Mapping quantum-classical Liouville equation: Projectors and trajectories
Mapping quantum-classical Liouville equation: Projectors and trajectories
Publication Type:
Journal ArticleSource:
Journal of Chemical Physics, Volume 136, Number 8 (2012)URL:
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84857839392&doi=10.1063%2f1.3685420&partnerID=40&md5=78421e5619228626233f4805bef4cb84Keywords:
Dynamical instabilities, Dynamics, Evolution equations, Evolution operator, Liouville equation, Liouville operator, Mapping, Mapping formalism, Phase space methods, Phase spaces, Poisson brackets, Projection Operator, Quantum electronics, Quantum optics, Quantum state, Quantum system, Quantum-classical, Quantum-classical systems, Trajectories, Trajectory-basedAbstract:
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. © 2012 American Institute of Physics.
