Linearly Coupled Bose-Einstein Condensates: From Rabi Oscillations and Quasi-Periodic Solutions to Oscillating Domain Walls and Spiral Waves

TL;DR

This paper introduces an exact transformation for coupled nonlinear Schrödinger equations with linear coupling, enabling the derivation of complex solutions like oscillating domain walls and spiral waves relevant to Bose-Einstein condensates.

Contribution

It provides a novel unitary transformation that constructs solutions of coupled equations with linear coupling from uncoupled solutions, especially applied to the Manakov system.

Findings

Derived periodic and quasi-periodic solutions including domain walls and spiral waves.

Established a method to generate solutions with linear coupling from uncoupled systems.

Applied the transformation to Bose-Einstein condensates of hyperfine states.

Abstract

In this paper, an exact unitary transformation is examined that allows for the construction of solutions of coupled nonlinear Schr{\"o}dinger equations with additional linear field coupling, from solutions of the problem where this linear coupling is absent. The most general case where the transformation is applicable is identified. We then focus on the most important special case, namely the well-known Manakov system, which is known to be relevant for applications in Bose-Einstein condensates consisting of different hyperfine states of $^{87}$Rb. In essence, the transformation constitutes a distributed, nonlinear as well as multi-component generalization of the Rabi oscillations between two-level atomic systems. It is used here to derive a host of periodic and quasi-periodic solutions including temporally oscillating domain walls and spiral waves.