Quantum-classical hybrid dynamics: coupling mechanisms and diffusive approximation

TL;DR

This paper classifies all Markovian quantum-classical hybrid evolutions into four coupling types, analyzes their diffusive limits, and identifies conditions for maintaining positivity, contributing a comprehensive framework for hybrid dynamics.

Contribution

It provides a complete characterization of coupling mechanisms in quantum-classical hybrid systems and derives conditions for diffusive approximations ensuring positivity.

Findings

Four basic coupling mechanisms identified

Diffusive limits characterized for each mechanism

Conditions for positivity in hybrid dynamics established

Abstract

In this paper we demonstrate that any Markovian master equation defining a completely positive evolution for a quantum-classical hybrid state can always be written in terms of four basic coupling mechanisms. Each of them is characterized by a different "backaction" on each subsystem. On this basis, for each case, we find the conditions under which a diffusive limit is approached, that is, the time evolution can be approximated in terms of the first and second derivatives of the hybrid state with respect to a classical coordinate. In this limit, the restricted class of evolutions that guaranty the positivity of the hybrid state at all times (quantum Fokker-Planck master equations) emerges when the coupling mechanisms lead to infinitesimal (non-finite) changes in both the quantum and classical subsystems. A broader class of diffusive evolutions is obtained when positivity is only granted after a transient time or alternatively is granted after imposing an initial finite width on the state of the classical subsystem. A set of representative examples support these results.