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Using the uncertainty principle against itself to gain precision

Accurate measurement underlies a huge amount of modern technology. Atomic clocks, fiber optical communications systems, and many other types of hardware require accurate and precise measurements. The laws of quantum mechanics, on the other hand, are designed to annoy anyone obsessed with precision. In some cases, it's impossible to increase precision—not because the laws of physics prohibit knowledge but because the probe with which we measure is limited by quantum mechanics.

This limit is often referred to as the standard quantum limit. However, you can, with a great deal of pain, prepare special probes that beat the standard quantum limit. In this case, a different limit applies, called the Heisenberg limit.

You can't beat the Heisenberg limit. So the big question is "can we find a method that reduces the amount of pain required to approach it?" The answer, it seems, is yes.

As usual, before we get to the cool stuff, we have to deal with some background—otherwise known as "older cool stuff." Light is a wave, so it has periodically spaced maximums and minimums. In light from a lamp, the maximums and minimums have no particular pattern, either in time or space. A laser produces light of a very particular character that produces minimums and maximums that are neatly arranged in space and time.

But lasers also have some limits. No matter how well I control a laser, the light field it emits isn't a pure color; it has a small range of colors. This is because the quantum description of a light field tells us that the phase and amplitude of the light field has some minimum joint uncertainty.

What does this all mean? Picture a light field frozen in time. Stretching out before you, you see the maximums and minimums in the waves. They all appear exactly as they should: the spacing between the maximums are exactly the same... or are they? A closer examination reveals tiny fluctuations: some maximums occur slightly earlier than expected, and others appear slightly later. The size of the maximums aren't exactly the same, either.

These small quantum fluctuations all average to the perfect value, though.

The fluctuations limit how accurately you can make a measurement. Imagine that I want to measure the distance to an object. To do that, I might measure how many wavelengths of light fit between me and the object. But those tiny fluctuations mean that I can't measure the distance with perfect accuracy.

Notice that when I discussed the uncertainty in the phase and amplitude of the light field, I said they had a joint uncertainty. In the case of laser light, this means we can reduce the uncertainty of the phase at the expense of increasing the uncertainty of the amplitude. This process, called squeezing, takes coherent light (such as a laser) as its input, and outputs squeezed light, which allows us to increase the accuracy of my distance measurement by reducing the uncertainty in the phase.

Unfortunately, these measurements work out great in theory but don't come anywhere close to so awesome in practice (I think a factor of ten has been demonstrated, while a factor of 100-1,000 is possible). The problem lies in two areas: first, hidden in the description above is the role of entanglement. Every phase-squeezed photon has a partner that is amplitude squeezed, and it's the pairing and joint measurement of the two that allow us to increase our precision. However, the increase is not proportional to the number of pairs of entangled photons but instead to the total number of photons that are entangled with each other.

To gain a lot of precision, it's not enough to make a light source that produces a lot of entangled photon pairs per second. Instead, you have to make something much trickier: a source that produces a group of photons that are all entangled with each other. The more precision you need, the more particles you need to simultaneously and tightly entangle, which is technically very difficult.

The second limitation is that of the measurement process itself. I send my light into the system, and the thing I want to measure is very tiny, so the light is only modified by a tiny amount. Even though this change can be measured thanks to the preparation of a special state of light, I still need to actually measure the tiny change.

Unfortunately, no measurement system is perfect, so my precision falls off rapidly as the imperfection in the technology increases. Think of it like this: I send in 100 photons to measure something. To obtain the perfect, theoretically predicted result, I need to detect all 100 photons at the end. But the best light detectors might only detect half the photons, and my precision falls dramatically with every lost photon. In the end, nature wins—apparently.

A trio of physicists from Stanford have suggested a new way of making these measurements. And confusingly, we're going to change how we measure. Instead of light, we're going to use atoms (you can do this with light, too, but the description is easier with atoms).

Atoms have a component of angular momentum called spin. You can think of it as the rotation of the atom about an axis (this isn't correct, but it's helpful). Now, let's imagine that we set the spin of the atom purely along a direction that we choose (along the x-axis, for instance). The laws of quantum mechanics tell us that this action will fail. We can maximize it, but the atom will also have some amount of spin spread randomly between the two remaining axes (y- and z-axis). This is our coherent state (a group of these atoms would be the equivalent to a laser beam of matter).

Now let's imagine that our measurement involves an interaction that causes a tiny rotation of the spin from the x-axis in the direction of the z-axis. Instead of pointing in the x direction, it sits at a tiny angle with respect to the x-axis. Thanks to the uncertainty cloud of the spin in the y- and z-axis, we can't see this rotation.

To imagine how squeezing works in this context, let's say we increase the certainty along the z-axis at the expense of certainty in the y-axis. Our measurement might succeed because any rotation toward the z-axis is easier to discern. But it's still a difficult measurement—each atom has to have its spin measured individually and very precisely.

In a clever bit of thinking, the researchers considered the following idea: squeezed states are very sensitive to small changes, but coherent states are easy to measure. Is there some way to probe with a squeezed state but measure a coherent state? The answer seems to be yes. And it all comes down to choosing the right form of squeezing.

From the point of view of math, squeezing is just a mathematical operation. Associated with that idea is an undo operation that returns the squeezed state to a coherent state. This isn't usually that interesting because a small change to the squeezed state will produce a small change to the coherent state when you undo the squeezing,  But this isn't the case if you use a twisted squeezed state.

A twisted squeezed state is a bit complicated to visualize, but it basically forms an ellipse. The ellipse is centered on the x-axis but is rotated so that the short and long axes are not aligned to the z- and y-axes ellipse. The twist refers to the angle between the orientation of the ellipse and the axes.

When the atoms are used to make a measurement, the ellipse still shifts slightly, moving its center off the x axis and toward the z-axis. When we untwist, the off-center ellipse is rotated strongly toward the y-axis (in other words, it rotates in the x-y plane).

Essentially, the thing we're measuring causes a very small rotation toward the z-axis, which is amplified to a large rotation around the z-axis by us attempting to return the atoms to their original state.

Even better, the state is now coherent, which means that we can measure the atoms en masse rather than individually. The researchers estimate that their scheme can get to within a factor of two to three of the Heisenberg limit. Practically speaking, that's pretty impressive.

Normally, we should be suspicious of papers without any experimental demonstrations because they often require unrealistic experimental conditions. But the researchers also calculated that the same impressive gains could be achieved for some realistic scenarios. That means that this measurement technique really should be feasible, and it may yet gain us a large amount of precision in the finest of measurements (hello, atomic clocks).

For those of us plebs who don't require such precision, just thinking about doing this work causes a certain amount of pain. As with all other technology, however, this sort of measurement will eventually filter down to generate spectacular new instrumentation for us all.

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