The Statistical Dynamic of the System, Consisting of the Classical and Quantum Subsystems

Luciano Colombo

Nikola Burić, Istok P. Mendaš, Duska B Popovic +2 more

General statistical ensembles in the Hamiltonian formulation of hybrid quantum-classical systems are analyzed. It is argued that arbitrary probability densities on the hybrid phase space must be considered as the class of possible physically distinguishable statistical ensembles of hybrid systems. Nevertheless, statistical operators associated with the hybrid system and with the quantum subsystem can be consistently defined. Dynamical equations for the statistical operators representing the mixed states of the hybrid system and its quantum subsystem are derived and analyzed. In particular, these equations irreducibly depend on the total probability density on the hybrid phase space.

Eugeni P. Gordov

И. В. Александров

Abstract For a system consisting of a quantum and a classical subsystem, an equation of the Liouville type is obtained starting with the density matrix equation. The validity condition and the transformation properties of this quantum-classical equation as well as the application of the adiabatic approximation and the perturbation theory within the frame of this equation are discussed. Some general expressions are obtained for the probability of the transition between the quasi-equilibrium states of the system. Special attention is given to the electron-nuclei system, as the general formalism was developed first of all in order to obtain a foundation for the theoretical analysis of elementary processes in liquids. An example is given to demonstrate that the quantum-classical approach in the range of its validity gives the same result as the corresponding quantum solution. It is also shown that both the cases of weak and strong interaction (leading to the transition) can be described by the quantum-classical equation in question.

V. I. Gerasimenko, D. Ya. Petrina

José Luis Alonso, Pierpaolo Bruscolini, Alberto Castro +3 more

We investigate decoherence, pointer states and the equilibrium distribution in the Ehrenfest Statistical Dynamics (ESD) model by considering ensembles of trajectories of simple but realistic molecular models, consisting of two classical cores and one quantum electron. The Ehrenfest model is sometimes discarded as a valid approximation to non-adiabatic coupled quantum-classical dynamics because it does not describe the decoherence in the quantum subsystem, which should exist due to its interaction with the classical subsystem. However, any rigorous statistical analysis of the Ehrenfest dynamics, such as the described ESD formalism described, proves that some decoherence-like effects, in particular purity decreasing, exist when ensembles of trajectories are considered. In this article, decoherence in ESD is studied by measuring the change in the quantum subsystem purity and by analysing the appearance of pointer states to which the system decoheres, which for our example are the eigenstates of the electronic Hamiltonian. Furthermore, following the formalism presented in previous works, the true equilibrium distribution for the ESD determined following the Balescu approach is used in the definition of potential energy surfaces dependent on the temperature and in the computation of the temperature dependence of the internuclear distance.