Statistical mechanics of one-dimensional quantum droplets

TL;DR

This paper investigates the statistical mechanics and dynamical relaxation of one-dimensional quantum droplets using a modified Gross-Pitaevskii equation, employing the transfer integral operator technique to analyze their emergent behavior and equilibrium properties.

Contribution

It introduces a semi-analytical transfer integral operator approach to study quantum droplet dynamics and compares it with Langevin dynamics, revealing insights into droplet formation and coalescence.

Findings

Quantum droplets form spontaneously due to modulational instability.

Droplets tend to collide and coalesce over time.

The TIO and Langevin methods agree well except at low temperatures with wide droplets.

Abstract

We study the statistical mechanics and the dynamical relaxation process of modulationally unstable one-dimensional quantum droplets described by a modified Gross-Pitaevskii equation. To determine the classical partition function thereof, we leverage the semi-analytical transfer integral operator (TIO) technique. The latter predicts a distribution of the observed wave function amplitudes and yields two-point correlation functions providing insights into the emergent dynamics involving quantum droplets. We compare the ensuing TIO results with the probability distributions obtained at large times of the modulationally unstable dynamics as well as with the equilibrium properties of a suitably constructed Langevin dynamics. We find that the instability leads to the spontaneous formation of quantum droplets featuring multiple collisions and by which are found to coalesce at large evolution times. Our results from the distinct methodologies are in good agreement aside from the case of low temperatures in the special limit where the droplet widens. In this limit, the distribution acquires a pronounced bimodal character, exhibiting a deviation between the TIO solution and the Langevin dynamics.