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arXiv:2606.21068

Finite-Time Electrometry with a Quantum-Regime Single-Ion Phonon Laser

Pei-Dong Li, Yuan-Zhang Dong, Zhuo-Zhu Wu, Jia-Wei Wang, Ji Li, Jian-Qi Zhang, Zhi-Jiao Deng, Liang Chen, Mang Feng

quant-phphysics.atom-phSubmitted Jun 19, 2026• Announced Jun 22, 2026• 0 citations
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Abstract

The phonon laser realized in a trapped ion, i.e., a self-sustained mechanical oscillator, has demonstrated the unique characteristics in practically detecting externally applied electric signals without the prerequisite of sideband cooling. Entering the quantum regime via sideband cooling is expected to further improve its sensing performance. Here we report the first experimental realization of a quantum-regime single-ion phonon laser ($\bar{n}<10$) using a trapped $^{40}\mathrm{Ca}^+$ ion and demonstrate electrometry based on its phase-space symmetry-breaking response to weak resonant electric fields. By tuning the phonon-laser parameters, we reveal that the sensing performance is fundamentally governed by the finite-time relaxation dynamics of the underlying open quantum system. We find that a slow Liouvillian relaxation, correlated with the finite experimental interaction window, effectively enhances the dynamic susceptibility while maintaining the structural robustness of the limit cycle. This regime, when applied to the detection of electric fields, produces a shot-noise-limited peak sensitivity of $14.15 \pm 0.77~μ\mathrm{V/m}/\sqrt{\mathrm{Hz}}$ and a minimum detectable field variation of $δE_{\mathrm{min}} \approx 1.83~μ\mathrm{V/m}$. Our results establish quantum phonon lasers as a practical platform for advanced sensing and highlight the central role of Liouvillian dynamics in non-equilibrium electrometry.

Comments
6 pages, 4 figures; plus Supplementary Materials

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The phonon laser realized in a trapped ion, i.e., a self-sustained mechanical oscillator, has demonstrated the unique characteristics in practically detecting externally applied electric signals without the prerequisite of sideband cooling. Entering the quantum regime via sideband cooling is expected to further improve its sensing performance. Here we report the first experimental realization of a quantum-regime single-ion phonon laser (nˉ<10\bar{n}<10nˉ<10) using a trapped 40Ca+^{40}\mathrm{Ca}^+40Ca+ ion and demonstrate electrometry based on its phase-space symmetry-breaking response to weak resonant electric fields. By tuning the phonon-laser parameters, we reveal that the sensing performance is fundamentally governed by the finite-time relaxation dynamics of the underlying open quantum system. We find that a slow Liouvillian relaxation, correlated with the finite experimental interaction window, effectively enhances the dynamic susceptibility while maintaining the structural robustness of the limit cycle. This regime, when applied to the detection of electric fields, produces a shot-noise-limited peak sensitivity of 14.15±0.77 μV/m/Hz14.15 \pm 0.77~μ\mathrm{V/m}/\sqrt{\mathrm{Hz}}14.15±0.77 μV/m/Hz​ and a minimum detectable field variation of δEmin≈1.83 μV/mδE_{\mathrm{min}} \approx 1.83~μ\mathrm{V/m}δEmin​≈1.83 μV/m. Our results establish quantum phonon lasers as a practical platform for advanced sensing and highlight the central role of Liouvillian dynamics in non-equilibrium electrometry.
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The characterization of externally applied signals is essential in order to improve performance and design appropriate engineering strategies in cold-atom experiments. Here, we present a practical technique to detect the artificially introduced electric signals by an exquisite sensor made of a single 40{}^{40}40Ca\mathrm{Ca}Ca+{}^{+}+ ion behaving as the phonon laser. The sensor can work as an oscilloscope, detecting spectra of the signals in the time domain, and also as a frequency spectrograph for the signal probe with respect to the frequency. The high precision of such an atomic sensor relies on the unique characteristics of the phonon laser. In particular, the sensor is practical since it works without the prerequisite of sideband cooling and the performance of the sensor can be further improved depending on better trapping conditions and a higher collection efficiency of the spontaneous emission fluorescence.
In this article we present a comprehensive theoretical investigation of phonon lasing with mixed-species trapped ions, as demonstrated in [T. Behrle, Phys. Rev. Lett. 131 (2023)], employing both a semi-classical mean-field description and a full quantum theory. We derive an analytic expression for the second-order coherence function, confirming the experimental observation of the system's lasing behaviour above threshold. Building on the successful implementation of the two-ion lasing scheme, we propose a novel approach for achieving phonon lasing with a single trapped ion, offering significant experimental advantages and making the implementation of multiple phonon lasers within a single setup feasible. Furthermore, we explore lasing in a squeezed basis and in different regimes of the Lamb-Dicke approximation, highlighting the potential to produce non-classical states with promising applications in precision sensing. Our analysis of a sensing protocol based on squeezed states, using experimentally feasible parameters, shows a sensitivity enhancement of up to two orders of magnitude.