Chip scale superconducting quantum gravimeter based on a SQUID transmon mechanical resonator

TL;DR

The paper introduces a chip-scale superconducting gravimeter that couples a flux-tunable transmon to a nanomechanical resonator embedded in a SQUID loop, enabling a longitudinal qubit–mechanics interaction in which gravity imprints a geometric phase on the qubit. It develops a closed-form, polaron-based evolution and a quantum Fisher information framework to quantify gravity-estimation precision, showing that operating at mechanical revival times enhances sensitivity while mitigating which-path dephasing. The analysis reveals two practical regimes—a near-term, lithographic device and a heavier, lower-frequency design—both achieving competitive or surpassing granular performance with kilohertz sampling and SI traceability via microwave spectroscopy. The work provides concrete design rules (e.g., larger $k=g_0/\omega_m$ improves QFI but increases decoherence) and highlights the potential for multiplexed, on-chip gravimeters for geoscience, navigation, and tests of fundamental physics, with performance approaching that of atomic sensors but in a chip-scale platform.

Abstract

Precise gravitational measurements are vital for geophysics and inertial navigation, but current platforms struggle to combine absolute accuracy with high-bandwidth tracking. We address this challenge with a chip-scale superconducting gravimeter that couples a flux-tunable transmon qubit to a high-$Q$ mechanical resonator. We embed the mechanical element inside the qubit's SQUID loop. This allows us to exploit the Josephson potential's nonlinearity, creating a motion-dependent inductance that maps gravitational displacement onto the qubit's geometric phase. Using a stroboscopic measurement protocol, we suppress mechanical decoherence at revival times. This yields a predicted sensitivity of $10^2\,\mathrm{nGal}/\sqrt{\mathrm{Hz}}$, approaching the performance of atomic sensors but with kilohertz-rate sampling. With electrical {in situ} tunability and SI traceability via microwave spectroscopy, this architecture offers a practical route to high-speed, quantum-limited on-chip gravimetry.

Figures (5)

• Figure 1: Schematic of the hybrid SQUID--transmon gravimeter. A nanomechanical oscillator (vertical displacement $z$) is embedded in a dc-SQUID loop. Its motion modulates the magnetic flux $\Phi$, tuning the transmon transition frequency (red crosses: Josephson junctions; $C$: shunt capacitor).
• Figure 2: Operational principle of the hybrid SQUID--transmon gravimeter. Gravitational acceleration $g$ deflects the beam, shifting the qubit frequency via a longitudinal interaction. A stroboscopic readout sequence encodes this shift into a geometric phase to measure $g$.
• Figure 3: Quantum Fisher information (QFI) and linear entropy as functions of mechanical cycles $\tau/\pi$. (a) Ideal vacuum QFI ($\alpha = 0$) vs cycles for different longitudinal couplings $k \in \{0.10, 0.20, 0.30\}$. (b) QFI vs cycles over the first mechanical cycle for different coherent drive amplitudes $|\alpha| \in \{1, 5, 10, 50\}$ at fixed coupling $k = 0.20$. (c) Linear entropy $S_L$ over the first mechanical cycle for different couplings $k \in \{0.10, 0.20, 0.30\}$ at qubit angle $\theta = \pi/2$.
• Figure 4: Metrological performance comparison between device configurations. The left column (a)--(c) displays the results for Scenario I (Near-term Device), while the right column (d)--(f) displays Scenario II (High-mass Device). (a), (d) Evolution of the Quantum Fisher Information ($\mathcal{F}_Q$) over a large number of mechanical cycles. (b), (e) Detailed view of $\mathcal{F}_Q$ in the short-time regime. The insets illustrate the oscillatory structure resulting from the stroboscopic protocol. (c), (f) The corresponding estimation sensitivity, $\eta$, as a function of interaction time. The blue curves represent the ideal unitary dynamics, showing continuous improvement, whereas the red curves incorporate the effects of environmental decoherence, which leads to a loss of sensitivity at longer times.
• Figure 5: Comparison of longitudinal coupling strengths $k = 0.1$ and $k = 0.3$ for both scenarios (note that $k = 0.2$ is analyzed in detail in the main Scenario I/II results). The panels show the quantum Fisher information (QFI) as a function of mechanical cycles $\tau/\pi$ for different $k$. The "ideal" curves correspond to QFI in the absence of decoherence, while the "decohered" curves include both intrinsic qubit relaxation/dephasing and interaction-induced mechanical noise. Insets show the visibility, defined as the residual qubit–mechanical coherence after intrinsic decay and polaron-induced dephasing. Increasing $k$ raises the ideal QFI but also amplifies decoherence, accelerating visibility loss and suppressing the amplitude of QFI revivals. An intermediate coupling $k \simeq 0.2$ optimally balances these competing effects in both scenarios.