The Exercise

An atom with a transition of frequency $\omega_0$ and dipole moment $\mu$ is prepared in its excited state and left alone in free space. Every mode of the electromagnetic field is coupled to it, and the atom can decay by emitting a photon into any one of them. The exercise asks for the probability $P(t)$ that the atom has decayed by time $t$, summed over all the modes, and for the rate at which this probability grows.

In the library, $P(t)$ is assembled from the probability of decay into a single mode, and the claim gives $P(t)$ and its time derivative in closed form.

Decay Into One Mode

Besides $\omega_0$ and $\mu$, the squared coupling depends on the reduced Planck constant $\hbar$, the permittivity of the vacuum $\varepsilon_0$ and the speed of light $c$. The squared coupling $g^2(\omega)$ of the atom to a field mode of frequency $\omega$ is $$g^2(\omega) = \frac{\omega_0^2\,\mu^2}{2\hbar\,\omega\,\varepsilon_0\,c^2}.$$(1)^✦

To lowest order in the coupling, the decay probability $p_{\mathrm{decay}}(g^2,\omega,\omega_0,t)$ of an atom of transition frequency $\omega_0$, coupled with squared strength $g^2$ to a single mode of frequency $\omega$ initially empty, is the probability that it has decayed into that mode by time $t$: $$p_{\mathrm{decay}}(g^2,\omega,\omega_0,t) = g^2\,\frac{4\sin^2\!\big((\omega-\omega_0)\,t/2\big)}{(\omega-\omega_0)^2}.$$(2)

Decay Into Free Space

The total decay probability $P(t)$ of the atom at time $t$ is the decay probability into one mode, with the coupling taken at resonance, integrated over all frequencies $\omega$: $$P(t) = \frac{1}{(2\pi c)^3}\,\frac{8\pi}{3} \int_{-\infty}^{\infty} \omega_0^2\; p_{\mathrm{decay}}\big(g^2(\omega_0),\omega,\omega_0,t\big)\,d\omega.$$(3)^✦

Carrying out the integral gives a constant, the atom’s spontaneous emission rate $\gamma_{\mathrm{rad}}$: $$\gamma_{\mathrm{rad}} = \frac{\omega_0^3\,\mu^2}{3\pi\,\hbar\,\varepsilon_0\,c^5}.$$(4)

The Claim

What Is Checked

The claim is stated over the total decay probability $P$ of (3), a real function of time once the five parameters are fixed, and over the rate $\gamma_{\mathrm{rad}}$ of (4); $P$ in turn rests on the coupling (1) and the decay probability into one mode (2).