The single-particle density matrix and the momentum distribution of dark "solitons" in a Tonks-Girardeau gas

TL;DR

This paper investigates the properties of dark soliton states in a Tonks-Girardeau gas, revealing unique correlation patterns and momentum distribution features linked to their quantum many-body nature.

Contribution

It provides a detailed analysis of the reduced single-particle density matrix and momentum distribution of dark soliton states, highlighting their distinctive correlation and oscillation behaviors.

Findings

Dark soliton states exhibit a characteristic momentum distribution with two sharp spikes.

High correlations between mirror points cause the spikes in the momentum distribution.

Correlations oscillate and do not decay with distance, unlike typical expectations.

Abstract

We study the reduced single-particle density matrix (RSPDM), the momentum distribution, natural orbitals and their occupancies, of dark "soliton" (DS) states in a Tonks-Girardeau gas. DS states are specially tailored excited many-body eigenstates, which have a dark solitonic notch in their single-particle density. The momentum distribution of DS states has a characteristic shape with two sharp spikes. We find that the two spikes arise due to the high degree of correlation observed within the RSPDM between the mirror points ($x$ and $-x$) with respect to the dark notch at $x=0$; the correlations oscillate rather than decay as the points $x$ and $-x$ are being separated.