Optimal control problems
Given a molecule in its ground state, interaction with a laser pulse can result in an excitation to a state of interest. This may, for instance, trigger a chemical reaction. The probability of reaching the desired state depends on the shape of the laser pulse. The problem of designing an optimal laser pulse, which maximises the probability of reaching the desired state, can be posed as an optimal control problem. Our ansatz for solving this problem is to reformulate the optimization problem by Fourier-transforming the electric field of the laser and narrow the frequency band. In this way, we reduce the dimensionality of the control variable. This allows for storing an approximate Hessian, which in a time-domain representation would have been prohibitively big. Thereby, we can solve the optimization problem with a quasi-Newton method. Such an implementation provides superlinear convergence. Moreover, such a reduction of the frequency band can make sure that the laser pulse can be realized in the laboratory.
Our frequency-domain optimisation algorithm is described in the following publication.
- A Fourier-coefficient based solution of an optimal control problem in quantum chemistry. In Journal of Optimization Theory and Applications, volume 147, pp 491-506, 2010. (DOI).
Currently, we are applying the optimisation algorithm to the higher harmonic generation of noble gases. The noble gas is exposed to a laser pulse, which is strong enough to ionise the atoms. The free electron is then accelerated by the electric field of the laser pulse. Since the electric field oscillates, the electron may return to a bound orbit around the nucleus. It will then have taken excess energy from the laser, which will be released in the form of a photon. This process is a standard tool for the experimental realisation of short, coherent pulses of light in the extreme ultra violet part of the spectrum.
The search for a laser pulse which maximises the yield in a certain part of the spectrum can be posed as an optimal control problem. As a first step to solving this, we have developed a new, efficient method for the solution of the Schrödinger equation with non-smooth potential.
- Accelerated convergence for Schrödinger equations with non-smooth potentials. In BIT Numerical Mathematics, volume 54, pp 729-748, 2014. (DOI, fulltext:postprint).