Two-photon exchange interaction from Tavis-Cummings Hamiltonian under parametric modulation

We consider the interaction of a single-mode cavity field and $N$ identical qubits in the scenario when one or several system parameters undergo a weak external modulation. It is shown that in the dispersive regime of light-matter coupling one can engineer the effective two-photon exchange interaction when the frequency of harmonic modulation is $\sim 2|\Delta_-|$, where $\Delta_-$ is the bare atom-field detuning. The effective Hamiltonian and associated transition rate are derived analytically for $N\gg 1$, and a good agreement is found with numerical data even when $N \sim {\it O}(1)$.