# Back-of-the-Envelope Gravitational Which-Way

There’s a new Science Express paper on interfering clocks today, which is written up in Physics World, with comments from yours truly. The quote is from a much longer message I sent– with no expectation that it would end up as anything other than a pull quote, I might add, but I thought the background would be helpful. Since I ended up doing a back-of-the-envelope estimate for that, though, I thought I would reproduce some of the reasoning here.

The basic proposal idea here is to do an atom interferometer inside a Ramsey interferometer for making an atomic clock. That is, before sending the atoms into the beamsplitter, you prepare them in a superposition state, like the first step in making an atomic clock. This gets you a superposition state with a phase that oscillates at the frequency associated with the atomic transition, which is what you use to make the clock.

In this case, though, the claim is that a different rate of “ticking” of the clocks along the different paths of the interferometer– say because one is at a higher altitude than the other, and thus subject to a gravitational time dilation from general relativity– could serve as a “which-way” measurement that would destroy the quantum interference effect. That is, the fact that the upper clock ticks more rapidly than the lower would let you distinguish which of the two paths the atom “really” followed on its way through, by making a clock measurement after you recombined the two paths. This would destroy the interference, which would reduce the contrast of the interference pattern. As a demonstration, they applied an artificial shift to the “clock” on one arm of their (horizontal) interferometer, and showed that when they make the resulting phase shift an odd multiple of π, the interference pattern gets wiped out.

As I said to Physics World, you would need to talk to a real atom interferometrist to clarify the difference between what they’re doing with the clock superposition state and a Ramsey-Bordé interferometer, and also to make sure there’s a sharp distinction between the gravitational shift they’re talking about and the phase shift people doing gravitational measurements with interferometers already measure. Assuming they’re right, though, you can try to estimate whether this would really be measurable.

The gravitational time dilation they’re talking about as making the “which-way” distinction is, near the surface of the Earth, approximately:

$T \approx T_0 (1+\frac{gR}{c^2} )$

where T is the time between ticks for the clock a distance R from the center of the Earth (something not too different from the radius of Earth), T0 the time for a clock far away from anything massive, and g the strength of gravity near the surface of the Earth. If you plug numbers in for two clocks at different elevations, this is a shift of about one part in 1016 per meter of difference.

(As a sanity check, that’s about what they see in the aluminum-ion clock experiment at NIST: they raised one clock above the other by about 33cm, and see a shift of a bit under 5 parts in 1017. So I’m not completely off base, here…)

The largest separation between paths I’m aware of in an atom interferometer is the 10-meter tower interferometer in the group of my old boss, Mark Kasevich. That’s from 2013, with a separation of a centimeter and a half. I have heard, but not seen solid documentation of, that they’ve expanded this to half a meter or so.

To get the interference-destroying effect, they applied a phase shift of π to one arm, which would correspond to half a “tick” of the clock– that is, half the oscillation period. To see this gravitationally, you would need to have that part-in-1016 shift amount of a difference of one oscillation period over the time in the interferometer (a couple of seconds for the 10-m tower). For a microwave clock transition like you have in the rubidium used in the Kasevich group, you’re a factor of a million away– the frequency is about 7,000,000,000Hz, so the shift would be on the short side of a microhertz. That’s not going to do much.

You might, however, get somewhere with one of the optical clock atoms, like strontium. the “clock” transition in Sr is in the visible region, at around 400,000,000,000,000Hz, so a part-in-1016 shift is close to 1Hz. Over a couple of seconds, that’s probably enough phase shift to significantly degrade the contrast, based on the graph in the new paper.

How plausible is that? Well, it’s not ridiculous. The 10-m tower experiments use a BEC of rubidium, and strontium has also been Bose condensed. So if you adapted the giant tower to use Sr rather than Rb (a challenge, but probably not impossible), you might be able to see something. Assuming you could distinguish this effect from the many, many other things that can degrade the contrast of an atom interferometer signal. (For that, I think you’d want to see a revival of the contrast, which means getting to a phase shift of 2π, and you could map the effect out by gradually increasing the separation through changing the momentum imparted by the laser beamsplitters in the interferometer.)

Does this sort of thing have anything to say about the interaction of gravity and quantum mechanics? Probably not, in my semi-informed opinion. It’s a much more clearly defined mechanism than you see in most theories invoking gravity as a reason for a loss of “quantum-ness” in macroscopic experiments (which tend to be of the form “We don’t understand the quantum-to-classical transition, and we don’t understand gravity, therefore they’re related”), so it’s at least something you could probe experimentally. It’s a really small effect, though, even in the most impressive interference experiments done to date, and seems to require a rather special set of experimental conditions (both a vertically oriented interferometer and a superposition of internal states), so I think the implications for quantum foundations are probably minimal.

It’s a clever idea, though, and it would be interesting to see somebody give it a try.

## Comments

1. #1 Adrian Morgan
Adelaide, SA
August 8, 2015

Not too long ago I tweeted you a link to a thing I wrote, on the off-chance that it might catch your eye. To no-one’s surprise it didn’t, but the topic is relevant to this post.

Here’s the backstory. Earlier in the year I decided to see, just for fun, if I could guess a formula for gravitational time dilation near the surface of the earth. My tools were a crude notion of what general relativity entails, some high school Newtonian physics, and Ockham’s Razor. I did not expect my result to be correct, not even approximately so, and was therefore astonished later to find that it checked out against results I found online.

Until now, that is. Applying my formula to the same sanity check you used, I get a shift of 3.6 parts in 10^17. That’s the right order of magnitude, but 3.6 isn’t quite “a bit under 5”, so perhaps my formula is not all that accurate after all.

As I was saying, however, I decided to write up an explanation of how I acquired my result, and to present it as a faux popular science article aimed at an audience as scientifically literate as myself (i.e. not all that). I did this for three reasons:

(1) You learn by doing, and I figured the exercise would benefit me.
(2) An experienced science communicator might be persuaded to have a look and point out all the things I got wrong, and then I’d learn even more.
(3) If I was actually onto something, then said science communicator might be inspired to write a blog post of their own, covering the same ground only better.

My article (which is 1000 words long) is here: https://outerhoard.files.wordpress.com/2015/06/relativity.pdf

Chad, if there’s an idea in there that you can use for a blog post, you are welcome to it. My only request is to be notified if you do.

2. #2 Chad Orzel
August 8, 2015

3.6 sounds about right, actually. My “a bit less than 5” is based on remembering that the version of the figure in my book has tick marks every 5e17, and the shift is not a full tick mark.

3. […] [Update: I linked to my article in a comment here.] […]

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