# Experimentally Verifying a Clock's Accuracy

So recently one of my professors went off on a tangent, and we ended up discussing atomic clocks and how they work, which is something I've always been fascinated with and thoroughly enjoyed. But it left me with a question he couldn't answer.

After you've built an atomic clock (or any clock for that matter), how do you verify that it is, in fact, working at a specified level of accuracy?

Let me be clear here. I understand that by knowing the properties of the atoms and the lasers and all the other parts, you can calculate what frequency it should work at, but once it is working, how do you verify that it is working at that level, without needing a more accurate clock?

Once again, to be clear:

I'm NOT asking about how they build/design the clock.


I want to know how you can experimentally prove that a clock is working at its theoretical accuracy (or within an acceptable level of accuracy).

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This is an excellent question. At the heart of it is this: you compare the clock to another copy of the same clock. Well, actually you need to compare three identical clocks to each other to make a strong statement about the clock noise, but lets not worry about that for now. If the noise of your clock is stationary (which it better be for a good clock), then the noise you measure will be given by $$S_t=\sqrt{S_1+S_2}=\sqrt{2}S_c,$$ where $S_t$ is the total noise and $S_c$ is the noise of the identical clocks.

On a technical note, you might be interested to know how one can compare two clocks. The answer to this lies in the fact that clocks are just oscillators producing waves of some sort. Some examples of devices which can be considered clocks are: mechanical oscillations of quartz crystals, atomic transition lines (in this sense a laser is a clock too), or the orbital period of a dead ultradense star. Comparing two oscillators is as simple as summing the two waves. This produces a beat note which tells you the difference in frequency between the two, i.e. the difference in timing between the two.

An interferometer is very similar in this regard. How do you measure a length much smaller than any ruler you have? You compare it to another length which is measured just as accurately.

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Chris Mueller: "If the noise of your clock is [...]" -- Please define "noise of a clock" (as a quantity to be measured). "Comparing two oscillators is as simple as summing the two waves. This produces a beat note which tells you the difference in frequency between the two" -- How should be determined whether any one of those oscillators had been oscillating at some particular "frequency" at all, i.e. such that its "oscillation periods" were all equal, and not of varying durations? – user12262 Mar 27 '14 at 23:21
Noise of a clock: $\frac{\delta f}{f}$ the amount which the frequency changes relative to the mean frequency over a given timespan, typically expressed in the frequency domain. How to determine that it has a particular frequency: beat it against another identical one. I don't understand the last part of your question, but maybe you're asking how to tell if the noise of the clock is stationary. Uncommon non-stationarity will show up in the measurement, but common non-stationarity will not; you have to rely on theory to rule this out. – Chris Mueller Mar 27 '14 at 23:36
Chris Mueller: "How to determine that it has a particular frequency: beat it against another identical one." -- Suppose you had two clocks; distinguishable, but co-located (to avoid other "complications"). Suppose they'd not tick (or, as required, oscillate) "in unison", so there could be some non-zero "beat" at all. Might they then be called "identical"?? "common non-stationarity [...] you have to rely on theory" -- I'm not sure which or how much such "theory" the OP would tolerate; but note the discussion of "ideal clocks" in MTW §16.4 – user12262 Mar 28 '14 at 6:31
Quantum mechanics forbids identical clocks from having exactly the same frequency all of the time. Heres a paper (published in Nature, this is the preprint copy on the arXiv) where they claim accuracy and stability at the $10^{-18}$ level. How did they measure it? From the section titled Frequency Comparison: "To demonstrate the improved performance of lattice clocks, we have built two Sr clocks in JILA" – Chris Mueller Mar 28 '14 at 14:14
Chris Mueller: "["An Optical Lattice Clock with Accuracy and Stability at the 10^{-18} Level", B.J. Bloom et al., arxiv.org/ftp/arxiv/papers/1309/1309.1137.pdf ...] How did they measure it? [...]" -- They wrote: [N]eutral atom clocks with many ultracold atoms confined in magic-wavelength optical lattices have the potential for much greater precision than ion clocks. This potential has been realized only very recently [...] We have used this measurement precision to systematically evaluate important effects How to relate such precision (good for them!) to accuracy (MTW §16.4) ?? – user12262 Mar 29 '14 at 7:16

That's a fundamental problem with all measurement devices.

What you can do is to compare your clock with other clocks. If yours differs significantly from the average then you know your clock is inaccurate.

Of course, this does not protect you from overall errors (i.e. a fundamental flaw in the mechanics/physics behind the device).

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Kvothe: "[...] compare your clock with other clocks. If yours differs significantly from the average then you know your clock is inaccurate." -- No, that's not a measure of accuracy (or rather trueness) of the one clock in question, but (merely) of its precision wrt. the other clocks that might be available or considered for comparison. – user12262 Mar 27 '14 at 21:15

Let's first recall that there is a strict definition for (how to determine) whether a given clock $A$ is "accurate" (or "good"); namely:

if for any three of its indications, $A_J$, $A_K$ and $A_Q$,

the durations of clock $A$ between pairs of those indications, say $\Delta \tau_A [ \small{\text{from }} A_J \small{\text{ until }} A_K ]$ and
$\Delta \tau_A [ \small{\text{from }} A_K \small{\text{ until }} A_Q ]$,

and the real numbers $t[ A_J ]$, $t[ A_K ]$ and $t[ A_Q ]$ which are associated to the clock indications as readings

satisfy the relation

$$\frac{(t[ A_K ] - t[ A_J ])}{(t[ A_Q ] - t[ A_K ])} == \frac{\Delta \tau_A [ \small{\text{from }} A_J \small{\text{ until }} A_K ]}{\Delta \tau_A [ \small{\text{from }} A_K \small{\text{ until }} A_Q ]}$$

then this clock $A$ is "accurate", or "good"; and otherwise it is not.

As far as the readings "$t$" are considered given (in the simple case of a "ticking" clock for instance as "the number of consecutive ticks" counted after a suitable "starting tick") and the left-hand-side of the equation is readily calculated, the remaining task is therefore to evaluate the ratio of durations on the right-hand-side. (In the described case of a "ticking" clock the question becomes "simply", whether this clock's duration from one (initial) "tick" indication to the next was and remained equal, for any initial "tick" indication; or how those durations varied.)

The short, perhaps superficial answer to your question is consequently that the accuracy of any given clock cannot be determined without comparison to a clock of known accuracy; ideally such as the "ideal clocks" considered by MTW §16.4.

A more careful answer would not only present the constructions of various such "ideal clocks", but also discuss the assumptions (or "construction pieces") which they require, and whether they comply to Einstein's maxim (concerning experimental determinations) that

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Identical clocks can be compared to each other to measure their noise. If you indeed needed a better reference to establish the accuracy of a clock, how would we ever establish the accuracy of any clock? – Chris Mueller Mar 27 '14 at 23:43
Chris Mueller: "Identical clocks [...]" -- Clocks which are distinguishable but equal by some (or several) particular measure(s). But by which measure(s), in particular?? "how would we ever establish the accuracy of any clock?" -- How about MTW Box 16.4 ("Ideal Clocks Built from Geodesics")? But surely there's a bit more to it (which makes the OP's question so interesting), such as "Characterizability of Free Motion ...", U. Schelb, FP 30(6):867 (2000) which (unfortunately) still needs/presumes "clock readings $t_A$ differentiable wrt. $\tau_A$". – user12262 Mar 28 '14 at 6:47