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Journal of Beijing Institute of Graphic Communication

A Numerical Value Method for Evaluation of BEC Initial Value

Dan Wang

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

It needs the initial value of eddy to discuss Bose-Einstein Condensate by a numerial value method,and various initial values of eddy for different parameters.We can seek solutions by void time G-P equation,which rapidly convergense to the solution of G-P equation when time tends to infinite.

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The subject of the present thesis is the study of the interaction between a Bose-Einstein condensate (BEC) and a superconductor.The BEC performs center-of-mass oscillations perpendicular to the superconducting surface.Each atom in the BEC carries a magnetic dipole moment, which induces eddy currents in the surface.These eddy currents are a source of a magnetic field which modifies the potential in which the BEC is trapped.This leads to a change of the dynamical behavior of the BEC.In particular the BEC center-of-mass oscillation frequency is shifted compared to the case without the surface interaction.Additionally, the potential generated by the superconductor excites shape oscillations of the BEC.To investigate these effects, the Gross-Pitaevskii equation for a dipolar BEC close to a superconducting surface is solved numerically.Also analytical approximations are presented and compared to the numerically obtained results.It is shown that the interaction with the superconducting surface generates a frequency shift, that is large enough to be detected in an experiment.Furthermore, different methods to identify the eddy current effect are presented.It is also discussed how the effects on the BEC can be enhanced by choosing the proper dipole orientation and by adjusting the geometry of the setup.The BEC shape oscillations can be additionally enhanced by making use of a resonance phenomenon.4 Center-of-mass frequency shift 4.1 Frequency shift within TF approximation . . . . . . . . . .4.1.1Frequency shift based on a Thomas-Fermi ellipsoid 4.1.2Frequency shift based on the column density model 4.2 Numerical results for the frequency shift . . . . . . . . . . .4.3 Different polarizations of the BEC . . . . . . . . . . . . . .4.3.1 Polarization in the y-direction . . . . . . . . . . . .4.3.2Polarization in the x-direction (perpendicular to the surface) . . . . . . . . . . .

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Journal of Beijing Institute of Graphic Communication•Jan 1, 2009•0 citations

Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation

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arXiv:cond-mat/0303239•Journal of Computational Physics•Apr 23, 2003•638 citations
We study the dynamics of a dipolar Bose-Einstein condensate, like, for example, a 52Cr^{52}\mathrm{Cr}52Cr or 164Dy^{164}\mathrm{Dy}164Dy condensate, interacting with a superconducting surface. The magnetic dipole moments of the atoms in the Bose-Einstein condensate induce eddy currents in the superconductor. The magnetic field generated by eddy currents modifies the trapping potential such that the center-of-mass oscillation frequency is shifted. We numerically solve the Gross-Pitaevskii equation for this system and compare the results with analytical approximations. We present an approximation that gives excellent agreement with the numerical results. The eddy currents give rise to anharmonic terms, which leads to the excitation of shape fluctuations of the condensate. We discuss how the strength of the excitation of such modes can be increased by exploiting resonances, and we examine the strength of the resonances as a function of the center-of-mass oscillation amplitude of the condensate. Finally, we study different orientations of the magnetic dipoles and discuss favorable conditions for the experimental observation of the eddy-current effect.
The subject of the present thesis is the study of the interaction between a Bose-Einstein condensate (BEC) and a superconductor.The BEC performs center-of-mass oscillations perpendicular to the superconducting surface.Each atom in the BEC carries a magnetic dipole moment, which induces eddy currents in the surface.These eddy currents are a source of a magnetic field which modifies the potential in which the BEC is trapped.This leads to a change of the dynamical behavior of the BEC.In particular the BEC center-of-mass oscillation frequency is shifted compared to the case without the surface interaction.Additionally, the potential generated by the superconductor excites shape oscillations of the BEC.To investigate these effects, the Gross-Pitaevskii equation for a dipolar BEC close to a superconducting surface is solved numerically.Also analytical approximations are presented and compared to the numerically obtained results.It is shown that the interaction with the superconducting surface generates a frequency shift, that is large enough to be detected in an experiment.Furthermore, different methods to identify the eddy current effect are presented.It is also discussed how the effects on the BEC can be enhanced by choosing the proper dipole orientation and by adjusting the geometry of the setup.The BEC shape oscillations can be additionally enhanced by making use of a resonance phenomenon.4 Center-of-mass frequency shift 4.1 Frequency shift within TF approximation . . . . . . . . . .4.1.1Frequency shift based on a Thomas-Fermi ellipsoid 4.1.2Frequency shift based on the column density model 4.2 Numerical results for the frequency shift . . . . . . . . . . .4.3 Different polarizations of the BEC . . . . . . . . . . . . . .4.3.1 Polarization in the y-direction . . . . . . . . . . . .4.3.2Polarization in the x-direction (perpendicular to the surface) . . . . . . . . . . .