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Classical microwave circuit theory is incapable of representing somephenomena at the quantum level. To include quantum statistical effects whentreating microwave networks, various theoretical treatments can be employedsuch as quantum input-output network (QION) theory and SLH theory. However,these require a reformulation of classical microwave theory. To make thesetopics comprehensible to an electrical engineer, we demonstrate someunderpinnings of microwave quantum optics in terms of microwave engineering.For instance, we equate traveling-wave phasors in a transmission line ($V_0^+$)directly to bosonic field operators. Furthermore, we extend QION to include astate-space representation and a transfer function for a single port quantumnetwork. This serves as a case study to highlight how microwave methodologiescan be applied in open quantum systems. Although the same conclusion could befound from a full SLH theory treatment, our method was derived directly fromfirst principles of QION.

An engineer's brief introduction to microwave quantum optics and a single-port state-space representation