I linked to a gravitational redshift experiment that was recently published, and have had a chance to read the paper. It’s quite cool. I had been dismayed at the first couple of popular summaries, but this one is pretty good and as I had indicated, the press release is pretty good for a press release. So there’s not a lot to add.
The original experiment was designed to measure gravitational acceleration; the two trajectories have a different potential energy, mgh, and will accumulate a phase difference as compared to the other trajectory. You can think of this as the difference in deBroglie wavelengths, where the difference in momentum leads to a slightly different wavelength, and this gives a phase difference when the atoms recombine. Since the mass and the relative trajectory are known, measuring the phase difference will allow you to determine g.
The special insight presented in this paper was the interpretation in terms of relativity. The energy, rather than being the classical kinetic and potential terms, is mc^2. The “oscillation” that allows for interference is now at a much higher frequency, and the accumulated phase will be gh/c^2, which is the gravitational redshift. To do this, you need to independently know g, which was determined using a corner-cube gravimeter.
There is also a difference in the implications by reinterpreting the results. The first measurement assumes the classical physics is correct — the phase difference is proportional to g is the result of an equation that is assumed to be true at this level of precision — and that there is a phase to measure, i.e. the atoms have a wave nature, which is an early prediction of quantum mechanics. The answer in the form of a value for g only makes sense if we assume these theories are correct. And that’s not really a problem, since we have independently tested those theories many times over the years.
But the other interpretation is a direct test of relativity — the theory predicts an answer which can be directly compared to the result. And that allows one to put limits on how “wrong” this aspect of relativity might be. You add another term onto the time dilation term, with a perturbation expansion (we already know relativity is pretty good and have results from the Vessot rocket experiment, so any deviation has to be small). So we write the time dilation as (1+B)(gh/c^2), and then assume the worst case, that all of the discrepancy in the measurement is not experimental error, but rather a flaw in the theory. And they get a result of B = 7±7 x 10^-9, which is consistent with relativity being correct (B = 0), and limits any problems with this aspect of it to a part in a hundred million.