The Physical Spectrum of a Driven Jaynes-Cummings Model

We investigate the time-dependent physical spectrum of the driven Jaynes-Cummings model, where both the atom and the quantized field are simultaneously driven by an external classical field. By leveraging the mapping of the time-dependent Hamiltonian onto the standard stationary Jaynes-Cummings form via unitary transformations, we determine the exact two-time correlation functions for both the atomic and field subsystems. These are then employed to compute the time-dependent physical spectrum using the Eberly-Wódkiewicz formalism. Our results demonstrate that the atomic spectral features are significantly reshaped by the external driving, exhibiting tunable asymmetries and shifts. Notably, we find that the driving parameters can be tuned to exactly cancel the initial coherent field amplitude, leading to an effective vacuum limit that recovers the fundamental vacuum Rabi splitting. This provides a clear interpretation of the emission dynamics in terms of the coherent displacement of the cavity field induced by the external drive.