  The dynamical backaction from a periodically driven optical or microwave
cavity can reduce the damping of a mechanical resonator, leading to parametric
instability accompanied by self-sustained oscillations. Fundamentally, the
driving breaks the continuous time-translation symmetry and replaces it with
the symmetry with respect to time translation by the driving period. This
discrete symmetry should be reflected in the character of the oscillations.
Here, we perform experimental and theoretical study of new aspects of the
backaction and the discrete time-translation symmetry using a micromechanical
resonator designed to have two nonlinearly coupled vibrational modes with
strongly differing frequencies and decay rates. We find that self-sustained
oscillations are induced not only in the low frequency mode as measured in
previous experiments, but also in the high frequency mode. The vibration
frequencies and amplitudes are determined by the system nonlinearity, which
also leads to bistability and hysteresis. A remarkable consequence of the
discrete time-translation symmetry is revealed by studying the vibration
phases. We find that the phase fluctuations of the two modes are nearly
perfectly anti-correlated. At the same time, in each mode the phase undergoes
anomalous diffusion, where the time dependence of the phase variance follows a
superlinear rather than the standard linear power law. Our analysis shows that
the exponent of the power law is determined by the exponent of the 1/f-type
intrinsic frequency noise of the resonator. We demonstrate the possibility of
compensating for these fluctuations using a feedback scheme to generate stable
oscillations that could prove useful in frequency standards and resonant
detection.
