Theory of interactions between cavity photons induced by a mesoscopic circuit

We use a quantum path-integral approach to describe the behavior of a microwave cavity coupled to a dissipative mesoscopic circuit. We integrate out the mesoscopic electronic degrees of freedom to obtain a cavity effective action at fourth order in the light/matter coupling. By studying the structure of this action, we establish sufficient conditions in which the cavity dynamics can be described with a Lindblad equation. This equation depends on effective parameters set by electronic correlation functions. It reveals that the mesoscopic circuit induces an effective Kerr interaction and two-photon dissipative processes. We use our method to study the effective dynamics of a cavity coupled to a double quantum dot with normal metal reservoirs. If the cavity is driven at twice its frequency, the double-dot circuit generates photonic squeezing and nonclassicalities visible in the cavity Wigner function. In particular, we find a counterintuitive situation where mesoscopic dissipation enables the production of photonic Schr\"odinger cats. These effects can occur for realistic circuit parameters. Our method can be generalized straightforwardly to more complex circuit geometries with, for instance, multiple quantum dots and other types of fermionic reservoirs such as superconductors and ferromagnets.