Effective mean-field equations for cigar-shaped and disk-shaped Bose-Einstein condensates

TL;DR

This paper derives simplified, accurate effective 1D and 2D equations for the dynamics of cigar-shaped and disk-shaped Bose-Einstein condensates, improving upon previous models and validated through numerical solutions.

Contribution

It introduces new effective equations that are simpler and more accurate for describing BECs in different geometries, including vortex states, with analytical solutions for key properties.

Findings

Effective 1D equation accurately models axial dynamics.

Effective 2D equation accurately models transverse dynamics.

Validated by numerical solutions of the 3D Gross-Pitaevskii equation.

Abstract

By applying the standard adiabatic approximation and using the accurate analytical expression for the corresponding local chemical potential obtained in our previous work [Phys. Rev. A \textbf{75}, 063610 (2007)] we derive an effective 1D equation that governs the axial dynamics of mean-field cigar-shaped condensates with repulsive interatomic interactions, accounting accurately for the contribution from the transverse degrees of freedom. This equation, which is more simple than previous proposals, is also more accurate. Moreover, it allows treating condensates containing an axisymmetric vortex with no additional cost. Our effective equation also has the correct limit in both the quasi-1D mean-field regime and the Thomas-Fermi regime and permits one to derive fully analytical expressions for ground-state properties such as the chemical potential, axial length, axial density profile, and local sound velocity. These analytical expressions remain valid and accurate in between the above two extreme regimes. Following the same procedure we also derive an effective 2D equation that governs the transverse dynamics of mean-field disk-shaped condensates. This equation, which also has the correct limit in both the quasi-2D and the Thomas-Fermi regime, is again more simple and accurate than previous proposals. We have checked the validity of our equations by numerically solving the full 3D Gross-Pitaevskii equation.