Howard Wiseman, a theoretical quantum physicist, is a Queen Elizabeth II Research Fellow in the School of Science at Griffith University in Queensland. He did his BSc and PhD in the Department of Physics at the University of Queensland. From 1994 to 1996 he worked on quantum and atom optics at the University of Auckland, before returning to Queensland with an Australian Research Council Fellowship to undertake more work on quantum feedback. He has had a permanent position at Griffith University since 1999. He has almost one hundred refereed publications and is currently writing a textbook for Cambridge University Press with Gerard Milburn. He is on the editorial board of the American Physical Society journal Physical Review A and is manager of the Griffith University Program in the ARC Centre for Quantum Computer Technology. He is the recipient of the 1995 Bragg Medal of the Australian Institute of Physics, the 2003 Pawsey Medal and Prize of the Australian Academy of Science, and the 2003 Malcolm McIntosh Prize for Physical Scientist of the Year.

Quantum feedback control – from the playground to the laboratory
by Associate Professor Howard Wiseman

I thought I should begin by explaining why I chose the title I did and just going through the terms. What does 'quantum' mean? Perhaps the simplest way to look at a quantum is that it is a sort of fundamental graininess in nature. It is the minimum amount of a certain physical quantity that you can get. We need to use quantum physics to describe physical systems that typically are very small and well-isolated from outside disturbance. One of the very early examples of the successes of quantum physics was to understand the structure of atoms – that was almost a century ago now. What has happened since then is that there has been a move more and more to applying quantum physics not just to naturally occurring systems like atoms, but to devices which we build. The reason for this is that as technology is becoming miniaturised there are more and more devices which actually have these characteristics of being small and well isolated, and so for which we have to apply quantum physics to understand how they work.

Something which is happening very recently is that this is actually leading to completely new sorts of technologies that could not have been conceived in the past. The leading example of this is the quantum computer.

I am lucky, in following Peter Bartlett, that you now have some idea about control problems. Flying a helicopter in a wind is an excellent example of a control problem. So a general definition of control would be to optimise the performance of a device – you could think of a helicopter as being a device – by observing it and then feeding back the results of the observation to alter its future properties, to make it do what you want.

I have a simpler example, that of cruise control in a car, where you decide what you want your device to do. You may have an optimum speed, say, of 60 kmh, and if the speedometer measures that the car is going more slowly than that, then it will automatically kick in the accelerator to bring you up to the desired speed.

Putting feedback control together with quantum, we have quantum feedback control, which just means applying feedback control to quantum systems. This has been one of my main research interests for over 10 years now, and what I find interesting about it is that where it really differs from ordinary feedback control theory is that with quantum systems you have to worry about the effect of the act of observation on the system itself. That is because of something called the Heisenberg uncertainty principle, which says that you can't measure a quantum object without affecting it.

I guess my first major contribution in this area, in my PhD, was to emphasise that really to understand quantum feedback you should begin by understanding how the state of your quantum system changes when you measure it and get a particular result, including this Heisenberg disturbance.

For the rest of this talk I am going to concentrate on one particular example of quantum feedback control. I don't really have time to give a full survey of the field, so I have just picked my favourite example, adaptive measurement.

The basic idea of it is this: theoreticians often assume that you can measure anything you like when you go into the laboratory, but as experimentalists well know, that is not the case because your technology limits you to the actual physical things that you can measure. And because of that, many measurements done in the laboratory are sub-optimal. What I mean by that is that the measurement introduces extra 'noise' into the measurement result which was not there in the system that you are actually measuring.

The idea of adaptive measurements is that you make a partial measurement of the thing you are interested in on your system, and so this is going to be a sub-optimal measurement because there is a limit to your technology. And then on the basis of the result you get from that partial measurement you change the way you are measuring the system for the next time step. Now you have two pieces of information on which to change the way you make the measurement for the time step after that, and so on. And once you cumulate all of those measurements at the end, you find that you have made a measurement which is much closer to an optimal measurement than you would have been able to do without having those adaptive steps in between.

The first practical example of this was adaptive measurements for optical fields, in particular measuring the phase of optical fields. That was some work which I did in 1995 and it has some applications which I will talk about later.

The Playground (1995)
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For now I want to continue explaining my title. I have chosen the term 'the playground' to emphasise something about the process of science which I thought might be interesting and an important point to make. A lot of good ideas happen not when you are trying to solve a problem but just when you are playing around – at least for theoretical physicists, I can speak personally – with ideas and equations to see where they lead.

This was what I was doing in 1995, just thinking about phase measurements, and I thought about this idea of adaptive measurements, which I had not thought about before. Then, after a couple of pages of maths, I came up with this equation here, which is an algorithm for doing an adaptive phase measurement. And as you can see [from the annotation: Hooray!], I was quite pleased with how it worked.

A lot of work happened after this to flesh out the details – many years of work – but the nice thing is that actually this very simple algorithm which I came up with back then was what was implemented in the laboratory.

The Laboratory (Caltech, 2002)
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As the final part of my title, the laboratory in question is that of Hideo Mabuchi, who is a professor at Caltech. And these [in photograph] are the other people in the team that actually did this experimentally. I am going to explain now a little bit about exactly what this experiment was, so you can appreciate it a bit more.

Optical phase is seen through interference
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To begin I think I need to explain what is this optical phase, which is the thing which we are trying to measure here. Phase in general is a property that waves have. Light is a wave, and so it has this property of amplitude, which is how big it is, and phase, which tells you whether you are in a peak or a trough at any particular point.

Phase is not something which our eyes are sensitive to. We can't directly see the phase of a light when it hits our eye. The way to 'see' it is actually to get a light wave to interfere with another light wave. Interference effects are very important in physics – they lead to some beautiful things like the colours in a soap bubble, for example. But let us consider a simpler example of where you have just two sources of light which are going to be interfering.

In what I have drawn here, the colours have nothing to do with real colours. I am just using, say, red for peaks and green for troughs. So they represent different phases.

If you have just one source of light, over here [under heading: Left wave only] what you would actually see with your eye is just a light beam of more or less constant intensity, just dropping off in intensity as you go away from the source. And the same thing if you had just one source of light over here [under heading: Right wave only]. But if you put the two together [under heading: Left and Right] you get what is called an interference pattern. Wherever you get a peak here and a trough here, they actually cancel and give you dark lines. So these dark lines going up here [still under heading: Left and Right] are characteristic of the interference pattern and they tell you something about the phase relation between those two sources.

Measuring optical phase
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If we want to measure optical phase in the laboratory, we want to make things even simpler. The way it is done is to actually have two beams of light and combine them at a beam splitter, which is just a half-silvered mirror. The idea is that we have a signal beam here with an unknown phase θ, and we have a beam called the local oscillator because we know its phase, φ. They combine at this beam splitter and we actually look at the light here using these detectors. The interference is manifest in the fact that the current which we get out of this detector depends on the phase difference [θ-φ] between these two light beams in some manner like that.

For example, then, if you do not know what θ is but if you measure, say, the current to be this, so here is your measurement of θ...

...The catch is that quantum mechanics says that there is always going to be some noise in that signal. This is another consequence of the uncertainty relation, that you always have noise.

For example, if you measured a current up here somewhere, because of the noise you are uncertain to a large degree about what phase it is that gave you that signal, so you are not going to be able to estimate that unknown phase very well. On the other hand, if you got an intensity about this level, you are going to be much more certain about the phase, because the slope of this curve here is much greater.

Obviously, the best thing to do is to make sure that you are always measuring at this point here, which is called a homodyne measurement of the phase quadrature. But the problem is that if this is really an unknown phase here, then how can you make yourself work at this point? You don't know what the phase is, so how can you arrange for this phase difference to be a particular value?

That was a problem that existed in measuring optical phase, and my idea about how to solve that problem was to do adaptive measurements.

Adaptive homodyne measurement
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To reiterate: the basic idea is that you just randomly pick some local oscillator phase to start with, you measure a little bit of your light signal and that gives you some idea about what the system phase is. Then, based on that information, you change the local oscillator phase for the next time step, so that it is as close to optimal as you can make it, given what you think about the system phase. And then you keep doing this continuously, with a feedback loop like this, and at the end, once you have absorbed all of the light signal, you have an estimate of the phase which is much better than you could have done without using an adaptive technique like that.

I should say that most of these slides I am using here were actually prepared by Hideo, and as he said, you can actually do quite well, even with very weak fields.

Closing the loop
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This is just to show that this really was done in an experiment. This is the experimental data, showing the local oscillator phase being controlled by the feedback loop.

The power of quantum feedback
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Then this is the analysed data. This axis here is looking at the uncertainty in your phase estimate versus the number of photons in the pulse, which is basically the intensity of the pulse. The red line here is the best which you could do without using an adaptive technique, or at least with the current technology available, and as you can see the experiment agrees very well with that. The grey line [Pegg-Barnett] is the theoretical best possible measurement of phase which you could possibly do, with technology which nobody has. And the blue curve, then, is what was actually achieved in the experiment with the adaptive technique. So you can see that it goes a fair way from what you can do without an adaptive technique towards the best possible that you could do. In theory, of course, it will get much closer to that, but that is the experimental results showing that.

Now I just want to say a little bit about the potential applications for this work. There are really two that come to mind. The first is in generation-after-next communication in optical fibres. The reason it might be important here is that there is a problem in using optical fibres for communication, in that you tend to get cross-talk between pulses in different fibres, or different pulses in the same fibre. A way to minimise that is to actually use very weak pulses, because that will minimise the amount that they interact with each other. But then you get into the problem that weak pulses mean more noise.

Current schemes for communication all rely on amplitude modulation of the pulses. The problem with that when going to weak pulses is that it turns out that for technical reasons it is very difficult to produce weak pulses with very well-defined amplitudes. For that reason you might want to consider instead using phase encoding – that is, putting the information on the phase of the pulses – rather than amplitudes. In that case, to read that information out, clearly you want to have a good wave measuring phase, and that is where this adaptive phase technique would come in.

The second application, which is probably even more exciting, is that we have done some recent work in the Centre for Quantum Computer Technology showing that this adaptive phase measurement is actually useful for preparing special states of light that could be applied in optical quantum computers. I probably should explain, then, basically what a quantum computer is. It is still at the moment a hypothetical device. Certainly experiments are under way but they are at a very early stage. The promise is that if we were able to build a full-scale quantum computer it could solve certain problems much faster than any conventional computer. That it is the point: it is not just faster than the computers we have now, but actually faster than any conceivable conventional computer that we could build. So that is the importance of that work.

Spin squeezing via real-time feedback (Caltech)
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I want to mention some other experiments that have been done, just to say that this is certainly not the only avenue of exploration in quantum feedback control. Another one also happens to be one done at Caltech, again with Hideo Mabuchi's lab. This was something that I did some theory on with a couple of other people, fairly recently. It is something called spin squeezing, and it is spin squeezing via real-time feedback.

Spin squeezing via real-time feedback (Caltech, 2004)
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I don't have time to actually explain what spin squeezing is, but I put up this picture to show the rough idea of how this works, just to give you an idea about another sort of quantum feedback.

This is actually a more conventional sort of feedback, in that we have a sample of atoms here, which is our quantum system, we have a probe beam of light going through that, and then we have a measurement device over here which measures some property of those atoms. The signal from that again goes through a feedback loop to control this magnetic field here, so that in the end we change some property of those atoms to produce a particular desired state of those atoms, which is called the spin squeeze state. Again I think it would be too complicated to explain what that is.

But again just to prove that it really has been done experimentally, here are some data, still unpublished [slide unavailable]. For anyone who happens to know anything about squeezing, this is actually pretty impressive data, showing that you can get a reduction. Essentially, what this is showing is a reduction in the uncertainty in a particular property of these atoms, below what you would be able to do by just preparing all the atoms independently. By measuring the atoms collectively and doing feedback, you can do much better than that.

I should have pointed out that the really big application for this is in atomic clocks, because essentially the accuracy of atomic clocks is directly related to the uncertainty on this axis [the y-axis]. So the fact that you can make that much smaller there is a good thing.

To summarise what I have been talking about: this quantum feedback control, which as I said I have been working in now for over a decade, I'm really excited about where it is heading now, because it is really into a new era characterised by three things.

First of all, the effect of the measurement on the system is now being incorporated from the very beginning in the design of the feedback control loop.

The second point is that experiments, as I have talked about, can now be done demonstrating these quantum effects in feedback control problems.

And then the third thing is that some of these experiments definitely have practical applications, like the spin squeezing and potentially like the adaptive phase measurement. I think that this is just the beginning, and that there are many more practical applications for these techniques likely in the near future.

To finish, I would like to thank the other theorists in the Centre for Quantum Dynamics, at Griffith University – many of whom have worked with me on these sorts of areas.

Questions/discussion

Question: One of the things that a number of us know is that there are very considerable sums of government money going into the whole quantum computing/quantum optics area in Australia. I think it is seen as a high national priority. Can you perhaps tell people what it is about?

Specifically at the Centre for Quantum Computer Technology?

Question (cont'd): I was thinking that there is also a New South Wales branch and also the Victorian and Australian National University connections.

To give a bit of an overview: quantum computing is a subset of an even broader field of research called quantum information, which has really only come about in the last decade or so. This is one of the areas where Australia is very strong, compared with its population size.

Part of that is the Centre for Quantum Computer Technology, which now has six universities involved in it. Its principal aim, I guess – where most of the money goes – is towards trying to build a solid-state quantum computer at the University of New South Wales, with Melbourne University also contributing very substantially to that. I have not said anything about that in my talk, but there are a lot of reasons to want to be able to build a quantum computer like that because it could interface more naturally with conventional computers.

The sort of quantum computer which I briefly mentioned was an optical one, and there is experimental work going on at the University of Queensland towards constructing one of those.

They both have the potential to be superpowerful computers, as I said, but they are both in the experimental stage. And so at Griffith we are involved in working on theory which relates both to the solid state and to the optical quantum computer.

As to the ANU connection, there are other experiments; many of you may have heard of the quantum teleportation of a year or so ago. That is definitely an example of a quantum information experiment, but not strictly quantum computing. That was an ANU experiment with a UQ link in that.

There are lots of universities in Australia interested in this sort of thing, and lots of collaboration is going on. So it is a great environment to be working in.

Question: The central problem of classical feedback theory is stability. In the case of a simple control system you might overshoot and then come back and go too slowly and go too fast and possibly even go completely wild. What is the equivalent thing in the quantum feedback systems, and have you addressed it?

Yes, exactly the same problem can arise in the quantum feedback systems. For linear systems much the same theory can be used to address the question, 'Will this feedback loop be stable or not?' So there are a lot of analogies there.

For non-linear systems it is a difficult problem and, I would guess, an even more difficult problem quantum-mechanically. So I would say that that is not a solved problem at all. That is something where the work is still going on.

But I should say, just as a general comment, that engineering principles, and especially the modern control theory in engineering which has developed in the last generation of engineering about so-called optimal control and then, moving beyond that, even to something called robust control, is something that we in the field of quantum control are very interested in and trying to incorporate in our work. So in that sense it is an interdisciplinary area where I think it is taking from engineering and also taking from physics, obviously, to understand how the quantum nature affects things.