Unifying Floquet Theory of Longitudinal and Dispersive Readout

We devise a Floquet theory of longitudinal and dispersive readout in circuit quantum electrodynamics (cQED). By studying qubits coupled to cavity photons and driven at the resonance frequency of the cavity ω_{r}, we establish a universal connection between the qubit ac Stark shift and the longitudinal and dispersive coupling to photons. We find that the longitudinal coupling g_{∥} is controlled by the slope of the ac Stark shift as function of the driving strength A_{q}, while the dispersive shift χ depends on its curvature. The two quantities become proportional to each other in the weak drive limit (A_{q}→0). Our approach unifies the adiabatic limit (ω_{r}→0)-where g_{∥} is generated by the static spectrum curvature (or quantum capacitance)-with the diabatic limit, where ω_{r} is large and the static spectrum plays no role. We derive analytical results supported by exact numerical simulations. We apply them to superconducting and spin-hybrid cQED systems, showcasing the flexibility of faster-than-dispersive longitudinal readout.