Bound-state formation and thermalization within the Lindblad approach

TL;DR

This paper investigates bound-state formation and thermalization in open quantum systems using the Lindblad approach, applying hydrodynamical reformulations to analyze dynamics and bound state properties in a thermal bath.

Contribution

It extends Lindblad dynamics to include bound state formation analysis with a hydrodynamical formulation, providing new insights into thermalization processes in open quantum systems.

Findings

Bound states can form in a Lindblad framework with a P"oschl-Teller potential.

Thermalization timescales for bound states are analyzed.

Multiple bound states can exist in deeper potentials, similar to quarkonia.

Abstract

The Lindblad equation, as one approach to open quantum systems, describes the density matrix of a particle or a chain of interacting particles, which are in contact with a thermal bath. Still, it is not fully understood yet, how arbitrary systems evolve towards a stationary distribution, which guarantees thermalization in a thermodynamical context, and how to systematically incorporate the variety of assumptions that are made in this approach in order to preserve thermal Gibbs states. Despite these shortcomings, Lindblad dynamics was successfully employed in heavy-ion physics (quarkonia) and also became of interest in quantum-computer applications. In this paper, we consider a problem borrowed from heavy-ion collisions, namely the formation of bound states, as for example the deuteron, in the non-relativistic regime by using the already well understood techniques of Lindblad dynamics. However, only recently, we were able to extend this toolbox by showing, that the position-space Lindblad equation can be reformulated in terms of a diffusion-advection equation with sources and therefore provides a hydrodynamical formulation of a dissipative quantum master equation. Making use of this advanced machinery and insights, we describe the possible formation of a bound state, which is realized by a P\"oschl-Teller-like potential, of a particle in interaction with a heat bath in a 1-dim setting. We analyse the possibility of a thermalization and the time-scale of the formation, population and depopulation of the bound state. Finally, we also show an example of a much deeper potential, where we allow for three bound states, just in the spirit of quarkonia. Besides this, we discuss general aspects of open quantum systems, like decoherence, entropy production etc.