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In this report we present preliminary results about the tunneling problem fora magnetic Schr\"odinger operator. As a motivation we consider the 3-Dtime-dependent Schr\"odinger operator $H(t)=-h^2\Delta+V+E(t)\cdot x$ where $V$is a radial potential and $E(t)$ a circularly polarized field with uniformfrequency $\omega$. The quantum monodromy operator (QMO) that takes the systemthrough a complete period $T=2\pi/\omega$, turns out to be unitarily equivalentto $e^{iTP_A(x,hD_x)/h}$, where $P_A(x,hD_x))$ identifies with a magneticSchr\"odinger operator. When $V$ is sufficiently confining, $P_A(x,hD_x))$presents a double magnetic well. Then we construct its semi-classical groundstate and examine the splitting between its two first eigenvalues.

On semi-classical spectral series for an atom in a periodic polarized electric field