  Phonon lasers, exploiting coherent amplifications of phonons, have been a
cornerstone for exploring nonlinear phononics, imaging nanomaterial structures,
and operating phononic devices. Very recently, by levitating a nanosphere in an
optical tweezer, a single-mode phonon laser governed by dispersive
optomechanical coupling has been demonstrated, assisted by alternating
mechanical nonlinear cooling and linear heating. Such levitated optomechanical
(LOM) devices, with minimal noises in high vacuum, can allow flexible control
of large-mass objects without any internal discrete energy levels. However,
untill now, it is still elusive to realize phonon lasing with levitated
microscale objects, due to much stronger optical scattering losses. Here, by
employing a Yb3+-doped active system, we report the first experiment on
nonlinear multi-frequency phonon lasers with a micro-size sphere governed
instead by dissipative LOM coupling. In this work, active gain plays a key role
since not only 3-order enhancement can be achieved for the amplitude of the
fundamental-mode phonon lasing, compared with the passive device, but also
nonlinear mechanical harmonics can emerge spontaneously above the lasing
threshold. Furthermore, for the first time, coherent correlations of phonons
are observed for both the fundamental mode and its harmonics. Our work drives
the field of LOM technology into a new regime where it becomes promising to
engineer collective motional properties of typical micro-size objects, such as
atmospheric particulates and living cells, for a wide range of applications in
e.g., acoustic sensing, gravimetry, and inertial navigation.
