Input-output theory for superconducting and photonic circuits that contain weak retro-reflections and other weak pseudo-cavities

Input-output theory is invaluable for treating superconducting and photonic circuits connected by transmission lines or waveguides. However, this theory cannot in general handle situations in which retro-reflections from circuit components or configurations of beam-splitters create loops for the traveling-wave fields that connect the systems. Here, building upon the network-contraction theory of Gough and James [Commun. Math. Phys. 287, 1109 (2009)], we provide a compact and powerful method to treat any circuit that contains such loops so long as the effective cavities formed by the loops are sufficiently weak. Essentially all present-day on-chip superconducting and photonic circuits will satisfy this weakness condition so long as the reflectors that form the loops are not especially highly reflecting. As an example we analyze the problem of transmitting entanglement between two qubits connected by a transmission line with imperfect circulators, a problem for which the new method is essential. We obtain a full solution for the optimal receiver given that the sender employs a simple turn on/turn off. This solution shows that near-perfect transmission is possible even with significant retro-reflections.