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International Journal of Theoretical Physics

On the Zero-Energy-Limit Solution for the Modified Gross–Pitaevskii Equation

Kwang‐Hua W. Chu

Published Jun 5, 2007• 2 citations
Publisher
DOI
10.1007/s10773-007-9440-0

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We consider a dilute and ultracold bosonic gas of weakly-interacting atoms. Within the framework of quantum field theory, we derive a zero-temperature modified Gross–Pitaevskii equation with beyond-mean-field corrections due to quantum depletion and anomalous density. This result is obtained from the stationary equation of the Bose–Einstein order parameter coupled to the Bogoliubov–de Gennes equations of the out-of-condensate field operator. We show that, in the presence of a generic external trapping potential, the key steps to get the modified Gross–Pitaevskii equation are the semiclassical approximation for the Bogoliubov–de Gennes equations, a slowly-varying order parameter and a small quantum depletion. In the uniform case, from the modified Gross–Pitaevskii equation, we get the familiar equation of state with Lee–Huang–Yang correction.
We study vortex ring solutions of the Gross-Pitaevskii equation for Bose-Einstein condensates confined in toroidal traps. Analogously to the Feynman-Onsager ansatz for a single vortex line, we assume a condensate wavefunction that describes a quantized vortex ring. This leads to a modified Gross-Pitaevskii equation that incorporates vortex ring solutions. By numerically solving this equation, we have obtained the density profiles and the vortex core structure. We have also calculated the nucleation energy of a vortex ring, and have analyzed its dependence on the ring position. Finally, a qualitative description of the vortex ring dynamics is presented.