Credible-interval-based adaptive Bayesian quantum frequency estimation for entanglement-enhanced atomic clocks

TL;DR

Entanglement-enhanced sensors face a fundamental trade-off between precision and dynamic range. The authors introduce a credible-interval-based adaptive Bayesian estimation (CI-adaptive) for GHZ-state atomic clocks, where interrogation times are chosen so that the next likelihood matches the current posterior credible interval, enabling dual Heisenberg-limited scaling $\Delta f_{est} \propto 1/(N t)$ before saturation. By combining either individual GHZ states with variable interrogation times or cascaded GHZ ensembles, the approach extends dynamic range by up to a factor of $T_{\max}/T_{\min}$ (and more with size-varied cascades) while maintaining Heisenberg-like precision, and remains robust to dephasing and detection noise. The method outperforms Fourier-coefficients-based adaptive protocols under fixed resources, provides a real-time posterior-based feedback loop, and offers a general framework for entanglement-enhanced quantum sensing beyond optical clocks.

Abstract

Entanglement-enhanced quantum sensors encounter a fundamental trade-off: while entanglement improves precision to the Heisenberg limit, it restricts dynamic range. To address this trade-off, we present a credible-interval-based adaptive Bayesian quantum frequency estimation protocol for Greenberger-Horne-Zeilinger (GHZ)-state-based atomic clocks. Our method optimally integrates prior knowledge with new measurements and determines the interrogation time by correlating it with the period of the likelihood function, based on Bayesian credible intervals. Our protocol can be implemented using either individual or cascaded GHZ states, thereby extending the dynamic range without compromising Heisenberg-limited sensitivity. In parallel with the cascaded-GHZ-state protocol using fixed interrogation times, the dynamic range can be extended through an interferometry sequence that employs individual GHZ states with variable interrogation times. Furthermore, by varying the interrogation times, the dynamic range of the cascaded-GHZ-state protocol can be further extended. Crucially, our protocol enables dual Heisenberg-limited precision scaling $\propto 1/(Nt)$ in both particle number $N$ and total interrogation time $t$, surpassing the hybrid scaling $\propto 1/{(N\sqrt {t}})$ of the conventional cascaded-GHZ-state protocol. While offering a wider dynamic range, the protocol is more stable against noise and more robust to dephasing than existing adaptive schemes. Beyond atomic clocks, our approach establishes a general framework for developing entanglement-enhanced quantum sensors that simultaneously achieve both high precision and broad dynamic range.

Figures (9)

• Figure 1: Schematics of GHZ-state-based Ramsey interferometry and credible-interval-based adaptive Bayesian estimation. (a) Ramsey interferometry with a GHZ state using parity measurement. (b) Ramsey interferometry with a GHZ state using one-axis twisting interaction-based readout. (c) A Bayesian estimation process demonstrating likelihood calculation based on measurement outcomes and subsequent posterior distribution updating. (d) Left: Implementation of the CI-adaptive protocol with $(K+1)$ ensembles of cascaded GHZ states. Each group (labeled by $k$) includes $M_k$ copies of $N_k$-particle GHZ states. The auxiliary phase $\theta$ is applied only to the group with $k=0$. The likelihood distribution $\tilde{\mathcal{L}}^{(j)} (f_c)$ is obtained via measurement and then is used to update the posterior distribution $\tilde{\mathcal{P}}^{(j)} (f_c)$. The current posterior distribution $\tilde{\mathcal{P}}^{(j)} (f_c)$ is then set as the next prior distribution $P^{(j+1)} (f_c)$ and gives the next interrogation time $T_{j+1}$. Right: Credible-interval update. The next interrogation time $T_{j+1}$ is determined by the credible interval (orange region) of the current posterior distribution (orange curve). This constrains the likelihood period (gray region) to match twice the credible interval width, expressed as $1/(N_0 T_{j+1}) = 2g \Delta f_{\text{est}}^{(j)}$.
• Figure 2: Performances of our credible-interval-based adaptive Bayesian protocol for entanglement-enhanced atomic clocks. (a) Root-mean-square error (RMSE) versus total interrogation time for cascading of two groups of GHZ states under different credible level $\zeta$. The corresponding $\alpha(\zeta)$ are $\alpha(99.999\%)=0.963$, $\alpha(99.99\%)=1.3237$, $\alpha(99.95\%)=1.6778$, and $\alpha(99.9\%)=1.86951$. The particle number and copy number are $(N_0=4,M_0=4)$ and $(N_1 = 4, M_1 = 5)$, respectively. The three colored regions indicate the Cramér-Rao lower bounds (CRLB) for: the frequentist scheme with fixed $T_{\text{min}}=0.75$ ms (gray, top region), the frequentist scheme with fixed $T_{\text{max}}=3$ ms (blue, bottom region), and the adaptive scheme for $T_{\text{min}} \rightarrow T_{\text{max}}$ (purple). The intermediate position of the purple region illustrates the transition from SQL to Heisenberg-limited scaling. The black dashed line corresponds to the theoretical dual Heisenberg scaling (Eq. 10). Inset: Bias versus iteration steps with different credible level $\zeta$. (b) RMSE versus initial detuning $\delta$ for frequentist schemes (fixed $T_{\text{min}}$: $40$ steps; fixed $T_{\text{max}}$: $10$ steps) and CI-adaptive scheme in (a) ($\zeta=99.999\%$, $13$ steps). Dashed line is the theoretical lower bound. (c) RMSE versus total interrogation time for cascaded GHZ states ($N_k = \{1,1,2,4\}$, $M_k = \{7,7,7,2\}$) under different credible level $\zeta$. The corresponding $\alpha(\zeta)$ are $\alpha(99.999\%)=2.64611$, $\alpha(99.995\%)=3.00035$, $\alpha(99.99\%)=3.1812$, and $\alpha(99.9\%)=3.97311$. Hybrid scaling (Heisenberg in $N$, SQL in $t$) emerges when $T_j$ saturates at $T_{\text{max}}$ (blue region), while adaptive protocol (purple) maintains dual scaling until saturation. (d) RMSE versus initial detuning $\delta$ for frequentist schemes (fixed $T_{\text{min}}$: $40$ steps; fixed $T_{\text{max}}$: $10$ steps) and CI-adaptive scheme in (c) ($\zeta=99.999\%$, $11$ steps). (e) Relative dynamic range (frequency range where RMSE $\leq$ 1.1$\times$CRLB) versus total interrogation time. (f) Overlapping Allan deviation $\sigma_y(\tau)$ versus averaging time $\tau$ showing clock stability. Error bars indicate $\pm$1 standard deviation from $1000$ simulations. Data in (a-e) are averaged with $R=5000$ simulations. Data in (f) are averaged with $R=1000$ simulations. The $^{88}$Sr clock transition $f_c \approx 4.295\times10^{14}$ Hz.
• Figure 3: Influence of dephasing on the fractional uncertainty. Simulations are performed with the CI-adaptive protocol under different maximum interrogation time $T_{\text{max}}$ for $M=9$ copies of GHZ states ($N=4$) with $T_{\text{min}}=3$ ms, $\alpha=1$, and $R=1024$. The dashed lines represent the theoretical CRLB as Eq. \ref{['eq:uncertainty_dephasing']}, the solid lines represent simulation RMSE during the adaptive process, and the marks (${\circ}$$: T_{\text{max}}=12$ ms, $\square: T_{\text{max}}=24$ ms, $\Diamond: T_{\text{max}}=40.875$ ms, $\triangle: T_{\text{max}}=48$ ms, $\triangledown: T_{\text{max}}=81.75$ ms, $\rhd: T_{\text{max}}=96$ ms, $\lhd: T_{\text{max}}=192$ ms) indicate the positions where the corresponding maximum interrogation time is reached.
• Figure 4: Comparisons between the CI-adaptive protocol and the FC-adaptive protocol. The CI-adaptive protocol ($M=9$ copies, $n=20$ steps) and the FC-adaptive protocol ($M=18$ copies, $n=10$ steps) are simulated with $R=1024$ times under the same total time resource $t_M=M\sum_{j=0}^n T_j$. Here, $T_{\text{min}}=3$ ms, $T_{\text{max}}=2^n T_{\text{min}}$. (a) The mean error of estimation versus the total time resource. (b) The variation of the averaged mean-square error in logarithmic scale versus the total time resource. (c) The optimal averaged mean-square error versus the detuning $\delta$ within the dynamic range.
• Figure S1: Averaged RMSE versus credible level and measurement copies. (a) Averaged RMSE versus credible level $\zeta$ when $M=9$. (b) Averaged RMSE versus measurement copy number $M$ with scaling factor $\alpha=1$. All data are averaged over the theoretical dynamic range and $R=5000$ simulations with identical GHZ states ($N=4$).