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Physical Review A

Condensation of N bosons: Microscopic approach to fluctuations in an interacting Bose gas

Anatoly A. Svidzinsky, Marlan O. Scully

Published Dec 28, 2010• 9 citations
Publisher

Abstract

We present a microscopic derivation of the master equation for the condensate density matrix for an interacting Bogoliubov-Bose gas of N atoms. We choose the interaction Hamiltonian in a special way that substantially simplifies the master equation, yielding no coupling between diagonal and off-diagonal terms. The present formulation allows us to solve the problem analytically in a steady state and obtain the expression for the distribution function and equilibrium condensate fluctuations. For the first two central moments, our results are equivalent to those obtained in the canonical-ensemble quasiparticle formalism [V. V. Kocharovsky, Vl. V. Kocharovsky, and M. O. Scully, Phys. Rev. Lett. 84, 2306 (2000); Phys. Rev. A 61, 053606 (2000)], in the low-temperature range where these papers are valid, but also give an accurate description at high temperatures. The present analysis for an interacting Bose gas is as accurate as the master equation approach of Kocharovsky et al. [Phys. Rev. A 61, 023609 (2000)] is for an ideal gas.

DOI
10.1103/physreva.82.063630

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Physical Review E•May 31, 2011•14 citations
We present a microscopic derivation of the master equation for the condensate density matrix for an interacting Bogoliubov-Bose gas of N atoms. We choose the interaction Hamiltonian in a special way that substantially simplifies the master equation, yielding no coupling between diagonal and off-diagonal terms. The present formulation allows us to solve the problem analytically in a steady state and obtain the expression for the distribution function and equilibrium condensate fluctuations. For the first two central moments, our results are equivalent to those obtained in the canonical-ensemble quasiparticle formalism [V. V. Kocharovsky, Vl. V. Kocharovsky, and M. O. Scully, Phys. Rev. Lett. 84, 2306 (2000); Phys. Rev. A 61, 053606 (2000)], in the low-temperature range where these papers are valid, but also give an accurate description at high temperatures. The present analysis for an interacting Bose gas is as accurate as the master equation approach of Kocharovsky et al. [Phys. Rev. A 61, 023609 (2000)] is for an ideal gas.
We present a new method of calculating the distribution function and fluctuations for a Bose-Einstein condensate (BEC) of N interacting atoms. The present formulation combines our previous master equation and canonical ensemble quasiparticle techniques. It is applicable both for ideal and interacting Bogoliubov BEC and yields remarkable accuracy at all temperatures. For the interacting gas of 200 bosons in a box we plot the temperature dependence of the first four central moments of the condensate particle number and compare the results with the ideal gas. For the interacting mesoscopic BEC, as with the ideal gas, we find a smooth transition for the condensate particle number as we pass through the critical temperature.
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.
We study condensate statistics and thermodynamics of weakly interacting Bose gas with a fixed total number NNN of particles in a cubic box. We find the exact recursion relation for the canonical ensemble partition function. Using this relation, we calculate the distribution function of condensate particles for N=200N=200N=200. We also calculate the distribution function based on multinomial expansion of the characteristic function. Similar to the ideal gas, both approaches give exact statistical moments for all temperatures in the framework of Bogoliubov model. We compare them with the results of unconstraint canonical ensemble quasiparticle formalism and the hybrid master equation approach. The present recursion relation can be used for any external potential and boundary conditions. We investigate the temperature dependence of the first few statistical moments of condensate fluctuations as well as thermodynamic potentials and heat capacity analytically and numerically in the whole temperature range.
We re-examine the way in which Bogoliubov's theory of a dilute Bose gas at T=0T=0T=0 has been extended to describe the statistical mechanics of interacting bosons at finite temperature. We show explicitly that the field-theoretic calculation of the grand partition function in this formulation amounts to a canonical trace over the eigenfunctions of the Bogoliubov Hamiltonian at fixed total number of bosons NNN, and that the additional trace over NNN that is required in the grand-canonical formalism is never carried out. This implies that what usually passes as the grand-canonical treatment of the Bogoliubov Hamiltonian is not quite grand-canonical, and is in fact a canonical one. We also show that the discontinuity in the condensate density predicted by previous formulations of this theory as the temperature TTT goes past the critical transition temperature TcT_cTc​ is a direct consequence of an inappropriate generalization of the Bogoliubov prescription to finite temperatures, and that this discontinuity disappears when this prescription is either used as a zero temperature approximation or avoided altogether. Armed with the above findings, we reformulate the statistical mechanics of interacting bosons in the canonical ensemble and derive the thermodynamics of the system. We then show how the canonical treatment can be used to setup a truly grand-canonical description of the statistical mechanics of a weakly interacting Bose gas where the average number of bosons in the system varies with temperature, unlike existing formulations where the total number of bosons NNN is taken to be a constant that does not depend on TTT. Consequences on the physics of interacting bosons are briefly discussed.
Based on counting statistics and Bogoliubov theory, we present a recurrence relation for the microcanonical partition function for a weakly interacting Bose gas with a finite number of particles in a cubic box. According to this microcanonical partition function, we calculate numerically the distribution function, condensate fraction, and condensate fluctuations for a finite and isolated Bose-Einstein condensate. For ideal and weakly interacting Bose gases, we compare the condensate fluctuations with those in the canonical ensemble. The present approach yields an accurate account of the condensate fluctuations for temperatures close to the critical region. We emphasize that the interactions between excited atoms turn out to be important for moderate temperatures.