  We propose and demonstrate phonon-number-resolving detection of the multiple
local phonon modes in a trapped-ion chain. To mitigate the effect of phonon
hopping during the detection process, the probability amplitude of each local
phonon mode is mapped to the auxiliary long-lived motional ground states.
Sequential state-dependent fluorescence detection is then performed. In the
experiment, we have successfully observed the time evolution of two local
phonon modes in two ions, including the phonon-number correlation between the
two modes.
