Why don't atoms emit more than one photon during an energy level transition? If you look at the emission spectrum of an atom, there are sharp lines corresponding to the different energy level transitions.  That's because the single photon emitted during each transition carries the entire energy of the transition.
My question is: how does it happen that just a single photon carries all the energy?  Why isn't the energy sometimes split among two or more photons?
I understand how the rules of quantum mechanics constrain the energy states of the atom to a discrete set of levels, but I don't understand how they also constrain the number of produced photons to be equal to one.
 A: Two-photon emission does exist, or else the 2s state of hydrogen would be stable. You can get a pretty decent estimate for this kind of rate without any fancy math or physics, just using the energy-time uncertainty principle. The typical rate of emission for a photon, when not forbidden by parity, is $R \sim 10^9\ \text{s}^{-1}$. We can think of the two-photon decay as an energy-nonconserving jump up to some higher-energy state, with the emission of a photon, followed by the emission of a second photon leading down to the ground state. The first jump can happen because of the energy-time uncertainty relation, which allows the electron to stay in the intermediate state for a time t ∼ h/E, which is on the order of $10^{−15}$ s. The probability for the second photon to be emitted within this time is $Rt$, so the rate for the whole two-photon process is $R^ 2 t \sim 10\ \text{s}^{-1}$. Considering the extremely crude nature of this calculation, the result is in good agreement with the observed rate of about $0.1\ \text{s}^{-1}$ for two-photon decay of the 2s state in hydrogen.
A: Another form of two photon emission is two photon excited fluorescence.
Two photon absorption is the absorption of two photons by a molecule to excite from ground state to excited state.

Two-photon absorption can lead to two-photon-excited fluorescence where the excited state produced by TPA decays by spontaneous emission to a lower energy state.


First, there is a two photon absorption, then a non-radiative deexitation, and a fluorescent emission. The electron returns to ground state by another non-radiative deexitation.
https://en.wikipedia.org/wiki/Two-photon_absorption
Now you are asking about two photon emission, and phosphorescence is a type of multi-photon emission, where the absorbed energy is released by the emission of multiple photons.

Phosphorescence is a type of photoluminescence related to fluorescence. Unlike fluorescence, a phosphorescent material does not immediately re-emit the radiation it absorbs. The slower time scales of the re-emission are associated with "forbidden" energy state transitions in quantum mechanics. As these transitions occur very slowly in certain materials, absorbed radiation is re-emitted at a lower intensity for up to several hours after the original excitation.

https://en.wikipedia.org/wiki/Phosphorescence
Two photon emission (stimulated) is possible by semiconductors.

We report the first experimental observations of two-photon emission from
  semiconductors, to the best of our knowledge, and develop a corresponding theory for the
  room-temperature process. Spontaneous two-photon emission is demonstrated in
  optically-pumped bulk GaAs and in electrically-driven GaInP/AlGaInP quantum wells.

https://arxiv.org/abs/quant-ph/0701114
