  We report an observation of phononic Schwinger angular momenta, which fully
represent twomode states in a micromechanical resonator. This observation is
based on simultaneous optical detection of the mechanical response at the sum
and difference frequency of the two mechanical modes. A post-selection process
for the measured signals allows us to extract a component of phononic Schwinger
angular momenta. It also enables us to conditionally prepare two-mode squeezed
(correlated) states from a randomly excited (uncorrelated) state. The phononic
Schwinger angular momenta could be extended to high-dimensional symmetry (e.g.
SU(N) group) for studying multipartite correlations in non-equilibrium dynamics
with macroscopic objects.
