  Nanoparticles levitated in an optical trap provide a versatile platform to
study mechanical oscillators in a controlled environment with tuneable
parameters. Recently, it has become possible to couple two of these
optomechanical oscillators. Here, we demonstrate the collective non-Hermitian
dynamics of such a pair of non-conservatively coupled oscillators. We take
advantage of the tunability of the optical interactions between the particles
in our system and set the optical interaction between the particles to be
purely non-reciprocal. By continuously varying the relative power of the
trapping beams, we take the system through a transition, similar to a
parity-time phase transition. A Hopf bifurcation at a critical point results in
the formation of collective limit cycle oscillations, resembling those observed
in phonon lasers. These coupled levitated oscillators provide a platform for
exceptional point optomechanical sensing and can be extended to multi-particle
systems, paving the way for the development of topological optomechanical
media.
