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A thin film, clamped between two metal posts, trembles at the slightest nudge. Its movements are picked up by a microwave resonator — a cavity where radio waves bounce around like sound in an empty room. When the membrane shifts, the cavity's response changes. If the film sits exactly in the middle, the signal depends not on the shift itself but on its square, just as the loudness of a drum depends on the force of the strike, not on which way the drum is tilted. Spectroscopy has nothing to do with it: this is pure light–mechanics interaction.

The researchers moved the membrane with a piezoelectric actuator — a crystal that squeezes when voltage is applied. They measured how much the resonator's frequency changed at different membrane positions. At the center, the quadratic response was largest, and the signal appeared at twice the membrane's vibration frequency. Shifting the membrane to the side turned the quadratic response linear: the signal returned to the ordinary frequency. This transition is like tuning a guitar: while the string is exactly over the fret, the note is clean, but move your finger slightly and overtones appear.

This scheme is a step toward devices that can sense individual quanta of membrane motion. Then we could observe a mechanical object behaving according to quantum mechanics: being in two states at once or changing energy only in jumps. This is not science fiction but the near-term goal of labs worldwide.

Original abstract

as written by the authors arXiv:2602.07305

We investigate a microwave-cavity-based displacement readout employing a split-post geometry to measure the motion of a dielectric membrane. Due to symmetry, the cavity response to membrane displacement is inherently quadratic when the membrane is positioned at the centre of the posts. We characterise this behaviour by driving the membrane with a piezoelectric actuator at both central and off-centre positions and we estimate the drive-to-displacement transfer function using the independently calibrated frequency-to-voltage response of the interferometric readout. When the membrane is located at the centre of the cavity and driven, the system exhibits the largest quadratic output, measured at the second harmonic of the membrane acoustic frequency. As the membrane is moved away from the centre, the response transitions from predominantly quadratic to predominantly a linear response at the membrane acoustic frequency. Quadratic optomechanical coupling is a key requirement for displacement - squared readout and, in the quantum regime, for measurements sensitive to mechanical energy or phonon number. The present work therefore establishes the split-post geometry as a promising platform for microwave-mechanical transduction, providing a practical route toward future experiments aimed at probing quantised mechanical motion and energy-sensitive readout schemes.