Is the Moon in the sky when you’re not looking?

If you find quantum physics hard to understand (or accept), rest assured that you are not alone. Even many physicists (including Albert Einstein, one of its founding fathers) refused to acknowledge that our world can behave so strangely. That atoms or electrons can be at two places at once or that it does not always make sense to talk about properties of particles before they are measured.

Physicists are also only people. When a new theory challenges our worldview, we start to look for mistakes in said theory and not in our assumptions about the universe. Many physical theories had to fight against ingrained beliefs — heliocentric model of the Solar system, Einstein’s special and general relativity, or quantum mechanics are just a few examples. Eventually, the theory prevails, at least until it is replaced by a new, more precise one.

In quantum physics, the assumptions of locality and realism are challenged. The locality assumption — which comes from special relativity — tells us that all information can travel through the universe only at the speed of light and not faster. When we assume realism, we assume that the outcome of a measurement exists already before the measurement is performed. In other words, if I look at the thermometer to find out how warm the weather is today, the temperature is not decided the moment I look at the thermometer; the air has this temperature independent of me looking at the thermometer.

Let’s illustrate that on an example. Suppose I have two coins with a very peculiar link between them. Every time I flip these coins, I get opposite outcomes. If the first coin shows heads, the other one shows tails and vice versa. I never know which coin will give which outcome but whenever I look at one coin, I immediately know what the outcome of the toss of the second coin will be.

So far so good. Now, I will take one of the coins and fly to the Moon while leaving you with the second coin here on Earth. If we now flip our coins at the same time, I immediately know what the outcome of your toss is, and you know the outcome of mine.

What does that have to do with local realism? Since we are now about 400 000 km apart, it takes any information about 1.3 seconds to travel between us. My coin thus cannot know what the outcome of your toss is; similarly, your coin knows nothing about my toss. That is what the locality assumption tells us.

How do the coins know what side up to end to always give opposite outcomes? That’s where the realism comes in. In this situation, it tells us that the result of the measurement has existed before the toss and both coins therefore know what outcome the toss is supposed to give.

This is how anyone should expect two such coins to behave. But if the coins obey the laws of quantum mechanics, things are different. We cannot say that the outcome of the toss exists before we actually toss them. (This is actually a matter of interpretation of quantum physics — it is generally assumed that the measurement outcome is decided the moment the measurement is performed.) That’s why some physicists claimed quantum physics must be incomplete — there must be some underlying theory that explains what outcome every single coin flip will give. And such a theory must be local and realistic.1

What should we believe? Local realism or quantum physics? It turns out there is a simple test for that. Suppose that instead of a pair of coins we have two such pairs and I take one coin of each pair to the Moon and keep the other two coins here with you. If we now both flip coins from the same pair, we will always get opposite outcomes. But if we flip coins from different pairs, any combination of outcomes is possible.

While the experiment with a single pair of coins was largely a matter of interpretation, with two pairs of coins do local realism and quantum physics predict different results. All we have to do is toss the coins many times, each of us deciding randomly which of the two coins to toss each round and then writing down which coin we tossed and what the outcome was. Then, we can compare our data and see which of the two theories is right.2

Although the test of local realism is, in principle, rather simple, it is not easy to build an experiment that can confidently decide whether local realism is true or not. There are two main challenges that need to be solved: The first problem is to make sure that the two systems are spatially separated. Here, it is important that the time difference between the measurements is so small that no communication between the two sites at the speed of light is possible. Since the distance over which quantum systems can be reliably transmitted is strongly limited, there are strict requirements on the synchronisation and speed of the experiment.

The second main problem is an efficient measurement. Most experimental tests of local realism are done with single photons but it is extremely difficult to detect those. The efficiency is so low that often detectors do not notice when a photon arrives. This opens a loophole — the measurement does not grant us access to the whole statistics but only to its part. And we cannot be sure that the statistics of the sub-ensemble is the same as that of the whole ensemble.

It took over 30 years to build an experiment (more precisely, three experiments; one with electron spins and two more with photons) that really confidently refute local realism and show that quantum mechanics has to be taken seriously. One of our basic assumptions about this world thus has to be wrong; some signals are able to travel faster than light or it does not make sense to talk about objects we are not currently observing.


1 The experiment with two coins and the conclusion in this paragraph are a simplified version of the famous EPR paradox. It was formulated by Albert Einstein, Boris Podolsky, and Nathan Rosen in 1935 to show the problems of the Copenhagen interpretation of quantum mechanics.

2 Such an experiment — not with coins but with electron spins — was first proposed by John Stewart Bell in 1964. He showed that a particular correlation between the spins is bounded by the value of 2 for local realistic theories, whereas quantum mechanics allows to have stronger correlations, with values exceeding 2.

 

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.

How to measure time

Precise timekeeping is crucial for many of our daily activities. High-speed communication (on the internet or in a mobile phone network), satellite navigation, and many other tasks require time synchronisation over long distances to work properly. But how is time measured? And can quantum physics help reach better accuracies?

The basic idea behind measuring time is simple and similar to any other measurement — you simply compare the time duration with a reference. The reference has to be an event that regularly repeats itself so that a single repetition is the basic time unit and the number of repetitions gives the overall time. An example of such  process can be the Sun rising and setting every day. Its regular movement in the sky defines one of the most fundamental units of time: a day. It is a very simple and natural way to measure time but it has one disadvantage — it is long. If we want to measure shorter times, we need a better reference — a process that repeats itself faster than once a day.

For this, we can use a pendulum and let it swing. Its movement will be periodic and counting the number of swings, we can measure time. Since the period of the oscillations depends on the length of the pendulum, we can even tune it and choose how fast our reference should be. If the length of the pendulum is about 25 centimetres, its period will be 1 second. (Clocks normally use pendulums that are about 99.4 centimetres long resulting in a period of two seconds, or a half-period of one second.) Using a system of gears, the periods can be counted and transformed into movements of hands that then show time on a clock.

There is one problem with using pendulum, though. The exact period of the swinging depends on the local gravitational field which varies on different places on Earth, depending on their latitude and altitude. A pendulum that oscillates with 1 second frequency on the Equator will have a period of 997 milliseconds on the North Pole. That might not seem like such a big difference but in a single day, the North Pole clock will be faster by more than four minutes! Clearly, if we want a more precise time measurement, we need something that oscillates even faster.

Quartz crystal for a wristwatch or a clock.
Quartz crystal for a wristwatch or a clock. (Public domain, source.)

The most commonly used oscillator in today’s clocks and watches is a quartz crystal. It can be made very small and due to its mechanical properties, it can vibrate at much larger frequencies. Typically, the crystal vibrates more than 32 thousand times in a single second and is therefore much more precise than pendulum clocks. The accuracy is improved from about 15 seconds per day to half a second per day — an improvement by a factor of 30. (The improvement is not larger because quartz clocks — especially wristwatches — suffer from many technical imperfections that are not so strong in pendulum clocks.)

We can use even faster processes to further improve the accuracy of timekeeping. But it is difficult to make mechanical oscillators — pendulums, vibrating crystals, or anything else — that can oscillate at such high frequencies. We therefore need some natural oscillator with a very high frequency. For that, we can use atoms because their internal energy can only have certain discrete values and an energy difference between two levels corresponds to a certain frequency of electromagnetic field that can be emitted or absorbed by this energy transition.

On-chip atomic clock.
On-chip atomic clock. (Public domain, source.)

Atomic clocks have two advantages: they are natural oscillators (not human-made) so that atoms of a given species will always oscillate at the same frequency, and they oscillate very fast — billion times a second. There is a price to pay for this precision because, naturally, it is extremely difficult to count individual periods of a system oscillating so fast. It can still be done, though, and such clocks are now the most precise time standards we have — their error is about one second in 100 million years.

Some atomic transitions have even higher frequencies than a few gigahertz which are used in atomic clocks now. Transitions in the optical domain (in contrast to microwave transitions for gigahertz frequencies) oscillate million billion times in a single second. Those oscillations are, of course, even more challenging to count than the oscillations in current atomic clocks. Clocks based on the optical transitions — called optical clocks — are nevertheless being developed and promise incredible accuracy. With optical clocks, it is possible to measure the age of the universe (about 14 billion years) with error smaller than one second!

What are such highly precise measurements good for? Without well synchronised time across the Earth, internet communication (and any other form of high-speed communication, including mobile phone networks or TV and radio signals) would be much slower. Navigation systems (such as GPS) would not work with a few-metre precision. GPS receivers measure time delay in signals from satellites and determine the position from the delay and positions of the satellites. More precise time means better accuracy of the navigation system.

There are also many scientific applications. With precise time measurements, we can, for instance, test one of the predictions of Einstein’s general relativity which states that the flow of time is affected by a gravitational field. In a strong gravitational field, time passes slower than when the gravity is weaker. The effect is very weak in the conditions on Earth but it still has to be taken into account for satellite systems. Current atomic clocks are, in fact, so precise that this difference in passage of time can be measured in two places that are about ten centimetres above one another.

New page on the basics of quantum physics

Do you have absolutely no knowledge about quantum physics? Do you want to get at least a basic understanding about what it is, why it is important, or how it can be relevant to everyday life? I just added a new page that tries to explain some of these basic questions in (I hope) an attractive form. This page is a work in progress, so remember to check back now and then as more and more issues will be answered! Also, if you have some questions yourselves that you would like answered, just ask and I will try to add them to the list.

Is nature scared of emptiness?

There can never be a truly empty space. That was the opinion of many scholars from the times of ancient Greece up to the beginning of the twentieth century. When the idea of aether as a medium in which light can travel has been refuted, the existence of vacuum became widely accepted. But then the quantum revolution came, and nothing is ever simple with quantum physics.

The main obstacle in achieving space that is entirely empty is the Heisenberg uncertainty principle. It states that the position and momentum of an object can never be known exactly. This is, furthermore, not just due to technical imperfections in measuring these quantities; the object itself does not know them exactly.

Let us now take a glass cell and pump all air out. If we also leave it in complete darkness, there will be no light and, therefore, no electromagnetic field and no atoms or molecules inside, right?

Not quite. Light is an oscillating electromagnetic field and as such can be mathematically described as a harmonic oscillator, similar to a pendulum. And a harmonic oscillator has a position and momentum which, even at ground state (i.e., with no light), cannot be exactly zero but have some uncertainty. So there still is some electromagnetic field present, even in complete darkness!

But things can get even weirder because in quantum physics, virtually everything can be described as a harmonic oscillator. For every kind of particles, there can be defined a field whose excitations are the respective particles. For light, there is the electromagnetic field and the particles are photons, electrons are excitations in an electron field, and so on. And each harmonic oscillator has to follow the uncertainty principle. In our glass cell, we thus have a small bit of fluctuations of the electromagnetic field but also fluctuations for electrons and other particles. Vacuum is an endlessly boiling soup where every now and then an electron pops out and disappears again, then a quark, then something else.

Two metallic plates placed in vacuum will attract or repel each other due to vacuum fluctuations.
Two metallic plates placed in vacuum will attract or repel each other due to vacuum fluctuations.

Does all that sound ridiculous? It turns out that these phenomena have observable effects. Take, for instance, two metallic plates placed in vacuum. One would naively expect that nothing will happen to them since they are in vacuum. But we know better — there are always fluctuations, and these will be smaller in the space between the plates than everywhere around. As a result, the plates will attract each other; in a different configuration than parallel, they could even repel. This behaviour is known as Casimir effect (though I am stretching things a bit here — only the fluctuations of the electromagnetic field are important for the Casimir effect) and has already been observed in an experiment.

Another, even more important evidence of fluctuations of the vacuum is the existence of spontaneous emission. If you excite an atom (for example by shining light on it) it will eventually radiate the energy it absorbed and end up in its ground state. But from the point of view of classical physics, this happens only when there is electromagnetic field around the atom. This means that an excited atom in utter darkness should stay excited — but it does not! This can only be explained by quantum physics; fluctuations in the vacuum are strong enough to kick the atom to its ground state while emitting a photon, similarly to the presence of electromagnetic field in the classical picture.

So remember — vacuum (for instance the vast empty space between you and the nearest star when watching the skies at night) is not empty. It is alive with many particles that we can never directly see, swirling around. And nature maybe, after all, really is scared of emptiness.