Interacting Stochastic Schrödinger Equation

Abstract

Being the annihilation and creation operators on the space h of square integrable Bernoulli functionals, quantum Bernoulli noises (QBN) satisfy the canonical anti-commutation relation (CAR) in equal time. Let K be the Hilbert space of an open quantum system interacting with QBN (the environment). Then K⊗h just describes the coupled quantum system. In this paper, we introduce and investigate an interacting stochastic Schrödinger equation (SSE) in the framework K⊗h, which might play a role in describing the evolution of the open quantum system interacting with QBN (the environment). We first prove some technical propositions about operators in K⊗h. In particular, we obtain the spectral decomposition of the tensor operator IK⊗N, where IK means the identity operator on K and N is the number operator in h, and give a representation of IK⊗N in terms of operators IK⊗∂k*∂k, k≥0, where ∂k and ∂k* are the annihilation and creation operators on h, respectively. Based on these technical propositions as well as Mora and Rebolledo’s results on a general SSE, we show that under some mild conditions, our interacting SSE has a unique solution admitting some regularity properties. Some other results are also proven.