Split-Post Microwave Displacement Transducer with Quadratic Readout

TL;DR

This work introduces a split-post re-entrant microwave cavity that yields a predominantly quadratic displacement readout by placing a dielectric membrane at the cavity symmetry plane, cancelling the linear coupling while preserving a quadratic term. The authors develop a theoretical framework showing how the cavity frequency shift ω_c(x) exhibits purely quadratic behavior at symmetry and how strong driving reveals a QND phonon-number readout channel. Experimentally, they demonstrate a tunable crossover between quadratic and linear coupling by repositioning the membrane (center vs off-center) and characterize the displacement-to-voltage readout via an interferometric setup, achieving a transfer function of about 1 nm/mV and quantifying the quadratic coefficient G_2. The results establish a versatile platform for quantum-limited sensing and energy-resolved measurements, with potential extensions to ground-state cooling and gravitational-wave/dark-matter sensing using phonon counting in microwave-mechanical devices.

Abstract

We investigate a microwave cavity-based displacement readout employing a split-post geometry for measuring the motion of a dielectric membrane. The cavity response to membrane displacement is predominantly quadratic when the membrane is positioned at the centre of the posts. We characterise this behaviour by driving the membrane piezo electrically at both central and off-centre positions and calibrating the displacement using an independent interferometric measurement. The calibration reveals a linear coupling between the membrane displacement and the applied drive voltage, while the microwave response follows the static displacement dependence. When the membrane is driven at the centre, the system exhibits the highest displacement-to-voltage sensitivity and the largest quadratic output. As the membrane is moved away from the centre, the response gradually transitions from quadratic to linear. There is a difference of ~ 97 $\%$ in the quadratic coefficient from the central position and a difference of ~ 92 $\%$ in the linear coefficient from the off-centre position. This controllable crossover between quadratic and linear coupling is a key requirement for sensors capable of resolving energy quantisation. It establishes this platform as a promising candidate for a microwave-mechanical quantum transducer.

Figures (6)

• Figure 1: (a,b) Schematics of the split-post re-entrant resonator with an acoustic membrane. The cavity electric field is axial and localised between the post end faces, while the magnetic field $\vec{B}$ circulates around each post. The mechanical modes of the central membrane modulate the microwave field. The post-separation is $d_1$ in (a) and $d_2$ in (b). The membrane positions are $x_1$ (symmetric, a) and $x_2$ (asymmetric, b). (c) Shows the change in microwave resonant frequency versus membrane position from FEM simulations and experiment. Experimentally, a VNA tracks the resonant frequency by monitoring $S_{21}$ as the membrane is moved between the posts.
• Figure 2: Experimental setup used to observe the mechanical membrane mode inside the split-post microwave resonator. The interferometric measurement techniques ensure that the signal at the DP is minimised, enabling maximum suppression of the unwanted signals. A spectrum analyser (FFT) is used to excite the PZT actuator and read the output of the interferometer. $\partial V_{mix}/\partial \omega_c$ is the interferometer sensitivity and is maximum when the DP have zero signal.
• Figure 3: The microwave resonator: (a) cross-section of the microwave cavity with absolute electric field displacement of the microwave resonator represented by $\vec{D}$ (b) The microwave mode characterisation by transmission parameter S$_{21}$ for the d$_1$ and d$_2$ positions in the Fig. \ref{['fig:topology']} in terms of the mode frequency offset during the change in the post .
• Figure 4: The mechanical resonance of the sapphire membrane: (a) simulation of the fundamental drum mode, showing the crystal's absolute displacement. (b) The measured frequency response of this resonance at $\omega_m/2\pi=4.169$ kHz. The mode is excited using a PZT drive and read via the mixer output of the interferometer.
• Figure 5: Mixer output voltage ($V_{mix}$) as a function of applied voltage to the piezoelectric drive ($V_{PZT}$), when the membrane is driven with a chirp signal with $\omega_m$ as the central frequency. Blue data is for the membrane positioned at the central, symmetric position, whilst red shows the off-centre position.