# What is a "cycle time" in Allan deviation formula for atomic clock instability? And why does having more independent atoms reduce $\sigma_y(\tau)$?

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?

• $T_{c}$ 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.
– nzag
Jan 13 at 12:00