The beauty and importance of light in everyday life are clear to everyone, literally under our eyes. Not surprisingly, then, the same is true also in science, where light has always played a central role. On the one hand, the theory of vision was already studied by the Greek and Roman philosophers, as well as by the Arabs during the Islamic Golden Age. On the other hand, light itself is an excellent tool to study our universe, and its phenomena can be found in most of the investigations carried out in physics. Following the studies of Newton, Faraday and many other pioneers, our understanding was finally consolidated by Maxwell’s landmark works, which laid the foundation of modern optics. More recently, optical systems have become a ubiquitous component for technology, to the extent that it’s hard to overrate their essential contribution. In the wake of this enthusiasm, today light is about to deliver new and even deeper thrills under the magnifying lens of quantum mechanics!

The relevance of quantum mechanics for science - and beyond - is well known, with its counter-intuitive phenomena to deal with and figure out. These curious phenomena have been associated with the so-called first quantum revolution, where scientists tried to unveil the mechanisms hidden behind the new observations. One century has passed since then and we’re now entering the second quantum revolution, where scientists are no longer only spectators of weird phenomena but also use them for practical applications! This long-term vision sets a clear departure from the former, more descriptive approach. Only a few decades later, quantum technologies are now believed to dramatically improve classical approaches in several fields, ranging from the simulation and exploration of complex systems to computer science and communication. All these research areas, which come under the name of quantum information, have recently witnessed great achievements both from the theoretical and experimental side.

This notwithstanding, the stage in which quantum technologies outperform classical devices in relevant problems is still beyond our engineering skills. Indeed, for quantum technologies to tackle problems of practical interest (e.g. factorization), we need many more qubits (the quantum equivalent of the classical bit) and of better quality (less noise, longer coherence time, better connectivity etc.). For this reason, to keep everyone happy and motivated - including funding agencies! - scientists have set an intermediate goal whose relevance, albeit symbolic, has driven an enormous effort on a global scale. This goal consists in finding a problem where a quantum hardware outperforms the best classical counterpart with current technology. This means that it’s not important whether 1. the problem is useful (as long as it’s clear and the comparison is fair!) or 2. the classical algorithm is known to be optimal (i.e. in the future, more efficient algorithms could change the outcome of the challenge!). People refer to this goal as the race towards quantum supremacy or (here) towards quantum advantage. The first demonstration of quantum advantage was reported by Google in 2019 using superconducting qubits. This notwithstanding, we will see how light has played, and is still playing, a prominent role in this quest.

In this blog entry, we will briefly retrace the main steps that lead all the way from Maxwell’s equations to the recent, cutting-edge experiments that seek a photonic quantum advantage. First, we will sketch the derivation of single photons in non-relativistic quantum field theory (with integrals and derivatives for the enthusiasts). Then, we will outline the task that aims to unlock a quantum advantage with near-term technology (for the happiness of computer scientists). Finally, we will sketch recent experiments and the problem of validation (for experimentalists and down-to-earth philosophers). Let’s just make a quick remark before we start: even though qubits are the logical units at the core of quantum information, here we will not need them!

From Classical to Quantum

You have all probably seen the t-shirts with the four Maxwell equations. What if I tell you now that you don’t need 4 equations at all?

In the special theory of relativity we described electromagnetism as a field theory which follows from an action principle (or a so-called Lagrangian density). In the following we will derive the 4 Maxwell equations from an a-priori arbitrary Lagrangian. This will legitimize our principle of action a posteriori. We will then quantize this classical field theory into a so-called quantum field theory, where the “quanta” will be identified with photons.

Maxwell’s Theory from Classical Field Theory

Before we start looking for the quantized photon, a brief reminder about Maxwell’s theory from the point of view of classical field theory.

Let’s start with the Lagrangian density for Maxwell’s equations in the absence of any sources. This is simply

where the field strength is defined by

As you might know, we get the equations of motion by solving the Euler-Lagrange equations. By doing this we get two of the four famous Maxwell equations

Or in terms of $A^{\mu}$:

Okay, but where are the Maxwell equations we know from highschool? To see the equations we need to go from the 4-vector notation to the 3-vector notation. Let’s define $A^{\mu}=(\phi, \vec{A}),$ then the electric field $\vec{E}$ and magnetic field $\vec{B}$ are defined by

which, in terms of $F_{\mu \nu},$ becomes

Using the explicit form of $F_{\mu\nu}$ in the found equations of motion yields two Maxwell equations

The other two Maxwell equations do not result from the action principle. They are more of a geometrical consequence. We can observe that $F_{\mu \nu}$ satisfies the so called Bianchi identity.

Using again our explicit form of $F_{\mu \nu}$ and the Bianchi identity we get the second pair of Maxwell’s equations,

Radiation Field Quantization

We’ve seen, that our classical field theory reproduces electromagnetism. Now let’s quantize. Quantizing is the moment when you have to hand over the cookbook and get creative. The modern approach would of course be the path-integral formalism. But we’re staying old school and quantize canonically, but first things first:

For the quantization procedure we need to compute the the momentum $\pi^{\mu}$ conjugate to $A_{\mu}$