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arXiv:2410.20090 · Physical Review A

Nonlinear spin dynamics induced by feedback under continuous Larmor frequency distributions

Tishuo Wang, Zhihuang Luo, Shizhong Zhang, Zhenhua Yu

Published May 5, 2025• 4 citations
arXivHTMLPDF

Abstract

Nonlinear spin dynamics are essential in exploring nonequilibrium quantum phenomena and have broad applications in precision measurement. Among these systems, the combination of a bias magnetic field and feedback mechanisms can induce self-sustained oscillations at the base Larmor frequency due to nonlinearity. These features have driven the development of single-species and multiple-species spin masers. The latter, with multiple discrete Larmor frequencies, provides significant advantages for precision measurement by mitigating uncertainties in precession frequencies due to long-term drifts in experimental conditions. The self-sustained oscillations of single-species and multiple-species spin masers correspond to limit cycles and quasi-periodic orbits of the stable nonlinear dynamics of the systems, respectively; the correspondence is elucidated in a recent study on a related spin system featuring two discrete intrinsic Larmor frequencies under dual bias magnetic fields [T. Wang et al., Commun. Phys. 8, 41 (2025)]. Here, we extend the study to the case where the intrinsic Larmor frequencies of individual spins of the system, given rise to by an inhomogeneous bias magnetic field, form a continuum. We show that generically the stable dynamics of the system includes limit cycles, quasi-periodic orbits, and chaos. We establish the relation between the synchronization frequency of limit cycles and the field inhomogeneity and derive an equation determining the stability of limit cycles. Furthermore, detailed characteristics of different dynamical phases, especially the robustness of limit cycles and quasi-periodic orbits against experimental fluctuations, are discussed. Our findings not only encompass the case of discrete Larmor frequencies, but also provide crucial insights for precision measurement and the exploration of continuous time crystals and quasi-crystals.

DOI
10.1103/physreva.111.052807

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Nonlinear spin dynamics are essential in exploring nonequilibrium quantum phenomena and have broad applications in precision measurement. Among these systems, the combination of a bias magnetic field and feedback mechanisms can induce self-sustained oscillations at the base Larmor frequency due to nonlinearity. These features have driven the development of single-species and multiple-species spin masers. The latter, with multiple discrete Larmor frequencies, provides significant advantages for precision measurement by mitigating uncertainties in precession frequencies due to long-term drifts in experimental conditions. The self-sustained oscillations of single-species and multiple-species spin masers correspond to limit cycles and quasi-periodic orbits of the stable nonlinear dynamics of the systems, respectively; the correspondence is elucidated in a recent study on a related spin system featuring two discrete intrinsic Larmor frequencies under dual bias magnetic fields [T. Wang et al., Commun. Phys. 8, 41 (2025)]. Here, we extend the study to the case where the intrinsic Larmor frequencies of individual spins of the system, given rise to by an inhomogeneous bias magnetic field, form a continuum. We show that generically the stable dynamics of the system includes limit cycles, quasi-periodic orbits, and chaos. We establish the relation between the synchronization frequency of limit cycles and the field inhomogeneity and derive an equation determining the stability of limit cycles. Furthermore, detailed characteristics of different dynamical phases, especially the robustness of limit cycles and quasi-periodic orbits against experimental fluctuations, are discussed. Our findings not only encompass the case of discrete Larmor frequencies, but also provide crucial insights for precision measurement and the exploration of continuous time crystals and quasi-crystals.
Nonlinear spin systems exhibit rich and exotic dynamical phenomena, offering promising applications ranging from spin masers and time crystals to precision measurement. Recent theoretical work [T. Wang et al., Commun. Phys. 8, 41 (2025)] predicted intriguing nonlinear dynamical phases arising from inhomogeneous magnetic fields and feedback interactions. However, experimental exploration of these predictions remains lacking. Here, we report the observation of nonlinear spin dynamics in dual-bias magnetic fields with dual-cell alkali-metal atomic gases and present three representative stable dynamical behaviors of limit cycles, quasi-periodic orbits, and chaos. Additionally, we probe the nonlinear phase transitions between these phases by varying the feedback gain and the difference of dual-bias magnetic fields. Furthermore, we demonstrate the robustness of the limit cycle and quasi-periodic orbit against the noise of magnetic fields. Our findings establish a versatile platform for exploring complex spin dynamics and open new avenues for the realization of multimode spin masers, time crystals and quasi-crystals, and high-precision magnetometers.
Continuous time crystals (CTCs) are a phase of matter characterized by spontaneous breaking of continuous time-translation symmetry. Recently, CTCs have garnered interest due to breakthroughs in experimental implementation. Here we report the experimental observation of CTCs in noble-gas nuclear spins and uncover previously unexplored dynamical phenomena. We observe that the CTCs manifest as persistent limit cycle oscillations of nuclear spins, with coherence times exceeding hours. Notably, these oscillations are robust against noise perturbations and exhibit random time phases upon repetitive realization, epitomizing continuous time-translation symmetry-breaking intrinsic to CTCs. Additionally, we observe a dynamical phase featuring quasi-periodic oscillations and random time phases, indicating the emergence of the continuous time quasi-crystals proposed by recent theories. By varying the feedback strength and magnetic gradient, we observe complex dynamical phase transitions between time crystal phases and chaotic regimes. This work broadens the catalog of phases of spin gases and unlocks opportunities in precision measurements. The authors report on the observation of continuous time crystals of nuclear spins of 129Xe atoms within a Rb-Xe gaseous mixture in a magnetic feedback scheme. This unveils nontrivial role of magnetic fields and a new continuous time quasi-crystal phase.
Feedback locking of a probe frequency to a spin resonance is the standard operating mode of precision quantum sensors. Here we deliberately operate such a lock outside its stable regime: a continuous-wave nitrogen-vacancy (NV) ensemble magnetometer, frequency-modulation (FM) locked to one flank of its optically detected magnetic resonance (ODMR), is driven through the flip (period-doubling) bifurcation of its discrete feedback map by raising the software loop gain GGG. Beyond a critical gain $\Gc$ the lock becomes a self-sustained oscillator whose limit cycle is generated by the loop itself. We derive the threshold condition $\Gc = 2\,\Dcal/\Dtrue$, which identifies the measurable content of the threshold: the ratio of the transduction slope of the ODMR lock-in signal at calibration time DcalD_{cal}Dcal​ to its value at present DtrueD_{true}Dtrue​. We present an identifiability analysis showing which physical parameters this single scalar can and cannot distinguish, characterize the estimators of $\Gc$ under realistic noise, and report measurements on our current setup: an experimental bifurcation diagram with onset at $\Gc \approx 2$ as predicted for a self-calibrated loop, sub-threshold critical fluctuations following the predicted G/(2−G)\sqrt{G/(2-G)}G/(2−G)​ divergence.
Continuous time crystals represent a novel state in many-body systems that can self-organize into timeperiodic oscillations without external periodic driving. Recent experiments have achieved such systems in dissipative solid-state materials, where persistent temporal order is autonomously sustained. A decisive characteristic of time crystals is their robustness, meaning that despite various disturbances, including fluctuations in internal parameters and external noise, they can still maintain rhythmic behavior, which has scientific value and echnological application potential. Although previous studies have shown that specific experimental parameters have robustness, thare is a lack of a systematic framework for quantifying and predicting their resilience to disturbances, and the underlying physics of this robustnessis still not fully understood. The key unresolved problems include how nonlinear interactions and feedback mechanisms contribute to stability, and what the critical thresholds are for parameter variations beyond which temporal order collapses. This work addresses these gaps by systematically analyzing how internal parameters and external influences affect the oscillation period and overall stability. Internally, the dynamics are determined by dipole-dipole interactions and atomic transition strengths, which define the temporal symmetry breaking that occurs in the system. Externally, the response of the system is controlled by the strength of the optical driving field and the energy dissipation rate. A key finding is the determination of an intrinsic feedback mechanism for a dynamically stabile time crystal. This mechanism plays a role in restoring force, correcting deviations caused by minor disturbances, and maintaining the coherence of oscillatory phase. Moreover, the system displays nonlinear dynamical behavior, characterized by two different states: one is stable oscillation continuing under moderate disturbance, and the other is stronger disturbance causing dynamical phase transition, resulting in switching between disordered or dynamically unstable state and stable state. These results provide a comprehensive understanding of the various behaviors observed in continuous time crystals and lay an important theoretical foundation for utilizing their unique properties in advanced applications such as quantum information processing and precision metrology.
The spin self-sustaining atomic magnetometer has the advantage of 1/τ measurement and great development potential in many applications. In this paper, we investigated the main elements that affect the stability and accuracy of the self-sustaining magnetometer and proposed the methods to improve its performance based on the measurement results. The correlation coefficient between fluctuations of the magnetic field generated by coils and the spin Larmor precession frequency is 0.97, which mainly dominates the stability in a short term. The accuracy of the magnetometer is affected by the power and frequency of the pump light. The Larmor precession frequency coefficient related to the pump light power is 26 mHz/mW, and the effect on the Larmor precession frequency is minimized when the pump light frequency is red detuned by 200 MHz from the 85Rb transition D1 line F = 3 to F′ = 3. The 1/τ measurement time after these corrections can be extended to 10 s, and the sensitivity achieved is 149 fT/Hz, which is close to the quantum projection noise limit of the system.
Local pure dephasing suppresses single-spin coherence, yet we show that it sustains continuous time-crystalline order in a fully connected ferromagnetic spin-1/21/21/2 ensemble with local decay. An exact operator identity reveals the mechanism: dephasing raises the interaction energy, while energy relaxation on a finite timescale converts this charging into delayed feedback. Within homogeneous mean-field theory, we prove that dephasing is necessary for periodic motion in this model and that feedback exactly balances transverse damping over each cycle. Beyond mean field, the identity fixes the relative amplitude and phase of energy and longitudinal magnetization in oscillatory Liouvillian modes. Liouvillian spectra and stationary two-time correlations characterize temporal order through decay rates that decrease algebraically with system size, frequencies approaching nonzero mean-field values, and comparable per-spin oscillatory weights. Semiclassical trajectories further reveal recurrent switching between a limit-cycle region and two symmetry-related stationary regions: fluctuations interrupt and reactivate collective oscillations, and oscillatory episodes provide an intermediate channel between stationary regions. These results establish local dephasing as a resource for autonomous collective motion and show how the resulting temporal order can participate in metastable dynamics.