Interacting Stochastic Schrödinger Equation

Lu Zhang, Caishi Wang () and Jinshu Chen
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Lu Zhang: School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
Caishi Wang: School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
Jinshu Chen: School of Science, Lanzhou University of Technology, Lanzhou 730050, China

Mathematics, 2023, vol. 11, issue 6, 1-16

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 I K ⊗ N , where I K means the identity operator on K and N is the number operator in h , and give a representation of I K ⊗ N in terms of operators I K ⊗ ∂ 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.

Keywords: quantum Bernoulli noises; stochastic Schrödinger equation; open quantum system (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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