Simplifying quantum systems

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I already talked about using measurements and feedback in quantum physics and how these tools can be used to prepare interesting quantum states. But it is not an easy task — experimental realisations require ultrafast electronics to apply feedback in real time. And theoretical analyses? Those are not easy either.

Take a simple example — an atom that is placed inside an optical cavity. We measure what leaves the cavity and want to use the information we get to control the state of the atom. The first thing a theoretical physicist will do is write the equation that describes the time evolution of the whole system (i.e., the atom and the cavity field). But we do not really care what happens with the field. We only want to know what the field can tell us about the atom. If only there was a way to get an equation that describes only the dynamics of the atom…

There actually is a whole bunch of methods that can help us do just that — they are generally known under the name adiabatic elimination. We eliminate the uninteresting part of the system (the cavity mode), leaving an equation just for the relevant part (the atom). And why is it called adiabatic? Because all the methods assume that the uninteresting part evolves much faster than the interesting one — the cavity mode will thus quickly reach a steady state (i.e., a state that does not further evolve in time) and, as the atom slowly evolves, the steady state of the cavity will follow it. And physicists call such following of one system by another adiabatic.

All these methods generally suffer from two problems. Firstly, they work only if the cavity field is in a so-called pure state. These are some rather special quantum states that you can get if there is no thermal noise (i.e., the system is cold or uses high — typically optical — frequencies). You can imagine thermal noise as if you were shining inside the cavity with a regular light bulb. Its light is very chaotic (much more than that of a laser) so the state of the field inside the cavity will be chaotic as well. And that is more difficult to deal with than when the light entering the cavity a nice coherent laser beam.

Secondly, adiabatic elimination methods can work well if you need to eliminate a single field.  If you have a more complicated system that you need to get rid of, it is not that simple. You can, in principle, eliminate more fields one by one but that takes a long time. And the order in which you eliminate imposes additional conditions on the system. (You start by eliminating the fastest of the fields, then the second fastest, and so on.)

Imagine now that you want to work with a more complicated system — you want to entangle two superconducting qubits coupled to optomechanical transducers (like I do). You have the transducers — consisting of a microwave cavity, a mechanical oscillator, and an optical cavity — that you do not really care about and the qubits that are the important part of the system. So if you now want to eliminate the transducers, you have a problem because you have many fields (three for each transducer) and mechanical oscillators which will have thermal noise.

Here, the adiabatic elimination becomes more crucial than with a single atom and a single optical cavity. Whereas it is just a matter of convenience for the simple system, the two qubits with two optomechanical transducers cannot be numerically simulated exactly. You would need several terabytes of memory to store state of such a large system in a single point in time. And what should the feedback applied to the qubits look like? You cannot guess that well with such a complicated system.

In order to be able to deal with such big and complicated systems, we had to develop a brand new method of adiabatic elimination and we had to take a completely different approach than people usually take. We made a different assumption than the usual purity — instead, we assume that the eliminated system is Gaussian. This means that there are some quantities in the system (our optomechanical transducer) that behave as a classical Gaussian probability distribution. That is true for a large class of systems (including our optomechanical transducers) and makes it possible to describe the transducers using parameters of these Gaussian distributions which is much easier than using a full quantum state.

The applications of this method are much broader than this particular system. Measurements and feedback are often used in superconducting systems which typically interact with microwave fields. As a result, thermal noise can be present and standard methods of adiabatic elimination do not work. The way around this problem is to assume that the noise is so small that we can safely neglect it and apply standard methods of adiabatic elimination. This assumption usually works relatively well but our new method works even better (with the almost nonexistent noise!) and is not much more complicated to deal with.

There is more to adiabatic elimination than tractability of numerical simulations (which is still pretty important!). It can give us information about the evolution of the small part of system that we are really interested in. A trained scientist can make a good guess based on the evolution of the whole system (including, for instance, the cavity field) but understanding the exact role of various system parameters (such as the amount of the thermal noise) is not always so easy. Now, we have made an important step in understanding these issues.

This post summarises the main results of a paper I wrote with my colleagues on adiabatic elimination with continuous measurements. A free preprint can be found at arXiv.