Probing the quantum motion of a macroscopic mechanical oscillator with a radio-frequency superconducting qubit

TL;DR

The work demonstrates repeated, resonant quantum interactions between a 4 MHz macroscopic SiN membrane and a heavy-fluxonium qubit, achieving >300 excitation exchanges during the membrane’s 6 ms lifetime. By performing stroboscopic, weak qubit measurements after each interaction, the team reconstructs the membrane’s position-noise spectrum and observes a quantum-emission/absorption asymmetry tied to phonon non-commutativity. The platform bridges GHz superconducting qubits and MHz mechanical motion, enabling potential tests of gravity-induced collapse (Diósi–Penrose) and CSL in a mass/scale regime previously inaccessible. The work also lays out pathways to enhance coupling and coherence to reach strong coupling and direct interferometric bounds on collapse models, thereby providing a concrete route to test foundational quantum-mechanical principles in macroscopic systems.

Abstract

Long-lived mechanical resonators like drums oscillating at MHz frequencies and operating in the quantum regime are a powerful platform for quantum technologies and tests of fundamental physics. Yet, quantum control of such systems remains challenging, owing to their low energy scale and the difficulty of achieving efficient coupling to other well-controlled quantum devices. Here, we demonstrate repeated coherent interactions between a 4 MHz suspended silicon nitride membrane and a resonant superconducting heavy-fluxonium qubit. The qubit is initialized at an effective temperature of $21~\mathrm{μK}$ and read out with 77% single-shot fidelity. During the $6~\mathrm{ms}$ lifetime of the membrane the two systems swap excitations more than 300 times. After each interaction, a state-selective qubit detection is performed, implementing a stroboscopic series of weak measurements that provide information about the mechanical state. The accumulated records reconstruct the position noise spectrum of the membrane, revealing both its thermal occupation $n_\mathrm{th}\approx47$ at $10~\mathrm{mK}$ and the qubit-induced back-action. By preparing the qubit either in its ground or excited state before each interaction, we observe an imbalance between the emission and absorption spectra, proportional to $n_\mathrm{th}$ and $n_\mathrm{th}+1$, respectively-a hallmark of the non-commutation of phonon creation and annihilation operators. Since the predicted Diósi-Penrose gravitational collapse time is comparable to the measured mechanical decoherence time, our architecture enters a regime where gravity-induced decoherence could be tested directly.

Figures (14)

• Figure 1: Experimental principle and design.(a) Energy-level diagram of the mechanical resonator (left) and fluxonium circuit (right). The membrane mode at frequency $\omega_m$ is modeled as a quantum harmonic oscillator. The heavy fluxonium’s potential (black line) at $\varphi_\mathrm{ext} = \pi$ is shown with its four lowest eigenstates. The qubit transition (blue/red) is tunable into resonance with the mechanical oscillator. (b) 3D schematic of the mechanical-fluxonium device. A superconducting loop composed of a small Josephson junction (red) and a superinductor (purple), is threaded by an external flux $\varphi_{\rm ext}$ (brown). The small junction is shunted by a capacitor (blue). One capacitor electrode couples to a readout resonator (light pink) and the other to a control waveguide (cyan). The suspended SiN membrane, cut open for visibility, metallized with Al (yellow), and the two fluxonium electrodes form a vacuum‐gap capacitor. A charge gate (orange) provides DC bias and drives to the membrane. (c) Spatial profile of the $4.4\,$MHz mechanical mode of the membrane, plotted at an amplitude equal to its zero‐point fluctuations and overlaid on the fluxonium capacitor electrodes layout (dashed grey lines). (d) Colored optical and electronic micrograph of a twin fluxonium sample. (e) Lumped element circuit diagram of the sample.
• Figure 2: Qubit-mechanical coupling.(a) Ramsey spectroscopy under mechanical drive. Top: Pulse sequence—each cycle begins with a mechanical displacement of the membrane, immediately followed by a fixed‐delay Ramsey sequence, repeated to sample the qubit frequency shift $\delta\omega = \chi\,\langle \hat{a}^\dagger \hat{a}\rangle$. Bottom: Extracted qubit frequency shift versus bias voltage and drive detuning (color map). Dark squares mark the peak of the Lorentzian response at each $V_b$, with a quadratic fit (dashed line). The drive amplitude is adjusted at each voltage to keep the peak response level approximately constant (except for $V_b = V_\text{offset}$ where the driving force vanishes). (b) Mechanical lifetime. Top: Pulse sequence—after resonant membrane displacement, a series of $N=121$ Ramsey measurements are performed, sampling the instantaneous qubit frequency at discrete times $k\times T$. Bottom: Averaged qubit frequency shift $\delta\omega$ plotted versus delay $k\times T$ (for $k=1,\dots,N$), where data at each delay are ensemble‑averaged. An exponential fit yields a mechanical lifetime $T_1^m = 5.9$ ms ($Q = 1.62\times10^5$). The right‑hand axis translates $\delta\omega$ into the equivalent mechanical occupation in units of phonon number. (c) Coherent Rabi exchange. Top: Pulse sequence—after initializing the membrane in a coherent state $|\alpha\rangle$, with $|\alpha|^2\simeq1.6\cdot10^5$, the qubit is Stark‐shifted to $\omega_q^{\rm st}$ for an interaction time $\tau$, and measured along $\sigma_z$. Bottom: Ground‐state population as a function of interaction time and Stark‐shifted qubit frequency. The spheres on the top right of each figure show the qubit Bloch vectors immediately after the interaction for the data points highlighted by blue circles.
• Figure 3: Spectrum analyzer experiment in the semi-classical limit.(a) Phase-space distribution of the mechanical mode amplitude, with an overlaid classical trajectory (black curve), and the qubit Bloch sphere. After initialization in $|g\rangle$ (blue) or $|e\rangle$ (red), the membrane mode acts as a coherent drive on the qubit. The complex mechanical mode amplitude is mapped onto the $(\boldsymbol{\sigma_x}, \boldsymbol{\sigma_y})$ plane of the Bloch sphere. The average energy transferred from membrane to qubit during the interaction $\Delta E_{g/e}$ is displayed on the $\langle\boldsymbol{\sigma_z}\rangle$ histogram on the right‐hand inset. (b) Classical position $x(t)$ of the mechanical resonator, shown in a frame rotating at the qubit–mode detuning $\Delta$. The signal is an oscillation at frequency $\Delta$, with amplitude and phase undergoing random fluctuations at rate $\kappa_m=1/T_1^m$. Blue bars represent qubit $\boldsymbol{\sigma}_x$ measurement outcomes after interaction with the mechanical mode. (c) Experimental pulse sequence. The qubit, prepared in $|g\rangle$ or $|e\rangle$, is Stark-shifted to a detuning $\Delta$ from the membrane frequency for a duration $\tau$, followed by a $\boldsymbol{\sigma}_x$ measurement. (d) Single‐shot outcomes $m_k\in\{-1,1\}$ (for initial state $|e\rangle$) are recorded, grouped into batches of length $N$, and each batch is discrete‐Fourier‐transformed. (e) Fourier transform of the first 10 batches represented as a waterfall plot. (f) Position noise spectra obtained for the qubit prepared in $|g\rangle$ (blue) and $|e\rangle$ (red), by averaging the squared magnitudes of the Fourier transforms shown in (e). The two Lorentzian peaks on the left are centered at the membrane frequency and correspond to the membrane thermal noise. Additionally, a weak calibration tone produces a sinc‐shaped peak offset by 500 Hz towards higher frequencies. (g) Zoom on the Lorentzian peaks, with each spectrum normalized to the calibration‐tone amplitude and flat background subtracted. The broader, lower‐amplitude blue curve ($|g\rangle$-preparation) indicates qubit‐induced cooling, while the narrower, higher‐amplitude red curve ($|e\rangle$-preparation) indicates heating of the mechanical mode. Data in (a) and (b) are simulated for illustration purpose, whereas real data are displayed in (d), (e), (f), (g).
• Figure 4: Quantum position-spectrum asymmetry.(a) Position‐noise spectrum of a harmonic oscillator at thermal equilibrium, showing absorption (blue peak at $-\omega_m$, area $\propto \bar{n}_{\rm th}$) and emission (red peak at $+\omega_m$, area $\propto \bar{n}_{\rm th}+1$). (b) Experimental pulse sequence: to eliminate the average dynamical backaction, we alternately initialize the qubit in the ground state $|g\rangle$ and the excited state $|e\rangle$. From the resulting measurement records, we then reconstruct the emission spectrum $S_{xx}^g$ (when starting in $|g\rangle$) and the absorption spectrum $S_{xx}^e$ (when starting in $|e\rangle$). (c) Reconstructed spectra for $|g\rangle$ (blue) and $|e\rangle$ (red) preparations at cryostat temperatures of 10, 15, 20, 25, and 30 mK, as illustrated by the thermometer. (d) Lorentzian‑peak areas at $\omega_m$ versus cryostat temperature: blue and red symbols show the peak areas in $S_{xx}^g$ and $S_{xx}^e$ respectively. A linear fit to the mean-area of the 5 temperature series is shown (dashed line), and the right‑hand axis converts the fitted peak area into mean phonon number using this calibration. (e) Measured phonon‐number difference between $|e\rangle$ and $|g\rangle$ spectra versus cryostat temperature. The horizontal dashed line at 1.37 quanta is the theoretical prediction, which differs from 1 due to qubit preparation infidelity and finite lifetime. All points include error bars representing the standard deviation from bootstrap fits to random FFT‐batch sub‐ensembles (Sec. S4.6 of Supplementary Information).
• Figure 5: CSL exclusion graph. In the ($r_C$, $\lambda$) plane, filled areas denote regions excluded by non-interferometric tests, while hatched areas denote interferometric exclusions. Filled: the green area derives from spontaneous X-ray emission tests Arnquist2022; the blue area from the translation-noise spectrum of LISA Pathfinder Altamura2025; and the red area from optomechanical measurements of Brownian-noise spectra in cantilevers Vinante2021. Hatched: The cyan region corresponds to atom interferometers (an atomic cloud prepared in a metre-scale spatial superposition) Kovachy2015, the blue region to molecular interferometry (near-field Talbot–Lau with $\sim10^{4}\,\mathrm{amu}$ molecules) Eibenberger2013, and the blue region to a GHz cQAD experiment using a high-overtone bulk acoustic wave resonator prepared in a Schrödinger-cat state Bild2023. The gray band indicates a macro-objectification prior Toros2017, i.e. a phenomenological requirement that macroscopic superpositions be suppressed on human timescales (not an experimental bound). The central white (rhombus-shaped) region remains unconstrained by current measurements, interferometric or otherwise. The black line shows the expected reach of the current 4 MHz device. Colored lines indicate the projected reach for soft-clamped phononic-crystal membranes at 4 MHz, 1 MHz, 250 kHz, and 62 kHz, assuming the empirical $Q$-scaling Tsaturyan2017 $Q \simeq 5\times10^{7}\,(2\pi\!\cdot\!4~\mathrm{MHz}/\omega_m)^{2}$. All curves mark the locus where $\Gamma_{\rm CSL}=0.10\,\Gamma_{\rm QM}$ (see main text for modeling details).