Phonon Tomography with Ancillary Modes in Trapped Ion System*

Boson tomography is a fundamental physical problem that has the capacity to show the power of quantum computation. While it has been considered mostly for linear optical systems with photons, phonons for trapped ions are also a good candidate to realize the full tomography of their quantum states. A recent proposal by L.Banchi et al. (PRL 121, 250402 (2018)) shows that with ancillary vacuum modes, the number of measurement settings required for the experiment can be reduced. Here, we demonstrate, for the first time, a full phonon tomography of a two-mode number-restricted input phonon state with a beam splitter operation and projective measurements. Then we use up to two ancillary vacuum modes to realize the reduction of settings for the tomography of this two-mode input state. Our experiment demonstrates a new way to realize boson tomography with minimal resources for a trapped ion system which can be generalized to other physical systems.