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arXiv:cond-mat/0302331 · Physical Review A

Beyond the Fermi pseudopotential: A modified Gross-Pitaevskii equation

Haixiang Fu, Yuzhu Wang, Bo Gao

Published May 27, 2003• 73 citations
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Abstract

We present an effective potential and the corresponding modified Gross-Pitaevskii equation, which account for the energy dependence of the two-body scattering amplitude through an effective-range expansion. For the ground-state energy of a trapped condensate, the theory leads to what we call a shape-dependent confinement correction that improves agreements with diffusion Monte Carlo calculations. The theory illustrates, for relatively strong confinement and/or high density, how the shape dependence on atom-atom interaction can come into play in a many-atom quantum system.

DOI
10.1103/physreva.67.053612

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We present an effective potential and the corresponding modified Gross-Pitaevskii equation, which account for the energy dependence of the two-body scattering amplitude through an effective-range expansion. For the ground-state energy of a trapped condensate, the theory leads to what we call a shape-dependent confinement correction that improves agreements with diffusion Monte Carlo calculations. The theory illustrates, for relatively strong confinement and/or high density, how the shape dependence on atom-atom interaction can come into play in a many-atom quantum system.
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.
One of the assumptions leading to the Gross-Pitaevskii equation (GPE) is that the interaction between atom pairs can be written effectively as a $\ensuremath{\delta}$ function so that the interaction range of the particles is assumed to vanish. A simple model that takes into account the extension of the interparticle potential is introduced. The correction to the GPE predictions for the energy of a condensate confined by a harmonic trap in the Thomas-Fermi regime is estimated. Although it is found to be small, we believe that in some situations it can be measured using its dependance on the frequency of the confining trap. Due to the simplicity of the model, it may have a wide range of applications.
Using the density-functional theory, the Ginzburg Pitaevskii Gross (GPG) equation for Bose-Einstein (BE) condensate, confined in a magnetic trap, is modified to include contribution from three-body collisions in the strongly interacting regime a>>l, 'a' is the scattering length and 'l' being the characteristic low energy length scale. This generalized GPG equation has been solved numerically using the analytically derived Thomas-Fermi order parameter, which also includes three-body interactions. The order parameter, chemical potential, extent of correlation and other ground state properties are computed when the aspect ratio, λ, is varied from1.0 to 0.05 (λ represents the anisotropy of the magnetic trap). As λ is varied from 1.0 to 0.05, the condensate shape changes from isotropic three-dimensional (3-D) to highly anisotropic quasi one-dimensional (1-D). The stability of the BE condensate increases with decrease in λ, which is also borne out by the behavior of chemical potential and the total energy per particle, as there is a decrease of about four times for a=5000 a0 as well as for a=7000 a0, 'a0' being the Bohr radius. The extent of correlations, however, increases by more than five folds, showing that quasi 1-D BE condensate is highly correlated. Both two- and three-body interaction energies show a decrease with decrease in λ: three-body interaction energy staying below two-body interaction energy for a=5000 a0 while for a=7000 a0, a cross-over occurs between the two at λ ~ 0.35. As one goes from 3-D to quasi 1-D, the percentage difference for various physical quantities, computed between only two-body interactions and when both two- and three-body interactions are considered, shows a decrease, suggesting that the effect of three-body collisions become increasingly less significant in agreement with the recent study.
We consider many-body effects on particle scattering in one-, two-, and three-dimensional (3D) Bose gases. We show that at T=0T=0T=0 these effects can be modeled by the simpler two-body T matrix evaluated off the energy shell. This is important in 1D and 2D because the two-body T matrix vanishes at zero energy and so mean-field effects on particle energies must be taken into account to obtain a self-consistent treatment of low-energy collisions. Using the off-shell two-body T matrix we obtain the energy and density dependence of the effective interaction in 1D and 2D and the appropriate Gross-Pitaevskii equations for these dimensions. Our results provide an alternative derivation of those of Kolomeisky and co-workers. We present numerical solutions of the Gross-Pitaevskii equation for a 2D condensate of hard-sphere bosons in a trap. We find that the interaction strength is much greater in 2D than for a 3D gas with the same hard-sphere radius. The Thomas-Fermi regime is, therefore, approached at lower condensate populations and the energy required to create vortices is lowered compared to the 3D case.
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.