Koopman wavefunctions and classical states in hybrid quantum-classical dynamics

TL;DR

This paper develops a novel hybrid quantum-classical dynamics model using wavefunctions that maintains positivity and consistency, combining Koopman and van Hove methods, with a special focus on two-level quantum systems.

Contribution

It introduces a new closure model for quantum-classical dynamics that preserves positivity and consistency, utilizing a variational wavefunction approach and Poisson reduction.

Findings

The model retains positivity of quantum and classical states.

A noncanonical Poisson structure for hybrid dynamics is derived.

Application to quantum two-level systems demonstrates the model's utility.

Abstract

We deal with the reversible dynamics of coupled quantum and classical systems. Based on a recent proposal by the authors, we exploit the theory of hybrid quantum-classical wavefunctions to devise a closure model for the coupled dynamics in which both the quantum density matrix and the classical Liouville distribution retain their initial positive sign. In this way, the evolution allows identifying a classical and a quantum state in interaction at all times, thereby addressing a series of stringent consistency requirements. After combining Koopman's Hilbert-space method in classical mechanics with van Hove's unitary representations in prequantum theory, the closure model is made available by the variational structure underlying a suitable wavefunction factorization. Also, we use Poisson reduction by symmetry to show that the hybrid model possesses a noncanonical Poisson structure that does not seem to have appeared before. As an example, this structure is specialized to the case of quantum two-level systems.