# Is an ideal Cesium Standard atomic clock more accurate than Einstein's thought experiment light-clock?

This question comes to mind to clarify points in an earlier question. Basically my question is this:

According to current international standards the second is defined by the Cesium Standard which is the basis of really good atomic clocks. I am not questioning if these clocks are good enough, clearly they are. What I want to know is, would the theoretically ideal Cesium Standard clock be equal or better than Einstein's thought experiment light clock where time is measured by the number of times a photon bounces between two perfect mirrors.

I am aware that this clock cannot be built in real-life, much less perfectly. In fact no clock is absolutely perfect, but let's assume the ideal case for both designs, which would actually keep better time?

• Is the light clock using visible photons, which have a minimum wavelength of ~400 nm? Nov 18, 2021 at 20:05
• What do you mean by "ideal design" in each case? If you take this to an extreme, surely the best clock possible is an ideal mass oscillating with an ideal spring on a frictionless surface, or a device that can be perfectly synchronized with such a system. Nov 18, 2021 at 23:11
• There are different "inaccuracies" in a clock. There are systematic errors and then there is statistical clock noise. The physical limit for the latter is determined by the total amount of energy in the clock and how much energy it bleeds to infinity. On a fundamental basis, once we have eliminated all other error sources, the heavier clock will be better than the lighter clock in terms of noise. Light is therefor not a good choice for the clock mechanism. Oct 7, 2022 at 22:58
• @PM2Ring why does the wavelength matter? Oct 9, 2022 at 18:07
• @FlatterMann how do you know this? Is there a theorem someone has worked out? Oct 9, 2022 at 18:08

If we are considering real clocks then the caesium clock wins hands down.

A second is defined as the time taken for 9192631770 oscillations at the frequency of the caesium hyperfine line. So once you've matched your microwave source to the caesium absorption line you just need to count the number of oscillations of your microwave source, i.e. 9192631770, and this is trivially easy with modern electronics. So your accuracy is about one part in about $$10^{10}$$.

Suppose you make a light clock from two mirrors, then the accuracy depends on the distance between the mirrors being correct, so if your mirrors were a metre apart then to match the caesium clock you'd need the distance to be one metre to a precision of 1 Angstrom. This is certainly possible, but it's not trivial and you'd need careful control of vibrations and temperature fluctuations to retain the accuracy.

Caesium clocks are really quite simple devices. You can quite literally buy them off the shelf.

• Hmmmm. Still not sure if this is exactly what I was after, but I don't think I'm going to get a better answer from this exchange. Nov 18, 2021 at 18:47

Two ideal clocks of any mechanism will always agree with each other. If they are both ideal then they will both perfectly accurately measure proper time.

• Okay, it's hard to refute that one, but it's not exactly what I meant. What I meant to ask is to compare clocks ideal to their standard, that is the best measurement that can be taken with a light-clock versus an atomic clock. Is there any fundamental limit to the accuracy of either? Nov 18, 2021 at 16:39
• I think that you should ask about real clocks then, not ideal clocks. You want "best" real clocks.
– Dale
Nov 18, 2021 at 16:48
• What I am really trying to get at is which type of clock is considered definitive. Nov 18, 2021 at 18:36
• Corollary. An ideal clock is always more accurate than a non-ideal clock. Nov 18, 2021 at 18:51
• @DerekSeabrooke the caesium clock is the definitive clock for the SI second. You could make a unit of time like the jiffy and use a light clock to define that unit.
– Dale
Nov 18, 2021 at 19:55

As John Rennie mentions, the light clock relies on a precise measurement of the distance between the two mirrors, and that distance has to remain stable while the clock is running. It's much easier to measure time precisely than it is to measure distance, so we'd use an atomic clock to measure & maintain the mirror distance.

Also, we somehow need to precisely measure the moments when the light bounces off the mirrors, and the precision of that measurement is limited by the wavelength of the light; the shortest visible wavelength is around 400 nm, which corresponds to ~$$1.33×10^{-15}$$ seconds. We could use ultraviolet light for a shorter wavelength, and perhaps even x-rays, although you need to use very shallow reflection angles with x-rays.

It's simply not practically possible to make an Einstein / Langevin-style light clock that could have anywhere near the precision of a good atomic clock. The mirrors need to be positioned precisely, with no outside vibration. Of course, the bouncing photons will cause the mirrors to vibrate, presumably that's how we detect the reflections. The mirror temperature has to be kept constant, and the space between them has to be an ultrahard vacuum.

According to current international standards the second is defined by the Cesium Standard which is the basis of really good atomic clocks.

The caesium clock is good, but it's not the best timekeeper that we have. The first atomic clock was an ammonia maser built in 1949, but it was less accurate than the best quartz clocks available then. The caesium-133 clock was developed a few years later, in 1955. There have been some improvements in the design since then. The modern caesium fountain uses a small "cloud" of laser cooled atoms in freefall. Using the optical molasses technique, the atom temperature is reduced to ~40 microkelvin. However, even using a small population of such cold atoms, some thermal noise and atomic collisions are inevitable when the cloud is energised by microwaves. But even if we could use a single atom at picokelvin temperatures there would still be some width of the transition frequency, it cannot be a single number, due to time-energy uncertainty.

There's another issue lurking in your question. Time is what a clock measures. An SI second is defined as the time taken for 9192631770 oscillations at the frequency of the caesium-133 hyperfine line at 0 K. So it doesn't actually matter that each of those oscillations have a slightly different duration. You count 9192631770 of them and you have an official SI second. That is, the variation of the Cs-133 hyperfine transition is baked into the definition of the SI second.

We now have various clocks that have more precise ticks than the caesium clock. Such clocks can be used to calibrate a caesium clock, but to use such a clock as a primary time reference it has to reproduce the behaviour of the Cs-133 SI second, warts and all.

Here's a summary of the relative uncertainty of various atomic clocks, from Wikipedia.

Atom Type Uncertainty
Cs-133 Beam 1e-13
Rb-87 Beam 1e-12
H-1 Beam 1e-15
Cs-133 Fountain 1e-16
Sr-87 Lattice 1e-17
Mg+Al Lattice 8.6e-18
Yb-177 Lattice 1.6e-18
Al+ Lattice 9.4e-19
Sr-87 Fermi gas 2.5e-19

"Beam" refers to a standard off-the-shelf beam maser. "Fountain" is an atomic fountain, that value is for NIST-F2. "Lattice" is an optical lattice. "Fermi gas" is a 3D quantum gas optical lattice.

NIST-F1 (also an atomic fountain) has an uncertainty around 5e-16. Together, NIST-F1 and NIST-F2 form the primary time & frequency reference for the USA.

FWIW, the SI second was defined in terms of an integral number of Cs-133 hyperfine transitions which approximately matched the current definition of the astronomical ephemeris second. The original ephemeris second was defined as 1/86400 of a mean solar day, but due to the variations in the solar day length & the practical difficulties in measuring it, the ephemeris second was redefined in terms of the mean tropical year of 1900, and prior to the adoption of the Cs-133 standard, ephemeris time was determined from observations of the Moon.

However, the mean solar second length used in those definition was calculated using data gathered from 1750 to 1892, and corresponds to the mean solar second from around the middle of that period, i.e., ~1820. The actual mean solar second is somewhat longer now.

Steve Allen of the Lick Observatory has much more info on this topic. See A brief history of time scales and other articles on his page about leap seconds.

• Mar 18, 2022 at 16:17
• Thanks. This is the best answer so far. It's first that addresses the crux of my question of definition of time versus definition of the second. I wish I could articulate the question better. Oct 9, 2022 at 18:25