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Physical Review A

First-order coherence in an atomic condensate at finite temperature

A. Montina

Published Apr 10, 2003• 3 citations
Publisher

Abstract

The field fluctuations of a Bose-Einstein condensate can be studied in terms of the first-order correlation function, which obeys a Yukawa-like equation already introduced in a previous work and expressed here in a real and complex notation. When the equation is reformulated using the number-conserving Bogoliubov approximation, it yields results that present quantitative differences with respect to the standard symmetry-breaking approach. Finally, we introduce an approximation to evaluate the correlation function for sufficiently high temperatures.

DOI
10.1103/physreva.67.043604

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The field fluctuations of a Bose-Einstein condensate can be studied in terms of the first-order correlation function, which obeys a Yukawa-like equation already introduced in a previous work and expressed here in a real and complex notation. When the equation is reformulated using the number-conserving Bogoliubov approximation, it yields results that present quantitative differences with respect to the standard symmetry-breaking approach. Finally, we introduce an approximation to evaluate the correlation function for sufficiently high temperatures.
A Yukawa-like equation, introduced to evaluate the first order correlation function in a Bose-Einstein condensate, can be used to evaluate the atomic density fluctuations. It does not require a few mode approximation. We apply this approach to calculate the atomic fluctuations of a condensate of attractive atoms in a double-well potential. The atomic system undergoes a spatial symmetry breaking above a critical number Nc{N}_{c}Nc​ of atoms. Near Nc{N}_{c}Nc​ the atomic fluctuation $\ensuremath{\delta}N$ inside a single well becomes very large and experimentally observable by a phase-contrast imaging technique. $\ensuremath{\delta}N$ is related to the collective mode temperature, which may be different from the condensate temperature measured by means of the thermal tails. A one-dimensional array of wells is also considered.
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.
A Langevin equation for the complex amplitude of a single-mode Bose-Einstein condensate is derived. The equation is first formulated phenomenologically, defining three transport parameters. It is then also derived microscopically. Expressions for the transport parameters in the form of Green-Kubo formulas are thereby derived and evaluated for simple trap geometries, a cubic box with cyclic boundary conditions, and an isotropic parabolic trap. The number fluctuations in the condensate, their correlation time ${\ensuremath{\tau}}_{c},$ and the temperature-dependent collapse time of the order parameter as well as its phase-diffusion coefficient are calculated.
By extending Einstein's separation of wave and particle parts of the second order thermal fluctuation to encompass "generalized fluctuations" in any Bose field, P. E. Gordon has proposed alternative definitions for nth order coherence and nth order coherent states. The main point of this paper is to explore some of the physical insights to be gained by extending dualism to higher orders. Recent experiments have examined aspects of the coherence of Bose-Einstein condensates. It has been argued that the condensate state is coherent to (at least) second or third order, but the coherence properties of Bose-Einstein condensates remain somewhat controversial. Using probability distributions developed by M. O. Scully and V. V. Kocharovsky et. al., we apply Gordon's dualistic expression of the coherence conditions to investigate coherence properties in Bose-Einstein condensation. Via numerical calculations, we present a graphical survey of wave-like and particle-like fluctuations in condensed and uncondensed fractions. Near the critical point, we find a very marked peak in the ratio of nth order wave to nth order particle fluctuations in the condensate. Not surprisingly, n-point correlations between the positions of condensate atoms also peak near the critical temperature, and this apparently mirrors, to higher orders, the well-known relation between the integral of the 2-point correlation function over a certain volume and the rms fluctuation in the number of particles in that volume.
We use the quantum kinetic theory to calculate the steady state and fluctuations of a trapped Bose-Einstein condensate at a finite temperature. The system is divided in a condensate and a noncondensate part. A quantum-mechanical description based on the number-conserving Bogoliubov method is used for describing the condensate part. The noncondensed particles are treated as a classical gas in thermal equilibrium with temperature TTT and chemical potential $\ensuremath{\mu}$. We find a master equation for the reduced density operator of the Bose-Einstein condensate, calculate the steady state of the system, and investigate the effect of one-, two-, and three-particle losses on the condensate. Using linearized Ito equations, we find expressions for the intensity fluctuations and the amplitude fluctuations in the condensate. A Lorentzian line shape is found for the intensity correlation function that is characterized by a time constant ${\ensuremath{\gamma}}_{I}^{\ensuremath{-}1}$ derived in the paper. For the amplitude correlation function, we find ballistic behavior for time differences smaller than ${\ensuremath{\gamma}}_{I}^{\ensuremath{-}1}$, and diffusive behavior for larger time differences.
Bose–Einstein condensation in a finite one-dimensional atomic Bose gas trapped in an optical lattice is studied within Bogoliubov’s approximation and then beyond this approximation, within the static fluctuation approximation. A Bose–Hubbard model is used to construct the Hamiltonian of the system. The effect of the potential strength on the condensate fraction is explored at different temperatures; so is the effect of temperature on this fraction at different potential strengths. The role of the number of lattice points (the size effect) at constant number density (the filling factor) is examined; so is the effect of the number density on the condensate fraction. The results obtained are compared to other published results wherever possible.