Mapping quantum-classical Liouville equation: projectors and trajectories

TL;DR

This paper develops a phase space approach to quantum-classical dynamics using projection operators within the mapping formalism, introducing a trajectory-based solution and analyzing an approximate method with potential instabilities.

Contribution

It demonstrates that the mapping quantum-classical Liouville operator commutes with projection operators, confining dynamics to physical space, and constructs a trajectory-based solution for this evolution.

Findings

The exact dynamics are confined to physical space via projection operators.

An approximate evolution using only the Poisson bracket can be simulated with independent trajectories.

The approximate method may lead to dynamics outside the physical space and exhibit instabilities.

Abstract

The evolution of a mixed quantum-classical system is expressed in the mapping formalism where discrete quantum states are mapped onto oscillator states, resulting in a phase space description of the quantum degrees of freedom. By defining projection operators onto the mapping states corresponding to the physical quantum states, it is shown that the mapping quantum-classical Liouville operator commutes with the projection operator so that the dynamics is confined to the physical space. It is also shown that a trajectory-based solution of this equation can be constructed that requires the simulation of an ensemble of entangled trajectories. An approximation to this evolution equation which retains only the Poisson bracket contribution to the evolution operator does admit a solution in an ensemble of independent trajectories but it is shown that this operator does not commute with the projection operators and the dynamics may take the system outside the physical space. The dynamical instabilities, utility and domain of validity of this approximate dynamics are discussed. The effects are illustrated by simulations on several quantum systems.