Magnon-induced transparency and amplification in &#x1D4AB;&#x1D4AF;-symmetric cavity-magnon system

Bao Wang; Zeng-Xing Liu; Cui Kong; Hao Xiong; Ying Wu

doi:10.1364/OE.26.020248

1. Introduction

Yttrium-iron-garnet (YIG) sphere is a typical ferrimagnetic garnet material, recently, a huge amount of progress has been made in the growing field of magnetooptics. Cavity-magnon system (CMS) [1–4] provides a unique setting for realizing good quantum coherence and strong coupling between magnons and cavity photons. The fundamental aspect of magnon is the description of the quantized magnetization excitation [5]. The coherent interaction between magnons and photons is determined by the Magneto-optics (MO) effects, which can enter the strong and even ultrastrong coupling regime of cavity quantum electrodynamics thanks to the large spin density in YIG [2,6]. Moreover, due to the extraordinary robustness against temperature, we can observe the strong coupling between magnons and microwave photons even at room temperatures [6]. Various physical systems have been proposed and investigated to study magnons experimentally and theoretically, including the research on ferromagnetic magnons with a high-finesse cavity [7,8] and the observation of the magnon-polariton bistability [9]. In addition, it has achieved significant advances in quantum information processing, for example, a magnon coherently couples with a superconducting qubit [10], elastic waves [11] and phonon [12, 13]. The emerging field of cavity-magnon system is also experiencing rapid evolution that is benefiting from low damping rates of both magnons and cavity photons and strong coupling between them.

Recently, 𝒫𝒯-symmetric optical structure has recently attracted extensive research on the field of optics [14–17], which has emerged as an unique platform to achieve distinctive optical behaviour which is previously unattainable with ordinary optical system [18]. It is known that the transition between an unbroken 𝒫𝒯-symmetric phase (real eigenvalue spectra) to a spontaneously broken-𝒫𝒯-symmetric phase (complex eigenvalue spectra) emerges as the system parameters in the Hamiltonian are properly modified [16]. 𝒫𝒯-symmetric systems show the significant property when the 𝒫𝒯-phase transits from 𝒫𝒯-symmetric phase to broken 𝒫𝒯-symmetric phase [19,20]. For instance, optical non-reciprocity have been observed in 𝒫𝒯-symmetric whispering-gallery microcavities (WGM) experimentally [15], and the measurement sensitivity of detecting weak mechanical motion can be significantly enhanced near the 𝒫𝒯-phase transition point (the exceptional point) [21]. In very recent work, non-hermitian degeneracies known as exceptional points occur in open physical systems have been utilized to possess exceptional-point-enhanced sensitivity in an optical microcavity [22,23], which suggests that the sensitivity enhancement conforms to square-root and cube-root decay law. Moreover, J.Q. You et al. have reported that the exceptional point can be observed in cavity magnon-polaritons by a tunable cavity quantum electrodynamics system with a small ferromagnetic sphere in a microwave cavity [24]. However, CMS properties underlying the MIT effect in the 𝒫𝒯-symmetric regime with balanced gain and loss, remain largely unexplored.

Thanks to recent progress in experiment, 𝒫𝒯-symmetric whispering-gallery microcavities and cavity-magnon system have been proposed and demonstrated experimentally, moreover many unique optical effects such as low-power optical isolation and magnon-polariton bistability have been observed [2,9,15,18]. On this basis, it is now clear that a considerable gain can be introduced to the cavity in cavity-magnon system. In this work, we study the light transmission process in a 𝒫𝒯-symmetric CMS which is formed by an active microwave cavity and an embedded YIG crystal, and revealing that the significant impact of 𝒫𝒯 symmetry on manipulating light transmission. We find that (i) In contrast to passive CMS, the transmitted probe power in 𝒫𝒯-symmetric CMS is enhanced about four orders of magnitude and the bandwidth also becomes much narrower by controlling the cavity gain. (ii) The magnon-induced transparency and amplification effects can be well modulated by adjusting the applied magnetic field without changing other parameters. We note that the active magnon-induced transparency and amplification effects we studied here, is different from the EIT or OMIT phenomena in atomic systems [25] and optomechanics [26] with auxiliary optical gain, where only the coherent driving field itself is generally controlled. This magnetic-field-dependence feature provides our system with great potential value on developing photonic devices.

The remainder of the paper is organized as follows: In sec. 2, we establish physical model and utilize the Heisenberg-Langevin equations of motion of the 𝒫𝒯-symmetric cavity-magnon system to calculate the optical spectral responses. In sec. 3, we illustrate the magnon-induced transparency and amplification in details and show that the 𝒫𝒯-symmetry plays a key role in our regime which can feasibly regulate and enhance the MIT effect. Further, we focus on the role of the applied magnetic field in the light transmission and present the discussion on the experimental implementation of our scheme. Finally, we make a conclusion based on the result we obtained in sec. 4.

2. Physical setup and dynamical equation

Figure 1(a) gives the schematic of the system we proposed, a small YIG sphere is placed inside an active three-dimensional (3D) rectangular microwave cavity, whose eigenfrequency of the cavity-mode TE₁₀₂ we used is ω_c/2π = 10.1GHz [9]. According to recent research, it is viable to make a passive cavity couple to an active cavity via optical tunneling [15,18]. Moreover, the photon and magnon could be coupled with each other in the cavity-magnon system, which is implemented with an electromagnetic mode in the superconducting resonator and a magnon mode in a small yttrium iron garnet (YIG) sphere [6,9]. The YIG sphere is magnetized by a local static magnetic field generated by a superconducting magnet, which can make the YIG sphere a single-domain ferromagnet [5]. The Hamiltonian of the total system reads [6]

H = H c + H_{m} + H_{1}

where a (a^†) and b (b^†) are the annihilation (creation) operators of the active cavity mode and the magnon mode, respectively. The interaction Hamiltonian H_I is given by J_m(a^†b + ab^†), with J_m being the coupling coefficient between the cavity and the magnon mode, which describes the magnon mode interacting with the cavity mode a (with frequency ω_c). Here H_c = ħω_ca^†a characterizes the cavity mode a with the resonance frequency ω_c. And H_m = ħω_mb^†b denotes the Hamiltonian of magnon mode b with the frequency ω_m/2π = 10.1 GHz of the Kittle mode (i.e., the ferromagnetic resonance mode). To study the system in input-output theory, we extend this system to include gain and loss yields the semiclassical Heisenberg-Langevin equations in a rotating frame at the probe field frequency ω_p

\dot{ℑ} = ℜ ℑ + 𝒟 + ϱ

where

ℜ = (\begin{matrix} - i δ_{c} + κ / 2 & - i J_{m} \\ - i J_{m} & - i δ_{m} - γ_{m} / 2 \end{matrix}), 𝒟 = (\begin{matrix} \sqrt{η κ} s_{p} \\ 0 \end{matrix})

where δ_c = ω_c − ω_p, δ_m = ω_m − ω_p, ϱ = (a_in, b_in)^T and ℑ = (a, b)^T, T is the representation of matrix transposition, a_in and b_in are the noise operators, with

〈 a_{in}^{†} (t) a_{in} (t^{'}) 〉 = 0

and

〈 b_{in}^{†} (t) b_{in} (t^{'}) 〉 = 0

at optical frequency taking Markov approximation into consideration [27,28]. The cavity mode a is coherently driven by a probe field with amplitude s_p, frequency ω_p, where

s_{p} = \sqrt{P_{p} / (ℏ ω_{p})}

, P_p denotes the power of light field. In the present work, we study the mean response of the system, and all operators can be replaced by their expectation values, viz. a = 〈â(t)〉 and b = 〈b̂(t)〉, and using the mean field approximation, e.g., 〈AB〉 = 〈A〉〈B〉. κ and γ_m are the optical gain of active cavity and the decay of magnon, respectively. From Eq. (2) we can obtain the expectation value of the cavity operator a in steady state as

a (δ) = \sqrt{η κ} s_{p} / Θ (δ)

where

Θ (δ) = i δ - κ / 2 + J_{m}^{2} / (i δ + γ_{m} / 2)

, δ = ω_c(m) − ω_p, and we consider the cavity and the magnon are on resonance (ω_c = ω_m). The transmission power of probe field relative to the incident probe power is defined as P_T = 𝒜_T/s_p, where the complex field transmitted through the cavity is

𝒜_{T} = \sqrt{η κ} a (δ)

. The coupling parameter η can be continuously adjusted, which is chosen to be the critical coupling value 1/2.

Fig. 1 (a) Schematic diagram of 𝒫𝒯-symmetric cavity-magnon system, which includes an active rectangular 3D cavity and a small YIG sphere. κ is the cavity gain. B₀ denotes the applied static magnetic field, which is parallel to the long edge of the cavity and is perpendicular to the microwave magnetic field at the YIG sphere. The system is pumped by a probe field ω_p. (b) The energy-level diagram of the cavity-magnon coupling scheme, where |a〉, |b〉 denote the number states of the cavity photon and magnon, respectively. The probe field s_p can make a single-photon transition from |0, 0〉_a,b to |1, 0〉_a,b, |1, 0〉_a,b to |2, 0〉_a,b and |0, 1〉_a,b to |1, 1〉_a,b.

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3. Results and discussion

In this section, we analyze the properties of the transmission of the probe field in this 𝒫𝒯 symmetric cavity-magnon system. Due to the presence of active gain, both the mode splitting and linewidth of the supermodes changes. Before proceeding further, it is instructive to briefly illustrate the principal mechanism behind 𝒫𝒯 symmetry in our studied system. Specifically, the Hamiltonian of 𝒫𝒯-symmetric cavity-magnon system is

{\hat{H}}_{m} = (\begin{matrix} {\hat{a}}^{†} & {\hat{b}}^{†} \end{matrix}) (\begin{matrix} ℏ ω_{c} + i κ / 2 & J_{m} \\ J_{m} & ℏ ω_{m} - i γ_{m} / 2 \end{matrix}) (\begin{matrix} \hat{a} \\ \hat{b} \end{matrix})

In the following, in order to explore the physical process more clearly, the second-order matrix

R = (\begin{matrix} ℏ ω_{c} + i κ / 2 & J_{m} \\ J_{m} & ℏ ω_{m} - i γ_{m} / 2 \end{matrix})

can be transformed into a diagonal matrix R′ by the transformation matrix M , namely, we can study the physical process in the eigenstate representation. Therein

R^{'} = (\begin{matrix} ℏ ω_{c} + i γ_{+} & 0 \\ 0 & ℏ ω_{-} + i γ_{-} \end{matrix})

and the 𝒫𝒯-symmetric subsystem can be diagonalized as

{\hat{H}}^{'}_{m} = (\begin{matrix} {\hat{B}}_{1}^{†} & {\hat{B}}_{2}^{†} \end{matrix}) (\begin{matrix} ℏ ω_{+} + i γ_{+} & 0 \\ 0 & ℏ ω_{-} + i γ_{-} \end{matrix}) (\begin{matrix} {\hat{B}}_{1} \\ {\hat{B}}_{2} \end{matrix})

via a Bogoliubov transformation

\hat{A} = M \hat{B}

[29], which implies that

(\begin{matrix} {\hat{b}}_{1} \\ {\hat{b}}_{2} \end{matrix}) = M (\begin{matrix} {\hat{B}}_{1} \\ {\hat{B}}_{2} \end{matrix})

, where ω_± denote the corresponding eigenfrequencies of two supermodes B̂₁ and B̂₂ created by the coupling of cavity and magnon modes, that is

\begin{array}{l} ω_{\pm} = \frac{ω_{c} + ω_{m}}{2} \pm Re \sqrt{J_{m}^{2} - {[\frac{i (ω_{c} - ω_{m})}{2} + \frac{κ + γ_{m}}{4}]}^{2}} \\ γ_{\pm} = \frac{(κ - γ_{m})}{4} \pm Im \sqrt{J_{m}^{2} - {[\frac{i (ω_{c} - ω_{m})}{2} + \frac{κ + γ_{m}}{4}]}^{2}} \end{array}

where γ_± denotes the decay of the two supermodes. When J_m < (κ + γ_m)/4, the system is in 𝒫𝒯-symmetry breaking phase with no supermode splitting, i.e., ω₊ = ω₋ = ω_m, which leads to two nondegenerate dissipation coefficients

\frac{(κ - γ_{m})}{4} \pm \sqrt{{(\frac{κ + γ_{m}}{4})}^{2} - J_{m}^{2}}

. For J_m = (κ + γ_m)/4, the corresponding eigenvalues simultaneously coalesce and the dissipation rate is degenerate, which means that the degenerate supermodes possess the same linewidth. For the case of a strong optical tunneling rate, i.e., J_m > (κ + γ_m)/4, the system have a degenerated effective dissipation rate γ_± = (κ − γ_m)/4 and the supermodes frequencies

ω_{\pm} = ω_{m} \pm \sqrt{J_{m}^{2} - {(\frac{κ + γ_{m}}{4})}^{2}}

are nondegenerate. The noteworthy feature of 𝒫𝒯-symmetric cavity-magnon system is that it exhibits a phase transition when J_m passes through the value of (κ + γ_m)/4, which is the so-called exceptional point (EP) [30].

Figure 2(b) and 2(d) plot, respectively, the transmitted probe power |P_T|² versus the probe detuning δ_c and the coupling strength J_m with passive and active cavity. It is counterintuitive that the transmission effect of the magnon does not increase monotonically with increasing coupling strength J_m, instead it has a maximum at $J_{m}^{m}$ . It can be seen from Fig. 2(d) that the maximal value of |P_T|² is greatly increased neighboring $J_{m}^{m} = 30 MHz$ at the resonance condition δ = 0, which illustrates that the system reaches the vicinity of the exceptional point [31–33]. On the other hand, with respect to the system assembled by passive optical cavity, a non-amplifying transmission spectra profile can be shown in Fig. 2(b). As shown in Figs. 2(b) and Figs. 2(d), apparently the maximum value of the |P_T|² of 𝒫𝒯-symmetric structure is enhanced about four orders of magnitude in the vicinity of the coupling strength J_m = 30 MHz, compared to passive cavity system. Due to the enhancement of coherent coupling between the cavity field and the magnon induced by 𝒫𝒯 symmetry, not only is the transmission intensity greatly enhanced but also the narrower bandwidth is obtained. As illustrated in the inset of Fig. 2(d), the value of the coupling strength J_m/2π = 12.9 MHz and δ_m = γ_m, the linewidth of transmission spectrum is very narrow in this 𝒫𝒯-symmetric system, which is a manifestation of the intrinsic property of 𝒫𝒯-symmetric CMS. To emphasize that the narrow laser linewidth leads to the better coherence property, such that it offers an approach to achieve high-intensity and narrow-pulse laser [34]. Consequently, we stress that 𝒫𝒯-symmetric regime plays a crucial role in light transmission.

Fig. 2 The logarithm of the transmitted probe power |P_T|² varies with the applied static magnetic field and the coupling strength J_m when (a) κ/2π = −8.75 MHz and (c) κ/2π = 8.75 MHz. We use the parameters here are ω_c/2π = ω_m/2π = 10.1 GHz, and γ_m/2π = 8.75 MHz [9]. (b) The logarithm of the |P_T |² varies with the coupling strength J_m and frequency detuning δ with the passive cavity, i.e., κ = γ_m, other parameters are the same as (a). Inset: |P_T|² varies with frequency detuning δ at the case of J_m = 12.9 MHz. (d) The logarithm of the |P_T|² varies with the coupling strength J_m and frequency detuning δ, with the active cavity, i.e., κ = γ_m, other parameters are the same as (c). Inset: |P_T|² varies with frequency detuning δ at the case of J_m = 12.9 MHz. All the parameters are chosen based on the recent experiment and can be achieved under the currently existing experimental technique.

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As shown in Figure 2(a) and Figure 2(c), the transmitted probe power |P_T|² varies with the coupling strength J_m and the external magnetic field B₀ with passive and active cavity respectively. We find that the gain can also strongly affect light transmission of the system, as Fig. 2(c) shows, the transmitted probe power |P_T|² output from the active cavity tends to be maximized as J_m approaches the vicinity of the critical point, i.e., J_m = 61.2 MHz. In sharp contrast to the conventional MIT profile in a lossy cavity, a non-amplifying transmitted rate of the probe profile can be shown in Fig. 2(a). Physically, such strong enhancement of |P_T|² originates from the small optical-gain-induced strong anti-Stokes field. In other words, the anti-Stokes field are greatly enhanced around the critical point in our 𝒫𝒯-symmetric cavity-magnon system.

In what follows, we turn to discuss the the influence of the gain and the applied magnetic field on MIT and MIA effects. According to the cavity-magnon coupling, we perform magnon excitation spectroscopy in the presence of active gain in the cavity. Fig. 3 shows |P_T|² as a function of the gain-loss ratio κ/γ_m and the applied magnetic field B₀ with the coupling strength J_m/2π = 24.6 MHz. We can find the giant enhancement of the optical transmission emerges in the area of κ/γ_m ∈ [0.33, 0.41]. Moreover, the maximum value of the transmitted rate of the probe |P_T|² occurs at the case of B₀ = 310 mT and κ/γ_m ∼ 0.38 approximately. At a fixed value of the external magnetic field, the transmitted rate of the probe can be significantly enhanced by tuning only the gain-loss ratio in a broken-𝒫𝒯-symmetric phase. This gain-induced reinforced transmission leads to a new way to manipulate optical transparency and amplification with cavity engineered magnon ststem, which may open the research of the magnon-induced novel physics phenomena with gain and loss.

Fig. 3 The logarithm of the |P_T|² versus the gain-loss ratio κ/γ_m and the applied magnetic field B₀. The other parameters are J_m/2π = 24.6 MHz and γ_m/2π = 8.75 MHz and the frequency detuning δ_c = δ_m = γ_m.

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We proceed to study the remarkable influence of the optical gain on the transmission spectrum, the transmitted probe power |P_T|² output from the cavity-magnon system is shown in Fig. 4, featuring resonance peaks for the δ = 0. We can clearly see that two peak values of the red solid spectrum line are about 41% at the resonance condition δ = ±J_m when κ = −γ_m which means that the cavity is passive. Interestingly, when we replace the passive cavity with an active cavity, the maximum of the transmitted probe power can be significantly enhanced for the resonance case, namely, when κ = γ_m, the maximum of |P_T|² are 6700%, in which the efficiency of probe field generation improves about two orders of magnitude under the condition of the same other parameter configuration, compared to the passive cavity system. Furthermore, we continue to increase the optical gain strength κ = 1.2 γ_m and 1.4 γ_m, while the transmitted probe power output from the 𝒫𝒯-symmetric cavity-magnon system reduces to 1400%, 720% approximately exhibited by the blue and magenta dashed curve. This indicates that an optimal enhancement of transmitted spectra can be obtained around EP with the gain-loss balance. The underlying physical mechanism can be explained as follows: The dynamical accumulations of photon energy result in drastic optical gain of probe frequency, which can tremendously improve the transmitted probe power. It can be found that the optical amplification results from optical gain κ via enhanced feedback-backaction. Meanwhile, |P_T|² sharply decreases when the gain crosses the EP [26].

Fig. 4 Calculated transmission spectrum of the probe light as a function of the frequency detuning δ under different cavity gain κ. J_m/2π = 4.1 MHz and γ_m/2π = 8.75 MHz. Inset: |P_T|² varies with frequency detuning δ at the case of κ = −γ_m.

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To explore the profound impact of applied static magnetic field on the transmitted probe power |P_T|², in Fig. 5, we plot |P_T|² charactered by the magnetic field strength B₀. We find that the transmission spectrum has a Lorentzian-like shape dependence on the applied magnetic field, which presents an additional degree of freedom to control light transmission in 𝒫𝒯-symmetric cavity-magnon system. Such magnon-induced transparency (MIT) manifests a remarkable difference from conventional EIT and OMIT. For traditional EIT and OMIT in atom [35] or optomechanical system [36] and even in 𝒫𝒯-symmetric system [26, 33], the transparency window in the transmission spectrum of the output field must be triggered by the strong coherent driving field. In our regime, the transparency window can be manipulated by applying a static magnetic field. In addition, [37] reports that it might be possible to utilize controlled magnetic fields to create a variable-width transparency window, nevertheless, the maximum of the transmission spectrum is about 1, which can’t realize optical amplification. In our regime, the 𝒫𝒯-symmetric cavity-magnon system leads to an actively amplified MIT spectrum, whose maximum value can achieve 15 under the current parameters as shown in Fig. 5. We find that the applied magnetic field indirectly influences the light propagation behavior, where a remarkable transparency-to-amplification transition happens in the vicinity of B₀ = 360.7 mT. As the magnetic field continually increases, the amplification of probe power has a obvious peak value at B₀ = 360.81 mT and then it drops off rapidly until B₀ = 361.16 mT. The blue area denotes the appearance of magnon-induced transparency (MIT). When the magnetic field is within the range of [360.7, 361.16]mT, magnon-induced amplification (MIA) appears. That is to say, the transmitted probe power can be effectively controlled including transparency and amplification by adjusting the magnetic field without changing other parameters. This magnetic-field-dependent feature may inspire the extensive exploration of 𝒫𝒯-symmetric cavity-magnon system and the engineering of magnetic-field-related integrated optical apparatus.

Fig. 5 The change of transmission spectrum |P_T|² with adjusted magnetic field. We use the parameters here are J_m/2π = 11.3 MHz, δ_c = δ_m = γ_m, γ_m/2π = 8.75 MHz and κ = γ_m. Blue area denotes |P_T|² ≤ 1.

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Before ending this section, let us discuss the experimental implementation of our proposal. The 3D cavity we used is made of oxygen-free copper with dimensions 44 × 22 × 6 mm³ [9], the frequency of the cavity-mode TE₁₀₂ that we use is ω_c/2π = 10.1 GHz. The YIG sphere with a diameter of 1 mm is adhered to an aluminum oxide rod and installed near the antinode of the magnetic field of the TE₁₀₂ mode as recent experiments reported [9]. The coupling strength between magnon and photon can be well tuned by adjusting the bias magnetic field [38]. Furthermore, the frequency of the magnon mode is adjustable ranges from several hundreds of megahertz to 28 GHz due to the fact that the external magnetic field is tunable in the range of 0 to 1 T, such wide frequency range allows the resonance coupling between the magnons and microwave photons in the cavity-magnon system [39]. We envision an experiment for constructing the 𝒫𝒯-symmetric cavity-magnon system via engineering an active 3D microwave cavity. Firstly, A piezoelectric membrane pumped by optical field can generate electric current which can be sensed by the coil. Then the system can receive the acoustic feedback which can realize the acoustic gain [40]. This design method achieving nonlinear active acoustic metamaterials can inspire us to realize the active 3D microwave cavity. Secondly, 𝒫𝒯-symmetric optical microcavities have been realized experimentally [15]. The cavity gain can be well controlled by adjusting the pump laser and doped ion. Consequently, the 3D microwave cavity is fabricated from oxygen-free copper, which can be doped with the active metamaterials with inherent enhancing. And the active metamaterials can emit photons in the 10.1 GHz frequency arising from optical pump, which results in the amplification of weak signal light in the 10.1 GHz frequency. Such that an active 3D microwave cavity can be effectively realized by delicate feedback systems using the active metamaterials. In our scheme, the resonance frequency of the microwave cavity mode is ω_c/2π = 10.1 GHz, and the frequency of magnon mode is determined by ω_m = γH, where H is the external applied static magnetic field and γ = gμ_B/ħ is the gyromagnetic ratio with g being the g factor and μ_B the Bohr magneton (here we consider γ = 2π × 28GHz/T) [5].

4. Conclusion

To summarize, we have demonstrated the magnon-induced transparency (MIT) and amplification (MIA) phenomenon under the different parameter configuration in the 𝒫𝒯-symmetric cavity-magnon system. Due to 𝒫𝒯 symmetry, the intensity of transmission spectrum is enhanced and the bandwidth is narrower, compared to passive cavity system. We have discussed the gain-induced dramatic enhancement of MIT and MIA effects concretely. Furthermore, we have found that the transmitted probe power can be well controlled by adjusting the magnetic field without changing other parameters, which may provide a powerful platform to utilize controlled magnetic field to study slowing light and stopping light effect. Note that the previous works on EIT and OMIT were limited to the driving-field-controlled regulatory method, yet our work realizes a new way to control MIT and MIA effects that takes advantage of the magnetic-field-controlled feature. We envision that 𝒫𝒯-symmetric cavity-magnon system will open a new route for exploring fundamental magnetic-optics field and would serve as an interesting playground to research nonlinear effects of magnon in a parameter regime that is unavailable to traditional optical system.

Funding

National Natural Science Foundation of China (NSFC) (11375067, 11405061, 11574104); National Basic Research Program of China (2016YFA0301203).

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Magnon-induced transparency and amplification in 𝒫𝒯-symmetric cavity-magnon system

Abstract

1. Introduction

2. Physical setup and dynamical equation

3. Results and discussion

4. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (9)

Optics Express