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arXiv:cond-mat/9806375 · Physical Review A

Semiclassical corrections to a static Bose-Einstein condensate at zero temperature

Jens O. Andersen, Eric Braaten

Published Sep 1, 1999• 10 citations
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Abstract

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.

DOI
10.1103/physreva.60.2330

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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 mean-field approximation, the number density $\ensuremath{\rho}(\mathbf{r})$ for the ground state of a Bose-Einstein condensate trapped by an external potential V(r)V(\mathbf{r})V(r) satisfies a classical field equation called the Gross-Pitaevskii equation. We show that quantum corrections to $\ensuremath{\rho}$ are dominated by quantum fluctuations with wavelengths of order $1/\sqrt{\ensuremath{\rho}a}$, where aaa is the SSS-wave scattering length. By expanding the equations for the Hartree-Fock approximation to second order in the gradient expansion, we derive local correction terms to the Gross-Pitaevskii equation for $\ensuremath{\rho}$ that take into account the dominant effects of quantum fluctuations. We also show that the gradient expansions for the density and for the condensate break down at fourth order and at second order, respectively.
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 oscillation frequencies of collective excitations of a trapped Bose-Einstein condensate, when calculated in the mean-field approximation and in the Thomas-Fermi limit, are independent of the scattering length aaa. We calculate the leading corrections to the frequencies from quantum fluctuations around the mean field. The semiclassical correction is proportional to N1/5a6/5{N}^{1/5}{a}^{6/5}N1/5a6/5, where NNN is the number of atoms in the condensate. The correction is positive semidefinite and is zero for surface modes whose eigenfunctions have a vanishing Laplacian. The shift in the frequency of the lowest quadrupole mode for an axially symmetric trap is large enough that it should be measurable in future experiments.
The diffusion Monte Carlo method is applied to describe a trapped atomic Bose-Einstein condensate at zero temperature, fully quantum mechanically and nonperturbatively. For low densities, ${n(0)a}^{3}<~2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ [n(0),[n(0),[n(0), peak density; a, s-wave scattering length], our calculations confirm that the exact ground-state energy for a sum of two-body interactions depends to a good approximation on only the atomic-physics parameter a, and no other details of the two-body model potential. Corrections to the mean-field Gross-Pitaevskii energy range from being essentially negligible to about 20% for N=2−−50N=2--50N=2−−50 particles in the trap with positive s-wave scattering length a=100−−10000a.u.a=100--10000\mathrm{a}.\mathrm{u}.a=100−−10000a.u. Our numerical calculations confirm that inclusion of an additional effective potential term in the mean-field equation, which accounts for quantum fluctuations [see, e.g., E. Braaten and A. Nieto, Phys. Rev. B 56, 14 745 (1997)], leads to a greatly improved description of trapped Bose gases.
In the present chapter we consider the structure of the Bose–Einstein condensed state in the presence of interactions. Our discussion is based on the Gross–Pitaevskii equation, which describes the zero-temperature properties of the non-uniform Bose gas when the scattering length a is much less than the mean interparticle spacing. We shall first derive the Gross–Pitaevskii equation at zero temperature by treating the interaction between particles in a mean-field approximation (Sec. 6.1). Following that, in Sec. 6.2 we discuss the ground state of atomic clouds in a harmonic-oscillator potential. We compare results obtained by variational methods with those derived in the Thomas–Fermi approximation, in which the kinetic energy operator is neglected in the Gross–Pitaevskii equation. The Thomas–Fermi approximation fails near the surface of a cloud, and in Sec. 6.3 we calculate the surface structure using the Gross–Pitaevskii equation. In Sec. 6.4 we determine how the condensate wave function ‘heals’ when subjected to a localized disturbance. Finally, in Sec. 6.5 we show how the magnetic dipole–dipole interaction, which is long-ranged and anisotropic, may be included in the Gross–Pitaevskii equation and determine within the Thomas–Fermi approximation its effect on the density distribution. The Gross–Pitaevskii equation In the previous chapter we have shown that the effective interaction between two particles at low energies is a constant in the momentum representation, U 0 = 4π ħ 2 a/m .