# Accuracy and Error of Atomic Clocks

I'm quoting a passage from my notes:

The development of clocks based on atomic oscillations allowed measures of timing with accuracy on the order of $1$ part in $10^{14}$, corresponding to errors of less than one microsecond (one millionth of a second) per year.

I do not understand what the accuracy of $1$ part in $10^{14}$ means. Does it mean that the atomic clocks can tell us the time accurate and certain to $10^{-14}s$? How should I understand this? Moreover, what is meant by the error of one microsecond per year? Is it a kind of uncertainty in measurement? How should I understand it? I googled this topic and found information about the atomic clocks and also reviewed the definitions of accuracy and error; however, I'm not able to make any sensible connection between the concepts. Please help me, thank you.

It means that if the clock begins set to the correct time, then after time $t$ the clock will be wrong by no more that $(\pm 10^{-14}) t$.

Or as a physicist would be likely put it $$\frac{\delta t}{t} \le 10^{-14} \,.$$

This kind of expression of "fractional errors" is very common in many fields of quantitative science.

Now, to be concrete, a year is about $3.156 \times 10^7 \,\mathrm{s}$, so after one year the clock will be wrong by no more that $$(3.156 \times 10^7 \,\mathrm{s}) \cdot 10^{-14} = 3.156 \times 10^{-7} \,\mathrm{s} = 0.3156 \,\mathrm{\mu s} \,.$$

• An answer that tries to sound authoritative about error propagation should not fall into the mistake of quoting the error to 4 significant digits when the initial error estimate was given to just one. @dmckee since you have most of the credit for this community wiki, would you like to take another look? – Floris Jun 30 '15 at 20:47

The error would be in the order of 10^-14. This is mathematically similar to the sense of errors you have on your hand watch, caused by mechanical inaccuracy - probably in the range of 1 second per week, or 1 second per year if its a Rolex :)

One should note however, that such a very small inaccuracy in time measurement in atomic clocks is perhaps less than the error that would be caused by relativistic effects for a person who spends a lot of time driving. suppose that the average velocity between an observer and an atomic clock is 30 m/s, this would give a relativistic time dilation of about 1.5 microseconds per year, which is bigger than the inaccuracy of the atomic clock. For a pilot the time dilation would be about 15 microseconds per year :)

• In reality the best mechanical watches (including Rolex) are only accurate to 1-2 seconds per day. Most quartz watches have rated accuracies on the order of 15 seconds per month, better ones can achieve 2-5 seconds per month. The Bulova Precisionist watch is specified as no more than 10 seconds per year. I have a Citizen watch which is reset once a day by the NIST radio signal from Colorado so that its long term error is zero and its short term error is determined by its internal quartz oscillator which in practice has a negligible error between resets. – Barry Feb 9 '14 at 4:01
• I was joking about the Rolex, however, for the price you pay to get them they should be more accurate than atomic clocks :) – user40229 Feb 13 '14 at 4:06

I do not understand what the accuracy of $1$ part in $10^{14}$ means. [...] reviewed the definitions of accuracy and error [...]

In definitions of "accuracy" or "error" you should have noticed mentioning of

• the true value of some particular quantity, referring to the trial(s) under consideration, and

• the corresponding, commensurate value(s), concerning the same trial(s), whose "error" is to be qualified with respect to the true value.

If both sorts of values are for instance plain real numbers (or if they are scaled isometric to real numbers) then the "error" of the value to be qualified with respect to the true value is (usually) just their plain difference.

Such a difference may also be put in relation to the range (or to "a sensible, limited part of the range") of values of the operator by which the true value was obtained; yielding the relative error as a result.
The characterization "$1$ part in $10^{14}$" is an assertion of (a constraint on the magnitide of) relative error.

Now, in evaluating the (relative) error of a particular clock $\mathcal A := (A, \theta)$ under consideration, in a particular trial, the applicable true value is (simply) the duration of this clock $\tau A[~_{\text{finish}}, {}_{\text{start}}~]$, from its indication at the start of the trial, $A_{\text{start}}$, until its indication at the trial end, $A_{\text{finish}}$.
(If these indications are distinct, and the corresponding duration of the clock from one until the other is therefore nonzero, then this duration is also a "sensible" denominator for evaluating relative error, in this trial.)

The value to be qualified, on the other hand, has to do with the clock readings $\theta$ assigned to the indications of the clock; in particular the readings $\theta [~A_{\text{start}}~]$ and $\theta [~A_{\text{finish}}~]$.

As concrete example consider a trial which took the clock to be qualified exactly one year. The corresponding relative error, as a real number value, might then (naively) be expressed as $$\epsilon := \frac{\theta [~A_{\text{finish}}~] - \theta [~A_{\text{start}}~]}{\tau A[~_{\text{finish}}, {}_{\text{start}}~]} - 1 = \frac{\theta [~A_{\text{finish}}~] - \theta [~A_{\text{start}}~]}{1~\text{year}} - 1.$$

There are of course two problems with this attempt:

• that the (formal) fraction $\frac{\theta [~A_{\text{finish}}~] - \theta [~A_{\text{start}}~]}{1~\text{year}}$ should be a real number value at all.
Consequently the clock readings $\theta$ might be "understood" as including some "unit" of duration; for instance both $\theta [~A_{\text{start}}~]$ and $\theta [~A_{\text{finish}}~]$ representing some real-number multiple of "$\text{year}$"s. And:

• that it presumes some external definition to the unit "$\text{year}$". If it should be conceivable that a given clock in a given "exactly $\text{year}$"-long trial had not been perfectly accurate, whereby the difference "$\theta [~A_{\text{finish}}~] - \theta [~A_{\text{start}}~]$" is necessarily designated as (a duration of) "not exactly $1~\text{year}$", then this particular unit apparently appeals to some external artefact (whose accuracy is itself doubtful in the first place).

But in Physics, we require definitions and result values to be independent of any particular choice of units (such as "$\text{year}$"), and independent of references to unique artefacts (such as "Earth orbiting the Sun"); instead we seek intrinsic definitions.

This is foremost accomplished by evaluating the error of a given clock in terms of ratios: for any three distinct indications of the clock being qualified, $A_M$, $A_P$, and $A_Q$ we can evaluate separately

$$\frac{\tau A[~_M, {}_P~]}{\tau A[~_M, {}_Q~]}$$

(as the true value), and

$$\frac{\theta [~A_P~] - \theta [~A_M~]}{\theta [~A_Q~] - \theta [~A_M~]}$$

(as the value to be qualified with respect to the true value); and consequently we can evaluate the relative error (for the three selected indications of the clock to be qualified) as

$$\epsilon_{\mathcal A}[~{}_{M, P, Q}~] = \frac{\left(\frac{\theta [~A_P~] - \theta [~A_M~]}{\theta [~A_Q~] - \theta [~A_M~]}\right)}{\left(\frac{\tau A[~_M, {}_P~]}{\tau A[~_M, {}_Q~]}\right)} - 1.$$

A given clock is consequently accurate (a.k.a. "good"), thoughout a trial under consideration, if its errors for any three (distinct) indications within the trial are null.

In order to quantify the (finite) accuracy of clock $\mathcal A$ which is not (perfectly) accurate, throughout a trial under consideration, i.e. for a given set $A$ of its (subsequent) indications, beginning with "mark" indication $A_M$ up to some particular final (ultimate) indication $A_U$, physicists can use for instance the method of least squares and determine the corresponding "sum $S_{\mathcal A}$ of squared residuals" as $$S_{ \! \mathcal A } := \sum_{A_{ \mathcal N } \, = A_M}^{A_U} \left( \frac{\tau A[ _M, _{\mathcal N} ]}{\tau A[ _M, _U ]} \right)^2 - \frac{ \left( \sum\limits_{A_{ \mathcal N } \, = A_M}^{A_U} \left( \theta[ A_{ \mathcal N } ] - \theta[ A_M ] \right) \frac{\tau A[ _M, _{\mathcal N} ]}{\tau A[ _M, _U ]} \right)^2 }{\sum\limits_{A_{ \mathcal N } \, = A_M}^{A_U} \left( \theta[ A_{ \mathcal N } ] - \theta[ A_M ] \right)^2 }.$$

For the clock under consideration then $$\sqrt{S_{ \! \mathcal A } } \le 10^{-14},$$ or factoring out the common denominator $\tau A[ _M, _U ] := \tau_{ A }$: $$\frac{\delta \tau_{ \mathcal A }}{\tau_{ A }} \le 10^{-14},$$ for any set of (subsequent) indications $\mathcal A$ of the clock;
where $$\delta \tau_{ \mathcal A } := \sqrt{ \sum_{A_{ \mathcal N } \, = A_M}^{A_U} \left( \tau A[ _M, _{\mathcal N} ] \right)^2 - \frac{ \left( \sum\limits_{A_{ \mathcal N } \, = A_M}^{A_U} \left( \theta[ A_{ \mathcal N } ] - \theta[ A_M ] \right) \tau A[ _M, _{\mathcal N} ] \right)^2 }{\sum\limits_{A_{ \mathcal N } \, = A_M}^{A_U} \left( \theta[ A_{ \mathcal N } ] - \theta[ A_M ] \right)^2 } }.$$

Does it mean that the atomic clocks can tell us the time accurate and certain to $10^{-14}\text s$ ?

In practice, apparently not: For Cs133 clocks, for instance, the ostensible readings $\theta$ are the counts of their "oscillation periods"; and there are (ideally, accurately) only $9,192,631,770 \approx 10^{10}$ to be counted within the duration of one second. Therefore it might take a trial of several hours (in the "worst case") to find an actual one-count difference between a Cs133 clock whose relative error would be found of order $10^{-14}$ throughout a still much longer trial, and one which would be found perfectly accurate.

Similarly, looking at trails of $\approx 3 \cdot 10^{17}$ counts of "oscillation periods" (i.e. in the order of one year), the claim of "relative accuracy of $1$ part in $10^{14}$" means roughly, that trials of exactly equal duration should differ in at most a few thousand counts.