Randomness of photon's spontaneous emission

Question

I'd like to understand why the phase and the direction of a photon spontaneously emitted is random. Is there a "mathematical" rigorous proof of its randomness? I searched for a proof but couldn't find any on Internet.

Answer 1

Spatial randomness

Saying that the photon is emitted into a specific direction is not quite formally correct. Actually the atom emits an electromagnetic field like a classical oscillating dipole with exponentially decreasing amplitude.

The difference is that in the case of an atom there is exactly 1 photon in the emitted mode. If you perform a position measurement of that photon, for example by putting a photographic plate around the atom, the state collapses. This is the famous collapse of the wavefunction which is by the Kopenhagen interpretation of quantum mechanics described as a random process. By forcing the original wavefunction, which was a superposition of all possible directions of emission according to the dipole emission pattern, to give an answer to the position measurement, the wavefunction collapses onto a state where the photon ends up at the measured position. If you do the experiment many times in the same way the electromagnetic waves are equal in each of the repetitions, but the projective measurement has different outcomes.

Phase randomness

Although it would be easy to claim that the phase randomness is also due to the projection of a superposition I think it's worth noting that the phase is not in a superposition because of the spontaneous emission itself. If you go through the calculations in the Weisskopf-Wigner theory of spontaneous emission and additionally to summing over all possible wavevectors $\vec{k}$ and polarizations $s$ sum over all possible phases $\phi$ of the electromagnetic field modes you find that then all contributions would cancel each other out and no emission takes place. Therefore the phase of spontaneous emission is fixed.

This is of course contradictory to experimental observations. Because if you let the spontaneously emitted photons of two atoms interfere you don't get an interference pattern, so there is no fixed phase relation between them. The problem with Weisskopf-Wigner theory is that it assumes that at $t = 0$ the atom is fully in its excited state $\left| e \right\rangle$, which can only be true if it was excited within an infinitely short time interval. In a real setup the atoms are excited by pulses that are long compared to the optical cycle duration, so that the decay begins already when the pulse starts to bring the atom into a partially excited state $A(t) \left| e \right\rangle + \sqrt{ 1 - A(t)^2 } \left| g \right\rangle$. The excited part of the state starts decaying and accumulating a phase while the excitation pulse is still exciting the ground state part which then accumulates a different phase.

From the observation side of view it seems like there is no way of discriminating whether the phase randomness comes from the time uncertainty of the excitation or from some unknown random effect in the spontaneous emission. Except if you manage to make laser pulses significantly shorter than one optical cycle...

But there is an easier way: You can excite two atoms with the same laser pulse – like this you make sure that both atoms undergo the same time evolution. Now, if the randomness comes from the excitation the atoms should have a fixed phase relation, because they were prepared in the exact same way, so there should be interference between them. If instead the emission process randomizes the phase, there should be no interference. The experiment shows that indeed you see interference; see for example this paper. (In the paper they used ensembles of atoms instead of single atoms, but that actually emphasizes the fact that exciting multiple atoms with the same laser pulse yields a fixed phase relation.)

Answer 2

Randomness of photon's spontaneous emission is found in experimental testing.There is no evidence at all that shows its NOT random. I think you would need to show mathematical" rigorous proof of NON randomness.

Answer 3

One simple way to think of it is energy and momentum conservation. When an electron drops into a hole on a band diagram and emits a photon in a solid state, or in an excited state, plasma, etc. when an excited states drops into a ground state, energy is conserved by the emission of a photon. But those diagrams often neglect the importance of momentum conservation. What were the pi's of initial electron/hole? Likely a distribution, and the photon will need to carry their vector sum.
In the solid state, by contrast, stimulated emission is much more rigorous; a photon with certain momentum can only stimulate emission of an electron/hole pair with matching momentum difference. So when you get gain in such a system, the emission is in phase. In terms of rigorous mathematical proof, you'd need QED.