Motional quantum states of a trapped ion: Measurement and its back action

We develop a method for reconstructing the quantum mechanical state of a trapped ion by bichromatically irradiating it on a weak electronic transition and subsequently probing a strong electronic transition for resonance fluorescence. Based on this recently proposed scheme [S. Wallentowitz and W. Vogel, Phys. Rev. Lett. 75, 2932 (1995)] the density matrix of the vibrational motion can be readily obtained either in a generalized position representation or in the number-state representation. The method allows to uniquely define an ultimate classical noise level by a reference measurement with an ion cooled to its vibrational ground-state. Disturbations of the measurement and their suppression are considered. Moreover, we study the effect of the back action of the measurement principle on the motional quantum state. It consists in the splitting of the state to be measured into two substates, giving rise to quantum interference effects. © 1996 The American Physical Society.