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arXiv:1207.1278 · Physical Review A

Two-body anticorrelation in a harmonically trapped ideal Bose gas

Tod M. Wright, A. Perrin, Andrew Bray, Jörg Schmiedmayer, Karen V. Kheruntsyan

Published Aug 16, 2012• 8 citations
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Abstract

We predict the existence of a dip below unity in the second-order coherence function of a partially condensed ideal Bose gas in harmonic confinement, signaling the anticorrelation of density fluctuations in the sample. The dip in the second-order coherence function is revealed in a canonical-ensemble calculation, corresponding to a system with fixed total number of particles. In a grand-canonical ensemble description, this dip is obscured by the occupation-number fluctuation catastrophe of the ideal Bose gas. The anticorrelation is most pronounced in highly anisotropic trap geometries containing small particle numbers. We explain the fundamental physical mechanism which underlies this phenomenon, and its relevance to experiments on interacting Bose gases.

DOI
10.1103/physreva.86.023618

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We predict the existence of a dip below unity in the second-order coherence function of a partially condensed ideal Bose gas in harmonic confinement, signaling the anticorrelation of density fluctuations in the sample. The dip in the second-order coherence function is revealed in a canonical-ensemble calculation, corresponding to a system with fixed total number of particles. In a grand-canonical ensemble description, this dip is obscured by the occupation-number fluctuation catastrophe of the ideal Bose gas. The anticorrelation is most pronounced in highly anisotropic trap geometries containing small particle numbers. We explain the fundamental physical mechanism which underlies this phenomenon, and its relevance to experiments on interacting Bose gases.
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.
We analyze the general relation between canonical and grand canonical ensembles in the thermodynamic limit. We begin our discussion by deriving, with an alternative approach, some standard results first obtained by Kac and coworkers in the late 1970s. Then, motivated by the Bose-Einstein condensation (BEC) of trapped gases with a fixed number of atoms, which is well described by the canonical ensemble and by the recent groundbreaking experimental realization of BEC with photons in a dye-filled optical microcavity under genuine grand canonical conditions, we apply our formalism to a system of non-interacting Bose particles confined in a two-dimensional harmonic trap. We discuss in detail the mathematical origin of the inequivalence of ensembles observed in the condensed phase, giving place to the so-called grand canonical catastrophe of density fluctuations. We also provide explicit analytical expressions for the internal energy and specific heat and compare them with available experimental data. For these quantities, we show the equivalence of ensembles in the thermodynamic limit.
For a non-self-interacting Bose gas with a fixed, large number of particles confined to a trap, as the ground-state occupation becomes macroscopic, the condensate number fluctuations remain microscopic. However, this is the only significant aspect in which the grand canonical description differs from canonical or microcanonical in the thermodynamic limit. General arguments and estimates including some of the vanishingly small quantities are compared to explicit, fixed-number calculations for 102{10}^{2}102--106{10}^{6}106 particles. \textcopyright{} 1996 The American Physical Society.
Abstract The grand canonical catastrophe—characterized by order-unity relative fluctuations of the ground-state occupation in the grand canonical ensemble (GCE) at low temperatures—is a hallmark of ensemble inequivalence in ideal Bose gases and has been observed in photon Bose–Einstein condensates. In this work, we investigate an alternative class of observables defined via conditional ensemble averages , namely ⟨ n 0 / N ⟩ and ⟨ δ n 0 2 / N 2 ⟩ , where the average excludes the empty-system sector ( N = 0 ). Using the exactly solvable model of an ideal Bose gas in a one-dimensional harmonic trap, we show that these conditional averages reproduce microcanonical predictions even for modest particle numbers ( N ∼ 10 ) and exhibit vanishing relative fluctuations as temperature approaches zero. We stress that this does not contradict the standard GCE result ⟨ δ n 0 2 ⟩ → ⟨ N ⟩ 2 , which remains thermodynamically valid for open systems. Instead, our framework provides a natural description for experimental scenarios involving post-selection or implicit conditioning on non-empty realizations—common in few-body quantum gas experiments. Thus, the absence of large fluctuations in these conditional averages reflects a change in the observable, not a failure of ensemble theory.
We investigate numerically the momentum correlations in a two-dimensional, harmonically trapped interacting Bose system at T=0T=0T=0 temperature, by using a particle number preserving Bogoliubov approximation. Interaction-induced quantum fluctuations of the quasicondensate lead to a large anticorrelation dip between particles of wave numbers k\mathbf{k}k and $\ensuremath{-}\mathbf{k}$ for $|\mathbf{k}|\ensuremath{\sim}1/{R}_{c}$, with Rc{R}_{c}Rc​ the typical size of the condensate. The anticorrelation dip found is a clear fingerprint of coherent quantum fluctuations of the condensate. In contrast, for larger wave numbers, $|\mathbf{k}|\ensuremath{\gg}1/{R}_{c}$, a weak positive correlation is found between particles of wave numbers k\mathbf{k}k and $\ensuremath{-}\mathbf{k}$, in accordance with the Bogoliubov result for homogeneous interacting systems.