i-SPin 2: An integrator for general spin-s Gross-Pitaevskii systems

TL;DR

This paper introduces a versatile, second-order symplectic algorithm for simulating general spin-$s$ Gross-Pitaevskii systems, applicable to both laboratory BECs and astrophysical dark matter scenarios, accommodating complex interactions and external potentials.

Contribution

The work presents a novel, extensible numerical method for evolving general spin-$s$ Gross-Pitaevskii equations with diverse interactions and external fields, including spin-orbit coupling.

Findings

Validated with spin-1 BECs under magnetic fields

Simulated spin-orbit coupling effects in spinor condensates

Modeled soliton collisions in dark matter scenarios

Abstract

We provide an algorithm for evolving general spin-$s$ Gross-Pitaevskii / non-linear Schr\"odinger systems carrying a variety of interactions, where the $2s+1$ components of the `spinor' field represent the different spin-multiplicity states. We consider many nonrelativistic interactions up to quartic order in the Schr\"odinger field (both short and long-range, and spin-dependent and spin-independent interactions), including explicit spin-orbit couplings. The algorithm allows for spatially varying external and/or self-generated vector potentials that couple to the spin density of the field. Our work can be used for scenarios ranging from laboratory systems such as spinor Bose-Einstein condensates (BECs), to cosmological/astrophysical systems such as self-interacting bosonic dark matter. As examples, we provide results for two different setups of spin-$1$ BECs that employ a varying magnetic field and spin-orbit coupling, respectively, and also collisions of spin-$1$ solitons in dark matter. Our symplectic algorithm is second-order accurate in time, and is extensible to the known higher-order accurate methods.