The Quantum Stochastic Limit for Non-Markovian Driving Processes

Abstract

We express the stochastic limit (weak-coupling limit) of a quantum system interacting with its environment as the limit of solutions to a parametrized family of quantum stochastic differential equations (QSDEs). We prove the convergence of such solutions in the case where the driving processes may be non-Markovian and the system may be infinite-dimensional with unbounded Hamiltonian. We achieve this by imposing a particular condition on the n-point functions of the driving process which we verify for annihilation, creation, and gauge operators on a bosonic Fock space with sufficient mixing. Additionally, we include not only the usual fast-evolving system Hamiltonian which is averaged out in the limit, but also a slow-evolving system Hamiltonian which survives the limit. We also relax the requirement for the fast-evolving Hamiltonian to have a purely discrete spectrum allowing accumulation points under some circumstances. Moreover, our convergence mode for the processes is convergence of all time-centered n-point functions which is stronger than the typical semigroup convergence for the stochastic limit.