Nonreciprocal high-order sidebands induced by magnon Kerr nonlinearity

We propose an effective approach for creating robust nonreciprocity of high-order sidebands, including the first-, second-, and third-order sidebands, at microwave frequencies. This approach relies on magnon Kerr nonlinearity in a cavity magnonics system composed of two microwave cavities and one yttrium iron garnet (YIG) sphere. By manipulating the driving power applied on YIG and the frequency detuning between the magnon mode in YIG and the driving field, the effective Kerr nonlinearity can be strengthened, thereby inducing strong transmission nonreciprocity. More interestingly, we find the higher the sideband order, the stronger the transmission nonreciprocity marked by the higher isolation ratio in the optimal detuning regime. Such a series of equally spaced high-order sidebands have potential applications in frequency comblike precision measurement, besides structuring high-performance on-chip nonreciprocal devices.

Figure 1

(a) Schematic diagram of a three-mode cavity magnonics system consisting of two coupled microwave cavities and a YIG sphere. (b) Frequency spectrogram of the cavity magnonics system. Here the driving field (with amplitude $\varepsilon$) applied on YIG sphere and the control fields (with amplitudes ${\varepsilon }_a$, ${\varepsilon }_b$) injected from microwave cavities have the same frequency ${\omega }_d$. Frequencies of the control fields and the probe fields (with amplitudes $s_a$, $s_b$) are detuned from the cavities' frequency by ${\mathrm{\Delta }}_c$ and $\delta$, respectively. And the frequency detuning between the probe field and the control field is ${\mathrm{\Delta }}_p$. There are higher-order sidebands at frequency ${\omega }_d+n{\mathrm{\Delta }}_p$, where $n$ is an integer representing the sideband order.

Figure 2

The steady-state microwave photon numbers ${\left|\alpha \right|}^2$, ${\left|\beta \right|}^2$ and magnon number ${\left|\varpi \right|}^2$ versus with the resonant driving power $P$ on the magnon mode in two cases. (a) Case 1: $P_a=5\mathrm{mW}$, $P_{s_a}=1n\mathrm{W}$, $P_b=P_{s_b}=0$; (b) case 2: $P_b=5\mathrm{mW}$, $P_{s_b}=1n\mathrm{W}$, $P_a=P_{s_a}=0$. Other parameters are given as ${\omega }_c/2\pi =10.1$ GHz, ${\omega }_m/2\pi ={\omega }_d/2\pi =10.06$ GHz, ${\mathrm{\Delta }}_c/2\pi =40$ MHz, ${\mathrm{\Delta }}_m=0$, ${\kappa }_a/2\pi =3.8$ MHz, ${\kappa }_b/2\pi =5$ MHz, ${\kappa }_m/2\pi =20$ MHz, $\delta /2\pi =-12$ MHz, $K/2\pi =3.9\times {10}^{-7}$ Hz, $J/2\pi =12$ MHz, and $g/2\pi =14.4$ MHz.

Figure 3

The logarithmic amplitudes $\mathrm{lo}{g}_{10}\mathrm{S}\left(\omega \right)$ of the transmitted high-order sidebands vs frequency $\omega /{\mathrm{\Delta }}_p$ in case 1 and case 2. Here $P=15\mathrm{mW}$, ${\mathrm{\Delta }}_p/{\mathrm{\Delta }}_c=0.7$; other parameters are the same as described in the caption of Fig. 2.

Figure 4

Transmission coefficients $|t_{ba}{|}^2$ and $|t_{ab}{|}^2$ of the first-order sideband vs frequency detuning $\delta /{\mathrm{\Delta }}_c$. To observe transmission coefficients clearly, $|t_{ab}{|}^2$ in (b) and $|t_{ba}{|}^2$ in (c) have been increased by one order of magnitude. The driving powers $P=0\mathrm{mW}$, $20\mathrm{mW}$, and $30\mathrm{mW}$ in (a), (b), and (c), respectively. Other parameters are the same as described in the caption of Fig. 2.

Figure 5

Transmission coefficients ${\tau }_{ba}$, ${\tau }_{ab}$ and ${\ell }_{ba}$, ${\ell }_{ab}$ of the second- and third-order sidebands vs frequency detuning $\delta /{\mathrm{\Delta }}_c$. To clearly observe the transmission coefficients, the values of ${\tau }_{ab}$ in panel (b) and ${\ell }_{ab}$ in panel (e) have been magnified by three and five orders of magnitude, respectively. Here $P=0$ in panels (a) and (d), $P=20\mathrm{mW}$ in panels (b) and (e), and $P=30\mathrm{mW}$ in panels (c) and (f). Other parameters are the same as described in the caption of Fig. 2.

Figure 6

The isolation ratio $I_t$ of the first-order sideband vs the resonant driving power $P$ and frequency detuning $\delta /{\mathrm{\Delta }}_c$. Parameters are the same as described in the caption of Fig. 2.

Figure 7

The isolation ratios (a) $I_{\tau }$ of the second-order sideband and (b) $I_{\ell }$ of the third-order sideband vs the resonant driving power P and frequency detuning $\delta /{\mathrm{\Delta }}_c$. Parameters are the same as described in the caption of Fig. 2.

Figure 8

The isolation ratios (a) $I_t$, (b) $I_{\tau }$, and (c) $I_{\ell }$ vs $\delta /{\mathrm{\Delta }}_c$ with different detunings ${\mathrm{\Delta }}_m$ between the magnon mode and the correspong driving field. Here $P=25\mathrm{mW}$, and the other parameters are the same as described in the caption of Fig. 2.

Figure 9

The isolation ratios $I_t$, $I_{\tau }$, and $I_{\ell }$ vs $\delta /{\mathrm{\Delta }}_c$ when the magnon mode is resonant with the driving field i.e., ${\mathrm{\Delta }}_m=0$. Here $P=25\mathrm{mW}$ and the other parameters are the same as described in the caption of Fig. 2.