# Relativity on a Human Scale: "Optical Clocks and Relativity"

As mentioned in yesterday's post on ion trapping, a month or so back Dave Wineland's group at NIST published a paper in Science on using ultra-precise atomic clocks to measure relativistic effects. If you don't have a subscription to Science, you can get the paper for free from the Time and Frequency Division database, because you can't copyright work done for the US government.

This paper generated quite a bit of interest when it came out, because it demonstrates the time-slowing effects of relativity without any need for exotic objects like black holes or particle accelerators-- they deal with objects moving small distances at low speeds, and the results agree very nicely with the predictions of relativity.

I'm a little late for the buzz, but it's a cool enough experiment that it's worth unpacking a little in the usual Q&A format for ResearchBlogging:

OK, what's the deal with this? Well, they used a pair of identical atomic clocks of exceptional precision to measure relativistic effects at everyday scales. Their clocks are good enough to be able to detect shifts on the order of a few parts in 1016, which means they can see the slowing of time due to motion at a walking pace, and due to elevation changes of less than a meter.

Back up a bit-- what's this "atomic clock" business? Well, as I explained a few years ago, an atomic clock measures time by making use of quantum physics. Atoms will only absorb light of certain very specific frequencies, so you can use an atom as a perfect frequency reference to determine the frequency of a light source-- if it absorbs the light, you're at the right frequency, and if it doesn't, you can correct the frequency until it does. If you keep comparing your light to the atoms, and correcting the frequency, you can make a light source whose frequency can be used as a reference to mark the passage of time.

And this lets you test relativity? If your clock is good enough, yes.

The specific clocks they're using use ultraviolet light with a frequency of about 1.12 x 1015 Hz (wavelength of 267 nm), which is absorbed or emitted when an ion of aluminum makes a transition between two specific states. They use single trapped ions as their reference, and use a clever "quantum logic" scheme to make their measurements.

How does quantum logic enter into this? They trap not one but two ions in their ion trap-- one aluminum ion to serve as the clock reference, and one "logic" ion of either beryllium or magnesium that is connected to the clock ion through their collective motion. As the two atoms sit next to one another in the trap, the motion of one affects the other, and that motional state can be used to connect the states of the two.

What's the point of that? Basically, it's easier to deal with beryllium and magnesium than aluminum, but aluminum has a better "clock" transition. If you wanted to make a clock of just aluminum, you would need a bunch of hard to work with lasers, but by coupling the two atoms together, you just need one laser for aluminum, and can do all the rest of the operations on the other ion.

Such as what? They lay the whole scheme out in another Science paper, but the key steps are laser cooling of the trapped ions and the input and readout of the states. Because the states of the two ions affect each other, you can cool them both to the ground state by interacting only with the logic ion, which will "sympathetically" cool the clock ion through their interactions in the trap. Then you can shine in the clock laser to excite the clock ion, in a way that will map its state onto its motion-- if it absorbs the laser, it starts moving more, otherwise it remains still. The motion of the clock ion affects the state of the logic ion, which you can then read out using the laser you did the cooling with.

It's a neat scheme, and avoids a lot of the hassles of working with aluminum ions directly, allowing them to use the exceptionally good laser systems they already had for working with beryllium and magnesium.

OK, so this is a clever scheme. How good is it as a clock? They measure stability on the order of parts in 1018. Or, to write it out with lots of impressive-looking zeros, they find that the clock is stable to within 0.008 Hz out of 1,120,000,000,000,000 Hz.

That's some clock, all right. Amazing, isn't it?

So, what's relativity got to do with this? Well, special relativity says that a moving clock will "tick" at a slower rate than one that is standing still. One of the two things they did in this paper was to set one of the ions moving, and measure the difference between the oscillation frequency of the light absorbed by the moving ion and the frequency of the light absorbed by the stationary ion.

So, what, they put it in the car and went for a drive? No, it was both simpler and more complex than that. The way they put the ion in motion depended on the operation of the ion trap. They just applied a small electric field that pushed the ion out of the center of the trap.

When the ion is exactly at the center of the trap, it experiences zero force, and thus doesn't move very much as the field switches back and forth. When they push the ion out to one side, though, it finds itself between two electrodes that are switching polarity very rapidly. This means that the ion is tugged back and forth between them, and so oscillates back and forth in the trap. The farther they push it out, the faster it moves, and the more the oscillation frequency changes.

So, how fast did the ion move? The key graph is here:

This shows the change in the clock frequency as a function of the average speed of the ion. The speeds range from zero to about 30 m/s, which is roughly highway speed for a car. The motion produces a measurable shift in the clock frequency for speeds as low as only a few m/s, which is walking pace. The solid line is the prediction from special relativity, and agrees very nicely with the data points.

So, how bis a shift is that? Really, really small. The shift goes like the speed squared divided by the speed of light squared, which is around a part in 1015 Put another way, in one second as measured by the stationary clock, its light oscillates 1,120,000,000,000,000 times while the moving clock's light oscillates 1,119,999,999,999,999 times.

Wow. Exactly.

What was the other part of this? In addition to the shift due to the motion, which is predicted by special relativity, they also measured the change in the clock due to gravity, which is general relativity. General relativity predicts that a clock deeper in a gravitational field will tick more slowly than one higher up, so they jacked one of the clocks up by about a foot, and measured the shift. Their results showed a small but unambiguous shift:

This shows a series of different measurements of the difference between the two clocks, 13 before the move, and 5 after one of the clocks was raised by 33cm. There's some scatter in the results, but the step is very clear. Moving the clock up by 33cm made it tick slightly faster than the clock that wasn't raised.

Can you explain the smallness of this shift in some way that involves lots of zeroes? Sure. The stationary clock ticks 1,120,000,000,000,000 times every second. If you added up all the ticks for 100 seconds, the total for the elevated clock would be greater than that for the stationary clock by right around 5 ticks.

Yeah, that's pretty amazing, all right. Yes it is. And it's in good agreement with what you would expect from general relativity.

So, what's the take-home message from all this? Basically, that Einstein was right. The predictions that relativity makes about the slowing of clocks due to motion or gravitation, as weird as they may seem, are really true, and verified even at everyday speeds and elevations. We don't notice these changes because they're so tiny, but with good enough measuring apparatus, we can pick them up, and they agree nicely with the predictions of relativity.

So crazy people trying to argue that relativity is wrong have their work cut out for them? Even more than they did before. Not that this will stop them, being crazy people, but for the non-crazy among us, this is a beautiful confirmation of the theory.

Chou, C., Hume, D., Rosenband, T., & Wineland, D. (2010). Optical Clocks and Relativity Science, 329 (5999), 1630-1633 DOI: 10.1126/science.1192720

Schmidt, P. (2005). Spectroscopy Using Quantum Logic Science, 309 (5735), 749-752 DOI: 10.1126/science.1114375

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Thanks for bringing this to our attention. This is beautiful.

That last plot isn't all that convincing to me. The step only looks obvious because there are lines fit to the different sections. I'm not sure you'd be able to tell which trial the height was changed at if the green lines and yellow bands weren't there.

Or, put another way, a fit to the null hypothesis is probably not all that bad a fit to the data. By eye, it seems that the 3-sigma confidence intervals for both sets would overlap by a fair amount.

Ok I played around with the plot a bit and put together this image. Assuming the yellow bands are 1-sigma uncertainties, the shaded regions are 3-sigma.
http://imgur.com/By1wP.jpg