Multimode Strong Coupling in Superconducting Cavity Piezoelectromechanics

High-frequency mechanical resonators subjected to low thermal phonon occupancy are easier to be prepared to the ground state by direct cryogenic cooling. Their extreme stiffness, however, poses a significant challenge for external interrogations. Here we demonstrate a superconducting cavity piezoelectromechanical system in which multiple modes of a bulk acoustic resonator oscillating at 10 GHz are coupled to a planar microwave superconducting resonator with a cooperativity exceeding $2\times 10^3$, deep in the strong coupling regime. By implementation of the noncontact coupling scheme to reduce mechanical dissipation, the system exhibits excellent coherence characterized by a frequency–quality-factor product of $7.5\times 10^{15}\mathrm{Hz}$. Interesting dynamics of classical temporal oscillations of the microwave energy is observed, implying the coherent conversion between phonons and photons. The demonstrated high-frequency cavity piezoelectromechanics is compatible with superconducting qubits, representing an important step towards hybrid quantum systems.

Figure 1

(a) An illustrative configuration of a cavity piezoelectromechanical system where a piezoelectric BAR (green) is sandwiched between the parallel capacitor plates (yellow) of an $LC$ resonator. Blue dashed lines: The electric field lines. Orange lines: Amplitude distributions of the $z$-component mechanical displacement of different orders of acoustic thickness modes (only the first 3 orders of modes are drawn). (b) A schematic of the piezoelectromechanical device (not to scale). The planar superconducting $LC$ resonator (“Ouroboros”) on a fabricated sapphire substrate is flipped over and suspended on top of the BAR which consists of a thin aluminum nitride layer (green) deposited on top of a thick oxidized high-resistivity silicon substrate. (c) A cross-sectional view of the device with the inductor of the Ouroboros indicated as an equivalent circuit. The device parameters are as following: $R_C=40\mu \mathrm{m}$, $s=30\mu \mathrm{m}$, $w=10\mu \mathrm{m}$, $d=550\mathrm{nm}$, $t=500\mu \mathrm{m}$. (d) An optical micrograph of the Ouroboros resonator. The long arc-shaped inductor wire has a width of $25\mu \mathrm{m}$ and an average bending radius of $593\mu \mathrm{m}$. (e) Simulated mode profile of perpendicular electric field ($E_z$) of the Ouroboros.

Figure 2

(a) Microwave reflection spectrum of the device measured at 1.7 K. Orange dashed lines: Unperturbed frequencies of the acoustic thickness modes of the BAR. (b) Close-up spectrum of the regime where the microwave mode strongly hybridizes with the acoustic thickness modes. Red lines: Fittings using the coupled-mode formula. Green and orange dashed lines: Fitted center resonant frequencies of the uncoupled microwave mode $f_a=({\omega }_a/2\pi )$ and the thickness mode $f_{b{n}_0}=({\omega }_{b{n}_0}/2\pi )$, respectively. (c) Fitted intrinsic quality factors of the microwave mode and the different orders of acoustic thickness modes of the BAR.

Figure 3

(a) Temperature dependence of the microwave reflection spectra. Blue dashed line: The uncoupled Ouroboros resonant frequency obtained from the fittings using the coupled-mode formula. (b) Line plots of the microwave reflection spectra at different temperatures. Orange dashed lines: Unperturbed frequencies of the acoustic thickness modes of the BAR. (c) Coupling strength and intrinsic $Q$ factors extracted at temperatures when the Ouroboros is on resonance with a thickness mode. (d) Microwave reflection in the time domain. The center frequency of the input microwave pulse is 10.1737 GHz. Red line: The fitting of the exponentially decaying oscillation with background signals. Inset: Temporal oscillations at different input frequencies detuned from 10.1737 GHz (white dashed line).