Two-port quantum model of finite-length transmission lines coupled to lumped circuits

We show that, in the framework of quantum circuit electrodynamics in the Heisenberg picture, a finite-length transmission line can be described as a two-port lumped element of Th\'evenin type. Each port consists of a resistor connected in series to a controlled voltage source. The resistance of the resistors is equal to the characteristic impedance of the line. The controlled voltage sources are governed by linear equations with delay that take into account the reflections at the line ends. We apply this model to a transmission line capacitively coupled to two lumped circuits and obtain the reduced system of Heisenberg equations that governs them. Then, we show these equations can be reformulated as a pair of quantum Langevin-like equations that are coupled through the controlled voltage sources. Finally, we apply our approach to an analytically solvable network. This approach may be useful for the modeling of quantum links between superconducting circuits.