This piece has been written as background support for Two photons everywhere. E. Zubizarreta Casalengua, F. P. Laussy and E. del Valle in Phil. Trans. R. Soc. A 382:20230315 (2024). 
; see this 𝕪eet for background information.
The inaugural call (in 2009) of the Royal Society's Newton fellowship welcomed one of us (EdV) to Southampton, to work with the Chair of Nanophysics and Photonics, Prof. Alexey Kavokin. This was to study quantum features of polariton Bose-Einstein condensation (BEC). In this group was also the other author of this blog (FPL).
Polaritons are particles that strongly couple light and matter. Strong coupling here is often understood as a quantum superposition of photons and electrons, similarly to how Schrödinger's cat is a superposition of a live and dead cat. But while the cat is macroscopic—which was the point of Schrödinger—polaritons are microscopic. They can become macroscopic, say when they undergo BEC. But are they, then, still quantum? This question which we started to entertain in Southampton—this was 15 years ago—accompany us to this day, since our wedding (also in 2009) strongly coupled us, and since the field of quantum polaritonic still prospects for its spot in various quantum initiatives.
What have we learned or found in 15 years? The Royal Society seized the opportunity of time passing so quickly to compile a series of texts from various Newton fellows, recently published as their 2281th Issue in Volume 382 of their Philosophical Transactions A, at http://bit.ly/TransA2281. Our contribution is an exposition of our main results, steadily gathered over the years. We wrote it as a self-contained overview of what we found and understood when trying to figure out what is quantum about complicated condensed phases of light and matter.
Our first observation is that this brought us down the rabbit hole. A bit like Hal trying to fix the lightbulb in the American sitcom "Malcolm in the Middle", that brings him to fix the broken cupboard, itself calling attention to a creaky drawer, which reveals that the oil can is empty, which cannot be replaced because the car is broken... When his wife in the evening asks if he replaced the lightbulb in the kitchen, the reply comes from under the car as "What does it look like I'm doing?" Similarly, in our quest to describe the quantum dynamics of polariton BEC, we passed from simple problems to simpler ones, from understanding the condensation buildup to what it means for a field to be quantum. This brought us to get rid more and more of sophistications from the models, going from two-dimensional condensed matter to zero-dimensional objects, then getting rid of the light-matter structure to describe a single mode, then reducing the said mode to a two-level system, and finally looking at the perturbative regime of vanishing driving as opposed to lasing under strong excitation. Each simplification brought us closer to a more fundamental problem, whose resolution we felt was a prerequisite to the more complicated one that kicked us there. This led us to follow Einstein's dictum: make it as simple as possible. But not simpler.
And what did we find at the bottom of the Rabbit hole? Something as far-reaching as our final problem was remote from the initial one: that even the emission of the simplest emitter, the two-level system, opens the door to a mind-boggling rich universe of multiphoton physics, which is invisible at the classical—or one-photon—level, and that needs to be explored with $n$-photon goggles. What we found is that there is a deeply ingrained classical prejudice, even from quantum opticians, to restrict their attention to the photons that they detect in photoluminescence, or that are scattered, or that they see directly, one way or the other, under the denomination of "signal". Nobody thought of looking at the photons where the system does not emit them. But if there is no emission, surely there is nothing to look at? This is classical thinking. In quantum optics, "photons" are but a particular case, albeit the building-brick one, of the type of light that can be emitted. What is usually meant by such "photons" is anything that in quantum theory is described by a one-photon observable. The intensity (or number operator) is such an observable. It has a classical counterpart, which also has the same name, in effect, is the same thing, and is thus intuitive. What people did so far is to characterize the quantum properties of those "photons" that show up as an intensity. They can be detected in time with different statistics, which is the tenet of quantum optics since Glauber related photon statistics to quantum coherence.
But, we argue, those photons are a tiny bit of the story, and are essentially of a classical character, even when their statistics violate some classical inequality. We say this because, now, let us put our two-photon goggles, i.e., a two-photon observable like coincidences, then one can detect pairs of photons which do not produce a signal for a single-photon observable, but exist nonetheless in the sense that they can be measured experimentally, and follow a pattern theoretically, exhibiting resonances, also enjoying various types of two-photon statistics, etc. Where are these photons which are not visible? Everywhere. They are distributed over all frequencies and, in particular, away from the frequency where the system emits its "classical-like" photons, what opticians call a "spectral peak". There is much less of these photons but this is precisely what one expects of a strongly quantum signal: to be scarce. They are inconspicuous at the one-photon level precisely because they are diffused all over. To see them, one needs to face them in their two-photon space. At the same time, though, one expects such photons to be strongly correlated, entangled, nonlocal, etc. And those two-photon everywhere are that, in a way that two "classical" photons are not, or much less.
We have now charted in very fine detail the full landscape of this two-photon space for the weak driving of the two-level system (a problem known as resonance fluorescence). These provide a literally new picture of quantum emission, revealing two-photon physics that was not even suspected before, in the form of geometrical patterns. Some features we understood a long time ago, such as lines of photon bunching, i.e., the photons coming together. In this case, this is a general manifestation of a particular case identified by Göppert-Mayer as Doppelemission. This is valid for any number of photons, so instead of introducing Dreifachemission und Vierfachemission und mehr, we called them "leapfrog processes", because our more general case arises from a virtual jump over real transitions. They form regions of the multiphoton hyperspace that can be exploited for quantum technology. Other features, we understood only recently, and in fact our latest discovery, the final charting of the two-photon space, is published for the first time in The Royal Society commemorative issue. Namely, we can now explain a mysterious circle of two-photon antibunching, a perimeter of the two-photon space where photons avoid each other. Why on a circle? This has puzzled us for a long time, until, with the help of Dr. Zubizarreta Casalengua, whom we could fruitfully interest to these multiphoton spaces, we finally understood what brings photons to such anti-bosonic extremes. This involves the most famous and basic optical concept: interferences. But in the multiphoton space. Details are, for the first time, provided in the paper: Two photons everywhere. Phil. Trans. R. Soc. Lond. A, 382:20230315, 2024.
So that's what we found down the rabbit hole in 15 years of digging: quantum opticians should go to the $n$-dimensional hyperspace of multiphotons, even if those come from a zero-dimensional single mode. One should stand on the shoulders of photons, and also step on their tail (events), but in the quantum multiphoton space, not only look at where they let their head emerge in the classical world.
Elena del Valle and Fabrice P. Laussy