Reconstructing the time evolution of a quantized oscillator

A method is proposed that allows one to reconstruct the unitary time-evolution matrix of a quantized oscillator. It combines coherent splitting and displacement of the quantum state after the interaction of interest with a subsequent measurement of the occupation of the oscillator ground state. The time-evolution matrix in number basis is then obtained by a twofold Fourier transform of the measured data. It is shown how to realize such a method for the case of the quantized center-of-mass motion of a trapped atom. Simulations for particular examples of interactions demonstrate the feasibility of the reconstruction method even in the presence of noisy data.