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Journal of Physics A Mathematical and Theoretical

A Bose–Einstein condensate in a double-well and periodic potential: a variational study

Kasturi Saha, Ajit Kumar

Published Jul 16, 2009• 1 citation
Publisher

Abstract

The dynamics of a Bose?Einstein condensate under the influence of a confining double-well potential and a periodic potential (optical lattice) is studied by the well-known variational method. The variational principle leads to coupled nonlinear differential equations for the amplitude (width), chirp, center position and the center frequency of the condensate. The interplay between the condensate parameters leads to various regimes of its time evolution which depend on the relative strengths of the double-well and the periodic potentials. In particular, we show that by adjusting the strengths of the applied potentials appropriately, one can control and manipulate the condensate in several ways including tunneling from one well to the other and the to-and-fro motion between the two wells.

DOI
10.1088/1751-8113/42/31/315303

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The dynamics of a Bose?Einstein condensate under the influence of a confining double-well potential and a periodic potential (optical lattice) is studied by the well-known variational method. The variational principle leads to coupled nonlinear differential equations for the amplitude (width), chirp, center position and the center frequency of the condensate. The interplay between the condensate parameters leads to various regimes of its time evolution which depend on the relative strengths of the double-well and the periodic potentials. In particular, we show that by adjusting the strengths of the applied potentials appropriately, one can control and manipulate the condensate in several ways including tunneling from one well to the other and the to-and-fro motion between the two wells.
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A variational method is developed to describe the dynamics of a Bose-Einstein condensate (BEC) trapped in an applied external potential consisting of both a harmonic and periodic component. Using this variational method, the BEC dynamics is shown to be well approximated by four coupled nonlinear differential equations, which describe the fundamental interactions in the system arising from the interplay of amplitude (width), chirp, center position, and center frequency. The simplified analytic theory allows for an efficient and convenient method for characterizing the experimental BEC behavior when localized condensates are generated. It further gives the critical strength ratio of harmonic to periodic potential necessary to support multiple stable lattice sites for the condensate and demonstrates that there can be an underlying chaotic behavior in the condensate system.
An extended variation method for describing the dynamical evolution of Bose-Einstein condensate solitons in the presence of optic-lattice and harmonic potentials was developed.This method agreed well with full numerical simulations and highlighted the dominant physical interactions in the system,namely,amplitude,width,chirp,center-position,and center-frequency dynamics.It further presented the critical strength ratio of optic-lattice to harmonic potential necessary to support multiple stable lattice sites for the Bose-Einstein condensate solitons and demonstrated that the Bose-Einstein condensate solitons could move selectively from one stable lattice site location to arbitrary final position by simply manipulating the optic-lattice potential.In addition,some interesting results for the experiments and applications of the Bose-Einstein condensates were also obtained.
We study dynamical behaviors of the weakly interacting Bose-Einstein condensate in the one-dimensional optical lattice with an overall double-well potential by solving the time-dependent Gross-Pitaevskii equation. It is observed that the double-well potential dominates the dynamics of such a system even if the lattice depth is several times larger than the height of the double-well potential. This result suggests that the condensate flows without resistance in the periodic lattice just like the case of a single particle moving in periodic potentials. Nevertheless, the effective mass of atoms is increased, which can be experimentally verified since it is connected to the Josephson oscillation frequency. Moreover, the periodic lattice enhances the nonlinearity of the double-well condensate, making the condensate more “self-trapped” in the π -mode self-trapping regime.
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The properties of a Bose-Einstein condensate in a two-well potential can be manipulated by periodic modulation of the potential parameters. We study the effects arising from modulating the barrier height and the difference in well depth. At certain modulation frequencies the system exhibits resonances, which may show up in an enhancement of the tunneling rate between the wells. Resonances can be used to control the particle distribution over the wells. Some of the effects occurring in the two-well system also arise for a Bose-Einstein condensate in an optical lattice.