Endoscopy in the Paul Trap: Measurement of the Vibratory Quantum State of a Single Ion

We reconstruct the density operator of the center-of-mass motion of an ion stored in a Paul trap by mapping the dynamics of the motion onto the internal dynamics of the ion. Our technique takes into account the explicit time dependence of the trap potential, operates outside the Lamb-Dicke limit, and is not restricted to pure states. We demonstrate the feasibility of this method using the example of a damped Schr\"odinger cat state.