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arXiv:cond-mat/0012101 · Physical Review A

Ground state of a trapped Bose-Einstein condensate in two dimensions: Beyond the mean-field approximation

Jens O. Andersen, Hårek Haugerud

Published Feb 20, 2002• 16 citations
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Abstract

We consider the ground state of a trapped Bose-Einstein condensate in two dimensions. In the mean-field approximation the ground-state density profile satisfies the Gross-Pitaevskii equation. We compute the leading quantum corrections to the density profile to second order in an expansion around the Thomas-Fermi limit. By summing the ladder diagrams, we obtain a modified equation that includes gradient corrections.

DOI
10.1103/physreva.65.033615

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We consider the ground state of a trapped Bose-Einstein condensate in two dimensions. In the mean-field approximation the ground-state density profile satisfies the Gross-Pitaevskii equation. We compute the leading quantum corrections to the density profile to second order in an expansion around the Thomas-Fermi limit. By summing the ladder diagrams, we obtain a modified equation that includes gradient corrections.
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.
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.
By incorporating the zero-point energy contribution we derive simple and accurate extensions of the usual Thomas-Fermi (TF) expressions for the ground-state properties of trapped Bose-Einstein condensates that remain valid for an arbitrary number of atoms in the mean-field regime. Specifically, we obtain approximate analytical expressions for the ground-state properties of spherical, cigar-shaped, and disk-shaped condensates that reduce to the correct analytical formulas in both the TF and the perturbative regimes, and remain valid and accurate in between these two limiting cases. Mean-field quasi-one-dimensional (quasi-1D) and -2D condensates appear as simple particular cases of our formulation. The validity of our results is corroborated by an independent numerical computation based on the 3D Gross-Pitaevskii equation.
Abstract We study the mean-field phase-space distribution of the ground state of an anisotropic, harmonically trapped Bose–Einstein condensate at zero temperature. First, we derive the Fourier transform of the ground-state wave function in the Thomas–Fermi limit in d dimensions. Second, we obtain analytical approximations for the Thomas–Fermi Wigner function in one and three dimensions with a factorization ansatz of the coherence function. These approximate expressions are compared with exact numerical simulations of the Gross–Pitaevskii equation.
We study the ground-state properties of dilute Bose gas confined to both isotropic and anisotropic traps to assess the accuracy of Gross-Pitaevskii (GP) theory. To go beyond GP approximation we use Huang-Yang theory of interatomic interaction energy for hard-sphere Bose gas and use a variational method to solve the resulting modified GP equation. We also make an analytic estimate of the corrections due to higher-order terms in the interatomic interaction energy. We find that the corrections are of the order of 1 percent. However, there is a qualitative change in the density profile due to presence of logarithimic term in the interaction energy for large N (number of atoms)