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.

Simple position measurement. A beam of light is reflected off of a mirror and the phase the light acquires during its transmission can be used to determine the precise position of the mirror.
Simple position measurement. A beam of light is reflected off of a mirror and the phase the light acquires during its transmission can be used to find the precise position of the mirror.

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.

Schematic depiction of a speed meter. The double reflection off the mirror (with a little time delay between them) can be used to determine the speed with which the mirror moves; moreover, the second reflection counteracts the momentum kick due to the first one.
Schematic depiction of a speed meter. The double reflection off the mirror (with a little time delay between them) can be used to find the speed with which the mirror moves; moreover, the second reflection counteracts the momentum kick due to the first one.

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.

 

Seeing ripples in spacetime

One hundred years after Albert Einstein shared it with the world, the general relativity is waiting for its last confirmation: direct observation of gravitational waves. These ripples in the curvature of spacetime are created when a massive object accelerates. Typical examples of such systems are binary neutron stars or black holes; as the two stars (or black holes) orbit each other, they gradually lose energy which gets emitted in the form of gravitational waves until, eventually, they collide.1

How large are these waves? That depends on how far their sources are but scientists generally assume that the relative size of waves we can expect to see can be about 10-20. This means that, due to a passing gravitational wave, a one-metre long rod will expand and shrink by 0.000 000 000 000 000 000 01 metre, which is a hundred thousand times smaller than a proton. In other words, if a proton were the size of a football field, a gravitational wave would be as small as a grain of sand. Said proton is, at the same time, just a grain of sand compared to a football-field sized atom; if atoms where as large as grains of sand, a single human hair would have one kilometre in diameter.

Michelson
Scheme of a Michelson interferometer. Light from a laser (left) is split on a beam splitter (middle) and travels through two arms of the interferometer (top and right); after reflection, the light recombines on the beam splitter. If both arms are equally long (top scheme), all light is reflected back to the laser. If the lengths of the two arms differ (bottom scheme), part of the light will escape through the bottom port where it can be detected.

To detect something that small, scientists build large interferometers. If both arms of such an interferometer are exactly the same length, all light that we send in will come out through the same port it came in. But if the length of the arms differs slightly (for example, because of a gravitational wave), part of the light will leak through the other port where it can be detected.

The interferometers have to be large because it is then easier to detect small changes in the arm length. With metre-long arms, the small change in length that has to be detected is 10-20 m. The interferometers of the LIGO detector are each four kilometres long so their length changes by about 10-17 m. For the eLISA interferometer (which will be sent to space to detect gravitational waves from there), arms long one million kilometres are planned; their length will change by 10-11 m which is just ten times smaller than the size of atoms.

The precision needed in LIGO still cannot be achieved with a simple interferometer. The solution is to place a set of mirrors into the arms so the light bounces back and forth many times. If light travels million times between the mirrors, it is as if the arms where 4 million km long and the required measurement precision is similar to eLISA. Further improvement can be achieved by adding another mirror into the input of the interferometer. Light leaving through this port then returns back into the interferometer and the intensity in the interferometer grows. And the more light is circulating through the interferometer, the more will leak through the output port where it can be detected. Final improvement is achieved by placing another mirror into the output that we are trying to measure.

The effective length of the arms can be extended by letting the light travel many times through them (top). The sensitivity can be further improved by adding mirrors into the input and output ports (bottom).
The effective length of the arms can be extended by letting the light travel many times through them (top). The sensitivity can be further improved by adding mirrors into the input and output ports (bottom).

Unfortunately, gravitational waves are not the only thing that can cause such small shifts in arm length. Any sort of vibrations can distort the measurement. Therefore, gravitational-wave detectors are built in remote locations where there is little or no human activity. Still, the occasional lorry driving by or even a person stomping near one of the mirrors will disturb the measurement. Other noise comes from various technical imperfections in the interferometer, such as fluctuations in laser frequency and intensity, presence of residual gas in the arms (which are supposed to be in vacuum), or heating of the mirrors due to light absorption. Finally, there is the quantum noise which ultimately limits the precision when all other imperfections are eliminated.

The hunt for gravitational waves will not be over once they are detected and Einstein’s theory confirmed. Once we are able to detect them with sufficient precision in a large frequency window (ranging from fractions of hertz to tens of kilohertz), we can use them to learn more about the universe. They can, for instance, tell us more about black holes than electromagnetic radiation can. Cosmic inflation is another source of gravitational waves which could tell us more about the universe shortly after the Big Bang. With a successful detection of gravitational waves, we will open a new window into the universe.


1 The loss of energy by such objects, in perfect agreement with Einstein’s predictions, has already been observed and was awarded the Nobel Prize in physics 1993. Scientists are therefore confident that such waves do exist.

Connecting superconducting quantum computers with light

Entanglement is a peculiar feature of quantum systems that makes them behave as if they were sitting directly next to each other even if they are kilometres away. Such behaviour does not occur in classical physics. Classical particles can affect each other through fields — such as the gravitational or electromagnetic field — but these fields propagate with the speed of light; the interaction between entangled particles, however, is instantaneous. Simply as if the distance between them did not exist.

Classical particles do not affect each other when they are far apart. In quantum physics, entangled particles do influence one another; it is as if they were always close to each other.
Classical particles do not affect each other when they are far apart. In quantum physics, entangled particles do influence one another; it is as if they were always close to each other.

Entanglement started to interest scientists in the early days of quantum physics (it was first mentioned by Erwin Schrödinger in 1935) but only in the 1990s, it was realised that entanglement can also be used as a resource for quantum communication. After some 20 years of intensive research, not much has been achieved in creating entanglement (over long distances as is needed for quantum communication) in the laboratory. The main problem is that the only system that we can send over such distances is light; if a system does not interact with light, it cannot be entangled with another system that is sitting far away.

One particular example of such a system is a superconducting circuit. These typically work at energies corresponding to microwave fields and microwaves cannot be transmitted as easily as light (at least in the quantum regime; the world is too hot for them and the signal they carry does not survive). But superconducting systems seem to be very well suited for quantum computing. And having a (quantum) computer which cannot communicate with other computers over (quantum) internet… well, what’s the point?

A superconducting system with a bus resonator (black line) and two quantum bits (in red squares). © Schoelkopf lab, Yale University
A superconducting system with a bus resonator (black line) and two quantum bits (in red squares). © Schoelkopf lab, Yale University.

Naturally, scientists started looking for a way to connect superconducting circuits and light using a third system that can interact with both. There are several candidates for such an interface — sort of a quantum network card, if you like — and one of the most promising options is to use a mechanical oscillator for the task. Those can be relatively easily manufactures, well controlled, and they can strongly interact with both light and superconducting circuits.

A scheme for entangling superconducting qubits that uses their interaction with mechanical oscillators and optomechanical measurement.
A scheme for entangling superconducting qubits that uses their interaction with mechanical oscillators and optomechanical measurement.

What can we do with all that? Let us start with the simplest possible task — entangling two quantum bits formed by superconducting circuits1 and connected by light. There are many ways this can be done; we will use an approach where the light is used to measure the two qubits. A well-chosen measurement which reveals some joint property of the qubits can result in an entangled state; furthermore, it has the advantage that the right measurement outcome signals that the entangled state has been successfully created. (This is also a reminder of the importance measurements have in quantum physics that I wrote about before.)

Suppose we now start by preparing the qubits in such a state that each qubit has values 0 and 1 at the same time. Now, we let each qubit interact with a mechanical oscillator and the oscillators interact with a beam of light that we measure. If we build the system the right way, the measurement of light will tell us how many qubits have the value 1. It can happen that both or none have this value, which is uninteresting. But if exactly one of the qubits has the value 1 (the other, naturally, has the value 0), they are entangled because we cannot tell which qubit has which value. No matter how far apart they are, if we now measure one of the qubits to be in the state 0, the other will immediately end up having the value 1 and vice versa.

There is are many things one can do once the qubits are entangled — transfer quantum states using quantum teleportation, send encrypted messages using quantum key distribution, or try to confirm quantum mechanics by violating Bell’s inequality, for example.

Ultimately, people are interested in creating entanglement in more complicated systems; with superconducting circuits, it would be interesting to have many qubits entangled. There is one very practical reason for that: Superconducting quantum computers need to work at very low temperatures (only about 0.01 °C above absolute zero) and it is very difficult to cool things to such low temperatures. As a result, only small things can be successfully cooled. Future superconducting quantum computers therefore cannot be very large; to have one large, powerful quantum computer, it is then necessary to connect several such computers using entanglement. Then, the many small computers will behave as if they all were in the same large fridge, forming parts of a large quantum computer.

This post aims to summarise the main results of a paper I wrote with my PhD advisor on the topic of generating entanglement of superconducting qubits using optomechanical systems. A free preprint can be found at arXiv.



1 Superconducting qubits work similar as classical bits in a computer — there is a current running through a circuit and the value of this current determines the value of the bit. The only principal difference is that a classical bit has a value of either 0 or 1 whereas a quantum bit can also be in their superposition, having values 0 and 1 simultaneously.

 

Through the looking glass

Studying physics ultimately changes the way one sees the world. This is probably true for any subject but with physics, this change goes deeper than with biology or history. One starts to see some very basic things very differently. At least that is what I think.

Take the simple act of measurement, for example. You want to know what the weather is like? You check the thermometer. Want to know whether you lost weight? You step on the scale. In any case, the act of measurement is just a way of obtaining some information that already exists. There was a particular temperature outside before you looked and it does not depend on whether you look or not.

Then you start learning quantum physics and your understanding of measurements dramatically changes. You want to measure a position of a particle? Sure, you can do that. But unlike in normal world, it does not make much sense to talk about the position before you measure. The particle was not here or there before you measured, there was only certain probability for it to be there.

Classical particles (left) are simply little balls but quantum particles (right) are just a cloud of probability.
Classical particles (left) are simply little balls but quantum particles (right) are just a cloud of probability.

At first, this might seem similar to your everyday experience — when you are looking for lost keys, you do not know where they are so there is only a certain probability to find them at a particular place. But there is an important difference; although you are unaware of the exact position of your keys, they are lying at a particular place. A quantum particle, however, is literally at several places at once. Only by measuring its position you localise it at a particular place. It is as if you are looking at the thermometer changed the temperature outside.

When the position of a quantum particle is measured, its cloud of probability is squashed, representing the gain in information about its position we get.
When the position of a quantum particle is measured, its cloud of probability is squashed, representing the gain in information about its position we get.

Since the particle was not at a particular position before the measurement but it is at a specific position afterwards, the measurement changes the behaviour of the particle. If you now let the particle move freely (i.e., without observing it), it will behave differently than if you did not look at it in the first place. To use the analogy with measuring temperature, it is as if the weather during a day depended on whether you looked at the thermometer in the morning.

Since the measurement affects the state of the particle, its evolution is depends on whether a measurement was performed or not.
Since the measurement affects the state of the particle, its evolution is depends on whether a measurement was performed or not.

If that is still not enough for you, you can go deeper and ask how the measurement process works. First of all, you will find that people know surprisingly little about that. They will tell you that the system you are measuring (such as the particle whose position you want to measure) interacts with a second, meter system in such a way that some variable of the meter contains information about the measured system and can give a strong, classical measurement signal. But what determines whether a system is classical and can be used to measure other systems or whether it is quantum and can be measured by other systems? Not a clue.1

Even with this little knowledge about measurements, people can describe what is going  on surprisingly well. Because the system and the meter have to interact for some time, a lot can happen during the measurement. If you try to measure the position of a particle, the particle will continue to move while you are measuring. Measure too quickly and you will not know where exactly the particle is because you do not collect a strong enough signal. Measure too long and the particle will move too much during your measurement.

In the end, you can never measure as precisely as you would like. There will always be a small uncertainty in the position of your particle. And this gets even weirder when you try to look at the position later again. Quite surprisingly, the better you know the position at an early time,the more blurred the measurement will be at a later time. This is the result of Heisenberg uncertainty relation between position and momentum but that is a story for another time.2

Another thing you can do is measure really slowly so you need a long interaction time between your system and the meter. At any given time, you do not have a complete information about the state of your system, i.e., you never know exactly where your particle is, all you can have is a guess. The measurement then becomes an inherent part of the evolution of your system and can be used to steer it. There is now certain randomness in the evolution (remember, all we can talk about before the measurement are only probabilities of each outcome and the measurement is thus random at heart) but that does not matter that much since you know what the random measurement outcome is.

You can imagine a simple feedback loop as a sequence of a measurement, a feedback force, and a free evolution. This way, you can, e.g., stabilise the position of a particle.
You can imagine a simple feedback loop as a sequence of a measurement, a feedback force, and a free evolution. The measurement outcome is random but the feedback ensures that the particle stays frozen at a fixed position.

If you do not like this randomness, you can use the information you get from the measurement to control your system. You can, for instance, use the result of the position measurement to keep a particle pinned to a particular position. Every time it tries to move a bit (and everything moves a lot in the quantum world), your measurement will tell you so and you can push it back. We thus came to the notion of measurement feedback I already talked about before.

Realisations that such simple things as measurements have such rich and complex internal structure are one of the things I love about physics. Where most people see a simple (and a little boring) way to get some information, I see an incredibly complex process people still don’t understand after studying it for decades. More than that, measurements are for me a tool that we can use to control and manipulate quantum systems. And there is nothing boring about that!


1 I am, of course, simplifying things a bit here. There is a lot that we know about measurements (and a lot we don’t!) but it all involves a lot of counterintuitive things and complicated maths. There is no room for the details in a blog.

2 Here, I am mixing the notion of single-shot measurements (i.e., measurements you only do once) and repeated measurements (which can be used to obtain statistics). But since even a single-shot measurements takes up a finite time, it is, in a way, a statistical matter. I will try to get to this problem in a later post.