We investigate the coherent control of the transmission spectrum in a cavity magnetomechanical system consisting of microwave photon, magnon, and phonon modes, where the microwave cavity is driven by a strong pump field and a weak probe field, and the magnon is driven by a weak microwave source. Different from a single transparency window in the absence of the phonon–magnon interaction, two transparency windows and three absorption dips can be observed in the presence of the phonon–magnon interaction, which originates from the joint interaction of phonon–magnon and photon–magnon. In addition, two absorption dips located at both sides of the central absorption dip can be modulated asymmetrically into amplification and absorption by varying the magnetic field amplitude of the magnon driving field. Interestingly enough, the relative phase of applied fields could have profound effects on both the transmission spectrum and the group delay of the output field by choosing the appropriate magnetic field amplitude of the magnon driving field. The transmission group delay can be switched between positive to negative and vice versa by adjusting the relative phase between the applied fields. The present results illustrate the potential to utilize the relative phase for controlling the microwave signal in the cavity magnomechanical system, as well as guidance in the design of information transduction and quantum sensing.

Cavity magnomechanics,1 similar to the cavity optomechanics,2 has emerged as an important new frontier in quantum optics and attracted extensive attention recently3–14 due to the great potential for building a highly adjustable hybrid coherent information processing system.15–21 Different from the cavity optomechanics, the magnetostrictive forces instead of the radiation pressure forces play an important role in the cavity magnomechanics, and the magnon oscillator can be flexibly adjusted through the magnetic field.1,4,6,7

In particular, the cavity magnomechanical system consisting of magnons in a single-crystal yttrium iron garnet (YIG) sphere strongly coupled to the cavity mode has been theoretically proposed and experimentally demonstrated in the past few years, in which the phonon–magnon and photon–magnon interactions are introduced and observed.1,17,18 The magnon-induced absorption, as well as the magnetomechanically induced transparency, has been investigated,1,22,23 which originates from the internal constructive (destructive) interference that can be interpreted by analogy with the optomechanical induced absorption (transparency)24–26 in the cavity optomechanics. Meanwhile, the modification in the light response of the system resulting from the coupling of the magnon and the cavity microwave photon brings a significant transformation in the phase dispersion of the transmitted field.22,27,28 Tunable slow and fast light has also been demonstrated in the cavity magnomechanics,22,23,29,30 which provide a potential application value for quantum memories,31 quantum information processing,32 tunable microwave filters and amplifiers,33 signal sensing, and microwave-to-optical conversion.34 

In this paper, we investigate the coherent control of the transmission spectrum in a cavity magnetomechanical system consisting of microwave photon, magnon, and phonon modes, where the microwave cavity is driven by a strong pump field and a weak probe field and the magnon is driven by a weak microwave source. In order to clearly show the dependence of the transmission spectrum on the driving parameters of the magnon, we provide a comparison of the transmission spectra of the cavity magnetomechanical system with and without including the phonon–magnon interaction. Different from a single transparency window in the absence of the phonon–magnon interaction, it finds that two transparency windows and three absorption dips can be observed in the presence of the phonon–magnon interaction. Furthermore, the effect of the relative phase of applied fields and the magnetic field amplitude of the magnon driving field on the transmission characteristics is explored. It demonstrates that two absorption dips located at both sides of the central absorption dip can be modulated asymmetrically into amplification and absorption by adjusting the magnetic field amplitude of the magnon driving field. More interestingly, it is found that the relative phase of applied fields could have profound effects on both the transmission spectrum and the group delay of the output field by choosing the appropriate magnetic field amplitude of the magnon driving field. As a result, the transmission of the output signal can be switched between slow to fast light and vice versa by modulating the relative phase between the applied fields.

As depicted in Fig. 1, we consider a cavity magnomechanical system consisting of a 3D microwave cavity with the frequency ωa and a YIG sphere with the diameter 250 μm. The cavity is driven by a strong pump field and a weak probe field with the driving frequencies ωl, ωp, and the phases ϕl, ϕp, respectively. The YIG sphere is directly driven by a weak microwave source with the driving frequency Ωm and the phase ϕm. When the fields are applied to the system, due to the excellent magnetic properties of the ferrimagnet, the magnon with the frequency ωm is embodied by a large number of collective spin motions in the YIG and can be tuned by adjusting the external bias magnetic field H.17,38 Meanwhile, the variation of magnetization induced by the magnon excitation leads to the geometric deformation of the YIG sphere so that the mechanical mode with the frequency ωb is produced in the system. In this system, the magnon mode can couple the photon mode through the magnetic dipole interaction with the magnon–photon coupling strength gma. The mechanical mode couples the magnon mode through magnetostrictive interaction, and the magnomechanical coupling strength is gmb. We consider the dimensions of the YIG sphere, in this work, is much smaller than the wavelength of the microwave so that the influence of light radiation pressure in the system is negligible.17,18 The total Hamiltonian of the system reads1,17,22

Ht=H0+H1+Hd+Hp+Hpm,
(1)

where

H0=ωaaa+ωmmm+ωbbb,H1=gma(a+a)(m+m)+gmbmm(b+b),Hd=iηκa(εinaeiωltiϕlεinaeiωlt+iϕl),Hp=iηκa(εpaeiωptiϕpεpaeiωpt+iϕp),Hpm=i(εmmeiΩmtiϕmεmmeiΩmt+iϕm).
(2)

Here, a(a), m(m), and b(b) are the annihilation(creation) operators of the cavity mode, the magnon mode, and the mechanical mode, respectively. H0 represents the free energy of the cavity, the magnon, and the mechanical oscillator. H1 represents the interaction Hamiltonian where the first term is the coupling term of the magnon and the cavity field, and the second term is the coupling term of the magnon and the mechanical oscillator. Hd represents the coupling Hamiltonian of the cavity field with the pump field where the amplitude εin=P1ωl. η is the coupling efficiency of the cavity and the fields that can be continuously adjusted in the experiment,39,40 and κa is the cavity decay rate. Hp represents the coupling Hamiltonian of the cavity field with the weak probe field where the amplitude εp=Ppωp. P1 and Pp are the pump and probe field powers, respectively. Hpm describes the interaction of the magnon mode with the weak microwave source where εm=54γNB0 is the coupling strength of the magnon and the microwave field; γ denotes the gyromagnetic ratio; N is the total number of spins inside YIG sphere; and B0 is the drive magnetic field amplitude.

FIG. 1.

(a) Schematic diagram of a cavity magnetomechanical system. The rectangular 3D microwave cavity21,35,36 is driven by a microwave strong pump field (MS) and a probe field. The magnon as an oscillator is directly driven by a microwave source with driving frequency Ωm and magnetic field amplitude B0. The amplitude and phase of the transmission field are measured by a vector network analyzer (VNA).35,37H represents external bias magnetic field and dominates the frequency of magnon. (b) The equivalent coupling model. The magnon mode(m) couples the photon mode(a) with coupling strength gma, which simultaneously interacts with mechanical mode(b) with coupling strength gmb. κa, κb, and κm are the total loss rates of the cavity mode, mechanical mode, and magnon mode, respectively.

FIG. 1.

(a) Schematic diagram of a cavity magnetomechanical system. The rectangular 3D microwave cavity21,35,36 is driven by a microwave strong pump field (MS) and a probe field. The magnon as an oscillator is directly driven by a microwave source with driving frequency Ωm and magnetic field amplitude B0. The amplitude and phase of the transmission field are measured by a vector network analyzer (VNA).35,37H represents external bias magnetic field and dominates the frequency of magnon. (b) The equivalent coupling model. The magnon mode(m) couples the photon mode(a) with coupling strength gma, which simultaneously interacts with mechanical mode(b) with coupling strength gmb. κa, κb, and κm are the total loss rates of the cavity mode, mechanical mode, and magnon mode, respectively.

Close modal

Furthermore, we consider the rotating-wave approximation of the system, i.e., gma(a+a)(m+m)gma(am+am) (which is valid when ωa, ωmgma, κa, κm1,17), and combine the transformation HUHtUiUtU written into the interaction picture with respect to U=exp[iωl(aa+mm)t], the Hamiltonian can be obtained afresh,

H=Δaaa+Δmmm+ωbbb+gma(am+am)+gmbmm(b+b)+iηκa(εpaeiδpltiϕplεpaeiδplt+iϕpl)+i(εmmeiδmltiϕmεmmeiδmlt+iϕm)+iηκa(εinaεina),
(3)

where Δa=ωaωl and Δm=ωmωl represent the detunings of the cavity-pump and magnon-pump fields, respectively. δpl=ωpωl represents the detuning of the probe and pump fields, and δml=Ωmωl represents the detuning of magnon driving field and pump field. ϕpl=ϕpϕl represents the relative phase of the probe and pump fields.

According to the Hamiltonian of the system and taking into account the decay of the cavity mode and the dissipation of magnon mode and the mechanical mode existing in the system, we can derive the Heisenberg–Langevin equations of the motion as follows:

a˙=(iΔaκa)aigmam+ηκaεin+ηκaεpeiδpltiϕpl,b˙=(iωbκb)bigmbmm,m˙=(iΔmκm)migmaaigmbm(b+b)+εmeiδmltiϕm,
(4)

where κa, κb, κm are the total loss rates of the cavity field, the mechanical oscillator and the magnon, respectively. The quantum input noise and thermal noise terms of the system have been eliminated due to we are interested in the mean response of this system to the probe field.41,42

Assuming that the amplitudes of the applied fields satisfy {εp,εm}εin and setting the value of the difference equation in the right half of Eq. (4) equal to 0, the steady state equations of the system can be obtained

as=igmamsηκaεiniΔaκa,bs=igmb|ms|2iωbκb,ms=igmaasiΔ~mκm,
(5)

where Δ~m=Δm+gmb(bs+bs) is the effective detuning between the magnon and the pump field due to the magnomechanical interaction.

Then, we divide each operator of Eq. (4) into a small fluctuation and its steady state expectation value, i.e., O=Os+δO(O=a,b,m). Through a series of mathematical calculations and ignoring all the higher order terms(δOδO), the fluctuation equations of the system can be written as

δa˙=(iΔaκa)δaigmaδm+ηκaεpeiδpltiϕpl,δb˙=(iωbκb)δbiGδmiGδm,δm˙=(iΔ~mκm)δmigmaδaiG(δb+δb)+εmeiδmltiϕm,
(6)

with G=gmbms is the effective magnomechanical coupling strength of the system.

Considering the perturbation produced by the probe field, the fluctuation terms of the system can be expanded as22,43

δa=A1eiδt+A1+eiδt,δb=B1eiδt+B1+eiδt,δm=M1eiδt+M1+eiδt,
(7)

with δ=δpl=δml. By substituting Eq. (7) into Eq. (6), we obtain the amplitude of the first-order sideband of the cavity magnomechanical system,

A1=i[β3gma2β1+h2+h2β2]ηκaεpeiϕpl+gma[h1β1+h2+h2(gma2+h1h3)]εmeiϕmi[β1(gma2h1h1+β4)+h2+h2(gma4+gma2h1h3+h1+β2)],
(8)

with h1±=±iΔa+κaiδ, h2±=±iωb+κbiδ, h3±=±iΔ~m+κmiδ, β1=|G|2(h2+h2), β2=h3+(gma2+h1h3), β3=h1β1(h3+h3), β4=gma2h3+h1+h1h3.

By using input–output relation Sout=Sinηκaa,44 in the rotating frame with respect to the pump frequency ωl, the output of the probe field can be given by

Sout=c0+c1eiδt+c1+eiδt,
(9)

where c0=εinasηκa, c1=εpeiϕplηκaA1, and c1+=ηκaA1+, respectively. We define the transmission of the probe field as tp=c1/(εpeiϕpl).45 The transmission rate of the cavity magnomechanical system can be derived as

|tp|2=|1ηκaA1εpeiϕpl|2.
(10)

It can be seen that the transmission rate |tp|2 depends on the amplitude of magnon driving field εm and the phases of fields ϕpl, ϕm, where εm is closely related to the drive magnetic field amplitude B0 and the relative phase of applied fields Φ=ϕmϕpl can be controlled in the experimental platform.21,46,47 In the following, we focus on analyzing the influence of these parameter values on the characteristics of transmission of the system.

In this section, we numerically evaluate the transmission rate of the output field based on Eqs. (8) and (10) and, in detail, analyze the effect of the magnon driving parameters, such as the magnetic field amplitude B0 of magnon driving field and the relative phase Φ=ϕmϕpl, on the transmission spectrum. Here, we choose the parameters of the cavity field,1,19 i.e., the frequency of the microwave cavity ωa/2π=7.86 GHz, the cavity decay rate κa/π=3.35 MHz, and the coupling efficiency of the cavity and the fields η=1/2. We consider the parameters of the YIG sphere1 with the diameter D=250μm, the spin density ρ=4.22×1027m3, the magnon linewidth κm/π=1.12 MHz, the gyromagnetic ratio γ/2π=28 GHz/T, and the magnon–photon coupling strength gma/2π=3.2 MHz.17 In addition, we set the phonon frequency ωb/2π=11.42 MHz,1 the mechanical dissipation rate κb/π=300 Hz,1 and the magnon–phonon coupling strength gmb/2π=1 Hz.15 In terms of the parameters of the fields applied in the system, we choose the pump power of cavity P1=10 mW, the amplitude of the probe field εp=0.0001εin. The detuning between the cavity field and pump field Δa=ωb, the detuning between the magnon driving field and pump field Δm=ωb.

In order to clearly display the influence of magnon driving parameters on the characteristics of the output field of the system, in the absence of the additional magnon driving field of the system, we first plot in Fig. 2 the transmission rate |tp|2 as a function of the probe detuning δ with several different values of the magnon–phonon coupling strength gmb. From Fig. 2, one can find that in the absence of the magnon–phonon coupling gmb=0, there is a transparency window in the position of δ=ωb and the bilateral symmetric absorption dips in the positions of δ=0.71ωb(left) and δ=1.29ωb(right) correspond to the magnon-induced transparency and the magnon-induced absorptions,22,23 respectively, which results from the formation of magnon–polaritons between magnon and photon coupling in the system.48 In contrast, considering the magnon–phonon coupling gmb0, an absorption dip appears at the point δ=ωb and becomes deep as the increasing of gmb. Thus, two transparency windows and three absorption dips can be observed in the presence of the magnon–phonon coupling, which is different from the conventional magnetiomechanically induced transparency (MMIT). The physical interpretation of this phenomenon shown in Fig. 2 is rather clear. The ultimate physical mechanism of this phenomenon can be explained by the energy level diagram in Fig. 3. In the absence of the magnon–phonon coupling, i.e., gmb=0, under the influence of the pump field and the cavity decay, the photon emission transition can be achieved via exciting the magnon from the state |Na,Nm+1,Nb|Na+1,Nm,Nb|Na,Nm,Nb, which degenerates with the probe photon generated from the state |Na,Nm,Nb|Na+1,Nm,Nb|Na,Nm,Nb excited by the probe field. In this case, the destructive interference effect emerges in the system resulting in a magnon-induced transparency window and the bilateral symmetric absorption dips. When the magnon–phonon coupling is included (gmb0), there is an additional transition path |Na,Nm+1,Nb|Na,Nm,Nb+1|Na,Nm+1,Nb|Na+1,Nm,Nb|Na,Nm,Nb. This transition induced by magnon–phonon interaction, along with the transition caused by photon–magnon coupling, forms a constructive interference and leads to a sharp dip of the magnetomechanically induced absorption at the point δ=ωb.

FIG. 2.

The transmission rate |tp|2 as a function of the probe detuning with and without the phonon–magnon coupling strength. The parameter values we selected are ωa/2π=7.86 GHz, κa/π=3.35 MHz, η=1/2, ωb/2π=11.42 MHz, κb/π=300 Hz, D=250μm, ρ=4.22×1027m3, κm/π=1.12 MHz, P1=10 mW, εp=0.0001εin, εm=0, Φ=0, B0=0 T, Δa=ωb, Δm=ωb, gma/2π=3.2 MHz, γ/2π=28 GHz/T.

FIG. 2.

The transmission rate |tp|2 as a function of the probe detuning with and without the phonon–magnon coupling strength. The parameter values we selected are ωa/2π=7.86 GHz, κa/π=3.35 MHz, η=1/2, ωb/2π=11.42 MHz, κb/π=300 Hz, D=250μm, ρ=4.22×1027m3, κm/π=1.12 MHz, P1=10 mW, εp=0.0001εin, εm=0, Φ=0, B0=0 T, Δa=ωb, Δm=ωb, gma/2π=3.2 MHz, γ/2π=28 GHz/T.

Close modal
FIG. 3.

Energy level diagram of the system. |Na,|Nm,|Nb denote the states of microwave photon, magnon, and phonon.

FIG. 3.

Energy level diagram of the system. |Na,|Nm,|Nb denote the states of microwave photon, magnon, and phonon.

Close modal

In the view of the discussion of Fig. 2, the positions of the magnon-induced absorption and the magnetomechanically induced absorption of the system are determined, respectively. We impose an additional magnon driving field to investigate, in detail, the effect of the magnon driving parameters on the characteristics of output filed of the cavity magnomechanical system. In the absence of the phase modulation, i.e., Φ=0, the transmission rate |tp|2 as a function of the probe detuning with the different drive magnetic field amplitude is plotted in Fig. 4. It is shown that the central absorption at the point δωb becomes deeper slightly with the increasing magnetic field amplitude B0 of magnon driving field, while the bilateral symmetric absorption dips located at both sides of the central absorption dip coupling become asymmetry. In the red-sideband regime (δ0.71ωb), it is found that with the increase of magnetic field amplitude, the dip induced by the magnon–photon interaction evolves into transparency. Furthermore, when the magnetic field amplitude becomes a little larger, the transmission rate of the output field is greater than 1, and thus, the amplification of the weak probe field can be obtained. On the contrary, it can find that in the blue-sideband regime(δ1.29ωb), with the increasing magnetic field amplitude, there is a lower absorption dip, which decreases gradually and reaches saturation. The result we have shown is analogous to the Autler–Townes effect where the coupled transition in a three-level -type system is perturbed by a microwave field.49–51 In the system, the modulation of the magnon driving field amplitude can control the magnon-induced absorption into amplification, which has potential applications in quantum computation and quantum sensing.

FIG. 4.

The transmission rate |tp|2 as a function of the probe detuning with the different magnetic field amplitude of magnon driving field, the values of the parameter are chosen as gmb/2π=1 Hz, Φ=0, and the other parameter values we select are the same as in Fig. 2.

FIG. 4.

The transmission rate |tp|2 as a function of the probe detuning with the different magnetic field amplitude of magnon driving field, the values of the parameter are chosen as gmb/2π=1 Hz, Φ=0, and the other parameter values we select are the same as in Fig. 2.

Close modal

It is worth noting that the relative phase of applied fields has been used for the coherent control of OMIT in the cavity optomechanics systems,43 coined as the phase control technology. Furthermore, we analyze the influence of the relative phase between the cavity and the magnon driving fields on the transmission rate of the system. We choose the magnetic field amplitude B0=10.3×1010 T and plot the transmission rate of the output field as a function of the probe detuning with the different relative phase Φ in Fig. 5. The curves of Fig. 5 demonstrate that the bilateral asymmetric absorption induced by magnon–polaritons interaction has a sensitive dependence on the relative phase. On the contrary, the central absorption dip corresponding to the magnetomechanically induced absorption at the point δ=ωb remains almost unchanged when the relative phase varies from 0 to π. The relative phase dependence of the transmission spectrum in Fig. 5 can readily be explained physically in the following. According to the energy level diagram depicted in Fig. 3, when the magnon driving field is applied in the system, there are three transition paths can form a closed-loop Δ-type transition structure in the system. Specifically, the transition path caused by the magnon driving field |Na,Nm,Nb|Na,Nm+1,Nb joins the transition paths caused by the cavity probe and the pump fields, i.e., |Na,Nm,Nb|Na+1,Nm,Nb and |Na,Nm+1,Nb|Na+1,Nm,Nb, lead to the phase-sensitive effect of the cavity magnomechanical system. It is similar to the phase-dependent gain absorption spectra when three coherent driving fields simultaneously drive a three-level atomic system.52,53 The present results suggest that when an additional magnon driving field is applied in the cavity magnomechanical system, the magnon-induced absorption and the magnon-induced transparency of the cavity magnomechanical system are switchable by adjusting the value of the magnetic field amplitude of magnon driving field and the relative phase of the applied fields.

FIG. 5.

The transmission rate |tp|2 as a function of the probe detuning with the different relative phase of the cavity and the magnon driving fields. The values of the parameters are chosen as gmb/2π=1 Hz, B0=10.3×1010 T, and the other parameter values we select are the same as in Fig. 2.

FIG. 5.

The transmission rate |tp|2 as a function of the probe detuning with the different relative phase of the cavity and the magnon driving fields. The values of the parameters are chosen as gmb/2π=1 Hz, B0=10.3×1010 T, and the other parameter values we select are the same as in Fig. 2.

Close modal

From Figs. 4 and 5, it is clear that the transmission rate of the system is significantly dependent on the amplitude and phase of the magnon driving field. To demonstrate the feasibility of our scheme, we present a sensitive tool for direct measurement of the perturbing magnetic field amplitude B0. By setting d(|tp|2)/dδ=0 in the probe detuning δ0.71ωb, the maximum values of absorption tmaxAB and amplification tmaxAM of transmission rate corresponding to the unit magnetic field amplitude B0 are plotted in Fig. 6. It is well known that the sensitivity of the measurement can be evaluated by the slope of maximum values of the transmission spectrum with respect to the perturbing magnetic field, which shows the identification ability for the change of magnetic field.41,54 In the absorption and amplification regime, we define sensitivity SAB=|d(tmaxAB)/dB0| and SAM=|d(tmaxAM)/dB0|, respectively. From Fig. 6, it is evident that the variation of maximum values of the transmission rate is distinct with different relative phases. In the case of Φ=0, the maximum values of the transmission rate increases linearly with increasing magnetic field amplitude B0, which experiences absorption and amplification regions. Correspondingly, S1AB<S1AM indicates the sensitivity is higher in weak perturbing magnetic field. In contrast to Φ=0, when the relative phase Φ=π, the maximum values of transmission rate decreases linearly with increasing perturbing magnetic field amplitude and only appear in the absorption region. With S3AB<S2AB, it demonstrates that sensitivity is higher in strong perturbing magnetic field. Compared with traditional sensors in cavity optomechanics, we calculate sensitivity coefficient η=tmaxtmaxtmax+tmax×100%, where tmax and tmax represent two adjacent maximum values of transmission rate when perturbing magnetic field amplitude changes in units of 1010 T. It verifies in present system ηS1AB=38.46%, ηS1AM=21.38%, ηS2AB=63.02%, and ηS3AB=97.85%, which is a significant improvement over the Refs. 55 and 56. These results provide a new possibility to improve the detection sensitivity of the magnetic field strength by adjusting the value of the relative phase of the applied fields.

FIG. 6.

The maximum value of the absorption and amplification in probe detuning δ0.71ωb corresponding to unit magnetic field amplitude B0, the values of parameters are chosen as Φ=0 in (a), Φ=π in (b), gmb/2π=1 Hz, and the other parameter values we select are the same as in Fig. 2.

FIG. 6.

The maximum value of the absorption and amplification in probe detuning δ0.71ωb corresponding to unit magnetic field amplitude B0, the values of parameters are chosen as Φ=0 in (a), Φ=π in (b), gmb/2π=1 Hz, and the other parameter values we select are the same as in Fig. 2.

Close modal

As discussed above, we have investigated how to manipulate the transmission spectrum of the present system by adjusting appropriately parameters of magnon and driving fields and have found that the transmission has sensitive dependence on the relative phase of the applied fields. As is known that the transmission in OMIT and MMIT system experiences a considerable group delay. Thus, it reminds us of another question: whether the group delay can be observed in the present cavity magnomechanical system with magnon driving. In the following, we study the effect of the group delay of the cavity magnomechanical system. The group delay of the output signal field τg22,57,58 can be defined as

τg=ϕ(ωp)ωp=Im[1tptpωp],
(11)

where ϕt(ωp)=arg[tp(ωp)] is the phase dispersion of the transmitted light field. When the group delay is a positive value, i.e., τg>0, there is a slow light in the system. When the group delay is a negative value, i.e., τg<0, a fast light can be observed in the system.

In the case of Φ=0, we show in Fig. 7, the group delay τg as a function of probe detuning δ with the different drive magnetic field amplitude B0. Clearly, it can be found that the group delay can be tuned by changing the magnetic field amplitude of magnon driving field. In the probe detuning δ0.71ωb and δ1.29ωb corresponding to the magnon-induced absorption dips of Fig. 4, either slow light or fast light can be achieved with the appropriate value of the magnetic field amplitude B0 in the present system. Moreover, Fig. 7 also shows that the group delay can be switched from a negative value to a positive value so that the fast light in the probe detuning δ0.71ωb and δ1.29ωb can be converted to the slow light. The maximal value of the group delay induced by the magnon–polaritons interaction can reach 3.7μs in the probe detuning δ1.29ωb. In the probe detuning δ1.00ωb, a negative group delay (τg<0) is achievable, which implies that the magnon–phonon interaction leads to fast light. Furthermore, the effect of negative group delay becomes pronounced with increasing the value of magnetic field amplitude B0.

FIG. 7.

The group delay as a function of probe detuning with the different magnetic field amplitude of the magnon driving field. The values of parameters are chosen as gmb/2π=1 Hz, Φ=0, and the other parameter values we select are the same as in Fig. 2.

FIG. 7.

The group delay as a function of probe detuning with the different magnetic field amplitude of the magnon driving field. The values of parameters are chosen as gmb/2π=1 Hz, Φ=0, and the other parameter values we select are the same as in Fig. 2.

Close modal

Figure 8(a) shows that the group delay can be switched by modulating the relative phase of the applied fields. Compared with the curve of group delay in the case of Φ=0, Fig. 8(a) shows that the positive group delay in the probe detuning δ0.71ωb corresponding to the mganon-induced absorption has a giant increase in the case of Φ=π. However, in the probe detuning δ1.29ωb, the positive group delay decreases dramatically when the relative phase Φ is tuned from 0 to π. Interestingly enough, it can be seen that the curve of the group delay in the case of Φ=π is the mirror image symmetric with regard to the case of Φ=0. In the probe detuning δ1.00ωb, Fig. 8(b) displays that the group delay vs the magnetic field amplitude of the magnon driving field with the different relative phase of the applied fields. It demonstrates that for a fixed value of the magnetic field amplitude B0, the dispersion property of the system can be modulated and the slow and the fast light can be achievable and switchable in the cavity magnomechanical system by using the phase modulation of the applied fields. Besides, the slow and the fast light can be significant boosted with the increasing magnetic field amplitude. Thus, we note that the present work provides an efficient scheme for controlling dispersion as well as slow light in the specific frequency. The results we investigated in the cavity magnomechanical system may provide a potential application for the fields of telecommunications and microwave data processing.

FIG. 8.

(a) The group delay as a function of the probe detuning with the different relative phase of the cavity and the magnon driving fields, the values of parameters are chosen as gmb/2π=1 Hz, B0=10.3×1010 T; (b) the group delay in probe detuning δ1.00ωb as a function of the magnetic field amplitude of the magnon driving field with the different relative phase, the values of parameter are chosen as gmb/2π=1 Hz, the other values of the parameters are the same as in Fig. 2.

FIG. 8.

(a) The group delay as a function of the probe detuning with the different relative phase of the cavity and the magnon driving fields, the values of parameters are chosen as gmb/2π=1 Hz, B0=10.3×1010 T; (b) the group delay in probe detuning δ1.00ωb as a function of the magnetic field amplitude of the magnon driving field with the different relative phase, the values of parameter are chosen as gmb/2π=1 Hz, the other values of the parameters are the same as in Fig. 2.

Close modal

In conclusion, we investigate the coherent control of the transmission spectrum in a cavity magnetomechanical system consisting of microwave photon, magnon, and phonon modes, where the microwave cavity is driven by a strong pump field and a weak probe field and the magnon is driven by a weak microwave source. We mainly focus on the relative phase of the applied fields and the magnetic field amplitude of magnon driving field on the transmission spectrum and the dispersion property of the system. The present results clearly show the dependence of the transmission spectrum on the driving parameters of the magnon by comparing the transmission spectra of the cavity magnetomechanical system with and without including the magnon–phonon coupling. It is worth noting that, different from a single transparency window in the absence of the magnon–phonon interaction, two transparency windows and three absorption dips can be observed in the presence of the magnon–phonon interaction, which originates from the joint interaction of magnon–phonon and magnon–photon. The present work also illustrates that two absorption dips located at both sides of the central absorption dip can be modulated asymmetrically into amplification and absorption by varying the magnetic field amplitude of magnon driving field. More interestingly, the relative phase of applied fields could have profound effects on both the transmission spectrum and the group delay of output field by choosing the appropriate magnetic field amplitude of magnon driving field so that the transmission spectra and the dispersion property could be controllable. To demonstrate the feasibility of our scheme, we present a sensitive tool for direct measurement of the perturbing magnetic field strength by adjusting the value of the relative phase of the applied fields, which demonstrate a significant improvement over previous research in cavity optomechanics. These present results illustrate that manipulating the relative phase of the applied fields in the cavity magnetomechanical system has a potential application in the field of quantum sensing and microwave signal processing.

The research is supported in part by National Natural Science Foundation of China under Grant Nos. 11774054 and 12075036.

The data that support the findings of this study are available within the article.

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