Reflections on quantum Reflectometry: Quantum and tunneling capacitances as well as sisyphus and hermes resistances

When a quantum electronic device is coupled to an electrical resonator, admittance changes of the quantum subsystem may be detected. The effective reactance may include capacitive and inductive terms that incorporate geometric, quantum, and tunneling components; while the effective resistance may be composed of Sisyphus and Hermes terms linked to relaxation and decoherence, respectively. Such reflectometry is usually studied when all characteristic times of the quantum system are much shorter than the resonator's period, in which case only stationary quantum states are probed. We present a rigorous description of a driven-dissipative qudit-resonator system. Our approach demonstrates how to strictly introduce quantum and tunneling capacitances as well as Hermes and Sisyphus resistances, and how these values are modified when the dynamics of the subsystems becomes mutually dependent. We present the cases of a Cooper-pair box, a single-Cooper-pair transistor, a double quantum dot, and a single-electron box. Our approach can be applied to describe any quantum system coupled to any classical resonator.