  Trapped atomic ions are a versatile platform for studying interactions
between spins and bosons by coupling the internal states of the ions to their
motion. Measurement of complex motional states with multiple modes is
challenging, because all motional state populations can only be measured
indirectly through the spin state of ions. Here we present a general method to
determine the Fock state distributions and to reconstruct the density matrix of
an arbitrary multi-mode motional state. We experimentally verify the method
using different entangled states of multiple radial modes in a 5-ion chain.
This method can be extended to any system with Jaynes-Cummings type
interactions.
