Direct characteristic-function tomography of quantum states of the trapped-ion motional oscillator

We implement direct readout of the symmetric characteristic function of quantum states of the motional oscillation of a trapped calcium ion. Using suitably chosen internal electronic state-dependent displacements based on bi-chromatic laser fields we map the expectation value of the real or imaginary part of the displacement operator to the internal states, which are subsequently read out using fluorescence detection. Combining the two readout results provides full information about the symmetric characteristic function. We characterize the performance of the technique by applying it to a range of archetypal quantum oscillator states, including displaced and squeezed Gaussian states as well as two and three component superpositions of displaced squeezed states which have applications in continuous variable quantum computing. For each, we discuss relevant features of the characteristic function and Wigner phase-space quasi-probability distribution. The direct reconstruction of these highly non-classical oscillator states using a reduced number of measurements is an essential tool for understanding and optimizing the control of these systems for quantum sensing and quantum information applications.