Split-Post Re-entrant Microwave Displacement Transducer with Quadratic Readout
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.