Soliton dynamics in the Gross–Pitaevskii equation: splitting, collisions and interferometry

Abstract

Bose–Einstein condensates with attractive interactions have stable 1D solutions in the form of bright solitary-waves. These solitary waves behave, in the absence of external potentials, like macroscopic quantum particles. This opens up a wide array of applications for the testing of quantum mechanical behaviours and precision measurement. Here we investigate these applications with particular focus on the
interactions of bright solitary-waves with narrow potential barriers.
We first study bright solitons in the Gross–Pitaevskii equation as they are split on Gaussian and δ-function barriers, and then on Gaussian barriers in a low energy
system. We present analytic and numerical results determining the general region in which a soliton may not be split on a finite width potential barrier. Furthermore,
we test the sensitivity of the system to quantum fluctuations.
We then study fast-moving bright solitons colliding at a narrow Gaussian potential barrier. In the limiting case of a δ-function barrier, we show analytically that the relative norms of the outgoing waves depends sinusoidally on the relative phase of the incoming waves, and determine whether the outgoing waves are bright solitons. We use numerical simulations to show that outside the high velocity limit nonlinear effects introduce a skew to the phase-dependence.
Finally, we use these results to analyse the process of soliton interferometry. We develop analyses of both toroidal and harmonic trapping geometries for Mach–Zehnder interferometry, and then two implementations of a toroidal Sagnac inter- ferometer, also giving the analytical determination of the Sagnac phase in such systems. These results are again verified numerically. In the Mach–Zehnder case, we again probe the systems sensitivity to quantum fluctuations.