a The reflection ∣Sjj∣2 (j = e, o) and bidirectional transduction ζ: = ∣Seo∣∣Soe∣ parameters as a function of signal tone detuning δj = ω − (ωd,j + ωm) for fixed pump powers Pe = 601 pW and Po = 625 pW. The dots represent the experimental data while the solid lines show the theoretical prediction with γm as the only fit parameter. b Measured photon number transduction efficiency as a function of microwave and optical pump powers. c Measured transduction efficiency with respect to optical pump power for fixed Pe = 601 pW. d Measured transduction efficiency with respect to microwave pump power for fixed Po = 625 pW. In (c) and (d), the error bars are the standard deviation of three independent measurement runs and solid lines are theory with interpolated γm (from (f)) and no other free parameters. e The coupling efficiency of the microwave resonator to the waveguide ηe = κex,e/κe, extracted from broadband reflection measurements, as a function of optical (blue, Pe = 601 pW) and microwave (red, Po = 625 pW) pump tones. f Intrinsic mechanical decoherence rate γm versus optical (blue, Pe = 601 pW) and microwave (red, Po = 625 pW) pump powers extracted from (c) and (d) with Eq. (3) and γm as only fit parameter. Solid lines in (f) are linear fits to the data.