A microgram acoustic mode stops guessing the signal phase: Fock states push displacement sensitivity to five times vacuum

A microgram-scale mechanical acoustic mode now works as a phase-insensitive quantum-enhanced sensor whose displacement-amplitude estimate beats any classical probe. Previously the phase-insensitive metrological advantage of non-Gaussian quantum states was realized only in lighter systems such as trapped ions, while mechanical oscillators beating the standard quantum limit used squeezed states whose advantage is restricted to a known quadrature. Two independent analyses of the measured phonon-number distributions, based on the curvature of the squared Hellinger distance and on the width of the Bayesian posterior, agree on a Fisher information five times the vacuum value, the maximum attainable by any mixture of coherent states.

The sensor works by using optimal-control pulses to prepare the acoustic mode from vacuum into a Fock state, letting a weak resonant drive imprint a phase-space displacement of unknown phase, and reading out with a phonon-number-resolving measurement. Readout uses resonant phonon-number measurement, in which an initially excited transmon qubit interacts with the mode through a Jaynes-Cummings coupling of variable duration before qubit readout, reconstructing the phonon-number distribution indirectly. The experiment runs on a transmon qubit coupled to a high-overtone bulk acoustic wave resonator whose acoustic mode has an effective mass of about one microgram; prepared Fock states reach seven phonons, and zero-displacement fidelities fall as phonon number rises.

A neural network trained directly on calibration records replaces the parametrized response model and delivers the same quantum advantage without an analytical description of the device. Bayesian estimation with the fitted model relies on a parametrized assumption about the device response, and residual mismatch between model and true response enters the displacement estimate as a systematic error. On the same measurement records, the network posterior matches the fitted-model posterior in peak and width, reduces residuals by up to an order of magnitude for the outcomes carrying most statistical weight, and roughly halves the mean absolute bias across the calibrated range.

The reported scope is displacement-amplitude sensing with a microgram-scale acoustic mode of a high-overtone bulk acoustic wave resonator, read out by phonon-number-resolving measurement, for a weak resonant drive of unknown phase. The demonstrated sensitivity gain peaks for the probe prepared in five phonons, and increasing the phonon number no longer improves the accessible sensitivity. Directions stated for the next step include raising the qubit-phonon coupling to shorten the preparation pulse, speeding up conditional reset to improve the duty cycle, and operating a lower-frequency overtone of the same resonator to improve force and acceleration sensitivity.

The observed sensitivity gain does not grow indefinitely with probe energy; it peaks for the probe prepared in five phonons and then saturates. This saturation is set by state preparation fidelity at high phonon number, not by the energy relaxation time of the acoustic mode or by the resonant phonon-number detection. The model-free estimator still deviates from the measurement for rare outcomes at the edges of the distribution, where calibration statistics are scarce. The text notes that once the probe state is mixed, phonon-number measurements do not necessarily saturate the quantum Fisher information.