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The usual formula for clock instability is given as

$\sigma_y(\tau)\approx\frac{\Delta f}{f_0\sqrt{N}}\sqrt{\frac{T_c}{\tau}}$

First off what do each of these symbols really mean? What is $T_c$? The literature mentions it is cycle time. What "cycle" is it? Is it the period of oscillation $1/f_0$? But, that can't be right as it is already present in the formula.

Next, as per my intuition, the more independent atoms $N$ you have more scrambled the phase of the clock would be. So I don't understand why it is in the denominator. Same for the averaging time $\tau$. Why is it in the denominator? Wouldn't the phase of a clock made of independent atoms scramble more and more with time?

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    $\begingroup$ $T_{c}$ denotes sampling time for white frequency noise. I'll cite a paper doi.org/10.1109/TUFFC.2021.3061005 maybe, you can find answer to other half of your question. $\endgroup$
    – nzag
    Jan 13 at 12:00

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Partial answer: Each atom is an independent measurement of the local oscillator frequency so you get more statistics on your frequency estimate each shot. However, you’re right that if different atoms are frequency shifted relative to each other due to, for example, a non zero velocity distribution or a inhomogeneous zeeman shifts due to an inhomogeneous magnetic field, then yes this can impact clock stability. The standard quantum limit for clock performance that you cited assumes there a no such broadening mechanisms like these.

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