Open Access
Add to BibSonomy
Abstract
Nonreciprocal sideband responses in a spinning microwave magnomechanical system consists of a spinning resonator coupled with a yttrium iron garnet sphere are proposed. We show that the efficiency of sideband generation can be enhanced in one driving direction but restrained in the opposite. This nonreciprocity results from Sagnac effect induced by the spinning resonator, leading to asymmetric magnonic responses in two different driving directions. Beyond the conventional linearized description, the properties of nonreciprocal two-color second-order sideband are demonstrated. By adjusting Sagnac-Fizeau shift and the power of control field, the degree of asymmetric magnonic responses can be strengthened, therefore causing stronger nonreciprocity of sideband. Especially, for the case of strong Sagnac-Fizeau shift and the control field, high level of efficiency and isolation ratio of sideband are achieved simultaneously and the operational bandwidth of strong nonreciprocity can be expanded. Our proposal provides an effective avenue for the manipulation of the nonreciprocity of sideband and has potentially practical applications in on-chip microwave isolation devices and magnon-based precision measurement.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
There has been significant study activity on nonreciprocity because it is crucial for integrated information technology and has important applications in quantum network [1]. Time reversal symmetry (TRS) breaking is a fundamental physical mechanism and necessary condition for the realization of nonreciprocal system [2]. A conventional method for creating nonreciprocity is to use Faraday-rotation to break TRS of the system in the magneto-optic crystal [3]. However, the material usually tends to be bulky and incompatibility, which is unsuitable for on-chip integration. For the past few decades, various schemes for enabling nonreciprocity have been implemented in microscale integrable devices via adopting different mechanisms, such as optomechanical interaction [4], spinning resonator [5–7], chiral coupling [8,9], and nonlinearity [10–12]. Based on these approaches, numerous theoretical and experimental efforts have also been devoted to the developments of nonreciprocal quantum devices [10–13].
Recently, magnon, as collective spin excitations of magnetic insulator yttrium iron garnet (YIG), has attracted extensive attentions and become a building block of quantum information science due to its low damping rate, high spin density, and long coherent time [14–16]. The strong and ultrastrong couplings between magnon and microwave photon through magnetic-dipole interaction have been realized experimentally [17,18]. Owing to the widespread frequency tunability and excellent compatibility of magnons, they can also be coupled to optical photon directly via magneto-optical effect [19] or superconducting qubits indirectly via the virtual photon excitation [20]. In addition, cavity magnomechanics, similar to cavity optomechanics, has been proposed and demonstrated experimentally, where magnon-phonon interaction is introduced through magnetostrictive force resulting in magnomechanically induced transparency (MMIT) [21–23]. MMIT, an analog of optomechanically induced transparency [24], which arises from the interference of sidebands generated by the parametric coupling to phonons and further promotes the study of magnetically controlled slow light [25,26]. Besides, these pioneering works [17–23] have also bright about a series of novel effects and applications in classical and quantum regimes [27–36]. Especially, it is worth noting that the nonreciprocal transmission based on cavity magnonic system has attracted intense interest and been realized by utilizing numerous different ways [37–40]. As a new platform to explore nonreciprocity, cavity magnonic system has also been extended to achieve the one-way control of diverse physical phenomena [41–46] including entanglement, phonon or magnon laser, magnon blockade, and so on.
The generation of high-order sideband is an essentially nonlinear phenomenon and important foundation for frequency comb since the production of frequency comb can be regarded as parametric processes, in which either external or self-induced time-harmonic modulation of a parameter leads to the generation of abundant equidistant frequency sidebands [47]. The study of high-order sideband thereby becomes an indispensable part in the field of precision measurement and has been originally reported in hybrid optomechanical systems [48–52], atom-cavity coupling system [53], Kerr resonator [54], and non-Hermitian system [55]. Meanwhile, it is also put forward that magnon-induced high-order sideband can be fulfilled in hybrid cavity magnonic system [56–59], which opens a new way to produce frequency comb in the area of magnon spintronics [60–62] and can be applied in magnon-based precision measurement [63] and even accurate detection of nonlinear energy spectrum [64]. To further develop high-order sideband effect, its nonreciprocity has been explored by using spinning cavity optomechanics [65] and magnon Kerr nonlinearity [66]. Despite such success, the improvement of both efficiency and nonreciprocal strength of sideband based on hybrid magnonic system remain in great demand since the investigation of nonreciprocal sideband effect not only could be used to design multi-frequency microwave isolators but also deepen our understanding of the optomagnonic nonlinear physics and may be instructive for the realization of a new type of frequency comb, nonreciprocal optomagnonic frequency comb.
In this paper, we propose a scheme to realize high tunability and strong nonreciprocity of sideband in a spinning microwave magnomechanical system comprised of a YIG sphere and a spinning resonator which is coherently driven by a control field and a probe field. The Sagnac effect induced by the spinning resonator can give rise to the significant difference in sideband responses for two different driving directions. The efficiency of sideband can be enhanced in one driving direction and restrained in another direction due to asymmetric magnonic responses with different directions, which eventually results in nonreciprocal sideband responses. The properties of nonreciprocal two-color second-order sideband are demonstrated for the first time in cavity magnomechanics, which is absent in previous schemes of nonreciprocity [37–41,65,66]. The nonreciprocity of sideband is tunable and distinctly strengthened by regulating Sagnac-Fizeau shift. The influence of the power of the control field on the nonreciprocal sideband responses is also considered in our scheme. With the increase of the control field power, the efficiency and nonreciprocal strength of sideband can be greatly boosted and even extreme efficiency ($>60\%$) [48,51,65] and isolation ratio ($>80$dB) [37,40,66] are simultaneously realized, and the frequency region of achieving strong nonreciprocity is expanded. In short, the manipulation of strong nonreciprocity can be turned to a wider band. Our results may be applied to implement multi-frequency microwave isolator, diode, and circulator [10,11].
2. Model and equations
The model under investigation is depicted in Fig. 1, it consists of a spinning microwave resonator and a YIG sphere placed at the maximum of magnetic field. The magnon can be embodied by the collective excitations of a large number of spins in YIG. The phonon is formed through the vibration of deformation of the sphere that derives from the variation of magnetization induced by the magnon excitation inside the YIG sphere [21]. The magnon is coupled to the photon via magnetic dipole interaction and simultaneously coupled to the phonon via the magnetostrictive force. The spinning resonator is driven by a control field $\xi _l$ with frequency $\omega _l$ and a probe field $\xi _p$ with frequency $\omega _p$. The Hamiltonian of system under rotating wave approximation and in a frame rotating at the control field frequency $\omega _l$ can be expressed as [21,22,41,42]
Fig. 1. Schematic diagram of cavity magnonic system which consists of a spinning resonator and a YIG sphere, where the magnon and phonon modes can be supported by YIG sphere. By fixing rotation of the resonator along clockwise direction, the microwave circulating in the resonator undergos a Sagnac-Fizeau shift $\Delta _F$. When the driving fields are input from the left, which corresponds to $\Delta _F>0$. $\Delta _F<0$ denotes the driving fields input from the right.
In this work, we are interested in the mean response of the system, thereby, the operators can be reduced to their expectation values, viz, $a(t)=\langle \hat {a}(t)\rangle$, $m(t)=\langle \hat {m}(t)\rangle$, $b(t)=\langle \hat {b}(t)\rangle$. In this case, the operator equations can be reduced to the mean value equations by using the mean-field approximation via factorizing averages, and the quantum noises can be safely ignored because their mean values are zero [24,48]. The dynamic evolutions of the system can be described by the Heisenberg-Langevin equations
By solving Eq. (5), we can obtain the mean magnon number as a function of the control field power $P_l$ for $\Delta _F=0$, $\Delta _F=0.12\omega _b$, and $\Delta _F=-0.12\omega _b$, as shown in Fig. 2(a). Obviously, the bistability of the system can be modulated via the Sagnac-Fizeau shift $\Delta _F$. When the resonator is stationary ($\Delta _F=0$), the bistable property of the system is similar to cavity optomechanics [48]. For the situation where the system is driven from the left ($\Delta _F>0$) or right ($\Delta _F<0$) direction, the bistability can be greatly different, thus the nonreciprocal bistable property is realized. When the control field power $P_l$ $\in$ [0,1.5] mW, compared to the stationary situation ($\Delta _F=0$), the magnon number $|m_s|^2$ decreases when $\Delta _F>0$ and it increases when $\Delta _F<0$, as shown in Fig. 2(b). This variation of magnon number associated with directionality induces asymmetric nonlinear responses in different driving directions. It therefore causes sideband responses exhibiting nonreciprocal behavior. In addition, there is a linear relationship between the mean number of magnon and the control field power, which hints the nonlinear effect and nonreciprocity of system can be also flexibly manipulated via $P_l$. It is worth pointing out that the results of all subsequent discussions are obtained in stable regime of system when $P_l\in$ [0,16] mW. The time evolutions of magnon number can be obtained by solving numerically Eqs. (2)–(4) and we plot the magnon number versus the time for $P_l=1.5$ mW in Fig. 2(c)-(e). The system reaches a stable oscillation after a transient process, the same steady state excitation of magnon number emerges when $\Delta _p=\omega _b$ and $\Delta _p=-\omega _b$ under condition of same value of $\Delta _F$. However, the steady oscillation of the magnon number with time shows different behaviors for different values of $\Delta _p$, which will induce different sideband responses in different frequency regions. For the excitations of photon and phonon, they also exhibit nonreciprocal behavior, which can be seen in Fig. 2(f) and (g). Furthermore, the magnon number of numerical simulation in Fig. 2(c)-(e) is fitted well with the analytical result in Fig. 2(b) (see the black, blue, and red solid dots) whether $\Delta _F=0$, $\Delta _F=0.12\omega _b$ or $\Delta _F=-0.12\omega _b$, which verifies the credibility of analytical solution and validity of perturbation method. The equilibrium mean thermal photon/magnon/phonon number can be described as [31,32] $n_{th}=[{\rm exp}(\frac {\hbar \omega _n}{K_BT})-1]^{-1}$ ($n=a,m,b$), with the frequency $\omega _n$ of the corresponding mode, the Boltzmann constant $K_B$, and the ambient temperature $T$. Under room temperature ($300$ K), the mean numbers of thermal photons $a_{th}$, magnons $m_{th}$, and phonons $b_{th}$ can be estimated to be about $10^3$, $10^3$ and $10^5$, respectively. They are far less than the steady-state photon, magnon, and phonon numbers [see Fig. 2(c)-(g)]. Thus, even at room temperature, thermal noises have very little effect on the results of current scheme and even can be safely ignored [62]. In the recent experiments of cavity magnonic system carried out at room temperature [18,21,23], the photon, magnon, and phonon numbers respectively reach about $10^{10}$, $10^{12}$, and $10^{10}$, which are approximately consistent with our results [see Fig. 2(c)-(g)]. These experiments performed high quality measurements under the above conditions. Therefore, our results are feasible to be measured under the experimental noise.
Fig. 2. (a), (b) The mean magnon number $|m_s|^2$ as a function of the control field power $P_l$ for different values of $\Delta _F$. Time evolution of magnon number $|m|^2$ when (c) $\Delta _F=0$, (d) $\Delta _F=0.12\omega _b$, and (e) $\Delta _F=-0.12\omega _b$ for $P_l=1.5$ mW and $\Delta _p=\pm \omega _b$. Time evolutions of (f) photon number $|a|^2$ and (g) phonon number $|b|^2$ when $P_l=1.5$ mW and $\Delta _p=\omega _b$. Here, the parameters are used [21,31,42]: $\omega _a/2\pi =7.86$ GHz, $\omega _b/2\pi =11.42$ MHz, $2\kappa _a/2\pi =3.35$ MHz, $2\kappa _b/2\pi =300$ Hz, $2\kappa _m/2\pi =1.12$ MHz, $g_{ma}/2\pi =3.2$ MHz, $g_{mb}/2\pi =1$ Hz, $\Delta _a=0.7\omega _b$, $\Delta _m=1.8\omega _b$, and $\xi _p=0.05\xi _l$.
In what follows, we turn to consider the perturbation generated by the probe field. Submitting the expressions $a=a_s+\delta a$, $b=b_s+\delta b$, and $m=m_s+\delta m$ into Eqs. (2)–(4). We can obtain the equations of fluctuation terms
By using the input-output relations $S_{out}=S_{in}-\sqrt {\eta _a\kappa _a}a$, the output field can be expressed as
3. Results and discussions
In this section, we study the nonreciprocity and tunability of FSG and SSG in present system and the complete underlying physical mechanism is given.
3.1 Analysis of the corresponding FSG and the modulation of enhanced SSG
We firstly show the properties of FSG in the situation that the magnon mode resonates with the cavity mode ($\Delta _a=\Delta _m=\Delta$) and the resonator is stationary ($\Delta _F=0$). From Fig. 3(a), one can observe that there is only a sideband peak located in $\Delta _p=\Delta$ for small magnon-photon coupling strength and the splitting occurs when $g_{ma}$ is large, which can be seen more clearly in two insets. To explain this interesting phenomenon, the eigenvalues $\omega _{\pm }$ and the decay $\gamma _{\pm }$ of two supermodes created by the coupling of magnon-photon can be expressed (in a frame rotating with $\omega _l$)
Fig. 3. (a) The efficiency $\eta _1^{u}$ varying with the detuning $\Delta _p$ and magnon-photon coupling strength $g_{ma}$ for the situation of the stationary resonator ($\Delta _F=0$). The insets of (a) show $\eta _1^u$ versus $\Delta _p$ under the condition of $g_{ma}=0.01\omega _b$ and $g_{ma}=0.3\omega _b$, respectively. (b) $\eta _1^{u}$ varying with the detuning $\Delta _p$ and $\Delta _F$ when $g_{ma}/2\pi =3.2$ MHz. The inset of (b) shows $\eta _1^u$ versus $\Delta _p$ at the case of different $\Delta _F$. (c) The eigenvalues of the system varying with $g_{ma}$ and $|\Delta _F|$. The inset of (c) shows eigenvalues versus $g_{ma}$ for $\Delta _F=0$. Here, $\Delta _a=\Delta _m=\Delta =0.7\omega _b$, $P_l=0.015$ $\mu$W and the other parameters are same with Fig. 2.
In what follows, we discuss the modulation of the enhanced second-order sideband generation. We firstly study the properties of the probe field transmission $T$ and the efficiency of SSG $\eta _2^u$ for different powers of the control field. From Fig. 4(a), it is observed that the transmission spectrum only displays two absorption dips that corresponds to two supermodes [see Fig. 3(c)]. Under same condition of the system parameters, SSG appears at $\Delta _p=\omega _b$ and $\eta _2^u$ is relatively small, as shown in Fig. 4(d). If the control field is increased, an additional transparent window emerges in the transmission spectrum that can be seen in Fig. 4(b), which implies the occurrence of MMIT [21]. It is because strong input field is a important factor in magnetomechanical interaction and consequently gives rise to the one of supermodes to split. Simultaneously, SSG shows a pair of splitting peaks and a local minimum near $\Delta _p=\omega _b$ [see Fig. 4(e)]. It shows SSG is subdued when MMIT arises. Compared with Fig. 4(d), however, the maximal efficiency of SSG has been enhanced, which is the consequence of the strong control field causing stronger the excitation of magnon, an intuitive perspective can be found in Fig. 2(b). In order to better illustrate sideband characteristic of cavity magnomechanics, the value of $\Delta _m$ is adjusted ($\Delta _m=1.8\omega _b$), which can be implemented by regulating external magnetic field $H$ in experiment. The effect of MMIT disappears and the spectrum of $T$ shows three absorption dips, as shown in Fig. 4(c). One is magnomechanically induced absorption (MMIA) located at $\Delta _p=\omega _b$. The other two asymmetric dips remain correspond to two supermodes. For SSG, it is significantly enlarged and even more than $20\%$ [see Fig. 4(f)]. The properties similar to those of the higher-order sideband in cavity optomechanics system [48] are first demonstrated in the cavity magnomechanics. In conclusion, the SSG can be enhanced by properly adjusting the system parameters, which allows us to implement high efficiency and strong nonreciprocity of sideband responses.
Fig. 4. The transmission probability $T$ of probe field and the efficiency $\eta _2^u$ of SSG versus $\Delta _p$ for (a), (d) $P_l=0.015$ $\mu$W, (b), (e) $P_l=1.5$ mW and (c), (f) $\Delta _m=1.8\omega _b$. $\Delta _m=0.7\omega _b$ in (a), (b), (d), and (e). $P_l=1.5$ mW in (c) and (f). $\Delta _a=0.7\omega _b$, $\Delta _F=0$ and the other parameters are same with Fig. 3(b).
3.2 Enhanced nonreciprocity of sideband
We focus on the nonreciprocity of sideband in the case of non-resonance magnon-photon in this subsection. In order to illustrate how the sideband generation is influenced by resonator rotation for non-resonance case, we plot that the efficiency $\eta _1^u$ of FSG as a function of $\Delta _p$ for different $\Delta _F$ in Fig. 5(a). It can be found that $\eta _1^u$ shows three visible peaks which are consistent with Fig. 4(c). In addition, if one zooms sufficiency at $\Delta _P=-\omega _b$, there is a tiny peak created by MMIA appears in frequency spectrum of FSG, which does not occur in a generic optomechanical system [48,65] but can be observed in atom-assisted cavity optomechanics [53]. It can be attributed to the cavity coupled to the additional medium (magnon of our scheme) and is consistent with the results of recent experiments [21]. Furthermore, the efficiency of FSG at $\Delta _P=-\omega _b$ is much smaller than the other at $\Delta _P=\omega _b$ since the magnon number with time shows stronger steady oscillation for $\Delta _p=\omega _b$ compared with $\Delta _p=-\omega _b$, as shown in Fig. 2(c)-(e). The degrees of frequency shift of two peaks that correspond the two supermodes are matched with the changes of the eigenvalues in Fig. 5(b). Two sharp peaks induced by MMIA, their values and frequency shifts are also sensitive to the driving direction, will match the generation of nonreciprocal second-order sideband. From insets of Fig. 5(a) (density plots), we find that the difference between two peak values and the degrees of frequency shifts in the two directions increase with the increase of $|\Delta _F|$. Thus, the nonreciprocity of FSG can become more prominent by regulating $\Delta _F$.
Fig. 5. (a) The efficiency $\eta _1^{u}$ of FSG versus the detuning $\Delta _p$ for different $\Delta _F$. The insets show enlarged view of $\eta _1^{u}$ at approximately $\Delta _p=\pm \omega _b$ and $\eta _1^{u}$ varying with $\Delta _p$ and $\Delta _F$. (b) The eigenvalues of the system versus $|\Delta _F|$. Here, we have chosen $P_l=1.5$ mW and the other parameters are same with Fig. 2.
Figure 6 displays the efficiency $\eta _2^u$ as a function of $\Delta _p$ for different values of $\Delta _F$, i.e., different driving directions. From Fig. 6(a), one can observe that the SSG can be generated not only for $\Delta _p=\omega _b$, but also for $\Delta _p=-\omega _b$, which suggests so-called two-color second-order sideband generation (TSSG) is actually achieved in our scheme. TSSG is tremendously asymmetrical, the right peak is four orders of magnitude than the left. This difference of two peaks is more obvious than FSG. It stems from the fact that the different behaviors of steady oscillation of the magnon number with time in different frequency region affect the properties of each sideband generation [see Fig. 2(c)-(e)]. Moreover, it can be seen that the efficiency $\eta _2^u$ located at $\Delta _p=\omega _b$ is enhanced when the driving fields are input from the left and is restrained for the opposite driving direction. In contrast, the SSG at $\Delta _p=-\omega _b$ is weakened for $\Delta _F>0$ and strengthened for $\Delta _F<0$. As far as we know, this is the first time that nonreciprocal TSSG is achieved, which is unrealized in previous schemes of nonreciprocal sideband [65,66] and also indicates our study of nonreciprocity is extended beyond the linearized description compared to nonreciprocal transmission [37–41]. We also can see that the frequency shifts of the peaks of TSSG emerge for different driving direction and their shifts are coincident with two sharp peaks of FSG from Fig. 6(b) and (c) that show the magnified peaks of $\eta _2^u$ near $\Delta _p=\mp \omega _b$, respectively. In particular, $\eta _2^u$ can reach to the maximal value in one driving field while it is suppressed to near zero in the opposite one for special frequencies including $\Delta _p=\pm 0.9997\omega _b$ and $\pm 0.9999\omega _b$. This indicates typical near-perfect nonreciprocity of TSSG is realized, which may underpin the multi-frequency diode behavior. The frequency shifts and difference values of the peaks for two input directions profoundly enhancing trend with the increase of $\Delta _F$, as exhibited in the insets of Fig. 6(a). Thereby, $\Delta _F$ can be used as a mean to control the nonreciprocal property of high-order nonlinear effect.
Fig. 6. (a) The efficiency $\eta ^{u}_2$ of SSG versus the detuning $\Delta _p$ for different $\Delta _F$. The insets show $\eta _2^{u}$ varying with $\Delta _p$ and $\Delta _F$ near $\Delta _p=-\omega _b$ and $\Delta _p=\omega _b$. Magnified SSG peak near (b) $\Delta _p=-\omega _b$ and (c) $\Delta _p=\omega _b$. The other parameters are same with Fig. 5.
In order to better illustrate the nonreciprocity of sideband, a dimensionless quantity, called as isolation ratio, is introduced as [37,39,41]
In the following, we investigate quantitatively the nonreciprocal strengths of each sideband by applying the isolation ratio. The isolation ratio $I_1^u$ of FSG as a function of $\Delta _p$ is plotted in Fig. 7(a). As the increase of $|\Delta _F|$, $I_1^u$ grows gradually whether the FSG corresponding to two supermodels or the FSG corresponding to MMIA. The maximum value of $I_1^u$ can reach to $10$dB located at $\Delta _p=\omega _b$. When we turn our attentions to the nonreciprocity of SSG, both the red ($\Delta _p<0$) and blue ($\Delta _p>0$) detuning regimes have two bright regions are displayed in Fig. 7(b) and (c). This is consistent with Fig. 6, i.e., there are four optimal nonreciprocal frequency regions. The isolation ratio of SSG is sensitive sufficiently to Sagnac-Fizeau shift. As the increase of $|\Delta _F|$, $I_2^u$ is significantly enhanced from $0$ to $25.4$ dB for red detuning regime and from $0$ to $40$ dB for blue detuning regime. By comparing Fig. 7(a)-(c), the optimal isolation ratio of SSG is approximatively by $4$ times as large as FSG. One can summary that the higher order of sideband, the stronger nonreciprocity. This derives from the stronger degree of asymmetric nonlinear response for higher order sideband. The FSG is related to the steady-state amplitude and the first-order perturbation terms of magnon and phonon. However, the SSG, in addition to the above terms, is also associated with the second order perturbation terms. The nonreciprocity of each sideband results from the aggregate asymmetric nonlinear effect of its associated perturbation terms. Thus, for higher order sideband, the asymmetric nonlinear response is stronger, which induces higher isolation ratio.
Fig. 7. (a) The isolation ratio $I_1^{u}$ of FSG versus the detuning $\Delta _p$, the insets show magnified view at $\Delta _p=\omega _b$ and $I_1^{u}$ varying with $\Delta _p$ and $|\Delta _F|$ when $\Delta _p$ around $-\omega _b$ and $\omega _b$. The isolation ratio $I_2^{u}$ of SSG varying with the detuning $\Delta _p$ and $|\Delta _F|$ when $\Delta _p$ is closed to (b) $-\omega _b$ and (c) $\omega _b$. The other parameters are same with Fig. 5.
It is worth emphasizing that the power of the control field plays an important role on the modulation of sideband generation [48,51]. Taking the blue detuning regime as an example, the influence of the control field on the nonreciprocity of sideband is investigated. The efficiency and isolation ratio of FSG varying with $\Delta _p$ and $P_l$ are drawn in Fig. 8(a) and (b). $\eta _1^u$ in two directions can be enhanced concurrently when the power of the control field is increased. In addition, the control field can be used to regulate the located position of maximum value of $\eta _1^u$, i.e., the degree of frequency shift of peak. These results can lead to the improvement in the level of nonreciprocity, a direct insight is shown in Fig. 8(b). The above phenomena are prominently displayed for SSG. When the control field is weak ($P_l=1$ mW), both efficiencies of SSG in two driving directions are less than $20\%$. i.e., $\eta _2^u=18.3\%$ for left driving direction and $\eta _2^u=15.2\%$ in right [see Fig. 8(c)]. For stronger control field with the power $8$ mW, $\eta _2^u$ are increased to $57.2\%$ and $49.8\%$ in two input directions, respectively. If we continue to enhance the control power to $P_l=15$ mW, the maximum value of $\eta _2^u$ can reach to $56.1\%$ for $\Delta _F<0$ and even more than $60\%$ ($63.9\%$) for $\Delta _F>0$, which are quite high efficiencies for SSG and the maximum efficiency is improved by two times compared to spinning cavity optomechanics [65]. In more detail, as the enhancement of $P_l$, the difference between the peak values of $\eta _2^u$ in the two directions increases. Furthermore, the peaks become wider and their frequency shifts get bigger. Thus, the frequency scope, in which strong nonreciprocity occurs, is expanded. More importantly, the maximum isolation ratio in Fig. 8(d) is improved to $80$dB that is twice as large as Fig. 7(c). These results illustrate that the properties of sideband including high-efficiency, strong nonreciprocity, and broadband are implemented, which plays an important role in the realization of isolation devices within broad operational bandwidth [11,13].
Fig. 8. (a) The efficiency $\eta _1^{u}$ and (b) the isolation ratio $I_1^{u}$ of FSG varying with $\Delta _p$ and $P_l$. (c) The efficiency $\eta _2^{u}$ of SSG versus $\Delta _p$ for different $P_l$ when the driving fields are input from different directions. (d) The isolation ratio $I_2^{u}$ varying with $\Delta _p$ and $P_l$. Here, we set $|\Delta _F|=0.12\omega _b$ and the other parameters are same with Fig. 5.
To better embody controllability of the nonreciprocal characteristic of sideband, the maximum isolation ratio $I_{2max}^u$ of SSG versus $|\Delta _F|$ for different control power $P_l$ is exhibited in Fig. 9(a). $I_{2max}^u$ has a growing trend under condition of increasing $|\Delta _F|$ or $P_l$. To give a physical explanation, the magnon number $|m|^2$ varying with the time $t$ and $|\Delta _F|$ is plotted in Fig. 9(b). Obviously, the time evolutions of magnon number can be regulated by modulting Sagnac-Fizeau shift. From Fig. 9(c), we can observe that $|\Delta _F|$ can improve the excitation of magnon number in one direction but reduce excitation in the opposite direction. It causes the different efficiencies of sideband are generated in two different driving directions. We define the difference $\Delta _N=||m|^2_{+}-|m|^2_{-}|$ ($|m|^2_{\pm }$ denotes the magnon number in left or right driving direction) of magnon number of two directions to describe the degree of asymmetric magnon responses and it versus $|\Delta _F|$ is displayed in Fig. 9(g). As the increase of $|\Delta _F|$, the value of $\Delta _N$ grows gradually, which brings the enhancement of sideband nonreciprocity. From Fig. 9(d) and (e), we can obtain the magnon number with time evolutions can simultaneously be enlarged in two driving directions when increasing power $P_l$, whereas, the slopes of two growth lines are different, as shown in Fig. 9(f). Thereby, for stronger control power, the degree of asymmetric magnon responses $\Delta _N$ is greater [see Fig. 9(h)], which allows us to achieve a higher isolation ratio of sideband. In a word, $\Delta _N$ can be modulated via Sagnac-Fizeau shift and the power of the control field. So, they can be used as two method of modulating sideband nonreciprocity.
Fig. 9. (a) The maximum isolation ratio $I^{u}_{2max}$ of SSG versus $|\Delta _F|$ for different $P_l$. (b) The magnon number $|m|^2$ varying with time $t$ and $\Delta _F$ (left) and the corresponding two dimension projection (right). (c) $|m|^2$ versus $\Delta _F$ for $P_l=1.5$ mW. The magnon number $|m|^2$ varying with time $t$ and $P_l$ (left) and the corresponding two dimension projection (right) when (d) $\Delta _F=0.12\omega _b$ and (e) $\Delta _F=-0.12\omega _b$. (f) $|m|^2$ versus $P_l$ for $|\Delta _F|=0.12\omega _b$. The difference $\Delta _N$ of the magnon number in different directions versus (g) $|\Delta _F|$ for $P_l=1.5$ mW and (h) $P_l$ for $|\Delta _F|=0.12\omega _b$. The other parameters are same with Fig. 2.
4. Conclusions
In conclusion, we explore theoretically the nonreciprocal sideband responses in a spinning microwave magnomechanical system. Compared to the case of stationary resonator, when the resonator is rotating, the efficiency of sideband can be enhanced in one input direction but suppressed in opposite direction because the splitting of the resonance frequencies of the counter-circulating modes is created by Sagnac effect. Meanwhile, the opposite frequency shift of the peak of sideband emerges. These contribute to the production of nonreciprocal sideband responses. What’s interesting is that the nonreciprocal TSSG is realized for the first time, which plays a very important role in the design of multi-frequency isolation devices. This also shows that our scheme extends the study of nonreciprocity beyond the linear description. The nonreciprocity of sideband is greatly sensitive to the the Sagnac effect. As the enhancement of Sagnac-Fizeau shift, the isolation ratio of sideband can be visibly increased. It is also verified that the nonreciprocal properties of sideband have a high tunability through the adjustment of the power of the control field. The strong input control field can not only enhance the efficiency and isolation ratio of sideband, but also enlarge operational frequency bandwidth of observing nonreciprocity. The high-efficiency, high isolation ratio, and broadband nonreciprocal sideband responses are implemented in our scheme. These methods of modulating nonreciprocity benefit from the fact that the degree of asymmetric magnonic responses in two different driving directions can be regulated flexibly via Sagnac-Fizeau shift and the power of the control field. Our results have important applications in on-chip microwave control, chiral quantum engineering, and frequency comb-like precision measurement [8,60,63].
Appendix
The corresponding parameters in Eq. (10) are
Funding
National Key Research and Development Program of China (2021YFA1400700); National Natural Science Foundation of China (11774113, 12022507); Fundamental Research Funds for the Central Universities (2019kfyRCPY111).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008). [CrossRef]
2. C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z. L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10(4), 047001 (2018). [CrossRef]
3. F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008). [CrossRef]
4. K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017). [CrossRef]
5. S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018). [CrossRef]
6. R. Huang, A. Miranowicz, J. Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018). [CrossRef]
7. Y. H. Lai, Y. K. Lu, M. G. Suh, and K. Vahala, “Observation of the exceptional-point enhanced Sagnac effect,” Nature 576(7785), 65–69 (2019). [CrossRef]
8. P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017). [CrossRef]
9. Y. Zhou, D. Y. Lü, and W. Y. Zeng, “Chiral single-photon switch-assisted quantum logic gate with a nitrogen-vacancy center in a hybrid system,” Photonics Res. 9(3), 405–415 (2021). [CrossRef]
10. L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335(6067), 447–450 (2012). [CrossRef]
11. K. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-kerr nonlinearity,” Phys. Rev. Lett. 121(20), 203602 (2018). [CrossRef]
12. T. Shui, W. X. Yang, M. T. Cheng, and R. K. Lee, “Optical nonreciprocity and nonreciprocal photonic devices with directional four-wave mixing effect,” Opt. Express 30(4), 6284–6299 (2022). [CrossRef]
13. X. Wang, W. X. Yang, A. X. Chen, L. Li, T. Shui, X. Li, and Z. Wu, “Phase-modulated single-photon nonreciprocal transport and directional router in a waveguide-cavity-emitter system beyond the chiral coupling,” Quantum Sci. Technol. 7(1), 015025 (2022). [CrossRef]
14. D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami, and Y. Nakamura, “Hybrid quantum systems based on magnonics,” Appl. Phys. Express 12(7), 070101 (2019). [CrossRef]
15. K. Wang, Y. P. Gao, R. Jiao, and C. Wang, “Recent progress on optomagnetic coupling and optical manipulation based on cavity-optomagnonics,” Front. Phys. 17(4), 42201 (2022). [CrossRef]
16. H. Y. Yuan, Y. Cao, A. Kamra, R. A. Duine, and P. Yan, “Quantum magnonics: when magnon spintronics meets quantum information science,” Phys. Rep. 965, 1–74 (2022). [CrossRef]
17. Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113(8), 083603 (2014). [CrossRef]
18. D. Zhang, X. M. Wang, T. F. Li, X. Q. Luo, W. Wu, F. Nori, and J. Q. You, “Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere,” npj Quantum Inf. 1(1), 15014 (2015). [CrossRef]
19. X. Zhang, N. Zhu, C. L. Zou, and H. X. Tang, “Optomagnonic whispering gallery microresonators,” Phys. Rev. Lett. 117(12), 123605 (2016). [CrossRef]
20. Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Coherent coupling between a ferromagnetic magnon and a superconducting qubit,” Science 349(6246), 405–408 (2015). [CrossRef]
21. X. Zhang, C. L. Zou, L. Jiang, and H. X. Tang, “Cavity magnomechanics,” Sci. Adv. 2(3), e1501286 (2016). [CrossRef]
22. C. A. Potts, E. Varga, V. A. S. V. Bittencourt, S. V. Kusminskiy, and J. P. Davis, “Dynamical backaction magnomechanics,” Phys. Rev. X 11(3), 031053 (2021). [CrossRef]
23. R. C. Shen, J. Li, Z. Y. Fan, Y. P. Wang, and J. Q. You, “Mechanical bistability in Kerr-modified cavity magnomechanics,” Phys. Rev. Lett. 129(12), 123601 (2022). [CrossRef]
24. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]
25. T. X. Lu, H. Zhang, Q. Zhang, and H Jing, “Exceptional-point-engineered cavity magnomechanics,” Phys. Rev. A 103(6), 063708 (2021). [CrossRef]
26. J. Zhao, L. Wu, T. Li, Y. X. Liu, F. Nori, Y. Liu, and J. Du, “Phase-controlled pathway interferences and switchable fast-slow light in a cavity-magnon polariton system,” Phys. Rev. Appl. 15(2), 024056 (2021). [CrossRef]
27. Y. P. Wang, G. Q. Zhang, D. Zhang, T. F. Li, C. M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120(5), 057202 (2018). [CrossRef]
28. B. Wang, Z. X. Liu, C. Kong, H. Xiong, and Y. Wu, “Magnon-induced transparency and amplification in PT-symmetric cavity-magnon system,” Opt. Express 26(16), 20248–20257 (2018). [CrossRef]
29. Z. X. Liu, H. Xiong, and Y Wu, “Magnon blockade in a hybrid ferromagnet-superconductor quantum system,” Phys. Rev. B 100(13), 134421 (2019). [CrossRef]
30. X. Li, X. Wang, Z. Wu, W. X. Yang, and A. Chen, “Tunable magnon antibunching in a hybrid ferromagnet-superconductor system with two qubits,” Phys. Rev. B 104(22), 224434 (2021). [CrossRef]
31. M. Yu, H. Shen, and J. Li, “Magnetostrictively induced stationary entanglement between two microwave fields,” Phys. Rev. Lett. 124(21), 213604 (2020). [CrossRef]
32. J. Li and S. Gröblacher, “Entangling the vibrational modes of two massive ferromagnetic spheres using cavity magnomechanics,” Quantum Sci. Technol. 6(2), 024005 (2021). [CrossRef]
33. D. W. Luo, X. F. Qian, and T Yu, “Nonlocal magnon entanglement generation in coupled hybrid cavity systems,” Opt. Lett. 46(5), 1073–1076 (2021). [CrossRef]
34. D. Kong, J. Xu, C. Gong, F. Wang, and X. Hu, “Magnon-atom-optical photon entanglement via the microwave photon-mediated Raman interaction,” Opt. Express 30(19), 34998–35013 (2022). [CrossRef]
35. C. Z. Chai, Z. Shen, Y. L. Zhang, H. Q. Zhao, G. C. Guo, C. L. Zou, and C. H. Dong, “Single-sideband microwave-to-optical conversion in high-Q ferrimagnetic microspheres,” Photonics Res. 10(3), 820–827 (2022). [CrossRef]
36. A. Kani, B. Sarma, and J. Twamley, “Intensive cavity-magnomechanical cooling of a levitated macromagnet,” Phys. Rev. Lett. 128(1), 013602 (2022). [CrossRef]
37. Y. P. Wang, J. W. Rao, Y. Yang, P. C Xu, Y. S. Gui, B. M. Yao, J. Q. You, and C. M Hu, “Nonreciprocity and unidirectional invisibility in cavity magnonics,” Phys. Rev. Lett. 123(12), 127202 (2019). [CrossRef]
38. Y. L. Ren, S. L. Ma, J. K. Xie, X. K. Li, and F. L. Li, “Non-reciprocal optical transmission in cavity optomagnonics,” Opt. Express 29(25), 41399–41408 (2021). [CrossRef]
39. C. Kong, X. M. Bao, J. B. Liu, and H. Xiong, “Magnon-mediated nonreciprocal microwave transmission based on quantum interference,” Opt. Express 29(16), 25477–25487 (2021). [CrossRef]
40. C. Kong, J. Liu, and H. Xiong, “Nonreciprocal microwave transmission under the joint mechanism of phase modulation and magnon Kerr nonlinearity effect,” Front. Phys. 18(1), 12501 (2023). [CrossRef]
41. Z. B. Yang, J. S. Liu, A. D. Zhu, H. Y. Liu, and R. C. Yang, “Nonreciprocal transmission and nonreciprocal entanglement in a spinning microwave magnomechanical system,” Ann. Phys. 532(9), 2000196 (2020). [CrossRef]
42. Y. Xu, J. Y. Liu, W. Liu, and Y. F. Xiao, “Nonreciprocal phonon laser in a spinning microwave magnomechanical system,” Phys. Rev. A 103(5), 053501 (2021). [CrossRef]
43. Y. J. Xu and J. Song, “Nonreciprocal magnon laser,” Opt. Lett. 46(20), 5276–5279 (2021). [CrossRef]
44. K. W. Huang, Y. Wu, and L. G. Si, “Parametric-amplification-induced nonreciprocal magnon laser,” Opt. Lett. 47(13), 3311–3314 (2022). [CrossRef]
45. Y. Wang, W. Xiong, Z. Xu, G. Q. Zhang, and J. Q. You, “Dissipation-induced nonreciprocal magnon blockade in a magnon-based hybrid system,” Sci. China Phys. Mech. Astron. 65(6), 260314 (2022). [CrossRef]
46. S. Y. Guan, H. F. Wang, and X. Yi, “Cooperative-effect-induced one-way steering in open cavity magnonics,” npj Quantum Inf. 8(1), 102 (2022). [CrossRef]
47. M. A. Miri, G. D’ Aguanno, and A. Alù, “Optomechanical frequency combs,” New J. Phys. 20(4), 043013 (2018). [CrossRef]
48. H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012). [CrossRef]
49. H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015). [CrossRef]
50. S. Liu, W. X. Yang, Z. Zhu, T. Shui, and L. Li, “Quadrature squeezing of a higher-order sideband spectrum in cavity optomechanics,” Opt. Lett. 43(1), 9–12 (2018). [CrossRef]
51. D. G. Lai, X. Wang, W. Qin, B. P. Hou, F. Nori, and J. Q. Liao, “Tunable optomechanically induced transparency by controlling the dark-mode effect,” Phys. Rev. A 102(2), 023707 (2020). [CrossRef]
52. J. H. Liu, Y. F. Yu, Q. Wu, J. D. Wang, and Z. M. Zhang, “Tunable high-order sideband generation in a coupled double-cavity optomechanical system,” Opt. Express 29(8), 12266–12277 (2021). [CrossRef]
53. B. Chen, L. Shang, X. F. Wang, J. B. Chen, H. B. Xue, X. Liu, and J. Zhang, “Atom-assisted second-order sideband generation in an optomechanical system with atom-cavity-resonator coupling,” Phys. Rev. A 99(6), 063810 (2019). [CrossRef]
54. Y. F. Jiao, T. X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a Kerr resonator,” Phys. Rev. A 97(1), 013843 (2018). [CrossRef]
55. L. Y. He, “Parity-time-symmetry-enhanced sideband generation in an optomechanical system,” Phys. Rev. A 99(3), 033843 (2019). [CrossRef]
56. Z. X. Liu, B. Wang, H. Xiong, and Y. Wu, “Magnon-induced high-order sideband generation,” Opt. Lett. 43(15), 3698–3701 (2018). [CrossRef]
57. S. N. Huai, Y. L. Liu, J. Zhang, L. Yang, and Y. X. Liu, “Enhanced sideband responses in a PT-symmetric-like cavity magnomechanical system,” Phys. Rev. A 99(4), 043803 (2019). [CrossRef]
58. W. L. Xu, Y. P. Gao, T. J. Wang, and C. Wang, “Magnon-induced optical high-order sideband generation in hybrid atom-cavity optomagnonical system,” Opt. Express 28(15), 22334–22344 (2020). [CrossRef]
59. C. Zhao, Z. Yang, R. Peng, J. Yang, C. Li, and L. Zhou, “Dissipative-coupling-induced transparency and high-order sidebands with Kerr nonlinearity in a cavity-magnonics system,” Phys. Rev. Appl. 18(4), 044074 (2022). [CrossRef]
60. Z. Wang, H. Y. Yuan, Y. Cao, Z. X. Li, R. A. Duine, and P. Yan, “Magnonic frequency comb through nonlinear magnon-skyrmion scattering,” Phys. Rev. Lett. 127(3), 037202 (2021). [CrossRef]
61. Z. X. Liu and Y. Q. Li, “Optomagnonic frequency combs,” Photonics Res. 10(12), 2786–2793 (2022). [CrossRef]
62. H. Xiong, “Magnonic frequency combs based on the resonantly enhanced magnetostrictive effect,” Fundamental Research (In Press) [CrossRef]
63. C. A. Potts, V. A. S. V. Bittencourt, S. V. Kusminskiy, and J. P. Davis, “Magnon-phonon quantum correlation thermometry,” Phys. Rev. Appl. 13(6), 064001 (2020). [CrossRef]
64. V. A. S. V. Bittencourt, I. Liberal, and S. V. Kusminskiy, “Optomagnonics in dispersive media: magnon-photon coupling enhancement at the Epsilon-near-zero frequency,” Phys. Rev. Lett. 128(18), 183603 (2022). [CrossRef]
65. W. A. Li, G. Y. Huang, J. P. Chen, and Y. Chen, “Nonreciprocal enhancement of optomechanical second-order sidebands in a spinning resonator,” Phys. Rev. A 102(3), 033526 (2020). [CrossRef]
66. M. Wang, C. Kong, Z. Y. Sun, D. Zhang, Y. Y. Wu, and L. L. Zheng, “Nonreciprocal high-order sidebands induced by magnon Kerr nonlinearity,” Phys. Rev. A 104(3), 033708 (2021). [CrossRef]
