  Classical microwave circuit theory is incapable of representing some
phenomena at the quantum level. To include quantum statistical effects when
treating microwave networks, various theoretical treatments can be employed
such as quantum input-output network (QION) theory and SLH theory. However,
these require a reformulation of classical microwave theory. To make these
topics comprehensible to an electrical engineer, we demonstrate some
underpinnings 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 a
state-space representation and a transfer function for a single port quantum
network. This serves as a case study to highlight how microwave methodologies
can be applied in open quantum systems. Although the same conclusion could be
found from a full SLH theory treatment, our method was derived directly from
first principles of QION.

Quantum Physics;Mesoscale and Nanoscale Physics;Superconductivity