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In 1957, Littlewood wrote that these are measurements of time in astronomy. Astronomers operate with times intervals between astronomical events 1000 years apart with accuracy 1/1000 of a second. This gives 14 significant digits. Has this situation changed since?

I mean ONLY the numbers related to physical phenomena, not pure mathematical calculations outcomes like, for example, the digits of $\pi$.

To make it more precise, I am not asking about the numbers like $3.256\cdot 10^{-19}$. I am asking about numbers with 14 or more significant digits (that is all zeros on the left of the dot must be discarded).

Ref. J. E. Littlewood, Mathematical Miscellany, London 1957, "Large numbers".

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    $\begingroup$ See physics.stackexchange.com/questions/497087/… $\endgroup$
    – Jagerber48
    Mar 25, 2023 at 2:52
  • $\begingroup$ Thanks for the references. The examples they give are comparable but do not exceed the number of digits in the Littlewood example. $\endgroup$ Mar 25, 2023 at 12:59
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    $\begingroup$ Littlewood claims that astronomers can calculate 1000-year time intervals to millisecond accuracy, not that they can measure them to that accuracy. There aren't any 1000-year-old measurements with that accuracy that could be used as a reference. $\endgroup$
    – benrg
    Mar 25, 2023 at 17:42
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    $\begingroup$ A bit silly, but any integer measurement, like the number of protons in a hydrogen atom or the number of stars in the solar system, can be measured with infinite precision. $\endgroup$ Mar 27, 2023 at 14:08
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    $\begingroup$ @AlexandreEremenko It's 1.00, repeating infinitely - integers have infinite precision, not precision limited by the number of digits. To say the number of protons in a hydrogen atom has only 1 significant figure allows the possibility that there are tenths or hundredths or any fractional quantities of a proton. But I can say with complete confidence that there is exactly one proton and not 1.01 or 1.00001 or anything else, to any level of precision you like. I certainly agree it's not really in the spirit of the question, though. $\endgroup$ Mar 27, 2023 at 20:18

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The periods of millisecond pulsars are measured to 15 significant figures. E.g., here's a paper that gives the period of PSR J1853+1303 as 4.09179744490025(2) ms (the figure in parentheses is the uncertainty in the final digit.)

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  • $\begingroup$ Thanks! But let me notice that it is again a "time measurement in astronomy", as in Littlewood's example:-) $\endgroup$ Mar 26, 2023 at 0:36
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    $\begingroup$ I'm pretty sure no time or frequency can be measured with more precision than the precision of the TAI timescale which is set by the precision with which we measure the Cs transition frequency. I think individual Cs fountain clocks have accuracy at the $10^{-15}$ level and an order of magnitude is gained by averaging many clocks around the world. $\endgroup$
    – Jagerber48
    Mar 26, 2023 at 3:13
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The answer is optical atomic clocks which are now pushing 19 digits of precision on the accuracy of the estimation of atomic transition frequencies. Citations to follow when I have time to add them. These technologies were enabled by advanced in lasers, laser cooling, atomic physics, and optical frequency combs.

Note these are state of the art atomic clocks. The atomic clocks used for establishing the TAI timescale are still sitting around 15 digits of precision on their accuracy of realizing the SI definition of the second.

The most accurate measurement which I am aware of is of the clock transition frequency in one of the $\text{Al}^+$ quantum logic ion atomic clocks at NIST. See Brewer et. al. (2019). Here the authors interrogate a $\text{Al}^+$ ion with a laser beam and they estimate that the ratio of their uncertainty in the lasers frequency $\Delta \nu$ to the laser frequency $\nu$ is $$ \frac{\Delta \nu}{\nu} = 9.4\times 10^{-19} $$

Here is their error budget: enter image description here

We see the leading term is excess micromotion. This refers to the fact that the ion feels a force due to the RF fields used to trap it in place. This force causes the ion to oscillate, and this oscillation of the ion results in Doppler shifts of the ions actual transition frequency. The authors estimate the magnitude of this shift to be at the $-45.8 \times 10^{-19}$ level but their uncertainty is at the $5.9 \times 10^{-19}$ level, leaving it as the leading term in their error budget.

One question that jumps to my mind that I'm still researching the answer on: Why report the fractional uncertainty only without reporting that actual $\text{Al}^+$ clock transition frequency out to 19 digits? The answer is related to the fact that our current absolute frequency standards are based on $\text{Cs}$ atomic clocks which only have a fractional frequency uncertainty at the $10^{-16}$ level or so. If you want to know the absolute frequency of any oscillator you must have some chain of clock frequency comparisons eventually tying you back to $\text{Cs}$ clocks. But I believe, because the $\text{Cs}$ clocks are limited to $10^{-16}$ stability, it is not possible to know ANY absolute frequency to better than $10^{-16}$. Folks at NIST are considering a road map towards transitioning from $\text{Cs}$ based definition of the second towards a optical-clock based definition of the second so that we could all reap the benefits of a realization of the SI second with better fractional frequency stability. This is the same case that is always made to redefine the second to take advantage of more stable and accurate clocks. e.g. when we switched from quartz oscillators to atomic clocks in the first place, or from astronomical measures of time towards clock-based measures. See here for more details about the roadmap to redefine the second going forward. Right now it is not easy to make an optical atomic clock, this is a major impediment towards adopting this technology as the primary frequency reference.

LIGO gravitational wave detection is probably a good honorable mention for most precise measurements in science, but I think optical clocks still have it beat. Experts in both fields may have more to say.

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The most precise measurements tend to be measurements of ratios or differences. For example, we know the ratio of the charge on the proton to the charge on the electron to 21 decimal places, despite only knowing what the charge on an electron is* to nine decimal places. Similarly, the LIGO observatories can measure changes in the distance between the mirrors of one part in 1022, despite only having a rough idea of how far apart the mirrors actually are.

*If you're working in the 2019 version of the SI system, the elementary charge is a fixed numeric value, and you're actually measuring a combination of other quantities.

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  • $\begingroup$ Any reference? I did not find in the Wikipedia article that some distance was measured to one part in $10^{22}$. $\endgroup$ Mar 25, 2023 at 3:47
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    $\begingroup$ The last sentence of the answer states that the distance was not measured to one part in $10^{22}$; merely that it is possible to deduce that the distance has changed by such a precise amount. $\endgroup$ Mar 25, 2023 at 12:07
  • $\begingroup$ A mirror is made of protons, neutrons and electrons. The size of proton is approximately $10^{-15}$ meters. So what the expression "the distance between mirrors" can mean in this context? $\endgroup$ Mar 26, 2023 at 0:39
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    $\begingroup$ @AlexandreEremenko LIGO measures relative changes in the distance travelled between a beam of light split along two orthogonal paths. "distance" is maybe a funny thing when you're talking about gravity waves warping space, but I don't think it's wrong. $\endgroup$
    – mbrig
    Mar 26, 2023 at 1:45
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    $\begingroup$ @AlexandreEremenko the arms are nearly (in order-of-magnitude terms) 10000 meters long, so a movement of 10^-15 meters is 10^-19. As for the rest - why can't a proton move by 1/1000 of its own size? $\endgroup$
    – user253751
    Mar 27, 2023 at 11:11
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See Weyers et. al., Advances in the accuracy, stability, and reliability of the PTB primary fountain clocks (2018). Here they report uncertainties on the PTB CSF2 Cs fountain clock at the $1.71\times 10^{-16}$ level.

In atomic clocks an electromagnetic oscillator (e.g. a laser beam or microwave tone) is made to interrogate an atomic transition. In this way the frequency of the electromagnetic oscillator can be compared to the frequency of the atomic transition.

The definition of the SI second is

The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency, ∆νCs, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s−1.

Now the Cs atoms in a real experiment do not exhibit an "unperturbed ground-state hyperfine transition". There are various systematic effects that shift these frequencies. But these systematic effects are studied so that they can be quantified and subtracted off. But the estimates of the systematic effects have uncertainty. This is why atomic clocks have finite uncertainty, despite the Cs hyperfine frequency literally being the definition of time.

In the linked reference, apparently the net uncertainty on these systematic effects was limited to $1.71 \times 10^{-16}$ fractional frequency uncertainty for CSF2. We have $$ 9,192,631,770 \text{ Hz }\times 1.71 \times 10^{-16} \approx 1.57 \text{ }\mu\text{Hz} $$

This means we can take this to mean that we can consider the atomic clock to be a measurement of the frequency of the microwave oscillator which yields a frequency measurement of $$ 9,192,631,770.0000000(16) \text{ Hz} \approx 9,192,631,770.000000(2) \text{ Hz} $$ The proper way to express how precise this measurement is is using the fractional uncertainty reported as $1.71 \times 10^{-16}$. I don't like sig figs since they can lead to confusion, but nonetheless, printed this way we can see this measurement of the microwave oscillation frequency was performed with a precision of 16 significant figures.

I don't claim this to be the most precise measurement ever since optical atomic clocks have yielded much lower fractional uncertainties. But, unless the definition of the second is changed, those clocks suffer from a technicality that prevents us from printing out the result of their measurement as a unitful physical quantity with more than 16 significant figures.

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If you want something less exotic than the above, I currently work at a company where we are using laser interferometry to measure abnormalities on the surface of optics with nanometer accuracy.

That's only 9 significant digits, but this is commercially applied science, and it is already a scale at which you can observe astonishing things, for example how a massive 2-inch thick disk of glass, supported at 3 points, sags in the unsupported regions due to its own weight. We have to use various exotic (to me) mechanisms to subtract the error due to gravity and be left with the real abnormality.

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Another good candidate is the measure of the electron's electric dipole moment $d_e$. Within the Standard Model, such a dipole is predicted to be non-zero but very small, at most $10^{−38} e \cdot \rm{cm}$. The current best measured upper bound is in Roussy et al., A new bound on the electron's electric dipole moment (2022) where the author give the following bound: $$|d_e|<4.1 \times 10^{−30} e\cdot \rm{cm}$$ at $90\%$ confidence.

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