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Atomic clocks use cesium-beam frequency to determine the length of a second. This has shown that the period of orbit of the earth is decreasing.

But what experiment showed that cesium-beam's period was so terribly consistent?

Did they just run several atomic clocks and note that there was no drift? Couldn't many things that might change the frequency of one clock also impact the other so this would not work?

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The frequency is determined by the energy spacing between two configurations of the caesium atom. Caesium has a single electron in the outermost $6s$ orbital, and this electron can be aligned with or against the nuclear spin. These two configurations differ in energy by about 0.000038 eV, and transitions between them produce/absorb light with a frequency of 9,192,631,770 Hz. This is the frequency used to measure time.

The only way this frequency could change is if the energy spacing of the two configurations of the caesium atom changed. But these energies are dependent only on fundamental constants such as the electron charge, mass, coupling constant, etc., and these constants are, well, constant. That means the energy levels must be constant and hence the frequency of the light emitted must be constant as well.

It's not impossible that the fundamental constants actually aren't constant, but if they were changing the effects of the changes would be wide reaching and affect far more than just the caesium atom. We would certainly have noticed by now :-)

I'm not sure what experiments have been done to measure the constancy of the caesium energy levels, but note that international atomic time is derived from around 400 different atomic clocks in different parts of the world. If all the clocks were subject to the same change we wouldn't see it, but random changes in clocks would be immediately detected because the clocks would get out of sync.

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But what experiment showed that cesium-beam's period was so terribly consistent?

They compared it with other clocks. That frequency is terribly consistent, because it isn't actually constant, because it varies with gravitational potential. Moreover this frequency is quoted in Hertz, which is cycles per second. And it's used to define the second. Spot the problem? What's really going on here is something like this: you sit there counting these Caesium-beam microwaves coming past you. Then when you get to 9,192,631,770 you say that's a second. Then the frequency of those microwaves is 9,192,631,770 Hz by definition. Hence the frequency is always the same.

As for what John Rennie said about energies being dependent on things like the electron charge, mass, coupling constant etc, one important expression is the fine structure constant α = e²/2εₒhc. See NIST. It's a "running" constant. Which means it isn't constant. People tend to say that e is effective charge, and that this changes whilst εₒ h and c don't, but that isn't the whole story because of conservation of charge and because the "coordinate" speed of light varies with gravitational potential. See this paper where Magueijo and Moffat talked about the tautology: we use the local motion of light to define the second and the metre, then we use them to measure the local motion of light. Duh! Which is why two NIST optical clocks, one 30cm above the other do get out of sync. And for the cherry on top of this can of worms, search the Einstein digital papers on "speed of light" for stuff like this:

enter image description here

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The problem with the question is the lack of definition for "same."
In the "real/'practical" world, nothing is just the same, you have to specify the measurement and the precision (accuracy) associated/required for the measurement!
Because of the higher precision/accuracy required for our modern scientific experiments, a need for higher precision standards has been created. So, if we can find something that gives us a high precision reference, who's inaccuracies due to all possible "defects" is less than the accuracy required for the experiments, then we have our "standard." In the specific case of cesium, if the inaccuracy of the frequency is less than $1x10^{-10}$ then two measurements with this standard would be "the same" to an accuracy of 1 part in $10^{10}$.

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