Factorization of Linear Quantum Systems with Delayed Feedback

TL;DR

This paper develops a method to factorize transfer functions of linear quantum systems with delayed feedback into products of realizable components, extending previous cascade realizations to non-Markovian systems with delays.

Contribution

It introduces a factorization approach for non-Markovian quantum transfer functions with delays, generalizing past cascade realizations to include infinite products.

Findings

The transfer functions can be factorized into physically realizable components.

Under certain conditions, the product converges and can be approximated.

The approach extends cascade realizations to systems with delays and non-Markovian dynamics.

Abstract

We consider the transfer functions describing the input-output relation for a class of linear open quantum systems involving feedback with nonzero time delays. We show how such transfer functions can be factorized into a product of terms which are transfer functions of canonical physically realizable components. We prove under certain conditions that this product converges, and can be approximated on compact sets. Thus our factorization can be interpreted as a (possibly infinite) cascade. Our result extends past work where linear open quantum systems with a state-space realization have been shown to have a pure cascade realization [Nurdin, H. I., Grivopoulos, S., & Petersen, I. R. (2016). The transfer function of generic linear quantum stochastic systems has a pure cascade realization. Automatica, 69, 324-333.]. The functions we consider are inherently non-Markovian, which is why in our case the resulting product may have infinitely many terms.