  The radiation-pressure interaction between one or more laser fields and a
mechanical oscillator gives rise to a wide range of phenomena: from sideband
cooling and backaction-evading measurements to pondermotive and mechanical
squeezing to entanglement and motional sideband asymmetry. In many protocols,
such as dissipative mechanical squeezing, multiple lasers are utilized, giving
rise to periodically driven optomechanical systems. Here we show that in this
case, Floquet dynamics can arise due to presence of Kerr-type nonlinearities,
which are ubiqitious in optomechanical systems. Specifically, employing
multiple probe tones, we perform sideband asymmetry measurements, a macroscopic
quantum effect, on a silicon optomechanical crystal sideband-cooled to 40%
ground-state occupation. We show that the Floquet dynamics, resulting from the
presence of multiple pump tones, gives rise to an artificially modified
motional sideband asymmetry by redistributing thermal and quantum fluctuations
among the initially independently scattered thermomechanical sidebands. For
pump tones exhibiting large frequency separation, the dynamics is suppressed
and accurate quantum noise thermometry demonstrated. We develop a theoretical
model based on Floquet theory that accurately describes our observations. The
resulting dynamics can be understood as resulting from a synthetic gauge field
among the Fourier modes, which is created by the phase lag of the Kerr-type
response. This novel phenomenon has wide-ranging implications for schemes
utilizing several pumping tones, as commonly employed in backaction-evading
measurements, dissipative optical squeezing, dissipative mechanical squeezing
and quantum noise thermometry. Our observation may equally well be used for
optomechanical Floquet engineering, e.g. generation of topological phases of
sound by periodic time-modulation.
