Cryogenic rf-to-microwave transducer based on a dc-Biased electromechanical system

TL;DR

This work introduces a cryogenic two-stage rf-to-microwave transducer that uses a dc-biased electromechanical membrane as a pre-amplifier in series with a superconducting LC cavity. The electrostatic pre-amplification, tunable by the dc bias $V$, multiplies the mechanical transduction gain $\,\mathcal{G}_V$, which is then upconverted by the electromechanical gain $\mathcal{G}_{em}$ to yield a total gain $\\mathcal{G}_{tot}$. In a 1.5 μm-gap flip-chip device at 10 mK, the authors observe dc-tunable anti-spring shifts and achieve a charge sensitivity of about $87 \,\\mu e/\\sqrt{Hz}$ (corresponding to $0.9$ nV/√Hz), with predictions of sub-200 fV/√Hz sensitivity for sub-micron gaps and $Q>10^8$ membranes. The results establish dc-biased electromechanics as a practical route toward quantum-grade rf electrometers and low-noise, modular heterodyne links compatible with superconducting microwave circuits.

Abstract

We report a two-stage, heterodyne rf-to-microwave transducer that combines a tunable electrostatic pre-amplifier with a superconducting electromechanical cavity. A metalized Si$_3$N$_4$ membrane (3 MHz frequency) forms the movable plate of a vacuum-gap capacitor in a microwave LC resonator. A dc bias across the gap converts any small rf signal into a resonant electrostatic force proportional to the bias, providing a voltage-controlled gain that multiplies the cavity's intrinsic electromechanical gain. In a flip-chip device with a 1.5 $\mathrmμ$m gap operated at 10 mK we observe dc-tunable anti-spring shifts, and rf-to-microwave transduction at 49 V bias, achieving a charge sensitivity of 87 $\mathrmμ$e/$\sqrt{\mathrm{Hz}}$ (0.9 nV/$\sqrt{\mathrm{Hz}}$). Extrapolation to sub-micron gaps and state-of-the-art $Q>10^8$ membrane resonators predicts sub-200 fV/$\sqrt{\mathrm{Hz}}$ sensitivity, establishing dc-biased electromechanics as a practical route towards quantum-grade rf electrometers and low-noise modular heterodyne links for superconducting microwave circuits and charge or voltage sensing.

Figures (7)

• Figure 1: Top: illustration of the transducer amplification chain, where the electrical spectrum $S_\mathrm{VV}$ (signal $\delta V_s$ at frequency $\Omega_\mathrm{RF}$ plus white noise of spectral density $S_\mathrm{VV}^n$, indicated in purple) undergoes transduction to position spectrum $S_\mathrm{xx}$ with the electrostatic gain $\mathcal{G}_\mathrm{V}$ (electrical noise contribution appears in purple, and thermal noise contribution in orange) followed by transduction to microwave sideband spectrum $S_\mathrm{SB}$ via the electromechanical gain $\mathcal{G}_\mathrm{em}$ (detection noise contribution is indicated in light green). Bottom: proposed implementation of the transducer. An LC microwave resonator (red) of resonance frequency $\omega_\mathrm{c}$ is coupled to a metalized free standing membrane acting as a mechanically compliant capacitor. The red part of the circuit is electrically floating. The membrane electrode (yellow) is biased through a capacitor $C_\mathrm{b}$, connected to a bias tee (pink rectangle) that combines the dc bias and an rf signal to drive membrane modes on resonance. The two halves of the membrane electrode behave as $C_m(x)/2$ capacitors that are connected in parallel with the microwave LC resonator, and in series with the bias-line, so that the bias circuit sees the full membrane capacitance $C_m(x)$ whereas the LC resonator sees only $C_m(x)/4$ in parallel with the stray capacitance $C_\mathrm{s}$. The LC resonator is inductively coupled to a read-out line (green). A probe tone $\alpha_\mathrm{in}$ at the red‑detuned frequency $\omega_\mathrm{d}=\omega_\mathrm{c}-\Omega_\mathrm{m}$ is sent into the resonator, and the reflected field $\alpha_\mathrm{out}$ is monitored with a vector spectrum analyser. In this scheme, the mechanical mode of amplitude $\delta x$, is driven on resonance by the rf source, and amplified by the dc source with a gain $\mathcal{G}_V$. It is then upconverted and amplified with a gain $\mathcal{G}_\mathrm{em}$ at microwave frequencies by the electromechanical system operating in the resolved sideband regime. It appears as a sideband of the reflected microwave carrier $\alpha_\mathrm{out}$.
• Figure 2: (a) LC resonator chip design: LC resonator (red), biasing line (blue), the read-out line (green), the ground plane (gold), and the aluminum pillars (light blue) used as spacers in the flip-chip process. For visualization, the inductor track (width: 8 µ m) has been thickened. Inset: close-up on the read-out line which is a single-sided coplanar waveguide terminating into the ground plane. (b) Optical microscope image of the capacitor pads of the LC resonator, overlaid with the membrane electrodes (pink). (c) The LC resonator (red) overlaid with simulated mode profiles of the first to membrane modes 11 and 12. (d) Optical microscope image of the membrane sample, standing on a rectangular pedestal 25 $\mu$m above the silicon substrate. The pedestal with the membrane and electrodes is in focus, the substrate is not in focus. (e) Optical microscope image of the membrane sample, seen from the other side. The released metalized SiN membrane can be seen at the top. Mirror images on the four edges of the membrane come from the smooth inclined silicon crystalline planes obtained after etching with KOH trough the substrate. (d) and (e) figures are taken from Ref. Gerashchenko2025.
• Figure 3: (a) Illustration of the custom mask holder used in the flip-chip process, showing the membrane chip (blue) being picked up via a vacuum suction tube. A viewing hole allows visual alignment. (b) We use the 3 axis micropositioning of the MJB4 mask aligner system under binocular microscope to align the membrane chip with the LC resonator chip that is already wirebonded in a microwave-shielding box, and bring them in contact. (c) Cross-sectional schematic of the final device. The membrane pedestal at a distance of 25 µ m from the substrate can be seen in the center. On the LC resonator chip, the 600 nm-tall aluminum pillars (not to scale) are used as spacers to set the inter-chip distance. Glue drops are also shown at the chip edges. (d) Photograph of the completed flip-chip assembly in the mask aligner, with chips glued together (white drops indicate adhesive). The microwave-shielding box is indicated.
• Figure 4: (a) Sideband noise amplitude from mode 12, on the read-out reflected signal, with the microwave pump at $\omega_\mathrm{d} = \omega_\mathrm{c} - \Omega_\mathrm{m}$, and with a non zero dc bias $V=30~$V. (b) and (c) show the frequency shift of the two mechanical modes for $V\in [-49, 49]$ V, highlighting the anti-spring effect. (d) and (e): ring-down measurements for mode 11 and 12 performed at 10 mK near zero dc bias. The membrane mode is driven by a strong resonant rf tone through the biasing line, and its exponentially decaying amplitude after turning off the rf tone, is monitored versus time. By fitting the decay with an exponential curve (yellow line) we extract $\Gamma_\mathrm{m}$.
• Figure 5: Membrane modes thermalization and electrical noise transduction. Left column: transduced sideband spectrum area versus cryostat temperature at zero bias voltage. (a): mode 11. (b): mode 12. From a linear fit (red), we extract a slope of $3.57\times10^{-10}$ V$^2$/K for mode 11, shown in (a), and $9.8\times10^{-11}$ V$^2$/K for mode 12, after correcting for power broadening. The 10 mK area value and 500 mK area value of each mode defines the corresponding thermal noise floor highlighted in two different gray colors. Right column: area versus $V$ at two different temperatures. Eq. \ref{['eqn:area_v_mode']} is fitted to the data (solid lines).