1. Introduction

Coherent interaction between laser radiation and multilevel atoms exhibits fascinating phenomena such as electromagnetically induced transparency (EIT) [1–6]. EIT is an approach for turning an opaque medium to a transparent one around resonance. EIT was first observed in cold atomic ensembles [1] and has attracted a great deal of interest due to its distinctive application in nonlinear optics and information processing. A remarkable characteristic of EIT is considerable reduction in the group velocity of light passing through the material, accomplished inside a practically lossless transparency window [2,3]. A noteworthy example of light beam control in optomechanical systems is optomechanically induced transparency (OMIT), which was initially theoretically predicted in April 2010 [6] and experimentally observed a few months later [4]. In analogy to the EIT and OMIT, magnomechanically induced transparency (MIT) is one of the promising phenomena in a Hybrid magnonic system [7–9], which allows us to manipulate light and explore the slow light in cavity magnomechanical systems by a nonlinear phonon-magnon interaction.

Recently, magnetic materials have offered a promising alternative to achieve strong light-matter interaction. It means a hybrid magnonic system is a novel class of hybrid quantum system that is based on magnons (collective spin excitations) in a YIG–Y$_3$ Fe$_5$ O$_{12}$ – crystal that has a low damping rate and high spin density. It should be noted that YIG is ferrimagnetic. However, YIG has a high Curie temperature as $559 K$, so it is ferromagnetic below this temperature [10]. Magnons have been extensively investigated because of their long coherence times [11–14] and their potential for coupling with other quantum excitations, such as other magnons [15], deformation vibration phonons through magnetostrictive force [16,17], spin qubits [18,19], optical [20,21] and microwave photons [12,13,19,22] which may open up a new way to thoroughly study quantum phenomena and promising applications in quantum communication, computing, and information. In other words, various magnonic-based hybrid quantum systems have been led to some intriguing effects, e.g. spin current control [23,24], magnon-phonon entanglement, and magnon-squeezed states [25–29], coherent optical-to-microwave conversion [19,30], ground-state cooling of the mechanical vibration mode [31,32], quantum state storage and retrieval [33], and magnon-induced nonreciprocity [34,35]. Additionally, interesting applications have been investigated in such hybrid systems, such as precision measurements [36–38], long-time memory [39,40], ultra slow light engineering [8,41,42], magnon laser [43,44], quantum thermometry [45], and magnon and photon manipulating using exceptional points [46–49].

Levitated objects provide an excellent platform for discovering the classical-quantum boundary with massive objects [50,51] and sensing weak forces [52,53], due to their weak coupling to the environment and the ability to cool their center of mass motion. Intriguing possibilities have been proposed for interacting systems comprising levitated particles, including quantum friction measurement [54], dark matter detection [55], and measurement of quantum gravity [51]. Additionally, levitated systems are a great platform for investigating non-equilibrium physics [56], as well as, employing these systems for ultrasensitive force detection opens up new possibilities for commercial sensing applications [57].

Motivated by the above-mentioned features of magnonic and levitated systems, we are interested in achieving controllable MIT and changing dispersive properties for manipulating light propagation in a hybrid magnonic system with a levitated particle. It should be noted that a magnet particle can be trapped in a number of ways, for example with an ion trap [58], magnetic trap [59,60], optical trap [61], Paul trap [62], or by being clamped to an ultrahigh-Q mechanical resonator [63,64]. Moreover, other ways of levitation such as magnetic levitation in a microwave cavity [65] or floating above a superconductor in free space [66], have the potential to levitate and cool millimeter-sized spherical magnets [50,67,68]. However, cooling and trapping a large particle, is still extremely challenging for the levitation of large objects [69,70]. Another advantage of a hybrid magnonic system is that the magnon excitations could couple to the center of mass motion (CM) [16,71]. Notably, a microwave cavity-magnomechanical system has been proposed [71], in which by effectively cooling the center of mass motion of a macroscopic YIG sphere, the phonon occupation 6 orders of magnitude smaller than standard cooling is obtained for large particles. Therefore, in the present contribution, we consider a microwave cavity magnomechanical system, composed of a levitated YIG sphere in which, the magnonic excitation in microwave regimes is coupled to the external CM quantum excitations of levitated particle. We illustrate how the coupling strength, the particle position, and the frequency of the control laser lead to the enhancement of the transparency window’s width. Besides, we find that MIT occurs for different coupling strength regimes i.e., the strong coupling and the intermediate coupling regimes. In addition, we demonstrate tunable and controllable slow and fast light emerges as a natural consequence of tunable MIT window. These properties demonstrate the capability of the cavity-magnomechanical systems with levitated ferromagnets as the manipulating and engineering tools for MIT.

The paper is organized as follows. In Sec. 2 the physical system is introduced, then in Sec. 3 the dynamics of the system is investigated through the quantum Langevin-Heisenberg equations of motion and the basic theory of MIT is presented. The results and parameters that play an important role in MIT are provided in Sec. 4. In Sec. 5 we discuss the group delay for slow and fast light propagation. Finally, the conclusion and outlook are mentioned in Sec. 6.

2. System and Hamiltonian

In this section, we consider a cavity-magnomechanical system with a levitated YIG sphere whose center of mass (CM) is trapped in a harmonic oscillator potential. In addition, a homogeneous magnetic field is applied on the sphere to excite the magnon modes [72–74]. Hamiltonian of the studied hybrid system is given by

The free Hamiltonian of the cavity field and the magnonic modes are described by the first two terms, with the bosonic annihilation operator $\hat {a}$ ($\hat {b}$) corresponds to the cavity (magnon) modes with frequency $\omega _a$ ($\omega _b$). In third term, $x$ and $p_x$ are the position and momentum of the magnetic particle CM that oscillates at $\omega _c$, the frequency of the trap. The fourth term describes the interaction between the magnon and photon modes. This coupling is characterized by the magnetic dipole interaction [9–11,75] where $g_{ab}=\frac {\gamma }{2}\sqrt {\frac {\hbar \omega _a \mu _0}{ V_a}}\sqrt {2\rho _s V \, s }$ is the coupling strength in which $\gamma$ is the gyromagnetic ratio, $V_a$ and $V$ are the mode volume of the cavity mode and volume of the YIG sphere, $\mu _0$ is the vacuum permeability, $\rho _s=4.22\times 10^{27} m^{(-3)}$ is the spin density of the YIG sphere, and $s=\frac {5}{2}$ is the spin number of the YIG’s ground-state. As is known, the light-matter interaction is realized in different regimes depending on the comparison between coupling constant and damping rates. In this study, we demonstrate induced transparency in the strong coupling (SC) as well as intermediate coupling (IMC) regimes that the rotating wave approximation (RWA) is valid in both of these regimes. Moreover, we assume that the cavity mode along $\hat {y}$ direction has the form as $\vec {B}_{cav}=\hat {y} B \, \cos (kx)$ where $k$ is the microwave field wave number [71]. The magnetic dipole interaction $H_{int}=-\gamma \vec {S}\,.\,\vec {B}_{cav}$, in which $\vec {S}$ being the collective spin angular momentum, can be used to describe the coupling between the microwave cavity $\vec {B}_{cav}$ and Kittel magnon. It is worth mentioning that the magnon frequency is determined by the external magnetic field $B_0$ and the gyromagnetic ratio $\gamma /2\pi =28\, \rm {GHz}/{\rm T}$, that is, $\omega _m=\gamma B_0$. Here, we focus on the Kittle magnon mode which is the homogeneous ground state of the magnonic mode and can be simply tuned by $B_0$. The last term in Hamiltonian represents the driving of the cavity mode with a time-dependent classical field, $s_{in}(t)$. In the following, we shall investigate the magnomechaniclly induced transparency (MIT) in our hybrid system. Two situations are considered: first, as a general form, we assume that the cavity is driven by a control laser, and also, by a weak probe laser, as illustrated in Fig. 1. Second, as a limiting case, we suppose that the YIG sphere is fixed and the cavity mode is only coherently driven by a weak probe field.

 

Fig. 1. Schematic of the cavity which consists of the levitated YIG sphere and is driven by a control field and a weak input probe field.

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3. Dynamics of the system

3.1 MIT via a levitated particle under the control laser

In this section, we shall proceed to investigate the MIT in our hybrid system. First we note that for a YIG sphere that is trapped with low-frequency trap $(\sim 50 kHz)$ and with a YIG sphere with the size of $R>1 nm$, the thermal variance in it’s CM position $\Delta x=\sqrt {(k_b T/(m \omega _c^2 )}$ is much smaller than the microwave wavelength [71]. Therefore, by expanding $\cos (kx)$ up to first order in $x$ around the minimum position of the trap, $x_0$, the Hamiltonian (1) can be obtained as follows:

3.1.1 Dynamics of the system

The dynamics of cavity, magnonic and levitated particle CM subsystems is described by quantum Langevin equations through adding damping and noise terms to the Heisenberg equations as follows:

Clearly, the fifth-order equation of the $I_a=\vert a_0\vert ^2$ is satisfied by the coupled equations stated above, and the magnonic system may exhibit multistability for some of the parameters. In this paper, we have chosen the parameters such that a stable solution exists. To achieve the results, we have used the following steady-state values (which are valid because of the small value of $G_{abc}$):

Moreover, the time evolution of the quantum fluctuations are obtained as:

In the above set of equations we have defined: $\Gamma =G_{ab}+2G_{abc}Re[c_0]$ and $\delta =\omega _p-\omega _d$. In the following, we are interested in the mean response of the hybrid system, so the mean values of noise operators are zero. In order to solve this set of coupled equations, we use the following ansatz for the solutions [5,76]:

By inserting these relations into Eqs. (10) and equating the coefficients with the same frequency, we arrive at the following relations:

Here, $(C^-+C^{-,*})=Q^-$. Since we are going to calculate the amplitude of the cavity output field, by using the input-output relation $\hat {a}^{out}+\hat {a}^{in}=\sqrt {2 \gamma _a} \hat {a}$ [77], we have

By substituting Eq. (11) in Eq. (14), it is clear that $A^-$ and $A^+$ oscillate at the frequencies $\omega _p$ and $2\omega _d-\omega _p$, respectively. The total output field $\varepsilon _t$, at the probe frequency, is given by

Now, as a limiting case, if we suppose our system is driven only with an input probe field and assume that we are dealing with a mounted YIG sphere (rather than a levitated sphere, i.e. $g_{abc}\ll g_{ab}$), the mentioned hybrid system reduces to an electromagnonic system [12,21]. As it is clear, in this simple case, the output entire fields $\varepsilon _t$, at the probe frequency is simplified to the following form as

The output field’s absorptive and dispersive behaviors at the probe frequency are given by the real part $\varepsilon _r$ and imaginary part $\varepsilon _i$ of the field amplitude $\varepsilon _t$, respectively. In the following, we investigate the MIT in both cases.

4. Magnomechanically induced transparency

In this section, we investigate the induced transparency in the hybrid magnomechanical system consisting of a levitated YIG sphere in the microwave cavity in interaction with magnon, photon and CM. It should be noted that the hybrid magnomechanical system stability is controlled by the Routh-Hurwitz criteria [78], i.e. real part of the eigenvalues of the drift matrix should be negative. According to this criteria, we have selected the experimentally realizable parameters so that the cavity-magnomechanical system with a levitated YIG sphere is stable in our studies.

4.1 MIT via a levitated particle

In this section, we explore MIT phenomena in the presence of a control field and demonstrate how to enhance the controllability of the system by choosing appropriate values for the magnomechanical system parameters. Similar to the optomechanical systems [5], we assume that the cavity field is driven by control and probe fields with different frequencies and amplitudes. Based on the input-output relation Eq. (14) and total output field Eq. (15), the real and imaginary parts of the field’s amplitude $\varepsilon _t$ versus the normalized frequency $\delta /\omega _c$ have been displayed in Fig. 2 for $\omega _a/2\pi =\omega _b/2\pi =30 \, GHz$, $\omega _c/2\pi =50\,kHz$, $\gamma _a/2\pi =1\, MHz$, $\gamma _b/2\pi =1\, MHz$, $\gamma _c/2\pi =10^{-6}\,Hz$, $kx_0=2\pi /3$, and $P=1\, \mu W$ [71]. In addition, for a bias field $B_0\sim 1 T$, the Kittel magnon can resonantly interact with microwave magnetic field of wavelength $\sim 1 cm$. Figure 2(a), corresponds to the absence of the magnon-photon coupling, i.e., $g_{ab}=0$, where the hybrid magnomechanical system releases to a standard bare cavity system. In Fig. 2(b), we see that the two symmetric Lorentzian peaks are also appeared. In other words, by the magnon-photon-CM interaction, a transparent window (i.e., $\varepsilon _r\simeq 0$) is appeared.

 

Fig. 2. The real (red line), $\varepsilon _r$, and the imaginary (blue line), $\varepsilon _i$, parts of the field amplitude, $\varepsilon _t$, versus the normalized frequency $\delta /\omega _c$ (a) without magnon-photon coupling $g_{ab}=0$ (b) with the magnon-photon coupling $g_{ab}/2\pi = 15.6\,MHz$. The parameters are selected as follows: $\omega _a/2\pi =\omega _b/2\pi =30 \, GHz$, $\omega _c/2\pi =50\,kHz$, $\gamma _a/2\pi =1\, MHz$, $\gamma _c/2\pi =10^{-6}\,Hz$, $k x_{0}=2\pi /3$, and $P=1\, \mu W$. To obtain this result, we have selected $\Delta _a=\Delta _b\equiv \Delta =\omega _c$ [71].

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Here, to acquire further insight, we demonstrate the role of magnon-photon-CM interaction in occurrence of the MIT in the introduced system. For this purpose, we assume that magnon-photon coupling $g_{ab}$ is small ($g_{ab}/2\pi \sim 1.5 MHz$) in such a manner that is not strong enough to lead to a perfect transparency window. In Fig. 3 the absorption profile, $\varepsilon _r$, has been plotted versus the normalized frequency $\delta /\gamma _b$ for different values of magnon-photon-CM coupling $g_{abc}/2\pi =10^{-4}Hz$ (green line), $12\times 10^{-2}Hz$ (blue line), $13\times 10^{-2}Hz$ (red line), $14\times 10^{-2}Hz$ (purple line). It should be noted that we assumed the experimentally realizable parameters in such a way that keep the system stable. As is shown in Fig. 3 , increasing the magnon-photon-CM coupling $g_{abc}$ enables us to achieve a wider transparency window. Thus, the additional coupling to the CM can play an important role to enhance the width of the transparency window. In order to obtain more insight, in the following we study three parameres that have important effects on the transparency window’s plots:

 

Fig. 3. The real part of the field amplitude, $\varepsilon _r$, versus the normalized frequency $\delta /\gamma _b$ with $\gamma _c/2\pi =10 Hz$, and $P=100\, mW$, and the magnon-photon coupling $g_{ab}/2\pi = 1.5\,MHz$ for different magnon-photon-CM couplings: $g_{abc}/2\pi =10^{-4}Hz$ (green line), $12\times 10^{-2}Hz$ (blue line), $13\times 10^{-2}Hz$ (red line), $14\times 10^{-2}Hz$ (purple line). Other parameters are selected the same as Fig. (2).

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Fig. 4. The real $\varepsilon _r$ part of the output field versus the normalized frequency $\delta /\omega _c$ and (a) magnon-photon coupling $g_{ab}$ and (b) particle position ${kx_0}/{\pi }$. The parameters are selected the same as Fig. (2)

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Fig. 5. (a), (b) The real part, $\varepsilon _r$, of the output field versus the normalized frequency $\delta /\omega _c$ and particle position ${kx_0}/\pi$, (a) $R=100 \, \mu m$ and (b) $R=50 \, \mu m$. Other parameters are selected the same as Fig. (2)

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Fig. 6. The real $\varepsilon _r$ part of the output field versus the normalized frequency $\delta /\omega _c$ and the normalized detuning $\Delta /\omega _c$. Other parameters are selected the same as Fig. (2)

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Fig. 7. The real part, $\varepsilon _r$, of the output field versus the normalized frequency $\delta /\omega _c$ and (a) particle position ${kx_0}/{\pi }$ and (b) magnon-photon coupling $g_{ab}$ for the intermediate coupling (IMC) regime $\gamma _b<g_{ab}<\gamma _a$ and $\gamma _a/2\pi =14\,MHz$. Other parameters are selected the same as Fig. (2)

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4.1.1 Limiting case: MIT without the control laser

Now, supposing a mounted YIG sphere in the absence of the control field, the output field $\varepsilon _t$ is given by Eq. (17). In Fig. 8, by considering the effective magnon-photon coupling, the absorption profile of the output field is plotted versus the normalized frequency $\Omega /\omega _b$ ($\Omega =\omega _a-\omega _p$) for $g_{ab}/2\pi =12.6,14.6,16.6,18.6 \,MHz$. Consequently, as a result of the effective magnon-photon interaction, a transparent window (i.e., $\varepsilon _r\simeq 0$) in the on-resonance condition $\omega _a=\omega _b$ is appeared. As it is seen, the transparency window for $g_{ab}/2\pi =18.6 \,MHz$ (green line) is larger than the transparency window for $g_{ab}/2\pi =12.6$ (pink line), which means that by increasing the magnon-photon coupling strength, the width of the transparency window is increased. In the following section, we present the numerical result of slow and fast light propagation based on the obtained MIT results.

 

Fig. 8. The real part, $\varepsilon _r$, of the output field versus the normalized frequency $\Omega /\omega _b$ and different magnon-photon couplings $g_{ab}/2\pi =12.6 MHz$ (pink line), $14.6 MHz$ (blue line), $16.6 MHz$ (red line), $18.6 MHz$ (green line). The parameters are selected the same as Fig. (2).

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5. Slow and fast light

The previous sections described MIT in various regimes (SC & IMC) as well as adjustable parameters that impact on changing the width of MIT window in magnomechanical system. As we know, MIT allows us to control and manipulate light propagation. Therefore, in this section we investigate the group delay of the total output field, $\varepsilon _t$, at the probe frequency. For this purpose, we use rapid phase dispersion $\Phi _t(\omega _p)=arg\left [\varepsilon _t(\omega _p)\right ]$, leading to define a group delay as [8]:

The slow light is indicated by a positive group delay, $\tau _g>0$, and the fast light is implied by a negative group delay, $\tau _g<0$. Before investigating the group delay, in Fig. 9, the probe field phase $\Phi _t(\omega _p)$ has been plotted versus the normalized frequency $\delta /\omega _c$ and magnon-photon coupling $g_{ab}/\gamma _b$. It is clear, when the coupling strength is small, the rapid normal phase dispersion results in a positive group delay, corresponding to the generation of the slow light. Intriguingly, an anomalous dispersion of the probe field phase appears when we increase the coupling strength, which is corresponding to the fast light. Additionally, a critical point is illustrated in the vicinity of the magnon-photon coupling strength $g_{ab}/\gamma _b\sim 2$. The phase dispersion critical point induced by magnon-photon-CM interaction reminds us the possibility of realizing the transformation between the slow and the fast light in our levitated system.

 

Fig. 9. The output field phase versus the normalized frequency $\delta /\omega _c$ and magnon-photon coupling $g_{ab}/\gamma _b$. A critical point in the vicinity $g_{ab}/\gamma _b\sim 2$. The parameters are selected the same as Fig. (2).

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In Figs. 10(a)-(c), respectively, we have plotted the group delay, $\tau _g$, versus the magnon-photon coupling, the particle position and the detuning frequency, under the various values of the damping rate of microwave cavity field $\gamma _a$. Figure 10(a) illustrates that in the regime of $g_{ab}/\gamma _b \in (0,2)$, the group delay is positive which is corresponding to the generation of slow light. Furthermore, it shows that for suitable values of $g_{ab}$, it is possible to switch the group delay from superluminal to subluminal propagation and vice versa. In addition, the maximum value of the group delay $\tau _g$ is inversely proportional to the damping rate of the cavity field $\gamma _a$. It can be seen that from $\gamma _a/2\pi =1.1MHz$ to $0.27 MHz$, the maximum value of the group delay increases from $\tau _g=0.15\, \mu s$ (the blue solid line) to $\tau _g=0.6 \, \mu s$ (the purple solid line). Another important result of our study, as is illustrated in Fig. 10(b), is that the YIG sphere position in two critical points allows us to manipulate light propagation. Also, Fig. 10(c) clearly shows that, we can change the group delay and have slow light, in the wide range of the detuning frequency. According to these results, the magnomechanical system with a levitated particle is a promising candidate for manipulating the propagation of light.

 

Fig. 10. Group delay $\tau _g$ versus (a) the magnon-photon coupling strength $g_{ab}/\gamma _b$ (b) the particle position $kx_0/\pi$ (c) the detuning frequency $\Delta /\gamma _b$, under the various of the damping rate of microwave cavity field $\gamma _a$, respectively. The parameters are selected the same as Fig. (2).

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By comparing this result with other studies on slow-fast light in mounted particles [7–9], we find that apart from magnon-photon coupling and control field, the particle position enables us to manipulate the light propagation. On the other hand, the magnetic and electrostatic levitation development made it possible to control particle position. Hence, the ability to couple the motion of the levitated system to magnon and photon modes provides an opportunity for more control on light propagation, via controlling the particle position.

According to the obtained results, effective manipulation of slow-fast light can be accomplished by varying the control laser’s frequency, YIG sphere position, and coupling strength. As expected, if the damping rate can be further reduced, the maximum value of the group delay will be increased, which reminds us of the possibility of achieving a larger group delay in an ultrahigh-Q cavity photon-magnon coupling system.

6. Conclusion

We have studied theoretically the realization of MIT and light propagation in the cavity magnomechanical system with magnon, photon, and CM nonlinear interaction. It has been shown that both of them extremely depend on the coupling strength. Furthermore, the particle position and the frequency of the control laser provide us controllable parameters for modifying the transparency window. Another advantage of the introduced model is that MIT can occur in both SC and IMC coupling regimes. Interestingly, we obtained the slow and fast light effects and a transformation between subluminal and superluminal propagation of the output probe field. On the other hand, a comparison between the introduced system and an optomechanical system reveals an analogy between magnons and mechanical modes in the optomechanical system, and significantly influence the MIT and light propagation.

The obtained results of the cavity-magnomechanical system, with a levitated YIG sphere, open new perspectives for quantum interaction between the microwave, the magnetic, and the mechanical systems. Applications of these findings include optical data processing and communication. In addition, the introduced system in the present contribution has potential applications such as the sensing of weak forces or magnetic signals by assuming some nonlinearities in future investigations, such as the Kerr effect or magnon squeezing, and achieving more adjustable parameters [28].