Fri. Mar 24th, 2023
You can't beat the Heisenberg limit, but with enough math you can get close.

You can’t beat the Heisenberg limit, but with enough math you can get close.

Accurate measurements underlie a vast amount of modern technology. Atomic clocks, fiber optic communication 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 is impossible to increase the 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 default quantum limit. However, with great effort you can create special probes that beat the standard quantum limit. In this case, another limit applies, 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 it takes to approach it?” The answer, it seems, is yes.

All your limits are default

As usual, before we get to the cool stuff, we’re dealing with some background – otherwise known as “older cool stuff.” Light is a wave, so it has periodically spaced maxima and minima. In the light of a lamp, the maxima and minima have no definite pattern, neither in time nor in space. A laser produces light of a very special character that produces minima and maxima that are neatly ordered in space and time.

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

What does all this mean? Imagine a field of light frozen in time. Stretched out in front of you, you can see the maxima and minima in the waves. They all appear exactly as they should: the distance between the maxima is exactly the same… or not? Closer examination reveals small fluctuations: some maxima occur a little earlier than expected, others appear a little later. The size of the maxima is also not exactly the same.

However, these small quantum fluctuations are all averaged to the perfect value.

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

Beat the default limit

Note that when I discussed the uncertainty in the phase and amplitude of the light field, I said they were a joint uncertainty. In the case of laser light, this means that we can reduce phase uncertainty at the cost of increasing amplitude uncertainty. This process, called pinching, takes coherent light (like a laser) as an input and outputs compressed light, allowing us to increase the accuracy of my distance measurement by reducing phase uncertainty.

Unfortunately, these measurements work great in theory, but come nowhere near in practice (I think a factor of ten has been demonstrated, while a factor of 100-1000 is possible). The problem lies in two areas: first, hidden in the description above is the role of entanglement. Each photon squeezed in phase has a partner squeezed in amplitude, and it is the coupling 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 to the total number of photons entangled.

To get a lot of precision, it’s not enough to make a light source that produces many entangled pairs of photons per second. Instead, you need to create something much trickier: a source that produces a group of photons that are all entangled together. The more precision you need, the more particles you have to entangle simultaneously and tightly, 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 small, so the light is only changed a tiny bit. Even though this change can be measured thanks to the preparation of a special light condition, I still need to actually measure the small change.

Unfortunately, no measurement system is perfect, so my precision quickly decreases as imperfection in the technology increases. Think of it this way: I send in 100 photons to measure something. To get the perfect, theoretically predicted result, I need to detect all 100 photons at the end. But the best light detectors may detect only half of the photons, and my precision drops dramatically with each lost photon. In the end, nature wins, apparently.

Can we reduce the pain?

A trio of Stanford physicists have proposed a new way to make these measurements. And confusingly, we’re going to change the way we measure. Instead of light we are going to use atoms (you can also do this with light, 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 around an axis (this is not correct, but it is useful). Now let’s imagine setting the atom’s spin purely along a direction we choose (for example, along the x-axis). The laws of quantum mechanics tell us that this action will fail. We can maximize it, but the atom will also have a certain amount of spin randomly distributed over the two remaining axes (y and z axes). This is our coherent state (a group of these atoms would be the equivalent of a laser beam of matter).

Now let’s imagine that our measurement involves an interaction that causes a small rotation of the spin from the x-axis toward the z-axis. Instead of pointing in the x direction, it is at a slight angle to the x axis. Thanks to the uncertainty cloud of the spin in the y and z axes, we cannot see this rotation.

To imagine how squeezing works in this context, let’s say we increase certainty along the z-axis at the expense of certainty in the y-axis. Our measurement could succeed because any rotation to the z-axis is easier to discern. But it remains a difficult measurement: the spin of each atom must be measured separately and very precisely.

Smart thinking led the researchers to the following idea: compressed states are very sensitive to small changes, but cohesive states are easy to measure. Is there a way to investigate with a compressed state but measure a coherent state? The answer seems to be yes. And it all comes down to choosing the right kind of squeeze.

From a mathematical point of view, squeezing is just a mathematical operation. Associated with that idea is an undo operation that returns the compressed state to a coherent state. This is usually not very interesting because a small change in the compressed state will cause a small change in the coherent state when you uncompress it. But this is not the case if you use a twisted compressed state.

A twisted compressed state is a bit complicated to visualize, but it actually forms an ellipse. The ellipse is centered on the x-axis but rotated so that the minor and major axes are not aligned with the z- and y-axis 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 the center of the x-axis toward the z-axis. When we unscrew, the eccentric ellipse is strongly rotated towards the y axis (in other words, it rotates in the xy plane).

Essentially, the thing we’re measuring causes a very small rotation in the direction of the z-axis, which is amplified into a large rotation around the z-as by us trying to return the atoms to their original state.

In fact, the state is now coherent, meaning we can measure the atoms en masse rather than individually. The researchers estimate that their schedule could come within a factor of two to three of the Heisenberg limit. Practically, that’s pretty impressive.

More musings from theoretical physicists?

Normally we should be suspicious of papers without experimental demonstrations, as they often require unrealistic experimental conditions. But the researchers also calculated that the same impressive gains could be made for some realistic scenarios. That means that this measurement technique should be really feasible, and that it might still give us a great deal of precision in the finest measurements (hello, atomic clocks).

For those of us plebs who don’t need such precision, the mere thought of this work causes a certain amount of pain. As with all other technology, this kind of measurement will eventually filter to generate spectacular new tools for all of us.

Physical assessment letters2016, DOI: 10.1103/PhysRevLett.116.053601

By akfire1

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