Superconducting Cavity Electromechanics on a Silicon-on-Insulator Platform

Paul B. Dieterle, Mahmoud Kalaee, Johannes M. Fink, and Oskar Painter

Phys. Rev. Applied 6, 014013 – Published 22 July 2016

Abstract

Fabrication processes involving anhydrous hydrofluoric vapor etching are developed to create high- $Q$ aluminum superconducting microwave resonators on free-standing silicon membranes formed from a silicon-on-insulator wafer. Using this fabrication process, a high-impedance 8.9-GHz coil resonator is coupled capacitively with a large participation ratio to a 9.7-MHz micromechanical resonator. Two-tone microwave spectroscopy and radiation pressure backaction are used to characterize the coupled system in a dilution refrigerator down to temperatures of $T_{f} = 11 mK$ , yielding a measured electromechanical vacuum coupling rate of $g_{0} / 2 π = 24.6 Hz$ and a mechanical resonator $Q$ factor of $Q_{m} = 1.7 \times 1 0^{7}$ . Microwave backaction cooling of the mechanical resonator is also studied, with a minimum phonon occupancy of $n_{m} \approx 16$ phonons being realized at an elevated fridge temperature of $T_{f} = 211 mK$ .

Received 15 January 2016

DOI:https://doi.org/10.1103/PhysRevApplied.6.014013

Physics Subject Headings (PhySH)

Optomechanics

Cavity resonators Etching Microwave techniques

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Paul B. Dieterle, Mahmoud Kalaee, Johannes M. Fink^*, and Oskar Painter^†

Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA

^*Present address: Institute of Science and Technology Austria (IST Austria), 3400 Klosterneuburg, Austria.
^†opainter@caltech.edu

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Vol. 6, Iss. 1 — July 2016

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Images

Figure 1
(a) Schematic of the electromechanical circuit and measurement setup. The electromechanical circuit (yellow) is inductively coupled to a $2 − μ m$ -wide wire (turquoise) which shorts to ground and reflects the signal. Acronyms: ${SG}_{i} = microwave signal generator$ , $VNA = vector network analyzer$ , $SA = spectrum$ analyzer, $LNA = low − noise amplifier$ , $HEMT = high − electron − mobility transistor amplifier$ . (b) FEM simulation of the differential mechanical mode. In this work, $l = 13.5 μ m$ , $w_{t} = 440 nm$ , and $l_{t} = 4 μ m$ . These values give a simulated mechanical-mode frequency of $ω_{m} / 2 π = 9.76 MHz$ . (c) Plot of FEM simulation values of $C_{m}$ versus slot size $d$ . (d) Plot of FEM simulation values of $g_{0}$ versus slot size $d$ for (i) $C_{s} = C_{l} = 0 fF$ corresponding to an ideal $η = 1$ (blue squares) and (ii) $C_{l} = 3.05 fF$ and $C_{s} = 1.13 fF$ from FEM simulations of the circuit (black diamonds). For these plots, the resonance frequency is fixed at the measured frequency of $ω_{r} / 2 π = 8.872 GHz$ . At the estimated capacitor gap of $d \approx 60 nm$ from SEM images, the theoretical values of the motional capacitance and the vacuum coupling rate are $C_{m} = 2.76 fF$ and $g_{0} / 2 π = 29.3 Hz$ , respectively.
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Figure 2
(a) SEM image of the fabricated microwave coil resonator and $H$ -slot mechanical resonator. The $H$ -slot resonator region is colored red and the undercut region is outlined in yellow. The coupling wire is colored turquoise. (b) An enlarged SEM image of the $H$ -slot mechanical resonator. Inset: a close-up of the 60-nm-wide capacitor gap formed by a 250-nm-wide Al electrode on the photonic-crystal slab and a 550-nm-wide Al electrode on the outer Si support membrane. Both wires are 60-nm thick, as is the ground plane. (c) Cross-section image showing the suspended membrane with a coil on top. The Al forming the coil is 120-nm thick. The $3 − μ m$ -thick dark area underneath the Si membrane is the undercut region where ${SiO}_{2}$ has been etched away. The bottom layer is the Si handle wafer.
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Figure 3
(a) Phase and amplitude response of the microwave resonator at fridge temperature $T_{f} \approx 11 mK$ and on-resonance cavity photon number of $n_{p} = 3.3$ . The intrinsic loss rate $κ_{i}$ and external coupling rate $κ_{e}$ are extracted by fitting the curves with a modified Lorentzian cavity model to take into account the asymmetry in the background frequency response. (b) Schematic showing two-tone EIT measurement procedure. A strong drive tone at frequency $ω_{d}$ is placed on the red sideband of the microwave cavity and the cavity response is swept by a weak VNA probe at $ω_{p}$ . (c) Plot of the measured EIT spectra at a series of drive intracavity photon numbers for a fridge temperature of $T_{f} \approx 11 mK$ . From top to bottom: $n_{d} = 484$ (orange curve), $1.20 \times 1 0^{5}$ (maroon curve), and $2.38 \times 1 0^{6}$ (blue curve). Note for the blue curve at $n_{d} = 2.38 \times 1 0^{6}$ , a weakly coupled auxiliary mechanical mode can be observed. The frequency range between the vertical red dashed lines, surrounding the auxiliary mechanical resonance, is omitted for fitting purposes. (d) Plot of the fit values from the measured EIT spectra using Eq. (2) for cavity coupling rates (top), intrinsic mechanical damping rate (middle), and parametrically enhanced coupling rate (bottom). Error bars correspond to a 95% confidence interval in the estimated fit parameter.
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Figure 4
(a) Schematic showing the time-domain mechanical ring-down protocol, wherein a strong blue pulse at $ω_{pulse} = ω_{r} + ω_{m}$ populates the mechanics and a weak probe tone at $ω_{p} = ω_{r} - ω_{m}$ is used to monitor the energy in mechanical resonator. Inset: schematic showing the frequency and scattering of the applied tones used to ring up and monitor the mechanical resonator. Here, $Γ_{S} \approx (4 n_{pulse} g_{0}^{2} / κ) (n_{m} + 1)$ [ $Γ_{AS} \approx (4 n_{p} g_{0}^{2} / κ) n_{m}$ ] is the Stokes [anti-Stokes] scattering rate of the pulse [probe] tone, where $n_{pulse}$ [ $n_{p}$ ] is the intracavity pulse [probe] tone photon number. (b) Time-domain mechanical ring-down measurement at $T_{f} \approx 11 mK$ . A steep decay resulting from the leakage of photons from the cavity, followed by a slow decay due to the intrinsic mechanical damping of the resonator, is observed.
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Figure 5
(a) Schematic showing thermometry measurement scheme, wherein a red-detuned drive tone is used to simultaneously cool and measure the average energy in the mechanical resonator through anti-Stokes scattering proportional to phonon occupancy of the resonator. $Γ_{AS} \approx (4 n_{d} g_{0}^{2} / κ) n_{m}$ is the cavity-enhanced anti-Stokes scattering rate of the drive tone proportional to $n_{m}$ . $Γ_{S} \approx (4 n_{d} g_{0}^{2} / κ) (κ / 4 ω_{m})^{2} (n_{m} + 1)$ is the cavity-suppressed Stokes scattering rate of the drive tone proportional to $n_{m} + 1$ . (b) Cooling curve obtained by fitting the measured microwave noise spectrum using a model which includes noise squashing and heating effects due to thermal noise in the microwave cavity and the input coupler. Spectra are taken at a fridge temperature of $T_{f} = 211 mK$ . Blue circles correspond to the inferred average mechanical-mode occupancy ( $n_{m}$ ) from fits to the measured noise (see inset). Gray triangles are the fit input waveguide ( $n_{b, wg}$ ) and cavity ( $n_{b, r}$ ) thermal-noise occupancies from the measured noise background level. The dashed line indicates the predicted occupancy as given by $n_{f, m} / (1 + C)$ , where $C$ is determined from the EIT fit values for the vacuum coupling rate ( $g_{0} / 2 π = 25.1 Hz$ ) and the intrinsic damping rate ( $γ_{i} / 2 π = 25.7 Hz$ ) taken at a fridge temperature of $T_{f} \approx 211 mK$ . (c) Plot of the measured noise spectral density (black curve) and modeled noise background (green curve) at $P_{d} = 22 dBm$ . The orange curve corresponds to the expected spectral noise density due to the waveguide bath ( $n_{b, wg}$ ) alone while the navy curve shows the expected contribution from the resonator bath ( $n_{b, r}$ ). The global offset of $n_{add} + 1$ shown as a gray dashed line.
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