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Europhysics Letters (EPL)

Finite-temperature correlations in the trapped Bose-Einstein gas

N. M. Bogoliubov, R. Bullough, V. S. Kapitonov, Cyril Leonidovich Malyshev, JUSSI TIMONEN

Published Sep 1, 2001• 8 citations
Publisher

Abstract

New analytical results are reported for the two-point correlation function at nonzero temperatures in d = 3,2 and 1 dimensions. Only for d = 3 are correlations below T c both long range and coherent—at long range consistent with the existence of a coherent state of the condensate in the trap. Asymptotically these correlations are found to decay as (ρ( S ) ·∣ r 1 − r 2 ∣) −1 towards a nonzero value. The condensate density ρ( S ), with S = ( r 1 + r 2 )/2, is a slowly varying function and introduces a small amplitude variation in the decay of correlations when considered as a function of ∣ r 1 − r 2 ∣. The prediction of a ∣ r 1 − r 2 ∣ −1 -dependence at least has very recently been confirmed in a beautiful experiment (Bloch I., Hänsch T. W. and Esslinger T. REVIEWNature4032000166). The two-dimensional condensate is marginally stable in the sense that for 0 < T < T c correlations decay by a power law.

DOI
10.1209/epl/i2001-00345-2

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New analytical results are reported for the two-point correlation function at nonzero temperatures in d = 3,2 and 1 dimensions. Only for d = 3 are correlations below T c both long range and coherent—at long range consistent with the existence of a coherent state of the condensate in the trap. Asymptotically these correlations are found to decay as (ρ( S ) ·∣ r 1 − r 2 ∣) −1 towards a nonzero value. The condensate density ρ( S ), with S = ( r 1 + r 2 )/2, is a slowly varying function and introduces a small amplitude variation in the decay of correlations when considered as a function of ∣ r 1 − r 2 ∣. The prediction of a ∣ r 1 − r 2 ∣ −1 -dependence at least has very recently been confirmed in a beautiful experiment (Bloch I., Hänsch T. W. and Esslinger T. REVIEWNature4032000166). The two-dimensional condensate is marginally stable in the sense that for 0 < T < T c correlations decay by a power law.
Previous functional integral methods for translationally invariant systems have been extended to the case of a confining trap potential. Essentially all finite-temperature properties of the repulsive Bose gas in a paraboloidal trap can be determined this way. New analytical results reported here are for the finite-temperature two-point correlation functions below the critical temperatures in d=3,2 and 1 dimensions. Only for d=3 are correlations both long range and coherent - at long range consistent with the existence of a coherent state of the condensate in the trap. The two-dimensional condensate is marginally stable in the sense that correlations decay by a power law.
We calculate the dynamic single-particle and many-particle correlation functions at nonzero temperature in one-dimensional trapped repulsive Bose gases. These correlation functions decay towards zero for increasing distance, and this decay is governed by a scaling exponent that has a universal expression in terms of observed quantities. This expression is valid in the weak-interaction Gross-Pitaevskii as well as in the strong-interaction Girardeau-Tonks limit, but the observed quantities involved depend on the interaction strength. The confining trap introduces a weak center-of-mass dependence in the scaling exponent. Based on previous exact results for a related untrapped Bose system and the results reported here, we also suggest the proper asymptotic form for the density-density correlation function in the trapped Bose gas as considered here.
At any finite temperature, the correlation function $〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(\mathcal{r})\ensuremath{\psi}(0)〉$ of a two-dimensional Bose system is known to approach zero as the separation r\mathcal{r}r approaches infinity. This paper is devoted to supporting the point of view that superfluid phenomena may still persist in such a system. To describe the low-temperature behavior of a condensed two-dimensional Bose system, we use a coherent-state representation of the low-lying phonon excitations. The expectation of the particle field operator $〈\ensuremath{\psi}(\mathcal{r})〉$ in a coherent phonon state of the system is shown to be a well-defined function. This function has only small fluctuations in its magnitude and, therefore, it is possible to make the usual arguments for superfluid behavior based on the existence of the condensate wave function. The large phase fluctuations of the function, when the coherentphonon states are collected into the thermal ensemble, give rise to the spatial decay of the correlation function or reduced density matrix $〈{\ensuremath{\psi}}^{+}(\mathcal{r})\ensuremath{\psi}(0)〉$. It is concluded that two-dimensional Bose systems are superfluid at sufficiently low temperatures, even in the inifinite-volume limit. An analogous argument would indicate the possibility of superconductivity in a two-dimensional Fermi system.
We study first- and second-order coherence of trapped dilute Bose gases using appropriate correlation functions. Special attention is given to the discussion of second-order or density correlations. Except for a small region around the surface of a Bose-Einstein condensate the correlations can be accurately described as those of a locally homogeneous gas with a spatially varying chemical potential. The degrees of first- and second-order coherence are therefore functions of temperature, chemical potential, and position. The second-order correlation function is governed both by the tendency of bosonic atoms to cluster and by a strong repulsion at small distances due to atomic interactions. In present experiments both effects are of comparable magnitude. Below the critical temperature the range of the bosonic correlation is affected by the presence of collective quasiparticle excitations. The results of some recent experiments on second- and third-order coherence are discussed. It is shown that the relation between the measured quantities and the correlation functions is much weaker than previously assumed.
The fact that two-dimensional interacting trapped systems do not undergo Bose-Einstein Condensation (BEC) in the thermodynamic limit, though rigorously proved, is somewhat mysterious because all relevant limiting cases (zero temperature, small atom numbers, noninteracting particles) suggest otherwise. We study the possibility of condensation in finite trapped two-dimensional systems. We first consider the ideal gas, which incorporates the inhomogeneity and finite size of experimental systems and can be solved exactly. A semiclassical self-consistent approximation gives us a feel for the temperature scales; diagonalization of the one-body density matrix confirms that the condensation is into a single state. We squeeze a three-dimensional system and see how it crosses over into two dimensions. Mean-field theory, our main tool for the study of interacting systems, prescribes coupled equations for the condensate and the thermal cloud: the condensate receives a full quantum-mechanical treatment, while the noncondensate is described by different schemes of varying sophistication. We analyze the T = 0 case and its approach to the thermodynamic limit, finding a criterion for the dimensionality crossover and obtaining the coupling constant of the two-dimensional system that results from squeezing a three-dimensional trap. We next apply a semiclassical Hartree-Fock approximation to purely two-dimensional finite gases and find that they can be described either with or without a condensate; this continues to be true in the thermodynamic limit. The condensed solutions have a lower free energy at all temperatures but neglect the presence of phonons within the system and cease to exist when we allow for this possibility. The uncondensed solutions, in turn, are valid under a more rigorous scheme but have consistency problems of their own. Path-integral Monte Carlo simulations provide an essentially exact description of finite interacting gases and enable us to study highly anisotropic systems at finite temperature. We find that our two-dimensional Hartree-Fock solutions accurately mimic the surface density profile and predict the condensate fraction of these systems; the equivalent interaction parameter is smaller than that dictated by the T = 0 analysis. We conclude that, in two-dimensional isotropic finite trapped systems and in highly compressed three-dimensional gases, there is a phenomenon resembling a condensation into a single state.
We investigate the behavior of the anomalous correlation function in two-dimensional Bose gas. In the local case, we find that this quantity has a finite value in the limit of weak interactions at zero temperature. The effects of the anomalous density on some thermodynamic quantities are also considered. These effects can modify, in particular, the chemical potential, the ground-state energy, the depletion, and the superfluid fraction. Our predictions are in good agreement with recent analytical and numerical calculations. We show also that the anomalous density presents a significant importance compared to the noncondensed one at zero temperature. The single-particle anomalous correlation function is expressed in two-dimensional homogenous Bose gases by using the density-phase fluctuation. We then confirm that the anomalous average accompanies in analogous manner the true condensate at zero temperature, while it does not exist at finite temperature.