Indirect coupling of magnons by cavity photons

The interaction between two magnetic spheres in microwave cavities is studied by Mie scattering theory beyond the magnetostatic and rotating wave approximations. We demonstrate that two spatially separated dielectric and magnetic spheres can be strongly coupled over a long distance by the electric field component of standing microwave cavity modes. The interactions split acoustical (dark) and optical (bright) modes in a way that can be mapped on a molecular orbital theory of the hydrogen molecule. Breaking the symmetry by assigning different radii to the two spheres introduces “ionic” character to the magnonic bonds. These results illustrate the coherent and controlled energy exchange between objects in microwave cavities.

Figure 1

A plane electromagnetic wave illuminates a large spherical cavity from an arbitray direction. The latter is modeled by a dielectric spherical shell of radius $R$, thickness $\delta$, and permittivity ${\epsilon }_c$. Two magnetic spheres of radius $a_1$ and $a_2$ are located at antinodes of the ac magnetic field of the (2,2) and (2,$-2$) confinement modes of the cavity, i.e., at ${\mathbf{d}}_1$ and ${\mathbf{d}}_2$ on the $\mathbf{x}$ axis. A constant magnetic field $H_0$ saturates the equilibrium magnetizations. The scattered waves are measured by a detector in the far field as a function of the scattering angles, here $\left(\theta ,\phi \right)=\left(\pi /2,\pi \right)$.

Figure 2

(a) The scattering efficiency factor Eq. (33) for two nonmagnetic dielectric spheres of radius $a_1=a_2=1$ mm, cavity radius $R=4$ mm, and asymmetry $\delta n=1$ plotted as a function of frequency $\omega /2\pi$ and average refractive index $n_{\mathrm{sp}}$. (b), (c) The scattering intensity $|S_1{|}^2$ as function of scattering angle $\theta$ and frequency $\omega /2\pi$ plotted for the same spheres ($n_{\mathrm{sp}}=7,\delta n=1$) without and with cavity, respectively, while (d) and (e) are the corresponding scattering efficiencies. The anticrossing in (e) reveals the interaction with the cavity field by the coupling strength $2g_{\mathrm{eff}}^{\mathrm{di}}$, i.e., the frequency splitting of the modes at $\delta n=0$. The dashed lines are guides for the eye.

Figure 3

Scattering efficiency factor $Q_{\mathrm{sca}}$ as function of magnetic field $H_0/M_s$ and frequency $\omega /2\pi$ for two YIG spheres of radius $a_1=a_2=0.5$ mm and relative permittivity ${\epsilon }_{\mathrm{sp}}/{\epsilon }_0=15$ in a spherical cavity of radius $R=4$ mm on the two antinodes of the cavity mode at ${\omega }_2/2\pi \sim 7.05$ GHz is shown in bottom panel. The field at each sphere is detuned by $\left|\delta H\right|/M_s\sim 0.2$ with opposite signs. $g_{\mathrm{eff}}^{\mathrm{mag}}$ is the magnon-cavity coupling strength. The radial component of microwave magnetic field $h_r$ for the cavity mode frequency ${\omega }_2$ in the equator plane is shown in the top panel, the black circles indicate two spheres.

Figure 4

(a), (b) Scattering efficiency factor $Q_{\mathrm{sca}}$ as function of $\omega /2\pi$ and $\delta H/M_s$ for the same two spheres as Fig. 3 without and with cavity, respectively, but the detuning is much smaller than in Fig. 3. $H_0/M_s=1$ is fixed such that the magnetostatic modes of each sphere are detuned from ${\omega }_1$. The anticrossing in (b) illustrates the coupling of the two YIG spheres; the nonlocal magnon-magnon coupling strength $g_{\mathrm{eff}}^{\mathrm{ind}.\mathrm{mag}}$ is the frequency splitting of the modes at $\delta H=0$. The azimuthal component of microwave magnetic field $h_{\varphi }$ for the cavity mode frequency ${\omega }_1$ in the equator plane is shown in top panel, the black circles indicate two spheres.

Figure 5

Scattering efficiency as function of frequency $\omega /2\pi$ and normalized bias field $\delta H/M_s$ for two YIG spheres (${\epsilon }_{\mathrm{sp}}/{\epsilon }_0=15$) with radius $a_1=0.5$ mm and $a_2=1$ mm (a) without cavity and (b) in a spherical cavity of radius $R=4$ mm. The white arrow in (b) illustrates the “indirect gap” induced by the radiative coupling.

Figure 6

Energy-level diagram describing the magnon hybridization in analogy with chemical bonds resulting from the interaction between two spheres via microwave cavity modes. (a) Magnonic homodimer consists of two similar magnetic spheres subjected to local magnetic fields $H_{1\left(2\right)}=H_0\pm \delta H$, and (b) magnonic heterodimer consists of two dissimilar magnetic spheres. In (a), magnon hybridization only occurs between magnonic states of the same angular momentum denoted by $n_i$, while the reduced symmetry in a heterodimer introduces coupling between all modes. In a homodimer, the bonding level is dark since it has no dipole moment, while the antibonding level is bright. In a heterodimer, all modes are visible. The arrows in circles indicate the relative magnonic phase (not spin or equilibrium magnetization).

Figure 7

Same as Figs. 3 and 4 but for relatively large YIG spheres of radius $a_1=a_2=1.25$ mm. The cavity modes are now strongly mixed with those confined in the two YIG spheres. In (a), the modes are shifted relative to each other by $\delta H/M_s\sim 0.7$ and $H_0/M_s\sim 6$ in panels (b) (no cavity) and (c).