Photoelectric Effect, Why can't two quanta interact with an electron at the same time?

I understand that assuming light is quantized implies that if a lower energy interacts with metal, it is possible that that quanta will not have enough energy to eject the electron. What prevents two quanta from interacting with the electron at the same time, though, and summing to have enough energy to release the particle. Indeed, what if two separate light rays of the same frequency impact the electron, why can't that free it?

Answer

The answer is that electrons will sometimes be emitted when two photons, each of less than the work function energy, hit the surface. But there is no special significance to this so you shouldn't leap out of your bath and run naked down the street just yet.

The electrons in a metal have a natural oscillation frequency called the plasma frequency. Even in total darkness lattice vibrations will randomly transfer energy to the plasma oscillations, and especially at high temperatures random redistributions of this energy may concentrate enough energy in a small volume to eject an electron. This is the phenomenon of thermionic emission, and while the probability of emission is only significant at high temperatures there is in principle a finite probability of electron emission at room temperature.

Incident photons also transfer energy to plasma oscillations, but they transfer a lot of energy to a small volume so it's much more probable the energy can end up ejecting an electron. Even so, the quantum yield for photoemission is typically much less than one because in most cases the energy transferred to the plasma oscillations just leaks away into the bulk. A quick Google suggests typical quantum yields are around $10^{-6}$ i.e. only one photon in a million ends up ejecting an electron.

The point of all this preliminary rambling is that there is a small probability than even a single sub work function photon will eject an electron because you could get a combined photo/thermionic transfer of energy resulting in emission. The probability of this is exceedingly low, but then there are an exceedingly large number of electrons in metals.

Given the above, it should be obvious that there will also be a finite probability of two sub work function photons ejecting an electron. But is this really photoemission? You could argue that the first photon just causes local heating and this increases the chance of the second photon ejecting an electron through combined photo/thermionic emission.

So the pedant in me would have to concede that two sub work function photons can eject an electron. However the experimental scientist in me would regard an unmeasurably small probability as being zero, and would suggest that the pedant should get out more.