Resolve the coupled atom–cavity doublet

The electric dipole interaction couples the atomic dipole to the mode field at the atom’s position. A two-level approximation retains one selected transition and a single-mode approximation excludes other cavity modes. The coupling includes the transition dipole projected onto local polarization and the normalized mode amplitude. It is therefore position and orientation dependent. A small cavity mode volume can increase the coupling, but geometry alone does not guarantee strong coupling once losses and unwanted transitions are included.

Before the rotating-wave approximation, the interaction contains terms that simultaneously create an atomic and field excitation or remove both. Near resonance, when g and relevant detunings and decay scales are much smaller than optical frequencies, these terms rotate rapidly and can be neglected. The retained terms exchange one excitation between atom and field. This approximation fails in ultrastrong coupling and can fail for unsuitable drive or spectral conditions; the Jaynes–Cummings Hamiltonian is not an exact arbitrary-coupling light–matter theory.

The retained interaction conserves cavity photon number plus excited atomic occupation. For an initially excited atom and n photons, the motion stays within the pair of states consisting of that initial state and a ground-state atom with n plus one photons. The ladder matrix element supplies square root of n plus one. Diagonalizing this two-dimensional block gives two dressed states separated by the generalized angular frequency. The vacuum block remains coupled because the square root is one when n is zero.

On resonance the two dressed states are equal-weight superpositions, with energies shifted oppositely by g times square root of n plus one. Their relative phase determines the oscillation between bare states. Off resonance, the energy mismatch changes the mixing angle and prevents complete transfer from the initial excited atom. A spectral doublet and a time-domain population oscillation describe related physics but use different observables, and their interpretation in a dissipative experiment also requires linewidth information.

Starting in the excited bare state, the ground-atom plus extra-photon probability is a sine squared multiplied by a detuning-dependent amplitude. The generalized frequency alone does not determine how much population transfers. At large detuning oscillations become faster but their transfer amplitude becomes small. On resonance the first full swap occurs when g times square root of n plus one times time equals pi over two. Vacuum exchange is thus a finite, coherent process.

A coherent cavity state contains many n blocks with different square-root frequencies, so their population contributions can dephase and later revive in the ideal model. A single two-state geometric lab cannot reproduce that multi-block evolution. It can display equal-weight dressed superpositions within one declared excitation block and show their phase distinction. Its playback is fixed relative-phase precession, not a tunable Jaynes–Cummings exchange, and it contains neither detuning-dependent dynamics nor cavity decay.