Scheme for direct observation of the Wigner characteristic function in cavity QED

We suggest a quantum state reconstruction scheme. Our proposal is applicable to the field inside a microwave cavity as well as to the harmonic motion of a trapped atom. It will be shown that the inversion of two-level atoms after a resonant interaction with a coherently displaced quantum state is directly related to the Wigner characteristic function of the initial state. This method neither requires the preparation of atoms in a quantum superposition of the upper and lower energy levels nor heavy numerical processing of the measured data. We demonstrate the reconstruction scheme for the example of a Schr\"odinger-cat superposition state.