Reconstruction of the two-mode vibronic quantum state of a trapped atom

We present a method for the direct measurement of the Wigner-function matri x for complex vibronic states of a trapped atom, that is suited to analyse the entanglement between two motional degrees of freedom and the internal e lectronic dynamics. It is a generalisation of the method for the determinat ion of vibronic quantum states [S. Wallentowitz, R.L. de Matos Filho, W. Vo gel, Phys. Rev. A 56, 1205 (1997)] in conjunction with the scheme for the d irect observation of the Wigner function of a single motional degree of fre edom [L.G. Lutterbach, L. Davidovich, Phys. Rev. Lett, 78, 2547 (1997)]. Th e major advantage of the present method is that it reduces the experimental efforts substantially. On the other hand, it is demonstrated that the nonl inear vibronic coupling necessary for this method turns out to be its main limitation.