Quantum feedback networks with uniform time delays

Quantum feedback networks with non-negligible propagation delays in the interconnecting edges have recently gained attention. Grimsmo has analysed the evolution of a Markov quantum system coupled to a single field via a doublepass interaction with a time delay and derived the evolution equation for the state (density operator) of the system (after tracing out the field) using a super-operator notion of partial tracing together with a tensor network representation. However, for the purposes of feedback control (both measurement-based and coherent feedback), the propagating output fields of the system can be used to facilitate feedback interconnections. Thus it is desirable to have an equivalent input-output description of quantum systems that incorporates time delays, without tracing out the field. Here, we first extend the analysis to the setting of a single system coupled to optical fields in multiple channels with a feedback structure in which the fields makes infinite repeated interactions with the system. Then we generalise this to the setting of quantum feedback networks with both instantaneous interconnections and uniform time delays in the interconnection