  We demonstrate that optomechanical non-reciprocity is equivalent to the
anti-resonance, often discussed in the context of coupled driven harmonic
oscillators. We show that that suitable phase-difference between the cavity
driving fields make the relevant optomechanical couplings complex, which leads
to non-reciprocity in the field fluctuations and anti-resonance in average
field amplitudes. This analogy with anti-resonance demonstrates that only for a
particular frequency (the so-called isolation frequency) of input signal,
maximum non-reciprocity can be achieved. In contrast to the previous studies,
we here show that one can dynamically control this isolation frequency by
applying a mechanical drive of suitable frequency to the membrane in the
optomechanical setup.
