Practical robust Bayesian spin-squeezing-enhanced quantum sensing under noises

Abstract

Spin-squeezed states constitute a valuable entanglement resource capable of surpassing the standard quantum limit (SQL). However, spin-squeezed states only enable sub-SQL uncertainty within a narrow parametric window near some specific points. Identifying optimal measurement protocols for spin-squeezed states remains an outstanding challenge. Here we present an adaptive Bayesian quantum estimation protocol that achieves optimal measurement precision with spin-squeezed states under noises. Our protocol operates by maintaining measurements near the optimal point and employing Bayesian inference to sequentially perform phase estimation, enabling robust high-precision measurement. To account for realistic experimental conditions, we explicitly incorporate phase noises into the Bayesian likelihood function for more accurate estimation. Our protocol can be applied to various scenarios, such as quantum gravimeters and atomic clocks, achieving precision enhancement over conventional fitting methods under noises. Our approach offers superior precision and enhanced robustness against noises, making it highly promising for squeezing-enhanced quantum sensing.

Figures (10)

• Figure 1: (color online). (a) Schematic of adaptive Bayesian spin-squeezing-enhanced phase estimation. Given a prior distribution $p_{l-1}$ and the likelihood function, the posterior distribution $p_l$ is given by the Bayes' formula. The estimate $\phi_{est}^ {l}$ and its standard deviation $\sigma^{\phi}_l$ can be computed with $p_l$. Then, $\Phi_l$ is given to perform optimal measurement for each step. Generally, the initial prior $p_0$ can be chosen as an uniform distribution. (b) Phase precisions for different spin squeezing parameters $\xi$.
• Figure 2: (color online). Performances of Bayesian spin-squeezing-enhanced phase estimation under noises. The precision $\sigma^{\phi}$ versus the iteration times $M$ under (a) depolarization noise and (c) phase noise. The error $\phi_{est}-\phi$ versus $M$ under (b) depolarization noise and (d) phase noise. In our calculations, we choose $M$=50, $N$=200, and $100$ repetitions for each point. Under phase noise of strength $\sigma_{n} = 0.03$, the $5$-th to $7$-th likelihood functions are calculated using (e) the ideal uncertainty $\sigma_l^{m^z}$ and (f) the reshaped uncertainty $\bar{\sigma}_l^{{m}^z}$, and the corresponding $5$-th to $7$-th posterior distributions are shown in (g) and (h).
• Figure 3: (color online). Bayesian gravimetry with spin-squeezed states under noises. (a) The distributions of white, flicker and random walk noises over $60$ trials. (b) The precision $\Delta g_{est}$ versus the total interrogation time $\tilde{T}$ for different spin-squeezed states under noises. Here, $B$ and $F$ denote Bayesian and Frequentist protocols, respectively. We set $N = 6000$, $T_{max} = 455~\mathrm{\mu s}$ and $C = 0.98$ based on Ref. PhysRevX.15.011029.
• Figure 4: (color online). Bayesian clock locking with spin-coherent state (SCS) and spin-squeezed state (SSS) under noises. (a) Distributions of white noise $(\sigma_w)$, flicker noise $(\sigma_f)$ and random walk noise $(\sigma_r)$ over $400$ trials. (b) PSD of $S_y(f)\propto f^{-\alpha}$ exhibiting different slopes, where $\alpha = \{0, 1, 2\}$ correspond to {white, flicker, random walk} noises. Allan deviations under (c) white noise, (d) flicker noise, and (e) random walk noise. Here, $T_{max} = 141$ ms, $N = 30000$, $C = 0.91$, and $\xi^2 = -5.1$ dB are chosen based on Ref. yang2025clockprecisionstandardquantum.
• Figure 5: (color online) The variation of the spin squeezing parameter versus the rotation angle for different one-axis twisting time $t$. Here, the optimal twisting time $t_{opt}=3^{1/3}N^{-2/3}/\chi$.