Compare coherent exchange with cavity and atomic loss

In weak excitation, the atom’s polarization behaves linearly and couples to the cavity mean amplitude. A photon-number decay rate kappa gives cavity amplitude decay kappa divided by two. Atomic coherence decays at gamma two, equal to half Gamma only when there is no pure dephasing. These amplitude rates enter the response matrix. Using population rates directly in that matrix changes pole positions, linewidths, and any numerical criterion by factors of two.

At resonance the matrix has two eigenvalues. Their real parts set damping and imaginary parts set oscillation frequencies in the rotating frame. With no pure dephasing the frequency splitting is twice the square root of coupling squared minus one sixteenth the squared difference of population decay rates. A nonzero imaginary splitting is a mathematical property of these poles, but observing two spectral peaks also requires enough separation relative to widths and a suitable drive-and-readout channel.

Define single-atom cooperativity without pure dephasing as four times coupling squared divided by the product of cavity and atomic population decay rates. It compares a cavity-induced emission scale with competing atomic emission. In a bad cavity, kappa much larger than coupling allows the cavity amplitude to be eliminated, leaving an additional atomic population decay rate four coupling squared divided by kappa on resonance. This can be useful when coherent intracavity exchange is not resolved.

Strong coherent exchange instead needs coupling large enough relative to both cavity and polarization damping that several exchange cycles survive. It must still remain small compared with optical frequencies for the rotating-wave model. Large cooperativity alone does not guarantee this hierarchy: making atomic free-space decay extremely small can raise cooperativity while the cavity remains very lossy. Detuning, pure dephasing, spatial mismatch, and mode-dependent output coupling can further weaken the connection between one dimensionless figure and an observed spectrum.

A time-domain swap experiment tests reversible exchange before excitation disappears. A weak-probe spectrum tests transmission or reflection poles with widths and interference determined by the measurement channel. An emission-collection experiment tests branching and extraction efficiency. These are related but operationally different measurements. A convincing claim should report the actual rates, detuning, relevant dephasing, and the observed quantity rather than invoke strong coupling solely from a large normal-mode frequency or a high cooperativity value.

The oscillator lab illustrates competition between oscillation and damping in an ordinary analytic second-order system. It starts with unit displacement and zero velocity, unlike a general atom–cavity initial state, so it is not a quantitative model of the coupled response matrix. Use it only to distinguish a damped oscillatory regime from critical relaxation. Its stiffness stays positive, and changing a control recomputes the full history at the retained time rather than implementing a continuous change of cavity losses.