# Clock Synchronization Done Right: "A 920-Kilometer Optical Fiber Link for Frequency Metrology at the 19th Decimal Place"

I've been busily working on something new, but I'm beginning to think I've been letting the perfect be the enemy of the good-enough-for-this-stage, so I'm setting it aside for a bit, and trying to get caught up with some of the huge number of things that have been slipping. Which includes getting the oil changed in my car, hence, I'm sitting in B&N killing time, which is a good excuse to do some ResearchBlogging.

Last week was a banner week for my corner of physics, with three really cool experiments published. Two of those are on the arxiv, which means I can use images from the paper (but those take longer to write). The third was in Science and isn't available in preprint form, and since the AAAS are bastards about permissions, and I'm not paying them \$30 for the sake of a blog post, we'll do that one first.

The paper in question has the incredibly sexy title "A 920-Kilometer Optical Fiber Link for Frequency Metrology at the 19th Decimal Place", and got a pretty good write-up in Physics World, but there's still room for some Q&A:

Seriously? Optical fibers and frequency metrology? What do those even mean? Seriously. This is a significant advance for people who care about precision measurement of time. The authors took two ultra-precise atomic clocks in labs at opposite ends of Germany, and were able to compare their frequencies at the level of a few parts in 1019. That's 0.0000000000000000004 times the original frequency, or, since they started with an optical frequency, 0.00008Hz out of 194,000,000,000,000.

OK, I admit, that's a lot of zeroes. But why does that matter? Isn't the whole point of ultra-precise atomic clocks that they all work exactly the same way? Why do you need to compare them? All atomic clocks using a given type of atom have the same basic frequency, but not all clocks are equally well made. The only way to determine the performance of a new one is to compare it to one you know works, but they're also not very portable, so you need to be able to do the comparison remotely.

There's also the fact that the local conditions for different clocks will be different, which can lead to some small shifts in the frequency due to things like general relativity, which means you can study some interesting physics using a distributed network of clocks. You can also check to see if the constants of nature are changing by comparing clocks based on different atoms, and again, these are not easy to make or move around, so being able to reliably do the comparison over long distances is a bug deal.

So, let me guess, they did this using optical fibers? Yep. They ran a special two-fiber line between the PTB standards lab in Braunschweig and the Max Planck Institute near Munich, and sent light from their optical clocks down the fiber both ways.

Isn't running a dedicated fiber line a lot of work? Why not just send it through the air? The industry standard, as it were, for this sort of thing has been to do time transfer via satellites, but that method is limited by fluctuations in the atmosphere. If you want to get to the 19th decimal place, you need to have better control of the transmission than that, which is something you can do with optical fibers.

Yeah, but how bright does their laser need to be to go 920 km through a fiber? It would need to be ridiculously bright, or to have ridiculously good detectors at the other end. Fortunately, though, they don't need to do that-- they set up nine repeater stations along the way, where erbium-doped fiber lasers of the sort used in telecom links amplify the signal. These are, of course, a little better than off-the-shelf versions, but the operation is well understood and boosts the signal to where they don't need super-powerful lasers or detectors.

OK, so they string some fiber, and shoot some light through it, and they compare clocks. This still doesn't seem like that big a deal. It's not that simple. The fiber gives you better control of the transmission conditions, but it's still subject to a lot of drift-- the glass expands and contracts as the temperature changes, and there are slight shifts in the refractive index, all of which needs to be corrected for.

So, what, they temperature-stabilized the whole 920km fiber? No, they work on the light directly. What they do is they send a bit of light down the fiber, then bounce it right back down the same fiber in the opposite direction. They compare the frequency of the light that comes out to the frequency of the light they started with, and then use an acousto-optic modulator (AOM) to shift the frequency of the light that goes in so that the frequency of the light that comes out is he same as the light from their clock. Anything involving temperature changes happens pretty slowly, so they can adjust the frequency of the AOM as needed to keep everything stable.

Isn't that cheating? No, not really. It's just a way of anticipating the known effect of imperfections in the transmission line, and correcting for them. My old group at NIST had to do this sort of thing to correct for polarization shifts in some optics, and referred to it by the colorful term "pre-fucking the polarization."

So, you said they had two fibers. What's the second one for? The same thing, in the other direction. They do the clock comparison twice, once from Braunschweig to Munich, and then from Munich to Braunschweig.

What's the point of that? It lets you determine the uncertainties in the system extremely well. They measure the difference between the clocks using each fiber, then they take the difference between the differences, which, mathematically looks sort of like this:

Δ1 = f(PTB)-f(MPQ) + δ(fiber 1)

Δ2 = f(PTB)-f(MPQ) - δ(fiber 2)

where the f(PTB) represents the actual frequency from the clock at PTB in Braunschweig, and so on. Because in one case the PTB clock was sent through the fiber, and in the other the MPQ clock was sent through the fiber, when you take the difference in the measured frequencies, the shift due to the fiber (given the symbol δ) is positive in one case and negative in the other. When you subtract the two differences (i.e., take Δ1-Δ2), the difference between the actual frequencies cancels out, while the two fiber shifts add. This lets you characterize the performance of your transmission system extremely well, which is what they're really doing in this paper.

And once they've done all this, they get something good to four parts in 1019? Yep. The thing that precision measurement people track is the "Allan Deviation" for the system, which characterizes the difference between two frequencies measured over some time interval. The details of how you get this aren't too important, but the important thing is that it improves as you average over longer periods, up to the point where some other source of noise comes into play. If they average their clock comparisons over a period of several hours, they get down to the part-in-1019 level. Which, according to their graph, is actually substantially better than the performance of their clocks over the same period.

Wait, what? How does that make any sense? Well, it's kind of the whole point. What you want for a system like this is something where the error introduced by the transmission is small compared to any other effect you might be interested in, and that's what they've done: the transmission error is better than 100 times smaller, and thus not really an issue for the purpose of comparing clocks in different locations.

So, for example, if you were to compare the frequencies of these two clocks, and found that they differ by a part in 1017, you can be confident that it's coming from a gravitational redshift, or a change in the fine-structure constant, and not just some flakiness in your transmission system.

Assuming the fiber-optic connectors at each end are tightened down all the way, anyway... Sigh. Yes, assuming the connections are all made correctly.

But, in a way, this is a demonstration of why people were surprised by the weird result from OPERA's loose fiber. If you work fairly hard, you can send signals through optical fibers and have everything work perfectly to 19 decimal places. Which is why nobody would ever expect a relatively short fiber link between a clock and a detector to produce a delay of tens of nanoseconds-- fibers are intrinsically pretty darn good, and only a really weird combination of factors could give you that big an error. Unfortunately for OPERA, they hit the weird combination of factors jackpot.

Predehl, K., Grosche, G., Raupach, S., Droste, S., Terra, O., Alnis, J., Legero, T., Hansch, T., Udem, T., Holzwarth, R., & Schnatz, H. (2012). A 920-Kilometer Optical Fiber Link for Frequency Metrology at the 19th Decimal Place Science, 336 (6080), 441-444 DOI: 10.1126/science.1218442

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Is there anything intrinsically better about doing this measurement over such a long distance? I mean, what kind of precision can you achieve if both clocks are in the same room? Does this set any kind of "most precise measurement ever" record?