Cavity-mediated dissipative spin-spin coupling

We study dissipative spin-spin coupling in a dispersive regime mediated by virtual photons in a microwave cavity. Dissipative coupling between spin of each magnetic material and the cavity photons is established by means of two phase-shifted driving forces acting on each spin. We show that mode-level attraction between two spin modes can be reached when one of the spins is dissipatively coupled to the cavity. By tuning the phase parameter at each ferromagnetic insulator, we can shift the order of “dark” and “bright” collective modes with the phase difference equal to 0 or $\pi$. Moreover, by selectively applying the phase-shifted field, it is possible to construct dark and bright collective modes with the phase difference equal to $\pm \pi /2.$

Figure 1

Schematic picture of the system. Two YIG films (blue) are on top of strip lines (yellow). The green arrows show the direction of ${\mathbf{H}}_{0,j}.$ The total driving field is the sum of the magnetic component of the microwave in the cavity ($\mathbf{h}{e}^{-i\omega t}$) and local fields acting on the YIGs with controllable phase and amplitude (${\mathbf{h}}_j^{\mathrm{\Phi }}={\mathbf{h}}_j{e}^{i{\mathrm{\Phi }}_j}$).

Figure 2

(a)–(c) The transmission amplitude in the case of opposite detuning of the external magnetic field at each FI, where ${\omega }_{r,1}/{\omega }_c={\omega }_r/{\omega }_c+\delta \omega /{\omega }_c,{\omega }_{r,2}/{\omega }_c={\omega }_r/{\omega }_c-\delta \omega /{\omega }_c$, with $\delta \omega =0.05{\omega }_c.$ The phase parameters are for (a) ${\mathrm{\Phi }}_1={\mathrm{\Phi }}_2=0,$ (b) ${\mathrm{\Phi }}_1=\pi ,{\mathrm{\Phi }}_2=0,$ and (c) ${\mathrm{\Phi }}_1={\mathrm{\Phi }}_2=\pi .$ The dashed lines represent the normalized spectrum ($\mathrm{Re}\left(\omega \right)/{\omega }_c$). (d)–(f) The normalized damping of the system $[\mathrm{Im}\left(\omega \right)/\mathrm{Re}\left(\omega \right)]$ corresponding to the parameters for (a)–(c), respectively.

Figure 3

The first column shows the transmission amplitude dependence on the applied field detuning when ${\omega }_{r,1}=1.05{\omega }_c+\delta \omega$ and ${\omega }_{r,2}=1.05{\omega }_c-\delta \omega .$ The dashed lines depict the normalized spectrum $[\mathrm{Re}\left(\omega \right)/{\omega }_c]$. The second column is the zoom of the white dotted boxes in the corresponding plots in the first column, where the arrows show the relative phases of the precessing spins. The dotted line marks $\mathrm{Re}\left(\omega \right)={\omega }_c.$ The third column shows the normalized damping $[\mathrm{Im}\left(\omega \right)/\mathrm{Re}\left(\omega \right)]$, where the labels “$m_1$” and “$m_2$” stand for spins in FI-1 and FI-2, and “p” indicates cavity photon. The sign $\leftrightarrows$ shows the coupling between them. The parameters are (a)–(c) ${\mathrm{\Phi }}_1={\mathrm{\Phi }}_2=0,$ (d)–(f) ${\mathrm{\Phi }}_1=\pi ,{\mathrm{\Phi }}_2=0,$ (g)–(i) ${\mathrm{\Phi }}_1=0,{\mathrm{\Phi }}_2=\pi ,$ and (j)–(l) ${\mathrm{\Phi }}_1={\mathrm{\Phi }}_2=\pi .$

Figure 4

Dependence of the phase shift on detuning. The upper frame ranges from 0 (EP1) to $2g^2/\mathrm{\Delta }$ (EP2) for (iii) and the lower frame ranges from $-2g^2/\mathrm{\Delta }$ (EP1) to 0 (EP2) for (iv). The arrows in the circles demonstrate the phase difference between the two magnetizations in the collective mode. The red dots correspond to values for the circles.