It is very nice to see more and more attempts to measure quantum phase noise separate from Shannon noise. The recent LISA Pathfinder resulted in an upper limit decay rate of 1e-8/s for the CSL. This was measured at 1.5 million km at L1 for a 3 hour measurement with two test masses and an interferometer. They managed to measure noise of 1e-14 m/s^2/sqrt(Hz) with this very stable craft far away from Earth's gravity.
It is very unlikely that any terrestrial measurement will better reveal the CSL limit, which others have suggested may be as small as 1e-17/s...another nine orders of magnitude. But have at it. My money is on 8.1e-18/s since that is the aether decay that ties gravity and charge together and is an intrinsic constant of aethertime. This is also the decay of the IPK mass standard and a bunch of other decays as well.
The only chance for such a measurement over several months at L1. Very high precision clocks show these kinds of phase decays for several hours before their phases begin to wander off due to local gravity noise. In a sense, this experiment is just a precision acoustic clock which simply will not be as sensitive as atomic lattice clocks.