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Journal of Physics B Atomic Molecular and Optical Physics

Stochastic field equation for the canonical ensemble of an ideal Bose gas

Sigmund Heller, Walter T. Strunz

Published Mar 30, 2009• 8 citations
Publisher

Abstract

We present a novel norm preserving stochastic evolution equation for a Bose field at finite temperature. Ensemble averages are quantum expectation values in the canonical ensemble. This numerically very stable equation suppresses high-energy fluctuations exponentially, which prevent cutoff problems from occurring. We present 3D simulations for an ideal gas in various trapping potentials and focus 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.

DOI
10.1088/0953-4075/42/8/081001

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We present a novel norm preserving stochastic evolution equation for a Bose field at finite temperature. Ensemble averages are quantum expectation values in the canonical ensemble. This numerically very stable equation suppresses high-energy fluctuations exponentially, which prevent cutoff problems from occurring. We present 3D simulations for an ideal gas in various trapping potentials and focus 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.
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.
We present a novel quantum stochastic evolution equation for a matter field describing the canonical state of a weakly interacting ultracold Bose gas. In the ideal gas limit our approach is exact. This numerically very stable equation suppresses high-energy fluctuations exponentially, which enables us to describe condensed and thermal atoms within the same formalism. We present applications to ground-state occupation and fluctuations, density profile of ground-state and thermal cloud, and ground-state number statistics. Our main aim are spatial coherence properties which we investigate through the determination of interference contrast and spatial density correlations. Parameters are taken from actual experiments.
Stochastic field equations represent a powerful tool to describe the thermal state of a trapped Bose gas. Often, such approaches are confronted with the old problem of an ultraviolet catastrophe, which demands a cutoff at high energies. In [arXiv:0809.1002, Phys. B 42, 081001 (2009)] we introduce a quantum stochastic field equation, avoiding the cutoff problem through a fully quantum approach based on the Glauber-Sudarshan P-function. For a close link to actual experimental setups the theory is formulated for a fixed particle number and thus based on the canonical ensemble. In this work the derivation and the non-trivial numerical implementation of the equation is explained in detail. We present applications for finite Bose gases trapped in a variety of potentials and show results for ground state occupation numbers and their equilibrium fluctuations. Moreover, we investigate spatial coherence properties by studying correlation functions of various orders.
We present a new exact method to numerically compute the thermodynamical properties of an interacting Bose gas in the canonical ensemble. As in our previous paper (Carusotto I, Castin Y and Dalibard J 2001 Phys. Rev. A 63 023606), we write the density operator ρ as an average of Hartree dyadics | N :ϕ 1 ⟩⟨ N :ϕ 2 | and we find stochastic evolution equations for the wavefunctions ϕ 1,2 such that the exact imaginary-time evolution of ρ is recovered after averaging over noise. In this way, the thermal equilibrium density operator can be obtained for any temperature T . The method is then applied to study the thermodynamical properties of a homogeneous one-dimensional N -boson system: although Bose-Einstein condensation cannot occur in the thermodynamical limit, a macroscopic occupation of the lowest mode of a finite system is observed at sufficiently low temperatures. If k B T >>µ, the main effect of interactions is to suppress density fluctuations and to reduce their correlation length. Different effects such as a spatial antibunching of the atoms are predicted for the opposite k B T ⩽µ regime. Our exact stochastic calculations have been compared with existing approximate theories.
We analyse occupation number fluctuations of an ideal Bose gas in a trap which is isolated from the environment with respect to particle exchange (canonical ensemble). We show that in contrast to the predictions of the grandcanonical ensemble, the counting statistics of particles in the trap ground state changes from monotonously decreasing above the condensation temperature to single-peaked below that temperature. For the exactly solvable case of a harmonic oscillator trapping potential in one spatial dimension we extract a Landau–Ginzburg functional which–despite the non-interacting nature of the system–displays the characteristic behaviour of a weakly interacting Bose gas. We also compare our findings with the usual treatment which is based on the grand-canonical ensemble. We show that for an ideal Bose gas neither the grand-canonical and canonical ensemble thermodynamically equivalent, nor the grand-canonical ensemble can be viewed as a small system in diffusive contact with a particle reservoir.
The mean ground-state occupation number and condensate fluctuations of interacting and noninteracting Bose gases confined in a harmonic trap are considered by using a canonical ensemble approach. To obtain the mean ground-state occupation number and the condensate fluctuations, an analytical description for the probability distribution function of the condensate is provided directly starting from the analysis of the partition function of the system. For the ideal Bose gas, the probability distribution function is found to be a Gaussian one for the case of the harmonic trap. For the interacting Bose gas, using a unified approach the condensate fluctuations are calculated based on the lowest-order perturbation method and on Bogoliubov theory. It is found that the condensate fluctuations based on the lowest-order perturbation theory follow the law $〈{\ensuremath{\delta}}^{2}{N}_{0}〉\ensuremath{\sim}N,$ while the fluctuations based on Bogoliubov theory behave as N4/3.{N}^{4/3}.N4/3.