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In a famous experiment, reported in

Optical Clocks and Relativity. CW Chou, DB Hume, T Rosenband and DJ Wineland. Science 329 no. 5999 pp. 1630-1633 (2010); NIST eprint.

and nicely described in this press release, the NIST ion-traps group built two identical optical atomic clocks, raised one by 33cm, and observed the resultant change in the frequency ratio between the two clocks due to the difference in gravitational time dilation over that height difference.

As pointed out here, optical clocks can hope for a fractional frequency stability of $\delta\nu/\nu\approx10^{-18}$, and this corresponds to the gravitational time dilation caused by a height difference of 1 cm. There is, then, room for a finer experiment at even smaller height differences.

Has the experiment been repeated at smaller height differences? If so, what's the smallest height difference at which gravitational time dilation has been experimentally confirmed in this way?

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As far as I know the 33 cm difference is the current champ for the measurement. The stability of part(s) in $10^{18}$ is only achieved after running for a certain length of time, and averaging out white frequency noise. Their actual result was obtained over a period of time (40,000s and 100,000s for the high and low measurements) and was 37 ± 15cm. They would have to run longer to decrease the error, and these research devices generally don't run for much longer than this — people have to sleep.

A few cm isn't possible without running for a much longer amount of time, to shrink the uncertainty

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