Watching Wavefunctions Collapse

In a comment to the book announcement, “HI” makes a request:

Would you be able to summarize the recent paper “Progressive field-state collapse and quantum non-demolition photon counting” (Nature. 2007 Aug 23;448(7156):889-93) for non-specialists? How do you interpret it?

This probably would’ve slipped by me if not for this comment, but it’s a really nice paper, and I’m happy to give it a shot. There’s also a commentary by Luis Orozco that you won’t be able to read without a subscription because Nature are bastards that way.

The basic idea of this paper is that they prepare a quantum system in a superposition of several different states, and then use an ingenious measurement technique to make repeated measurements of the state of the system. This lets them follow the state as it moves from a quantum superposition of several different states to a more classical state where it has one and only one value. As they say in the abstract, the experiment “illustrates all the postulates of quantum measurement (state collapse, statistical results, and repeatability),” making it a really impressive piece of work.

So, how do they do this?

The basic scheme is very similar to a paper I blogged back in March, where the same group looks at the spontaneous appearance and disappearance of thermal photons. Their quantum system is a superconducting cavity– basically, two extremely good mirrors facing one another– in which small numbers of microwave photons can be trapped for long periods of time– nearly a second. In the previous experiment, they set the system up so there were no photons in the cavity, while here, they load it with an average of 3.8 photons, and watch what happens to the number of photons in the cavity.

Now, that may sound like an obvious thing to do, but if you think about it a bit, you’ll see that it’s actually very difficult. Photons aren’t like billiard balls that you can look at from a distance, and count without perturbing them. If you directly detect a photon, you destroy it– it gives up its energy to your detector, causing the canonical “click.” So while you can count the number of photons in the cavity directly, doing so immediately destroys the state, and you need to start over.

This is the ingenious part of their experiment: they have a way to detect the presence of photons in the cavity without destroying the photons. They do this by passing the atoms from an atomic clock through the center of the cavity. The photons in the cavity aren’t at the right frequency to be absorbed by the atoms, but they do shift the energy levels of the atoms by a tiny amount, causing the clock to run a tiny bit faster.

If you think of the atoms like little analog clocks, imagine that an atom passing through an empty cavity emerges with its hands showing exactly 12:00 noon. If there’s one photon in the cavity, though, the clock “ticks” a little faster, and the atom comes out showing 12:08. Two photons gets 12:15, three photons 12:23, and so on. In principle, this lets you distinguish any number of photons in the cavity, by looking at the time on the emerging clocks. In practice, it’s a little more complicated– atoms don’t have hour hands, so there’s no way to distinguish between 12:00 noon and 1 pm, but the important thing is that they have a way to distinguish between states of up to seven photons.

What they do, then, is to load the cavity with a burst of light that leaves an average of 3.8 photons in the cavity. Of course, photons are discrete objects, so you can’t really have four-fifths of a photon, so what you’ve really got is a collection of a bunch of different numbers, ranging between 0 and 8 or so. There’s about a 16% chance of finding three photons, a 10% chance of finding five, a 5% chance of finding seven, and so on.

i-29963d138aa70a9452ce1f6038a47984-sm_collapse_fig2b.jpgThey load the cavity up, and then they start sending atoms in. But this is a quantum system, so the initial state of the system isn’t exactly three, or five, or seven photons, but a superposition of all the photon numbers from 0 to 7 (and beyond, though the probability of 8 or more is very small) at the same time. There is no definite photon number in the cavity at the start of the experiment, so when they send the first atom in, they get an indeterminate answer– the most probable number is three, say, but there’s a pretty good chance of it being two or four, or even six. This produces the large, spread-out initial distribution seen in the figure at left (cropped from Figure 2b of the Nature paper).

When they send in a second atom, though, they get a bit more information about the state, and the distribution gets a little narrower. A third atom gets still more information, and a fourth, a fifth, and so on. What they see is that, as time goes on, the system evolves from a superposition of lots of different photon numbers into a single definite number– by the time they’ve sent 50 atoms through, the state has pretty much converged to a single number, say five photons, as seen in the figure. And they can track the evolution of the state by looking at each of the individual atoms as it comes out.

i-ee16858a2d3ad0deadfe368c2a63b3e5-sm_collapse_fig3.jpgThis is exactly the sort of thing that people talking about quantum measurement are always looking at: somehow, the process of measurement takes a quantum superposition of several different states, and causes it to “collapse” into a single value that we measure with our classical apparatus. If you repeat the experiment many times, you get a random result every time, but when you put all your results together, you find that they follow a predictable probability distribution– in this case, a comb of integer values (0, 1, 2, 3, 4, 5, 6, 7), with a very small background level of runs where an incomplete “collapse” ends up being read as 3.5 photons, as seen in the figure at right (Figure 3 from the Nature paper.).

(“Collapse” gets scare quotes not just because it’s jargon, but because it’s a loaded term, implying that the quantum wavefunction actually changes in a discontinuous and irreversible way. In the Many-Worlds Interpretation, the wavefunction continues to evolve in a smooth and mathematically satisfying manner, and for some reason, we only perceive one of the possible outcomes. These views are operationally indistinguishable– whatever interpretation you favor, at the end of the experiment, you only see one number– but the terminology may upset some people.)

They can also follow the state after the “collapse” to a single value, and what they see there is pretty cool, as well: the number of photons in the cavity decreases through discrete and random jumps, as individual photons slowly leak out of the cavity, over a few tenths of a second. The transition from, say, five photons to four happens very quickly, in a hundredth of a second or so, and then the cavity will sit at that state for a short time before dropping to three photons, and so on. They tracked 2000 states, and from that can put together a nice description of the photon lifetime, including some odd states where photons hang around for anomalously long times, or where thermal fluctuations cause the number to actually increase for a short time.

It’s really an outstanding piece of work. The one thing that it would be nice to see that isn’t there is a demonstration that this is a real “collapse,” and not just a refinement of the measurement– some experiment to show that the initial state is really a superposition of many different numbers, and not a state with a definite number that just isn’t measured very accurately. I’m not quite sure how one would go about that, though.

Even without that, though, this is an extremely cool paper in quantum optics, offering a really cool way of watching the process of quantum measurement in action.