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arXiv:1802.02555 · Physical Review A

Correlations of occupation numbers in the canonical ensemble and application to a Bose-Einstein condensate in a one-dimensional harmonic trap

Olivier Giraud, Aurélien Grabsch, Christophe Texier

Published May 21, 2018• 7 citations
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Abstract

We study statistical properties of $N$ noninteracting identical bosons or fermions in the canonical ensemble. We derive several general representations for the $p$-point correlation function of occupation numbers $\overline{{n}_{1}\ensuremath{\cdots}{n}_{p}}$. We demonstrate that it can be expressed as a ratio of two $p\ifmmode\times\else\texttimes\fi{}p$ determinants involving the (canonical) mean occupations $\overline{{n}_{1}}$, ..., $\overline{{n}_{p}}$, which can themselves be conveniently expressed in terms of the $k$-body partition functions (with $k\ensuremath{\le}N$). We draw some connection with the theory of symmetric functions and obtain an expression of the correlation function in terms of Schur functions. Our findings are illustrated by revisiting the problem of Bose-Einstein condensation in a one-dimensional harmonic trap, for which we get analytical results. We get the moments of the occupation numbers and the correlation between ground-state and excited-state occupancies. In the temperature regime dominated by quantum correlations, the distribution of the ground-state occupancy is shown to be a truncated Gumbel law. The Gumbel law, describing extreme-value statistics, is obtained when the temperature is much smaller than the Bose-Einstein temperature.

DOI
10.1103/physreva.97.053615

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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 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.
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We consider Bose-Einstein condensation of noninteracting homogeneous three-dimensional gas in canonical ensemble when both particle number NNN and total momentum P\mathbf{P}P of all particles are fixed. Using the saddle-point method, we derive the large-NNN analytical approximations for partition function, free energy, and statistical distributions of occupation numbers of different single-particle energy levels. At temperatures below the critical point of phase transition, we predict, in some ranges of P\mathbf{P}P, fragmentation of the condensate, when more than one single-particle level is macroscopically occupied. The occupation number distributions have approximately Gaussian shapes for the levels hosting the condensate, and exponential shapes for other, noncondensate levels. Our analysis demonstrates breaking of Galilean invariance of moving finite-temperature many-particle system in the presence of Bose-Einstein condensation and extends the theory of moving and rotating quantum systems to the finite-temperature large-NNN limit.
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The choice of statistical ensemble has a measurable impact on the statistical description of Bose-Einstein condensates beyond the well-known grand canonical anomaly. We develop a segmented-contour approach for the ideal Bose gas, which refines the saddle-point approximation by including non-Gaussian corrections in the canonical and microcanonical ensembles. The method applies to a broad class of noninteracting spectra, and provides efficient access to condensate fluctuations, higher moments, and correlation functions. For a three-dimensional harmonic trap with fixed anisotropy, we find that the ratio of the maxima of the microcanonical and canonical condensate-number variances approaches the theoretically predicted asymptotic value 1−3ζ(3)2/[4ζ(2)ζ(4)]≈0.3911-3ζ(3)^2/[4ζ(2)ζ(4)]\approx 0.3911−3ζ(3)2/[4ζ(2)ζ(4)]≈0.391 only extremely slowly with increasing particle number. We also show that ensemble effects are visible in the second-order correlation function g(2)(x)g^{(2)}(x)g(2)(x) near the condensation crossover. After ballistic release from a three-dimensional harmonic trap, the time evolution of g(2)(x)g^{(2)}(x)g(2)(x) preserves the differences between statistical ensembles.