Criteria for being able to work with gravity in quantum mechanics, without a full theory of quantum gravity? It's common to see people oversimplify by saying that physics currently lacks the tools to describe any situation involving both quantum mechanics and gravity. Clearly this is not the case. For example, the Pound-Rebka experiment involves inescapably quantum-mechanical phenomena such as the Mossbauer effect, but is really just a test of the equivalence principle. Less trivially, Colella et al. did an experiment demonstrating interference between two beams of neutrons that had traveled through different gravitational potentials. It seems to me that there are probably a bunch of different levels of difficulty we could consider:


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*Experiments, such as the ones above, involving quantum mechanics, that can be described in flat spacetime using the equivalence principle. Curvature of spacetime is negligible.

*Experiments in which curvature is nonnegligible, but the analysis is still trivial. For example, I could imagine, at least in principle, doing gravitational lensing with neutrons and observing quantum interference effects between different parts of the beam. (In reality, I'd guess this example wouldn't work due to decoherence.)

*Semiclassical gravity, e.g., Hawking radiation.

*Planck-scale physics.
Can anyone comment on whether this 4-level classification seems right, or give a more rigorous set of criteria for distinguishing the levels?
 A: As you say, Colella et al already demonstrated (or confirmed, depending on one's perspective) that gravity affects a quantum system. Lately, there has been an effort to come up with an experimental apparatus that would exhibit the opposite: a gravitational effect from a quantum system in a low energy regime. Some of the thrust comes from the work by Bose et al; and Marletto and Vedral (conceived as "table-top" experiments).
Markus Aspelmeyer in https://www.youtube.com/watch?v=hjSUF42F6qQ discusses some of the recent experimental approaches.
Re 2: I believe https://arxiv.org/abs/1610.03832 goes towards it, showing the curvature-induced phase shift of an atomic wave.
Re 1 and 3: https://arxiv.org/abs/1807.11494 has an overview of some of the newer proposals (in particular, how the outcomes of the experiments might disambiguate among varying quantum gravity contenders, in particular, falsification of semiclassical gravity if entanglement observed).
Re 4: This is probably a stretch for the foreseeable future. Jake Taylor discusses some of the difficulties in https://www.youtube.com/watch?v=DDP5vL9OlaU.
