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Journal of Research of the National Institute of Standards and Technology

Aspect ratio analysis for ground states of bosons in anisotropic traps

N. Ilinski, Alexander Moroz

Published Jul 1, 1996• 2 citations
Publisher

Abstract

Characteristics of the initial condensate in the recent experiment on Bose-Einstein condensation (BEC) of 87 Rb atoms in an anisotropic magnetic trap are discussed.Given the aspect ratio R , the quality of BEC is estimated.A simple analytical ansatz for the initial condensate wave function is proposed as a function of the aspect ratio which, in contrast to the Baym-Pethick trial wave function, can be used for any interaction strength, reproduces both the weak and the strong interaction limits, and which is in better agreement with numerical results than the latter.

DOI
10.6028/jres.101.056

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Characteristics of the initial condensate in the recent experiment on Bose-Einstein condensation (BEC) of 87 Rb atoms in an anisotropic magnetic trap are discussed.Given the aspect ratio R , the quality of BEC is estimated.A simple analytical ansatz for the initial condensate wave function is proposed as a function of the aspect ratio which, in contrast to the Baym-Pethick trial wave function, can be used for any interaction strength, reproduces both the weak and the strong interaction limits, and which is in better agreement with numerical results than the latter.
Introduction Bose-Einstein condensation (BEC) is a phenomenon where a macroscopic number of particles is in the ground state of the system at finite temperature. The phenomenon of BEC plays a significant role in many branches of physics [1]. Because of the presence of strong interactions, BEC has been inferred rather than directly observed so far. Recently, however, three different groups have reported the direct evidence of BEC in weakly interacting systems of atoms such as rubidium [2], lithium [3], and sodium [4], confined in anisotropic magnetic traps and cooled down to very low temperatures. These experiments show promise of becoming a new laboratory for quantum statistical phenomena that are inaccessible to other conventional techniques and of enabling us to experimental study phenomena that have been addressed only theoretically, such as spontaneous symmetry breaking and decay of unstable macroscopic states. They may also advance our understanding of superconductivity and
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.
Based on the energy functional and variational method, we present a new method to investigate the ground state properties for a weakly interacting Bose-condensed gas in an anisotropic harmonic trap at zero temperature. With this method we are able to find the analytic expression of the ground-state wavefunction and to explore the relevant quantities, such as energy, chemical potential, and the aspect ratio of the velocity distribution. These results agree well with previous ground state numerical solutions of the Gross–Pitaevskii equation given by Dalfovo et al. [Phys. Rev. A 53 (1996) 2477] This new method is simple compared to other methods used to solve numerically the Gross–Pitaevskii equation, and one can obtain analytic and reliable results.
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.
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.