Realization of Versatile and Effective Quantum Metrology Using a Single Bosonic Mode

PRX Quantum 6, 010304 (2025) · DOI: 10.1103/prxquantum.6.010304

License: CC BY 4.0.

Abstract

Quantum metrology offers the potential to surpass its classical counterpart, pushing the boundaries of measurement precision toward the ultimate Heisenberg limit. This enhanced precision is normally attained by utilizing large squeezed states or multiparticle entangled quantum states, both of which are often challenging to implement and prone to decoherence in real quantum devices. In this work, we present a versatile and on-demand protocol for deterministic parameter estimation that leverages two efficient state-transfer operations on a single bosonic mode. Specifically, we demonstrate this protocol in the context of phase estimation using the superposition of coherent states in the bosonic circuit quantum electrodynamics (cQED) platform. With low average photon numbers of only up to 1.76, we achieve quantum enhanced precision approaching the Heisenberg scaling, reaching a metrological gain of 7.5(6) dB. Importantly, we show that the gain or sensitivity range can be further enhanced on the fly by tailoring the input states, with different superposition weights, based on specific system constraints. The realization of this versatile and efficient scheme affords a promising path toward practical quantum enhanced sensing, not only for bosonic cQED hardware but also readily extensible to other continuous-variable platforms.

Figures

35 panels with data across 12 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

Illustrative figure, no extractable data. Shown in the paper PDF.

Fig. 2

• panel (c): Fisher information FI against $\theta$ (rad) for the SCS (solid blue) and the CS (solid purple), calculated from the polynomial fits of panel c-inset, with a dashed simulation curve for each. The shaded bootstrap bands and the gray dynamical-range region of the printed panel are not drawn. data.csv
• panel (c-inset): Inset of printed panel (c): the measured $p_{g0}(\theta)$ against $\theta$ (rad) for the SCS (blue circles) and the CS (purple diamonds), with error bars, and the dotted polynomial fit to each. The dashed line joining the printed markers is not drawn. data.csv
• panel (d): FI$_{\mathrm{max}}$ against average photon number $\bar{n}$ for the SCS (blue circles) and the CS (purple diamonds), with error bars, the standard deviation from bootstrapping. Dashed curves are the simulations, the dotted curve is the fit to the SCS and the pink curve is the ideal QFI of the SCS. The caption's dotted fit for the CS is not drawn. data.csv

Fig. 3

• panel (a): Average FI $\widetilde{\mathrm{FI}}$ of the SCS within the dynamical range (blue squares, left axis) and the $\theta$ offset (rad) at which the FI is maximum (purple triangles, right axis), against $\bar{n}$, with error bars. The dashed curves are simulations and the dotted curve is the fit to $\widetilde{\mathrm{FI}}$. data.csv
• panel (b): FI against $\theta$ (rad) for initial states prepared with three offsets in $\theta$. The hosted curves are named offset 1 (teal), 2 (blue) and 3 (brown); the printed legend gives these offsets as 0.25, 0.00 and $-0.20$. The shaded band of the printed panel is not drawn. data.csv
• panel (c): Phase precision $\Delta\theta$ against $\bar{n}$ on log-log axes for the SCS (blue circles) and the CS (purple diamonds), with error bars; the dotted lines are linear fits. The two solid Heisenberg-scaling lines of the printed panel, $1/2\bar{n}$ and $1/\bar{n}$, and its 7.5 dB marker are not drawn. data.csv

Fig. 4

• panel (b): FI against $\theta$ (rad) for the SCS$_w(\alpha)$ with three weights $w$ at the same average photon number $\bar{n} = 1$: SCS$_{0.3}$(1.95), SCS$_{0.5}$(1.60) and SCS$_{0.7}$(1.39), the legend giving $w$ and, in brackets, $\alpha$. The shaded bootstrap bands of the printed panel are not drawn. data.csv
• panel (c): FI$_{\mathrm{max}}$ of the SCS$_w(\alpha)$ (blue, left axis, with error bars) and its dynamic range (rad) (purple, right axis) against the weight $w$, at $\bar{n} = 1$. Markers are experimental data and dashed curves are simulations. data.csv

Fig. 5

Illustrative figure, no extractable data. Shown in the paper PDF.

Fig. 6

• panel (1): FI against $\theta$ (rad) for the SCS, squeezed vacuum (Sqvac), the Fock superposition $(|0\rangle + |1\rangle)/\sqrt{2}$ and the CS, all at a fixed average photon number $\bar{n} = 0.5$. The shaded standard-deviation bands of the printed panel are not drawn. data.csv

Fig. 7

• panel (1): FI$_{\mathrm{max}}$ from full-Hamiltonian simulation against $\bar{n}$ for the SCS (solid curves), for no decoherence and self-Kerr, cavity decoherence, self-Kerr, qubit decoherence and all imperfections together, with the theoretical QFI in pink. The two dashed curves are the squeezed-vacuum QFI and its all-imperfections result. data.csv

Fig. 8

• panel (b): FI of the SCS against $\bar{n}$: the theoretical FI at $\theta \to 0$, equal to the QFI (pink), the theoretical FI at the maximum probability slope (red), the simulation (blue dashed) and the experimental values (blue circles, with error bars). data.csv

Fig. 9

• panel (a): FI for amplitude estimation against $\gamma$ at $\bar{n} = 1.76$ for the CS (solid purple), the SCS-v, displaced in the vertical direction (solid blue), and the SCS-h, displaced in the horizontal direction (solid gray), each with a dashed simulation curve. The shaded dynamical range of the printed panel is not drawn. data.csv
• panel (b): FI$_{\mathrm{max}}$ for amplitude estimation against $\bar{n}$ for the SCS-v (blue circles) and the CS (purple diamonds), with error bars. Dashed curves are the simulations, dotted curves the fits, and the solid pink curve is the ideal QFI of the SCS-v. data.csv

Fig. 10

• panel (a): Average FI $\widetilde{\mathrm{FI}}$ of the SCS where it is above the maximum FI of the CS (blue squares, left axis) and the $\gamma$ offset at which the FI is maximum (purple triangles, right axis), against $\bar{n}$, with error bars; dashed curves are simulations. The dotted fit curve of the printed panel is not drawn. data.csv
• panel (b): FI for amplitude estimation against $\gamma$ for initial states prepared with an offset of 0.0, $-0.4$ and 0.5, as the printed legend labels the three curves. data.csv
• panel (c): Amplitude precision $\Delta\gamma$ against $\bar{n}$ on log-log axes for the SCS (blue circles) and the CS (purple diamonds), with error bars; the dotted lines are linear fits of $\log(\Delta\gamma)$ against $\log(\bar{n})$. The 9.3 dB marker of the printed panel is not drawn. data.csv

Fig. 11

• panel (a): Metrological gain (dB) of the SCS over the CS for phase estimation against the number of measurement repetitions, star markers with error bars. data.csv
• panel (b): Metrological gain (dB) of the SCS over the CS for amplitude estimation against the number of measurement repetitions, star markers with error bars. data.csv

Fig. 12

• panel (a): Phase estimated by Bayesian inference, $\theta_{\rm Bayesian}$, against the true phase $\theta_{\rm True}$ for the SCS with $\alpha = 0.5$. Error bars are the standard deviation of the posterior; the solid line is equality. data.csv
• panel (b): Phase estimated by Bayesian inference, $\theta_{\rm Bayesian}$, against the true phase $\theta_{\rm True}$ for the SCS with $\alpha = 0.75$. Error bars are the standard deviation of the posterior; the solid line is equality. data.csv
• panel (c): Phase estimated by Bayesian inference, $\theta_{\rm Bayesian}$, against the true phase $\theta_{\rm True}$ for the SCS with $\alpha = 1.0$. Error bars are the standard deviation of the posterior; the solid line is equality. data.csv
• panel (d): Phase estimated by Bayesian inference, $\theta_{\rm Bayesian}$, against the true phase $\theta_{\rm True}$ for the SCS with $\alpha = 1.25$. Error bars are the standard deviation of the posterior; the solid line is equality. Two of the estimates carry no error bar. data.csv
• panel (e): Phase estimated by Bayesian inference, $\theta_{\rm Bayesian}$, against the true phase $\theta_{\rm True}$ for the SCS with $\alpha = 1.5$. Error bars are the standard deviation of the posterior; the solid line is equality. data.csv
• panel (f): Phase estimated by Bayesian inference, $\theta_{\rm Bayesian}$, against the true phase $\theta_{\rm True}$ for the SCS with $\alpha = 1.6$; the printed caption labels this panel $\alpha = 0.6$. Error bars are the standard deviation of the posterior; the solid line is equality. Seven of the estimates carry no error bar. data.csv
• panel (g): Phase estimated by Bayesian inference, $\theta_{\rm Bayesian}$, against the true phase $\theta_{\rm True}$ for the SCS with $\alpha = 1.75$. Error bars are the standard deviation of the posterior; the solid line is equality. Three of the estimates carry no error bar. data.csv
• panel (h): Phase estimated by Bayesian inference, $\theta_{\rm Bayesian}$, against the true phase $\theta_{\rm True}$ for the SCS with $\alpha = 2.0$. Error bars are the standard deviation of the posterior; the solid line is equality. 23 of the estimates carry no error bar. data.csv

Fig. 13

• panel (a): Uncertainty $\Delta\theta$ in the estimated phase against $\theta$ for the SCS with $\alpha = 0.5$, as predicted by the Cramer-Rao bound (blue) and by Bayesian inference (red). data.csv
• panel (b): Uncertainty $\Delta\theta$ in the estimated phase against $\theta$ for the SCS with $\alpha = 0.75$, as predicted by the Cramer-Rao bound (blue) and by Bayesian inference (red). data.csv
• panel (c): Uncertainty $\Delta\theta$ in the estimated phase against $\theta$ for the SCS with $\alpha = 1.0$, as predicted by the Cramer-Rao bound (blue) and by Bayesian inference (red). data.csv
• panel (d): Uncertainty $\Delta\theta$ in the estimated phase against $\theta$ for the SCS with $\alpha = 1.25$, as predicted by the Cramer-Rao bound (blue) and by Bayesian inference (red). The Bayesian curve leaves out two phases. data.csv
• panel (e): Uncertainty $\Delta\theta$ in the estimated phase against $\theta$ for the SCS with $\alpha = 1.5$, as predicted by the Cramer-Rao bound (blue) and by Bayesian inference (red). data.csv
• panel (f): Uncertainty $\Delta\theta$ in the estimated phase against $\theta$ for the SCS with $\alpha = 1.6$; the printed caption labels this panel $\alpha = 0.6$, as predicted by the Cramer-Rao bound (blue) and by Bayesian inference (red). The Bayesian curve leaves out seven phases. data.csv
• panel (g): Uncertainty $\Delta\theta$ in the estimated phase against $\theta$ for the SCS with $\alpha = 1.75$, as predicted by the Cramer-Rao bound (blue) and by Bayesian inference (red). The Bayesian curve leaves out three phases. data.csv
• panel (h): Uncertainty $\Delta\theta$ in the estimated phase against $\theta$ for the SCS with $\alpha = 2.0$, as predicted by the Cramer-Rao bound (blue) and by Bayesian inference (red). The Bayesian curve leaves out 23 phases. data.csv

Fig. 14

The available data cannot be matched to this figure unambiguously. The figure is shown in the paper PDF.

Fig. 15

• panel (a): Estimated interaction strength $\chi_{\mathrm{est}}/2\pi$ (MHz) against adaptive-estimation iteration for the SCS with $\alpha = 1$, SCS(1), and $\alpha = 2$, SCS(2), with error bars. The horizontal blue line is the experimental calibration value. data.csv

Cite

Xiaozhou Pan, Tanjung Krisnanda, Andrea Duina, Kimin Park, Pengtao Song, Clara Yun Fontaine, Adrian Copetudo, Radim Filip, Yvonne Y. Gao. Realization of Versatile and Effective Quantum Metrology Using a Single Bosonic Mode. PRX Quantum 6, 010304 (2025). https://doi.org/10.1103/prxquantum.6.010304

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