Tunable optical amplification and group delay in cavity magnomechanics

In our study, we primarily focus on the average response of the system; thus by considering the Hamiltonian of Eq. (2), and ignoring the input quantum noises [87], the Heisenberg-Langevin equations of our compound CMM system can be expressed as:

where $\sum_{j=1}^{2}\kappa_{mj}$ and $\kappa_{b}$ denotes the decay rates of magnons and phonons. Keep in mind that $\kappa_{1}>0$, $\kappa_{2}>0$ in Eq. (3) is associated to passive-active compound CMM resonator system, while $\kappa_{1}>0$, $\kappa_{2}<0$ is associated to a passive-passive system [88].

Relative to the intensity of the controlled field, we assume that the intensities of the external magnetic driving fields and the weak probe field satisfies the condition $|\varepsilon_{p}|\ll\varepsilon_{l}$ and $|\varepsilon|_{m}\ll\varepsilon_{l}$ [87, 89]. Thus in this case, we can linearize the set of above dynamical equations of our proposed model by assuming each operator is the sum of its mean value and fluctuation i.e., $O=O_{s}+O_{+}e^{-i\delta t}+O_{-}e^{i\delta t}$, where $O=a_{1},a_{2},b,m_{1},m_{2}$ [44, 90], the first term $O_{s}$ represents the steady-state values and $\delta{O}=O_{+}e^{-i\delta t}+O_{-}e^{i\delta t}$ depicts the small fluctuating terms. By using perturbation expansion ansatz in Eq. (3) and considering the time derivative equal to zero we can obtain the steady-state solution as:

Here we define: $\lambda_{1}=\frac{{(\kappa_{1}+i\Delta_{a})((\kappa_{2}-i\Delta_{a}))-J^{2}}}{{(\kappa_{2}-i\Delta_{a})}}$, $\lambda_{2}=(\kappa_{1}-i\Delta_{a})+(\frac{g_{1}^{2}}{-i\Delta_{m1}+\kappa_{m1}})+(\frac{g_{2}^{2}}{-i\Delta_{m2}+\kappa_{m2}})-(\frac{J^{2}}{\kappa_{2}+i\Delta a})$, $\lambda_{3}=(\frac{\lambda_{1}\lambda_{2}-4G^{2}}{\lambda_{2}})+(\frac{g_{1}^{2}}{i\Delta_{m1}+\kappa_{m1}})+(\frac{g_{2}^{2}}{i\Delta_{m2}+\kappa_{m2}})$. By performing a series of mathematical calculations and neglecting all higher-order terms $(\delta o\delta o)$ the evolution of the perturbation terms of Eq. (3) can be shown as:

here, $F=g_{mb}m_{2s}$, and $\Delta_{m_{2}}^{\prime}=\Delta_{m_{2}}+g_{mb}(b_{s}+b_{s}^{*})$ signifies the effective strength of the magnon-phonon interaction, and effective magnon detuning respectively.

To determine the amplitudes of the first-order side band, we expand the small fluctuation terms in Eq. (5) by using the following ansatz as [91, 87]:

We consider the magnon driving field to become resonant with the probe field frequency which leads us to consider $\delta=\delta_{pl}=\delta_{ml}$ [45, 92], and $A_{i+}$ and $A_{i-}$ denotes the $i$th cavity generated probe field and four-wave mixing field amplitudes, respectively.

Now by employing Eq. (6) into Eq. (5) and comparing the coefficients of similar order, one can get the amplitudes of the first ordered side-bands (detail calculations and the explicit constants are presented in §A.1) as [91, 87]: After solving Eqs. (12) we derive the coefficients for the first lower sidebands. This demonstrates the characteristics of our system. Therefore, the coefficient of the $A_{1+}$, which indicates the output probe field amplitude of the CMM resonator can be written as:

Here, $\Phi=\phi_{m}-\phi_{pl}$ corresponds to the relative phase of the applied fields. Furthermore, by employing the input/output relationship of the cavity $s_{\text{out}}=s_{\text{in}}-\sqrt{2\eta\kappa_{1}}a_{1}$ [93], we get the normalized output probe field intensity from the CMM resonator as:

The real and imaginary components of the $\alpha$ ($\alpha=t_{p}+1$) can be quantified as the absorption and dispersion of the probe field at the probe frequency, respectively.

In this subsection, we will explore the physics behind the formation of MMIT window profiles by systematically analyzing the influence of magnon-photon and magnon-phonon coupling strengths in our model. Figure 2 displays the absorption (dispersion) spectrum of the probe field versus probe detuning ($\delta/\omega_{b}$) for various coupling strengths. In Fig. 2(a), we consider the scenario where both the magnon-phonon coupling ($g_{mb}$) and the coupling strength of the magnon mode $m_{2}$ ($g_{2}$) with the cavity are absent. As a result, only magnon mode $m_{1}$ is coupled with the cavity. Following this scenario, we observe a typical Lorentzian-type absorption peak of the output spectrum (as shown by a blue solid line) in Fig. 2(a), and the corresponding dispersion curve is indicated by red dotted lines. Interestingly, when the coupling strengths of both magnon modes $m_{1},m_{2}$ ($g_{1},g_{2}$) are present, and only the magnon-phonon coupling ($g_{mb}$) is absent, we observe that the typical Lorentzian-type absorption peak at resonance $\delta=\omega_{b}$ transforms into a transparency window. Additionally, two symmetrical absorption dips occur at $\delta=0.7\omega_{b}$(left) and $\delta=1.3\omega_{b}$(right) as shown in Fig. 2(b). The central peak at $\delta=\omega_{b}$ represents magnon-induced transparency, while the two sideband absorption peaks indicate magnon-induced absorption (MIA) [66]. The physical reason for these peaks is the generation of magnon-polaritons due to the photon-magnon interaction in our system. As a result, our entire system is effectively reduced to a cavity magnonic (CM) system [94, 95, 96].

It is noteworthy that the width of the transparency window and the amplification in our system can be controlled by adjusting the magnon-photon coupling strength ($g_{2}$). However, the results are not shown here. Finally, we consider the scenario where all three couplings are present, i.e., $g_{1}=g_{2}=2\kappa_{1}$, and $g_{mb}=2\pi$ Hz. In this case, due to the nonzero magnetostrictive interaction, the single MMIT window at the resonance point ($\delta=\omega_{b}$) of the Fig. 2(b) splits into double MMIT windows with a central absorption peak, as shown in Fig. 2(c). The physical explanation is that the existence of phonon interactions can create additional pathways for destructive interference, leading to additional transparency windows and our entire system reduced to CMM system [69].

In Fig. 3, we plot the real and imaginary parts of the output probe field as a function of optical detuning $\delta/\omega_{b}$ for different selected OPA gain parameters $G$. It is evident that changes in the parametric gain influence the transparency window. In the absence of OPA ($G=0$), we observe clear absorption in our system. However, introducing OPA ($G\neq 0$) converts the positive absorption peak at resonance into a negative peak, indicating gain in our system. The gain provided by the OPA in CMM systems offers substantial benefits, including signal amplification, tunability, sensitivity, and the ability to explore complex physical phenomena. Given these advantages, our work provides greater benefits compared to previously reported studies in [44].

In the following, we investigate the impact of the microwave field’s amplitude, its relative phase, and the nonlinear parameter OPA on the transmission profiles of the output spectrum. Referring to our discussion about Fig. 2, the positions of the magnetomechanically induced absorption and the MIA peaks are determined, respectively. Here, we are interested in employing an extra magnon driving field to examine the detailed impacts of the magnon driving parameters on the features of the output field in our CMM system. In Fig. 4(a), We graph the transmission spectrum ($|t_{p}|^{2}$) of the output probe field versus optical detuning ($\delta/\omega_{b}$) for different magnetic field amplitudes ($B_{0}$), keeping the relative phase $\Phi=0$. It is observed that with an increase in the magnetic field amplitude ($B_{0}$), the central absorption peak at the resonance point ($\delta=\omega_{b}$) becomes slightly deeper. There is a slight enhancement in the transparency spectrum on the left side of the resonance point, coupled with a lower absorption dip on the right side of the resonance point ($\delta=\omega_{b}$). Interestingly, when we further increase the magnetic field amplitude ($B_{0}=10\times 10^{-10}$T), the symmetric absorption dips on both sides of the resonance point ($\delta=\omega_{b}$) become asymmetric. In the blue-sideband regime ($\delta=1.3\omega_{b}$), the absorption dip not only transforms into transparency but also exceeds an amplitude of 1, thereby enabling the amplification of the weak probe field. Conversely, in the red-sideband regime ($\delta=0.7\omega_{b}$), there is a lower absorption dip as shown by the blue solid line in Fig. 4(a). Controlling the magnitude of the magnon driving field can convert MIA into amplification, which has potential uses in quantum sensing and computation.

Notably, the relative phase of the applied fields has been employed to regulate OMIT in cavity optomechanics devices [97, 98]. We now study the effect of the relative phase of the cavity and magnon driving fields on the transmission rate of the system. In Fig. 4(b), we set the magnetic field amplitude $B_{0}=12\times 10^{-10}$T and display the transmission spectrum ($|t_{p}|^{2}$) of the output probe field versus the probe detuning with different relative phase ($\Phi$). Figure. 4(b) clearly shows that the absorption resulting from the combined action of magnons and polaritons is sensitive to the relative phase ($\Phi$). Setting $\Phi=0$ leads to an asymmetric transmission profile in the output spectrum. This configuration results in amplification on the right side of the resonance point ($\delta=\omega_{b}$) and absorption on the left side, as illustrated by the red dotted line in Fig. 4(b). Accordingly, when we set the relative $\Phi=\pi$ the switching of the asymmetry line shapes becomes noticeable [see blue solid line in Fig. 4(b)]. In contrast, the central absorption dip at $\delta=\omega_{b}$ remains fairly stable as the relative phase varies from 0 to $\pi$.

The physical mechanism behind this attribute is that by adjusting the relative phase ($\Phi$) of the cavity and magnon driving fields, periodic constructive and destructive interference occurs, significantly altering the cavity-field transmission spectra [97, 99]. As shown in Fig. 4(b), at the exact resonance point ($\delta=\omega_{b}$), destructive interference occurs, resulting in strong MIA. In contrast, on either side of the resonance point, varying $\Phi$ from 0 to $\pi$ leads to constructive interference and noticeable amplification. This behavior is referred to as magnon-induced amplification (MIAMP). This ability of selectively switching and amplifying the input probe signal could be highly desirable in practical optical communications [97, 100].

To better illustrate the impact of the relative phase $\Phi$ on the MMIT profile, we plot the contour map of the transmission spectrum against the optical detuning ($\delta/\omega_{b}$) and the relative phase $\Phi$ as shown in Fig. 4(c). The transmission of the probe light changes periodically. At the exact resonance point, we observe strong absorption. The transmission rate transitions from strong absorption to amplification on either side of the resonance point as the phase is tuned from $-2\pi$ to $2\pi$.

Therefore, based on the results from Fig. 4, we conclude that the application of an additional magnon driving field in the CMM system allows the MIA and the transparency of the CMM system to be switched by adjusting the magnetic field amplitude and the relative phase while keeping the other system parameters fixed.

To illustrate the significant effect of an OPA on the transmission rate ($|t_{p}|^{2}$) we depict the graph of transmission rate ($|t_{p}|^{2}$) versus detuning ($\delta/\omega_{b}$) as shown in Fig. 5. In Fig. 5(a), in the absence of OPA, specifically when the gain ($G$) along with phase difference ($\theta$) are both zero, we observed that our system displays an absorption dip at resonance position ($\delta=\omega_{b}$) as shown by the red dot curve in Fig. 5 (a). As anticipated, introducing the nonlinear gain ($G=0.5\kappa_{1}$) results in the MMIT spectrum displaying an asymmetric Fano line shape and an enhanced transparency window. The primary mechanisms generating such Fano-like peaks are the coupling of magnonic modes within cavities and the interference between direct transmission and resonant tunneling paths. Further increasing the nonlinear gain ($G=1\kappa_{1}$) of an OPA improves the transmission rate and wider transparency window, indicating amplification in our system, as indicated by the blue solid curve in Fig. 5 (a). This is because increasing the nonlinear gain ($G$) of an OPA increases the magnon number, resulting in a broader MMIT window [91] and amplification in our system. In Fig. 5(b), we plot the transmission rate ($|t_{p}|^{2}$) versus detuning ($\delta/\omega_{b}$) for different phases of an OPA ($\theta=0.5\pi$, $\theta=\pi$ and $\theta=1.5\pi$) we see that phase shift of an OPA also affects the window width as well as the transmission peak. The maximum transmission peak can be obtained when we set the OPA phase $\theta=0.5\pi$. Hence, we can effectively regulate the transmission rate by modifying the OPA settings.

It is commonly established that systems with $\mathcal{PT}$-symmetry can experience a phase transition, leading to several fascinating and novel phenomena [101, 102].To investigate the occurrence of a phase transition in our $\mathcal{PT}$-symmetric system, we focus primarily on the optical modes, taking into account both the optical loss and gain. Furthermore, we ignore the influence of OPA, and $g_{1}=g_{2}=g_{mb}=0$. This yields an effective non-Hermitian Hamiltonian, as in [103, 88, 104]:

Equation (9) can be straightforwardly diagonalized via a Bogoliubov transformation [105], and we obtain the complex frequency as:

To investigate the $\mathcal{PT}$-broken and $\mathcal{PT}$-symmetric regimes in our system we plot the real and imaginary parts of eigenfrequencies ($\lambda-\Delta_{a})/\kappa_{1}$ against the coupling strength $J/k_{1}$, as depicted in Fig. 6. First, we investigate the equal gain and loss scenario ($k_{1}=k_{2}$), as depicted by the blue-solid and red-dashed curves in Figs. 6(a$-$b). In this scenario, two distinct regimes can be observed: (i) when $J>|\kappa_{1}+\kappa_{2}|/2$ in this scenario the eigenfrequencies of the two supermodes exhibit two distinct frequencies ($\Delta_{a}\pm\sqrt{4J^{2}-(k_{1}+k_{2})^{2}/2}$), and an identical linewidth ($k_{1}-k_{2})/2$, indicating that our system is in a $\mathcal{PT}$-symmetry regime (as shown by the white region in Figs. 6((a)$-$(b))). (ii) when $J<|\kappa_{1}+\kappa_{2}|/2$ in this scenario, the two supermodes coalesce and have the identical frequency and different linewidths, indicating that our system is in a broken $\mathcal{PT}$-symmetry regime (as illustrated by the light pink shaded area in Figs. 6(a$-$b)).

The critical point, i.e., $J=|\kappa_{1}+\kappa_{2}|/2$, where the resonance frequencies and the linewidths of the two supermodes become degenerate, is known as the EP or $\mathcal{PT}$-phase transition point in our hybrid system. As demonstrated in Figs. 6(a$-$b), by enhancing the coupling rate $J$, one can observe the transition from a broken to an unbroken $\mathcal{PT}$-phase. Unlike the $\mathcal{PT}$-symmetric system with gain and loss, if both cavities are operating in a passive-passive mode with the same loss rate, i.e., $\kappa_{1}=-\kappa_{2}$, the supermodes always have different frequencies but identical linewidths. The phase transition point does not occur [see black-solid and green-dotted lines in Figs. 6(a$-$b)].

Now, we examine the scenario where the gain/loss are not perfectly matched i.e., $k_{1}=Rk_{2}~{}(R\neq 1)$, where $R$ shows the gain/loss rate. In Figs. 6(c$-$d) we consider two cases, i.e., $R=0.5$ ($k_{1}=0.5k_{2}$) and $R=2$ ($k_{1}=2k_{2}$). Compared to the equal gain/loss scenario (as in Figs. 6(a$-$b)), one can clearly see that by varying the gain/loss rate ($R=2$ or $R=0.5$) causes the EP to shift the right or left, which implies that either a weaker or a stronger coupling strength $J$ is necessary for $\mathcal{PT}$-phase transition.

The simplification to gain and loss in the cavities is done to isolate and study the fundamental $\mathcal{PT}$-symmetric behavior of the system. This approach allows us to focus on the key aspects of $\mathcal{PT}$-symmetry—gain and loss—without introducing additional complexities that could obscure the desired physical insights. Once we establish a clear understanding of $\mathcal{PT}$ -symmetry in this context, more complex interactions can be systematically added to extend the analysis.

In the above subsections §III.1 and §III.2, we examine the impact of various parameters on the transmission rate. However, we keep $J=0$, which means we work with a single-loss cavity system. Now, our main task is to incorporate the effect of the tunneling rate $J$ and observe its impact on the transmission of our CMM system.

Figure 7 (a$-$b) displays the plot of the optical transmission rate ($|t_{p}|^{2}$) as a function of optical detuning ($\delta/\omega_{b}$) in the passive-passive and active-passive CMM systems for different optical tunneling strengths $J$, respectively. In Fig. 7(a) for the passive-passive case, we observed that when the optical tunneling strength $J$ is zero, there is a higher absorption dip at the resonance point with a small transmission peak on the left side of the resonance point, as shown by the red dotted curve in Fig. 7(a). We noticed that increasing the optical tunneling strength ($J=0.5\kappa_{1},0.8\kappa_{1}$) for the passive-passive case in our CMM system reduced transmission. This occurs because the increased optical tunneling strength in a passive-passive scenario leads to the formation of two hybrid supermodes, $\hat{a}_{\pm}=(\hat{a}_{1}\pm\hat{a}_{2})/\sqrt{2}$, arising from the coupling between the two cavities [101, 106, 107]. The interference between these hybrid supermodes reduces the efficiency of light transmission through the system, thereby compressing the overall transmission.

Compared to the passive-passive scenario (Fig. 7(a)), the optical transmission rate in the passive-active CMM system (Fig. 7(b)) is significantly boosted. The reason for this enhancement is that as the optical tunneling strength $J$ increases, the probe light absorption by the passive cavity weakens at resonance, resulting in a transparency window in the spectrum. The increased photon tunneling strengthens the normal-mode splitting between the passive and active cavities, disrupting resonance absorption. Consequently, Stokes scattering is significantly suppressed while the anti-Stokes field (a two-phonon process) is amplified [108]. When the anti-Stokes field becomes degenerates with the probe field, destructive interference between the light pathways in the coupled cavities suppresses absorption, allowing more light to pass through and thus increasing transmission. Therefore, amplification in our CMM system can be tuned by adjusting the optical tunneling strength $J$ [108, 84].

For better comprehension, we provide the analytical results of the optical transmission rate $|t_{p}|^{2}$ versus optical detuning $\delta/\kappa_{1}$ in two types of systems by setting the optical tunneling strength $J=0.8\kappa_{1}$ as in Fig. 7(c). In Fig. 7(c), the black-dotted curve illustrates the conventional passive-passive scenario, whereas the red-solid curve represents the passive-active $\mathcal{PT}$-symmetric-like system. For the passive-passive scenario, an Autler-Townes-splitting-like spectrum is observed [109, 89]. However, unlike the passive-passive scenario, a strong signal amplification can be observed between the two Autler-Townes absorption dips in a passive-active scenario. The significant amplification peak is centered at $\delta=7\kappa_{1}$, as illustrated by the red solid line in Fig. 7(c). Figure. 7(d) depicts the logarithm of the transmission rate $|t_{p}|^{2}$ vs optical tunneling strength $J/\kappa_{1}$. For the $\mathcal{PT}$-symmetric system with gain and loss (passive-active scenario), as $J$ increases, the signal transmission gradually changes from absorption to amplification. Significant signal enhancement is achievable at the phase transition point (EP), indicated by the green solid line in Fig. 7(d). Moreover, perfect signal absorption can be achieved in the broken $\mathcal{PT}$-symmetric regime. Additionally, when the coupling strength is low, no signal amplification occurs in a system without gain, and significant signal absorption is observed. Consequently, the transmission rate is always less than 1 in the passive-passive system, as shown by the dotted-dashed curve in Fig. 7(d). As $J$ increases beyond the phase transition point, the logarithm of the transmission rate forms an approximately straight line near zero. Therefore, in the $\mathcal{PT}$-symmetric CMM system, the absorption and amplification of the probe light can be controlled by adjusting the optical tunneling strength $J$.

Correspondingly, Fig. 8 shows the contour (2D) map of transmission rate $|t_{p}|^{2}$ vs cavity detuning $\Delta_{a}/\omega_{b}$ and relative phase $\Phi$ for two different cases. From Fig. 8(a), it is observed that for the passive-passive case, due to the change of relative phase $\Phi$, a periodic change of the transmission rate takes place, and strong absorption is also observed near the $\Delta_{a}=\omega_{b}$ point. In Fig. 8(b), the passive-active scenario shows strong amplification at the $\Delta_{a}=\omega_{b}$ point. The reason for this is that in the passive-active scenario, our $\mathcal{PT}$-symmetric CMM system can add nonlinear effects that increase amplification. These nonlinearities can enhance the strength of the signal in the gain region, resulting in greater overall amplification than systems with solely passive components as in Fig.8(a).

In general, the CMM systems can not only achieve the MMIT effect but also cause the light to slow or advance [66, 91]. This characteristic can be explained through the optical group delay. The optical group delay of the output probe light is [110, 44, 111]:

A group delay greater than zero ($\tau_{g}>0$) indicates the phenomenon of slow light, whereas a negative group delay ($\tau_{g}<0$) corresponds to the fast light phenomenon. Specifically, the recent advancements in research utilizing slow light have led to a diverse range of applications, including telecommunications and optical data processing.

The optical group delay ($\tau_{g}$) of our system’s probing field is currently being examined. To illustrate this, in Fig. 9 we plot the optical group delay $\tau_{g}$($\mu s$) versus optical detuning $\delta/\omega_{b}$. In Fig. 9(a), we depict the group delay $\tau_{g}$($\mu s$) of the probe light for differed driving magnetic field amplitudes $B_{0}$ while keeping the relative phase at $\Phi=0$. It is evident that the group delay can be adjusted by varying the magnetic field amplitude of the magnon driving field. When the magnetic field amplitude $B_{0}=0$, a negative group delay is observed [as shown by the blue dashed-dotted line in Fig. 9(a)], indicating the fast light effect of the output probe field. Setting the magnetic field amplitude  $B_{0}=6\times 10^{-10}$T reduces the negative group delay at the resonance point and increases the positive group delay on the right side of the resonance point [as shown by the black dotted line in Fig. 9(a)], indicating that slow or fast light can be achieved. As the magnetic field amplitude increases to $B_{0}=12\times 10^{-10}$T, the positive group delay rises and reaches a maximum value of $3.8\mu s$ near the resonance point (refer to the red solid line in Fig. 9(a)), indicating that the magnon-phonon interaction leads to slow light. Figure. 9(b) illustrates that the group delay can be adjusted by modulating the relative phase $\Phi$ of the applied field. By comparing the group delay profiles for the relative phases $\Phi=0$, and $\Phi=\pi$ [as shown by the red and black solid lines in Fig. 9(b)], we observe that the positive group delay at the probe detuning $\delta\approx 1.002\omega_{b}$ corresponding to magnon-induced absorption, significantly decreases and shifts in the case of $\Phi=\pi$. This investigation demonstrates that for a fixed value of the magnetic field amplitude ($B_{0}$), the dispersion property of the system can be modulated, and the slow and the fast light can be achievable and switchable in the CMM system by using the phase modulation of the applied fields.

To further investigate the group delay of the probe light in our CMM system, we plot the group delay $\tau_{g}$($\mu s$) versus pump power $P_{l}(mW)$ for various values of OPA gains: $G=0$ (blue dashed-dot curve), $G=0.2\kappa_{1}$ (black dashed curve), and $G=0.3\kappa_{1}$ (red solid curve) as shown in Fig 10. Here, we consider two different scenarios, (i) passive-active system [as shown in Fig 10(a)], (ii) passive-passive system [as shown in Fig 10(b)].

(i) In Fig 10(a), for passive-active system ($\kappa_{1}=\kappa_{2}$). In the absence of a degenerate ($G=0$), the positive group delay of the probe light decreases sharply as the pump power $P_{l}(mW)$ increases, reaching $\tau_{g}=0$, and then remains constant with further increases in pump power. This behavior corresponds to the slow light effect [see the blue dashed-dot curve in Fig 10(a)]. In sharp contrast, introducing non-linear gain from an OPA ($G=0.2\kappa_{1}$, and $G=0.3\kappa_{1}$) into the system and increasing the pump power ($P_{l}(mW)$) causes the group delay ($\tau_{g}$) of the probe light to become negative [see the light yellow region in Fig 10(a)], resulting in the observation of fast light. One advantage of our CMM system is its ability to gradually produce either fast or slow light by adjusting the OPA gains.

(ii) In Fig 10(b) for passive-passive system ($\kappa_{1}=-\kappa_{2}$). When compared to the passive-active system, it is evident that the positive group delay has a lower value and decreases sharply as the pump power increases [blue dashed-dot curve in Fig 10(b)], indicating that most of the region exhibits negative group delay and the majority of the probe light demonstrates fast-light behavior. Surprisingly, in the presence of an OPA ($G=0.2\kappa_{1}$, and $G=0.3\kappa_{1}$), the behavior differs significantly from that of the passive-active system. The positive group delay peak increases with OPA gains, then sharply decreases as the pump power increases up to $P_{l}=4(mW)$, after which it remains constant. This suggests that the OPA gains can dramatically affect the dispersion of the system.

In summary, this subsection analyzes the optical group delay of the probe field against probe detuning and pump power, considering different system parameter settings. Our observations suggest that careful adjustment of system parameters can strongly influence the group delay, thereby modulating the speed of light transmission and enabling a transition from slow to fast light within the system. This tunability offers flexibility in controlling the optical properties and can be leveraged for various applications in optical communication and signal processing [52].

In the following subsection, we present numerical values to demonstrate that the assumptions in our analysis are reasonable and consistent with current state-of-the-art experimental achievements, thereby reinforcing the experimental feasibility of our proposal. We consider a microwave cavity containing YIG spheres and an OPA. This method can be experimentally implemented using a planar cross-shaped cavity or a coplanar waveguide made from high-conductivity copper. The choice of cavity structure can affect the frequency range of the magnomechanical interactions studied, as different designs may support various modes and resonances. Planar structures offer benefits in terms of integration with other components, such as magnetic materials or sensors, potentially enhancing the functionality of the device.

The YIG spheres, with a diameter of $D=250\mu$m, function as both phonon and magnon resonators, featuring a phonon frequency of $\omega_{b}=2\pi\times 11.42$ MHz and the spin density $\rho=4.22\times 10^{27}\textup{cm}^{-3}$ [69, 82]. Because of its exceptional material as well as geometrical characteristics, the YIG sphere is also a great mechanical resonator. The phonon along magnon modes is connected to the changing magnetization brought about by the magnon excitation, which deforms the YIG sphere’s spherical shape as well as vice versa.

To measure group delay experimentally, techniques like interferometry and pump-probe measurements can be employed. Interferometry compares a reference pulse with the output pulse to determine phase shifts related to group delay, while pump-probe methods use a short pump pulse to excite the system and a delayed probe pulse to analyze the response. High-speed photodetectors can capture the temporal characteristics of the output light, enabling precise determination of group delay and insight into the system’s dynamics. When comparing time-delay values with state-of-the-art setups, recent experiments have demonstrated highly controllable and tunable slow and fast light effects in various cavity magnonic systems [112, 99] achieving group delays on the order of nanoseconds. Notably, the cavity systems in Refs. [113, 114, 115] demonstrate group delays in the microsecond range, which are very close to the values achieved in our system. In our system, by utilizing experimentally realizable parameters, we achieve both pulse advancement and delay, specifically ranging from $\tau_{g}$ = -7$\mu s$ and $\tau_{g}$ = 9$\mu s$. This demonstrates excellent agreement with the experimental realizations of fast and slow light reported in Refs. [116, 117] and can be further improved with longer lifetimes in magnonic systems. Recent advancements in cavity magnonics make it feasible to realize the characteristics we have reported in state-of-the-art laboratory experiments, and we believe that the system parameters and structure we propose are achievable with current experimental techniques.