Equilibration in one-dimensional quantum hydrodynamic systems

TL;DR

This paper investigates how one-dimensional quantum hydrodynamic systems retain or lose initial information over time, highlighting the roles of ballistic excitations, dispersion, and interactions in the equilibration process.

Contribution

It demonstrates the conditions under which quantum hydrodynamic systems preserve memory of initial correlations and how dispersion and interactions influence this behavior, including the potential restoration of localization.

Findings

Memory of initial correlations persists in the Luttinger model due to ballistic dynamics.

Dispersion and interactions suppress non-Gaussian correlations, leading to Gaussian steady states.

Localization and soliton formation are suggested when dispersion and interactions are combined.

Abstract

We study quench dynamics and equilibration in one-dimensional quantum hydrodynamics, which provides effective descriptions of the density and velocity fields in gapless quantum gases. We show that the information content of the large time steady state is inherently connected to the presence of ballistically moving localised excitations. When such excitations are present, the system retains memory of initial correlations up to infinite times, thus evading decoherence. We demonstrate this connection in the context of the Luttinger model, the simplest quantum hydrodynamic model, and in the quantum KdV equation. In the standard Luttinger model, memory of all initial correlations is preserved throughout the time evolution up to infinitely large times, as a result of the purely ballistic dynamics. However nonlinear dispersion or interactions, when separately present, lead to spreading and delocalisation that suppress the above effect by eliminating the memory of non-Gaussian correlations. We show that, for any initial state that satisfies sufficient clustering of correlations, the steady state is Gaussian in terms of the bosonised or fermionised fields in the dispersive or interacting case respectively. On the other hand, when dispersion and interaction are simultaneously present, a semiclassical approximation suggests that localisation is restored as the two effects compensate each other and solitary waves are formed. Solitary waves, or simply solitons, are experimentally observed in quantum gases and theoretically predicted based on semiclassical approaches, but the question of their stability at the quantum level remains to a large extent an open problem. We give a general overview on the subject and discuss the relevance of our findings to general out of equilibrium problems.