Section: Application Domains

Laser control

The laser control of chemical reactions is today an experimental reality. Experiments, carried out by many groups of researchers and in many different contexts and settings, have demonstrated the feasibility of controlling the evolution of a quantum system using a laser field. All these experiments exploit the remarkable properties of quantum interactions (interferences) between one, or more, external interactions (e.g. lasers) and the sample of matter under study. In order to create the ad hoc interferences that will drive the system to the desired goal, one can play with the dephasing between two beams, conveniently choose the frequencies of the beams, or make use of the two aspects mixed together, which amounts to allowing for “all” possible laser fields as in optimal control schemes.

Whatever the strategy, the success of these numerous experiments not only validates the idea of manipulating and controlling quantum systems with lasers, but also motivates the need for further theoretical studies in this direction, in order to further improve the results and the range of their applicability. Interest in this research area has also been increasing in more applied communities. The standard modeling for the problem of the laser control of a molecular system involves the time-dependent Schrödinger equation which rules the evolution of the wavefunction describing the state of the system. On the basis of the Schrödinger equation, one then states a control problem, either in the framework of exact control or in the framework of optimal control.

The first fact to underline as a crucial feature of the problem of laser control is the orders of magnitude in time and space that are typically encountered here. The space scale is indeed that of an atom, say 10^-10  m, but more important than that, the time scale is of the order of the femtosecond (10^-15  s) and can even go down to the attosecond (10^-18  s). As surprising as it may seem, the laser fields can literally be “tailored” on these tiny timescales. They can involve huge intensities (10¹²  W/cm² and above), and their shots can be cycled at 1 KHz. One can do several thousands of experiments in a minute. This ability changes the whole landscape of the control problem, for making an experiment is here far cheaper than running a numerical simulation. This has motivated the paradigm of closed-loop optimization where the criterion to be optimized is evaluated on-the-fly on an experimental device. One of the current challenging issue for the mathematicians taking part into the field is to understand how to take advantage of a combined experimental/numerical strategy. In this respect, it is to be noted that the experimental side can come from on-the-fly experiments (how to decide what to do?), but may also come from the tremendous amount of data that can be (and actually is) stored from the billions of experiments done to this day (how to dig into this database?).

A second point is to remark on the way in which the control enters the problem: the control multiplies the state. Theoretically and numerically, this bilinear nature causes difficulties. Finally, open-loop control is dealt with, at least for two reasons: first, the timescale on which the phenomenon goes is too short for the current capabilities of electronic devices, which prevents closing the loop within one experiment; but secondly, feedback control means measuring something, which in a quantum framework means interacting with and thus perturbing the system itself. These two bottlenecks might be overcome in the future, but this will undoubtedly require a lot of theoretical and technical work.

A third peculiarity regards the choice of admissible laser fields as control : what types of E(t) should be allowed when setting up the control problem? This question leads to a dichotomy : one can choose either to restrict oneself to the experimentally feasible fields, or to basically let the field be free, therefore allowing for very general laser fields, even those out of reach for the contemporary technology. The two tracks may be followed. In particular, the second track, the most “imaginative” one (rather unusual in comparison to other contexts), can serve as a useful technical guide for building the lasers of tomorrow.

A final key issue is robustness. It is of course a standard fact in every control problem that the control obtained needs to be robust, for obvious practical reasons. The somewhat unusual feature in the present setting is that the experiments show that they are surprisingly robust with respect to all kinds of perturbations (noise, uncertainties in the measures, ...). Clearly, there is here something to be understood at the theoretical level, e.g. by envisioning new modeling strategies that incorporate undesirable perturbations.