# 4. Optimal Control

## 4.1 Optimal control of time-averaged quantities in quantum systems

Questions addressed: Development of a theory for optimal control over finite time-intervals?

We developed a variational theory for the optimal control of quantum systems with relaxation over a finite time interval. In our approach, which is a nontrivial generalization of previous formulations and which contains them as limiting cases, the optimal control field fulfills a high order Euler-Lagrange differential equation, which guarantees the uniqueness of the solution. We solve this equation numerically and also analytically for some limiting cases. The theory is applied to two-level quantum systems with relaxation, for which we determine quantitatively how relaxation effects limit the control of the system.

In a further work we have extended the method to treat multiphoton excitations. For that we have used the Floquet formalism and the adiabatic approximation and developed a theory for optimal control of a multilevel quantum system interacting with an external electric field under multiphoton resonance conditions. We derived a differential equation to determine the shape of the optimal pulse, and showed how the order of the multiphoton resonance affects the solution. In particular, we demonstrate that optimal control in the case of two-photon resonance and using only adiabatically shaped control pulses represents a special degenerate case, for which the shape of the pulse cannot be optimized.

Methods: Analytical theory. Variational principle.

Publications: [49], [62] (see list of publications).