Derive decay into a continuum of vacuum modes

Begin with the atom excited and the electromagnetic field in vacuum. In the one-excitation sector the joint state contains a survival amplitude for excited atom plus vacuum and an amplitude for each ground-atom one-photon mode. The rotating-wave dipole interaction couples these components. No classical incident amplitude is needed: photon creation has a nonzero matrix element from vacuum. The coupling to a given mode depends on its field normalization and on the dipole projection onto its polarization.

For free space, count modes in a large quantization volume and sum over two transverse polarizations. The inverse-volume factor in the squared coupling cancels the volume in mode density, so the observable decay rate cannot depend on an arbitrary normalization box. The mode density contributes frequency squared and the field-per-photon coupling contributes another frequency, giving a frequency-cubed rate. The angular and polarization integral produces the stated SI coefficient for an isolated electric-dipole transition.

Eliminating the photon amplitudes produces a memory integral for the atomic survival amplitude. A broad smooth reservoir spectrum and weak coupling let that amplitude vary slowly during the reservoir correlation time. Replacing its past value by its present slowly varying value gives the Markov approximation. Extending the relevant frequency integration near resonance then gives an amplitude damping term and a transition-frequency shift. The resulting exponential is useful when the bath memory is much shorter than the lifetime.

The survival amplitude decays at half the population rate, so probability decays with Gamma while amplitude has Gamma divided by two in its exponent. A coherent phase also evolves with the shifted transition frequency. This survival amplitude is a component of the joint atom–field state, not automatically the atom’s off-diagonal density-matrix element. An initially excited atom has zero atomic optical coherence, even though its excitation later decays through a nonzero survival amplitude.

At sufficiently short times, unitary evolution of a state with finite energy variance gives a quadratic survival-probability change, rather than an exact exponential with a nonzero initial slope. At extremely late times, spectral thresholds can generate nonexponential tails. Structured photonic environments, band edges, strong atom–cavity coupling, and delayed feedback can make bath memory important even on experimentally relevant scales. These cases require the reservoir spectrum or explicit field modes rather than a constant free-space decay rate.

A branching transition adds partial population decay rates, and collection efficiency determines what fraction of emitted photons is recorded. A reduced count rate alone does not demonstrate a longer intrinsic lifetime. For independent identically prepared atoms, the number still excited at a specified time is binomial with survival probability given by the assumed lifetime law. The probability lab matches that ensemble count, not a coherent atomic trajectory, a bath memory integral, or the arrival times of emitted photons.