Unifying Floquet theory of longitudinal and dispersive readout

We devise a Floquet theory of longitudinal and dispersive readout in circuit QED. By studying qubits coupled to cavity photons and driven at the resonance frequency of the cavity $\omega_{\rm r}$, 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 $g_\parallel$ is controlled by the slope of the AC Stark shift as function of the driving strength $A_{\rm q}$, while the dispersive shift $\chi$ depends on its curvature. The two quantities become proportional to each other in the weak drive limit ($A_{\rm q}\rightarrow 0$). Our approach unifies the adiabatic limit ($\omega_{\rm r}\rightarrow 0$) -- where $g_\parallel$ is generated by the static spectrum curvature (or quantum capacitance) -- with the diabatic one, where 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.