Entanglement and Localization of a Two-Mode Bose-Einstein Condensate

TL;DR

This paper models a two-mode Bose-Einstein condensate using SU(2) algebra, analyzing entanglement, localization, and phase transitions through variational states, with results showing singular behavior at phase boundaries.

Contribution

It introduces a simple second quantization model for two-mode BECs and explores entanglement and localization properties using coherent states and catastrophe theory.

Findings

Entanglement entropy correlates with localization in phase space.

Phase transitions are identified by singularities in entanglement and overlap.

Localization properties are linked to the Husimi function's second moment.

Abstract

A simple second quantization model is used to describe a two-mode Bose-Einstein condensate (BEC), which can be written in terms of the generators of a SU(2) algebra with three parameters. We study the behaviour of the entanglement entropy and localization of the system in the parameter space of the model. The phase transitions in the parameter space are determined by means of the coherent state formalism and the catastrophe theory, which besides let us get the best variational state that reproduces the ground state energy. The entanglement entropy is determined for two recently proposed partitions of the two-mode BEC that are called separation by boxes and separation by modes of the atoms. The entanglement entropy in the boxes partition is strongly correlated to the properties of localization in phase space of the model, which is given by the evaluation of the second moment of the Husimi function. To compare the fitness of the trial wavefunction its overlap with the exact quantum solution is evaluated. The entanglement entropy for both partitions, the overlap and localization properties of the system get singular values along the separatrix of the two-mode BEC, which indicates the phase transitions which remain in the thermodynamical limit, in the parameter space.