) are simulated from first principles. The model uses a second-quantized Hamiltonian with two-particle interactions (possibly ranged), external potential, and interactions with an environment, with no further approximations. Simulations are of a set of stochastic equations that in the limit of many realizations correspond exactly to the full quantum evolution. These are obtained using the stochastic gauge method (derived here), an extension of the positive P phase-space representation. The systems investigated are: 1) Dynamics of spatial correlations in uniform 1D and 2D Bose gases after the rapid appearance of significant two-body collisions (e.g. after entering a Feshbach resonance). 2) Dynamics of stimulated Bose enhancement of scattered atom modes during the collision of two Bose-Einstein condensates with a mean of 150 000 atoms. 3) Dynamics of trapped bosons, where the size of the trap is of the same order as the range of the interparticle potential. 4) Grand canonical thermodynamics of uniform 1D Bose gases for a variety of temperatures and collision strengths. Observables calculated include 1st-3rd order spatial correlation functions (including finite separation) and momentum distributions. The stochastic gauge method is derived, and its application to interacting Bose gases investigated in detail. It is found to improve simulation effectiveness under many conditions, and to be capable of overcoming instability and boundary term problems. Additionally, conditions under which very generalized phase-space represntations can be used to obtain tractable many-body simulations are analysed.
We present a method aimed at a stochastic derivation of the equilibrium distribution of a classical/quantum ideal gas in the framework of the canonical ensemble. The time evolution of these ideal systems is modelled as a series of transitions from one system microstate to another one and thermal equilibrium is reached via a random walk in the single-particle state space. We look at this dynamic process as a Markov chain satisfying the condition of detailed balance and propose a variant of the Monte Carlo Metropolis algorithm able to take into account indistinguishability of identical quantum particles. Simulations performed on different two-dimensional (2D) systems are revealed to be capable of reproducing the correct trends of the distribution functions and other thermodynamic properties. The simulations allow us to show that, away from the thermodynamic limit, a pseudo-Bose–Einstein condensation occurs for a 2D ideal gas of bosons.
We present a novel norm preserving stochastic evolution equation for a Bose field. Ensemble averages are quantum expectation values in the canonical ensemble. This numerically very stable equation suppresses high-energy fluctuations exponentially, preventing cutoff problems to occur. We present 3D simulations for an ideal gas in various trapping potentials focussing on ground state occupation numbers and spatial correlation functions for a wide range of temperatures above and below the critical temperature. Although rigorously valid for non-interacting Bosons only, we argue that weakly interacting Bose gases may also be amenable to this approach, in the usual mean-field approximation.
Using the method of nonequilibrium statistical operator by Zubarev, an approach is proposed for the description of kinetics which takes into account the nonlinear hydrodynamic fluctuations for a quantum Bose system.Non-equilibrium statistical operator is presented which consistently describes both the kinetic and nonlinear hydrodynamic processes.Both a kinetic equation for the nonequilibrium one-particle distribution function and a generalized Fokker-Planck equation for nonequilibrium distribution function of hydrodynamic variables (densities of momentum, energy and particle number) are obtained.A structure function of hydrodynamic fluctuations in cumulant representation is calculated, which makes it possible to analyse the generalized Fokker-Planck equation in Gaussian and higher approximations of the dynamic correlations of hydrodynamic variables which is important in describing the quantum turbulent processes.
Condensation of the ideal Bose gas in a closed volume having the shape of a rectangular parallel-epiped of length L with a square base of side length l ( L ≫ l ) is theoretically studied within the framework of the Bose-Einstein statistics (grand canonical ensemble) and within the statistics of a canonical ensemble of bosons. Under the condition N ( l/L ) 4 ≪ l, where N is the total number of gas particles, dependence of the average number of particles in the condensate on the temperature T in both statistics is expressed as a function of the ratio t = T/T 1 , where T 1 is a certain characteristic temperature depending only on the longitudinal size L . Therefore, the condensation process exhibits a one-dimensional (1D) character. In the 1D regime, the average numbers of particles in condensates of the grand canonical and canonical ensembles coincide only in the limiting cases of t → 0 and t → ∞. The distribution function of the number of particles in the condensate of a canonical ensemble of bosons at t ≤1 has a resonance shape and qualitatively differs from the Bose-Einstein distribution. The former distribution begins to change in the region of t ∼ 1 and acquires the shape of the Bose-Einstein distribution for t ≫ 1. This transformation proceeds gradually that is, the 1D condensation process exhibits no features characteristic of the phase transition in a 3D system. For N ( l/L ) 4 ≫ 1, the process acquires a 3D character with respect to the average number of particles in the condensate, but the 1D character of the distribution function of the number of particles in the condensate of a canonical ensemble of bosons is retained at all N values.