An Example of Explicit Generators of Local and Nonlocal Quantum Master Equations

A simple example of completely positive dynamics is considered for which both the generator of a nonlocal integro-differential equation leading to such dynamics and the time-local generator can be calculated explicitly. It is demonstrated that introducing a small parameter in this simple example allows one to reproduce some nonperturbative phenomena that occur in more realistic models. In addition, a special case of fermionic dynamics is considered, and it is shown that one can actually find families of moments whose dynamics is linear but satisfies non-Markovian equations.