Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates

TL;DR

This paper derives and proves the validity of non-linear effective equations describing the dynamics of pseudo-spinor Bose-Einstein condensates in the mean-field limit, including external magnetic fields and spin interactions.

Contribution

It extends the derivation of Gross-Pitaevskii equations to pseudo-spinor condensates with spin interactions and external fields, using an adapted projection counting method.

Findings

Complete condensation persists over time.

The condensate evolves according to coupled non-linear Schrödinger equations.

Quantitative convergence rates are established.

Abstract

We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schr\"odinger equation at the leading order in the number of particles. The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction. The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling. In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schr\"odinger equations coupled among themselves through linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case. Quantitative rates of convergence are available, but not made explicit because evidently non-optimal. In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.