Fig. 1.

Optomechanical nonreciprocal transmission via interference of two asymmetric dissipative coupling pathways. a Two microwave modes ${â}_1$ and ${â}_2$ are coupled via two mechanical modes ${\widehat{b}}_1$ and ${\widehat{b}}_2$ through optomechanical frequency conversion (as given by the coupling constants g ₁₁, g ₂₁, g ₁₂, g ₂₂). Nonreciprocity is based on the interference between the two optomechanical (conversion) pathways g ₁₁, g ₂₁ and g ₁₂, g ₂₂, in the presence of a suitably chosen phase difference ϕ between the coupling constants as well as the deliberate introduction of an asymmetry in the pathways. b, c The symmetry between the pathways can be broken by off-setting the optomechanical transmission windows through each mechanical mode (dashed lines in dark and light green) by a frequency difference 2δ, resulting in different |S ₂₁| and |S ₁₂| (solid lines). Each single pathway, in the absence of the other mode, is described by Eq. (2). In the forward direction b, the two paths interfere constructively, allowing transmission and a finite scattering matrix element S ₂₁ on resonance with the first microwave cavity. In contrast, in the backward direction c, the paths interfere destructively, such that S ₁₂ ≈ 0, thereby isolating port 1 from port 2 on resonance with the second microwave cavity. The isolation bandwidth is determined by the intrinsic dissipation rate of the mechanical modes