Experimental Determination of the Motional Quantum State of a Trapped Atom

We reconstruct the density matrices and Wigner functions for various quantum states of motion of a harmonically bound ${}^9{\mathrm{Be}}^{+}$ ion. We apply coherent displacements of different amplitudes and phases to the input state and measure the number state populations. Using novel reconstruction schemes we independently determine both the density matrix in the number state basis and the Wigner function. These reconstructions are sensitive indicators of decoherence in the system.