While this is technically true, it doesn't help much, since the quantum theory described is not considered a viable candidate for a fundamental theory of gravity for a number of reasons.
> We did observed gravitational waves of GR. A force field with waves mediating that force interactions
We have observed classical gravitational waves. We are many, many orders of magnitude away from being able to measure any possible quantum properties of those waves.
> that requires just a bit of math to "quantize" it
Math (more than "a bit") can quantize it in theory. But theory is not enough. We would actually have to observe the quantum properties experimentally to confirm the theory.
While not directly quantum properties of gravitational waves yet, we've already observed quantization of gravitational potential (and that naturally suggests "gravitational quantas" at least by similar machinery as with EM->QM) - at small scales/energies when the spacetime curvature can be not paid attention to, the gravity potential of an ultra-cold neutron quantizes just nice using the Schrodinger equation in full agreement with experimental observation - https://www.physi.uni-heidelberg.de/Publications/dipl_krantz... . In my view, giving the amount of clarity in gravitation as well as in QM that experiment produced, it is worth of a Nobel.
No, we haven't. We've observed that, under appropriate conditions, gravitational potential has to be included in the potential term in the Hamiltonian. But the potential term in the Hamiltonian is not quantized; it's not part of the quantum state and it does not exhibit quantum properties. The only things in the experiment that exhibit quantum properties are the neutrons themselves.
It is like saying that discrete orbits of electron say nothing about quantization of associated EM potential.
>The only things in the experiment that exhibit quantum properties are the neutrons themselves.
And those properties are position and momentum.
They specifically chosen neutrons to avoid effect of other forces and the paper is pretty clear that position and motion of neutrons were primarily affected by gravitational field, and as result the neutron position got quantized :
"... we conclude that the measurement manifests strong evidence for quantisation of motion in the gravitational field as is expected from quantum mechanics
...
the quantum theory we derived neglecting gravity is unable to reproduce the shape of the neutron height distribution even on the largest scale.
..."
Do you think the height of observed "steps" in neutron position would be the same or different if the experiment were repeated in different gravity, say Moon?
That's right, they don't. There is no such thing as "quantization of EM potential". The electromagnetic field is quantized, if you're using quantum field theory, but in QFT there is no "EM potential". In non-relativistic QM, the EM potential term in the Hamiltonian is not quantized and does not exhibit quantum properties. Even in models which add relativistic corrections, such as the models used in the late 1940s to predict the Lamb shift, the "quantization of the EM field" only applies to the "quantum electromagnetic field" external to the atom; it does not apply to the EM potential due to the nucleus.
> the paper is pretty clear that position and motion of neutrons were primarily affected by gravitational field, and as result the neutron position got quantized
No, the neutron positions were not quantized "as a result" of the gravitational field. They are already quantized in the absence of a gravitational field, as shown by numerous previous experiments. The only thing this experiment shows, as I've already said, is that under appropriate conditions, you have to include the gravitational potential in the potential term in the Hamiltonian.
Obviously since the gravitational potential would be different, the neutron position "steps" would be different. That still doesn't change the fact that the experiment is not showing "quantized gravitational potential". It's only showing that "quantized neutron position", which is already well established by other experiments, is affected by the gravitational potential in the same way as it would be by any other classical potential in the Hamiltonian.
i think we're on a right way in general :
QM -> GR : https://en.wikipedia.org/wiki/Quantum_gravity#Quantum_mechan...
" In particular, contrary to the popular claim that quantum mechanics and general relativity are fundamentally incompatible, one can demonstrate that the structure of general relativity essentially follows inevitably from the quantum mechanics of interacting theoretical spin-2 massless particles (called gravitons)"
GR -> QM : We did observed gravitational waves of GR. A force field with waves mediating that force interactions - that requires just a bit of math to "quantize" it. It is a bit more complicated when classic EM -> QM because of added complication of non-stationary spacetime and just vastly different scales of spacetime/energy to run the experiments to establish/verify the gravitons parameters.
For the "true deep" nature of graviton I subscribe to the view that - given gravitational force not being the "real" force and the gravitational waves being really a ripples of spacetime - graviton is more like phonon, ie. quasi-particle from the point of view of QM/SR theories.