Entanglement in the Bose-Einstein condensate phase transition

TL;DR

This paper investigates how entanglement develops in a Bose gas during the Bose-Einstein condensation phase transition, linking spatial coherence with entanglement and quantifying it in specific geometries.

Contribution

It introduces a purity-based entanglement witness for spatial regions in a Bose gas and analyzes how entanglement correlates with temperature and spatial probabilities.

Findings

Entanglement increases as temperature decreases.

Spatial coherence is necessary for entanglement.

Maximum entanglement occurs with equal probabilities in three regions.

Abstract

We determine the behaviour of entanglement between regions of space in a Bose gas of fixed particle number around the critical temperature condensation. Long-range correlations develop in the Bose-Einstein condensate (BEC) phase transition and the aim here is to find out whether spatial coherence alone implies entanglement. We use a purity measure of entanglement to derive an entanglement witness that detects entanglement between two regions of space in the BEC. It is shown that spatial coherence between the two regions is necessary for entanglement with coherence and entanglement becoming equivalent only when the regions occupy the entire confining volume of the gas. The probabilities for Bosons to occupy the regions is the only other parameter that influences the amount of entanglement. We calculate explicitly the amount of entanglement between two regions for a cigar-shaped harmonic potential and find that it increases with decreasing temperature. A second entanglement witness is derived for entanglement between three regions of space, where again spatial coherence between each pair of regions is precursor to entanglement. It is shown that the state with the maximum amount of entanglement occurs when the probabilities for the Bosons to occupy the three regions are equal.