PARIS SCIENCES & LETTRES (PSL)

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In September 2011, Nature published a paper about the first experiment involving quantum state feedback. This breakthrough offers many perspectives: the ability to monitor in real-time typical quantum states without disrupting them opens new paths, both in terms of basic research and practical applications. Gathered around the specialist of quantum mechanics, Serge Haroche, the experiment was carried out in combination with other disciplines. Here's the story of this collaboration.

11

April 2017

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**Paris Innovation Review – You are a specialist in dynamic systems and more precisely control: the engineering issues you usually deal with fall far from basic research or quantum physics. How did you get involved?**

**Pierre Rouchon – **By a series of meetings that obeyed both chance and their own internal logic. The stabilization of dynamic systems is an engineering issue as old as physics itself. Over time, it experienced a high degree of mathematical formalization, but the principle remains fairly simple: everything is based on the concept of feedback i.e. feedback loops that allow to stabilize the system.

Quantum systems greatly complicate matters, not only because of their extreme sensitivity but also because in quantum physics, observing a system disrupts it. Feedback, on the hand relies on measurement and therefore, on a disturbance of the system: the so-called “wave function collapse,” in the scientific jargon. What is more, this is a random disturbance. In this regard, “quantum control” is like squaring the circle.

In the early 2000s, I became interested in feedback problems at quantum scale, while pursuing my research and teaching activities. The subject was both difficult and exciting, and I was forced to refresh my knowledge of quantum physics. I read books, articles and followed the lectures of Serge Haroche at Collège de France – extraordinary lectures in which he explained, using simple models, current experiments on the manipulation and observation of quantum states.

But things would have remained there without our first meeting. I had an outstanding student, Mazyar Mirrahimi, who started a thesis on the application of systems theory to the quantum environment. His work was mainly theoretical. But we had good relationships with a colleague in the Department of Mechanical Engineering of Caltech, Richard Murray, who also worked on this type of subject with Hideo Mabuchi, a physicist, and Ramon van Handel, a PhD student. Mazyar spent two months in their lab, which gave him the opportunity to translate our assumptions into something much closer to what could be realized experimentally.

In September 2008, at a symposium in Otranto, Italy, I met Jean-Michel Raimond, who worked with Serge Haroche within the Laboratoire Kastler Brossel. I told him that if they had problems regarding control, it was an issue we were interested in. “The timing couldn’t be any better,” he said, “we just bought a new, powerful, real-time computer and we have a post-doctoral researcher, Igor Dotsenko, who is working on feedback problems. Any backup would be most welcome.” We agreed to meet again in October.

This is how it all started: a chance encounter based on mutual intellectual interest, without any research assignment from the National Research Agency. Mazyar and I had been investigating these issues for several years so we were able to quickly understand the challenges facing the team of the Laboratoire Kastler Brossel. We understood what they said, we understood their models and ideas about possible feedback and we were quickly able to build on and adapt Mazyar’s work at Caltech. So we helped them develop the feedback loop.

**Can you tell us, in a few words, about this experiment which is referred to in the Scientific Background of the Nobel Prize shared by Serge Haroche and David Wineland in 2012?**

The purpose of the experiment is to stabilize the light around the so-called photon number states (or Fock states). These quantum states are very different from classical states used to describe light. Quantum states, where light can be reduced to a few photons (three or four, for example), are extremely fragile. They are hard to obtain and even harder to stabilize. The challenge of the experiment is twofold: directly observe the behavior of photons, something Einstein dreamed of and was deemed impossible to put in practice for a long time; but also learn to stabilize a quantum state, fighting against “decoherence” i.e. disruptions that result in the loss of information into the environment. In this regard, controlling decoherence appears as a significant step towards the quantum computer, allowing for the “protection” of quantum states.

Before exploring any further the details of this experience, let us remind here that the design, development, physical modeling as well as the real-time implementation of the controller are entirely the work of Laboratoire Kastler Brossel.

The controlled light is formed by photons that correspond to an electromagnetic field at a frequency of approximately 10 GHz. The latter is confined between two superconducting mirrors facing each other and forming an open cavity on both sides. This cavity acts as a trap for the photons that “bounce” on the mirrors.

The aim of the operation is to count the photons without destroying them. Atoms go through the cavity, one by one, and interact with the photons. The energy of each atom – that can take only two different values – is measured at the exit. Therefore, the detector provides a binary measure: 0 for the lowest energy and 1 for the highest energy.

The laws of quantum mechanics provide two types of crucial information for this experiment. On one hand, even if they don’t predict the value of 0 or 1, they provide the *probability* of detecting 0 or 1 based on the quantum state of the photons. This probability is valuable for the feedback summary.

Moreover, they predict the changes experienced by photons once the detection occurred. These modifications are generally different for a detection of 0 and a detection of 1. But for this experiment, this doesn’t apply if photons are in a Fock state. The measurement process is nondestructive for Fock states: Quantum Non Demolition (QND) measurement of photon(s).

**Does this open possibilities of control?**

Yes, precisely. For control, a conventional electromagnetic source is used, at the same frequency as the photons confined in the cavity. This source is used between the passage of two atoms, in the form of a small pulse with an adjustable duration and phase which illuminates the cavity and thereby modifies the state of the photons in a deterministic way.

To avoid any doubts: this type of control isn’t enough to obtain Fock states. But by combining it with the measure outlined above, we can obtain and stabilize these states.

The control algorithm proceeds in two steps.

In the first, the quantum state of photons is estimated. This estimate is updated in real time by the computer, at each detection, approximately every 80 microseconds. It is based on a filtering algorithm of input signals (“conventional” electromagnetic pulses) and output signals (detections of 0s or 1s). The algorithm incorporates the measurement feedback outlined above. This quantum filter provides in real-time the quantum state of the photons.

In the second step, we calculate in real-time quantum-state feedback law. This feedback law provides the period and phase of the next conventional impulse that is applied, the control input. This pulse will bring the photon state to the desired state i.e. a state with target number of photons(s) (the controller set-point), typically either one, two, three or four photons.

Here, the feedback has a specificity: it is a non-linear function of the full quantum state of the system – a state given by the filter mentioned above. This experiment is the first experimental realization of a state feedback at quantum scale.

It revealed that when one of the mirrors absorbs a photon, the estimated state begins to deviate significantly from the target after approximately 3 milliseconds. The feedback is then activated and it takes about 15 milliseconds to bring the field back to its set-point, the state before the absorption of the photon by the mirror. In total, the process takes less than 20 milliseconds, shorter than the lifetime of a photon in the current cavity (around 70 milliseconds).

**What was your contribution to the realization of the control?**

Mazyar and I contributed to two aspects that both fall within our field of research: systems theory.

Regarding “measurement and estimation,” we helped develop the filtering algorithm to estimate the quantum state of the system in real time; and in the “feedback and stabilization” part, we realized a Lyapunov synthesis, a mathematical method often used to stabilize non-linear deterministic dynamical systems. This method relies on a Lyapunov function, which measures the deviation between the estimated field and the target: by feedback, it can be made to strictly decrease over time, ensuring an asymptotic convergence towards the goal (a strict supermartingale).

We also helped to take into account the delay in the feedback loop. A probe atom passes every 80 microseconds and there are three or four atoms between the cavity containing the photons and the detector. Hence, there is a significant time delay in the feedback loop and this delay needs to be taken into account in the quantum filter and by the feedback law.

**Adapting the control theory to this experience is a major innovation. Were any theoretical models available?**

Yes, because without even mentionning the Lyapunov functions, mathematicians developed sophisticated tools to handle these situations: the so-called hidden Markov model. Let’s imagine a system that contains ten variables: you only have one sensor, which provides information about one single variable. If you know the equations of the system, their evolution over time, you are able to infer all nine variables from the information on this isolated variable. It’s an asymptotic reconstruction, a simple filter – nonetheless effective. For classical physics systems, it’s the Kalman filter. For quantum systems, it’s the quantum filter that integrates the theory of quantum measurement and also the imperfections of measurement and delay.

But to adapt these tools, it isn’t enough to be a good specialist in systems theory, a good automation engineer. You also need to understand quantum equations, the rules of the game, so to speak. Because in the quantum world, rules are different. And this is where chance served us well: if we hadn’t explored the subject by our own over the last decade, without Serge Haroche’s lectures, we would have been much less useful to his team.

It must also be said that on their side, they were committed to fully understanding what we were doing, including intuitively. This is crucial from a scientific standpoint. Besides, they already had ideas about solutions they could bring, even if they weren’t familiar with our tools at the start.

It was a genuine meeting: a meeting between two disciplines that allows one of them to achieve a major breakthrough, but also a meeting between two teams.

There were certainly many scientific contact points: for example, the Schrödinger equations – that describe the interaction between the atoms that pass through the cavity and the photons trapped inside – are deterministic differential equations, easily understandable by any good mathematician and not very different from those used by our automation colleagues. What really changes is the measure.

And also the fact that, in composite quantum systems i.e. formed by several subsystems, the dimensions will not add, but multiply. Take a situation with ten degrees of freedom on one side, two degrees on the other: in the classical world, they add up and make twelve; in the quantum world, they make twenty. The quantum world is much larger than the classical world. Everything can be formalized. But to fully understand its implications, one needs to know these rules.

**Is this why everything went to fast?**

Yes, indeed. We started in 2008 and in mid-2009, we published our first theoretical paper with realistic simulations in *Physical Review A*. Then, despite some problems, the experiment went pretty quickly: our conclusions were published in 2011 by *Nature*.

**What about afterwards? The conquest of the additional photon, for example?**

It largely depends on the imperfections of the experiment: the defects of the cavity mirrors, but also the detection efficiency – for example, in 2011, we only detected one of three atoms, not to mention 10% of false detections.

But the progress realized since 2011, those to come, don’t depend on the addition of an extra photon. Superconducting circuits offer a very interesting lead that is currently being explored by several laboratories such as Quantronics in the CEA and the Pierre Aigrain laboratory of the École Normale Supérieure. These are electronic circuits that operate at low temperature. A lot of progress has been made since the 2000s, with a dramatic increase in coherence time. Despite the differences in the physical supports, the underlying dynamics and models are very similar to the photons and Rydberg atoms studied by Serge Haroche. These so-called spin/spring models have, to some extent, a universal value.

Experiments on these superconducting circuits are much simpler to implement. They rely on microwave technologies from the telecom sector. This explains the growing number of laboratories but also manufacturers who invest in this field. These include the Canadian D-Wave company that builds and sells circuits with hundreds of degrees of freedom i.e. several hundred qubits (quantum bits). Do these circuits really work in the quantum regime? Do they solve algorithmic problems, such as combinatorial optimization, much faster than our conventional computers? For now, these questions have not been clearly resolved. But they raise interesting issues concerning scalability*,* that is to say, the leap from a logic circuit of several qubits, with an experimentally well-controlled quantum behavior, to a logic circuit of several hundred of qubits, with the same degree of control. The future will tell us whether automatic control and systems theory, partly adapted and redesigned for the quantum world, will be effective in addressing these fascinating questions.