How well can we measure position?

It is a well-known fact in quantum physics that the position and momentum of an object (e.g., a single atom or a vibrating mirror) cannot be known with an arbitrary precision. The more we know about the position of a mirror, the less we know about how fast it is moving and vice versa. This fact — sometimes misattributed to the Heisenberg uncertainty principle¹ — has far-reaching consequences for the field of optomechanics, including the efforts to detect gravitational waves.

Imagine one of the most basic tasks in optomechanics: using light to find the position of a mechanical oscillator. The simplest scenario assumes that you just bounce light off of the oscillator; its position determines the phase shift the light gets. Measuring this shift, you can infer the position of the mirror.

If the measurement is very precise, the momentum of the oscillator will become very blurry. This means that although we know where the mirror is now, we cannot predict how it will move in the future because we do not know how fast it is moving. If we attempt a second position measurement after the first one, its result will be very imprecise.² The uncertainty relation connecting position and momentum thus gets translated into an uncertainty relation between positions at two different times.

There is a simple explanation for this behaviour: the change in the statistics (i.e., in the variances of position and momentum) is due to the interaction with light. Precise measurement of the light’s phase results in our precise knowledge of the mirrors position. Correspondingly, there must be something in the light pulse that disturbs the momentum.

Indeed, there is a part of the interaction that is responsible for this reduced knowledge of the momentum. The light that interacts with the mirror kicks it — similar to a person jumping on a trampoline pushes the trampoline downwards. That in itself would be perfectly fine if we knew how strongly the mirror gets kicked but there is no way for us to know. There is another uncertainty relation at play (this time it is a true Heisenberg uncertainty relation), namely between the amplitude (or intensity) and phase of the light beam.

At first, it might seem that overcoming this backaction of the measurement on the mirror can be done by using a light pulse with a specific number of photons. If we know exactly how many photons kicked the mirror, we can (at least in principle) determine how strong this kick was. The mirror thus gets a well specified kick and the momentum uncertainty does not grow. The problem with this approach is that such states of light have random phase and we do not learn anything about the mirror’s position from the measurement. If, however, we use a state with a precisely specified phase, its amplitude is completely random and we cannot know anything about the size of the momentum kick.

In any practical setting, scientists do not have such precise control over the state they can use to probe the mirror. In most cases, they will use a coherent state — the state of light you get out of a laser, characteristic by having equal uncertainty in amplitude and phase. The overall amplitude of the pulse is the only thing that can be controlled. Using a very weak pulse does not give a very good measurement because the signal from the mirror is weak as well. While the precision improves with growing power, the backaction grows too because the uncertainty in amplitude increases. When the intensity is neither too large nor too small, the joint error of successive measurements is minimised. When this is the case, the measurement reached the standard quantum limit.

Is this the best measurement of mechanical motion we can do? As the name suggests, there is another, non-standard limit on quantum measurements. The problem with the current setting is that we are trying to measure two incompatible observables (mirror positions at two different times). If we try to measure the velocity of the mirror instead, this problem does not arise. Velocity is the momentum divided by the mass of the mirror; its knowledge tells us how the mirror will move in the future. This is in stark contrast to a position measurement where a better knowledge leads to a less precise prediction of the future movements of the mirror.

Another option is to disregard the fast, periodic oscillations of the mirror. Since we know that the mirror is oscillating at a particular frequency, we can work in a reference frame where the oscillations do not play a role. The mirror is then almost motionless while the rest of the universe is now oscillating around us. The slowly changing position of the mirror in this frame can be measured precisely since it is not affected by the momentum uncertainty. The momentum in this rotating frame, of course, becomes more blurry as the precision is increased but it does not influence future positions of the mirror.³

Both these approaches to measurement of mechanical motion are more complicated than a simple position measurement. But since various tasks require very precise measurements — often more precise than the standard quantum limit allows — there is a lot of scientific activity around these alternative strategies. Who knows, they might even find their way to real-world applications in a few years time.

¹Strictly speaking, the Heisenberg uncertainty relation concerns the property of a quantum state that we prepare. Here, on the other hand, we are asking how our position measurement affects the momentum which is not the same. As we will se, however, there is a close connection between the two.

²This is true in the statistical sense — we assume that we do many such pairs of measurements and look at the variance of the second one.

³If you find it strange that momentum does not affect future position, it is because we are not talking about position and momentum in the standard sense. Their usual relationship therefore does not hold.