From Repeated to Continuous Quantum Interactions

Abstract.We consider the general physical situation of a quantum system $$\mathcal{H}_{0} $$ interacting with a chain of exterior systems $$ \otimes _{{\mathbb{N}^{ * } }} \mathcal{H},$$ one after the other, during a small interval of time h and following some Hamiltonian H on $$\mathcal{H}_{0} \otimes \mathcal{H}.$$ We discuss the passage to the limit to continuous interactions (h → 0) in a setup which allows to compute the limit of this Hamiltonian evolution in a single state space: a continuous field of exterior systems $$ \otimes _{{\mathbb{R}^{ + } }} \mathcal{H}.$$ Surprisingly, the passage to the limit shows the necessity for three different time scales in H. The limit evolution equation is shown to spontaneously produce quantum noises terms: we obtain a quantum Langevin equation as limit of the Hamiltonian evolution. For the very first time, these quantum Langevin equations are obtained as the effective limit from repeated to continuous interactions and not only as a model. These results justify the usual quantum Langevin equations considered in continual quantum measurement or in quantum optics. Physically, the typical Hamiltonian allowing this passage to the limit shows up three different parts which correspond to the free evolution, to an analogue of a weak coupling limit part and to a scattering interaction part. We apply these results to give an Hamiltonian description of the von Neumann measurements. We also consider the approximation of continuous time quantum master equations by discrete time ones; in particular we show how any Lindblad generator is naturally obtained as the limit of completely positive maps.Communicated by Christian Gérard