Emergent quantum and tunneling capacitances from multi-timescale quantum dynamics

Physical systems often involve multiple characteristic timescales, making their theoretical description challenging. A quantum system under a drive-and-probe setup naturally evolves on three distinct timescales: the intrinsic timescale of the quantum system, the intermediate timescale of Rabi oscillations induced by the drive, and the slowest timescale associated with rf-probing via a coupled classical resonator. This combination forms a multi-timescale quantum–classical problem that cannot be treated with conventional perturbation theory or the rotating-wave approximation [1].

A consistent analytical description remains an open problem, although the system’s effective response can often be phenomenologically captured by a single measurable quantity—the effective capacitance [2,3]. Here, we apply renormalized perturbation theory known as the Quantum Averaging Theory [1] to systematically separate dynamics on all relevant timescales. This allows us to rigorously derive the effective capacitance as the slowest-timescale manifestation while also predicting modifications to the quantum dynamics at faster timescales. We also discuss whether this framework can be extended to describe open quantum systems, exploring its potential applicability to a broader class of hybrid quantum–classical devices.