Non-local synchronization of continuous time crystals in a semiconductor

TL;DR

This work demonstrates non-local synchronization of continuous time crystals formed by auto-oscillating electron–nuclear spin systems in a GaAs semiconductor. Mutual synchronization occurs over mesoscopic distances (up to about $38\pm3\ \mu$m) via spin diffusion of electron spins, with a diffusion length $L_s$ of approximately $18\ \mu$m, and can be driven across wide areas using flat-top optical pumping. A Bloch-diffusion model, complemented by time-resolved Kerr microscopy measurements, shows that spin diffusion is the principal mechanism mediating the coupling and yielding a single, robust synchronized state despite inhomogeneities. This collective coherence enhances the stability of the auto-oscillations and points toward scalable coherent spin networks and neuromorphic spintronic architectures.

Abstract

Synchronization resulting in unified collective behavior of the individual elements of a system that are weakly coupled to each other has long fascinated scientists. Examples range from the periodic oscillation of coupled pendulum clocks to the rhythmic behavior in biological systems. Here we demonstrate this effect in a solid-state platform: spatially remote, auto-oscillating electron-nuclear spin systems in a semiconductor. When two such oscillators separated by up to 40 $μ$m are optically pumped, their individually different frequencies lock to a common value, revealing long-range coherent coupling. For larger separations, the synchronization breaks. The interaction distance matches the electron spin diffusion length, identifying spin transport as the coupling-mediating mechanism and establishing phase coherence over mesoscopic distances. As a consequence, a wide-area optical pump drives all oscillators within the illuminated spot into a single synchronized state, despite their inhomogeneity. This synchronization accounts for the exceptional stability of the resulting auto-oscillations, enabling collective motion in distributed spin systems and paving the way toward coherent spin networks in spintronics.

Figures (13)

• Figure 1: Inhomogeneity.a, Examples of time traces measured at different sample positions. Pump and probe spots are overlapped, and the sample is shifted in the range of $-150\,\mu$m to $+150\,\mu$m along the horizontal direction ($\text{x}$). Pump spot diameter at $1/e^2$ intensity level is equal to $22\,\mu$m (FWHM of 13$\mu$m). Probe spot size at $1/e^2=12\,\mu$m (FWHM of 7$\mu$m). Pump/Probe powers are $0.15/1$ mW, respectively. b, Corresponding fast Fourier transform spectra of the traces from panel a measured for 10 minutes at the arbitrary position $\text{x}=0$, the top black trace; to the right side of the previous position at $\text{x}=140\,\mu$m, red trace; and to the left side at $\text{x}=-130\,\mu$m, blue trace. The difference in frequencies is related to the intrinsic sample variation.
• Figure 2: Mutual synchronization.a, Sketch of experiment: red pump spot is prepared by a $\pi$Shaper to produce a flat-top intensity distribution on the sample, as seen in the simulated bottom part of the figure. Pump spot diameter is about 200 $\mu$m. The Gaussian probe beam is focused down to $12\,\mu$m at $1/e^2$ (FWHM of 7$\mu$m) and can be shifted horizontally within the pump beam by the same distances as in the case of Fig. \ref{['fig1']}a. The bottom part shows the simulated profiles of the lasers. Pump/Probe powers $= 10/1$ mW. b, Examples of the fast Fourier transform spectra of the traces measured for 10 minutes at the center $\text{x}=0$, the top black trace; on the right side of the pump center at $\text{x}=154\,\mu$m, red trace; and on the left side at $\text{x}=-118\,\mu$m, blue trace, and at $\text{x}=-154\,\mu$m, light blue trace. The difference in frequencies is now negligible compared to Fig. \ref{['fig1']}b, suggesting complete synchronization within the pump-spot excitation. c, Summarized data for the FFT peak positions of the first harmonics for both experimental cases versus shift of the sample for tightly focused pump (blue squares) or the probe shift within a single widened pump (red circles).
• Figure 3: Range of synchronization.a, Experimental time traces of auto-oscillations for three different cases: the top blue one shows the pump-1 only, the red one in the middle is the pump-2 only, shifted horizontally by $-25\,\mu$m away from the pump-1 position. Finally, the last green time trace at the bottom represents the combined application of pump-1 and pump-2. See Fig. \ref{['fig3']}c for the labeling of the pumps. b, First harmonic range of the fast Fourier transform spectrum of the traces presented in panel a, measured for 10 minutes. As one can see, each separate pump beam at its sample position induces auto-oscillations with a characteristic frequency. If both pumps are applied together, one can see a single frequency, a hallmark of synchronization. Laser powers for Pump-1/Pump-2/Pr$= 0.1/0.1/1$ mW. c, Sketch of three Gaussian laser spots on the sample with the wide blue spot representing the probe ($1/e^2=150\,\mu$m, FWHM of 88.3$\mu$m), the central red spot is the pump-1 ($1/e^2=17\,\mu$m, FWHM of 10$\mu$m), and the red spot on the right is the pump-2 ($1/e^2=9\,\mu$m, FWHM of 5.3$\mu$m) that can be shifted horizontally. The top sketch shows a frontal view of the sample, and the bottom sketch shows the normalized intensity profile simulation of the involved beams of 38 $\mu$m separation. d, Experimental time traces of auto-oscillations for the case of the pump-2 spot shifted horizontally by 50 $\mu$m away from the pump-1 position. e, First harmonic range of the fast Fourier transform spectrum of the traces presented in panel d, measured for 10 minutes. Each separate pump beam induces auto-oscillations with different frequencies. When both pumps are applied together (green, bottom), two frequencies are observed at positions close to those of the single-pump beam cases, indicating no averaging and thus no synchronization. f, The measured synchronization range presented for a pump-2 shift from $-50\,\mu$m to $+50\,\mu$m relative to pump-1 position. The synchronization radius is about $38\pm 3\,\mu$m measured between the pump-spot centers. The precision of the rotation screw determines the error bars.
• Figure 4: Simulation of synchronization.a, Simulated time traces of auto-oscillations for three cases. The top (blue) trace corresponds to pump-1 only, the middle (red) trace to pump-2 only, shifted by $0.5L_\text{s}$ from the pump-1 position. The spatial width (FWHM) of both pump-1 and pump-2 is $0.25 L_\text{s}$. The bottom (green) trace shows the combined excitation by pump-1 and pump-2. See Fig. \ref{['fig3']}c for pump labeling. Detection is performed with a spatially wide probe covering the range $0-10 L_\text{s}$. b, First harmonic of the fast Fourier transform spectra of the traces in panel a. c, Simulated time traces of auto-oscillations for three cases: the top (blue) trace corresponds to pump-1 only, the middle (red) trace to pump-2 only, shifted by $2 L_\text{s}$ from the pump-1 position, and the bottom (green) trace to their combined excitation. d, First harmonic of the FFT spectra of the traces in panel c. e, Extracted synchronization range for pump-2 shifts between $0$ and $2 L_\text{s}$ relative to pump-1. f, Synchronization in simulation under flat-top pump excitation. The blue curve shows the first harmonic in the FFT for the spatially narrow pump-1 and narrow probe, the red curve corresponds to pump-2 and probe shifted by $9 L_\text{s}$, and the green curve shows the resulting signal under flat-top pump excitation spanning $0-10 L_\text{s}$. The detection is performed with a spatially narrow probe at $9 L_\text{s}$.
• Figure S1: PL and Sample. Normalized PL of two samples. The red dashed line shows the PL of the sample on the thick GaAs substrate. The blue solid line shows the sample with the reduced thickness of the GaAs substrate. The black arrow shows the position of the probe wavelength. Pump is at $E_\text{exc}=1.579\,$eV (785 nm), and the probe $E_\text{pr}=1.426\,$eV (869.4 nm). The insert shows the sample structure with 10 $\mu$m epilayer and GaAs substrate.