A microchip optomechanical accelerometer – Supplementary Information

The oscillator susceptibility χ(ω) given in the main text follows from the differential equation of the harmonic oscillator: m ¨ x + mγ ˙ x + mω 2 m x = F appl. (S1) Transforming to Fourier space, this reads −ω 2 x + iωγx + ω 2 m x = F appl (ω) m. (S2) With F appl (ω)/m = a appl , this yields the accelerometer response x(ω) = χ(ω)a appl (ω) = 1 ω 2 m − ω 2 + i ωω m Q m a appl (ω). (S3) This function has the following properties: χ(0) = 1 ω 2 m = m k , (S4) χ(ω m) = −i Q m ω 2 m = −iQ m χ(0), (S5) χ(ω ω m) ∝ 1 ω 2. (S6) For the device studied here with ω m = 2π × 27.5 kHz, this gives an acceleration sensitivity of χ(0) = 329 pm/g with g = 9.81 m/s 2. In order to calculate the intensity transmission profile T (ω) of a photonic-crystal resonator side-coupled by a fiber-taper waveguide, we start from the equation of motion ofâ, the annihilation operator of the cavity field: d ˆ a dt = − i∆ + κ 2 ˆ a + κ e 2 ˆ a in + √ κ i ˆ a i + κ e 2 ˆ a −. (S7) Here, ∆ = ω l − ω c is the laser-cavity detuning, κ e is the total taper-cavity coupling rate, κ = κ i + κ e is the total cavity decay rate, with κ i the intrinsic cavity damping rate, andâ in is the taper input field, which together with the output fieldâ out obeys the boundary condition ˆ a in + ˆ a out = κ e 2 ˆ a. (S8) The last two terms on the right-hand-side of eq. (S7) represent the vacuum inputs due to coupling with the intrinsic (loss) bath of the cavity and the backward fiber taper waveguide mode, respectively (these input terms are ignored going forward as they are in the vacuum state and do not modify the classical field equations). In steady state, where d ˆ a dt ≡ 0, the intracavity field operator is ˆ a 0 = κ e 2 ˆ a in i∆ + κ 2 .