Quantum dark solitons in the 1D Bose gas: From single to double dark-solitons

TL;DR

This paper constructs and analyzes quantum double dark-soliton states in the 1D Bose gas, demonstrating their close resemblance to classical solutions and proposing effective mean-field approximations for these quantum states.

Contribution

It introduces novel quantum states for single and double dark solitons in the Lieb-Liniger model and shows their agreement with classical solutions in the weak coupling regime.

Findings

Quantum states of single dark solitons are explicitly constructed and analyzed.

Density and phase profiles of double dark-solitons match classical solutions in weak coupling.

Mean-field product states with Gaussian weights effectively approximate quantum double dark-solitons.

Abstract

We study quantum double dark-solitons by constructing corresponding quantum states in the Lieb-Liniger model for the one-dimensional Bose gas. Here we expect that the Gross-Pitaevskii (GP) equation should play a central role in the long distance mean-field behavior of the 1D Bose gas. We first introduce novel quantum states of a single dark soliton with a nonzero winding number. We show them by exactly evaluating not only the density profile but also the profiles of the square amplitude and phase of the matrix element of the field operator between the $N$-particle and $(N-1)$-particle states. For elliptic double dark-solitons, the density and phase profiles of the corresponding states almost perfectly agree with those of the classical solutions, respectively, in the weak coupling regime. We then show that the scheme of the mean-field product state is quite effective for the quantum states of double dark solitons. Assigning the ideal Gaussian weights to a sum of the excited states with two particle-hole excitations we obtain double dark-solitons of distinct narrow notches with different depths. We suggest that the mean-field product state should be well approximated by the ideal Gaussian weighted sum of the low excited states with a pair of particle-hole excitations. The results of double dark-solitons should be fundamental and useful for constructing quantum multiple dark-solitons.