$\mathcal{PT}$-symmetric magnon laser in cavity optomagnonics

Nonlinear magnonics based on the coupling of magnons and optical photons has become a crucial topic in the quantum magnonic field. Here, we demonstrate the magnon laser, the analog of the optical laser in magnonics, in a parity-time ($\mathcal{PT}$) symmetric cavity optomagnonical system involving active and passive optical whispering gallery modes and the magnon mode. We show that the stimulated emitted magnon number can be coherently amplified above a threshold driving power in the $\mathcal{PT}$-symmetric regime where the splitting of optical supermodes is resonant with the magnon frequency, indicating the occurrence of the magnon laser. And this magnon laser action can work with low threshold when the system approaches the optical gain-loss balance and when increasing the optomagnonic coupling. Furthermore, the frequency of the magnon laser can be continuously tuned by adjusting the applied magnetic field strength, which is a distinct feature compared to the photon or phonon amplification by stimulated emission. The study of the $\mathcal{PT}$-symmetric magnon laser may promote the intersection and merging of various disciplines like non-Hermitian physics and quantum magnonics, and provide the theoretical basis and reference for the research and development of new frequency-tunable laser devices.

Figure 1

(a) Schematic diagram of the $\mathcal{PT}$-symmetric cavity optomagnonical system. The ferrimagnet YIG sphere supports optical WGM ${\widehat{a}}_2$ and magnon mode ${\widehat{S}}_x$ and couples to an active optical WGM microcavity ${\widehat{a}}_1$. The input light evanescently couples to the active optical WGM via a tapered fiber. The two optical supermodes ${\omega }_{+}$ and ${\omega }_{-}$ caused by the coupled optical WGMs interact with the magnon mode with frequency ${\omega }_m$, i.e., ${\omega }_{+}\leftrightarrow {\omega }_{-}+{\omega }_m$. (b) The physical model of three-mode interaction.

Figure 2

(a) Mode splitting $({\omega }_{\pm }-{\omega }_c)/{\kappa }_2$ and (b) linewidth ${\gamma }_{\pm }/{\kappa }_2$ of the eigenmodes vary with the optical tunneling rate $J/{\kappa }_2$ under the condition of ${\kappa }_1={\kappa }_2$. Inset map in (b): The phase diagram displays the EP, $\mathcal{PT}$-symmetry phase (PTSP), and $\mathcal{PT}$-symmetry-breaking phase (PTBP).

Figure 3

The magnon gain $\mathcal{G}$ as a function of the incident power $P_{\mathrm{in}}$ and (a) optical gain-loss ratio ${\kappa }_1/{\kappa }_2$ and (b) the optomagnonic coupling strength $G$. The black solid lines represent the threshold power $P_{\mathrm{th}}$ evolved by (a) ${\kappa }_1/{\kappa }_2$ and (b) $G$ subject to $\mathcal{G}={\gamma }_m/2$. We use $G=1\mathrm{Hz}$ [53, 54, 55] in (a) and ${\kappa }_1/{\kappa }_2=0.9$ in (b), and the other system parameters are ${\omega }_m/2\pi =1\mathrm{GHz}, {\kappa }_2/2\pi =15\mathrm{MHz}, J=\frac{1}{2}\sqrt{{\omega }_m^2+{({\kappa }_1+{\kappa }_2)}^2/4}, {\gamma }_m/2\pi =0.1\mathrm{MHz}, S=1\times {10}^{10}$ [20, 21, 38].

Figure 4

Magnon laser as seen through the stimulated emitted magnon number $M\left(\tau \right)$ varied with the input power $P_{\mathrm{in}}$ for the different gain coefficients ${\kappa }_1$ under the matching condition $\mathrm{\Delta }\omega ={\omega }_m$. The other system parameters are the same as Fig. 3.

Figure 5

The magnon emission line shape function below and above the threshold value in the context of gain-loss ratio ${\kappa }_1/{\kappa }_2=0.9$ [(a) and (c)] and ${\kappa }_1/{\kappa }_2=0.95$ [(b) and (d)]. The input powers in (a)–(d) are 10, 7, 70, and $47\mathrm{mW}$ respectively. The other system parameters are the same as Fig. 3.

Figure 6

(a) Stimulated emitted magnon number $M\left(\tau \right)$ (in logarithmic form) as a function of the optomagnonic coupling strength $G$ and the laser frequency. (b)–(d) plot the magnon line shape function in the cases of $G=1$, 2, and $1.5\mathrm{Hz}$, respectively. We choose $P_{\mathrm{in}}=20\mathrm{mW}, {\kappa }_1/{\kappa }_2=0.9$, and the other system parameters are the same as Fig. 3.

Figure 7

The line shape function of stimulated emitted magnon number $M\left(\tau \right)$ in the case of different magnetic field strengths $H_0$. The curves with different colors, denoting the magnon lasers with different frequencies, correspond to the different applied magnetic fields. Here ${\kappa }_1/{\kappa }_2=0.9, P_{\mathrm{in}}=50\mathrm{mW}$, and the other system parameters are are the same as Fig. 3.