# Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured?

The definition of the SI unit "second" is stated as

with the explicitly added note that

In referring to a caesium atom in its "ground state", does this definition pertain to caesium atoms that are plainly and exactly unperturbed, whether by black body (ambient) radiation or due to any known or unknown perturbation?

If so, is there any requirement to determine (and possibly correct for) the perturbation, or "shift", of any and all primary frequency standards, besides the described "shift due to ambient radiation"?

In particular, is there any requirement to measure whether the durations of 9 192 631 770 periods of different primary frequency standards and/or of the same primary frequency standard in different trials, had been and remained equal to each other, by (presumably) unambiguous means (such as the "ideal clocks" described in MTW ยง16.4) ?

EDIT
In response to comments, the following are excerpts of two relevant sections of MTW, "Gravitation":

Box 16.4: Ideal Rods and Clocks Built from Geodesic World Lines; Based on Marzke and Wheeler (1964)

Each geodesic clock is constructed and calibrated as follows:

(1) A timelike geodesic $\mathcal{ AC }$ (path of a freely falling particle) is passing through $\mathcal{ A }$.

(2) A neighboring world line, everywhere parallel to $\mathcal{ AC }$ [...] is constructed by the method of Schild's ladder (Box 10.2). [...]

(3) Light rays (null geodesics) bounce back and forth between these parallel world lines; each round trip constitutes one "tick". [...]

(4) The proper time lapse, $\tau_0$, between ticks is related [...] $s^2[~\mathcal{ AB }~] = -(N_1~\tau_0)~(N_2~\tau_0)$, where $N_1$ and $N_2$ are the number of ticks between events shown in the diagram.

$$~$$ $$\textbf{ <Insert MathJax source code for generating an appropriate diagram here>} .$$

Spacetime is filled with such geodesic clocks. Those that pass through $\mathcal{ A }$ are calibrated as above against the standard interval [...] and are used subsequently to calibrate any other clocks they meet.

Any interval [... $s^2[~\mathcal{ PQ }~]$ ... with event $\mathcal{ P }$] along the worldline of a geodesic clock can be measured by the same method [...]

To achieve a precision of measurement good to one part in $N$, where $N$ is some large number, take two precautions: [...]

The M-W construction makes no appeal whatsoever to rods and clocks of atomic constitution. [...]

Box 10.2: From Geodesics to Parallel Transport to Covariant Differentiation to Geodesics to ...

A. Transport any sufficiently short stretch of a curve $\mathcal{ AX }$ [...] parallel to itself along curve $\mathcal{ AB }$ to point $\mathcal{ B }$ as follows:

(1) Take some point $\mathcal{ M }$ along $\mathcal{ AB }$ close to $\mathcal{ A }$. Take geodesic $\mathcal{ XM }$ through $\mathcal{ X }$ and $\mathcal{ M }$.
[...] define a unique point $\mathcal{ N }$ [on geodesic $\mathcal{ XM }$] by the condition [...] "equal stretches of time in $\mathcal{ XN }$ and $\mathcal{ NM }$".

[...]

(4) Repeat process over and over [...] Call this procedure "Schild's Ladder" from Schild's (1970) similar construction [...].

• This seems like two different questions to me. Most of the question is about the perturbing effect of blackbody radiation, but the final paragraph seems to be about anomalous shifts in clock rates over time. – user4552 Aug 26 '14 at 18:49
• @Ben Crowell: "This seems like two different questions to me." -- Well: the present OP question text contains (even) three separate question marks. "Most of the question is about the perturbing effect of blackbody radiation" -- Not at all. I'll accept for the purpose of my question that "the perturbing effect of blackbody radiation" is settled, e.g. with results as seen in arxiv.org/abs/1107.2412 (Tab. 2). Instead, I am (only) trying to ask about "the perturbing effect of anything else" not listed e.g. in that table. Is that called "(due to) anomalous (reasons)"? – user12262 Aug 26 '14 at 20:44
• Your last paragraph is unclear to me. There's no way to compare a clock's output with what it gave out yesterday, and definitely not without external assumptions (as discussed e.g. here). You should clarify it but it probably deserves its own post. – Emilio Pisanty May 20 '15 at 10:17
• @Emilio Pisanty: "There's no way to compare a clock's output with what it gave out yesterday," -- MTW Box 16.4, "Ideal Rods and Clocks Built from Geodesic Worldlines", seems to suggest otherwise. Though I haven't been able to discern whether and how MTW supposed to distinguish "geodesic worldlines" from "non-geodesic worldlines", or how to distinguish an "affine parametriziation" of a worldline (required in Box 10.2) from any "non-affine parametriziation", before and without having "ideal rods and clocks" available already. "external assumptions" -- Hardly; only coincidence determinations. – user12262 May 20 '15 at 21:23
• Apologies, I don't have easy access to MTW. I stand by what I said on external assumptions, but if you want me to address a passage in MTW then you will need to provide it in full. (This is sort of what happens if you ask an atomic physics question and then expect the answerers to be fully conversant in GR.) That is probably the subject of another question, though - feel free to ping me here if you do post it separately. – Emilio Pisanty May 20 '15 at 21:40

is there any requirement to determine (and possibly correct for) the perturbation, or "shift", of any and all primary frequency standards, besides the described "shift due to ambient radiation"?

Yes. These are called "systematic errors" and they are the order of business pretty much all day, every day, at the metrology labs that implement frequency standards. This includes the effect of thermal radiation, called the blackbody radiation shift, as well as a number of other effects which can perturb the atoms. Chief among these are Zeeman shifts from stray magnetic fields in the lab, but you also get Stark shifts, light shifts, and a bunch of implementation-specific effects. If you're trapping your atoms in place, for instance, you need to account for any perturbations this might make to the ground state; if you're shooting them up in a fountain clock then you need to worry about things like gravitational redshift between the bottom and the top of the fountain.

The definition of the SI second is for caesium in its true, ideal ground state, with no perturbations whatsoever. This is obviously an idealization, and any implementation simply needs to estimate the accuracy with which it conforms to the idealization, report it, and move on.

To quote a terrible, terrible man, when you try to build a frequency standard you have two types of perturbations to deal with: known unknowns and unknown unknowns.

• With some perturbations, you know they exist, and you do your best effort to minimize them and to estimate their values (which will always be nonzero). Any serious metrological paper will contain an error budget in which they report all the different significant perturbations on the experiment and estimations on their magnitude.

As an example, here is the one from arXiv:1505.03207, and you can find more examples via e.g. this arXiv search.

The blackbody radiation shift often makes an appearance in these budgets, and it is one of the hardest to estimate and reduce. It is generally quite small, which means it was swamped by all the other systematics from the development of the atomic clock in the 50s until technology brought down all the other systematics, in the 90s, to the point where the blackbody radiation shift became an important consideration.

However, as you can see, it is only one factor among many that need to be estimated and controlled.

• Once you've done your very best effort to identify all the possible effects that might perturb your ground state, or affect the accuracy of your results in some way, you still don't know whether there are any other effects that you might have missed. Unfortunately, by definition, the perturbations which you missed are impossible to quantify - if you could, they'd be in the known unknowns category.

There is really no way to deal with this within a single experiment. The possibility of such effects can only be mitigated by producing many different implementations which are based on different physical principles and are as different from each other as possible. If we build two clocks with low known systematics using completely different schemes and they tick in step within the known uncertainties, this increases our confidence that we have indeed identified all the relevant systematics, at least at that level of precision.

Nevertheless, the possibility of systematic errors which we're not yet aware of is taken very seriously by metrologists. Each lab as a unit can only really implement one clock at a time, but the field as a whole does its best effort to ensure we have many different approaches which can validate each other's values and uncertainties.

At your (rude and abrasive) insistence I have taken a look at MTW's ideal clocks, as described in box 16.4. (Take this as a good example of how "provide in full" is not hard, requires minimal technical skills, and enables everyone involved to know what is being talked about.)

It is completely unclear to me how you propose to implement this, based as it is on free-falling mirrors which are somehow meant to maintain perfect alignment though they have no connection to each other. There are finer points such as radiation pressure on the mirrors and shot noise on the laser (both of which you should understand thoroughly before attempting to reply) but simply put, MTW's ideal clocks are just that: an idealization.

I stand by what I said:

There's no way to compare a clock's output with what it gave out yesterday, and definitely not without external assumptions.

Any clock you build will be made of atoms and it will remain on a non-freefalling worldline on Earth's surface (unless you send it to orbit, of course). Show me an actual, physical setup that can compare the frequency of an oscillator now with its frequency yesterday, and I will show you the assumptions it rests on.

As far as MTW's ideal clocks are concerned, the assumptions are essentially the temporal constancy of the speed of light, but this is because MTW completely redefine the basic unit of time.

Suppose that you could implement an MTW clock: you launch into orbit two mirrors and a caesium clock. You position your mirrors a length $L$ apart and you set them on a free-falling orbit, with a ray of light bouncing between them. By a happy coincidence, you can choose $L=\tfrac12 c/9 192 631 770\:\mathrm{Hz}\approx 18\:\mathrm{cm}$ as a convenient length at which one cavity round trip will correspond to one Rabi oscillations for the caesium atoms. The light and the atomic oscillators should therefore be completely in step.

Suppose, further, that for some reason after a number of orbits (probably quite large) the light in the cavity is no longer in step with the caesium atoms, and does $n$ round trips for every $n'\neq n$ caesium oscillations. Can you conclude that the frequency standard has changed?* As it turns out, this is an ill-defined question. How can you distinguish that conclusion (i) from the alternative interpretations that (ii) the distance between the mirrors changed, or (iii) the speed of light did?

To decide whether (ii) is true, you can accompany your setup by a ruler (i.e. an actual ruler made of atoms) which can measure the distance between the two mirrors. There are two possible outcomes to this: the distance either changes or it doesn't. If the distance changes, do you conclude that space expanded? Or that the atoms in the ruler contracted? Either is a perfectly valid explanation, and the only difference between the two is what you define 'length' to be.

If the distance between the mirrors, as measured by the ruler, doesn't change, then you are exactly on the setup discussed here. The speed of light, as measured in atomic units, changed. Or, the atomic unit of time (as a geometric length in spacetime) changed. You are attempting to measure the change of a dimensionful constant, and this is impossible without external assumptions: constancy of atomic time for the first, and constancy of the speed of light for the second. Both interpretations are indistinguishable even in principle.

However, in practice, the conclusion to draw is in fact that the speed of light changed. The reason for this is that us humans, and everything around us, are made of atoms, and therefore we are built on (fixed) multiples of the atomic length and we operate on (fixed) multiples of the atomic timescale. If the atomic unit of time expands by two, we cannot notice, as our brains and clocks will slow down to match. Indeed, the only observable effect is that light now covers twice the distance in the same amount of time. As far as humans are concerned, it's light that sped up.

* Note that I am leaving measurement error out of this discussion. If you actually tried to implement this, it should be clear which of the two components - the optical clock and the magical set of freefalling perpetually-aligned mirrors - is more susceptible to experimental error.

• Emilio Pisanty: Very sorry, only just now I noticed that you had posted this lengthy answer. (Btw., AFAIU its URL is specificly " physics.stackexchange.com/a/184999 "). So, thanks for that; I'll need to take a while to read it ... And, due to weekend etc. I may be able to submit some additional comment (as I surely may want to) only late on Monday. – user12262 May 21 '15 at 22:47
• Yes, that was intentional. Where else would I have posted an edit that I would expect you to have seen? This is the kind of thing that makes you come across as abrasive, by the way. Take your time if you need to. – Emilio Pisanty May 21 '15 at 22:57
• Emilio Pisanty: "Yes, that was intentional." -- Are you suggesting that I was lying in my preceding comment, and/or that I didn't in good conscience carry our correspondence right below the OP question since yesterday?? (<outrage streniously suppressed>) Well, no. Think again about the working conditions and habits of PSE contributors. I had been curious (and I had asked) when you mentioned "your edit" the first time (I considered whether you had referred again to that); the second time "your edit" came up I discovered this answer here right away. – user12262 May 22 '15 at 6:02
• p.s. On first looks this answer of yours, especially the "Addendum" seems very interesting, thoughtful, substantial, and ... well ... very much debatable/contestable. (Indeed closely along the lines of physics.stackexchange.com/q/78684 where I put my two cents already). Indeed too interesting for me to get hung up too much about your preceding comment. (And, therefore, it may take rather until mid next week to express a reply; perhaps best in form of an own separate answer to my question; of which I plan to notify you explicitly, because, thankfully, you seem interested in the Physics.) – user12262 May 22 '15 at 6:02
• Simply put, I expect you to do your due diligence and check whether I have edited my answer when I say I have. I responded on the OP because, for reasons best known to yourself, you chose to reply to my answer there. That said, please keep the discussion to the point, and about the physics. – Emilio Pisanty May 22 '15 at 9:36

I don't know if I'm right, but here is an attempt to estimate one effect that might be relevant. If a 133Cs atom of mass $m$ is in thermal equilibrium with blackbody radiation at temperature $T$, then it has an average kinetic energy $(1/2)mv^2=(3/2)kT$. This will cause Doppler shifts. The longitudinal Doppler shift cancels out on the average, but the transverse Doppler shift, which is by a factor of $\gamma$, doesn't. The average effect is $\gamma-1=3kT/2mc^2$. I suppose the cesium has to be a gas, so the minimum actual temperature would be 944 K. Putting this in, I get $\gamma-1\sim 10^{-12}$. This seems below the $\sim10^{-10}$ precision implied by the number of sig figs in the standard, but maybe it's anticipated that future improvements in technology would make it relevant.

• Ben Crowell: "I don't know if I'm right, but here is an attempt to estimate one effect that might be relevant." -- Well, you've completely missed the intended point of my question ... Perhaps it's helpful to contrast rather symbolically: Your answer seems to be concerned with "reasoning out" $$\frac{\partial}{\partial~T}[~f~]~|_{~T =0~ \text K,~f = f_S} \times (T - 0~\text K)$$ while I like to know about (how to determine experimentally) $$f - f_S,$$ where $f_S$ denotes the "standard transition frequency of a plainly and exactly unperturbed Cs133 atom", and $f$ is "of the given sample". – user12262 Aug 26 '14 at 23:00
• p.s. Btw., some more or less rigorous derivation of how to "correct for the shift due to ambient radiation" might be gathered for instance from "Black Body Radiation Shift of the 133Cs Hyperfine Transition Frequency". But, again, that's not at all what I like to know ... – user12262 Aug 26 '14 at 23:11
• p.p.s. I appreciate that you let your answer stand despite my criticism; thus allowing my comments to stand. – user12262 Aug 28 '14 at 18:23
• This is not really what the provision is addressing. Doppler shifts are eliminated by having the atom at rest. The additional requirement that the ambient temperature be 0K is to eliminate light shifts (i.e. dynamic Stark shifts) from the thermal radiation itself. – Emilio Pisanty May 20 '15 at 9:41

is there any requirement to measure whether the durations of 9 192 631 770 periods of different primary frequency standards and/or of the same primary frequency standard in different trials, had been and remained equal to each other, by (presumably) unambiguous means (such as the "ideal clocks" described in MTW ยง16.4) ?

There's no explicit mentioning of such a requirement that I know.

However, if some unambiguous means of comparing durations (such as the "ideal clocks" described in MTW ยง16.4) were taken into consideration then the comparison of durations of oscillation periods allows(1) to conclude

• whether any one given instance of an oscillator (such as a "primary frequency standard") had been constantly "perturbed" (even possibly including having been constantly "un-perturbed"), or variably perturbed; for whatever expected or unexpected "reasons", and

• whether or not any two given primary frequency standards had been equally perturbed (even possibly including having been constantly "un-perturbed"), or not; for whatever expected or unexpected "reasons".

Related considerations are conveniently presented in response to issues raised by this answer which had been posted earlier, from which the following quotes were taken:

how [...] to implement this

There's no strict requirement to actually implement an "ideal clock" according to MTW's description; but merely to judge and quantify (or even only to estimate) how the relation between given setup participants differs from having constituted such an "ideal clock", to be "corrected" as suitable.

(Similarly there's no strict requirement to actually implement "caesium atoms unperturbed by black body radiation"; but the requirement presents a definitive ideal relative to which the given primary frequency standards should be "corrected".)

Still it may be asked which sort of observational data might be the basis of such a quantification at all(2). Looking at the illustration of MTW Box 16.4 I'd think foremost of the (possible) appearance (or disappearance) of "interference patterns" involving the relevant setup constituents.

external assumptions [...] Suppose that you could implement an MTW clock: you launch into orbit two mirrors and a caesium clock. [...] The light and the atomic oscillators should therefore be completely in step.

(To simplify the discussion: let's say this is found during an "initial setup phase" of at least $n'$ caesium oscillations.)

Suppose, further, that for some reason after a number of orbits (probably quite large) the light in the cavity is no longer in step with the caesium atoms, and does $n$ round trips for every $n' \ne n$ caesium oscillations.

First to note regarding "orbits" is that the MTW prescription (as quoted in the excerpt) involves certain necessary "precautions", or (arguably) "corrections". If $n$ represents the corresponding "precise" number, and correspondingly $n' \ne n$ caesium oscillations were found during the "trial phase" then:
the mean oscillation frequency of the caesium atomic oscillators had changed in comparison to the "setup phase"; it had been "perturbed" differently in the "trial phase", in comparison to the "setup phase".

How can you distinguish that conclusion (i) from the alternative interpretations that (ii) the distance between the mirrors changed,

Certainly the (relevant, "cautious" or suitably "corrected") tick duration of the orbiting MTW clock remained constant; as a matter of definition.

Of course, the tick duration(s) (or "ping duration(s)", or "signal roundtrip duration(s)"), either as "corrected", or "uncorrected" ("raw"), may be taken as measures of "spatial separation" between relevant setup constituents;
typically (for formal distinction from all sorts of other durations) with some fixed symbolic non-zero prefix attached, such as "$c_0$", or "$c_0/2$".

or (iii) the speed of light did?

It is certainly absurd that a mere and supposedly fixed (non-zero) symbol such as "$c_0$" should have changed from "setup phase" to "trial phase"; at all, and especially by a real number value "$n / n' \ne 1$".

you can accompany your setup by a ruler (i.e. an actual ruler made of atoms)

But relevant is surely not just any actual ruler made of atoms, but only such actual rulers made of atoms for which the "spatial separation between its two ends" (or "between two relevant marks") remained equal to a significantly better ratio than the real number value "$n / n' \ne 1$".
So how to determine which of all given or even all imaginable rulers satisfy this requirement? (The SI "metre" definition should give a valuable clue.)

The definitions (idealized thought-experimental descriptions) of how to measure "duration" and "spatial separation" are of course human conventions.
But there are very explicit and useful guidances on which of all imaginable conventions are preferrable, namely Einstein's assertion:

in fulfillment of Bohr's requirement:
{W}e must employ common language {...} to communicate what we have done and what we have found.

in practice [...] us humans, and everything around us, are made of atoms, and therefore [...]

... therefore we may want to determine their possible "perturbations", trial by trial, even if we had not expected them, and even if we have not yet pinned down their possible "reasons".

Notes (added after the initial posting):

1: Utilizing the notion of "ideal clocks" as described in MTW ยง16.4, a concrete definition of a duration unit (fittingly called here one "artefact second") might be the following:

"The artefact second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atoms of the NIST primary frequency standard, referred to being at rest, at a temperature of 0 K, starting at UTC date January 1st, 2000, 00:00:00 .
This definition refers specificly to the artefact second being distributed in reference to the Marzke-Wheeler procedure."

2: That is, only as far as it is considered impractical to gather observational data which is explicitly required for carrying out the Marzke-Wheeler procedure, and thus to identify a suitably dense network of "ideal clocks"; including explicit coincidence determinations such as at the tick event preceding event $\mathcal{ C }$ in the first illustration of MTW box 16.4.

• Overall it's not very clear to me what you're saying, but I do have some comments. – Emilio Pisanty May 27 '15 at 10:44
• 1. I don't understand why you're so hung up about the mention of blackbody radiation. The bulk of the systematics corrections go into the part of the definition which asks for "the ground state of the caesium 133 atom". This is never achievable in the lab because of various Zeeman, Stark and light shifts. Demanding ground-state atoms is equally (if not more) contentious than asking for blackbody radiation shifts not to be present. – Emilio Pisanty May 27 '15 at 10:48
• 2. It should be obvious that no implementation could possibly use "a caesium atom at rest at a temperature of 0 K", particularly since that would violate the third law of thermodynamics. You seem to be confused by the distinction between the standard and its implementation. – Emilio Pisanty May 27 '15 at 10:50
• 3. The mirrors in your MTW implementation seem to be exempt from experimental error (or, as you would call it, "perturbation"), which makes them even more magical than I thought. In practice there are fundamental limits to contend with, starting with the radiation pressure on them. You simply cannot postulate that the mirrors are on exact geodesics, particularly if you want to quibble at the susceptibility of everything else to perturbations. – Emilio Pisanty May 27 '15 at 10:57
• 4. Re "It is certainly absurd that a mere and supposedly fixed (non-zero) symbol such as "$c_0$" should have changed". It is, but by doing this you have decoupled the speed of light from that symbol. It is perfectly reasonable for the speed of light to change, and specifically it is exactly as reasonable as claiming that the atomic unit of time has changed (which you do claim). If the length of the ruler has changed with respect to the SI, speed-of-light meter, then the SI length of every other physical object has also changed. How useful is the SI meter in that scenario? – Emilio Pisanty May 27 '15 at 11:11