Description
Microwave cavity displacement readout provides a powerful route to high-sensitivity mechanical sensing and quantum transduction when the cavity frequency depends quadratically on membrane position. This quadratic response enables quantum non-demolition phonon-number measurements, suppresses linear back-action at symmetry, and provides a route to resolving mechanical energy quanta. We investigate a symmetric split-post microwave resonator containing a 0.5-mm thick sapphire membrane and show that its electromechanical response can be tuned between predominantly quadratic and linear coupling by positioning the membrane at or away from the geometric centre between the posts.
The static microwave response is obtained by measuring the resonant frequency shift as a function of post gap, while an independent dynamic calibration is performed by piezo-electrically driving the membrane at $4.169~\mathrm{kHz}$ and detecting the second-harmonic response at $8.35~\mathrm{kHz}$ using a microwave interferometer. At the centred position, the system gives the largest quadratic output and the highest displacement-to-voltage sensitivity. Moving the membrane off centre produces a gradual crossover to linear coupling, with a $97\%$ change in the quadratic coefficient relative to the centred configuration and a $92\%$ change in the linear coefficient relative to the off-centred configuration. The dynamic measurement gives $G_2 = 3.36 \times 10^{11}~\mathrm{Hz/mm^2}$, corresponding to a zero-point second-order coupling rate of $g_2 = 1.26 \times 10^{-25}~\mathrm{Hz}$, and a room-temperature thermally enhanced value of $91.7~\mu\mathrm{Hz}$. The present quadratic displacement sensitivity is $2.16 \times 10^{-16}~\mathrm{m^2/\sqrt{Hz}}$, with simulations indicating that cryogenic optimisation can improve this by 6-7 orders of magnitude.
This controllable quadratic microwave-mechanical coupling establishes the split-post platform as a promising transducer for precision displacement sensing, quantum optomechanics, and future tests requiring energy-resolving mechanical measurements.
[1] S. Parashar, W.M. Campbell, J. Bourhill, E.N. Ivanov, M. Goryachev, M.E. Tobar, APL Photonics, vol. 9, 111304 (2024).
[2] S. Parashar, J.F. Bourhill, M. Goryachev, M.E. Tobar, arXiv:2602.07305 (2026).
| I am the presenting author | Yes |
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