How to close an open system

When students encounter quantum physics for the first time, it is as simple as it gets — there are no unwanted interactions, no noise, particles do not get lost. In the real world, nothing is so easy, though.

Take a single atom placed in an optical cavity, for instance. (The cavity helps to enhance the interaction between the atom and the electromagnetic field, just like it did with the optomechanical interaction in the last post.) We would like to have just the interaction between the field inside the cavity and the atom but there is a lot more going on. The field can leak out from the cavity or the atom might lose energy. The atom and the cavity field thus represent an open system because they interact with the outside world.

Things would get better if we could somehow keep track of what is happening. We could, for example, place a detector outside the cavity so we can see every photon that leaves. Every time we register a photon, we know that there is one photon less inside the cavity. This approach brings us then more information about the cavity than we would have without the measurement.

This idea was originally developed as a simple numerical tool to solve dynamics of open quantum systems. Because the system fast becomes complex with growing size, only small systems can be analyzed directly. But if we randomly generate many possible measurement results we get from such a system and take the average, we end up with the same result we would get by solving the dynamical equation.

When monitoring the outside field of an optical cavity, we can  undo the effect of losses by conditionally affecting the system inside the cavity.
When monitoring the outside field of an optical cavity, we can undo the effect of losses by conditionally affecting the system inside the cavity.

At first, this was just a useful numerical tool but today experimentalists can indeed watch cavities lose photons in real time. They can do even more — if they see that a photon has been lost, they can inject a new one into the cavity and keep the cavity field at a constant intensity. Moreover, if the cavity field, interacts with an atom, the outgoing photons carry some information about the state of this atom and we can use more complicated feedback on the atom. In this way, the state of the atom and the cavity field can be stabilised and the effect of the losses (at least partly) undone.

Measurement and feedback have become a powerful tool in quantum physics. Apart from protecting quantum systems from losses, they can also be used to bring a system to a desired state. For example, in optomechanics one of the main problems is noise in the mechanical oscillations. Because of low frequencies of mechanical oscillations (usually of the order of megahertz up to a few gigahertz), the mechanical oscillator is full of random vibrations that degrade the interaction with light. Measuring the oscillator position and applying feedback, it is possible damp the random oscillations, leaving the mechanical oscillator in its ground state and ready for a truly quantum interaction with light.

Mixing output of two optical cavities on a beam splitter, it is possible to entangle atoms that sit inside these cavities even though the atoms never interact directly.
Mixing output of two optical cavities on a beam splitter, it is possible to entangle atoms that sit inside these cavities without their direct interaction.

The resulting state can even be more complicated than that. We can take two of the atom-cavity systems and mix the output fields on a beam splitter. (A beam splitter is a partially reflecting mirror, that lets part of the light go through and reflects the other part. It is then possible to send in two different light modes and get their combination at each output.) Using suitable interaction between the atoms and the cavity fields and a proper measurement, one can entangle the two atoms even though they never interact directly. The feedback is then used to ensure that the atoms always end in the same state. This can be important for some tasks because the measurement results are in principle random and the particular state of the atoms is then random as well.

The main advantage of measurement based feedback for preparing desired states lies in combatting losses. If you want to prepare a quantum system in a certain state by well controlled interactions excluding the outside world (i.e., in a closed system setting), any kind of losses will have a negative effect on the state. With measurement and feedback, however, you let losses work to your advantage because you learn information about the system by monitoring what comes out.

All that said, feedback is not all-powerful. There are usually more kinds of losses present and you typically cannot have them all reverted. Even then, detectors never work perfectly so the losses cannot be compensated for completely. It is also not always obvious what form the feedback should take to bring your quantum system to the state you want to reach. Nevertheless, it is a crucial instrument in studying quantum systems and their possible applications.

Of light and springs

Using light, we can achieve more than simply see the world around us. Spectroscopy can be used to find chemical composition of a sample, frequency of light interacting with atoms can be used to measure time. We can even move objects by shining light at them. Such manipulations are far from tractor beams of science fiction, but optical tweezers are commonly used to manipulate small objects in many labs around the world. And there are other ways how to control matter using light.

This is possible because photons, the particles of light, carry momentum (even though they are massless!), and during an interaction between any two objects, momentum has to be conserved. When a photon bounces off a surface — say, a mirror — it changes its momentum because it changes the direction of its movement. This change of momentum has to be compensated by exactly opposite change of momentum of the mirror.

Photon bouncing off a mirror
When photons reflects off a mirror, part of its momentum is transferred to the mirror, causing it to move.

So why don’t we see light moving objects around all the time? Because light’s momentum is tiny. If you turned on a laser pointer (which is the source of the most concentrated light you can get easily) for a single second, the momentum of the light beam would be about a hundred thousand times smaller than that of a flying mosquito.

That does not stop scientists from trying to see this effect, though. Many physics labs have lasers stronger than your average laser pointer. They can also produce tiny mirrors (as small as few hundred micrometers) that are very light and thus much easier to move. When suspended on a spring, such a mirror should start swinging when light is shining on it.

It turns out, however, that this is still not enough to see the mirror jump to motion when laser light reflects off it. But there is a simple trick how to enhance the effect. You can use a second mirror (this one fixed in place) and trap the light between these two mirrors — forming an optical cavity is a well-known way to enhance interactions in quantum optics. The light bounces many times back and forth and while a single kick to the movable mirror is not enough to make it move, kicking it again and again finally sets the mirror in motion. And since there is a cavity and light interacting with a mechanical oscillator, the field studying such systems is called cavity optomechanics.

Standard optomechanical setup
To enhance the interaction between light and a mirror, light is trapped inside a cavity so that each photon interacts with the mirror many times.

There is a very simple and intuitive explanation of what is going on in such a system. This is thanks to the fact that light can survive in the cavity only when the cavity length is a whole number multiple of the light’s half-wavelength. Such a light enters the cavity and starts pushing the moving mirror, lengthening the cavity. But that means that the cavity resonance (i.e., the wavelength or frequency of the light it supports) shifts. The light, no longer supported by the cavity, then begins to leak out which decreases the pressure it exerts on the mirror which thus moves back. The light intensity inside the cavity increases and the cycle start again.

This probably sounds like a neat toy to play with but it might seem that there is not much to do with such an apparatus. But if the system is built very carefully, the light, as well as the vibrating mirror, has to be described quantum mechanically. This means that the forms of vibration are not completely arbitrary but can only have certain discrete values. Like light (electromagnetic vibrations) is build from photons, the vibrations of the mirror come in form of quantum particles — phonons. And that opens up a whole new sea of possibilities.

For example, we can send in light that is not exactly at the cavity resonance but whose frequency is smaller by an amount corresponding to the mechanical frequency. What can happen is that a photon we sent in combines with a phonon and they create a photon at the cavity resonance. Similarly, if we shine both resonant and detuned light, the resonant photon can split into a phonon and a detuned photon. In this way, we can transfer resonant photons to phonons and vice versa, swapping the vibrational state of the cavity and the mirror.

Alternatively, we can also send in light with frequency higher than the resonance frequency. A photon of this frequency can now split into a phonon and a cavity photon. Because the photons and phonons are now created in pairs, the number of excitations in the cavity and the mirror are now correlated. These correlations are stronger than any correlations in classical physics could be and the cavity field and the mirror become entangled. (And trust me, physicists can have loads of fun with that!)

With these tools at hand, there is a lot one can do. Perhaps the most intriguing thing is trying to see how far quantum mechanics can go. If we use larger and heavier mirrors, will we still see quantum behaviour, or will they start behaving classically at some point? Will there be a slow transition from quantum to classical oscillations or will such a change be abrupt? Today, nobody really knows, and it is a question that has bothered physicists for a while.

A more practical option for such systems is to use them for frequency conversion. A single mirror can be coupled to several cavities (or a single cavity with several resonant light modes). Using state swapping as explained above, it is possible to transfer signal from one light field to another by swapping it first from the first light field to the mirror and then to the second light field. It is even possible to convert visible photons to microwave photons or the other way round. This can be used, for instance, to improve detection efficiency of microwave fields from which magnetic resonance imaging could greatly benefit. Since this conversion is something I am working on in my PhD, you will certainly here more about this problem later in more detail.