The Series Product for Gaussian Quantum Input Processes

We present the theory for connecting quantum Markov components into a network with quantum input processes in a Gaussian state (including thermal and squeezed), not necessarily vacuum fields.One would expect on physical grounds that the connection rules should be independent of the state of the input to the network. To compute statistical properties, we use a version of Wicks' Theorem involving fictitious vacuum fields (Fock space based representation of the fields) and while this aids computation, and gives a rigorous formulation, the various representations need not be unitarily equivalent. In particular, a naive application of the connection rules would lead to the wrong answer. We establish the correct interconnection rules, and show that while the quantum stochastic differential equations of motion display explicitly the covariances (thermal and squeezing parameters) of the Gaussian input fields We introduce the Wick-Stratonovich form which leads to a way of writing these equations that does not depend on these covariances and so corresponds to the universal equations written in terms of formal quantum input processes. We show that a wholly consistent theory of quantum open systems in series can be developed in this way, and as required physically, is universal and in particular representation-free.