Mathematics

In this talk we present how techniques of "subriemannian geometry" and "singular Riemannian geometry" can be applied to some classical problems of quantum mechanics and permit to improve some results. We study a 3--levels quantum system controlled by two laser pulses in resonance i.e. with frequencies $\omega_1=E_2-E_1,~\omega_2=E_3-E_2$, being $E_1,E_2,E_3$ the three energy levels ($\hbar=1$). Our aim is to transfer all the population from the state with energy $E_1$ to the state with energy $E_3$ minimizing the amplitude of the lasers pulses. Notice that we do not dispose of a laser with frequency $E_3-E_1$. This problem is described by a Schrodinger equation in $C^3$. We show that making a suitable unitary transformation depending on the time, this problem can be stated as a subriemannian problem on $S^5$. Then we prove that the orbit trough our initial point is a two dimensional sphere. Our problem is then reduced to the problem of finding the geodesics on the sphere, for a singular Riemannian metric. We compute the optimal synthesis and in particular the trajectory that reaches our final state. We would like to stress the fact that, since the optimal solution is invariant for time reparameterization, one can use very smooth optimal controls to go from the initial to the final state. This fact is important because smooth controls are the ones that can be realized in practice. A possible choice for the controls is presented. Finally we discuss the geometry of the manifold that one obtain by gluing all the spheres corresponding to all possible choices of (equivalent) initial conditions. Results for two level systems are also presented: this problem is simpler but, contrarily, to the 3--level system, leads to a problem in contact isoperimetric subriemannian geometry (and not singular Riemannian geometry). Moreover the "geometric situation" in this 2--level system is extremely nice.