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arXiv:2309.11855 · npj Quantum Information

Quantum locking of intrinsic spin squeezed state in Earth-field-range magnetometry

Peiyu Yang, Guzhi Bao, Jun Chen, Wei Du, Jinxian Guo, Weiping Zhang

Published Mar 2, 2025• 3 citations
arXivHTMLPDF

Abstract

In the Earth-field range, the nonlinear Zeeman (NLZ) effect has been a bottleneck limiting the sensitivity and accuracy of atomic magnetometry from physical mechanism. To break this bottleneck, various techniques are introduced to suppress the NLZ effect. Here we revisit the spin dynamics in the Earth-field-range magnetometry with the NLZ effect and identify the existence of the intrinsic spin squeezed state (SSS), generated from the coupling between nuclear and electron spins of each individual atom, with the oscillating squeezing degree and squeezing axis. Such oscillating features of the SSS prevent its direct observation and as well, accessibility to magnetic sensing. To exploit quantum advantage of the intrinsic SSS in the Earth-field-range magnetometry, it’s essential to lock the oscillating SSS to a persistent one. Hence we develop a quantum locking technique to achieve a persistent SSS, benefiting from which the sensitivity of the Earth-field-range magnetometer is quantum-enhanced. This work presents an innovative way turning the drawback of NLZ effect into the quantum advantage and opens a new access to quantum-enhanced magnetometry in the Earth-field range.

DOI
10.1038/s41534-025-00971-9

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In the Earth-field range, the nonlinear Zeeman (NLZ) effect has been a bottleneck limiting the sensitivity and accuracy of atomic magnetometry from physical mechanism. To break this bottleneck, various techniques are introduced to suppress the NLZ effect. Here we revisit the spin dynamics in the Earth-field-range magnetometry with the NLZ effect and identify the existence of the intrinsic spin squeezed state (SSS), generated from the coupling between nuclear and electron spins of each individual atom, with the oscillating squeezing degree and squeezing axis. Such oscillating features of the SSS prevent its direct observation and as well, accessibility to magnetic sensing. To exploit quantum advantage of the intrinsic SSS in the Earth-field-range magnetometry, it’s essential to lock the oscillating SSS to a persistent one. Hence we develop a quantum locking technique to achieve a persistent SSS, benefiting from which the sensitivity of the Earth-field-range magnetometer is quantum-enhanced. This work presents an innovative way turning the drawback of NLZ effect into the quantum advantage and opens a new access to quantum-enhanced magnetometry in the Earth-field range.
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=21/2f=21/2f=21/2 manifold of 161Dy^{161}\mathrm{Dy}161Dy, where the learned policy rapidly prepares strongly squeezed internal states and stabilizes more than 4 dB4\,\mathrm{dB}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 13.9 pT/Hz13.9\,\mathrm{pT}/\sqrt{\mathrm{Hz}}13.9pT/Hz​, approximately 3 dB3\,\mathrm{dB}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.
Atomic magnetometers based on Zeeman shift measurement have the potential for high sensitivity and long-term stability. Like other atomic sensors including atomic clocks and atom interferometers, the atomic magnetometer could in principle be augmented with spin squeezing for further sensitivity enhancement. However, existing atomic magnetometers are not compatible with spin squeezing because the atoms can hardly be in a pure quantum state during operation. A natural challenge is the arbitrary direction of the magnetic field. In this paper, we propose a cold-atom-based magnetometer with spin squeezing that can measure both the magnitude and the direction of an arbitrary magnetic field. For experimentally accessible parameters, we show that the technique described above could achieve a sensitivity nearly three orders of magnitude higher than that of the best existing magnetometers.
The nonlinear Zeeman effect can induce splittings and asymmetries of magnetic-resonance lines in the geophysical magnetic-field range. We demonstrate an all-optical scheme, based on spin locking, to suppress the nonlinear Zeeman effect. This scheme achieves spin locking via an effective oscillating magnetic field in the form of AC Stark shifts induced by an intensity- and polarization-modulated laser beam. This results in the collapse of the multicomponent asymmetric magnetic-resonance line with $\ensuremath{\sim}100\phantom{\rule{0.16em}{0ex}}\mathrm{Hz}$ width in the Earth-field range into a peak with a central component width of 25 Hz. The technique is expected to be broadly applicable in practical magnetometry, potentially boosting the sensitivity and accuracy of Earth-surveying magnetometers by increasing the magnetic-resonance amplitude and decreasing its width. An advantage of the all-optical approach is the absence of crosstalk between nearby sensors when they are used in a gradiometric or array arrangement.
The nonlinear Zeeman effect can induce splitting and asymmetries of magnetic-resonance lines in the geophysical magnetic-field range. This is a major source of "heading error" for scalar atomic magnetometers. We demonstrate a method to suppress the nonlinear Zeeman effect and heading error based on spin locking. In an all-optical synchronously pumped magnetometer with separate pump and probe beams, we apply a radio-frequency field which is in phase with the precessing magnetization. This results in the collapse of the multicomponent asymmetric magnetic-resonance line with ∼100 Hz width in the Earth-field range into a single peak with a width of 22 Hz, whose position is largely independent of the orientation of the sensor within a range of orientation angles. The technique is expected to be broadly applicable in practical magnetometry, potentially boosting the sensitivity and accuracy of Earth-surveying magnetometers by increasing the magnetic-resonance amplitude, decreasing its width, and removing the important and limiting heading-error systematic.
The sensitivity of an atomic spin sensor is fundamentally constrained by the quantum Fisher information (QFI) of the probe state. The pinched state, ∣F,0⟩|{F,0}\rangle∣F,0⟩, has an (F+1)(F+1)(F+1)-fold larger single-particle QFI for transverse-field sensing than the stretched state, ∣F,F⟩|{F,F}\rangle∣F,F⟩, corresponding to a predicted F+1\sqrt{F+1}F+1​ reduction in the spin-projection-noise-limited field uncertainty. We show that this advantage can be accessed in an rf sensing scheme by engineering a spectrum symmetric about m=0m=0m=0, yielding an ideal (F+1)(F+1)(F+1)-fold enhancement of magnetic-field response. For 87Rb^{87}\mathrm{Rb}87Rb with F=2F=2F=2, we observe up to 2.7-fold response enhancement. Implementing this approach in a compact 0.8 cm30.8~\mathrm{cm}^30.8 cm3 anti-relaxation-coated multipass cell, we realize a single-beam, dual-axis, zero-field magnetometer, achieving 131313--232323~fT/Hz\mathrm{fT}/\sqrt{\mathrm{Hz}}fT/Hz​ over 333--100100100~Hz at room temperature.
Measurements with quantum limited resolution have important applications in metrology and sensing, including atomic clocks, atom interferometry, and magnetometry. We perform a quantum-non-demolition measurement on a collective angular momentum of a million spin-4 Cs atoms with high optical depth in our work. Quantum mechanics limits the measurement's resolution to what is known as the standard quantum limit (SQL) when there is no entanglement between the individual atoms. We can generate entanglement between atoms to create spin squeezed states through quantum backaction, improving the measurement sensitivity by 5.5 dB relative to the SQL. We have recently rebuilt the entire experiment inside a state-of-the-art magnetic shield of mu-metal and aluminum. This upgrade has drastically mitigated classical control errors in the spin squeezing experiment. Furthermore, the ensemble's sensitivity to magnetic fields allows us to make precise measurements of the magnetic field environment. Using closed-loop feedback, we use our atomic magnetometer to measure magnetic fields 4.3 dB below the SQL. The shielded environment will allow us to optimize the magnetometer and sets the stage for future experiments to use the closed-loop control to explore more complicated quantum dynamics.