Learning Unified Control of Intrinsic Nonlinear Spin Dynamics in Atomic Qudits for Magnetometry

C. Z. Cao; J. Z. Han; M. Xiong; M. Deng; L. Wang; X. Lv; M. Xue

quant-phMar 30, 2026

Learning Unified Control of Intrinsic Nonlinear Spin Dynamics in Atomic Qudits for Magnetometry

Authors: C. Z. Cao, J. Z. Han, M. Xiong, M. Deng, L. Wang, X. Lv, M. Xue

Organizations: College of Physics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China · China Mobile (Suzhou) Software Technology Co., Ltd., Suzhou 215163, China · Yichang Testing Technique R&D Institute, Specialized Weak Magnetic Metering Station of NDM, Yichang 443003, China · Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China ·

Abstract

Generating and preserving metrologically useful quantum states is a central challenge in quantum-enhanced metrology. In low-field atomic magnetometry with multilevel atoms, the nonlinear Zeeman (NLZ) effect is both a resource and a limitation. It can generate internal spin squeezing within a single atomic qudit, but under fixed readout it also rotates and distorts the measurement-relevant quadrature, limiting the usable metrological gain. The problem is further complicated by the time dependence of both the squeezing axis and the nonlinear evolution itself. Here we show that reinforcement learning can transform NLZ dynamics from a source of readout degradation into a sustained metrological resource. Using only experimentally accessible low-order spin moments, a trained agent identifies a unified control policy for this class of intrinsically nonlinear sensing dynamics. We illustrate the approach in the $f=21/2$ manifold of $^{161}\mathrm{Dy}$ , where the learned policy rapidly prepares strongly squeezed internal states and stabilizes more than $4\,\mathrm{dB}$ of fixed-axis spin squeezing under continuous NLZ evolution. Including state-preparation overhead, the learned protocol yields a single-atom magnetic-field sensitivity of $13.9\,\mathrm{pT}/\sqrt{\mathrm{Hz}}$ , approximately $3\,\mathrm{dB}$ beyond the standard quantum limit. Our results establish learning-based control as an experimentally feasible route for converting unavoidable intrinsic nonlinear dynamics in multilevel atomic sensors into operational metrological advantage.

Figures & tables

Figure 1: Reinforcement-learning framework for controlling intrinsic spin dynamics in an atomic qudit. (a) A single ${}^{161}\mathrm{Dy}$ atom in the $f=21/2$ hyperfine manifold forms a $(2f+1)$ -dimensional qudit. A magnetic field induces an effective quadratic Zeeman contribution, producing intrinsic nonlinear spin dynamics. (b) The qudit evolves under interleaved nonlinear evolution and transverse control rotations. The red shaded region on the Bloch sphere denotes a metrologically useful spin-squeezed state, while the gray deformed region denotes its distortion under continued nonlinear evolution. At each step, the agent receives a reduced observation $\mathbf{o}_{k}\in\mathcal{O}$ , selects an action $a_{k}\in\mathcal{A}$ , and maximizes the cumulative reward $\mathcal{G}$ to learn a unified policy for squeezing generation and stabilization.

Figure 2: Learned pulse protocol and squeezing dynamics. (a) Pulse sequence selected by the RL agent. Colored bars denote discrete transverse rotations applied at each control step. (b) Time evolution of the Wineland squeezing parameter $\xi^{2}(t)$ (red solid) and the fixed-axis squeezing parameter $\xi_{y}^{2}(t)$ (green solid) versus the dimensionless time $\chi t$ . Gray solid and dotted curves denote $\xi^{2}(t)$ of the QZE (OAT) evolution and the effective TACT benchmark, respectively. The vertical gray line near $\chi t\simeq 0.15$ marks the reward-defined switching time $t_{r}$ . The inset Wigner distributions show representative snapshots at selected times $t_{i}$ ; the colored shaded regions on each Bloch sphere represent the spin Wigner distributions of the atomic state at the labeled times.

Figure 3: Fidelity evidence for the stabilization mechanism. Fidelity evolution for representative reference states. Solid and dashed curves denote evolution under

\hat{f}_{y}^{2}

and

\hat{f}_{z}^{2}

, respectively. The reference states are the coherent spin state

\lvert f,m_{x}{=}f\rangle

, the RL-generated state after the

\hat{R}_{x}(\pi/3)

pulse in Fig. 2 , and the OAT and TACT states evaluated at their respective minima of

\xi^{2}

. For the OAT and TACT states, an additional

\hat{R}_{x}

rotation is applied so that

\xi_{y}^{2}=\xi^{2}

Figure 4: Metrological analysis of the unified RL control strategy. (a) Phase sensitivity relative to the SQL, expressed in dB, as a function of the dimensionless interrogation time $\chi T_{e}$ . The red curve corresponds to the unified RL control starting from the stabilized state at time $t_{4}$ in Fig. 2 , the blue curve corresponds to bare QZE evolution of the TACT-optimal squeezed state without control, and the green curve corresponds to the DD protocol. The gray dashed line denotes the SQL. The SQL is defined by the CSS phase sensitivity $\Delta\phi_{\mathrm{SQL}}=1/\sqrt{2f}$ . (b) Readout distributions $P_{m}(\phi)$ for representative probe states and control protocols. Panels \scriptsize1⃝ and \scriptsize2⃝ show the distributions before ( $\phi=0$ , top) and after ( $\phi=0.2$ , bottom) phase encoding for the initial states used in the blue and red curves of panel (a), respectively. Panels \scriptsize3⃝ and \scriptsize4⃝ show the corresponding distributions at $\chi t\approx 0.09$ for the green and red curves in panel (a), respectively. (c) Single-atom magnetic-field sensitivity relative to the SQL, expressed in dB, as a function of the total protocol time $T_{\mathrm{tot}}$ . The shaded region indicates the preparation time $T_{\mathrm{p}}$ , and the inset shows the absolute sensitivity $\delta B(T_{\mathrm{tot}})$ in $\mathrm{nT}/\sqrt{\mathrm{Hz}}$ .

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Value
PPO
Actor learning rate	$3\times 10^{-4}$
Critic learning rate	$1\times 10^{-3}$
Discount factor $\gamma$	$0.9999$
GAE parameter $\hat{\lambda}$	$0.96$
Clip parameter $\epsilon$	$0.2$

Appendix

Table 1: Parameters used for PPO training and environment design.

Figure 5: Phase sensitivity with different pulse intervals

\delta t

. The red curve corresponds to the sequence with the same

\delta t

as that used in the RL protocol in the main text, while the other colored curves correspond to shorter

\delta t

Figure 6: Train performance across different spin-

f

systems. Squeezing parameters obtained from independently trained RL agents for different spin

f

. The red and green curves show the minimum

\xi^{2}

reached by the RL protocol and the average stabilized value of

\xi_{y}^{2}

. The gray solid and dashed curves denote the optimal

\xi^{2}

of the QZE and effective TACT models.

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Nitrogen-vacancy (NV) centers in diamond can serve as highly sensitive solid-state quantum sensors for high-sensitivity magnetometry. However, in the noisy intermediate-scale quantum (NISQ) era, extracting reliable information from noisy, finite-shot, and measurement-limited sensing data remains a considerable challenge. Whereas, quantum machine learning (QML) offers a potential path to improve parameter estimation by learning nonlinear relationships between quantum-sensing data and the underlying physical signal. In this work, we investigate the role of QML in magnetic-field estimation within an NV center-inspired magnetometry setting. We formulated magnetic field sensing as a supervised regression task. We compared the performance of several classical machine learning models trained on measurement-based classical data with that of quantum kernel-based models trained on pre-measurement coherent quantum states. Our objective is to isolate the impact of measurement-induced information loss and therefore provide a theoretical upper bound on the sensing performance. The upper bound is achievable only when coherent quantum information is directly available to the learning model. Our results show that QML-based sensing performance improves significantly with coherent quantum-state information, and not much with changes in model complexity or learning paradigm. This observation underscores the importance of learning pipelines that tightly integrate quantum sensors and QML models to enhance magnetic field sensing under realistic constraints.

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