Magnetically controllable slow light based on magnetostrictive forces

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

doi:10.1364/OE.27.005544

1. Introduction

Yttrium iron garnet (YIG) [1–3] as a high quality magnetic material, has attracted considerable attention. The Kittel mode [4] in a YIG sphere possesses several distinguishing advantages, including a low damping rate [2], a long coherence time [5], and rich magnetic nonlinearities owing to the Kerr effect of magnons [6]. By harnessing a high spin density in YIG exceeding previous spin ensembles by several orders of magnitude, it can be used to realize strong coupling [1,2,7] between magnons and microwave photons in a high-finesse cavity, leading to both magnon and cavity polaritons [8–11]. Also, the ultrastrong coupling regime [12,13] has become available by either utilizing a specially designed microwave cavity [3] or increasing the size of a YIG sampler. Moreover, we can observe the strong coupling between magnons and microwave photons even at room temperatures, due to the extraordinary robustness of the magnon mode against temperature [5]. Besides, in virtue of the frequency of magnon steered by the external bias magnetic field, which has a distinct merit of high tunability, a lot of intriguing phenomena have been implemented in integration of the magnetic systems with opto- or electromechanical elements, including the observation of bistability [14], the magnon gradient memory [15], magnon induced transparency [16] and the emerging field of spintronics [17–20]. In addition to the cavity-magnon coupling, it has been demonstrated the coherent coupling between a single-magnon excitation in a millimeter-sized ferromagnetic sphere and a superconducting qubit [21], as well as coherent phonon-magnon interactions depending on the effect of magnetostrictive deformation in a YIG sphere [22].

It is worth to note that owing to the superior material and geometrical features of the YIG, it can be also regarded as a perfect mechanical resonator, which introduces the phonon-magnon interaction resulting in magnetomechanically induced transparency (MMIT) [22]. MMIT is a coherent phenomenon, in analogy to the electromagnetically induced transparency (EIT) [23] and optomechanically induced transparency (OMIT) [24, 25], which arise from the quantum interference between different excitation paths. The initial pioneering demonstrations of slow light in various media all exploited narrow spectral resonances, typically created by EIT [26]. In addition, the slow light experiments have been completed by using OMIT observed in a nanoscale optomechanical crystal [27]. Slow light offers the opportunity for compressing optical signals and optical energy in space, which reduces the device footprint and enhances light-matter interactions [28–30]. Moreover, due to the remarkably low group velocity of the slow light, it makes optical buffer and control of optical signals in the time domain possible [31, 32]. The research on slow light has brought in a wealth of applications, such as telecommunications [33], quantum information processing [34], semiconductor quantum wells [35] and interferometer [36]. In the light of these advances, it is natural to explore the slow light in a cavity magnetomechanical system with a nonlinear phonon-magnon interaction, which roots in magnetostrictive forces.

In the present work, we primarily discuss the slow light effect in a cavity magnetomechanical system, which has intrinsically good tunability in contrast to other a variety of cavity electro- and optomechanical systems [37, 38]. In consideration of a strong photon-magnon coupling strength, this leads to the hybridization between magnon and photon, which shows up in the transmission of the probe field as a pair of split normal modes. For a special photon-magnon coupling strength, by adjusting the frequency of the control field to make the control field on the red and blue detuned sidebands of the two hybridized modes respectively, we can observe the phenomena of MMIT and magnetomechanically induced absorption (MMIA) simultaneously [22]. Accompanying the MMIT process, the slow light effect emerges featured by the optical group delay. Utilizing the flexible adjustability of the external bias magnetic field, it is easy to accomplish the transformation between slow and fast light. In addition, we show the control field including its intensity and frequency has an effect on the slow light effect. More importantly, the pump power is utilized to enhance the group delay and make long-lived slow light (group delay of millisecond order) achieve. These results may deepen our cognition of the fundamental principle of cavity magnetomechanical systems, and due to the controllability of the magnetic field in room temperature, it may expand to the quantum regime, in which we can find applications in quantum entanglement [39] and long-lifetime quantum memories [40].

The paper is organized as follows. In Sec. 2, we introduce the theoretical mode and show the MMIT based on a nonlinear phonon-magnon interaction. In Sec. 3, we discuss the slow light effect in the transmission of the probe field. In Sec. 4 a conclusion of the results is summarized.

2. Theoretical model and derivation of MMIT in a cavity magnetomechanical system

The image of our device is shown in Fig. 1(a). A highly polished single-crystal YIG sphere with 250-μm-diameter glued to a silica fiber with 125-μm-diameter served as a catch point is mounted in a three-dimensional copper cavity. A uniform external magnetic field (H) is applied along the z direction to bias the YIG sphere for magnon-photon coupling. In Fig. 1(b), the photon-magnon coupling strength (g) can be tuned by adjusting the direction of the bias magnetic field or the position of the YIG sphere inside the microwave cavity. In addition, the strong microwave photon-magnon coupling has been achieved, due to the extremely large spin density in single crystal YIG. The YIG sphere is also an excellent mechanical resonator, considering the effect of magnetostrictive deformation in a YIG sphere in response to the external magnetic field, which introduces the magnon-phonon interaction with coupling strength G. The Hamiltonian, including three modes of photon, magnon and phonon can be written as (ħ = 1) [22]

\begin{array}{l} {\hat{H}}_{0} = & ω_{a} {\hat{a}}^{†} \hat{a} + ω_{b} {\hat{b}}^{†} \hat{b} + ω_{m} {\hat{m}}^{†} \hat{m} \\ + g (\hat{a} + {\hat{a}}^{†}) (\hat{m} + {\hat{m}}^{†}) + G {\hat{m}}^{†} \hat{m} (\hat{b} + {\hat{b}}^{†}) . \end{array}

Here â (â^†) is the annihilation (create) operator of the cavity TE₀₁₁ mode with resonant frequency ω_a. The boson operators b̂ and m̂ are the phonon and magnon modes with respective frequencies ω_b and ω_m. g is the linear photon-magnon coupling strength. The form of last term is the phonon-magnon interaction arising from magnetostrictive forces, in which G represents the single-magnon magnetomechanical coupling strength.

Fig. 1 (a) Schematic diagram of a cavity magnetomechanical system, consisting of a three-dimensional copper cavity and a YIG sphere which is glued to a silica fiber. (b) Schematic of the linearly coupled photon (a) and magnon (m), and the nonlinearly coupled magnon and phonon (b). κ_a, κ_m, κ_b are the dissipation rates of microwave cavity, magnon and phonon modes, respectively, where κ_a includes an external loss rate κ and an intrinsic loss rate κ₀ [22].

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Applying a microwave driving (control) field and a microwave signal (probe) field to the cavity, the whole Hamiltonian can be described by

\hat{H} = {\hat{H}}_{0} + \sqrt{2 κ} [{\hat{a}}^{†} (ε_{d} e^{- i ω_{d} t} + ε_{p} e^{- i ω_{p} t}) + H . c .],

where κ is the external coupling rate of microwave cavity and

ε_{j} = \sqrt{P_{j} / ℏ ω_{j}}

(j = d, p) is the intensity of the control (probe) field, in which the corresponding input power and frequency are P_j and ω_j respectively. In a frame rotating at ω_d and applying the rotating-wave approximation [g(â + â^†)(m̂ + m̂^†) becomes g(âm̂⁺ + â⁺m̂) when ω_a, ω_m ≫ g, κ_a, κ_m], the whole Hamiltonian is given by

\begin{array}{l} \hat{H} = & Δ_{a d} {\hat{a}}^{†} \hat{a} + ω_{b} {\hat{b}}^{†} \hat{b} + Δ_{m d} {\hat{m}}^{†} \hat{m} \\ + g (\hat{a} {\hat{m}}^{†} + {\hat{a}}^{†} \hat{m}) + G {\hat{m}}^{†} \hat{m} (\hat{b} + {\hat{b}}^{†}) \\ + \sqrt{2 κ} [{\hat{a}}^{†} (ε_{d} + ε_{p} e^{- i Ω t}) + H . c .], \end{array}

where Δ_ad = ω_a − ω_d, Δ_md = ω_m − ω_d and Ω = ω_p − ω_d.

In the semiclassical limit, the operators can be reduced to their expectation values, viz. 〈â(t)〉 = a(t), 〈â^†(t)〉 = a^*(t), 〈m̂(t)〉 = m(t), 〈m̂^†(t)〉 = m^*(t), 〈b̂(t)〉 = b(t), 〈b̂^†(t)〉 = b^*(t). The averaged version of this system is valid for sufficiently large photon, magnon and phonon numbers. The quantum and thermal noise terms are eliminated due to 〈â_in(t)〉 = 〈m̂_in(t)〉 = 〈b̂_in(t)〉 = 0. Introducing classical decay rates of the cavity field (κ_a), magnon (κ_m) and photon (κ_b) modes, the Heisenberg-Langevin equations of motion can be written as:

\dot{a} = (- i Δ_{a d} - κ_{a}) a - i g m - i \sqrt{2 κ} (ε_{d} + ε_{p} e^{- i Ω t}),

\dot{b} = (- i ω_{b} - κ_{b}) b - i G m^{*} m,

\dot{m} = (- i Δ_{m d} - κ_{m}) m - i g a - i G (b + b^{*}) m .

Eqs. (4)–(6) describe the time evolution of the cavity magnetomechanical system. Considering the input control field being much stronger than the probe field, the solutions of these equations can be written by a = A₀ + δa, b = B₀ + δb and m = M₀ + δm, as a steady-state value steered by the control field plus a small fluctuation perturbed by the probe field.

When the probe field is absent and only the control field with amplitude ε_d is incident into the system, the evolution of Eqs. (4)–(6) admits a steady-state solution as follows:

A_{0} = \frac{i g M_{0} + i \sqrt{2 κ} ε_{d}}{- i Δ_{a d} - κ_{a}},

B_{0} = \frac{i G {| M_{0} |}^{2}}{- i ω_{b} - κ_{b}},

M_{0} = \frac{i g A_{0}}{- i Δ_{m d} - κ_{m} - i G (B_{0} + B_{0}^{*})} .

The linearized Heisenberg-Langevin equations of motion can be written in the following compact matrix form:

\dot{u} (t) = M u (t) + n (t) .

Here u(t) = (δa, δa^*, δb, δb^*, δm, δm^*)^T (the superscript T denotes the transposition) is a vector of the operators, and

n (t) = {(- i \sqrt{2 κ} ε_{p} e^{- i Ω t}, i \sqrt{2 κ} ε_{p} e^{i Ω t}, 0, 0, 0, 0)}^{T}

is a vector of the input probe field. In addition, the coefficient matrix

M = (\begin{matrix} - i Δ_{a d} - κ_{a} & 0 & 0 & 0 & - i g & 0 \\ 0 & i Δ_{a d} - κ_{a} & 0 & 0 & 0 & i g \\ 0 & 0 & - i ω_{b} - κ_{b} & 0 & - i G M_{0}^{*} & - i G M_{0} \\ 0 & 0 & 0 & i ω_{b} - κ_{b} & i G M_{0}^{*} & i G M_{0} \\ - i g & 0 & - i G M_{0} & - i G M_{0} & ζ & 0 \\ 0 & i g & - i G M_{0}^{*} & i G M_{0}^{*} & 0 & ζ^{*} \end{matrix}),

where

ζ = - i Δ_{m d} - κ_{m} - i G (B_{0} + B_{0}^{*})

, B₀ and M₀ correspond to the steady-state solutions of b, m respectively.

To study the MMIT based on this cavity magnetomechanical system with different photon-magnon coupling strengths, the system should work in a stable regime. The stability for this system can be determined by the eigenvalues of the matrix M [41]. When all the eigenvalues of matrix M have negative real parts, the system is stable. In general, the stability condition can be derived by applying the Routh-Hurwitz criterion [42], but its form is too cumbersome to give here. Therefore, we use numerical calculation of the eigenvalues of matrix M to distinguish the stability condition of the system. As shown in Fig. 2, it can be seen that the dynamics of the system becomes unstable only in a narrow region, due to the blue detuned driving signal to the system by virtue of the adjustment of the external bias magnetic field. In fact, when P_d = 10mW, the magnetomechanical parametric amplification (MMPA) [22] could appear, which easily leads to the parametric oscillatory instability of the system. As long as the intensity of the magnetic field is not in the range about from 280.3 mT to 280.5 mT, or in other words, we do not consider the induced MMIT from the blue detuned driving, the system can be stable. In the present work, we will choose appropriate parameters to make the system keep in a stable environment and work in the perturbative regime [43].

Fig. 2 Parameter regime of stability in the cavity magnetomechanical system with different external bias magnetic fields and photon-magnon coupling strengths. Here, we use a set of experimentally feasible values, i.e., ω_a/2π = 7.86GHz, ω_b/2π = 11.42MHz, G/2π = 4.1mHz, 2κ_a/2π = 3.35MHz, 2κ_m/2π = 1.12MHz, 2κ_b/2π = 300Hz [22], P_d = 10mW, Δ_ad = 0Hz.

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Now we turn to consider the perturbation made by the probe field. To calculate the optical transmission rate of the probe field, the perturbed solutions of the photon, magnon and phonon modes can be solved by using the following ansatz:

δ a = A^{+} e^{- i Ω t} + A^{-} e^{i Ω t},

δ b = B^{+} e^{- i Ω t} + B^{-} e^{i Ω t},

δ m = M^{+} e^{- i Ω t} + M^{-} e^{i Ω t} .

Dropping the nonlinear terms −iGδm^*δm and −iG(δb + δb^*)δm due to the condition |ε_p| ≪ |ε_d|, and substituting Eqs. (12)–(14) into Eq. (10), we can obtain the perturbed solutions of this system.

After a series of calculations, we could obtain the analytic expressions of both the A⁺ and A⁻, which originate from the upconverted and downconverted processes of the control field, respectively. However, in what follows, we only present a discussion of the transmission of the probe field (anti-Stokes field), that is A⁺. One also can discuss the Stokes field (A⁻) by using the same method. Therefore, we figure out the positive part of photon perturbation as follows:

A^{+} = \frac{(i g M^{+} + i \sqrt{2 κ} ε_{p})}{- i (Δ_{a d} - Ω) - κ_{a}},

where

M^{+} = \frac{\sqrt{2 κ} h_{1} (Ω) ε_{p}}{g [\frac{h_{2} (Ω) h_{2}^{*} (- Ω) {| M_{0} |}^{4}}{1 + h_{1}^{*} (- Ω) + {| M_{0} |}^{2} h_{2}^{*} (- Ω) + h_{3}^{*} (- Ω)} - β]},

h_{1} (Ω) = \frac{g^{2}}{[i (Δ_{a d} - Ω) + κ_{a}] [i (Δ_{m d} - Ω) + κ_{m}]},

h_{2} (Ω) = \frac{2 i G^{2} ω_{b}}{[i (ω_{b} - Ω) + κ_{b}] [i (ω_{b} + Ω) - κ_{b}] [i (Δ_{m d} - Ω) + κ_{m}]},

h_{3} (Ω) = \frac{i G (B_{0} + B_{0}^{*})}{i (Δ_{m d} - Ω) + κ_{m}},

β = 1 + h_{1} (Ω) + {| M_{0} |}^{2} h_{2} (Ω) + h_{3} (Ω) .

Then by making use of the input-output relation between the input and output fields [44,45], the output of the probe field is obtained as follows:

a_{out} = ε_{p} - i \sqrt{2 κ} A^{+} .

The transmission of the probe field is defined as:

{| t_{p} |}^{2} = {| \frac{a_{out}}{a_{in}} |}^{2} = {| 1 - i \frac{\sqrt{2 κ} A^{+}}{ε_{p}} |}^{2} .

This sets up the framework for our discussion of the transmission rate of the probe field in the cavity magnetomechanical system.

Under the condition of a strong photon-magnon coupling strength (g ≫ κ_a, κ_m), we can obtain two hybridized modes (two supermodes [46]) consisting of photons and magnons. In order to simplify, here we discuss the case that the magnon mode has the same resonance frequency as the photon mode. These two hybridized modes can be defined as:

A_{+} = (\hat{m} + \hat{a}) / \sqrt{2}, A_{-} = (\hat{m} - \hat{a}) / \sqrt{2 .}

In the representation of A_± and omitting the g(âm̂ +â⁺m̂⁺) due to the rotating-wave approximation, the Hamiltonian Eq. (1) becomes

\begin{array}{l} {\hat{H}}_{0} = & ω_{+} {\hat{A}}_{+}^{†} {\hat{A}}_{+} + ω_{-} {\hat{A}}_{-}^{†} {\hat{A}}_{-} + ω_{b} {\hat{b}}^{†} \hat{b} \\ + \frac{G}{2} ({\hat{A}}_{+}^{†} {\hat{A}}_{+} + {\hat{A}}_{-}^{†} {\hat{A}}_{-} + {\hat{A}}_{+}^{†} {\hat{A}}_{-} + {\hat{A}}_{-}^{†} {\hat{A}}_{+}) (\hat{b} + {\hat{b}}^{†}), \end{array}

with frequencies

ω_{\pm} - i κ_{\pm} = ω_{a} - i χ \pm \sqrt{g^{2} - τ^{2}},

where

χ = \frac{κ_{a} + κ_{m}}{2}

and

τ = \frac{κ_{m} - κ_{a}}{2}

. The hybridization of the magnon and photon modes strongly depends on the photon-magnon coupling strength which can be easily manipulated in the experiments, and the decay rates of both the photon and magnon. In the light of the ratio value of

\frac{g}{| τ |}

, we can quantify the weak- and strong-coupling regimes and make

α = \sqrt{g^{2} - τ^{2}}

. When

\frac{g}{| τ |} < 1

, the system is in a weak-coupling regime and α is an imaginary. As a result, the two supermodes have the same resonance frequency but different linewidths. When

\frac{g}{| τ |} > 1

, the system is in a strong-coupling regime and α is a real. Consequently, the two supermodes have different resonance frequencies with frequency space 2α and the same linewidth.

Figure 3(a) shows the transmission rate |t_p|² of the probe field as a function of optical detuning for different photon-magnon coupling strengths and the frequency detunings of the control field from the cavity field. In Fig. 3(a), we can observe two absorption points, which is the result of the strong photon-magnon coupling (g = 0.2ω_b > |τ|), leading to a pair of split supermodes with frequency space 2α. For three other Figs. 3(b)–3(d), the transparent windows occur, which is different from Fig. 3(a). This explains that when g = 0.2ω_b, the magnetomechanical interactions is too weak to produce MMIT due to a little light acting on the mechanical resonator. With the enhancement of the photon-magnon coupling strength, such as in Figs. 3(b)–3(d), the strong light plays a key role in magnetomechanical interactions, which makes the one of the supermodes split, MMIT is induced. In addition, it is seen that in Fig. 3 all the two dips are about central axisymmetric in virtue of the two supermodes possessing the photon and magnon of the same components, which roots in ω_a = ω_m. It is worth noting that when g = ω_b in Fig. 3(d), both the transparency peak (MMIT) and the absorption dip (MMIA) appear, due to the red- and blue-detuned driving to the sidebands of the two supermodes synchronously. If we continue to increase the intensity of the driving field, the transition from MMIA to MMIT and then to MMPA may be observed [22]. Furthermore, it is found that in Fig. 3(b) the effect of the MMIT is much better, because both the drive and probe photons are applied on-resonance with the two supermodes, resulting in a drastically enhanced magnetomechanical coupling [22].

Fig. 3 The transmission rate |t_p|² of the probe field as a function of the two-photon detuning Ω with different photon-magnon coupling strengths and the frequencies of the control field. Δ_md = Δ_ad (H = 280.7mT) and the other parameters are the same as in Fig. 2.

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3. Numerical analysis of the slow light

As we all know that the optical transmission in an EIT window experiences a dramatic reduction in its group velocity. The velocity change is the strongest for narrower spectral resonances. We now investigate the group delay of the optical signal in a cavity magnetomechanical system. Thanks to the dramatic variation of the refractive index with a MMIT window, the light transmitted in the MMIT window accompanies a rapid phase dispersion ϕ_t(ω_p) = arg[t_p(ω_p)], leading to a significant group delay defined in terms of an expression [27,47]

τ_{g} = \frac{\partial ϕ (ω_{p})}{\partial ω_{p}} = Im [\frac{1}{t_{p}} \frac{\partial t_{p}}{\partial ω_{p}}] .

The evolution of the τ_g includes two cases: a slow light (positive group delay: τ_g > 0) and the other fast light (negative group delay: τ_g < 0). Especially for slow light, the advent of research into it brings in a large number of applications, such as the fields of telecommunications and optical data processing.

Figure 4 shows that the group delay can be tuned by changing the bias magnetic field and the photon-magnon coupling strength. It should be noted that here we only discuss the slow light effect at Ω = ω_b, on account of a rapid phase dispersion in the region of the MMIT window of the probe light transmitted. It can be known from Fig. 4, that the value of τ_g can be tuned to be positive or negative with the adjustment of the bias magnetic field and the photon-magnon coupling strength. For any given g value (0 ≤ g/ω_b ≤ 1), the system can enter into the slow-light regime (τ_g > 0), but the corresponding value of H is only in a small range. For example, when the photon-magnon coupling strength is adjusted to g = 0.8ω_b, the transmission group delay τ_g is positive around H from 280.8 mT to 280.9 mT. In addition, it is found that only the value of H exactly satisfies the condition of the appearance of the transparency window, such as shown in Fig. 3(d) (H = 280.7mT), can the value of the group delay be positive, which also indicates that the probe field suffers a rapid phase dispersion in the region of the transparency window. Further, the positive group delay is proportional to the increase of g. By selecting the photon-magnon coupling strength as g = ω_b, the positive group delay reaches a maximum value about 0.39 ms. However, once the drive-resonance detuning ω_d − ω₊ ≠ −ω_b, that is, the value of the H is not satisfied the condition of the MMIT taking place, the system will complete the transition from slow to fast light (τ_g < 0).

Fig. 4 Group delay of the probe light τ_g (in units of ms) as a function of the photon-magnon coupling strength g and the bias magnetic field H. Ω = ω_b and the other parameters are the same as in Fig. 2.

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In addition to the photon-magnon coupling strength, the pump power also affects the group delay of the probe light. Figure 5 shows that the group delay varies with the bias magnetic field and the pump power. Here we discuss the slow light effect at g = ω_b and Ω = ω_b. Although the value of P_d reaches 0.3W in Figs. 5(a) and 5(b), the system also keep stable at g = ω_b for various bias magnetic fields. In Fig. 5(a), the value of τ_g can be tuned to be positive or negative with the adjustment of the bias magnetic field and the pump power. Increasing H from 280.1mT first pushes the system into the fast light effect, and then switches it to slow light for any fixed value P_d. Moreover, the transition is obtained quickly along with the increase of P_d. In order to observe the influence of the stronger bias magnetic field on the group delay, we plot Fig. 5(b) where the H ranges from 280.9mT to 281.3mT. The value of the group delay is proportional to the increase of the P_d in the range of the bias magnetic field less than 281.0mT. In stark contrast with Fig. 5(a), the increase of H is required to observe the transition from fast to slow light even at a very low pump power. Besides, the value of the negative group delay is larger. This indicates a new method of achieving slowing or advancing signals by tuning the frequency of magnon which can be conveniently tuned in a broad range, because it is determined by the external bias magnetic field.

Fig. 5 Group delay of the probe light τ_g as a function of the pump power P_d and the bias magnetic field H. In panels (a) and (b), the value of τ_g is in units of μs. g = ω_b, Ω = ω_b and the other parameters are the same as in Fig. 2.

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For completeness of the bias magnetic field continuously changing, we also plot the group delay of the probe light with H from 280.5mT to 280.9mT for different values of pump power in Fig. 5(c). When P_d = 0.017W, a negative group delay is obtained [seeing the black solid curve], which corresponds to fast light propagation at H < 280.6mT or H > 280.8mT. However, increasing P_d to 0.018W completes the transition from fast to slow light, and τ_g becomes positive for any value of H [seeing the blue solid curve]. Beyond the point, a further increasing P_d, such as P_d = 0.1W, pushes τ_g to negative or positive values [seeing the red solid curve], which is similar to the case of the black solid curve. Reducing or enhancing P_d, it is shown that only around H = 280.7mT, which corresponds to slow light propagation in the proximity of the transparency window in Fig. 3(d), can the group delay be positive. Otherwise, the negative group delay is acquired and the system enters fast light [seeing the black and red solid curves].

In Fig. 6, we plot group delay τ_g as a function of the frequency detuning between the control field and the cavity field for different photon-magnon coupling strengths. Both positive and negative group delays can be obtained for various g. When g = 0.2ω_b, the maximal value of the group delay is no more than 0.06 ms [seeing the blue solid curve], which corresponds to slow light propagation in the proximity of the right absorption point in Fig. 3(a). The ultimate value of the group delay is amplified with the increasing values of g, as shown by the three other solid curves. Specifically, the positive group delay with g = ω_b reaches about 0.4 ms [seeing the pink solid curve], which is increased by seven times in contrast with the case of g = 0.2ω_b. Furthermore, it is shown that, for a suitable value of g, it is possible to switch the group delay from superluminal to subluminal propagation in a wider range of the frequency Δ_ad [seeing both black and red solid curves] and vice vera. According to the analysis of Figs. 5 and 6, we can draw a conclusion that both the frequencies of the control field and the magnon can be used to achieve slowing or advancing signals, where the frequency of magnon is determined by the external bias magnetic field in a broad range. By applying the tunable delays of the optical data signals in fibres as well as integrating such devices into existing communication networks, it is certainly a very attractive approach for optimizing the flow of data traffic in future networks.

Fig. 6 Group delay of the probe light τ_g (in units of ms) as a function of the frequency detuning between the control field and the cavity field for different photon-magnon coupling strengths. Δ_md = Δ_ad, Ω = ω_b and the other parameters are the same as in Fig. 2.

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Figure 7 plots the group delay of the transmitted probe field versus the pump power of the control field for different values of the photon-magnon coupling strength and the frequency detuning between the control field and the cavity field. One can see that the group delays are positive with the pump power increasing, which represents the slow light effect. Here we adopt the data of Fig. 3 to research the slow light at Ω = ω_b. When g = 0.2ω_b or ω_b [seeing the blue and pink dotted curves], the group delay is monotonically increasing with the pump power varying and finally towards a stable value. The basic result indicates that under the influence of the strong control field, the propagation of the probe field in the MMIT window enormously modifies the dispersion of the system. Besides for g = ω_b, the maximum value of the group delay can reach 1 ms by continuing to enlarge the pump power, causing a long-lived slow light effect, which is strongly reminiscent of the maximum value of the group delay still staying on the microsecond scale [48,49]. The long-lived slow light promotes stronger light-matter interaction, and it allows to delay and temporarily store light in all-optical memories [50]. In addition, in the cases of g = 0.5ω_b or 0.8ω_b [seeing the red solid curve and the black dotted curve], with the P_d increasing, the value of τ_g first increases until its peak value, and then decreases approaching τ_g = 0.

Fig. 7 Group delay of the probe light τ_g (in units of ms) as a function of the pump power P_d for different photon-magnon coupling strengths and the frequency detunings between the control field and the cavity field. Δ_md = Δ_ad, Ω = ω_b and the other parameters are the same as in Fig. 2.

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4. Conclusion

In summary, we have theoretically discussed the MMIT and slow light in a cavity magnetomechanical system with a nonlinear magnon-phonon interaction. When the photon-magnon coupling strength is a strong (g > κ_a, κ_m), there are two hybridized modes consisting of photons and magnons, which shows up in the reflection spectrum as a pair of split normal modes. We place a strong control field on the red (blue) detuned sideband of the hybridized modes to produce coherent interference with a resonant probe beam, inducing a transparency (absorption) window for the probe. Then, we calculate the group delay to obtain the slow light. Making use of the intrinsically good tunability of the frequency of the magnon determined by the external bias magnetic field, we can obtain both the slow and fast light effects, and a mutual transformation between the subluminal and superluminal propagation of the transmitted probe field. In addition, the slow light is also affected by the control field including its intensity and frequency. Moreover, one may achieve long-lived slow light (group delay of millisecond order) via enlarging the pump power. These exotic features of the cavity magnetomechanical system greatly widen new perspectives for quantum interference among microwave, magnetic and mechanical systems, and it can be applied to quantum entanglement [39], long-lifetime quantum memories [40] and quantum information processing.

Funding

National Natural Science Foundation of China (11774113).

References

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Magnetically controllable slow light based on magnetostrictive forces

Abstract

1. Introduction

2. Theoretical model and derivation of MMIT in a cavity magnetomechanical system

3. Numerical analysis of the slow light

4. Conclusion

Funding

References

Cited By

Figures (7)

Equations (26)

Optics Express