  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.
