Collective photon echoes in the Tavis-Cummings model: distribution independence, two detuning regimes, and the Dicke ladder
An ensemble of $N$ two-level molecules prepared in its ground state and sharing a lossless cavity with a weak field does not simply absorb: the intensity collapses and then recurs, in a train of collective photon echoes. Working from the exact solution of the Tavis-Cummings model in the few-photon regime $\bar n<N$, we confirm the echo time $\tau_E=4\pi\sqrt{N+\Delta}/g$ numerically at resonance from $N=5$ to $400$, the small-$N$ end discriminating this form from the alternative $4\pi\sqrt{N-\bar n+\Delta}/g$ in its favor. The initial state requires no preparation, being the ground state. The echo time is independent of the initial photon distribution: coherent, thermal, squeezed and oscillatory-squeezed distributions spanning variances from 2 to 34 all recur together, as does a controlled pair with identical mean and variance differing only in the shape of $\rho_{nn}$. The echo amplitude is not: it varies at the tens-of-percent level at fixed mean, including a factor of 1.8 with the squeezing phase at fixed squeezing strength. Detuning organizes the dynamics into two clean regimes separated by a fragmented crossover, the dispersive-branch echo time approaching one-half the resonant one, and sufficient detuning removes the dependence on the initial Dicke state. For arbitrary initial Dicke state, emission replaces absorption at $m\simeq-N/2+\bar n$, and the echo envelope acquires one component per step up the ladder: a single-component echo occurs only at the ground state. A feasibility analysis against a five-qubit superconducting device, with Lindblad simulations of cavity decay and dephasing and full-Hilbert-space disorder simulations, shows the first echoes observable at $N\sim5$-$20$ on existing hardware: the echo survives the dominant loss channel with contrast $\sim e^{-\kappa\tau_E/2}$, photons being shielded from cavity decay while resident in the emitters.