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arXiv:quant-ph/9808002

Mean-density Bogoliubov description of inhomogeneous Bose-condensed gases

John A. Vaccaro

Published Aug 3, 1998• 0 citations
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Abstract

A mean-density description of spatially-inhomogeneous Bose-condensed gases based on Bogoliubov's method is introduced. The description assumes only a large mean atomic density and so remains valid when the mean field collapses due to phase diffusion. A spread in the number of particles in the condensate is shown to lead to an anomalous coupling between the condensate and excited modes. This coupling is due to the dependence of the condensate spatial wavefunction on particle number and it could, in principle, be used for reducing particle fluctuations in the condensate.

DOI
10.48550/arxiv.quant-ph/9808002

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A mean-density description of spatially-inhomogeneous Bose-condensed gases based on Bogoliubov's method is introduced. The description assumes only a large mean atomic density and so remains valid when the mean field collapses due to phase diffusion. A spread in the number of particles in the condensate is shown to lead to an anomalous coupling between the condensate and excited modes. This coupling is due to the dependence of the condensate spatial wavefunction on particle number and it could, in principle, be used for reducing particle fluctuations in the condensate.
The recent realization of Bose-Einstein condensation in atomic gases opens new possibilities for observation of macroscopic quantum phenomena. There are two important features of the system - weak interaction and significant spatial inhomogeneity. Because of this inhomogeneity a non-trivial "zeroth-order" theory exists, compared to the "first-order" Bogolubov theory. This theory is based on the mean-field Gross-Pitaevskii equation for the condensate ψ-function. The equation is classical in its essence but contains the ℏ constant explicitly. Phenomena such as collective modes, interference, tunneling, Josephson-like current and quantized vortex lines can be described using this equation. The study of deviations from the zeroth-order theory arising from zero-point and thermal fluctuations is also of great interest. Thermal fluctuations are described by elementary excitations which define the thermodynamic behaviour of the system and result in Landau-type damping of collective modes. Fluctuations of the phase of the condensate wave function restrict the monochromaticity of the Josephson current. Fluctuations of the numbers of quanta result in the quantum collapse-revival of the collective oscillations. This phenomenon is considered in some details. Collapse time for the JILA experimental conditions turns out to be of the order of seconds.
We develop a finite-temperature perturbation theory (beyond the mean field) for a Bose-condensed gas and calculate temperature-dependent damping rates and energy shifts for Bogolyubov excitations of any energy. The theory is generalized for the case of excitations in a spatially inhomogeneous (trapped) Bose-condensed gas, where we emphasize the principal importance of inhomogeneity of the condensate density profile and develop the method of calculating the self-energy functions. The use of the theory is demonstrated by calculating the damping rates and energy shifts of low-energy quasiclassical excitations, i.e., the quasiclassical excitations with energies much smaller than the mean-field interaction between particles. In this case the boundary region of the condensate plays a crucial role and the result for the damping rates and energy shifts is completely different from that in spatially homogeneous gases. We also analyze the frequency shifts and damping of sound waves in cylindrical Bose condensates and discuss the role of damping in the recent MIT experiment on the sound propagation.
We derive a description of the spatially inhomogeneous Bose-Einstein condensate, treating the system locally as a homogeneous Bose gas. This approach, similar to the Thomas-Fermi model for the inhomogeneous many-particle fermion system, is well suited to describe the atomic Bose-Einstein condensates that have recently been obtained experimentally through atomic trapping and cooling. In this paper, we confine our attention to the zero-temperature case, although the treatment can be generalized to finite temperatures, as we shall discuss elsewhere. Several features of this approach, which we shall call the Thomas-Fermi-Bogoliubov description, are very attractive. (i) It is simpler than the Hartree-Fock-Bogoliubov technique. We can obtain analytical results in the case of weakly interacting bosons for quantities such as the chemical potential, the local depletion, pairing, pressure, and density of states. (ii) The method provides an estimate for the error due to the inhomogeneity of the Bose-condensed system. This error is a local quantity so that the validity of the description for a given trap and a given number of trapped atoms can be tested as a function of position. We see, for example, that at the edge of the condensate the Thomas-Fermi-Bogoliubov theory always breaks down. (iii) The Thomas-Fermi-Bogoliubov description can be generalized to treat the statistical mechanics of the Bose gas at finite temperatures.
We study a system of trapped bosonic particles interacting by model harmonic forces. Our model allows for detailed examination of the notion of an order parameter (a condensate wave function). By decomposing a single particle density matrix into coherent eigenmodes we study an effect of interaction on the condensate. We show that sufficiently strong interactions cause that the condensate disappears even if the whole system is in its lowest energy state. In the second part of our paper we discuss the validity of the Bogoliubov approximation by comparing its predictions with results inferred from the exactl y soluble model. In particular we examine an energy spectrum, occupation, and fluctuations of the condensate. We conclude that Bogoliubov approach gives quite accurate description of the system in the limit of weak interactions.
Various widely-used mean-field type theories for a dilute Bose gas are critically examined in the light of the recent discovery of Bose-Einstein condensation of atomic gases in a confined geometry. By numerically solving the mean-field equations within the framework of the Bogoliubov approximation both stationary non-uniform case and the vortex case under rotation in a cylindrically symmetric vessel are investigated. We obtain spatial structures of condensate, non-condensate, anomalous correlation. The low lying excitation spectra, the local density of states and the circulating current density in a vortex corresponding to various levels of mean-field theories are predicted.
Abstract We define a formalism of a self-consistent description of the ground state of a weakly interacting Bose system, accounting for higher order terms in expansion of energy in the diluteness parameter. The approach is designed to be applied to a Bose–Bose mixture in a regime of weak collapse where quantum fluctuations lead to stabilization of the system and formation of quantum liquid droplets. The approach is based on the generalized Gross–Pitaevskii equation accounting for quantum depletion and renormalized anomalous density terms. The equation is self-consistently coupled to modified Bogoliubov equations. We derive well defined procedure to calculate the zero temperature renormalized anomalous density—the quantity needed to correctly describe the formation of quantum liquid droplet. We pay particular attention to the case of droplets harmonically confined in some directions. The method allows to determine the Lee–Huang–Yang-type contribution to the chemical potential of inhomogeneous droplets when the local density approximation fails.