1. Introduction

Photon blockade (PB) as an excellent candidate for the preparation of single-photon sources for various applications in quantum information has received more attention in recent years. The PB effect is the fact that the first photon passing through the system impedes the following photons and forms a phenomenon where photon is passed by one by one. Typically, the conventional photon blockade (CPB) attributing to the anharmonicity in eigenenergy spectrum resulted from the strong couplings has been studied theoretically and experimentally in various quantum systems, including atoms [1], circuit quantum electrodynamics systems [2], cavity quantum electrodynamics systems [3], quantum dots [4], optomechanical systems [5], coupled cavities [6], cavity-free systems [7], nonlinear cavity with an $n$-photon parametric drive [8], quantum nonreciprocal devices [9], etc. Most recently, the quantum interference induced unconventional photon blockade (UPB) plays a crucial role in generating the antibunching light even for weakly couplings, where two or more different quantum transition pathways are existent and the occurrence of destructive interferences between them inhibits the transfer from single-photon state to two-photons state of the system. The UPB effects have been extensively studied in different systems and were experimentally achieved in coupled optical [1] and superconducting resonators [10]. Note that the quantum properties of the cross correlations between photon and photon, photon and phonon are studied deeply in different systems, including the composite atom-cavity system [11], the optomechanical system [12], a qubit-plasmon-phonon ultrastrong-coupling system [13]. Furthermore, the importance of optical nonlinearity in photon blockade has been investigated both theoretically and experimentally in recent years. The PB effect has been demonstrated in strongly nonlinear systems [14,15]. The realisation of UPB with $\chi ^{(2)}$ nonlinearities has been reported [16,17]. In weakly nonlinear photonic molecules, photon antibunching of symmetric and antisymmetric modes has been proved [18].

On the other hand, the magnon-based hybrid quantum systems provide a powerful platform for focusing on various quantum effects and implementing multiple quantum tasks due to the advantages of complementing different subsystems [19–23], where magnons represent quanta of collective spin excitations in ordered ferrimagnetic materials such as yttrium iron-garnet (YIG) spheres and have the striking features including the low damping, high spin density and flexible tunability. It is for such reasons that the strong or even ultrastrong couplings occur between microwave photons and magnons [24], which could lead to many interesting phenomena [21,25,26]. The optical photon-magnon interaction via the Brillouin light scattering [27,28] and the Faraday effect [26,29] are observed to realize the microwave-optical conversion [30]. The hybrid ferromagnet-superconductor systems are recently presented and are applied to achieve some interesting phenomena such as the coherent coupling between a magnon and a qubit and entanglement-based single-shot detection of a magnon with a qubit [31,32]. It is found that cavity magnomechanics where magnons interact with the vibration phonons by the nonlinear magnetostriction has become a promising nonlinear system for showing the strong nonlinearity and producing multipartite entanglement and steering [33,34]. Moreover, the magnon Kerr effects are striking due to the strong drivings and could lead to the interesting nonlinear phenomena [21,35,36]. The recent researches show that magnon blockade (MB) analogous to PB is significant to the controllable generation of single magnon sources and has been carried out in the hybrid cavity-magnon system. For instance, the conventional magnon blockade (CMB) has been discussed in a hybrid ferromagnet-superconductor system for the cases of one qubit and two qubits [37–40]. Recently, the MB effect has been theoretically investigated in the hybrid microwave optomechanical-magnetic system [41]. And the unconventional magnon blockade (UMB) are also shown via the three-wave mixing process and the hybrid microwave optomechanical-magnetic system [40–42]. Multi-magnon blockade effect and magnon-induced tunneling effect could be obtainable in the system consisting of one three-level qubit and magnon mode [22]. And the dissipation-induced nonreciprocal magnon blockade has been exploited in a magnon-based hybrid system [43].

Here we investigate how to engineer the UPB, UMB and magnon-photon cross-correlation effects in a hybrid nonlinear cavity-magnon system, in which the second-order optical nonlinearity occurs between one pump cavity and signal cavity and meanwhile there has the linear magnon-photon coupling due to magnetic dipole-dipole interaction of signal cavity and magnon from the YIG sphere. It is found that the dispersive couplings lead to the nonlinear interaction between the pump cavity and the magnon mode, which could induce the magnon-photon anticorrelations. By the numerical simulation and analytical calculation, the statistical features represented by the second-order correlation functions appear in the system. In the weak coupling and driving regime, the magnon blockade, photon blockade and magnon-photon anticorrelations could be achieved on demand. The present work quite different from the previous schemes for magnon antibunching is based on the effective magnon-photon nonlinearity transferred from the optical nonlinear couplings, which may have significance for the exploration of precision metrology, quantum information processing and quantum simulation.

The paper is constructed as follows. In Section 2, we describe the physical model and the corresponding Hamiltonian, and give detailed derivation of the effective Hamiltonian of the subsystem consisting of the pump cavity and the magnon mode. The equal-time second-order correlation functions are also introduced. In Section 3, we study the UMB and cross-correlation effects via single magnon driving in such a nonlinear system. The numerical results, analytical solutions and the physical mechanism are presented. In section 4, we study the case of two driving and give the optimal parameters. And based on this, the numerical results of magnon blockade and magnon-photon anticorrelation effects related with various parameters are discussed. In Section 5, the summary is given.

2. Model and equation

We consider a hybrid cavity magnonic system consisting of two microwave cavities and one YIG sphere, whose spin is oriented along the $z$ axis, as illustrated in Fig. 1. Two cavities named as pump and signal cavities are nonlinearly coupled via $\chi ^{\left ( 2\right ) }$ nonlinearity, where the second-order coupling mediates the conversion between pump single photon and pairs of signal photons. Meanwhile, the magnon mode representing the collective excitations in the YIG sphere placed inside the signal cavity couples to the cavity mode via magnetic dipole interaction. Furthermore, the pump cavity and the magnon mode are respectively subjected to the coherent drivings. Therefore the Hamiltonian of the system reads as $\left ( \hbar =1\right )$

$H_{0}$ is associated with the free Hamiltonian of the pump cavity, signal cavity and magnon. $H_{I}$ refers to the nonlinear coupling between the pump and the signal cavities and the magnetic dipole interaction between the signal cavity and magnon mode. Strikingly, optical nonlinearities have been extensively studied [44,45] and applied to the realisation of some interesting phenomena such as harmonic entanglement, frequency comb and controlled-phase gate [46–48]. And $H_{d}$ is associated with the Hamiltonian of the driving of the pump cavity and magnon mode. Here $a_{j}^{\dagger }$ and $a_{j}$ $\left ( j=p,s\right )$ represent the creation and annihilation operators of the pump and signal cavities, and $m^{\dagger }\left ( m\right )$ is the creation (annihilation) operator of magnon mode. The parameters $\omega _{p},\omega _{s}$ and $\omega _{m}$ are the eigenfrequencies of two cavities and one magnon mode, respectively. And $J$ is the intercavity parametric coupling coefficient and $g_{ms}$ denotes the coupling rate between the magnon mode with the signal mode. The driving amplitudes and frequencies with respect to the pump cavity are given by $\Omega$ and $\omega _{d1}$ and these of the magnon mode are $F$ and $\omega _{d_{2}}$ with $\omega _{d1}=2\omega _{d2}=\omega _{d}$. Based on the unitary operator $U=$exp$\left [ \frac {i\omega _{d}t}{2}\left ( 2a_{p} ^{\dagger }a_{p}+a_{s}^{\dagger }a_{s}+m^{\dagger }m\right ) \right ]$, the Hamiltonian of the system in a frame rotating at the driving frequency $\omega _{d}$ can be written as

Based on the virtual-photon excitation of signal cavity mode, the second-order nonlinear coupling occurs between the pump and magnon modes, by which a pair of excited magnons can jointly emit a pump photon. In order to make the physics clear, we will perform the following steps. (i) Introducing the signal-magnon detuning $\Delta =\omega _{s}-\omega _{m}$, and then carrying on the picture transform in terms of $U=$exp$\left ( i\Delta ta_{s}^{\dagger }a_{s}\right )$, we obtain the following Hamiltonian $H_{I}=\Delta _{p}a_{p}^{\dagger } a_{p}+\Delta _{m}\left ( a_{s}^{\dagger }a_{s}+m^{\dagger }m\right ) +\Omega \left ( a_{p}+a_{p}^{\dagger }\right ) +F\left ( m+m^{\dagger }\right ) +V_{I}\left ( t\right )$ with $V_{I}\left ( t\right ) =g_{ms}\left ( a_{s}m^{\dagger }e^{-i\Delta t}+a_{s}^{\dagger }me^{i\Delta t}\right ) +J\left ( a_{p}a_{s}^{\dagger 2}e^{i2\Delta t}+a_{p}^{\dagger }a_{s}^{2}e^{-i2\Delta t}\right )$. (ii) Assuming the larger detuning with $\Delta {\gg }\{g_{ms} ,J\}$, we find from the above expression $V_{I}\left ( t\right )$ that the large-detuning couplings appear among three modes including magnon and two cavity photons with the different detunings $\Delta$ and $2\Delta$, and so we could deal with it based on the high-order perturbation theory. (iii) Using the effect Hamiltonian method introduced in the Refs. [49] and taking the Markovian approximation, the Schrödinger equation of the largely detuned interacting quantum system could be expressed as $\frac {\partial }{\partial t}\left \vert \Psi \right \rangle =-iH_{eff}\left \vert \Psi \right \rangle$ with $H_{eff}=H_{eff}^{(2)}+H_{eff}^{(3)}+\cdots +H_{eff}^{(n)}+\cdots$ being the effective Hamiltonian. (iv) Focusing on the second- and third-order cases and according to the formulas $H_{eff}^{(2)}=-iV_{I}\left ( t_{1}\right ) \int _{0}^{t}V_{I}\left ( t_{1}\right ) dt_{1}$ and $H_{eff}^{(3)}=-V_{I}\left ( t\right ) \int _{0}^{t}V_{I}\left ( t_{1}\right ) \int _{0}^{t_{1}}V_{I}\left ( t_{2}\right ) dt_{2}dt_{1}$, we have the second-order effective Hamiltonian $H_{eff}^{(2)}=-\left ( \frac {g_{ms}^{2}}{\Delta }m^{\dagger }m+\frac {J^{2} }{\Delta }a_{p}^{\dagger }a_{p}\right ) ,$ where we have assumed that the signal cavity is initially in the vacuum state. It is obvious that the second-order process only causes the additional dispersive resonance shifts for the pump and magnon modes, which can be compensated by properly detuning the pump cavity frequency from twice the magnon frequency. After neglecting the second-order effect, the third-order process becomes dominant. As a result, the third-order Hamiltonian is also derived as

 

Fig. 1. Schematic diagram of a hybrid cavity magnonic system. The pump and signal cavities are coupled via intercavity parametric coupling coefficient $J$, and the magnon mode of a YIG sphere with frequency $\omega _{m}$ is coupled to signal cavity mode. Here, the pump cavity is subject to a coherent drive with amplitude $\Omega$ and frequency $\omega _{d}$. $\kappa _{p}$ and $\kappa _{m}$ are the cavity and magnon decay rates, respectively.

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Correspondingly, the master equation for the density matrix of the magnon-pump cavity system can be described by

To obtain the statistical properties of the magnon and photon modes, we introduce the equal-time second-order correlation functions with the form:

3. UPB and magnon-photon anticorrelations

Here we concentrate on the simple case of magnon driving $\left ( \Omega =0\right )$. In order to obtain the perfect photon antibunching, we firstly look for the optimal conditions based on the method of the probability amplitudes [54]. The Hamiltonian including the decays of cavity and magnon modes can be written as $H^{^{\prime }}=H_{eff}-i\frac {\kappa }{2}a_{p}^{\dagger } a_{p}-i\frac {\gamma }{2}m^{\dagger }m$, where we assume that the pump cavity field and the magnon are connected with two individual vacuum reservoirs. In the weak driving and coupling regime, the wave function with time-dependent can be approximated expressed by

Next, we can obtain the steady-state solutions under the approximate relationship of probability amplitudes, i.e., $C_{00}\approx 1\gg \{C_{01} \}\gg \{C_{02},C_{10}\}\gg \{C_{03},C_{11}\}\gg \{C_{04},C_{12},C_{20}\}$ and other high-excitation states being neglected due to the weak magnon driving. The steady-state solutions of Eqs. (13–21) are given as

It is obvious that the condition $g_{aa}^{(2)}(0)=0$ could be obtained via setting $C_{20}=0$, which means that the perfect UPB effect is existent in the system. Clearly, the optimal coupling strength $g$ is

However, we find that there is a non-zero value of second-order correlation with $g_{mm}^{(2)}(0)\neq 0$ and it is difficult to achieve the perfect UMB only via using the single magnon driving.

Now we concentrate on the statistical properties of photon and magnon and the cross-correlation effects between them in the present nonlinear system. In order to do so, we plot the logarithms of second-order correlation functions log$_{10}g_{ij}^{(2)}(0)(ij=aa,am,mm)$ with the choice of $\Delta _{a}=0,\gamma =0.2\kappa$ and $\bar {n}_{th}=10^{-5}$. Figure 3(a) displays log$_{10} g_{ij}^{(2)}(0)$ as function of $\Delta _{m}/\kappa$ with $F=0.025\kappa$ and $g=0.458\kappa$. It is found that the cross-correlation log$_{10}g_{am} ^{(2)}(0)$ has the same change as the auto-correlations of magnon and photon. For the case of resonance with $\Delta _{m}=0$, the value of log$_{10} g_{am}^{(2)}(0)$ is about log$_{10}g_{am}^{(2)}(0)=1.748$ and the strong anticorrelation appears between magnon and photon, which means that the probability for the simultaneous presence of one photon and one magnon in both modes is suppressed. Obviously, the photon blockade is existent for the present parameters due to the smaller log$_{10}g_{aa}^{(2)}(0)$, which could be derived by the above analytical solutions. Figure 3(b) shows the functions log$_{10}g_{ij}^{(2)}(0)$ versus $F/\kappa$ with $\Delta _{m}=0$ and $g=0.458\kappa$. It is clearly seen that photon blockade and magnon-photon anticorrelations depend on the parameter $F$. For $F=0.025\kappa$, the function log$_{10}g_{aa}^{(2)}(0)$ has the minimum and the strong photon antibunching effect is obtained. The logarithms of second-order correlation functions log$_{10}g_{ij}^{(2)}(0)$ dependent on $g/\kappa$ are also discussed for the weak driving $F=0.05\kappa$ in Fig. 3(c). Both the cross-correlation function log$_{10}g_{am}^{(2)}(0)$ and the auto-correlation function log$_{10} g_{mm}^{(2)}(0)$ decrease with the increasing $g/\kappa$. However, the parameter log$_{10}g_{aa}^{(2)}(0)$ becomes smaller and then it gets bigger. The smallest log$_{10}g_{aa}^{(2)}(0)=3.442$ could be reached and the photon blockade is obtainable. Based on the single magnon driving, the UPB and magnon-photon anticorrelation effects are accessible via choosing different parameters.

The mechanism of the above nonclassical properties can be easily analyzed in terms of the energy-level transition processes of the effective Hamiltonian $H_{eff}$ of the hybrid system. In the weak coupling and driving regime, there are two pathways $\left \vert 0,1\right \rangle \rightarrow \left \vert 0,2\right \rangle \rightarrow \left \vert 0,3\right \rangle \rightarrow \left \vert 0,4\right \rangle \rightarrow \left \vert 1,2\right \rangle \rightarrow \left \vert 2,0\right \rangle$ and $\left \vert 0,1\right \rangle \rightarrow \left \vert 0,2\right \rangle \rightarrow \left \vert 1,0\right \rangle \rightarrow \left \vert 1,1\right \rangle \rightarrow \left \vert 1,2\right \rangle \rightarrow \left \vert 2,0\right \rangle$, as shown in Fig. 2(a). The photons coming from the two pathways would destructively interfere when the antibunching condition given by Eq. (28) is satisfied and the perfect UPB is achieved, which will be available for the realization of single photon resource. Correspondingly, the magnon-photon anticorrelation could be also achieved. In contrast, the magnon blockade will not occur due to the absence of two transition channels. To deeply understand the anticorrelation properties between magnon and photon, we discuss the influences of the thermal noise of magnon on the second-order cross-correlation function $g_{am}^{\left ( 2\right ) }\left ( 0\right )$. In Fig. 4(a), we plot the parameter log$_{10}g_{am}^{(2)}(0)$ related to $\Delta _{m}/\kappa$ for different mean thermal magnon numbers $\bar {n} _{th}=10^{-2},10^{-3},10^{-5}$, where the other parameters are given by $\Delta _{a}=0,\gamma =0.2\kappa,g=0.458\kappa$ and $F=0.013\kappa$. It can be seen that magnon-photon anticorrelation effects are sensitive to the thermal magnon numbers. With the increase of $\bar {n}_{th}$, the value of log$_{10}g_{am}^{(2)}(0)$ becomes larger and the anticorrelations between them weakens, and the optimal cross-correlation occurs at $\Delta _{m}=0$. Figure 4(b) displays their dependence on the weak driving $F/\kappa$ at full resonance $\Delta _{a}=\Delta _{m}=0$. For $\bar {n}_{th}=10^{-2}$, the log$_{10} g_{am}^{(2)}(0)$ less than $1$ could be present and there still has the magnon-photon anticorrelation effect in the system. The variance log$_{10}g_{am}^{(2)}(0)$ versus $g/\kappa$ with the different thermal magnon occupations are shown in Fig. 4(c) with the other parameters being the same as those in Fig. 4(a) for the case of full resonance. Choosing the larger coupling strength $g$ will induce the optimized anticorrelations between magnon and photon even in a thermal noise environment.

 

Fig. 2. The schematic energy spectrum of the hybrid cavity magnonic system show the transition pathways in magnon driving(a) and double driving(b) cases, respectively. Here red arrows represent the photon-magnon coupling with strength $g$, blue and green arrows represent the driven of the magnon mode and the pump cavity with amplitudes $F$ and $\Omega$, respectively.

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Fig. 3. log$_{10}g_{ij}^{(2)}(0)(ij = aa, am, mm)$ are plotted (a) as functions of the detuning $\Delta _{m}/\kappa$; (b) as functions of the magnon driving strength $F/\kappa$; (c) as functions of the coupling strength $g/\kappa$. $F = 0.025\kappa$ and $g= 0.458\kappa$ in (a); $\Delta _{m}$ = 0 and $g= 0.458\kappa$ in (b); $\Delta _{m}$ = 0 and $F= 0.025\kappa$ in (c). The other parameters are $\Delta _{a}$ = 0, $\gamma$ = 0.2$\kappa$ and $\bar {n}_{th}=10^{-5}$.

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Fig. 4. log$_{10}g_{am}^{(2)}(0)$ are plotted (a) as functions of the detuning $\Delta _{m}/\kappa$; (b) as functions of the magnon driving strength $F/\kappa$; (c) as functions of the coupling strength $g/\kappa$ for different mean thermal magnon numbers $\bar {n}_{th}=10^{-2},10^{-3},10^{-5}$. F = 0.013$\kappa$ and $g=0.458\kappa$ in (a); $\Delta _{m}$ = 0 and $g= 0.458\kappa$ in (b); $\Delta _{m}$ = 0 and F= 0.013$\kappa$ in (c). The other parameters are $\Delta _{a}$ = 0 and $\gamma$ = 0.2$\kappa$.

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4. UMB and magnon-photon anticorrelations

Now we consider the case of double driving of the magnon and pump cavity and the corresponding Hamiltonian $H_{eff}$ is shown in Eq. (7). For the case of the weak coupling and double driving, the diagram of energy-level transition processes is shown in Fig. 2(b). It is clearly seen that there are two simple transition pathways for the double magnon state $\left \vert 0,2\right \rangle$. One is the direct driving pathway $\left \vert 0,0\right \rangle \rightarrow \left \vert 0,1\right \rangle \rightarrow \left \vert 0,2\right \rangle$ and the other channel is $\left \vert 0,0\right \rangle \rightarrow \left \vert 1,0\right \rangle \rightarrow \left \vert 0,2\right \rangle$. The perfect UMB could be existent due to quantum destructive interference between the above two channels. Correspondingly, the magnon-photon anticorrelation effects are discussed based on such the optimal condition, which means the magnon blockade is available and the photon antibunching is absent. Using the similar method, the optical condition could be obtained with the form

To observe these phenomena, we plot the Fig. 5 by carrying out numerical simulation of the master equation with the parameters $\gamma =0.2\kappa,\Delta _{a}=0$ and $\bar {n}_{th}=10^{-5}$. In Fig. 5(a), the variable log$_{10}g_{ij}^{(2)}(0)(ij=aa,am,mm)$ depending on $\Delta _{m}$ is presented with parameters $g=0.225\kappa,F=0.015\kappa$ and $\Omega =-0.005\kappa$. Significantly, the logarithm of second-order correlation functions log$_{10}g_{mm}^{(2)}(0)$ reaches the minimum with the value less than $0$ at $\Delta _{m}=0$, which means that the phenomenon of UMB is observed and the single-magnon resource in analogy with single-photon resource could be obtainable. Furthermore, the magnon-photon cross-correlation functions $g_{am}^{(2)}(0)$ are also smaller than 1 nearby the resonant region and the anticorrelations between magnon and photon. However, the value of log$_{10}g_{aa}^{(2)}(0)$ is close to 0 and the photon blockade is impossible for the present parameters. The logarithm log$_{10}g_{ij}^{(2)}(0)$ is described as a function of coupling strength $g$ with the chosen parameters $\Omega =-0.005\kappa,g=0.225\kappa$ and $\Delta _{m}=0$, as shown in Fig. 5(b). When the driving strength $F$ is approximately equal to $0.016\kappa$, the variable log$_{10}g_{mm}^{(2)}(0)$ reaches the minimum. Figure 5(c) shows the parameter log$_{10}g_{ij}^{(2)}(0)$ versus coupling strength $g$. It is obvious that quantum statistical properties of UMB and magnon-photon anticorrelations are achievable, where these parameters $\Omega,F$ and $g$ are very weak.

 

Fig. 5. log$_{10}g_{ij}^{(2)}(0)(ij = aa, am, mm)$ are plotted (a) as functions of the detuning $\Delta _{m}/\kappa$; (b) as functions of the magnon driving strength $F/\kappa$; (c) as functions of the coupling strength $g/\kappa$. $F = 0.015\kappa$, $\Omega =-0.005\kappa$ and $g= 0.225\kappa$ in (a); $\Delta _{m}$ = 0, $\Omega =-0.005\kappa$ and $g= 0.225\kappa$ in (b); $\Delta _{m}$ = 0, $F= 0.015\kappa$ and $\Omega =-0.005\kappa$ in (c). The other parameters are $\Delta _{a}$ = 0, $\gamma$ = 0.2$\kappa$ and $\bar {n}_{th}=10^{-5}$.

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In Figs. 6(a) and 6(b), $g^{(2)}_{mm}(0)$ and $g^{(2)}_{am}(0)$ are shown on a logarithmic-scale plot as a function of the magnon detuning $\Delta _{m}/\kappa$ and the driving amplitude of the pump cavity $\Omega /\kappa$, respectively. In Fig. 6(a), when $\Delta _{m}$ is equal to 0 and $\Omega$ is approximately equal to $-0.01\kappa$, the smallest log$_{10}g_{mm}^{(2)}(0)$ could be reached and the magnon blockade is obtainable. Meanwhile, magnon-photon cross-correlation around these parameters can also be found with good effect in Fig. 6(b). In addition, in Figs. 6(c) and 6(d), $g^{(2)}_{mm}(0)$ and $g^{(2)}_{am}(0)$ are shown on a logarithmic-scale plot versus the driving amplitude $\Omega /\kappa$ and the coupling strength $g/\kappa$, respectively. In Fig. 6(c), the white dashed curves highlight the optimal relation between $\Omega$ and g for the strongest MB according to Eq. (29). The white dashed curves coincide with the numerical results. The result suggests that the strong MB effect arises from the destructive interference between the two transition paths. Meanwhile, the magnon-photon cross-correlations could found in Fig. 6(d).

 

Fig. 6. Contour plot of log$_{10}g_{mm}^{(2)}(0)$(a) and log$_{10}g_{am}^{(2)}(0)$(b) versus the magnon detuning $\Delta _{m}/\kappa$ and the driving amplitude $\Omega /\kappa$ with $g=0.45\kappa$ and $F=0.03\kappa$; contour plot of log$_{10}g_{mm}^{(2)}(0)$(c) and log$_{10}g_{am}^{(2)}(0)$(d) versus the coupling strength g/$\kappa$ and the driving amplitude $\Omega$/$\kappa$ with $\Delta _{m}$ = 0 and $F=0.03\kappa$. The other parameters are $\Delta _{a}$ = 0, $\gamma$ = 0.2$\kappa$ and $\bar {n}_{th}=10^{-5}$.

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5. Conclusion

In conclusion, we have theoretically investigated the magnon and photon statistical properties including UMB, UPB and magnon-photon cross-correlation effects in the hybrid nonlinear system including one YIG sphere and two coupled cavities. Fortunately, the effective $\chi ^{\left ( 2\right ) }$ nonlinear interaction occurs between the pump cavity and magnon mode via the transfer of optical nonlinearity between two cavities with the help of the virtual photon excitation of the signal cavity. Based on the analytical calculations and numerical simulations, we find that the strong cross-correlation between magnon and photon could be achievable via adjusting the physical parameters, by which the magnon blockade and photon antibunching selectively appear. Our scheme is focused on the system in the weak coupling regime and meets the requirement of experimental operations, which provides the possibility of realizing the antibunching behavior on demand and magnon-photon cross-correlation effect in the hybrid cavity magnonical system.