## Section: New Results

### Laser control

Participants : Andreea Grigoriu, Claude Le Bris, Gabriel Turinici.

The project-team's interest closely follows the recent prospects opened by the laboratory implementations of closed loop optimal control. This is done in collaboration with the group of H. Rabitz (Princeton University).

#### Control filtering by monotonic algorithms

The development of pulse-shaping techniques opens new ways to control atomic or molecular processes by laser fields. Many promising results have been obtained with a setup made of a pulse shaper controlled by the algorithm which, from the results of the preceding experiments, builds an improved new control field. Such algorithms lead to very efficient solutions but have some negative points. In particular, no insight into the control mechanism is gained from this approach since little to no knowledge about the system is needed and the control field is not optimal by construction. On the theoretical side, optimal control theory, OCT, is a powerful tool to design electric fields to control quantum dynamics. Monotonically convergent algorithms are other efficient approaches to solve the optimality equations. They have been applied with success to a large number of controlled quantum systems in atomic or molecular physics and in quantum computing. These methods are flexible and can be adapted to different nonstandard situations encountered in the control of molecular processes. Among recent developments, we can cite the question of nonlinear interaction with the control field and the question of spectral constraints on the field. The latter problem is particularly important in view of experimental applications since not every control field can be produced by pulse-shaping techniques. For instance, liquid crystal pulse shapers are able to tailor only a piecewise constant Fourier transform of the control field in phase and in amplitude. Experimentally, the spectral amplitude and phase are discretized, e.g., into 640 points which is the number of pixels in a currently used standard mask.

To address this issue, a monotonically convergent algorithm is proposed in [33] which can enforce spectral constraints on the control field and extends to arbitrary filters. The procedure differs from standard algorithms in that at each iteration, the control field is taken as a linear combination of the control field computed by the standard algorithm and the filtered field. The parameter of the linear combination is chosen to respect the monotonic behavior of the algorithm and to be as close to the filtered field as possible. We test the efficiency of this method on molecular alignment. Using bandpass filters, we show how to select particular rotational transitions to reach high alignment efficiency. We also consider spectral constraints corresponding to experimental conditions using pulseshaping techniques. We determine an optimal solution that could be implemented experimentally with this technique.

#### Beyond bi-linear control: feedback approaches

The control of quantum dynamics induced by an intense laser field continues to be a challenge to both experiment and theory. In this context, optimal control theory is an efficient tool for designing laser pulses able to control quantum processes. Different methods have been developed to solve the optimal equations such as the Lyapounov-like approaches.

A vast majority of works have considered a linear interaction between the quantum system and the electromagnetic field. This linear interaction corresponds, for molecular systems, to the first-order dipolar approximation of the permanent dipole moment. Due to the intensity of the field or to the particular structure of the problem, some systems need to go beyond this approximation e.g., for the control of molecular orientation and alignment of a linear molecule by non-resonant laser pulses. The natural question arises of whether one can apply these approaches to such systems interacting nonlinearly with the field. This question has been answered in a situation where a quadratic term in the control is present. The theory works as expected for systems with controllable linearization; however for systems globally but not locally controllable it is proven that no continuous feedback exists; for these situations we proposed two solutions: either a discontinuous feedback or an averaging procedure that weakens the monotonic property of Lyapounov approaches [44] , [24] .

#### Propagator space methods

Traditionally, the numerical simulations consider
some description of the interaction of the laser and the system, of which the most
used is the *dipole approximation* , and perform optimizations
considering the laser intensity as the main variable.
A different view of the problem has been taken: the evolution semigroup of unitary propagators is
considered and it is asked that the resulting Hamiltonian be consistent with the chosen approximation type.
The specific optimization algorithm is now implemented and the first encouraging numerical results
are presented in a submitted work [30] .