The single-particle density matrix of a quantum bright soliton from the coordinate Bethe ansatz

TL;DR

This paper introduces a new method to compute the single-particle density matrix of quantum bright solitons using Bethe ansatz, revealing high condensate fractions and enabling studies of time-dependent phenomena.

Contribution

A novel diagrammatic and integration-based approach for calculating reduced density matrices of Bethe-ansatz eigenstates, applicable to time-dependent and higher-order correlation analyses.

Findings

Condensate fraction exceeds 97% for up to 10 bosons.

Method efficiently computes density matrices with modest resources.

Applicable to time-dependent problems and higher-order correlations.

Abstract

We present a novel approach for computing reduced density matrices for superpositions of eigenstates of a Bethe-ansatz solvable model by direct integration of the wave function in coordinate representation. A diagrammatic approach is developed to keep track of relevant terms and identify symmetries, which helps to reduce the number of terms that have to be evaluated numerically. As a first application we compute with modest numerical resources the single-particle density matrix and its eigenvalues including the condensate fraction for a quantum bright soliton with up to $N=10$ bosons. The latter are constructed as superpositions of string-type Bethe-ansatz eigenstates of nonrelativistic bosons in one spatial dimension with attractive contact interaction. Upon delocalising the superposition in momentum space we find that the condensate fraction reaches maximum values larger than 97\% in the range of particles studied. The presented approach is suitable for studying time-dependent problems and generalises to higher-order correlation functions.