Input-output theory for superconducting and photonic circuits that contain weak retroreflections and other weak pseudocavities

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 retroreflections 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 our method is essential. We obtain a full solution for the optimal receiver given that the sender employs a simple turn-on–turn-off procedure. This solution shows that near-perfect transmission is possible even with significant retroreflections.

Figure 1

Two examples of superconducting circuits (networks) that contain weak loops, along with their corresponding reduced (loop-free) input-output (IO) models. The traveling-wave fields that are internal to the networks, and that are thus eliminated in obtaining the input-output descriptions, are denoted by solid gray arrows. The fields that are the external inputs and outputs to the networks are denoted by solid black arrows. The outgoing emissions from the systems are depicted by dashed arrows. (a) A superconducting qubit capacitively coupled to a terminated planar waveguide, forming a quasi-1D input output system. (b) The effective IO model for (a) derived by eliminating the internal field modes (gray arrows). (c) Two of the qubit IO models in (b) connected together via imperfect circulators. (d) The reduced two-qubit IO model for (c) obtained by eliminating all the internal fields.

Figure 2

A two-qubit “crossed waveguide” network. (a) A schematic depicting two superconducting circuits weakly coupled to a pair of crossed waveguides, with a general unitary scattering relation. (b) A block diagram of all field operators. Input field operators $b_j^{\text{in}}$ are scattered to the output fields $b_k^{\text{out}}$ via the local elements of $\mathbf{S}$, e.g., ${\mathbf{S}}_J$, and irrespective of the waveguides. The internal inputs are connected to the internal outputs by the connection matrix elements $w_{kl}$ (gray).

Figure 3

An empty cavity. A section of waveguide of length $z_3-z_2=l$ is capped by two partially reflecting boundary conditions: ${\mathbf{S}}_{\ell }$ and ${\mathbf{S}}_r$.

Figure 4

Dark state evolution. Numerical simulations for three different network parameters; see text. (a) $\left|\uparrow \downarrow \right.〉\mapsto \left|\uparrow \downarrow \right.〉$ transfer probability vs time. (b) Receiver coupling rate ${\kappa }_b$, relative to a fixed ${\kappa }_a={\kappa }_0$ in dB. (c) Bloch vector trajectories, $\vec{\mathsf{b}}\left(t\right)$, shown in the $x\text{−}z$ plane of the Bloch ball.