  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.
