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

Quantum state reconstruction is an important element enabling diagnosis and improvement of quantum control. As larger states come under experimental control the number of measurements required to perform state reconstruction becomes crucial. Here, significant gains can be found by choosing the appropriate basis in which to make measurements. In this talk I will present recent experiments analyzing direct phase-space tomography of the motion of a single trapped Ca⁺ ion. The method, which is based on ion internal state rotations combined with state-dependent shifts of the oscillator, is used to reconstruct displaced squeezed oscillator states as well as a three-component superposition thereof. Such states have applications in quantum information, quantum sensing and fundamental studies. We find reductions in measurement times of a factor 20 compared to methods we used previously, which we anticipate to improve future ion motion state preparation and control.