Coupled oscillators driven with difference-frequency parametric position feedback

We examine theoretically and measure experimentally the instability regions for a pair of parametrically driven mechanical oscillators on an air track. We take the position of the second mass, multiply it by cos (ω_Dt), and apply it to the first mass. We show that this leads to an instability for both normal modes of the coupled oscillator system when ω_D is close to the difference of the normal mode frequencies. Such an instability mechanism is of special interest because the input frequency, ω_D, can be much lower than the output frequencies of the system. In addition, we consider the case where cos (ω_Dt) is replaced by a negative constant, which also leads to instability, but at a single frequency. This study of the difference frequency oscillator system was motivated by the QASER system described in Svidzinsky, Yuan and Scully [Phys. Rev. X 3, 041001 (2013)].