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arXiv:1601.00446 · Physical Review A

Direct and inverse cascades of spin-wave turbulence in spin-1 ferromagnetic spinor Bose-Einstein condensates

Kazuya Fujimoto, Makoto Tsubota

Published Mar 11, 2016• 16 citations
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Abstract

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.

DOI
10.1103/physreva.93.033620

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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\ensuremath{-}7/3\ensuremath−7/3 and \ensuremath−5/3\ensuremath{-}5/3\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 theoretically and numerically study spin turbulence (ST) with small spin magnitude in spin-1 spinor Bose-Einstein condensates by using the spin-1 spinor Gross-Pitaevskii (GP) equations. This kind of ST is realized in two cases: (i) with antiferromagnetic interaction and (ii) with ferromagnetic interaction under a static magnetic field. The ST with small spin magnitude can exhibit two characteristic power laws in the spectrum of the spin-dependent interaction energy: $\ensuremath{-}1$ and $\ensuremath{-}7/3$ power laws in the low- and high-wave-number regions, respectively. These power laws are derived from a Kolmogorov-type dimensional scaling analysis for the equations of motion of the spin vector and nematic tensor. To confirm these power laws, we perform a numerical calculation of the spin-1 spinor GP equations in a two-dimensional uniform system. In case (i), the $\ensuremath{-}7/3$ power law tends to appear in the high-wave-number region, but the spectrum in the low-wave-number region deviates from the $\ensuremath{-}1$ power law. In contrast, both the $\ensuremath{-}1$ and $\ensuremath{-}7/3$ power laws tend to appear in case (ii).
We theoretically study counterflow instability and turbulence in a spin-1 spinor Bose-Einstein condensate by the Gross-Pitaevskii equation and the Bogoliubov-de Gennes equation. Our study considers (i) the dynamics induced by the counterflow of two components with different magnetic quantum numbers, which leads to turbulence with spin degrees of freedom, and (ii) the properties of the turbulence. For (i), the behavior of the condensate induced by the counterflow strongly depends on whether the spin-dependent interaction is ferromagnetic or antiferromagnetic, leading to different behaviors for the dispersion relation and the spin density vector, etc. For (ii), we numerically calculate the spectrum of the spin-dependent interaction energy, which also depends on the spin-dependent interaction. The spectrum of the spin-dependent interaction energy in the ferromagnetic case clearly exhibits a $\ensuremath{-}7/3$ power law, which can be explained by scaling analysis. The spectrum in the antiferromagnetic case seems to show some power law too, but the power exponent cannot be estimated as definitely as the ferromagnetic case.
Spin-superflow turbulence (SST) in spin-1 ferromagnetic spinor Bose-Einstein condensates is theoretically and numerically studied by using the spin-1 spinor Gross-Pitaevskii (GP) equations. SST is turbulence in which the disturbed spin and superfluid velocity fields are coupled. Applying the Kolmogorov-type dimensional scaling analysis to the hydrodynamic equations of spin and velocity fields, we theoretically find that the $\ensuremath{-}5/3$ and $\ensuremath{-}7/3$ power laws can appear in spectra of the superflow kinetic and the spin-dependent interaction energies, respectively. Our numerical calculation of the GP equations with a phenomenological small-scale energy dissipation confirms SST with the coexistence of disturbed spin and superfluid velocity field with two power laws.
We perform a numerical study of spin turbulence in a two-dimensional trapped spin-1 spinor Bose-Einstein condensate, focusing on the energy spectrum. The spin turbulence in the trapped system is generated by instability of the helical structure of the spin density vector in the initial state. Our numerical calculation finds that in the trapped system, the spectrum of the spin-dependent interaction energy in the ferromagnetic case exhibits a $\ensuremath{-}7/3$ power law, which was confirmed in a uniform system by our previous study. The relation between the $\ensuremath{-}7/3$ power law and the motion of the spin density vector is discussed by investigating the orbits of dynamical variables in the spin space.
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.
When a Bose-Einstein condensate (BEC) is driven out of equilibrium, density waves interact nonlinearly and trigger turbulent cascades. In a turbulent BEC, energy is transferred toward small scales by a direct cascade, whereas the number of particles displays an inverse cascade toward large scales. In this work, we study analytically and numerically the direct and inverse cascades in wave-turbulent BECs. We analytically derive the Kolmogorov-Zakharov spectra, including the log correction to the direct cascade scaling and the universal prefactor constants for both cascades. We test and corroborate our predictions using high-resolution numerical simulations of the forced-dissipated Gross-Pitaevskii model in a periodic box and the corresponding wave-kinetic equation. Theoretical predictions and data are in excellent agreement, without adjustable parameters. Moreover, in order to connect with experiments, we test and validate our theoretical predictions using the Gross-Pitaevskii model with a confining cubic trap. Our results explain previous experimental observations and suggest new settings for future studies.