The photoelectric effect

The photoelectric effect is usually explained (following Einstein, who received the Nobel price for this explanation) by saying that a sufficiently energetic photon falling on a photosensitive substance causes the latter to eject a single electron, which is then magnified by a photomultiplier to produce a macroscopic and hence observable effect - the ''click'' of the detector. This is commonly used in discussions of experiments on entangled photons carried out by Alice and Bob, who make statistics on clicks to prove or disprove things, or to communicate secret information.

In the semiclassical picture known to Einstein 1905, currents are produced by discrete electrons. In 1905, when Einstein proposed his explanation, the photoelectric effect was a clear indication of the particle nature of light, since no other model was available that could have explained the process. Einstein's explanation was so important for the development of the subject that he got 1921 the Nobel prize for it, a few years before modern quantum mechanics was born. The modern concept of a photon was created only later (Lewis 1926, Dirac 1927).

According to today's knowledge, just like Bohr's atomic model, Einstein's explanation of the photoeffect is too simplistic, and is not conclusive. Now, 100 years later, his picture is known to be approximate only, and that currents in metals are in fact produced by the continuous electron fields of QED. Discrete semiclassical particles are just very rough approximations.

Indeed, the argument of Einstein put forward for the discrete nature of radiation is spurious, since it ignores the quantum nature of the detector (which was of course completely unknown at the time). As one can read in the standard reference for quantum optics,

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995.

Mandel and Wolf write (on p.629, in the context of localizing photons), about the temptation to associate with the clicks of a photodetector a concept of photon particles:

''Nevertheless, the temptation to interpret the electronic signal registered by a photodetector as due to a photon that is localized in some sense is quite strong.''

Sections 9.1-9.5 show that the electron field responds to a classical external electromagnetic radiation field by emitting electrons according to Poisson-law probabilities, very much like that interpreted by Einstein in terms of light particles. Thus the quantum detector produces discrete Poisson-distributed clicks, although the source is completely continuous, and there are no photons at all in the quantum mechanical model. The state space of this quantum system consists of multi-electron states only. So here the multi-electron system (followed by a macroscopic decoherence process that leads to the multiple dot localization of the emitted electron field) is responsible for the creation of the dot pattern. This proves that the clicks cannot be taken to be a proof of the existence of photons.

Note that initially, only single photoelectrons are emitted, which would leave no experimental trace without being magnified. A macroscopic magnification is needed to make the photoelectrons observable. In a photodetector, a photomultiplier is used to produce an observable current. In the case of detection by a photographic plate, the detector is a photoemulsion, and the photoelectrons are magnified via a chemical reaction that produces tiny dots whose density is proportional to the incident intensity of the electromagnetic radiation.

If you are new to quantum optics and want to have a shortcut through this book of over 1100 pages: At first, you need enough classical background. To update your math, read or review Sections 2.1-2.3 and 3.1 and go back to the pieces from Chapter 1 that you need to make sense of these sections. Classical physics in a simplified setting without polarization starts in Chapter 4 and 5, where you need at first only 4.1-4.3 and 5.6-5.7 -- again, reading omitted stuff you need for understanding that as you go along. Full classical electromagnetism is covered in Chapters 6-8. You need 6.1-6.5. The quantum part starts in Chapter 9. You'd read 9.1-9.5, 10.1-10.5, 10.9, 10.10, 11.1-8, 11.13, 12.1-12.4, 12.10, 13.1-13.3, 14.1-14.6., 15.1-3, 18.1-4, 20.1-6, 22.4. Then you have an overview over the central part of quantum optics, and are well prepared to start a second, thorough reading of the whole book.)

Section 12.11 is about the problems with photon position, and that there is no associated operator, but only a POVM. It is in this section that they made the remark referred to above. Sections 14.1-14.5 show that the semiclassical picture of Chapter 9 holds with small corrections also in the quantum case, and is virtually unaltered in case of coherent light.
We conclude that the discreteness of the clicks must be caused by the quantum nature of matter, since there is nothing discrete in an incident classical external radiation field.

I discussed the situation in some more detail in a public lecture given in 2008. See Section 3 (pp.35-44); names in smallcaps are accompanied by references, given at the end of the slides.

See also The photoelectric effect and the abuse of the notion of photons by Hendrik van Hees.

Note that this holds even for very faint light. In deep-field astronomy, 'photographs' of perhaps several billion light years distant astronomical objects using CCD detectors is routine. The time interval between individual events on a CCD array of a few cm^2 can be several minutes or more in some cases.

To explain the image, it is enough that the detector elements on the plate respond locally according to a Poisson process with probability rate determined by the incident energy density. This means it fires randomly at the rate determined at each moment from the incident faint field. No memory is needed, and energy loss is irrelevant (except for the efficiency of the process). The local detector elements will respond independently and rarely but occasionally, and waiting long enough will precisely reproduce the averaged intensity profile - the goal of the imaging.

It doesn't make sense to somehow count photons classically and pretend that each of the myriads of photons created in a distant star is a spherical wave spreading out through space to be ''collapsed'' when entering the CCD detector. The detector doesn't see the myriads of these extremely faint spherical waves and decides to collapse just one of them. Instead, it ''sees'' the energy density; according to its value and the quantum-stochastic state of the environment, it is more or less disposed to respond, resulting in a Poisson statistics. The reason is that in QED, the local mean energy density is an observable field, whereas the concept of a photon number density cannot even be meaningfully defined.

This is not to say that the electromagnetic field can be considered to be purely classical. QED (and therefore photons) are of course needed to explain special quantum effects of light revealed in modern experiments. But not for the photoeffect.