Analysis of the quantum-classical Liouville equation in the mapping basis

TL;DR

This paper analyzes the quantum-classical Liouville equation in the mapping basis, identifying a complex term as back reaction, and proposes a simplified, trajectory-based approximation that effectively captures key dynamics in nonadiabatic systems.

Contribution

It introduces a simplified approximation of the quantum-classical Liouville equation in the mapping basis by neglecting the excess coupling term, enabling easier simulation of quantum dynamics.

Findings

Neglecting the excess coupling term still yields accurate expectation values.

The simplified approach effectively models nonadiabatic population and coherence dynamics.

Trajectory-based simulations are feasible with the approximate equation.

Abstract

The quantum-classical Liouville equation provides a description of the dynamics of a quantum subsystem coupled to a classical environment. Representing this equation in the mapping basis leads to a continuous description of discrete quantum states of the subsystem and may provide an alternate route to the construction of simulation schemes. In the mapping basis the quantum-classical Liouville equation consists of a Poisson bracket contribution and a more complex term. By transforming the evolution equation, term-by-term, back to the subsystem basis, the complex term (excess coupling term) is identified as being due to a fraction of the back reaction of the quantum subsystem on its environment. A simple approximation to quantum-classical Liouville dynamics in the mapping basis is obtained by retaining only the Poisson bracket contribution. This approximate mapping form of the quantum-classical Liouville equation can be simulated easily by Newtonian trajectories. We provide an analysis of the effects of neglecting the presence of the excess coupling term on the expectation values of various types of observables. Calculations are carried out on nonadiabatic population and quantum coherence dynamics for curve crossing models. For these observables, the effects of the excess coupling term enter indirectly in the computation and good estimates are obtained with the simplified propagation.