What is the experimental evidence for a quantized EM field? I've recently been trying to understand on a deeper level what is the physical meaning or existance of photons, and relatedly,  what is the experimental verification of their existance. We all learn that historically Einstein employed them to explain the photoelectric effect, after Planck discovered the quantization of energy allows correct derivation of blackbody radiation. Later, Compton's scattering experiments were also explained successfully using the photon picture and thus it became established that EM waves somehow propagate in quanta. A few years later Dirac correctly quantized the EM field, and the resulting theory explained the quantum nature of the EM field and light propagation, explaining the photon as the minimum excitation of the EM field, and QED began. (Correct me if this is false; I'm an undergrad and still don't have a formal knowledge in QFT.)
The resulting theory also predicted verified phenomena such as the Lamb shift and the anomalous magnetic moment of the electron. However, it seems that in the years since then, many of the phenomena whose existence supposedly proves QED were derived from semiclassical models. According to Scully & Zubairy in "Quantum Optics",

"There are many processes associated with the radiation-matter interaction which can be well explained by a semiclassical theory in which the field is treated classically and the matter is treated quantum-mechanically. Examples of physical phenomena which can be explained either totally or largely by semiclassical theory include  the photoelectric effect, stimulated emission, resonance flourescence.
Perhaps the most important example of a situation which is not covered by the semiclassical theory is the spontaneous emission of light. Furthermore, the Lamb shift is a good example of a physical situation which is only understood with the introduction of the vacuum into the problem... When vacuum fluctuations are included, we see that the Lamb shift is qualitatively accounted for and conceptually understood. Other phenomena, such as the Planck distribution  of blackbody radiation and the linewidth of the laser, can be understood by such smiclassical plus vacuum fluctuations arguments."

Indeed as they mention, a google search shows derivations of the photoelctric effect, the compton effect, blackbody radiation, spontaneous emission, the Lamb shift, and the anomalous magnetic moment of the electron - all the classical experimental verifications for the quantization of the EM field and QED (as far as I'm aware), from semiclassical models with vacuum fluctuations.
Scully & Zubairy do go on to describe a phenomenon in which the predictions of QED and semiclassical theory differ significantly: quantum beats in $\Lambda$-type systems. In QED such systems show no quantum beats, while semiclassical theory predicts existance of beats in these systems. However, Scully & Zubairy don't mention even a single experimental verification that indeed there are no beats in $\Lambda$ systems, and I wasn't able to find any in google, either.
And so I have to ask, what is, after all, the experimental verification that indeed the EM field is also quantized (and for the existance of photons). Were there experiments trying to detect the aforementioned quantum beats? Is there another effect or phenomenon whose predictions differ between QED and semiclassical theory (with vacuum fluctuations) and was shown experimentally to follow QED behavior, and thus the EM field is definitely quantized (I found the Hong–Ou–Mandel effect which might fit here, but I don't have enough knowledge to understand whether is really shows quantization of the field)? What are the experimental verifications for the existance of photons, which cannot be explained with semiclassical theory (in which there are no photons)? Are there, perhaps, theoretical or thought-experimental arguments for this quantization, if there's no definite experimental evidence? Even though QED is a well-established and well-tested theory, I feel like this is an important question to ask, because if we don't even have clear evidence that the EM field is definitely quantized, then who is to say gravity has to be quantized...
 A: To my mind, the core experimental benchmark that really requires a quantized electromagnetic field to be explained is, as mentioned tangentially in the question, the Hong-Ou-Mandel effect.
The wavelike nature of photons, as displayed in the Hong-Ou-Mandel effect, is rather subtle ground, but let me start off with a statement that sounds controversial but really isn't:

Interference effects such as double-slit interference are not representative of the quantum-wave nature of photons.

This is somewhat amplified when seen from a quantum-optics perspective, but it's important to keep in mind that photons aren't "particles", much: what they really are is discrete excitations of the classical modes of the electromagnetic field. (You can then go full-circle and argue that all particles, from electrons to atoms in a BEC, are also excitations in a matter field, but that's a separate argument.)
As such, you only really start to be able to talk about photons in a way that is quantum-mechanically meaningful when you talk about counting statistics of a given state of light. Those counting statistics are, in a sense, 'riding' on top of the classical modes that they inhabit, and any interference features that those modes exhibit (like, say double-slit fringes or diffraction rings) are not really "the photon interfering with itself", they're just a geometrical feature of the mode that's being excited.
The reason I discount those interference features as unrepresentative of the 'true' quantum wave of photons is that there's yet another layer of interesting interference, and it is when you make those excitations themselves interfere with each other, in both constructive and destructive ways. This is what the Hong-Ou-Mandel experiment does: it combines the probability amplitudes of different combinations of possible excitations to rule out some of them,


Image source 

in such a way that the light emerging from the beam splitter has a 50:50 split of the energy, on average, but this only ever comes in joint pairs of photons on each arm, and never as coincident photons on both arms (an outcome whose probability amplitude has vanished through destructive interference). There is simply no semiclassical model that can account for this.
Now, as OON has noted, you can also get experimental observations of photon-counting statistics that are not explainable by any semiclassical model by simpler configurations of sub-poissonian light, but for me the Mandel dip is much more striking, much more clearly recognizable, and not that much more recent historically speaking.

Also, I would like to address some the comments in anna v's answer. All light is quantum; we know this because we've tried repeatedly to find cracks in quantum mechanics, including its treatment of light, and we haven't found any. Light often 'looks' classical, but that is only because quantum mechanics, in its classical limit, looks like classical mechanics.
However, there is still a lot of value of which experiments can still be explained by having a classical electromagnetic field (so, e.g. replacing coherent states with just classical states, possibly with some shot noise) interacting with quantized matter, and this includes things from atomic absorption and emission spectra to the photoelectric effect, as well as the pointwise response of film in double-slit experiments (whose interference features, again, are just a classical-optics feature of the mode that photons are riding on).
Given what we know of the fundamentally quantum nature of the electromagnetic field (through experiments like the Hong-Ou-Mandel dips), we know that these semiclassical descriptions are just effective models that don't fully describe the core aspects of nature, but it is those experiments that are undescribable without quantized fields that really force us to adopt that perspective. Take those away, and saying "the EM field is quantized" becomes just an opinion without meaningful experimental support.
A: Most direct example I know of: You can simply measure the number of photons that there are in a cavity (a 'light box'), by sending an atom through and measuring its phase change.
This has been done and published in 2007 by Gleyzes et al.: https://www.nature.com/articles/nature05589
The Hong-Ou-Mandel effect mentioned by Max Tyler in the comments is another great example.
A: The classical electromagnetic field is emergent from an enormous number of photons, and that can be proven mathematically.
Experimentally the simplest demonstration is the emergence of the double slit interference pattern one photon at a time:


Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.

One sees the individual photon leaving a footprint that looks random, but is not as the phases of the photon wavefunctions in superposition build up the classical interference. One uses this sequence to stress the quantum mechanical, probabilistic wave nature  of the photon wavefunction
This pattern is explained by both QED and classical electromagnetism.
Yes, Maxwell's equations for light are very useful and it is not necessary to go to the underlying quantum level, because the mathematics is consistent. Only for absorption and emission spectra the existence of photons is absolutely necessary,  and the photoelectric and black body radiation of course.
As far as semi classical claims go, the "semi" is indicative of phenomenology fits, which is fine for fitting data, but not an argument against the underlying quantization.
A: The semiclassical model indeed works nice in many cases. You may look at the density matrix of the quantum field e.g. in the Glauber-Sudarshan representation,
\begin{equation}
\rho=\int d^2\alpha\, P(\alpha)|\alpha\rangle\langle\alpha|
\end{equation}
where $|\alpha\rangle$ are coherent states. Now, this $P(\alpha)$ function can often be treated simply as a probability distribution on the phase space.
But not always. There are states for which $P(\alpha)$ may become negative in some regions of the phase space and also there are states for which $P(\alpha)$ becomes more singular than $\delta$-function. Such states can't be described by semiclassical approach and because of that are known as nonclassical light.
The textbook example of the nonclassical light is a light possessing a sub-Poissonian photon statistics i.e. $\langle (\Delta n)^2\rangle<\langle n\rangle$. From this follows that the second-order intensity correlation function $g^{(2)}(0)<1$ (whereas semiclassical treatment expects $g^{(2)}(0)\geq 1$). This leads to the effect known as the photon anti-bunching that was first observed in 1977 by Kimble, Mandel and Dagenais.
Of course this direct measurement of various correlations of intensities is rather exquisite way to "discover" the quantum nature of light. Highly nonclassical states appear in particle interactions all the time and no semiclassical treatment would be able to describe them as QED does.
A: Re Hong Ou Mandel effect: Just in case someone is trying to understand the Hong Ou Mandel effect in super simple terms like I was, here is a very nice presentation:
https://www.mpq.mpg.de/5020845/0508b_two-photon_interference.pdf
For instance the relative phase shift of $\pi$ happens on the lower side due to the difference of the refractive indices. Once you know that (I didn't :-) it is obvious to see (without any math) why the two solutions cancel each other out.

(Sorry I am not able to comment yet, otherwise I would have added the comment to the Hong Ou Mandel answer)
