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.
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.
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.
5 thoughts on “How to close an open system”
“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.”
Cannot the lost photon be entangled with the cavity (it’s phonon) which would mean that measuring the photon would modify the cavity’s state?
That’s a very good question. And, in fact, that is exactly what happens. Since the interaction of the cavity field with the outside world is unitary, it creates entanglement between the light fields inside and outside the cavity.
The problem with this description is that the light field outside the cavity comprises many different light modes and we don’t know which particular mode gets entangled with the cavity field. We hence can describe only the cavity field decoupled from any outside influence, and the arising description brings the simple dynamics in terms of cavity losses.
We could, in principle, place a second cavity right after the first one and then the light leaking from the first cavity into the second one would generate an entangled state of the two cavities. But that happens only because we have a well defined light mode in each cavity and it is then possible to keep track of their evolution.