Fig. 3.

Experimental demonstration of nonreciprocity. a–c Power transmission between modes 1 and 2 as a function of probe detuning, shown in both directions for pump phases ϕ _p = −0.8π, 0, 0.8π radians (respectively a–c). Isolation of more than 20 dB in the forward c and backward a directions is demonstrated, as well as reciprocal behaviour b. d The ratio of transmission |S ₂₁/S ₁₂|², representing a measure of nonreciprocity, is shown as a function of pump phase ϕ _p and probe detuning. Two regions of nonreciprocity develop, with isolation in each direction. The system is reconfigurable as the direction of isolation can be swapped by taking ϕ _p → −ϕ _p. e Theoretical ratio of transmission from Eq. (3), calculated with independently estimated experimental parameters. The theoretical model includes effectively lowered cooperativities for the mechanical mode ${\widehat{b}}_1$ due to cross-damping (optomechanical damping of the lower frequency mechanical mode by the pump on the sideband of the higher frequency mechanical mode) acting as an extra loss channel