Floquet-Nambu theory of electron quantum optics with superconductors

We present a comprehensive Floquet-Nambu theory to describe the time-dependent quantum transport in mesoscopic circuits involving superconductors. The central object of our framework is the first-order correlation function, which accounts for the excitations that are generated by a time-dependent voltage and their coherent scattering off the interface with a superconductor. We analyze the time-dependent current generated by periodic voltage pulses and how it depends on the excitation energies of the voltage drive compared to the gap of the superconductor. Our general formalism allows us to identify the conditions for the excitations that are scattered off the superconductor to become coherent electron-hole superpositions. To this end, we consider the purity of the outgoing states, which characterizes their ability to carry quantum information. To illustrate our formalism, we apply it to a system composed of chiral quantum Hall edge states connected to a superconductor, and we calculate the current in the outgoing lead and the purity of the outgoing states for Lorentzian and harmonic voltage drives. Our framework paves the way for systematic investigations of time-dependent scattering problems involving superconductivity, and it may help interpret future experiments in electron quantum optics with superconductors.