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arXiv:cond-mat/9707199 · Physical review. B, Condensed matter

Quantum corrections to the ground state of a trapped Bose-Einstein condensate

Eric Braaten, Agustín Nieto

Published Dec 1, 1997• 74 citations
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Abstract

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 $V(\mathbf{r})$ 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 $a$ is the $S$-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.

DOI
10.1103/physrevb.56.14745

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In the mean-field approximation, the number density \ensuremathρ(r)\ensuremath{\rho}(\mathbf{r})\ensuremathρ(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 \ensuremathρ\ensuremath{\rho}\ensuremathρ are dominated by quantum fluctuations with wavelengths of order 1/\ensuremathρa1/\sqrt{\ensuremath{\rho}a}1/\ensuremathρa​, 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 \ensuremathρ\ensuremath{\rho}\ensuremathρ 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.
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.
We study the ground state of a system of Bose hard spheres trapped in an isotropic harmonic potential to investigate the effect of the interatomic correlations and the accuracy of the Gross-Pitaevskii equation. We compare a local-density approximation, based on the energy functional derived from the low-density expansion of the energy of the uniform hard-sphere gas, and a correlated wave-function approach, which explicitly introduces the correlations induced by the potential. Both higher-order terms in the low-density expansion, beyond Gross-Pitaevskii, and explicit dynamical correlations have effects of the order of percent when the number of trapped particles becomes similar to that attained in recent experiments $(N\ensuremath{\sim}{10}^{7}).$
[EN] In this work, we have studied a dilute and ultracold bosonic gas of weakly interacting atoms, \ni.e. a Bose-Einstein Condensate (BEC) by means of the Modi ed Gross-Pitaevskii equation \n(MGPE) taking into account beyond-mean- eld corrections due to quantum \nuctuations. We \nhave considered the case where the cloud of atoms is strongly con ned transversally (within a \ncigar-shaped trap) and therefore, the three-dimensional (3D) MGPE is reduced to an e ective \none-dimensional (1D) form by averaging over the transverse coordinates. Based on this approach, \nwe have performed numerical simulations for analyzing the behavior of the BEC under the \nin \nuence of quantum \nuctuations, which give rise to an additional nonlinear quartic term. After \nreviewing the mathematical theory of Bose-Einstein condensation based on the GPE and the \nnumerical methods used to integrate it we discuss the computation of the ground state and the \nsimulation of its dynamic evolution in a waveguide (free expansion).
The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local-density approximation. Consistency of the theory for temperatures through the Bose condensation point Tc{T}_{c}Tc​ requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length asc,{a}_{\mathrm{sc}},asc​, increases the number of condensate atoms at each temperature, and raises the value of Tc{T}_{c}Tc​ relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near Tc.{T}_{c}.Tc​. Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.