The Whittaker-Shannon sampling theorem for experimental reconstruction of free-space wave packets

A data analysis scheme is proposed for determining the quantum state of a one-dimensional wave packet describing atomic translational motion in free space. The Whittaker-Shannon sampling theorem is applied to reconstruct exactly a certain portion of the momentum-basis density matrix from Fourier transforms of measured position-probability densities. The sampling theorem is valid if the spatial wave correlation function equals zero ouside some finite range. It is most useful in cases where the range of available data is insufficient to allow a complete reconstruction of the state by phase-space tomography, as in recent experimental studies of atomic-beam state reconstruction by Pfau and Kurtsiefer.