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International Conference on Path Integrals

THERMODYNAMICAL PROPERTIES FOR WEAKLY INTERACTING DIPOLAR GASES WITHIN CANONICAL ENSEMBLES

Konstantin Glaum, Hagen M. Kleinert, Axel Pelster

Published Nov 1, 2008• 1 citation
Publisher

Abstract

We set up a recursion relation for the partition function of a fixed number of harmonically confined bosons. For an ideal Bose gas this leads to the wellknown results for the temperature dependence of the specific heat and the ground-state occupancy. Due to the diluteness of the gas, we include both the isotropic contact interaction and the anisotropic dipole-dipole interaction by an infinite-bubble sum of the lowest-order perturbative results. Due to the anisotropy of the dipole-dipole interaction, the thermodynamic quantities of interest crucially depend on the trap configuration. Keywords: Dipolar gas; Bose-Einstein condensation; Canonical ensemble. 1. Many-Body Path Integral Formalism The aim of this work is to give a path-integral description for a system of a fixed particle number N. To this end, we start with the imaginary-time evolution amplitude which is given by the following N-fold path integral (x1,..., xN; τb | x ′ 1,..., x ′ N; τa) B ≡ 1 ∑ N∏ [ ∫ xn(τb)=xP(n) N!

DOI
10.1142/9789812837271_0057

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We set up a recursion relation for the partition function of a fixed number of harmonically confined bosons. For an ideal Bose gas this leads to the wellknown results for the temperature dependence of the specific heat and the ground-state occupancy. Due to the diluteness of the gas, we include both the isotropic contact interaction and the anisotropic dipole-dipole interaction by an infinite-bubble sum of the lowest-order perturbative results. Due to the anisotropy of the dipole-dipole interaction, the thermodynamic quantities of interest crucially depend on the trap configuration. Keywords: Dipolar gas; Bose-Einstein condensation; Canonical ensemble. 1. Many-Body Path Integral Formalism The aim of this work is to give a path-integral description for a system of a fixed particle number N. To this end, we start with the imaginary-time evolution amplitude which is given by the following N-fold path integral (x1,..., xN; τb | x ′ 1,..., x ′ N; τa) B ≡ 1 ∑ N∏ [ ∫ xn(τb)=xP(n) N!
Based on the canonical description of a non-interacting Bose gas, we work out how both thermodynamic and statistical properties change perturbatively with respect to weak two-particle interactions. Up to first order, we obtain a recursion formula for the canonical partition function, which consists of the same Feynman diagrams as the grand-canonical description but with different Feynman rules. Resumming this recursion formula for the canonical partition function allows one to characterize the statistics of the ground-state occupancy by its respective cumulants. We demonstrate the applicability of this approach by analyzing a dilute Bose gas with contact interaction in a box trap. To this end, we used Dirichlet boundary conditions in view of their relevance for current experiments with atomic gases, where the box trap is implemented, for instance, with digital mirror devices.
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 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.
For ideal Bose atoms in an isotropic harmonic trap, we consider thermodynamic variables obtained from microcanonical, canonical, and grand canonical ensembles, each with certain variables specified and other variables fluctuating. For the first two of these ensembles, we derive recursion relations that link partition functions for different dimensions. We discuss fluctuations in general, and obtain expressions for variances of the atom number N, the chemical potential \ensuremath{\mu}, and the temperature T for small, Gaussian fluctuations in the grand canonical ensemble. Then from our recursion relations and others given elsewhere, we obtain probability distributions for N, for ground-state occupation number n0{n}_{0}n0​, for \ensuremath{\mu}, and for T. Below the critical temperature, the shape of the distributions for N, n0,{n}_{0},n0​, and \ensuremath{\mu} are definitely not Gaussian for the grand canonical ensemble. For given temperature and small N, we find that the chemical potential values pertaining to the three ensembles differ. We compare the specific-heat function CN(T){C}_{N}(T)CN​(T) for the three ensembles and propose to use the minimum of dCN/dT{\mathrm{dC}}_{N}/dTdCN​/dT to define the critical temperature to facilitate comparisons with similar configurations of interacting atoms.
We theoretically investigate various beyond mean-field effects on Bose gases at zero temperature featuring the anisotropic and long-range dipole-dipole interaction in addition to the isotropic and short-range contact interaction. Within the realm of the Bogoliubov--de Gennes theory, we consider static properties and low-lying excitations of both homogeneous and harmonically trapped dipolar bosonic gases. For the homogeneous system, the condensate depletion, the ground-state energy, the equation of state, and the speed of sound are discussed in detail. Making use of the local density approximation, we extend these results in order to study the properties of a dipolar Bose gas in a harmonic trap and in the regime of large particle numbers. After deriving the equations of motion for the general case of a triaxial trap, we analyze the influence of quantum fluctuations on important properties of the gas, such as the equilibrium configuration and the low-lying excitations in the case of a cylinder-symmetric trap. In addition to the monopole and quadrupole oscillation modes, we also discuss the radial quadrupole mode. We find that the latter acquires a quantum correction exclusively due to the dipole-dipole interaction. As a result, we identify the radial quadrupole as a reasonably accessible source for the signature of dipolar many-body effects and stress the enhancing character that dipolar interactions have for quantum fluctuations in the other oscillation modes.
The thermodynamic properties and finite-size scaling of weakly interacting Bose gases with a finite number of particles confined in a cubic box of volume L3{L}^{3}L3 with periodic boundary conditions are investigated theoretically based on an exact recursion relation for the canonical-ensemble partition function. In a similar manner to the case of an ideal Bose gas, the recursive scheme of the interacting finite system at any temperatures is established in the framework of the Bogoliubov theory. The temperature dependence of the condensate fraction and specific heat with different particles and interactions is obtained numerically. By defining two different transition temperatures of the finite systems, the effect of interactions on the transition temperatures yields the different and nonmonotonous behaviors. The finite-size scaling condensate fraction at the transition temperature for the systems is also discussed. It is shown that the calculated finite-size scaling is universally independent on the various system sizes and transition temperatures.