Radio-frequency optomechanical characterization of a silicon nitride drum

On-chip actuation and readout of mechanical motion is key to characterize mechanical resonators and exploit them for new applications. We capacitively couple a silicon nitride membrane to an off resonant radio-frequency cavity formed by a lumped element circuit. Despite a low cavity quality factor (QE ≈ 7.4) and off resonant, room temperature operation, we are able to parametrize several mechanical modes and estimate their optomechanical coupling strengths. This enables real-time measurements of the membrane’s driven motion and fast characterization without requiring a superconducting cavity, thereby eliminating the need for cryogenic cooling. Finally, we observe optomechanically induced transparency and absorption, crucial for a number of applications including sensitive metrology, ground state cooling of mechanical motion and slowing of light.


.1 Fit of the cavity transmission
The response of the cavity, Fig. 1 (b) of the main text, was characterized by fitting the measured transmission to the following expression: where α is the isolation of the directional coupler and φ a phase 1 . The cavity resonance frequency is f C , κ e is the external dissipation rate and κ the total dissipation rate of the cavity. From a fit with a fixed value of α = 0.001 from the directional coupler data sheet and correcting for an insertion loss of -16.4 dB, we extract φ = (0.9 ± 0.1)π rad, f C = 209.23 ± 0.01 MHz, κ/(2π) = 28.20 ± 0.01 MHz and κ e /(2π) = 15.80 ± 0.01 MHz. Figure 1. Circuit model. Capacitors C M and C D are taken as simple lumped elements, including any parasitic capacitances in parallel. Elements R L , R C and C L model parasitic contributions to the impedance of the inductor L. The effective resistance R models other losses in the circuit. The membrane is modelled by the combination of R M and C C , the capacitance between the metallised area of the membrane and both antenna electrodes. To each electrode, the membrane-electrode capacitance is 2C C , which when summed together in series gives C C .

Circuit simulation
To extract circuit parameters the cavity response can also be fit using the circuit model in Fig. 1, as shown in Fig. 1 (b). The capacitors C D and C M are taken as simple lumped elements. We consider the capacitor formed between the electrodes and the metalized membrane, C C , formed by two electrode-membrane capacitors of value 2C C , as well as the capacitance between antenna electrodes, C A . The inductor is modeled as a network of elements as shown, which simulate its self-resonances and losses. The membrane has a resistance R M and other losses in the circuit are modeled by an effective resistance R.
The reflection coefficient Γ is then equal to where Z tot is the total impedance from the circuit's input port and Z 0 = 50 Ω is the line impedance. We relate the measured transmission S 21 to Γ by assuming a constant overall insertion loss A, incorporating attenuation in the lines, the coupling of the directional coupler, and the gain of the amplifier, such that Fitting to Eq. (1), we take C D = 10 pF from the known component value, and L = 223 nH, R L = 3.15 × 10 −4 Ω × f P [Hz], R C = 25 Ω and C L = 0.082 pF from the datasheet of the inductor. For the resistance of the aluminium film, we estimate R M = 7.5 Ω using the resistivity of aluminium (15 µΩ cm) and the known film thickness (20 nm) 2 . We have estimated 1.4 pF from a COMSOL model of the antenna electrodes and we have added a parasitic capacitance of 0.7 pF based on previous work 3 making a total parasitic capacitance C P of 2.1 pF. Fit parameters are then A, C M , R, and C C . From the fit we obtain A = −16.657 ± 0.001 dB, C M = 20.93 ± 0.01 pF, R = 15.87 ± 0.01 Ω and C C = 1.6432 ± 0.0001 pF.
The cavity coupling to the membrane displacement is given by Although this approximation applies strictly only for a simple LC resonator, we confirmed numerically that this procedure gives a good approximation for ∂ f C ∂C T .

Mechanical quality factors
From Lorentzian fits as in Fig. 2

Mechanical sidebands as a function of cavity drive
In order to distinguish the mechanical sidebands in Fig. 2(a) from parasitic resonances we measure them as a function of P C (Fig. 3), as the frequency of the mechanical modes decreases for the highest values of P C . This might have to do with heating of the membrane surface and thereby a decrease in its tension. From the circuit model we can calculate the power dissipated in the membrane. For P C = 15 dBm, the power dissipated is ∼ 4 µW. We estimate the maximum temperature increase by assuming that all the dissipated power is emitted as thermal radiation. Taking the emissivity of aluminum as 0.09 and applying the Stefan-Boltzmann law leads to an estimated temperature increase of 4.5 K. For P C = 20 dBm, the temperature difference is ∼ 13.5 K. A similar calculation confirms that the injected noise does not change the temperature of the membrane significantly. -140 -120 P (dBm) Figure 3. Power spectrum as a function of frequency and P C for V M = 2.7 V rms . The cavity drive is at 209.5 MHz. Several mechanical sidebands can be observed. The first few mechanical modes are displayed in (a) and the higher frequency mechanical modes are displayed in (b). As P C increases, the mechanical sidebands are brighter and their frequency shifts to lower values for P C 15 dBm.

Electromechanical coupling
3.1 Extraction of g 0 from mechanical sidebands From the area below the sidebands in Fig. 2(a) of the main text, we extracted the values of g 0 plotted in Fig. 3(b) of the main text. In this Section we will derive the expression relating g 0 to the effective thermomechanical power (P side ) extracted from this area. We start with where u ZP = h/(4πm f i,j ) is the zero point motion of the membrane with effective mass m ∼ 4.5 × 10 −10 kg. We choose to normalize the mode eigenfunctions such that the effective mass equals the mass of the suspended segment for all modes 4 . The value of L is known, C T is obtained from the circuit model fit and u ZP can be estimated from the values of f i,j extracted from the Lorentzian fit of the mechanical sidebands. To estimate ∂C C ∂ u , we write P side , which obeys whereP C is the cavity drive P C having taken into account the overall insertion loss andn i,j is the phonon occupancy of the mode 5 . Replacing g 2 0 with Eq. (4), For the measurements in Fig. 2 of the main text P C = 5 dBm at port 1. The values of κ, κ e and A are obtained from the cavity characterization (Section 1.1) and the circuit model fit (Section 1.2).
We now writen i,j ,

3/6
where δ u i,j is the membrane displacement from its equilibrium position. In order to estimate the rms value δ u 2 i,j , we write the effective electromechanical force on the membrane, where V (t) = V DC + δV (t). The time-independent part V DC = 15 V is much larger than the time-dependent part δV (t). The time-dependent part of F(t) is to lowest order where we assumed this electronic fluctuating force is much larger than the thermal noise.
In the frequency domain, the displacement is where the mechanical susceptibility is The values of Q i,j can be extracted from the Lorentzian fit of the mechanical sidebands (Section 2.1). We calculate [δ u 2 i,j ] rms , where χ i,j ( f ) and δV ( f ) are uncorrelated. The fluctuating voltage δV ( f ) is assumed to be well approximated by white noise over the frequency range of interest, with δV where S V is the single-sided white noise power spectrum of the driving voltage V . We obtain given that We can now rewrite Eq. 7, Replacing Eq. 13 in Eq. 5, we obtain the following expression, With the amplitude of the driving noise set to V M = 2.7 V rms , the corresponding spectral density is measured as S 2 V = (3 ± 0.1) × 10 −10 V 2 /Hz and we calculate ∞ 0 d f |χ i,j ( f )| 2 numerically. Once we estimate ∂C C ∂ u , Eq. 4 gives us g 0 for each observed mechanical mode (Fig. 3 of the main text). The error in this quantity reflects uncertainty in the cavity characterization (A, κ and κ e ), the circuit model fit (C T ), the measurement of S 2 V , P side and the fit of the mechanical sidebands ( f i,j and Q i,j ). We also calculate the rms displacement [δ u i,j ] rms corresponding to the mechanical sidebands in Fig. 3 of the main text using Eq. 12. For the fundamental mode, [δ u 0 ] rms ∼ 14 nm (Fig. 4). For comparison, for a thermal state at 293 K [δ u 0 ] rms = u ZP k B T /h f 0 ∼ 4 pm. Therefore the white noise power spectrum of the electronic force is much larger than the power spectrum of the thermal noise V 2 DC S 2 V (∂C C /∂ u) 2 4πk B T f i,j /Q i,j , justifying the approximation in Eq. 9.

Mode profile correction to g 0
The mode profile modifies ∂C C ∂ u and thus g 0 . For modes with i × j even, the sections of the membrane moving away from the antenna and the sections moving towards the antenna are equal, and therefore the net ∂C C ∂ u is close to zero, and g 0 ∼ 0. For i × j odd, there is always a section of the membrane which does not have a counterpart moving out of phase. Therefore, for odd mode profiles ∂C C ∂ u , and thus g 0 , are reduced by a factor 1/(i × j).

Electromechanically induced transparency
The transmission of our circuit in the presence of a weak probe tone at f P and a strong drive tone at f D , as shown in the Fig.4(c-e) of the main text, is 6,8 : where We define χ 0 as the mechanical susceptibility of the fundamental mode and S in as the photon flux incident from the drive tone, whereP D is the power of the drive tone (P D ) having taken into account the overall insertion loss. For the measurements in Fig. 4 of the main text P D = 5 dBm.
Using the values of κ/(2π) and κ e /(2π) extracted in section 1.1, we fitted the curves in each panel of Fig. 4(c-e) of the main text with equation 15. In this way, we obtained the reported values for Q 0 , f 0 and g 0 /2π. The uncertainties in these quantities reflect the variance among the values obtained for each curve.
To estimate the amplitude of the membrane's motion we consider parametric amplification of the oscillator due to the beat frequency between the drive and probe tones 8 . For the OMIT measurement, the probe is 32.5 dB weaker than the drive (see Fig. 4 of the main text), hence the intracavity photon number is modulated by N ≈ 2 × 10 −32.5/20 n C . Here we have neglected any difference due to the cavity resonance as both tones are well within the linewidth. The circulating photon number is 5/6 estimated at n C ≈ 5.2 × 10 12 (at 5 dBm). For OMIT the beat frequency is at f 0 , hence the oscillator sees the parametric force F =hg 0 N cos(2π f 0 t)/u ZP . This force results in a coherent amplitude u max =¯h g 0 Q 0 N mu ZP (2π f 0 ) 2 ∼ 8.6 nm. This is significantly above the thermal rms motion.