Prediction of Attractive Level Crossing via a Dissipative Mode

The new field of spin cavitronics focuses on the interaction between the magnon excitation of a magnetic element and the electromagnetic wave in a microwave cavity. In the strong interaction regime, such an interaction usually gives rise to the level anticrossing for the magnonic and the electromagnetic mode. Recently, the attractive level crossing has been observed, and it is explained by a non-Hermitian model Hamiltonian. However, the mechanism of such attractive coupling is still unclear. We reveal the secret by using a simple model with two harmonic oscillators coupled to a third oscillator with large dissipation. We further identify this dissipative third party as the invisible cavity mode with large leakage in cavity-magnon experiments. This understanding enables one to design dissipative coupling in all sorts of coupled systems.

Figure 1

(a)(1) Harmonic oscillators with reactive coupling via a direct spring, (a)(2) dissipative coupling via viscous forces, and (a)(3) dissipative coupling via a third oscillator $m_0$ in contact with the friction surface. (b) The three-circuit model with three $RLC$ circuits. Both circuit 1 (blue) and circuit 2 (green) are coupled reactively to a highly dissipative third party, circuit 3 (red), realizing dissipative coupling between circuits 1 and 2. The circuits 0, 1, 2 here are equivalent to the high dissipative ${\mathrm{TE}}_x^{11}$ mode, the low dissipative cavity ${\mathrm{TE}}_y^{11}$ mode, and the YIG magnon mode in Fig. 4. (c) The eigenfrequencies for the three-circuit model with ${\omega }_0={\omega }_1$, ${\lambda }_i=1$, ${\kappa }_{12}=0$, ${\kappa }_{0i}=0.11$, ${\gamma }_0/{\omega }_1=0.35$, and ${\gamma }_1/{\omega }_1={\gamma }_2/2{\omega }_1=0.001$. (Insets) The imaginary part of the eigenfrequencies (bottom panel) and the relative phase (top panel) between circuits 1 and 2.

Figure 2

Transmission spectra $S_{21}$ calculated from input-output theory. (Left panel) Direct coupling between modes 1 and 2 (${\kappa }_{12,10}\ne 0$, ${\kappa }_{20}=0$) gives rise to repulsive coupling. (Right panel) Indirect coupling between modes 1 and 2 via dissipative mode 0 (${\kappa }_{10,20}\ne 0$, ${\kappa }_{12}=0$) leads to attractive coupling (see the Supplemental Material [22] for calculation details and parameters). The color scale is in decibels using $10\log |S_{21}{|}^2$. The white dashed lines are the eigenfrequencies calculated directly from the three-mode Hamiltonian (4).

Figure 3

(a) The magnetic field distribution for the $f=9.16\mathrm{GHz}$ cavity mode in a directional coupler with the microwaves injected from port 1. The length of the coupler is 40 mm, the width of each stripe is 2.5 mm, and the relative dielectric constant of the substrate is ${ϵ}_r=3.38$. (b) The simulated transmission spectrum when the YIG sphere is placed at points $A$, $B$, $C$, $D$, respectively. All material parameters are the same as those in Ref. [19].

Figure 4

(a) YIG sphere in a circular waveguide. (b) The magnetic field distribution across the cross section. The magnetic field vanishes along $\widehat{\mathbf{x}}$ for the cavity ${\mathrm{TE}}_y^{11}$ mode (left panel) and vanishes along $\widehat{\mathbf{y}}$ for the ${\mathrm{TE}}_x^{11}$ mode (right panel). When the YIG sphere is placed at point $A$ ($B$), it couples strongly with ${\mathrm{TE}}_x^{11}$ (${\mathrm{TE}}_y^{11}$) mode.