Approximate mean-field equations of motion for quasi-two-dimensional Bose-Einstein-condensate systems

We present a method for approximating the solution of the three-dimensional, time-dependent Gross-Pitaevskii equation (GPE) for Bose-Einstein-condensate systems where the confinement in one dimension is much tighter than in the other two. This method employs a hybrid Lagrangian variational technique whose trial wave function is the product of a completely unspecified function of the coordinates in the plane of weak confinement and a Gaussian in the strongly confined direction having a time-dependent width and quadratic phase. The hybrid Lagrangian variational method produces equations of motion that consist of (1) a two-dimensional (2D) effective GPE whose nonlinear coefficient contains the width of the Gaussian and (2) an equation of motion for the width that depends on the integral of the fourth power of the solution of the 2D effective GPE. We apply this method to the dynamics of Bose-Einstein condensates confined in ring-shaped potentials and compare the approximate solution to the numerical solution of the full 3D GPE.

Figure 1

A comparison of the vertical column density of a Bose-Einstein condensate, as determined by the 3D TDGPE and the HLVM equations of motion, after direct release from a ring-shaped trap plotted along a line through the trap center ($x$ axis) for various times of flight (TOFs) during expansion: (a) 0.0, (b) 2.0, (c) 4.0, (d) 6.0, (e) 8.0, and (f) 10.0 ms. The condensate is given one unit of angular momentum before release by phase imprint.

Figure 2

GPE vs LVM vertical column density comparison for different angular momenta applied to the initial state. The condensates are formed, stirred to add $m$ units of angular momentum (simulated by phase imprint), and then released and allowed to expand for 10 ms. (a) $m=0$, (b) $m=1$, (c) $m=2$, (d) $m=3$, (e) $m=4$, and (f) $m=5$.

Figure 3

GPE and LVM vertical column density comparison for the case where the LG potential is ramped down to 20$\%$ of its initial depth over a span of 50 ms. A phase imprint is applied that simulates one unit of angular momentum added by stirring. The plots show the comparison for ramp times of (a) 0 ms, (b) 10 ms, (c) 20 ms, (d) 30 ms, (e) 40 ms, and (f) 50 ms.