Coupled spin-light dynamics in cavity optomagnonics

Experiments during the past 2 years have shown strong resonant photon-magnon coupling in microwave cavities, while coupling in the optical regime was demonstrated very recently for the first time. Unlike with microwaves, the coupling in optical cavities is parametric, akin to optomechanical systems. This line of research promises to evolve into a new field of optomagnonics, aimed at the coherent manipulation of elementary magnetic excitations in solid-state systems by optical means. In this work we derive the microscopic optomagnonic Hamiltonian. In the linear regime the system reduces to the well-known optomechanical case, with remarkably large coupling. Going beyond that, we study the optically induced nonlinear classical dynamics of a macrospin. In the fast-cavity regime we obtain an effective equation of motion for the spin and show that the light field induces a dissipative term reminiscent of Gilbert damping. The induced dissipation coefficient, however, can change sign on the Bloch sphere, giving rise to self-sustained oscillations. When the full dynamics of the system is considered, the system can enter a chaotic regime by successive period doubling of the oscillations.

Figure 1

Schematic configuration of the model considered. (a) Optomagnonic cavity with homogeneous magnetization along the $z$ axis and a localized optical mode with circular polarization in the y-z plane. (b) The homogeneous magnon mode couples to the optical mode with strength $G$. (c) Representation of the magnon mode as a macroscopic spin on the Bloch sphere, whose dynamics is controlled by the coupling to the driven optical mode.

Figure 2

Spin dynamics (fast-cavity limit) at blue detuning $\mathrm{\Delta }=\mathrm{\Omega }$ and fixed $GS/\mathrm{\Omega }=2$, $\kappa /\mathrm{\Omega }=5$, ${\eta }_{\mathrm{G}}=0$. The left column depicts the trajectory (green solid line) of a spin (initially pointing near the north pole) on the Bloch sphere. The color scale indicates the optical damping ${\eta }_{\mathrm{opt}}$. The right column shows a stereographic projection of the spin's trajectory (red solid line). The black dotted line indicates the equator (invariant under the mapping), while the north pole is mapped to infinity. The stream lines of the spin flow are also depicted (blue arrows). (a) Magnetization switching behavior for light intensity $G{\alpha }_{\max }^2/\mathrm{\Omega }=0.36$. (b) Limit-cycle attractor for larger light intensity $G{\alpha }_{\max }^2/\mathrm{\Omega }=0.64$.

Figure 3

Full nonlinear spin dynamics and route to chaos for $GS/\mathrm{\Omega }=3$ and $G{\alpha }_{\max }^2/\mathrm{\Omega }=1\left({\eta }_{\mathrm{G}}=0\right)$. The system is blue detuned by $\mathrm{\Delta }=\mathrm{\Omega }$ and the dynamics, after a transient, takes place in the southern hemisphere. The solid red curves represent the spin trajectory after the initial transient on the Bloch sphere for (a) $\kappa /\mathrm{\Omega }=3$, (b) $\kappa /\mathrm{\Omega }=2$, (c) $\kappa /\mathrm{\Omega }=0.9$, (d) $\kappa /\mathrm{\Omega }=0.5$. (e) $S_z$ projection as a function of time for the chaotic case $\kappa /\mathrm{\Omega }=0.5$.

Figure 4

Bifurcation density plot for $GS/\mathrm{\Omega }=3$ and $\kappa /\mathrm{\Omega }=1$ at $\mathrm{\Delta }=\mathrm{\Omega }\left({\eta }_{\mathrm{G}}=0\right)$, as a function of light intensity. We plot the $S_z$ values attained at the turning points ($\dot{S}{}_z=0$). For other possible choices (e.g., $\dot{S}{}_x=0$) the overall shape of the bifurcation diagram is changed, but the bifurcations and chaotic regimes remain at the same light intensities. For the plot, 30 different random initial conditions were taken.

Figure 5

Phase diagram for blue detuning with $\mathrm{\Delta }=\mathrm{\Omega }$, as a function of the inverse coupling strength $\mathrm{\Omega }/GS$ and the optically induced field $B_{{\alpha }_{\max }}/\mathrm{\Omega }=G{\alpha }_{\max }^2/\mathrm{\Omega }$. Boundaries are qualitative. Switching, in white, refers to a fixed-point solution with the spin pointing near the south pole. Limit cycles in the $xy$ plane are shaded in blue, and they follow the optomechanical attractor diagram discussed in Ref. [40]. For higher $B_{{\alpha }_{\max }}$, chaos can ensue. Orange denotes the parameter space in which limit cycles deviate markedly from optomechanical predictions. These are not in the $xy$ plane and also undergo period doubling, leading to chaos. In red is depicted the area where pockets of chaos can be found. For large $B_{{\alpha }_{\max }}/\mathrm{\Omega }$, the limit cycles are in the $yz$ plane. In the case of red detuning $\mathrm{\Delta }=-\mathrm{\Omega }$, the phase diagram remains as is, except that instead of switching there is a fixed point near the north pole.

Figure 6

Bifurcation density plot for $G{\alpha }_{\max }^2/\mathrm{\Omega }=1$ and $\kappa /\mathrm{\Omega }=1$ at $\mathrm{\Delta }=\mathrm{\Omega }\left({\eta }_{\mathrm{G}}=0\right)$, as a function of the relative coupling strength $GS/\mathrm{\Omega }$. The dash-dotted blue line indicates $GS/\mathrm{\Omega }=3$, for comparison with Fig. 4. As in the main text, the points (obtained after the transient) are given by plotting the values of $S_z$ attained whenever the trajectory fulfills the turning point condition $\dot{S}{}_z=0$, for 20 different random initial conditions.

Figure 7

Bifurcation density plot for $GS/\mathrm{\Omega }=3$, $G{\alpha }_{\max }^2/\mathrm{\Omega }=1$ and $\kappa /\mathrm{\Omega }=1\left({\eta }_{\mathrm{G}}=0\right)$, as a function of the detuning $\mathrm{\Delta }/\mathrm{\Omega }$. The dotted blue line indicates $\mathrm{\Delta }/\mathrm{\Omega }=1$, for comparison with Fig. 4.

Figure 8

Attractor diagram for $\mathrm{\Delta }=1.5\mathrm{\Omega }$ and $\kappa /\mathrm{\Omega }=1$ with condition $G^{2}S{\left|{\alpha }_{\max }\right|}^2=n{\mathrm{\Omega }}^2$. Top, $n=1$; bottom, $n=10$. We plot the $S_x$ values attained at the turning points (${\dot{S}}_x=0$) for $S_x>0$. The diagram is symmetric for $S_x<0$, as expected for a limit cycle on the Bloch sphere. The diagram at the top coincides to a high degree of approximation with the predictions obtained for optomechanical systems (i.e., replacing the spin by a harmonic oscillator). In contrast, this is no longer the case for the diagram at the bottom, which involves higher light intensities.