Past exam of the atomic and laser physics course of the University of Oxford / 2022 / 3 / c

Let the upper-state population lifetime be $T_1=1/\Gamma$, and take the lower energy level to be stable. Spontaneous emission gives survival probability $e^{-\Gamma t}$, hence a field or excited-state probability amplitude proportional to $e^{-\Gamma t/2}{e}^{-i{\omega }_{0}t}$ for $t\ge 0$. Its Fourier transform at angular frequency $\omega$ is proportional toThe squared modulus therefore has the Lorentzian line shapeHere $L$ is normalized over the full mathematical frequency axis. The full width at half maximum in ordinary frequency is $1/(2\pi T_1)$. If both energy levels decay, their population decay rates add in the natural linewidth; pure dephasing adds to the amplitude-decay rate instead.

Impact-regime pressure broadening provides another example. Independent phase-randomizing collisions with rate ${\gamma }_c$ give optical correlation proportional to $e^{-{\gamma }_c|t|}$, whose Fourier transform is a Lorentzian line shape of full width at half maximum $2{\gamma }_c$. This assumes short, uncorrelated impacts: a general static distribution of perturbing fields need not give a Lorentzian line shape.