Bose-Einstein condensation dynamics in three dimensions by the pseudospectral and finite-difference methods

TL;DR

This work addresses the time-dependent dynamics of a three-dimensional Bose-Einstein condensate described by the GP equation in anisotropic traps. It develops a pseudospectral Runge-Kutta framework (PSRK) based on a Hermite spectral expansion and a differentiation matrix, and contrasts it with a finite-difference Crank-Nicholson approach (FDCN). The authors demonstrate PSRK's accuracy for stationary states and resonance dynamics under periodic modulation of the scattering length, and show FDCN's effectiveness for rapidly varying optical-lattice potentials, highlighting complementary applicability. The results provide efficient, stable tools for simulating complex 3D BEC dynamics with time-dependent interactions and structured traps, with potential impact on modeling experiments involving Feshbach resonances and optical lattices.

Abstract

We suggest a pseudospectral method for solving the three-dimensional time-dependent Gross-Pitaevskii (GP) equation and use it to study the resonance dynamics of a trapped Bose-Einstein condensate induced by a periodic variation in the atomic scattering length. When the frequency of oscillation of the scattering length is an even multiple of one of the trapping frequencies along the $x$, $y$, or $z$ direction, the corresponding size of the condensate executes resonant oscillation. Using the concept of the differentiation matrix, the partial-differential GP equation is reduced to a set of coupled ordinary differential equations which is solved by a fourth-order adaptive step-size control Runge-Kutta method. The pseudospectral method is contrasted with the finite-difference method for the same problem, where the time evolution is performed by the Crank-Nicholson algorithm. The latter method is illustrated to be more suitable for a three-dimensional standing-wave optical-lattice trapping potential.

Figures (2)

• Figure 1: The rms sizes $\langle x\rangle _{\hbox{rms}}$, $\langle y\rangle _{\hbox{rms}}$, and $\langle z\rangle _{\hbox{rms}}$ vs. time for a BEC in a harmonic trap with $\kappa =\sqrt 2, \nu =2$ subject to a sinusoidal variation of nonlinearity $n=0.3 [\sin (\Omega t)$ with (a) $\Omega =2$, (b) $\Omega =2\sqrt 2$, and (c) $\Omega =4$.
• Figure 2: Three-dimensional contour plot of the interior part of the BEC ground state wave function under the combined action of the harmonic and the optical trap for $n\equiv N_0a/l=10$, $\lambda _0 =1$ and $V_0=10$ on a cubic lattice of size $3\times 3 \times 3$ ($-1.5<x,y,z<1.5$): (a) view along the diagonal of the cube and (b) along one of the axes of the cube.