On the characterisation of fragmented Bose-Einstein condensation and its emergent effective evolution

TL;DR

This paper investigates the complex dynamics of fragmented Bose-Einstein condensates, highlighting challenges in characterisation and providing refined methods and quantitative convergence results for their effective evolution.

Contribution

It refines previous characterisations of fragmented condensates and establishes a quantitative rate of convergence to the effective dynamics in a double limit.

Findings

Characterisation of fragmentation in terms of reduced density matrices is inadequate.

The rank of reduced marginals generally increases over time, indicating occupation transfer.

A quantitative convergence rate to effective dynamics is established in the double limit.

Abstract

Fragmented Bose-Einstein condensates are large systems of identical bosons displaying \emph{multiple} macroscopic occupations of one-body states, in a suitable sense. The quest for an effective dynamics of the fragmented condensate at the leading order in the number of particles, in analogy to the much more controlled scenario for complete condensation in one single state, is deceptive both because characterising fragmentation solely in terms of reduced density matrices is unsatisfactory and ambiguous, and because as soon as the time evolution starts the rank of the reduced marginals generically passes from finite to infinite, which is a signature of a transfer of occupations on infinitely many more one-body states. In this work we review these difficulties, we refine previous characterisations of fragmented condensates in terms of marginals, and we provide a quantitative rate of convergence to the leading effective dynamics in the double limit of infinitely many particles and infinite energy gap.