Echoed Random Quantum Metrology

Abstract

Quantum metrology typically demands the preparation of exotic quantum probe states, such as entangled or squeezed states, to surpass classical limits. However, the need for carefully calibrated system parameters and finely optimized quantum controls imposes limitations on scalability and robustness. Here, we circumvent these limitations by introducing an echoed random process that achieves sensitivity approaching the Heisenberg limit while remaining blind to the random probe state. We demonstrate that by simply driving a Kerr nonlinear mode with random pulses, the emergence of sub-Planck phase-space structures grants high sensitivity, eliminating the need for complex quantum control. The protocol is statistically robust, yielding high performance across broad driving parameter ranges while exhibiting resilience to control fluctuations and photon loss. Broadly applicable to both bosonic and qubit platforms, our work reveals a practical, hardware-efficient, scalable, and optimization-free route to quantum-enhanced metrology in high-dimensional Hilbert spaces.

Figures (3)

• Figure 1: (a) Schematic illustration of the phase estimation procedure using the echoed random quantum metrology protocol. Probe states are prepared via dynamics $\mathcal{E}_{\hat{H}}$, driven by randomly modulated coherent pulses. A time-reversed evolution $\mathcal{E}_{-\hat{H}}$ is applied before measurement, effectively realizing a non-Gaussian measurement that accesses information stored in higher-order moments. (b) Evolution of the Wigner function, starting from the vacuum state, under the forward random channel, probing, and the corresponding time-reversed channel. The phase $\theta$ induces a rotation in phase space, and the fine structures of the distributions make the states highly sensitive to these rotations. As the magnified insets demonstrate, even a small rotation in phase space induces a sign flip in the Wigner function.
• Figure 2: (a) Probability $p_0(\theta_0)$ versus the bias point $\theta_0$ for a randomly generated probe state with $\epsilon/\chi = 100, \chi T = 2$. Inset: corresponding Wigner function with average photon number $\ev{\hat{n}}\approx 27$. (b) Corresponding metrological gain $G_\mathrm{c}(\theta_0)$ of (a), where the black dashed line marks the standard quantum limit (SQL), $G_\mathrm{SQL}=1$, and the red dashed line indicates the quantum Fisher information bound. (c) $G_\mathrm{c}(\theta_0)$ for $\epsilon/\chi = 100$ (light green) and $\epsilon/\chi = 200$ (pink), both at $\chi T = 2$. Shaded areas denote standard deviations over the realizations. Inset: corresponding histograms of $G_\mathrm{c,max}$ at the optimal bias point, overlaid with Gaussian fits. (d) Scatter plot of classical Fisher information (CFI) $I_\mathrm{c,max}$ versus $\ev{\hat{n}}$ at the optimal bias point for $\chi T=2$. The black dashed line marks the SQL $I_\mathrm{SQL}=4\ev{\hat{n}}$ while the red dashed curve shows a power-law fit, $I_\mathrm{c,max}=2.17\times\ev{\hat{n}}^{1.95}$, approaching the Heisenberg scaling $I_\mathrm{HL}\propto\ev{\hat{n}}^2$. Inset: corresponding optimal bias point $\theta_\mathrm{b}$ versus $\ev{\hat{n}}$, with weighted power-law fit: $\theta_\mathrm{b}=0.23\times \ev{\hat{n}}^{-0.89}$. (e) The mean of $G_\mathrm{c,max}$ over 1000 realizations, denoted as $\overline{G_\mathrm{c,max}}$. The light green and pink markers indicate the parameter sets $(\epsilon/\chi=100, \chi T=2)$ and $(\epsilon/\chi=200, \chi T=2)$ used in (c). (f) The standard deviation of $G_\mathrm{c,max}$ over 1000 realizations, denoted as $\sigma_{G_\mathrm{c,max}}$. The system exhibits a transient regime at early times, entering a stabilization phase for $\chi T \gtrsim 1.5$ where the variance remains low. Contour lines (black) are included in (e–f). Other parameters: $\tau = 0.1/\chi$, $d = 1050$, and $\epsilon_\mathrm{dp} = 10^{-3}$.
• Figure 3: Robustness to control fluctuations and photon loss. (a) Metrological gain $G_{\mathrm{c,max}}$ versus relative fluctuation strength $\Delta\epsilon/\epsilon$ for random single-photon-driven (blue, $\ev{\hat{n}}=34.8$) and two-photon-driven (orange, $\ev{\hat{n}}=3.4$) probes. Parameters are $(\epsilon/\chi, \chi T) = (40, 1.5)$ and $(6, 0.8)$, respectively, with step size $\tau=0.02/\chi$ for both. Shaded regions indicate standard deviations over 1000 random realizations of fluctuations $\delta u_{1,2}(t)$. (b) Scatter plot of CFI $I_{\mathrm{c,max}}$ versus mean photon number $\ev{\hat{n}}$ for single-photon drive under different photon-loss rates $\kappa$. The data are shown for $\chi T=1.5$ and various driving strengths $\epsilon/\chi \le 100$. Dashed curves are power-law fits to the data; the black dashed line marks the SQL $I_\mathrm{SQL}=4\ev{\hat{n}}$. Inset: power-law exponent $b$ from $I_\mathrm{c,max}=a\ev{\hat{n}}^b$ as a function of $\kappa$. (c) Scatter plot of $I_\mathrm{c,max}$ versus $\ev{\hat{n}}$ for two-photon drive under different photon-loss rates $\kappa$. The data are shown for $\chi T=0.8$ and various driving strengths $\epsilon/\chi \le 20$. Dashed curves are power-law fits to the data; the black dashed line marks the SQL $I_\mathrm{SQL}=4\ev{\hat{n}}$. Inset: power-law exponent $b$ from $I_\mathrm{c,max}=a\ev{\hat{n}}^b$ as a function of $\kappa$. Other parameters: $\tau=0.1/\chi$, $\epsilon_\mathrm{dp}=10^{-3}$.