Elena & Fabrice's Web

Take a light source. Any light source would do, but for convenience, let us consider a laser. They are handy for three attributes:

- they are intense,
- they are focused,
- they are coherent.

*Intense* means it's a lot of light. *Focused* means it's all concentrated in a narrow beam. As for what *coherent* means, everybody probably has a fair idea, but we will come back to that shortly.

Now, if we look close enough, we will find out that light is ultimately composed of single entities, the so-called photons, the fundamental grains of light.

*Coherence* is a rather vague notion, in physics, since it is so fundamental and widespread. It keeps popping up in a variety of contexts and it is not always clear what it refers to.

In the context of light, one of the cases where you would imagine it was pretty much nailed down, Glauber made the outstanding discovery that coherence means that photons are uncorrelated, rather than they are all of the same energy (energy is frequency for photons and a monochromatic field was understood as the archetype of a coherent field before this input).

Uncorrelated photons means that the stream they make as they fly by is random, i.e., Poisson distributed. So a close look at them it could look like this:

Glauber introduced the quantities $g^{(n)}(t_1,\ldots,t_n)$ that describe the correlations of $n$ photons detected at the times $t_1$, ..., $t_n$. The light is coherent when $g^{(n)}=1$ at all times and for all $n$. Along with the quantum state for the coherent state, this provided the quantum theory of (optical) coherence, for which he was awarded the Nobel prize in 2005.

Here it must be emphasized that $g^{(n)}$ is a measure of correlations. It is not a probability, as is found in an incredible number of top publications from first rate scientists (one day I'll compile a list of stunning examples of this careless simplification).

All of this is standard textbook material.

This post is to present two twists on this story. The first one is to present a light source that we have recently proposed~[1] (in June of this year), that emits light not with single photons, as do all sources so far, but by grouping them in *bundles* of $N$ photons, for integer $N$. That is to say, the basic unit of light is not a photon anymore but a group of $N$ of them. This is the corresponding sketch if you would look at our source in, say, the $N=2$ and the $N=4$ regimes of $N$-photon emission:

Such sources are great for a variety of purposes, for instance because they rescale Planck constant, $E=Nh\nu$, which is a nice thing if you want to penetrate biological tissues to do tomography at a given depth without burning everything in the process. Of course they also have countless uses for quantum information processing.

In the sketch above, you will have noticed that the bundles have the same statistics as the photons. In the sense of Glauber, they are coherent. This is the second twist I want to introduce here. It is presented accurately in version~2 of our manuscript: **Emitters of $N$-photon bundles**, by Carlos Sánchez Muñoz et al.

If you compute $g^{(n)}$ on our $N$-photon emitter, nothing special happens: it could be bunched, it could be antibunched, there is nothing striking. If you disrupt the device and it stops working, emitting light not in bundles anymore, you don't notice anything in its $g^{(n)}$ either.

This is because $g^{(n)}$ are photon correlation functions. What one needs are *bundle correlation functions*, that upgrade to the level of $N$-photon physics the conceptual framework of Glauber.

We have identified these functions. We call them $g^{(n)}_N$, the case $N=1$ being that of Glauber. They are defined as follows (in standard notations):

$$g^{(n)}_N(t_1,\ldots,t_n)= \frac {\langle\mathcal{T}_-\{\prod_{i=1}^na^{\dagger N}(t_i)\}\mathcal{T}_+\{\prod_{i=1}^na^{N}(t_{i})\}\rangle} {\prod_{i=1}^n\langle a^{\dagger N}a^N\rangle(t_i)}\,.$$

The two-photon correlation $g^{(2)}$ is by far the most common and most important one. Its two-bundle counterpart reads:

$$g^{(2)}_N(\tau)=\frac{\langle a^{\dagger N}(0)a^{\dagger N}(\tau)a^N(\tau)a^N(0)\rangle}{\langle(a^{\dagger N}a^N)(0)\rangle\langle(a^{\dagger N} a^N)(\tau)\rangle}\,,$$

with $\tau$ the delay between the two bundles. Note that we say "bundle" because here we would otherwise have to say, "between the two two-photons". Also, because a brand of two-photons arise naturally from Poisson processes, simply when you got two one-photon by chance. Our bundles present a stronger type of correlations in the way of what quantum state they realize. But that is another question that goes beyond not only this post, but also of our manuscript. Note however that the Poisson trick is how people currently get the long-wavelength, high energy photons in those fields where they need them: they use a normal laser, excite at half the energy, and wait that once in a while, two photons come together. That is not very practical when you can have a source that emits *all* of its light directly in the desired form, especially if you want the $N=3$ or even higher bundling, as the useful signal dies exponentially. Our source solves that problem.

The formula above is general. It can be applied to any system, where it will be usually meaningless because virtually all light sources in the universe emit at the single photon level. But when quantum technology will spread $N$-photon sources all around, then it is $g^{(2)}$—or should I write, $g^{(2)}_1$—that will become meaningless and the above $N$-photon version meaningful to quantify their coherence or other types of correlations. As a matter of fact, our $N$-photon emitter can indeed operate in the lasing regime, in the sense of Glauber that $g^{(n)}_N=1$, but for different system parameters, it can also release its bundles in a self-avoiding way, known as antibunching, that is, $g^{(2)}_N(\tau)=0$ at zero delay~$\tau$.

Various regimes of emission will likely be useful or relevant for various applications, such as coherent statistics for quantum lithography or medical tomography, and antibunching as an input to a quantum information processing unit of for cryptography.

The device is not yet realized in the lab, though. To do so, you need an atom in a cavity, and a conventional laser to excite them. If you excite at a particular energy, for which we provide a simple closed-form expression for all $N$, the classical light that goes in will go out in bundles.

*Update: these results have now been published in Nature Photon.*