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arXiv:2606.00766

Energy spectra and cascade in the spin turbulence of a driven spinor Bose-Einstein condensate

Junghoon Lee, Jongmin Kim, Donggyu Lee, Yong-il Shin

Published May 30, 2026• 0 citations
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Abstract

We investigate the spin-interaction energy spectrum of spin turbulence in a driven spinor Bose-Einstein condensate. Continuous spin driving of a spin-1 condensate produces a nonequilibrium steady state with spatially fluctuating magnetization. We observe a power-law scaling consistent with the $-7/3$ exponent predicted for spin-wave turbulence, which persists across our full range of drive strengths despite substantial changes in the spectral anisotropy. After switching off the drive, we track the free-decay evolution and find evidence consistent with a direct cascade of spin-interaction energy toward higher wavenumbers. These results establish an energy-spectral hallmark of spin turbulence and enable quantitative studies of cascade dynamics in spinor superfluids.

DOI
10.48550/arxiv.2606.00766

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We investigate the spin-interaction energy spectrum of spin turbulence in a driven spinor Bose-Einstein condensate. Continuous spin driving of a spin-1 condensate produces a nonequilibrium steady state with spatially fluctuating magnetization. We observe a power-law scaling consistent with the −7/3-7/3−7/3 exponent predicted for spin-wave turbulence, which persists across our full range of drive strengths despite substantial changes in the spectral anisotropy. After switching off the drive, we track the free-decay evolution and find evidence consistent with a direct cascade of spin-interaction energy toward higher wavenumbers. These results establish an energy-spectral hallmark of spin turbulence and enable quantitative studies of cascade dynamics in spinor superfluids.
We report the observation of stationary turbulence in antiferromagnetic spin-1 Bose-Einstein condensates driven by a radio-frequency magnetic field. The magnetic driving injects energy into the system by spin rotation and the energy is dissipated via dynamic instability, resulting in the emergence of an irregular spin texture in the condensate. Under continuous driving, the spinor condensate evolves into a nonequilibrium steady state with characteristic spin turbulence, while the low-energy scale of spin excitations ensures that the sample's lifetime is minimally affected. When the driving strength is on par with the system's spin interaction energy and the quadratic Zeeman energy, remarkably, the stationary turbulent state exhibits spin-isotropic features in spin composition and spatial spin texture. We numerically show that ambient field fluctuations play a crucial role in sustaining the turbulent state within the system. These results open up different avenues for exploring quantum turbulence in spinor superfluid systems.
We numerically investigate the stationary turbulent states of spin-1 Bose-Einstein condensates under continuous spin driving. We analyze the entanglement entropy and magnetization correlation function to demonstrate the isotropic nature of the intricate spin texture that is generated in the nonequilibrium steady state. We observe a $\ensuremath{-}7/3$ power-law behavior in the spin-dependent interaction energy spectrum. To gain further insight into the statistical properties of the spin texture, we introduce a spin state ensemble obtained through position projection, revealing its close resemblance to the Haar random ensemble for spin-1 systems. We also present the probability distribution of the spin vector magnitude in the turbulent condensate, which can be tested in experiments. Our numerical study highlights the characteristics of stationary turbulence in the spinor Bose-Einstein condensate system and confirms previous experimental findings [D. Hong et al., Phys. Rev. A 108, 013318 (2023)].
We present a turbulence-sustaining mechanism in a spinor Bose-Einstein condensate, which is based on the chaotic nature of internal spin dynamics. Magnetic driving induces a complete chaotic evolution of the local spin state, thereby continuously randomizing the spin texture of the condensate to maintain the turbulent state. We experimentally demonstrate the onset of turbulence in the driven condensate as the driving frequency changes and show that it is consistent with the regular-to-chaotic transition of the local spin dynamics. This chaos-assisted turbulence establishes the spin-driven spinor condensate as an intriguing platform for exploring quantum chaos and related superfluid turbulence phenomena. Published by the American Physical Society 2024
We theoretically and numerically study spin wave turbulence in spin-1 ferromagnetic spinor Bose-Einstein condensates, finding direct and inverse cascades with power-law behavior. To derive these power exponents analytically, the conventional weak wave turbulence theory is applied to the spin-1 spinor Gross-Pitaevskii equation. Thus we obtain the $\ensuremath{-}7/3$ and $\ensuremath{-}5/3$ power laws in the transverse spin correlation function for the direct and inverse cascades, respectively. To confirm these power laws, numerical calculations are performed that obtain results consistent with these power laws.
We numerically investigate statistically steady quantum turbulence in the cyclic phase of a spin-2 spinor Bose--Einstein condensate (BEC) driven by large-scale energy injection via a divergence-free external velocity field. In the full non-Abelian cyclic system, the mass-current spectrum exhibits an anomalous k−7/3k^{-7/3}k−7/3 scaling, whereas the spin-current spectrum approximately follows a k−5/3k^{-5/3}k−5/3 power law. To isolate the role of non-Abelian vortex dynamics, we compare this system with an Abelian-restricted cyclic system sharing the same Hamiltonian parameters and driving conditions. Strikingly, the Abelian restriction collapses both the mass- and spin-current spectra into a k−5/3k^{-5/3}k−5/3 scaling, identical to that of a scalar BEC, despite the three distinct turbulent states maintaining nearly identical vortex-line densities. A Helmholtz decomposition reveals that the mass- and spin-current spectra are predominantly incompressible and transverse, respectively, thereby confirming that the anomalous scaling cannot be attributed simply to compressible fluctuations or to the transverse nature of the energy injection. Local spectral exponents further quantify these distinct scaling regimes. The disappearance of the mass-spin spectral separation under the Abelian restriction, alongside the formation of a large-scale web of rung-connected non-Abelian vortices, provides compelling evidence that non-Abelian vortex dynamics fundamentally govern the organization of turbulent mass flow.
We review turbulence in ultracold quantum gases, using the scalar contact-interaction Bose-Einstein condensate as the reference system for quantized circulation, compressibility, vortices, sound, and cascades. We focus on the quantitative diagnostics that connect helium and classical phenomenology to microscopic wave-function dynamics: incompressible and compressible kinetic-energy spectra, wave-occupation spectra, spectral fluxes, vortex-resolved correlations, and velocity statistics. These diagnostics distinguish equilibrium vortex organization, decaying turbulent relaxation, forced cascade dynamics, and weak-wave turbulence, and show why power laws alone are insufficient evidence for a cascade. We survey experiments on two-dimensional Onsager clustering, three-dimensional vortex-line turbulence, box-trap wave cascades, engineered dissipation, and turbulent equations of state. We close by briefly placing the contact-interaction scalar superfluid system in a broader landscape of nonlocal, multicomponent, fermionic, and driven-dissipative quantum fluids, where turbulence concepts can be tested for universality.