The Orbital Angular Momentum of Azimuthal Spin-Waves

Abstract

In the context of a growing interdisciplinary interest in the angular momentum of wave fields, the spin-wave case has yet to be fully explored, with the extensively studied notion of spin transport being only part of the broader picture. Here we report experimental evidence for magnon orbital angular momentum, demonstrating that the mode exhibits rotation rather than remaining stationary. This conclusion is drawn from observations of the lifted degeneracy of waves with counter-rotating wave fronts. This requires an unambiguous formulation of spin and orbital angular momenta for spin waves, which we provide in full generality based on a systematic application of quantum field theory techniques. The results unequivocally establish magnetic dipole-dipole interactions as a magnetic-field controllable spin-orbit interaction for magnons. Our findings open a new research direction, leveraging the spectroscopic readability of angular momentum for azimuthal spin waves and other related systems.

Figures (2)

• Figure 1: a) Graphical representation of the precession pattern of ${\bm m} ({\bm x})$ for SW modes labeled $(n_R,n_J)$. b) MRFM (Magnetic Resonance Force Microscopy) spectroscopy as a function of normal magnetic field and frequency on a YIG disk. c) Line cuts at field values indicated by vertical lines of respective colors in b). The split between the $(0,2)$ and $(0,0)$ peaks defines the SOI. d) Magnetic field dependence of the SOI. The dots are the experimental points, while the solid lines are theoretical predictions calculated by Eq. (\ref{['eq:SOI']}) for 3 different values of $\mu_0 M_s=$0.167, 0.170, and 0.173 T corresponding to its uncertainty range.
• Figure 2: (a) Illustration of the azimuthal pattern of the dynamical DDI. The stray magnetic field profile at a fixed radius from the counterclockwise rotating magnetic dipole that generates it is a $3:1$ superposition of the $(n_L = +2,\, n_S = -1)$ and $(n_L = 0,\, n_S = +1)$ in the OAM eigenmode basis. (b) Coupling (wavy line) of the two OAM eigenmodes with $n_J =2$via dynamical DDI. The asterisk helps count $n_L$: the repetition of a given orientation (here at $0^\circ$, cf. color wheel). (c) Comparison of the dynamical DDI coupling for two states of opposite OAM $n_L=\pm 1$. The red(blue) dotted line shows the dispersion of the $n_R=0$ eigenstates of $\hat{\mathcal{O}}_0$ for $n_S=+1$($n_S=-1$). The unperturbed $(n_L=\pm 1,n_S=+1)$ eigenstates undergo energy lowering through hybridization mediated by $\hat{\mathcal{O}}_d^{(-)}$ with $n_S=-1$ states of different $|n_L|$. This gives rise to two distinct elliptically precessing states (magenta levels) depending on the polarity of $n_L$. The frequency shifts are inversely proportional to the gap between the $n_S =\pm 1$ branches, resulting in a SOI splitting that increases when the applied magnetic field approaches saturation.