Dynamics of quantum double dark-solitons and an exact finite-size scaling of Bose-Einstein condensation

TL;DR

This paper investigates the non-equilibrium dynamics of quantum double dark-soliton states in a 1D Bose gas, deriving exact finite-size scaling of BEC fraction and demonstrating soliton scattering using Bethe ansatz.

Contribution

It provides the first exact analysis of quantum double dark-soliton dynamics, including finite-size BEC scaling and soliton scattering in the Lieb-Liniger model.

Findings

Exact time evolution of density profiles shown

Macroscopic wave-function approximated from matrix elements

Exact demonstration of soliton notch scattering

Abstract

We show several novel aspects in the exact non-equilibrium dynamics of quantum double dark-soliton states in the Lieb-Liniger model for the one-dimensional Bose gas with repulsive interactions. We also show an exact finite-size scaling of the fraction of the Bose-Einstein condensation (BEC) in the ground state, which should characterize the quasi-BEC in quantum double dark-soliton states that we assume to occur in the weak coupling regime. First, we show the exact time evolution of the density profile in the quantum state associated with a quantum double dark-soliton by the Bethe ansatz. Secondly, we derive a kind of macroscopic quantum wave-function effectively by exactly evaluating the square amplitude and phase profiles of the matrix element of the field operator between the quantum double dark-soliton states. The profiles are close to those of dark-solitons particularly in the weak-coupling regime. Then, the scattering of two notches in the quantum double dark-soliton state is exactly demonstrated. It is suggested from the above observations that the quasi-BEC should play a significant role in the dynamics of quantum double dark-soliton states. If the condensate fraction is close to 1, the quantum state should be well approximated by the quasi-BEC state where the mean-field picture is valid.