Quantum-enhanced multiparameter sensing in a single mode

Sci. Adv. 11, eadw9757 (2025) · DOI: 10.1126/sciadv.adw9757 · arXiv: 2412.04865

License: CC BY-NC 4.0.

Abstract

Precise measurements underpin scientific and technological advancements. Quantum mechanics provides an avenue to enhance precision, but it comes with a restriction: Incompatible observables, such as position and momentum, cannot be simultaneously measured to arbitrary accuracy as decreed by Heisenberg’s uncertainty principle. This restriction can be bypassed by instead measuring commuting modular observables, which are counterparts to the naturally incompatible observables. Here, we measure modular observables to estimate small changes in position and momentum with a single-mode multiparameter sensor. We deterministically prepare grid states in the mechanical motion of a trapped ion and demonstrate uncertainties in position and momentum below the standard quantum limit (SQL). Further, we examine another pair of incompatible observables—number and phase. We prepare a different resource—number-phase states—and demonstrate a metrological gain over the SQL. These results introduce previously unidentified measurement capabilities unavailable to classical systems and mark a substantial step in quantum metrology.

Figures

13 panels with data across 3 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

Interactive data forthcoming. The figure is shown in the paper PDF for now.

Fig. 2

• panel (a-1): Experimentally reconstructed characteristic function $\mathrm{Re}[\chi(\beta)]$ of the grid state with target $\Delta = 0.61$, against $\mathrm{Re}[\beta]/\sqrt{\pi}$ and $\mathrm{Im}[\beta]/\sqrt{\pi}$, on a 49 by 49 grid. Map (i) of panel (a). data.csv
• panel (a-2): Experimentally reconstructed characteristic function $\mathrm{Re}[\chi(\beta)]$ of the grid state with target $\Delta = 0.37$, against $\mathrm{Re}[\beta]/\sqrt{\pi}$ and $\mathrm{Im}[\beta]/\sqrt{\pi}$, on a 49 by 49 grid. Map (ii) of panel (a). data.csv
• panel (a-3): Experimentally reconstructed characteristic function $\mathrm{Re}[\chi(\beta)]$ of the grid state with target $\Delta = 0.30$, against $\mathrm{Re}[\beta]/\sqrt{\pi}$ and $\mathrm{Im}[\beta]/\sqrt{\pi}$, on a 49 by 49 grid. Map (iii) of panel (a), with crosses at the points that give the visibilities $\eta_x$ (at $i\sqrt{\pi}$) and $\eta_p$ (at $-\sqrt{\pi}$). data.csv
• panel (b): Multiparameter variance $V(\epsilon_x) + V(\epsilon_p)$ of grid states against mean phonon number $\langle \hat{n} \rangle$ (log-log): measured variances (circles) with one-SD error bars and shaded band, the expected grid-state variance (red dashed), the squeezed state (purple dashed), SQL$^*$ (black dashed) and LB$^*$ (black). The grey shading below LB$^*$ and the (i)-(iii) arrows are not drawn. data.csv

Fig. 3

• panel (a): Multiparameter Holevo variance of a grid state against measurement repetitions $M$ (log-log) for nonadaptive QPE (circles, one-SD error bars and shaded band) and adaptive QPE (star), with SQL$^*$ (dashed). The inset is panel a-inset; the shading below SQL$^*$ and the zoom box are not drawn. data.csv
• panel (a-inset): Inset of panel (a): individual Holevo variances $V_H(\tilde{\epsilon}_x)$ (blue) and $V_H(\tilde{\epsilon}_p)$ (purple) against measurement repetitions $M$ (log-log), with one-SD error bars and shaded bands, and the dashed SQL$^*$ line, on the axes of the main panel. data.csv
• panel (b): Zoom of panel (a) at $M = 128$: multiparameter Holevo variance for nonadaptive and adaptive QPE as bars with one-SD error bars, against the dashed SQL$^*$ line (log axis). The circle and star at the bar tops are not drawn. data.csv

Fig. 4

• panel (a): Theoretical Fock-state weights of the NP state with spacing $N = 4$, cutoff $F = 18$ and offset 2 (bars), with the sine envelope that damps them (grey dotted), against Fock state $n$. The arrows marking $\epsilon_n$ and $l_\phi$ are not drawn. data.csv
• panel (b): Theoretical characteristic function $\mathrm{Re}[\chi(\beta)]$ of the NP state of panel (a) against $\mathrm{Re}[\beta]$ and $\mathrm{Im}[\beta]$, on a 51 by 51 grid. The arrows marking $S_n$, $l_n$ and $\epsilon_\phi$ are not drawn. data.csv
• panel (c): Experimentally reconstructed characteristic function $\mathrm{Re}[\chi(\beta)]$ of the NP state of panel (a) against $\mathrm{Re}[\beta]$ and $\mathrm{Im}[\beta]$, on a 49 by 49 grid. data.csv
• panel (d): Combined number and phase variance $V(\epsilon_\phi) \times 2\langle n \rangle + V(\epsilon_n)/2\langle n \rangle$ of NP states against average phonon number $\langle n \rangle$ (log-log): squares with one-SD error bars and shaded band, the expected variance of the target NP state (red dashed) and SQL$^*$ (black dashed). Insets are d-inset-1 and d-inset-2. data.csv
• panel (d-inset-1): Inset (i) of panel (d): phase variance $V(\epsilon_\phi)$ against average phonon number $\langle n \rangle$ (log-log), squares with one-SD error bars and shaded band, and SQL$_\phi$ (black dashed). data.csv
• panel (d-inset-2): Inset (ii) of panel (d): number variance $V(\epsilon_n)$ against average phonon number $\langle n \rangle$ (log-log), squares with one-SD error bars and shaded band, and SQL$_n$ (black dashed). data.csv

Cite

Christophe H. Valahu, Matthew P. Stafford, Zixin Huang, Vassili G. Matsos, Maverick J. Millican, Teerawat Chalermpusitarak, Nicolas C. Menicucci, Joshua Combes, Ben Q. Baragiola, Ting Rei Tan. Quantum-enhanced multiparameter sensing in a single mode. Sci. Adv. 11, eadw9757 (2025). https://doi.org/10.1126/sciadv.adw9757

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