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arXiv:1104.0690 · Physical Review A

Anomalous density for Bose gases at finite temperature

Abdelâali Boudjemâa, Mohamed Benarous

Published Oct 21, 2011• 30 citations
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Abstract

We analyze the behavior of the anomalous density as function of the radial distance at different temperatures in a variational framework. We show that the temperature dependence of the anomalous density agrees with the Hartree-Fock-Bogoliubov (HFB) calculations. Comparisons between the normal and anomalous fractions at low temperature show that the latter remains higher and, consequently, the neglect of the anomalous density may destabilize the condensate. These results are compatible with those of Yukalov. Surprisingly, the study of the anomalous density in terms of the interaction parameter shows that the dip in the central density is destroyed for sufficiently weak interactions. We explain this effect.

DOI
10.1103/physreva.84.043633

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We analyze the behavior of the anomalous density as function of the radial distance at different temperatures in a variational framework. We show that the temperature dependence of the anomalous density agrees with the Hartree-Fock-Bogoliubov (HFB) calculations. Comparisons between the normal and anomalous fractions at low temperature show that the latter remains higher and, consequently, the neglect of the anomalous density may destabilize the condensate. These results are compatible with those of Yukalov. Surprisingly, the study of the anomalous density in terms of the interaction parameter shows that the dip in the central density is destroyed for sufficiently weak interactions. We explain this effect.
Abstract Using the time-dependent Hartree–Fock–Bogoliubov approach, where the condensate is coupled with the thermal cloud and the anomalous density, we study the equilibrium and the dynamical properties of three-dimensional quantum-degenerate Bose gas at finite temperature. Effects of the anomalous correlations on the condensed fraction and the critical temperature are discussed. In uniform Bose gas, useful expressions for the Bogoliubov excitations spectrum, the first and second sound, the condensate depletion and the superfluid fraction are derived. Our results are tested by comparing the findings computed by quantum Monte Carlo simulations. We present also a systematic investigation of the collective modes of a Bose condensate confined in an external trap. Our predictions are in qualitative agreement with previous experimental and theoretical results. We show in particular that our theory is capable of explaining the so-called anomalous behavior of the m = 0 mode.
We critically examine the Hartree-Fock-Bogoliubov (HFB) solution of the equations of motion for condensate fluctuations in a weakly coupled plasma of charged bosons at finite temperature. Analytic expressions are derived for the first two infrared-divergent terms in both the momentum distribution of the noncondensate and the anomalous Bose correlation function at low momenta. Incorporation into the theory of the appropriate form of the Hugenholtz-Pines relation for the chemical potential is needed to cancel an unphysical divergence. Exact cancellation of infrared-divergent terms is demonstrated in the HFB shift of the single-particle excitation energy away from the Bogoliubov value at long wavelengths, with the residual terms raising it towards the plasma frequency at low temperature. Numerical illustrations are presented for a number of properties of the boson plasma as functions of temperature and density in the weak-coupling regime: these are the chemical potential, the condensate fraction, the normal and anomalous momentum distribution functions and the corresponding one-body density matrices, and the dispersion relation of single-particle excitations.
We derive and discuss the equations of motion for the condensate and its fluctuations for a dilute, weakly interacting Bose gas in an external potential within the self-consistent Hartree-Fock-Bogoliubov (HFB) approximation. Account is taken of the depletion of the condensate and the anomalous Bose correlations, which are important at finite temperatures. We give a critical analysis of the self-consistent HFB approximation in terms of the Hohenberg-Martin classification of approximations (conserving vs gapless) and point out that the Popov approximation to the full HFB gives a gapless single-particle spectrum at all temperatures. The Beliaev second-order approximation is discussed as the spectrum generated by functional differentiation of the HFB single-particle Green's function. We emphasize that the problem of determining the excitation spectrum of a Bose-condensed gas (homogeneous or inhomogeneous) is difficult because of the need to satisfy several different constraints.
We study a Bose-Einstein condensate of a dilute gas with dipolar interactions, at finite temperature, using the Hartree-Fock-Bogoliubov theory within the Popov approximation. An additional approximation involving the dipolar exchange interaction is made to facilitate the computation. We calculate the temperature dependence of the condensate fraction of a condensate confined in a cylindrically symmetric harmonic trap. We show that the biconcave-shaped condensates found in [Ronen et al. Phys. Rev. Lett. 98, 30406 (2007)] in certain pancake traps at zero temperature are also stable at finite temperature. Surprisingly, the dip in the central density of these structured condensates is actually enhanced at low finite temperatures. We explain this effect.
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The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local-density approximation. Consistency of the theory for temperatures through the Bose condensation point Tc{T}_{c}Tc​ requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length asc,{a}_{\mathrm{sc}},asc​, increases the number of condensate atoms at each temperature, and raises the value of Tc{T}_{c}Tc​ relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near Tc.{T}_{c}.Tc​. Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.