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"does the shift directly depend on being above or below resonance?" Phase of reflected light will depend on which side of resonance laser is(check out cavity phase response). "I would have expected you to mix the output signal with the initial laser signal, such that the shift in phase could be determined and hence accounted for." You could in theory do ...

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As you suggested in your comment, the $\mu^-$ and $\mu^+$ that stop in matter do not have the same lifetimes. The $\mu^+$ come to rest between the atoms of your stopper (eg: scintillator?) and decay into $\nu_{\mu}e^+\nu_e$ with the standard 2.2 usec lifetime. However, the $\mu^-$ get captured into Bohr orbits about the stopper nuclei. The $\mu^-$ then ...

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I know that you are explicitly asked about not equipment related answers. But when I learned something from experimental physics then that you should always consider equipment flaws. I could imagine a scenario where the events on which you trigger to start/stop the clock have different rise times depending on where they take place in the scintillator, ...

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