1. Introduction

In recent years, ferromagnetic materials have attracted increasing attention to the study of fundamental quantum physics and the implementation of quantum information protocols [1–6]. Especially, the yttrium iron garnet (YIG) is one of the most important representatives because it possesses a high spin density, low damping rate and high Curie temperature [7,8], below which all spins are aligned in the direction of an externally applied magnetic field. Due to the strong exchange interaction between neighboring spins in YIG, the energy excitations are not individual spin flips, but the collective precessional motions. In this sense, a quanta of spin excitation in YIG can be effectively described as a quasiparticle, known as magnons [9].

On one hand, the magnetic interaction between magnons in a YIG crystal and microwave photons in a superconducting resonator can reach the strong and ultra-strong coupling regimes [10–13], which benefits from the enhancement of the coupling strength by a factor $\sqrt {N}$. Experimentally, many quantum applications have been made in this field, such as magnon Kerr effect [14], magnon dark modes [15], bistability of cavity magnon polaritons [16], detection of single magnons [17], nonreciprocal microwave transmission [18–21], etc [22–32]. On the other hand, the ferromagnetic crystals fabricated with ellipsoidal geometries such as spheres or disks host the optical whispering gallery modes (WGMs) [33]. The emerging field of cavity optomagnonics, where optical photons can interact with magnons, is expected to provide a promising platform for developing new optical applications [34,35].

The optomagnonic interaction between photons and magnons can lead to elastic or inelastic light scattering. First, the magnetically induced birefringence causes the elastic photon scattering, known as magneto-optical effects [36–39]. Second, the optical WGMs can interact with the magnetostatic modes in the Brillouin light scattering, i.e., an inelastic scattering process. The photons exchange energy with the magnons via Stokes or anti-Stokes scattering, accompanying the angular momentum conservation [40,41]. However, the Brillouin scattering is either reciprocal or nonreciprocal depending on the total angular momentum of magnetic modes, including the spin and orbital angular momenta [42,43]. For the magnons with a zero $z$ component of the total angular momentum, the light scattering is reciprocal. However, it is nonreciprocal when the $z$ component of the total angular momentum of magnons is nonzero. The recent experiments have observed the nonreciprocal behavior of the Brillouin light scattering [44–48]. Therefore, it is desirable to design the chiral optical device with the cavity optomagnonic system, which has potential applications for signal processing and communication technology [49,50].

In this work, we propose to implement the one-way optical transmission in a ferromagnetic sphere. In our scheme, the Kittel mode with a nonzero $z$ component of the total angular momentum is exploited to unidirectionally couple with two optical modes. When the cavity is optically pumped in one direction, we show that a weak probe field from this direction can be transmitted or amplified, but completely absorbed in the opposite direction. The bandwidth and noise effect associated with this non-reciprocal optical transmission are analyzed in detail. The cavity optomagnonic system as a nonreciprocal optical device has the following advantages. First, the magnon with a magnetic tunable frequency can well satisfy the phase-matching requirement. Second, the high-frequency magnon has a low thermal occupation, which enables not only the large separability between optical pump and optical signal, but also the amplification of a weak optical signal with a high signal-to-noise ratio. Third, since there is no magnon-photon coupling in the opposite direction, the protection of the optical signal from extraneous noise can be realized. Moreover, the present work could stimulate interest for studying other nonreciprocal phenomena, such as nonreciprocal photon-magnon entanglement [51] and nonreciprocal magnon laser [52].

2. Mode

As illustrated in Fig. 1(a), we investigate the cavity optomagnonic system of a YIG sphere that supports the clockwise (CW) and counterclockwise (CCW) travelling-wave WGMs, as well as the magnetostatic modes. A waveguide is optically coupled to the ferromagnetic sphere via the overlap of the transverse evanescent light. The WGMs have transverse-electric (TE) modes and transverse-magnetic (TM) modes, which exchange energy and angular momentum with the magnetostatic modes via Brillouin scattering.

 

Fig. 1. (a) Schematic of the experimental architecture. The cavity optomagnonic system supports CW and CCW optical WGMs, as well as the magnetostatic modes. A uniform external magnetic field $\mathrm {B}$ is applied along the $+z$ axis to bias the YIG sphere, and an optical waveguide guides the CW propagating strong pump field and the two oppositely propagating weak probe fields. (b) Spectrum of the WGMs.

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The optical modes are fully characterized by three indices $n_{\sigma }$, $l_{\sigma }$, $q_{\sigma }$ ($\sigma =a$, $b$ for TE and TM modes, respectively) [53,54]. $n_{\sigma }$ is the number of nodes of the electric field amplitude in the radial direction, and $l_{\sigma }$ is the azimuthal number. $n_{\sigma }$ and $l_{\sigma }$ determine the spectrum of the optical modes. $q_{\sigma }$ is the $z$ component of the total angular momentum and satisfies $|q_{\sigma }|\leq {l_{\sigma }}$. The sign of $q_{\sigma }$ governs the orbit direction, i.e., $q_{\sigma }>0$ and $q_{\sigma }<0$ refer to the CW and CCW directions, respectively. For a fixed $n_{\sigma }$, the energy gap between $l_{\sigma }$ and $l_{\sigma }-1$ ($l_{\sigma }+1$) is the free spectral range $\Delta _{FSR}$, as shown in Fig. 1(b). In addition, the frequency separation between the TE and TM modes is $\Delta _{GB}$ for a given $l_{\sigma }$, which is caused by the geometrical linear birefringence [55]. To well illustrate the Brillouin light scattering, we only consider one TE mode ($l_{a}=l$) and three TM modes ($l_{b}=l-1,l,l+1$) with the maximal $z$ component of the total angular momenta $q_{\sigma }^{max}=-l_{\sigma }$, $l_{\sigma }$. So, the free Hamiltonian of the optical modes reads (hereafter $\hbar =1$)

 

Fig. 2. Transmission spectrums $T_{ccw}$ (a) and $T_{cw}$ (b) versus the detuning $\Delta /\kappa _{m}$. The relevant parameters are chosen to be $\eta _{a}=\eta _{b}=0.5$, $\kappa _{m}/2\pi =1$ MHz, $\kappa _{a}/2\pi =\kappa _{b}/2\pi =50$ MHz and $C$=0.5.

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The magnetostatic modes are also labeled by three indices ($n_{m}$, $l_{m}$, $q_{m}$) [55]. $n_{m}$ and $l_{m}$ are associated with the profiles in the radial and polar directions. $q_{m}$ is the most relevant index that represents the $z$ component of the total angular momentum. To realize the nonreciprocity, the magnon modes with $q_{m}\neq 0$ are applicable. Here we study the simplest Kittel mode with $q_{m}=1$, whose Hamiltonian is given by

The optical modes (TE and TM) interact with the Kittel mode through Brillouin light scattering process, obeying the phase-matching condition [42–48]. We now analyze the nonreciprocal photon-magnon interaction. Constrained by the conservation of angular momentum, the interaction takes either $g^{+}_{cw}(a_{l}b^{\dagger}_{l-1}m_{1}^{\dagger}+a^{\dagger}_{l}b_{l-1}m_{1})$ [$g^{-}_{ccw}(a_{-l}b^{\dagger}_{-(l-1)}m_{1}+a^{\dagger}_{-l}b_{-(l-1)}m_{1}^{\dagger})$] via Stokes scattering, or $g^{-}_{cw}(a_{l}b^{\dagger}_{l+1}m_{1}+a^{\dagger}_{l}b_{l+1}m^{\dagger}_{1})$ [$g^{+}_{ccw}(a_{-l}b^{\dagger}_{-(l+1)}m_{1}^{\dagger}+a^{\dagger}_{-l}b_{-(l+1)}m_{1})$] via anti-Stokes scattering, where $g^{+}_{cw,ccw}$ and $g^{-}_{cw,ccw}$ are the single-photon coupling strength. However, due to the energy conservation, the final interaction form is determined by the resonance frequency of Kittel mode. As seen in Fig. 1(b), the Stokes scattering occurs for $\omega _{m}-(\Delta _{FSR}-\Delta _{GB})=0$ [$\omega _{m}+(\Delta _{FSR}-\Delta _{GB})=0$], while the anti-Stokes scattering takes place for $\omega _{m}-(\Delta _{FSR}+\Delta _{GB})=0$ [$\omega _{m}+(\Delta _{FSR}+\Delta _{GB})=0$]. When the external magnetic field $\mathrm {B}$ is applied along the $+z$ axis, we have $\omega _{m}>0$. As a result, the Brillouin scattering can only occur in the CW direction, where the energy conservation can be satisfied. Instead, the magnon and photon modes are decoupled in the CCW direction. If we choose $\omega _{m}=\Delta _{FSR}-\Delta _{GB}$ by tuning the external magnetic field, the magnon-photon interaction Hamiltonian will yield

The total Hamiltonian is now written as

3. Non-reciprocal optical transmission

3.1 Transmission spectrum and bandwidth

To investigate the non-reciprocal optical transmission, we consider that two identical weak probe fields are applied to excite the TM optical modes $b_{cw}$ and $b_{ccw}$, respectively. In addition, the TE optical mode $a_{cw}$ is driven by a strong pump field, which is used to induce the desired photon-magnon coupling. After the unitary transformation $U=\mathrm {exp}(\omega _{c}a^{\dagger}_{cw}a_{cw}+\omega _{c}m^{\dagger}m)$, the Hamiltonian of the system yields

Under the situation of a resonant strong driving field $\delta _{a}=0$, we can apply the nondepletion approximation and treat the optical mode $a_{cw}$ as a classical number $\langle {a}_{cw}\rangle =\frac {\sqrt {\eta _{a}\kappa _{a}}\epsilon _{c}}{\kappa _{a}/2}$. By substituting this classical number into Eq. (5), we can get the linearized Hamiltonian

To give a more quantitative analysis of the nonreciprocal effect, we solve the quantum Langevin equations by taking into account the quantum noises. The equations of motion of the quantum operators are given by

 

Fig. 3. (a) Amplified transmission spectrum $T_{cw}$ versus the detuning $\Delta /\kappa _{m}$ and the cooperativity $C$. (b) Bandwidth $\Delta _{g}/\kappa _{m}$ as a function of the cooperativity $C$. The other parameters are chosen the same as those in Fig. 2.

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where the cooperativity parameter $C=4G^{2}/(\kappa _{b}\kappa _{m})$ is a dimensionless parameter that characterizes the photon-magnon coupling strength. It is noted that the parameter $C$ determines the stability of the system. For $\Delta \rightarrow 0$ and $C\rightarrow 1$, the system becomes unstable as $|B_{cw}^{out}|\rightarrow \infty$. Here, we focus on the stable regime $0{\leq }C<1$.

To quantify the nonreciprocal behavior of light propagation, we define the transmission spectrums $T_{ccw(cw)}=|B_{ccw(cw)}^{out}/\epsilon _{p}|^{2}$, which are given by

For $C=0$, $T_{ccw}$ and $T_{cw}$ are identical regardless of the direction of light propagation. The presence of the optomagnonic interaction will modify the transmission spectrum. For $0<C<1$, it is obvious that $T_{ccw}$ and $T_{cw}$ become asymmetric, corresponding to non-reciprocal optical transmission.

In Fig. 2, we plot the transmission spectrums $T_{ccw}$ and $T_{cw}$ as a function of the detuning $\Delta /\kappa _{m}$. Under the critical coupling condition $\eta _{b}=0.5$, we have $T_{ccw}=0$ at the resonance point, i.e., the CCW signal photons are completely dissipated into the environment. Conversely, the transmission $T_{cw}\approx 1$ is observed around the central resonance frequency, i.e., the CW probe field can fully transmit for $C=0.5$. It can be explained that the gain and loss are in balance, resulting in the perfect transparency of the signal.

To realize the optical signal amplification $T_{cw}>1$, we have to arrive the range $0.5<C<1$, where the nondegenerate parametric amplifier dominates the dynamics compared to the intrinsic dissipative process. Figure 3(a) displays the amplified region of the CW optical signal. It is observed that with the increase of cooperativity $C$, the transmission $T_{cw}$ is gradually increased. For $C=0.91$, we have $T_{cw}\approx 100$ at the resonance frequency, which means that the CW probe field is amplified by 100 times.

We now discuss the bandwidth related to this one-way optical transmission, which is defined as the full width at half maxima of the transmission spectrum. In the present scheme, the cavity has a larger decay rate ($\kappa _{b}\gg \kappa _{m}$), so the associated bandwidth can be simplified as

We can see that the bandwidth is determined by the damping rate of the Kittel mode $\kappa _{m}$ and the cooperativity $C$. Obviously, there is a trade-off between the bandwidth and the amplification, i.e., the bandwidth takes the opposite behavior of the amplification as the cooperativity $C$ gets larger. In Fig. 3(b), we plot the bandwidth $\Delta _{g}/\kappa _{m}$ versus the cooperativity $C$, which is consistent with the numerical result obtained by Eq. (10).

Finally, we investigate the feasible parameters [47,48]: $\omega _{a}/2\pi \approx \omega _{b}/2\pi =$193 THz (free space wavelength $\sim$ 1550 nm), $\omega _{m}/2\pi =7$ GHz, $\kappa _{a}/2\pi =\kappa _{b}/2\pi =50$ MHz and $\kappa _{m}/2\pi =1$ MHz. For a YIG sphere with a diameter of $D=110$ $\mu$m, the refractive index $n_{r}=$2.19, the Verdet constant $\mathcal {V} =377$ rad/m, the speed of light in vacuum $c=3\times 10^{8}$ m/s and the spin density $\rho =2.1\times 10^{28}$ m$^{-3}$, we can get the single-photon coupling strength $g_{cw}^{+}=\frac {\mathcal {V}c}{n_{r}}\sqrt {\frac {12}{\rho \pi {D^{3}}}}\approx 2\pi \times 96$ Hz. For $C=\{0.5,0.91\}$, the enhanced photon-magnon coupling strength is $G/2\pi \approx \{2.5,3.4\}$ MHz, and the pump power of the TE mode is calculated to be $P_{a}\approx${13.6, 24.7} mW. It is noted here that the chosen single-photon coupling strength has not been realized in current experiments. For a 700-$\mu$m-diameter YIG sphere [46], the experimentally measured single-photon coupling strength is about $2\pi \times 5$ Hz, which is consistent with the theoretical value $g_{cw}^{+}$. So, with the technological development of cavity optomagnonics, the above parameters are expected to be attainable in experiments.

3.2 Effect of the added noise

For the nonreciprocal optical transmission, the noises are inevitable and should be taken into consideration. To avoid the signal submerged by the noises, the mean photons of the input field have to be much bigger than the added noise photons. Thus, it is necessary to quantitatively study the effect of added noise. We now consider the quantum vacuum fluctuation terms of Eq. (7) while omitting the signal field. In this case, we can convert Eq. (7) into the frequency domain by using the Fourier transform $O(t)=\int _{-\infty }^{+\infty }\frac {d\omega _{p}}{2\pi }O(\omega _{p})e^{-i\omega _{p}{t}}$ ($O=b_{cw}$, $m^{\dagger}$) with

According to the input-output theory, the output noise of the cavity modulated by the Kittel mode can be described as

Then the corresponding noise spectrum takes the form

 

Fig. 4. (a) Added noise $n_{add}$ versus the detuning $\Delta /\kappa _{m}$ with $T=0$. (b) Added noise $n_{add}$ as a function of the bath temperature $T$ with $\Delta =0$. The other parameters are chosen the same as those in Fig. 2.

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4. Conclusion

In summary, we have proposed a method for the realization of non-reciprocal optical transmission in a cavity optomagnonic system. By utilizing the non-reciprocal Brillouin scattering between the travelling-wave optical modes and the Kittel mode with a nonzero $z$ component of the total angular momentum, we show that the weak optical signal can be transmitted or amplified in a given propagation direction, but fully absorbed in the reverse direction. The present work promotes the cavity optomagnonics as an alternative ingredient for implementing on-chip optical isolators with a high signal-noise ratio.