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Indian Journal of Pure & Applied Physics

Effect of three-body interaction in strongly interacting Bose-Einstein condensate in isotropic and highly anisotropic magnetic trap

Shri Prakash Tewari, Poonam Silotia, Aditya Saxena, Lokesh Kumar Gupta

Published Feb 1, 2007• 1 citation
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Abstract

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.

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Sep 11, 2007•0 citations
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.
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.
Various ground state properties such as chemical potential, differential and total energy per particle etc. of Bose-Einstein condensate of 10000 85 Rb atoms with varying repulsive self-interaction energy have been reported considering not only two-body interaction but also including the higher order terms of the low-density energy expansion of homogeneous Bose gas in the Ginzburg, Pitaevskii and Gross (GPG) equation. These include hard-core approximation of the bosons neglected earlier and the three-body interaction terms. The study is more general as it includes the terms beyond logarithm in energy density expansion. It is also shown that such a consideration does not violate the lower bound predicted earlier in which the 'constant' beyond logarithm term in the three-body interaction was neglected.
The stability of a Bose–Einstein condensed state of trapped ultra-cold atoms is investigated under the assumption of an attractive two-body and a repulsive three-body interaction. The Ginzburg–Pitaevskii–Gross (GPG) nonlinear Schrödinger equation is extended to include an effective potential dependent on the square of the density and solved numerically for the s-wave. The lowest collective mode excitations are determined and their dependences on the number of atoms and on the strength of the three-body force are studied. The addition of three-body dynamics can allow the number of condensed atoms to increase considerably, even when the strength of the three-body force is very small compared with the strength of the two-body force. We study in detail the first-order liquid–gas phase transition for the condensed state, which can happen in a critical range of the effective three-body force parameter. 1.
The stability of a Bose-Einstein condensed state of trapped ultra-cold atoms is investigated under the assumption of an attractive two-body and a repulsive three-body interaction. The Ginzburg-Pitaevskii-Gross (GPG) nonlinear Schrödinger equation is extended to include an effective potential dependent on the square of the density and solved numerically for the s-wave. The lowest collective mode excitations are determined and their dependences on the number of atoms and on the strength of the three-body force are studied. The addition of three-body dynamics can allow the number of condensed atoms to increase considerably, even when the strength of the three-body force is very small compared with the strength of the two-body force. We study in detail the first-order liquid-gas phase transition for the condensed state, which can happen in a critical range of the effective three-body force parameter.
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.
Bose-Einstein condensates have been, by now, observed in as many as nine different atomic assemblies of bosons. Such a condensate is quantum mechanical interacting system whose ground state properties can be studied theoretically by solving the appropriate non-linear Gross-Pitaevskii-Ginzburg GPG equation. One can now study the change in the behavior of Bose-Einstein condensate by introducing a localized impurity which interacts with the condensate as a function of position of impurity in the condensate. The introduction of such an impurity can be mimicked by simply allowing an intensely focused laser beam to interact with the condensate. This would lead to alteration of ground state properties of the condensate as it would now interact with a potential of type V Sech2(r/w) where, V and w are amplitude and width of the impurity potential, respectively. The modified GPG equation in the presence of localized impurity potential as function of position in the condensate, has been numerically solved to obtain its various ground state properties as function of position, such as total energy per particle, chemical potential, kinetic, harmonic trap potential and two-body interaction energies per particle in addition to energy associated with impurity potential, correlation length, healing length etc. We have studied the behavior of the above-mentioned ground state properties as the position of localized impurity is changed in the condensate from core to peripheral position. While the total, harmonic oscillator potential and impurity energies decrease as the position of localized impurity is displaced from core of the condensate to its periphery, the value of two-body inter-particle interaction energy increases. Further, the values of chemical potential and total energy per particle shows decrease by ~ 9% and ~ 17% respectively, leading to the inference that the stability of condensate increases as the localized impurity is moved away from the core of the condensate.