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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

General expression of chemical potential for Bose Einstein condensate in an anisotropic magnetic trap

S.P. Tewari, Poonam Silotia, Aditya Saxena

Published Feb 3, 2004• 0 citations
Publisher

Abstract

A general expression for the chemical potential, μ, of Bose Einstein condensate in any form of the magnetic trap has been derived without resorting to any approximation regarding the number of particles in the condensate. Specific expression for μ has been deduced when the form of the magnetic trap corresponds to a right circular cylinder. Computations on the chemical potential, the speed of sound, healing length, interparticle separation in the core of the condensate, energy and the correlation length of the condensate have been made for different number of 87 Rb atoms ranging from hundred to ten million particles in the Bose Einstein condensate and compared with the corresponding results of numerically evaluated GPG equation and other approximate results wherever possible.

DOI
10.1117/12.512323

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A general expression for the chemical potential, μ, of Bose Einstein condensate in any form of the magnetic trap has been derived without resorting to any approximation regarding the number of particles in the condensate. Specific expression for μ has been deduced when the form of the magnetic trap corresponds to a right circular cylinder. Computations on the chemical potential, the speed of sound, healing length, interparticle separation in the core of the condensate, energy and the correlation length of the condensate have been made for different number of 87 Rb atoms ranging from hundred to ten million particles in the Bose Einstein condensate and compared with the corresponding results of numerically evaluated GPG equation and other approximate results wherever possible.
We solve the Gross-Pitaevskii equations for a dilute atomic gas in a magnetic trap, modeled by an anisotropic harmonic potential. We evaluate the wave function and the energy of the Bose-Einstein condensate as a function of the particle number, both for positive and negative scattering length. The results for the transverse and the vertical size of the cloud of atoms, as well as for the kinetic and potential energy per particle, are compared with the predictions of approximated models. We also compare the aspect ratio of the velocity distribution with experimental estimates available for 87Rb^{87}\mathrm{Rb}87Rb. Vortex states are considered and the critical angular velocity for production of vortices is calculated. We show that the presence of vortices significantly increases the stability of the condensate in the case of attractive interactions. \textcopyright{} 1996 The American Physical Society.
Ground state properties of Bose-Einstein condensate of 20000 23 Na atoms confined in isotropic and highly anisotropic magnetic traps in the presence of three-body interaction, in addition to the two-body and hard-core interactions, have been theoretically studied by solving modified highly non-linear Gross-Pitaevskii-Ginzburg (GPG) equation. There is increasingly significant change in the chemical potential, total and differential energies per particle as the aspect ratio. λ, is increased from 0.1 to 1.0, with respect to the corresponding values when only two-body and hard-core interactions are considered. The condensate order parameter changes drastically and becomes quasi 1-dimensional, as λ is varied from 1 to 0.1.
We present results on the size, density, chemical potential, and lifetime of a Bose-Einstein condensate as a function of the number of condensate atoms for parameters appropriate to the time-averaged orbiting potential trap in which a Bose condensate of 87Rb^{87}\mathrm{Rb}87Rb atoms was recently formed [M. H. Anderson et al., Science 269, 198 (1995)]. These results were obtained by solving the time-independent nonlinear Schr\"odinger equation with anisotropic harmonic trapping potential. We also find vortex solutions for the trap and the critical rotation frequency required for their formation.
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.
For Bose–Einstein condensation of neutral atoms in anisotropic traps at zero temperature, we present simple analytical methods for computing the properties of ground state and single vortex of Bose–Einstein condensates, and compare those results to extensive numerical simulations. The critical angular velocity for production of vortices is calculated for both positive and negative scattering lengths a, and find an analytical expression for the large- N limit of the vortex critical angular velocity for a >0, and the critical number for condensate population approaches the point of collapse for a <0, by using approximate variational method.
In the present work we determine the speed of sound in a Bose-Einstein condensate confined by an isotropic harmonic oscillator trap. The deduction of this physical parameter is done resorting to the NNN-body Hamiltonian operator. The single-particle eigenfunctions that have been employed in this formalism are those stemming from the corresponding harmonic oscillator potential, and an expression for the dependence of this speed on the temperature is also deduced. These functions are used in the calculation of the scaterring length, etc. The situation for a Bose-Einstein condensate of sodium is evaluated and the corresponding speed of sound is obtained and compared against the known experimental outcomes. The possibility that the solution, to the existing discrepancy between experiment and theoretical predictions, could be given by the Zaremba-Nikuni-Griffin formalism is also explored.