Reconstructing the phonon distribution of trapped ions out of the Lamb-Dicke regime

Characterizing the phonon distribution of trapped-ion motional states is essential for many applications in quantum information processing, but existing methods become ill-conditioned beyond the Lamb-Dicke regime. We overcome this limitation by recasting phonon-state reconstruction as a filter-function inversion problem. Using composite pulses on the carrier and multiple sidebands, we engineer well-conditioned filters in phonon space to characterize pure and mixed states, study heating dynamics, and reconstruct Fock states up to $n=250$.