Quantum reflectometry:effective capacitance of two- and multi-level systems

When a classical resonator is coupled to a quantum system, its capacitance, inductance, and resistance are modified [1–3]. The corresponding terms include quantum and tunneling contributions. By probing the resonator's response, one can access the state of the quantum system. Such reflectometry is usually studied in situations where 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 explore how both isolated and open two- and multi-level systems interact with classical electric circuits and how they can be effectively represented by an impedance that is directly measurable. Our approach demonstrates how to rigorously define quantum and tunneling capacitances. To determine the effective capacitance, we quantize the qubit/qudit-resonator system and apply perturbation theory. We show that the quantum capacitance originates from the curvature of the energy levels, while the tunneling capacitance appears when relaxation, temperature or resonant microwave driving is taken into consideration. Our theoretical framework can be applied to describe any quantum system coupled to any classical resonator [4].