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arXiv:cond-mat/9704173 · Physical Review A

Beyond the Thomas-Fermi approximation for a trapped condensed Bose-Einstein gas

Alexander L. Fetter, David L. Feder

Published Oct 1, 1998• 61 citations
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Abstract

Corrections to the zero-temperature Thomas-Fermi description of a dilute interacting condensed Bose-Einstein gas confined in an isotropic harmonic trap arise due to the presence of a boundary layer near the condensate surface. Within the Bogoliubov approximation, the various contributions to the ground-state condensate energy all have terms of order ${R}^{\ensuremath{-}4}\mathrm{ln}R$ and ${R}^{\ensuremath{-}4},$ where R is the number-dependent dimensionless condensate radius in units of the oscillator length $\sqrt{\ensuremath{\Elzxh}/m{\ensuremath{\omega}}_{0}}.$ The zero-order hydrodynamic density-fluctuation amplitudes are extended beyond the Thomas-Fermi radius through the boundary layer to provide a uniform description throughout all space. The first-order correction to the excitation frequencies is shown to be of order ${R}^{\ensuremath{-}4}.$

DOI
10.1103/physreva.58.3185

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Corrections to the zero-temperature Thomas-Fermi description of a dilute interacting condensed Bose-Einstein gas confined in an isotropic harmonic trap arise due to the presence of a boundary layer near the condensate surface. Within the Bogoliubov approximation, the various contributions to the ground-state condensate energy all have terms of order R\ensuremath−4lnR{R}^{\ensuremath{-}4}\mathrm{ln}RR\ensuremath−4lnR and R\ensuremath−4,{R}^{\ensuremath{-}4},R\ensuremath−4, where R is the number-dependent dimensionless condensate radius in units of the oscillator length \ensuremath\Elzxh/m\ensuremathω0.\sqrt{\ensuremath{\Elzxh}/m{\ensuremath{\omega}}_{0}}.\ensuremath\Elzxh/m\ensuremathω0​​. The zero-order hydrodynamic density-fluctuation amplitudes are extended beyond the Thomas-Fermi radius through the boundary layer to provide a uniform description throughout all space. The first-order correction to the excitation frequencies is shown to be of order R\ensuremath−4.{R}^{\ensuremath{-}4}.R\ensuremath−4.
The low-lying hydrodynamic normal modes of a dilute Bose-Einstein gas in an isotropic harmonic trap determine the corresponding Bogoliubov amplitudes. In the Thomas-Fermi limit, these modes have large low-temperature occupation numbers and they permit an explicit construction of the dynamic structure function $S(\mathbf{q},\ensuremath{\omega})$. The total noncondensate number ${N}^{\ensuremath{'}}(0)$ at zero temperature increases like R6{R}^{6}R6, where RRR is the condensate radius measured in units of the oscillator length. The lowest dipole modes are constructed explicitly in the Bogoliubov approximation.
The Bogoliubov approximation is used to study the ground state and low-lying excited states of a dilute gas of N atomic bosons held in an isotropic harmonic potential characterized by frequency \ensuremath{\omega} and oscillator length d0{\mathit{d}}_{0}d0​=\ensuremath{\surd}\ensuremath{\Elzxh}/m\ensuremath{\omega}. By assumption, the self-consistent condensate has a macroscopic occupation number N0{\mathit{N}}_{0}N0​\ensuremath{\gg}1, with N-N0{\mathit{N}}_{0}N0​\ensuremath{\ll}N0{\mathit{N}}_{0}N0​. A linearized hydrodynamic description yields operator forms of the particle-conservation law and Bernoulli's theorem, expressed in terms of the small density fluctuation operator \ensuremath{\rho}^\ensuremath{'} and velocity potential operator \ensuremath{\Phi}^\ensuremath{'}, along with the condensate density n0{\mathit{n}}_{0}n0​ and velocity v0{\mathbf{v}}_{0}v0​. For positive scattering length a and large stationary condensate (N0{\mathit{N}}_{0}N0​\ensuremath{\gg}d0{\mathit{d}}_{0}d0​/a and v0{\mathbf{v}}_{0}v0​=0), the spherical condensate has a well-defined radius R0{\mathit{R}}_{0}R0​\ensuremath{\gg}d0{\mathit{d}}_{0}d0​, and the low-lying excited states are irrotational compressional waves localized near the surface. Approximate variational energies E0l{\mathit{E}}_{0\mathit{l}}E0l​ of the lowest radial modes (n=0) for successive values of orbital angular momentum l form a rotational band given by E0l{\mathit{E}}_{0\mathit{l}}E0l​\ensuremath{\approxeq}E00{\mathit{E}}_{00}E00​+1/2${\mathrm{\ensuremath{\Elzxh}}}^{2}$l(l+1)/mR02{\mathrm{mR}}_{0}^{2}mR02​, with radial zero-point energy E00{\mathit{E}}_{00}E00​\ensuremath{\propto}\ensuremath{\Elzxh}\ensuremath{\omega}(R0{\mathit{R}}_{0}R0​/d0{\mathit{d}}_{0}d0​)2/3{)}^{2/3})2/3=(${\mathrm{\ensuremath{\Elzxh}}}^{2}$ m${\mathrm{\ensuremath{\omega}}}^{4}$R02{\mathit{R}}_{0}^{2}R02​)1/3{)}^{1/3})1/3. \textcopyright{} 1996 The American Physical Society.
In the mean-field approximation, a trapped Bose-Einstein condensate at zero temperature is described by the Gross-Pitaevskii equation for the condensate or, equivalently, by the hydrodynamic equations for the number density and current density. These equations receive corrections from quantum field fluctuations around the mean field. We calculate the semiclassical corrections to these equations for a general time-independent state of the condensate, extending previous work to include vortex states as well as the ground state. In the Thomas-Fermi limit, the semiclassical corrections can be taken into account by adding a local correction term to the Gross-Pitaevskii equation. At second order in the Thomas-Fermi expansion, the semiclassical corrections can be taken into account by adding local correction terms to the hydrodynamic equations.
In the Thomas–Fermi (TF) regime, S. Stringari [Phys. Rev. Lett., 77, 2360 (1996)] established the hydrodynamic normal modes of an interacting dilute Bose-condensed fluid confined in an isotropic harmonic trap. Here, we extend this treatment beyond the TF condensate using the work of B. Hu, G. Huang and Y. Ma [Phys. Rev. A, 69, 063608 (2004)] as a starting point. Both numerical results for low-lying eigenfrequencies, for various angular momentum quantum numbers ℓ and n = 0, and also samples of corresponding eigenfunctions, are presented. Finally, eigenfrequencies are also given for some collective excitations in an axially symmetrical trap.
This set of four lectures reviews various aspects of the theory of a dilute low-temperature trapped Bose gas, starting with (I) a review of the Bogoliubov description of the elementary excitations in a uniform system. The treatment is then generalized (II) to include the new physical effects of a confining harmonic trap potential on the condensate and its normal modes. An equivalent hydrodynamic description (III) focuses directly on the density and velocity fluctuations. The physics of vortices (IV) in an incompressible fluid is summarized and extended to the case of a trapped Bose condensate.
We propose a method for finding analytical solutions of the Bogoliubov--de Gennes (BdG) equations for the low-lying collective excitations in a harmonically trapped Bose-Einstein condensate beyond the Thomas-Fermi limit. We first use a simple variational wave function for ground state to eliminate the divergence at the boundary layer of the condensate, which appears in the Thomas-Fermi approximation. We then solve the BdG equations analytically and obtain explicit and divergence-free expressions for the eigenvalues and eigenfunctions of the excitations for the traps with spherical and cylindrical symmetries. The solutions of the zero-energy mode of the BdG equations are also presented.