Probability of phase separation for the Bose gas with delta interactions

TL;DR

This paper analyzes the probability of phase separation in the Bose gas with delta interactions by studying the asymptotics of emptiness formation probability using Riemann Hilbert techniques.

Contribution

It introduces a method to evaluate the emptiness formation probability for the Bose gas with delta interactions using determinant representations and Riemann Hilbert analysis.

Findings

Derived asymptotics of the emptiness formation probability

Connected phase separation probability with quantum correlation functions

Provided insights into the large-distance behavior of the Bose gas

Abstract

We consider the quantum Non-linear Schr\"{o}dinger equation $i\partial_t\Psi=-\partial_x^2\Psi +2c\Psi^\dagger\Psi^2$ with positive coupling constant $c$ varying from zero to infinity. We study quantum correlation functions of this model using the determinant representation of these correlation functions. We consider the case of a finite density ground state and evaluate, using the Riemann Hilbert problem, the asymptotics of the probability that no particles are present in the space interval $[0,x]$ in the large $x$ limit. We call this the probability of phase separation or alternately the emptiness formation probability.