Superconducting Cavity Electromechanics: The Realization of an Acoustic Frequency Comb at Microwave Frequencies

We present a nonlinear multimode superconducting electroacoustic system, where the interplay between superconducting kinetic inductance and piezoelectric strong coupling establishes an effective Kerr nonlinearity among multiple acoustic modes at 10 GHz that could hardly be achieved via intrinsic mechanical nonlinearity. By exciting this multimode Kerr system with a single microwave tone, we further demonstrate a coherent electroacoustic frequency comb and provide theoretical understanding of multimode nonlinear interaction in the superstrong coupling limit. This nonlinear superconducting electroacoustic system sheds light on the active control of multimode resonator systems and offers an enabling platform for the dynamic study of microcombs at microwave frequencies.

Figure 1

(a) An optical image of the nonlinear frequency-tunable Ouroboros resonator. The scale bar is $200\mu \mathrm{m}$. Inset: Enlarged view of the nonlinear inductor (wire width: $1\mu \mathrm{m}$, spacing: $4\mu \mathrm{m}$). (b) Microwave reflection spectrum of the Ouroboros at different input power showing the nonlinear resonance shift. (c) Frequency tuning of the Ouroboros resonator under an external magnetic field. Red line shows a quadratic fitting. (d) A schematic of the nonlinear electroacoustic device (not to scale). The Ouroboros chip is flipped over and bonded with a piezo-bulk acoustic resonator (BAR) by cryogenic epoxy (black) at the four corners. A superconducting coil is placed above the Ouroboros to provide external magnetic field for frequency tuning. The Ouroboros is inductively coupled to a loop probe for microwave signals input and output. (e) Illustration of the nonlinear electroacoustic system with multimode strong coupling between the Ouroboros and the BAR. $g_n$: piezoelectromechanical coupling strength. $L_k$ and $L_m$: the nonlinear kinetic inductance and the magnetic inductance of the Ouroboros. $R$: equivalent resistance (intrinsic dissipation) of the microwave mode.

Figure 2

(a) Microwave reflection spectrum at different magnetic field. Gray dashed lines indicate the unperturbed acoustic resonant frequencies. Red arrows mark the corresponding pump frequencies in (b). The pink shades indicate the pump frequency range where a hybrid comb can be generated. (b) Microwave power spectrum of the hybrid comb at different magnetic field under a pump power of $-18\mathrm{dBm}$. Red arrows indicated the pump frequency. (c) Principle of comb generation via cascaded degenerate FWM. Under a single pump tone at $f_p$, two pump photons (green arrows in the upper inset) are annihilated to create one signal and one idler photon (red and blue arrows). In presence of multiple hybrid modes (gray shades), a cascaded process is initiated to form a frequency comb with an FSR of ${\mathcal{F}}_c$.

Figure 3

(a) Measurement diagram for the comb coherence. An rf pump source ($f_0\sim 10\mathrm{GHz}$) is used to generate a pump tone at $f_{\mathrm{p}}=f_0+\delta$, where $\delta =250\mathrm{MHz}$, via single-sideband modulation (SSBM). The 10-GHz frequency comb from the device is then demodulated using the same rf source to down-convert the comb to low frequencies centered at $\delta$. DUT: device under test. rf Amp: rf amplifier. LPF: low-pass filter. PD: photodiode. (b) Microwave power spectrum of the down-converted comb centered at 250 MHz. The top panel shows the relative phase (${\phi }_n$) of each comb line extracted from the fitting of the temporal oscillation in (c). (c) Time-domain oscillation of the down-converted comb. An enlarged section with a perfectly matched fitting (red line) is shown in the bottom panel. The red dashed lines in the top panel indicate the envelope of entire the fitting curve. (d) Power spectrum of the mixing signal of the down-converted comb after the rf photodiode.