1. INTRODUCTION

Cavity magnonics [1,2], which focuses on the combination of magnetism and optics, has become a new frontier for exploring fundamental physical phenomena and technological revolutions. In this subject, a vital but challenging question is how the optical and magnetic degrees of freedom display a strong interaction. Fortunately, by synthesizing a yttrium iron garnet (YIG) single crystal with high spin density into a microwave cavity [3,4], collective spin excitations in this ferromagnetic material couple strongly with the microwave cavity-photon [5–13]. Simultaneously, these spin excitations possess low magnetic damping [14] as well as a long time of quantum coherence [15,16]. Based on these fascinating optical properties, cavity magnonics interconnecting light with magnetism has exhibited considerable achievements, such as the coherent control of photon flow [17–20], cavity-mediated coupling of phonons and magnons [21], radio-frequency-to-optical transducers [22], and magnetism-based information storage and processing [23].

In addition to providing strong coupling between photon and magnon, an analogous Kerr nonlinearity of a magnon from the magnetocrystalline anisotropy in the YIG crystal would be a field of interest and plays an important role in researching the interactions between the magnons and photons [24–32]. By applying an external pumping to directly drive the YIG crystal, the enhanced magnon Kerr effect is experimentally confirmed by observing the frequency shift of the magnon mode [24]. This high-performance Kerr nonlinearity has also been used to induce various nonlinear phenomena in the cavity magnonical system, including magnon-induced nonreciprocity [33], the photon-blockade effect [34], the tristability of cavity-magnon polaritons [35–37], the quantum entanglement [38], the quantum phase transition [39], the dynamics of magnon chaos [40], and magnon-induced high-order sidebands [41,42]. The optical second-order sideband (OSS) is the first nonlinear sideband of the higher-order sidebands. Different from the first-order sideband of linear phenomenon, the OSS is a basic nonlinear phenomenon. Research on the generation of OSS can deepen our understand of magnon Kerr nonlinearity in cavity magnonical system. This cavity magnonical system is driven by a strong control field and a weak probe field. The dynamics evolution of system can be described by nonlinear Heisenberg–Langevin equations (HLEs). If the probe field is far weaker than the control field, the perturbation technique can be used to solve the nonlinear HLEs and obtain the cavity and magnonical modes at the steady state ${a_s}$ and ${m_s}$. The total analytical solution of the cavity and magnonical modes under both the control field and probe field can be written as $a = {a_s} + \delta a$ and $m = {m_s} + \delta m$.

In this work, we mainly focus on the control of the transmission spectra via magnon Kerr nonlinearity in a cavity magnonical hybrid system consisting of a single small YIG crystal sphere and a three-dimensional (3D) rectangular cavity driven with a weak probe and a strong control field. By using the perturbation technique, we solve the nonlinear HLEs and obtain the exact analytical forms of the transmission intensity. Interestingly, it is found that there are OSSs [43,44] in the present hybrid system due to the magnon Kerr nonlinearity. Concretely, the higher-order sideband processes are such that, when the control light with frequency ${\omega _d}$ and the probe light with frequency ${\omega _p}$ are incident upon the optical microcavity, the transmission spectra in Fig. 1 can be generated with frequencies ${\omega _d} \pm n\Omega$, where $n$ is an integer that represents the order of the sidebands and $\Omega$ is the control-probe field detuning. For example, the first-order upper sideband at the probe pulse frequency ${\omega _d} + \Omega$ originates from anti-Stokes processes, while the second-order upper sideband with frequency ${\omega _d} + 2\Omega$ stems from the magnon Kerr nonlinearity in the present system. Based on the experimentally achievable parameter settings, our simulated results show that the OSS generation can be significantly enhanced via increasing the magnon Kerr nonlinearity even if the coupling between the cavity and magnon is weak. When the cavity-magnon coupling is in the strong-coupling regime, it is shown that the hybrid of cavity and magnon modes induces the formation of the magnonical polaritons, resulting in the observance of the giant enhanced OSS generation and the splitting of the OSS spectra [45–47]. Our research results may also offer an attractive new prospects for the development of magnon frequency comb.

 

Fig. 1. (a) Schematic diagram of a cavity magnon system. This configuration contains a three-dimensional copper cavity and a YIG sphere. (b) The equivalent mode-coupling model of photon mode $a$ and magnon mode $m$ with coupling strength $g$, where ${\kappa _a}$, ${\kappa _m}$ are the dissipation rates of the microwave and magnon modes, respectively. (c) Frequency spectrogram of transmission spectra in the present system. The black and purple lines denote the control field with frequency ${\omega _d}$ and probe field with frequency ${\omega _p}$, respectively, while the brown line indicates the cavity (magnon) resonance frequency ${\omega _a} = {\omega _m}$. ${\Delta _{\textit{ad}}}$(${\Delta _{\textit{md}}}$) denotes the detuning between control field and the cavity mode (magnon mode), and ${\Delta _{\textit{pa}}}$ denotes the detuning between cavity mode and probe field.

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2. MODELS AND EQUATIONS

Let us consider a cavity-magnon hybrid system as shown in Fig. 1. A $250{ \text{-} }\unicode{x00B5}{\rm m}$-diameter highly polished YIG sphere is placed in a 3D copper cavity. Meanwhile, an uniform external magnetic field ($H$) is applied along the $Z$ direction to bias the YIG sphere. The photon-magnon coupling strength ($g$) can be modulated by the direction of bias magnetic field or the position of YIG sphere in the microwave cavity. Considering that the size of the sphere is much smaller than the microwave wavelength, the effect of radiation pressure can be ignored [48]. The cavity is driven by a microwave driving (control) field and a microwave signal (probe) field. The total Hamiltonian [49] of the present hybrid system ($\hbar = 1$) can be written as

Based on Eq. (2), the HLEs can be given by

As is known, it is very difficult to get a steady-state solution to the above Eqs. (3) and (4) since the magnon Kerr nonlinear terms (i.e., $i(2K{m^*}m + K)m$) are included. We will show that these nonlinear terms can give rise to some interesting phenomena in the present system, such as the efficient generation of OSS. As the control field ${\varepsilon _d}$ is much stronger than the probe field ${\varepsilon _p}$, we can use the perturbation method to solve Eqs. (3) and (4). To this end, the total solution of Eqs. (3) and (4) can be described by $a = {a_s} + \delta a$ and $m = {m_s} + \delta m$, where ${a_s}$, ${m_s}$ are the steady-state solutions. The steady-state solutions of the HLEs are obtained as

We now turn to consider the perturbation made by the probe field. The evolution of the perturbation terms $\delta a$ and $\delta m$ can be given by

We solve the problem of inputting a probe field, ${\varepsilon _p}{e^{- i\Omega t}}$, by using the following ansatz: $\delta a = \delta {a^1} + \delta {a^2}$, $\delta m = \delta {m^1} + \delta {m^2}$, where $\delta {a^1} = A_1^ - {e^{- i\Omega t}} + A_1^ + {e^{i\Omega t}}$, $\delta {a^2} = A_2^ - {e^{- 2i\Omega t}} + A_2^ + {e^{2i\Omega t}}$, $\delta {m^1} = M_1^ - {e^{- i\Omega t}} + M_1^ + {e^{i\Omega t}}$, $\delta {m^2} = M_2^ - {e^{- 2i\Omega t}} + M_2^ + {e^{2i\Omega t}}$. The physical picture of such an ansatz is that there are output fields with frequencies ${\omega _d} \pm n\Omega$ generated, where $n$ is an integer that represents the order of the sidebands and $\Omega$ is the control-probe field detuning. So we can simplify the ansatz as follows:

3. RESULTS AND DISCUSSION

In this section, we demonstrate the creation and control of the upper OSS in the cavity-magnon system. It is worth noting that we have defined the dimensionless efficiency of the OSS generation ${\eta _2}$. As a matter of fact, we scale the output intensity of the OSS generation with respect to the intensity of the input probe field. For example, the value of efficiency ${\eta _2} = 1\%$ indicates that the amplitude of the output OSS is 1% of the amplitude of the input probe field, not that 1% of the probe light is being converted into the OSS. First, we analyze the properties of the transmission and the OSS for the case of $g = 0.1{\kappa _a}$ corresponding to the weak photon-magnon coupling. We show in Fig. 2 that the transmission $|{t_p}{|^2}$ of the probe field and the efficiency ${\eta _2}$ of the upper OSS process vary with the detuning $\Omega$ for different magnon Kerr nonlinear strength $K$. It can be found from Fig. 2(a) that the spectrum of $|{t_p}{|^2}$ exhibits a single deep valley when the Kerr nonlinear strength is $K/{\kappa _a} = 5 \times {10^{- 17}}$, which indicates that the probe field is absorbed significantly at the position $\Omega = 80{\kappa _a}$. Accordingly, the OSS spectrum ${\eta _2}$ shown in Fig. 2(c) exhibits a sharp peak. When the Kerr nonlinear strength $K$ increases to $5 \times {10^{- 15}}$, it can be found from Figs. 2(b) and 2(d) that the transmission remains a same absorption valley and the OSS spectrum exhibits a higher peak, which implies that the OSS generation can enhanced significantly as the magnon Kerr nonlinearity increases. As shown in Fig. 2(d), the peak efficiency of ${\eta _2}$ can be enhanced with 2 orders of magnitude compared with that in Fig. 2(c). Figure 2(e) clearly illustrates that the larger magnon Kerr nonlinearity induces more pronounced OSS generation, which indicates a strong Kerr nonlinearity is beneficial for amplifying the OSS conversion efficiency even if the cavity photon-magnon coupling is in the weak-coupling regime.

 

Fig. 2. Transmission rate of the probe pulse $|{t_p}{|^2}$ and the logarithm of ${\eta _2}$ versus the detuning $\Omega /{\kappa _a}$. In panels (a) and (c), we use $K/{\kappa _a} = 5 \times {10^{- 17}}$. In panels (b) and (d), we use $K/{\kappa _\alpha} = 5 \times {10^{- 15}}$. (e) The logarithm of ${\eta _2}$ versus the detuning $\Omega /{\kappa _a}$ and the logarithm of magnon Kerr nonlinear coefficient $K/{\kappa _a}$. The other parameters are ${\omega _a}/2\pi = 7.86\;{\rm{GHz}}$, ${\omega _m}/2\pi = 7.86\;{\rm{GHz}}$, $2{\kappa _a}/2\pi = 3.35\;{\rm{MHz}}$, $2{\kappa _m}/2\pi = 1.12\;{\rm{MHz}}$ [50], ${P_d} = 1\;{\rm{W}}$, $g = 0.1{\kappa _a}$, ${\Delta _{\textit{ad}}} = {\Delta _{\textit{md}}} = 80{\kappa _a}$, $\kappa = 0.5{\kappa _a}$.

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When the coupling between magnon and the cavity is pushed into strong-coupling regime, i.e., $g = 12{\kappa _a}$, it can be seen from Fig. 3(a) that a wide transparent window near $\Omega = 80{\kappa _a}$ and two slightly asymmetry absorbed valleys appear on both sides of the transparent window. Accordingly, the enhanced OSS generation can be observed even if the magnon Kerr nonlinearity is weak ($K/{\kappa _\alpha} = 5 \times {10^{- 17}}$) and the OSS spectrum shown in Fig. 3(c) exhibits two-peak line shape. When the magnon Kerr nonlinear strength $K$ increases to $K/{\kappa _a} = 5 \times {10^{- 15}}$, Figs. 3(b) and 3(d) show that the transmission and OSS spectra still keep a similar spectral shape with Figs. 3(a) and 3(c). Further, it can be seen that the maximum efficiency of ${\eta _2}$ in Fig. 3(d) can be enhanced 2 orders of magnitude compared with that in Fig. 3(c). Direct comparison in Figs. 2 and 3 implies that the maximum value in the conversion efficiency of the OSS can be enhanced and the splitting of the transmission and OSS spectra occurs when the cavity photon-magnon coupling is in the strong-coupling regime. In other words, the strong coupling between cavity photon and magnon induces the two-color OSS generation in the present cavity magnonical system with magnon Kerr nonlinearity. Furthermore, Fig. 3(e) clearly illustrates that the larger magnon Kerr nonlinearity also leads to more pronounced OSS generation when the cavity photon-magnon coupling is in the strong-coupling regime.

 

Fig. 3. Transmission rate of the probe pulse $|{t_p}{|^2}$ and the logarithm of ${\eta _2}$ versus the detuning $\Omega /{\kappa _a}$. The other parameter is $g = 12{\kappa _a}$. In panels (a) and (c), we use $K/{\kappa _\alpha} = 5 \times {10^{- 17}}$. In panels (b) and (d), we use $K/{\kappa _\alpha} = 5 \times {10^{- 15}}$. (e) The logarithm of ${\eta _2}$ versus the detuning $\Omega /{\kappa _a}$ and the logarithm of magnon Kerr nonlinear coefficient $K/{\kappa _a}$. Other parameters are the same as in Fig. 2.

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The physical picture of such spectra splitting is rather clear. Under the condition of switching off the control (probe) field ${\varepsilon _j} = 0(j = d,p)$, the Hamiltonian in Eq. (2) can be rewritten as [33,35,36],

The Hamiltonian $\hat H^\prime $ can be diagonalized via a Bogoliubov transformation with ${\hat B_ +} = u\hat m - v\hat a$, $\hat B_ + ^\dagger = u{\hat m^\dagger} - v{\hat a^\dagger}$, ${\hat B_ -} = u\hat a + v\hat m$, and $\hat B_ - ^\dagger = u{\hat a^\dagger} + v{\hat m^\dagger}$, where ${\hat B_ \pm}$ represents polariton modes including both $\hat a$ (cavity mode) and $\hat m$ (magnon mode) components with the definition ${u^2} + {v^2} = 1$. Thus, we can conclude that $\hat m = u{\hat B_ +} + v{\hat B_ -}$, ${\hat m^\dagger} = u\hat B_ + ^\dagger + v\hat B_ - ^\dagger$, $\hat a = u{\hat B_ -} - v{\hat B_ +}$, and ${\hat a^\dagger} = u\hat B_ - ^\dagger - v\hat B_ + ^\dagger$. After a Bogoliubov transformation, the Hamiltonian $\hat H^\prime $ can be given by

 

Fig. 4. (a) The logarithm of ${\eta _2}$ versus the detuning $\Omega /{\kappa _a}$ and the coupling strength $g/{\kappa _a}$. (b) Efficiency of ${\eta _2}$ as functions of $\Omega /{\kappa _a}$ and the different coupling strength $g$ between photon and magnon. (c) The $d/{\kappa _a}$ changes with coupling strength $g/{\kappa _a}$. We use $K/{\kappa _\alpha} = 5 \times {10^{- 15}}$, and the other parameters are the same as in Fig. 2.

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Physically, the results shown in Figs. 2 and 3 can be understood as following. When the cavity photon-magnon coupling is in the weak-coupling regime, i.e., $g = 0.1{\kappa _a}$, Eq. (15) can be reduced to ${\omega _ +} = {\omega _ -} \approx {\Delta _{\textit{ad}}}$. In other words, the weak coupling between cavity photon and magnon leads to the degeneracy of the polariton modes, resulting in single valley in spectra of $|{t_p}{|^2}$ and single peak in spectra of the OSS ${\eta _2}$, as shown in Fig. 2. Both the valley of the transmission spectra and the peak of the OSS spectra located at $\Omega = 80{\kappa _a}$ correspond to the eigenfrequency ${\omega _ -} = {\omega _ +} = 80{\kappa _a}$. When the interaction between the cavity photon and magnon is pushed into the strong-coupling regime, i.e., $g = 12{\kappa _a}$, Eq. (15) cannot be reduced, and the degeneracy of the polariton modes does not occur. As a result, the splitting of the transmission and OSS spectra (two-color OSS) can be observed in Fig. 3, where the two peaks (valleys) of the OSS (transmission) are located at ${\omega _ \pm}$. The width between two OSS peaks can be calculated as

In the present parameter settings, one can readily find that $g$ is much larger than $K \langle {{{\hat m}^\dagger}\hat m} \rangle$; thus, the splitting width $d$ can be regarded as $d \approx 2g \approx 24{\kappa _a}$. Thus we can conclude, the splitting width of the OSS spectra depends on the coupling strength $g$ between the cavity photon and magnon in the strong-coupling regime.

To provide a better insight on the OSS spectra from the coupling strength $g$, in Fig. 4, we show the conversion efficiency ${\eta _2}$ as a function of the coupling strength $g$ and the probe-control detuning under the strong-coupling condition. We plot in Fig. 4(a) the logarithm of the upper OSS ${\eta _2}$ versus the detuning $\Omega$ and the coupling strength $g$. It can be found from Fig. 4(a) the splitting width between the two peaks becomes pronounced as the coupling strength increases. For direct insight into the influence of the coupling strength on the OSS spectra splitting, we present the spectral profiles of ${\eta _2}$ for several different values of $g$, as shown in Fig. 4(b). In addition, we can clearly observe that the splitting width $d/{\kappa _a}$ has a linear relationship with coupling strength $g/{\kappa _a}$ (i.e., $d \approx 2g$) in the strong-coupling regime, as shown in Fig. 4(c).

In order to clearly show the role of the coupling strength $g$ between cavity photon and magnon on the OSS conversion efficiency ${\eta _2}$ whether the cavity photon-magnon coupling is in the weak-coupling or the strong-coupling regime, the maximal values of ${\eta _2}$ versus $g$ for several different values of the magnon Kerr nonlinear coefficient $K$ are depicted in Fig. 5(a). One can find from Fig. 5(a) that the maximal conversion efficiency $\eta _2^{{\max}}$ first goes through exponential growth in the weak-coupling regime ($g \ll {\kappa _a}$) as the value of $g$ increases. When the value of $g$ increases further to the strong-coupling regime ($g \gg {\kappa _a}$), the maximal conversion efficiency $\eta _2^{{\max}}$ increases almost linearly. In order to reveal the physical mechanism of the dependence effect of $\eta _2^{{\max}}$ on the coupling strength $g$ shown in Fig. 5(a), we plot the time evolution of magnon number $|m{|^2}$ for several different values of $g$ in Fig. 5(b). It can be found that the system reaches a stable oscillation after a fast transient process. The increase of the coupling strength leads to the broadening of magnon oscillation amplitude and the increase of the magnon number in the cavity, which results in the broadening of OSS splitting and the enhancement of OSS generation. Besides, it can be found from Fig. 5(a) as the photon-magnon coupling intensity increases in the strong-coupling regime, the curve of the maximum value of ${\eta _2}$ tends to be flat. This is mainly because, when the photon-magnon coupling strength reaches a certain value, the number of magnon in the cavity is saturated. In addition, Fig. 5(a) also illustrates that the increasing magnon Kerr coefficient can significantly improve the conversion efficiency, which is consistent with the results in Figs. 2 and 3.

 

Fig. 5. (a) The logarithm of the maximum value of ${\eta _2}$ under different magnon Kerr nonlinear coefficients varies with the coupling strength $g/{\kappa _a}$. (b) Time evolution of magnon numbers with different photon-magnon coupling coefficients; we use $K/{\kappa _\alpha} = 5 \times {10^{- 15}}$. Other parameters are the same as in Fig. 2.

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It is worth noting that the frequency detuning between the cavity and the control field has been widely used for modulation of the optical nonlinear effect. As analyzed above, we have chosen the frequency detuning between cavity and the control field as ${\Delta _{\textit{ad}}} = {\Delta _{\textit{md}}} = 80{\kappa _a}$. In what follows, we show that the OSS spectrum can be controlled by tuning the frequency detuning ${\Delta _{\textit{ad}}}$ between the cavity and the control field. Figure 6(a) shows the simulation results of the OSS spectrum with several different values of ${\Delta _{\textit{ad}}}$ (i.e., ${\Delta _{\textit{ad}}} = 80{\kappa _a}$, $75{\kappa _a}$, and $70{\kappa _a}$). We use the coupling strength $g = 12{\kappa _a}$ between cavity photon and magnon with the fixed magnon Kerr nonlinear coefficient $K/{\kappa _\alpha} = 5 \times {10^{- 15}}$ in Fig. 6(a) corresponding, respectively, to the strong-coupling regime and the strong magnon Kerr nonlinearity. When the detuning ${\Delta _{\textit{ad}}}$ increases, the conversion efficiency of the two-color OSS decreases slowly, which can be understood by examining the time evolution of the magnon number in the cavity. We plot the time evolution of magnon number $|m{|^2}$ for several different values of ${\Delta _{\textit{ad}}}$ in Fig. 6(b). One can find from Fig. 6(b) that the increase of the detuning ${\Delta _{\textit{ad}}}$ could lead to the decrease of magnon number slightly in the cavity and decrease the conversion efficiency of the OSS. Interestingly, the right shift of the OSS spectra can be observed in Fig. 6(a) as increasing the detuning ${\Delta _{\textit{ad}}}$, which can be understood according to the shift of the eigenfrequency of the polariton modes. According to Eq. (15), the eigenfrequency ${\omega _ \pm}$ increases with the increasing of the ${\Delta _{\textit{ad}}}$ and, thus, the OSS spectrum shift to right.

 

Fig. 6. (a) Efficiency of ${\eta _2}$ as functions of $\Omega /{\kappa _a}$ and the different detuning ${\Delta _{\textit{ad}}}$. (b) Time evolution of magnon numbers with different detuning ${\Delta _{\textit{ad}}}$. We use $K/{\kappa _\alpha} = 5 \times {10^{- 15}}$, $g = 12{\kappa _a}$. Other parameters are the same as in Fig. 2.

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Before ending this section, we give a brief discussion on the possible experiment setup with the existing experimental techniques of a single small YIG crystal sphere and a 3D microwave cavity. The microwave cavity is machined from high-conductivity copper. The $250{ \text{-} }\unicode{x00B5}{\rm m}$-diameter YIG sphere is highly polished from pure single-crystal YIG to guarantee excellent magnonic performances. The higher spin density ${\rho _s} = 2.1 \times {10^{22}}\;{{\rm cm}^{- 3}}$ [52] YIG sphere is positioned at the magnetic field $H = 280.7\;{\rm mT}$ of the cavity mode, which resonates at ${\omega _a}/2\pi = {\omega _m}/2\pi = 7.86\;{\rm GHz}$. Other feasibility parameters $2{\kappa _m}/2\pi = 1.12\;{\rm MHz}$, $2{\kappa _a}/2\pi = 3.35\;{\rm MHz}$ are based on [50]. For the cavity magnonical system, the present proposal for enhancing OSS generation requires that the coupling between magnon and cavity is the strong-coupling regime. Recent experiments [24,26] have allowed the observation of strong-coupling regimes of interaction between magnon and cavity, and have a 1-mm-diameter YIG sphere correspond to the Kerr effect. These experiments provide both a similar method for probing the present system and a step toward the measurement of the OSS generation. In addition, the cavity-magnon coupling strength can be adjusted by varying the direction of the bias field or the position of the YIG sphere inside the cavity. We believe that the proposed structure is feasible in experimental realizations and deserves to be tested under the currently existing experimental technique.

4. CONCLUSION

In summary, we have theoretically put forward a practical scheme to the generation of the two-color OSS via magnon Kerr nonlinearity from a cavity magnonical hybrid system consisting of a single small YIG crystal sphere and a 3D rectangular cavity driven with a weak probe and a strong control field. Different from the linear interaction between the cavity photon and magnon, here we take account of magnon Kerr nonlinearity and use an effective perturbation method to solve the nonlinear HLEs for achieving the analytical solutions. It has been demonstrated that the OSS generation can be significantly enhanced via increasing the magnon Kerr nonlinearity even if the coupling between the cavity and magnon is weak. Furthermore, the splitting of the OSS spectrum (i.e., two-color OSS generation) can be observed when the cavity-magnon coupling is pushed into the strong-coupling regime, which results from the magnonical polaritons induced by the hybrid of cavity and magnon modes. Beyond their fundamental scientific significance, our results may provide measurement with higher precision in new degrees of freedom.