Hybrid quantum-classical systems: Quasi-free Markovian dynamics

Understand

In the case of a quantum-classical hybrid system with a finite number of degrees of freedom, the problem of characterizing the most general dynamical semigroup is solved, under the restriction of being quasi-free.

• This is a generalization of a Gaussian dynamics, and it is defined by the property of sending (hybrid) Weyl operators into Weyl operators in the Heisenberg description.
• The result is a quantum generalization of the L\'evy-Khintchine formula; Gaussian and jump contributions are included.
• As a byproduct, the most general quasi-free quantum-dynamical semigroup is obtained; on the classical side the Liouville equation and the Kolmogorov-Fokker-Planck equation are included.