Measuring entanglement-induced quantum correlations in the oscillatory motion of a trapped ion

The preparation of a wide class of entangled vibrational states involving two or three phononic modes of a three-dimensional trapped ion has been reported in the literature from both theoretical and experimental points of view. Here, the time evolution of such a system from an initial condition wherein two oscillatory modes (M _x, M _y) are reciprocally entangled and both are disentangled to the third mode (M _z) is studied. By coupling one of the entangled oscillators (M _x) with the third oscillator (M _z), a correlation between the two uncoupled modes (M _y, M _z) is induced, well visible when the mean value of a suitable operator is considered. A method of measuring the expectation value of a vibrational observable is briefly sketched and then exploited in order to reveal such nonclassical behavior.