A newly proposed table-top experiment might be able to demonstrate that gravity is quantized

Tl;dr: Experimentalists are bringing increasingly massive systems into quantum states. They are now close to masses where they might be able to just measure what happens to the gravitational field.

Quantum effects of gravity are weak, so weak they are widely believed to not be measurable at all. Freeman Dyson indeed is fond of saying that a theory of quantum gravity is entirely unnecessary, arguing that we could never observe its effects anyway. Theorists of course disagree, and not just because they’re being paid to figure out the very theory Dyson deems unnecessary. Measurable or not, they search for a quantized version of gravity because the existing description of nature is not merely incomplete – it is far worse, it contains internal contradictions, meaning we know it is wrong.

Take the century-old double-slit experiment, the prime example for quantum behavior. A single electron that goes through the double-slit is able to interact with itself, as if it went through both slits at once. Its behavior is like that of a wave which overlaps with itself after passing an obstacle. And yet, when you measure the electron after it went through the slit it makes a dot on a screen, like a particle would. The wave-like behavior again shows up if one measures the distribution of many electrons that passed the slit. This and many other experiments demonstrate that the electron is neither a particle nor a wave – it is described by a wave-function from which we obtain a probability distribution, a formulation that is the core of quantum mechanics.

Well understood as this is, it leads to a so-far unsolved conundrum.

The most relevant property of the electron’s quantum behavior is that it can go through both slits at once. It’s not that half of the electron goes one way and half the other. Neither does the electron sometimes take this slit and sometimes the other. Impossible as it sounds, the electron goes fully through both slits at the same time, in a state referred to as quantum superposition.

Electrons carry a charge and so they have an electric field. This electric field also has quantum properties and moves along with the electron in its own quantum superposition. The electron also has a mass. Mass generates a gravitational field, so what happens to the gravitational field? You would expect it to also move along with the electron, and go through both slits in a quantum superposition. But that can only work if gravity is quantized too. According to Einstein’s theory of General Relativity though, it’s not. So we simply don’t know what happens to the gravitational field unless we find a theory of quantum gravity.

It’s been 80 years since the question was first raised, but we still don’t know what’s the right theory. The main problem is that gravity is an exceedingly weak force. We notice it so prominently in our daily life only because, in contrast to the other interactions, it cannot be neutralized. But the very reason that planet Earth doesn’t collapse to a black hole is that much stronger forces than gravity prevent this from happening. The electromagnetic force, the strong nuclear force, and even the supposedly weak nuclear force, are all much more powerful than gravity.

For the experimentalist this means they either have an object heavy enough so its gravitational field can be measured. Or they have an object light enough so its quantum properties can be measured. But not both at once.

At least that was the case so far. But the last decade has seen an enormous progress in experimental techniques to bring heavier and heavier objects into quantum states and measure their behavior. And in a recent paper a group of researchers from Italy and the UK propose an experiment that might just be the first feasible measurement of the gravitational field of a quantum object.

Almost all researchers who work on the theory of quantum gravity expect that the gravitational field of the electron behaves like its electric field, that is, it has quantum properties. They are convinced of this because we have a well-working theory to describe this situation. Yes, I know, they told you nobody has quantized gravity, but that isn’t true. Gravity has been quantized in the 1960s by DeWitt, Feynman, and others using a method known as perturbative quantization. However, the result one gets with this method only works when the gravitational field is weak, and it breaks down when gravity becomes strong, such as at the Big Bang or inside black holes. In other words, this approach, while well understood, fails us exactly in the situations we are interested in the most.

Because of this failure in strong gravitational fields, perturbatively quantized gravity cannot be a fundamentally valid theory; it requires completion. It is this completion that is normally referred to as “quantum gravity.” However, when gravitational fields are weak, which is definitely the case for the little electron, the method works perfectly fine. Whether it is realized in nature though, nobody knows.

If the gravitational field is not quantized, one has instead a theory known as “semi-classical gravity,” in which the matter is quantized but gravity isn’t. Though nobody can make much sense of this theory conceptually, it’s infuriatingly hard to disprove. If the gravitational field of the electron remained classical, its distribution would follow the probability of the electron taking either slit rather than itself going through the slits with this probability.

To see the difference, consider you put a (preferably uncharged) test particle in the middle between the slits to see where the gravitational pull goes. If the gravitational field is quantized, then in half of the cases when the electron goes through the slit, the test particle will move left, in the other half of cases it would move right (it would also destroy the interference pattern). If the gravitational field is classical however, the test particle won’t move because it’s pulled equally to both sides.

So the difference between quantized and semi-classical gravity is observable. Unfortunately, even for the most massive objects that can be pushed through double slits, like large molecules, the gravitational field is far too weak to be measurable.

In the new paper now, the researchers propose a different method. They consider a tiny charged disk of osmium with a mass of about a nano-gram, held by electromagnetic fields in a trap. The particle is cooled down to some hundred mK which brings it into the lowest possible energy state. Above this ground-level there are now discrete energy levels for the disk, much like the electron orbits around the atomic nucleus, except that the level spacing is tiny. The important point is that the exact energy values of these levels depend on the gravitational self-interaction of the whole object. Measure the spacing of the energy levels precisely enough, and you can figure out whether the gravitational field was quantized or not.

Figure 1 from arxiv:arXiv:1510.01696. Depicted are the energy levels of the disk in the potential, and how they shift with the classical gravitational self-interaction taken into account, for two different scenarios of the distribution of the disk’s wave-function.

For this calculation they use the Schrödinger-Newton equation, which is the non-relativistic limit of semi-classical gravity incorporated in quantum mechanics. In an accompanying paper they have worked out the description of multi-particle systems in this framework, and demonstrated how the system approximately decomposes into a center-of-mass variable and the motions relative to the center of mass. They then calculate how the density distribution is affected by the gravitational field caused by its own probability distribution, and finally the energy levels of the system.

I haven’t checked this calculation in detail, but it seems both plausible that the effect should be present, and that it is large enough to potentially be measurable. I don’t know much about these types of experiments, but two of the authors of the paper, Hendrik Ulbricht and James Bateman, are experimentalists and I trust they know what current technology allows to measure.

Suppose they make this measurement and they do, as expected, not find the additional shift of energy levels that should exist if gravity was unquantized. This would not, strictly speaking, demonstrate that perturbatively quantized gravity is correct, but merely that the Schrödinger-Newton equation is incorrect. However, since these are the only two alternatives I am aware of, it would in practice be the first experimental confirmation that gravity is indeed quantized.