Jaynes-Cummings Model: Collapse and Revival

An interactive view of the resonant Jaynes-Cummings model: a two-level atom, initially excited, coupled to one quantised cavity mode in the rotating-wave approximation at exact resonance. Each photon-number doublet $\{|e,n\rangle, |g,n+1\rangle\}$ oscillates at the quantum Rabi frequency $\Omega_n = 2 g \sqrt{n+1}$, so for a field with photon distribution $P(n)$ the atomic inversion is the exact closed-form sum $W(t) = \sum_n P(n) \cos(2 g t \sqrt{n+1})$ with $P_e + P_g = 1$ by construction. For a coherent state the Poissonian spread of Rabi frequencies dephases the oscillation: it collapses on $t_c \sim \sqrt{2}/g$ (set by the frequency spread $\sim g$, hence independent of the mean photon number $\bar{n}$) and then, because the frequencies are discrete, rephases into a revival near $t_r = 2\pi\sqrt{\bar{n}}/g$. The playground draws the full analytic $W(t)$ over the window faint, with a bright sweep and playhead revealing it in time, alongside the Poissonian $P(n)$ and the coherent-field Wigner blob in phase space. The collapse time is set by the spread of Rabi frequencies (of order $g$, hence independent of the mean photon number) and the revival by their discreteness, recurring near $t_r = 2\pi\sqrt{\bar{n}}/g$; this is the exact quantum result of the model, a direct signature of field quantisation.