Fig. 2: Electromechanical detection of parametric, phase-sensitive mechanical amplitude amplification.

a Experimental scheme. The mechanical oscillator is coherently driven by a combination of DC and alternating voltage with frequency Ω ~ Ω_m, while the electrostatic spring constant is modulated with twice this frequency 2Ω ~ 2Ω_m. Via the optomechanical coupling, the mechanical oscillations generate sidebands to a microwave pump tone sent to the cavity with frequency ω = ω_c, which are used for homodyne detection of the mechanical amplitude. b Mechanical amplitude gain \(20{\mathrm{log}\,}_{10}{G}_{{\rm{p}}}\) vs. offset phase ϕ_p between resonant drive and parametric modulation. When the phase is swept, the amplitude is oscillating between amplification or de-amplification with a periodicity of π. Circles show data and the line shows a fit with the theoretical expression Eq. (2). c Maximum and minimum gain on resonance vs. parametric modulation strength. The maximum (ϕ_p = π∕2) and minimum (ϕ_p = 0) gain values on resonance follow the theoretical curves (lines) up to a maximum gain of ~22 dB. For stronger parametric modulation amplitudes close to the instability threshold (indicated as vertical line), the gain in our experiments is limited by resonance frequency fluctuations of the mechanical resonator. d Maximum and minimum gain vs. detuning from resonance. For a driving frequency slightly detuned from resonance, the maximum gain gets reduced compared to the resonant case. Points are extracted from phase-sweep curve fits. Lines show the corresponding theoretical curves and the shaded area contains all gain values achievable by changing ϕ_p.