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arXiv:cond-mat/9704234 · Physical Review A

Excited states of a dilute Bose-Einstein condensate in a harmonic trap

Alexander L. Fetter, Daniel S. Rokhsar

Published Feb 1, 1998• 62 citations
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Abstract

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 ${R}^{6}$, where $R$ is the condensate radius measured in units of the oscillator length. The lowest dipole modes are constructed explicitly in the Bogoliubov approximation.

DOI
10.1103/physreva.57.1191

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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(q,\ensuremathω)S(\mathbf{q},\ensuremath{\omega})S(q,\ensuremathω). The total noncondensate number N\ensuremath′(0){N}^{\ensuremath{'}}(0)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.
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 ${R}^{\ensuremath{-}4}.$
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 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.
A hydrodynamic description is used to study the normal modes of a vortex in a zero-temperature Bose-Einstein condensate. In the Thomas-Fermi limit, the circulating superfluid velocity far from the vortex core provides a small perturbation that splits the originally degenerate normal modes of a vortex-free condensate. The relative frequency shifts are small in all cases considered (they vanish for the lowest dipole mode with ∣m∣=1|m|=1∣m∣=1), suggesting that the vortex is stable. The Bogoliubov equations serve to verify the existence of helical waves, similar to those of a vortex line in an unbounded weakly interacting Bose gas. In the large-condensate (small-core) limit, the condensate wave function reduces to that of a straight vortex in an unbounded condensate; the corresponding Bogoliubov equations have no bound-state solutions that are uniform along the symmetry axis and that decay exponentially far from the vortex core.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.
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.