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 · Key Laboratory of Aerospace Information Sensing and Physics (NUAA), MIIT, Nanjing 211106, China
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 161Dy, where the learned policy rapidly prepares strongly squeezed internal states and stabilizes more than 4dB of fixed-axis spin squeezing under continuous NLZ evolution. Including state-preparation overhead, the learned protocol yields a single-atom magnetic-field sensitivity of 13.9pT/Hz, approximately 3dB 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 161Dy 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 ok∈O , selects an action ak∈A , and maximizes the cumulative reward 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 ξ2(t) (red solid) and the fixed-axis squeezing parameter ξy2(t) (green solid) versus the dimensionless time χt . Gray solid and dotted curves denote ξ2(t) of the QZE (OAT) evolution and the effective TACT benchmark, respectively. The vertical gray line near χt≃0.15 marks the reward-defined switching time tr . The inset Wigner distributions show representative snapshots at selected times ti ; 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 f^y2 and f^z2 , respectively. The reference states are the coherent spin state ∣f,mx=f⟩ , the RL-generated state after the R^x(π/3) pulse in Fig. 2 , and the OAT and TACT states evaluated at their respective minima of ξ2 . For the OAT and TACT states, an additional R^x rotation is applied so that ξy2=ξ2 .
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 χTe . The red curve corresponds to the unified RL control starting from the stabilized state at time t4 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 ΔϕSQL=1/2f . (b) Readout distributions Pm(ϕ) for representative probe states and control protocols. Panels \scriptsize1⃝ and \scriptsize2⃝ show the distributions before ( ϕ=0 , top) and after ( ϕ=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 χt≈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 Ttot . The shaded region indicates the preparation time Tp , and the inset shows the absolute sensitivity δB(Ttot) in nT/Hz .
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Parameter
Value
PPO
Actor learning rate
3×10−4
Critic learning rate
1×10−3
Discount factor γ
0.9999
GAE parameter λ^
0.96
Clip parameter ϵ
0.2
Appendix
Table 1: Parameters used for PPO training and environment design.
Figure 5: Phase sensitivity with different pulse intervals δt . The red curve corresponds to the sequence with the same δt as that used in the RL protocol in the main text, while the other colored curves correspond to shorter δt .
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 ξ2 reached by the RL protocol and the average stabilized value of ξy2 . The gray solid and dashed curves denote the optimal ξ2 of the QZE and effective TACT models.
Quantum State Preparation develops circuits and control strategies for creating target quantum states with limited gates, noise, and hardware resources.
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.
Sounak Bhowmik, Himanshu Thapliyal
Department of Electrical and Computer Engineering Southern Methodist University, Dallas, TX, USA
Learning the Hamiltonian of a many-body system from its dynamics is a central task in quantum science, yet the algorithms with the strongest provable guarantees assume some level of quantum control--fast, arbitrary single-qubit gates interleaved with time evolution, and measurements in arbitrary bases--that is beyond the capabilities of near-term analog quantum simulators. Motivated by analog atom- and ion-based platforms, we study Hamiltonian learning under minimal access models. Uniform state preparation and measurements: We first consider the setting where in every experiment, one can rotate each qubit to the same state, perform short-time evolution, and measure every qubit in the same basis. Surprisingly, we show that for generic 2-local Hamiltonians on any interaction graph, all of the parameters can be reconstructed from such experiments. Computational basis state preparation and measurements: We then consider a similarly constrained setting, but where state preparation and measurement are restricted to the computational basis. For nearest-neighbor Hamiltonians with only Pauli X/Z interactions, a class which captures contemporary Rydberg atom platforms, we show that over 1D and 2D rectangular lattices, all of the parameters can be reconstructed from such experiments up to unavoidable gauges. Our protocols introduce new techniques for solving structured polynomial systems over an extensive number of parameters. Taken together, our results suggest that one can learn a great deal from the dynamics of quantum many-body systems even under the most stringent experimental constraints.
Quantum technology has the potential to transform scientific discovery, but quantum advantages often require processing capabilities well beyond the reach of experimental platforms. We show that coupling a single controllable qubit to an otherwise conventional sensor can exponentially reduce the number of measurements required to learn classical signals. These rigorous quantum advantages apply to fundamental sensing tasks, including learning Fourier coefficients, extracting temporal correlations from time-varying signals, and estimating transformations of physical observables. Using a superconducting cavity--qubit architecture, we experimentally demonstrate 107-fold reductions in the number of measurements required for Fourier-amplitude and time-varying signal learning. Our quantum feature sensing algorithms further enable orders-of-magnitude improvements in simulations of weak-signal dark matter detection and wireless communication applications. These quantum advantages are derived from Quantum Phase-Space Inference (QΨ), a unifying theory of quantum-enhanced experiments that simultaneously converts a set of experimental objectives and constraints into tight lower bounds and optimal quantum-enhanced learning algorithms while producing a certificate of quantum advantage. QΨ extends beyond the regimes captured by quantum Fisher information and provides a framework for systematically identifying rigorous quantum advantages in practical experimental tasks. Together, our results establish that near-term quantum technology can exponentially enhance our ability to learn from classical signals.
Ishaan Kannan, Sridhar Prabhu, Saeed A. Khan +7
Harvard Quantum Initiative, Harvard University, Cambridge, MA 02138, USA · School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA · Department of Physics, Cornell University, Ithaca, NY 14853, USA +2