Clock uncertainty / building a better clock

Question

Let's say, hypothetically, I've built the best (lowest uncertainty) clock in the world. How is this proven?

We can bring my new clock in the next room to a caesium fountain, optical lattice, whatever, and they will drift apart, as all clocks do. After 1 week, how do we know which is "better", and what does better mean in this context?

Feels like I'm lacking knowledge of some basic part of the experimental process around quantifying uncertainty. Do I need to build 2 clocks and compare them to each other?

Answer 1

Typically you build an ensemble of at least three of your new clocks and at least three of the previous best clocks. You put them together to measure a set time. At the end of that time you read what all the clocks say. If the new clock is more precise than the previous clock, then the standard deviation of the new clocks will be smaller than the standard deviation of the previous best clocks.

You can also get similar information by comparing a single new clock to the mean of a larger group of the previous best clocks. If the new clock tracks the mean of the previous clocks better than any individual previous clock, then that also indicates that the new clock is more precise.

These are the usual strategies for all new metrological devices. Not just clocks.

Answer 2

At the bottom end of what you can do, there's the "three-cornered hat" measurement. You need your new clock and two other clocks. The other two clocks don't have to be better, or even comparable to, your new clock (although the better they are the more reliable the results are). What is important is that the errors of all three clocks are statistically uncorrelated. Then you measure the difference between all three pairs of clocks (A vs. B, A vs. C, B vs. C) many times over a period of time.

Under the assumption of uncorrelated errors, you can calculate:

$$\sigma^{2}_{A} = \frac{1}{2}\left(\sigma^{2}_{ab} + \sigma^{2}_{ac} - \sigma^{2}_{bc}\right) \\ \sigma^{2}_{B} = \frac{1}{2}\left(\sigma^{2}_{ab} + \sigma^{2}_{bc} - \sigma^{2}_{ac}\right) \\ \sigma^{2}_{C} = \frac{1}{2}\left(\sigma^{2}_{ac} + \sigma^{2}_{bc} - \sigma^{2}_{ab}\right) \\ $$

(source; I reformatted the equation in the introduction slightly). $\sigma^{2}$ represents "variance", and these equations are valid for the conventional statistical variance, but when measuring clocks you would probably be using the Allan variance (AVAR) or modified Allan variance (MVAR), and the equations are valid for them too, as long as all three $\sigma^{2}$s are of the same type (and the same $\tau$ for AVAR and MVAR).

In other words, you can calculate the quality of your clock — its variance against "perfect time", given the three sets of pairwise variances, even if the two other clocks are worse than yours.

In real life, the assumption of uncorrelated errors never holds exactly, and it's impossible for this method to tell the difference between an error that affects A, and an error that affects both B and C consistently. When the assumptions are violated, the variances that come out will be inaccurate, and will sometimes even be negative (which is physically impossible). Likewise if B and C are too much noisier than A, or if the collection period simply isn't long enough, A's true variance may be "lost in the noise". Those are downsides. Nonetheless, it's commonly assumed that if we design an experiment well, use the best standards we can get our hands on (but of different designs), and isolate from common environmental influences, we can get measurements that will be useful for some purpose (maybe while tweaking your new design to find the best stability).

Then when you need to characterize things even better you can use bigger ensembles, as in Dale's answer, to reduce your odds of getting fooled by two clocks that just happen to zig and zag in a coordinated way.

Answer 3

As the other answers have stressed you need to build a minimum of two of these same clocks and have them measure a time interval while monitoring the relative delay between the two clocks. This is only a measure of the stability of your oscillators and is a measure of the time it takes for your clock to average down to a certain level of uncertainty (in fractional frequency terms).

The other factor which has been only alluded to is the systematic effects which alter the frequency of your clock and are a limit on the accuracy of the device. You would be required to demonstrate in a rigorous manner that you have compensated for these frequency shifts due to e.g. gravitational potential, ac stark effect, magnetic fields.

If you pass all these hurdles and your clock does not fail to agree with the existing frequency standards within the error margins, then you will be able to contribute data to the BIPM for participation in the generation of international atomic time.