Generation and manipulation of phonon lasering in a two-drive cavity magnomechanical system

A simple and feasible scheme for the generation and manipulation of phonon lasering is proposed and investigated based on a generic three-mode cavity magnomechanical system, in which a magnon mode couples simultaneously with a microwave cavity mode and a phonon mode. In sharp contrast to all previous phonon lasering schemes with only a single drive, the input pump field for the system in the proposed scheme is split into two microwave driving fields to drive the microwave cavity mode and the magnon mode, respectively. The impact of changing relative phase and relative amplitude ratio of the two microwave drives on mechanical gain, stimulated emitted phonon number, threshold power, and phonon emission line shape are theoretically and numerically investigated. The results indicate that the phonon laser action can be effectively controlled simply by adjusting the relative phase and relative amplitude ratio, so additional and tunable degrees of freedom are introduced to control the phonon laser. Considering the experimental feasibility of the generic cavity magnomechanical system and the two-drive approach, the present scheme provides a potentially practical route for the development of tunable phonon lasering devices with low-threshold, high-gain, and narrow-linewidth properties based on the platform of cavity magnomechanics.


Introduction
Phonon lasers, as the counterpart of optical lasers, have drawn much attention in the past decades. Phonon lasering has been demonstrated to exhibit properties similar to that of an optical laser, including threshold and linewidth narrowing in the lasing regime [1,2]. Phonon lasers can hence provide coherent acoustic sources to drive functional phononic devices for fundamental researches as well as various applications, such as audio filtering, high-precision sensing, acoustic imaging, topological sound control, and study of nonlinear phononics [3][4][5][6][7]. So far, the possibility of phonon laser action has been explored in a wide range of physical systems [8][9][10][11][12][13][14][15], and several experimental demonstrations have been reported in recent years [16][17][18][19][20]. In particular, a phonon laser that operates in close analogy to a two-level optical laser has been demonstrated using a compound optical microcavity system coupled to a radio-frequency mechanical mode [16]. This approach uses tunable optical coupling induced by photon tunneling to generate two optical supermodes, and transitions between their corresponding energy levels are mediated by a phonon field [21]. So the compound system behaves as a two-level laser system, in which the optical supermodes play the role of laser medium and the laser field is provided by the phonon mode. This architecture has motivated many phonon lasering schemes which possess different properties, such as parity-time symmetry [22][23][24][25], exceptional points [26], nonreciprocity [27], and polarization dependence [28,29]. Moreover, ultralow-threshold phonon lasering can be realized by using two optical parametric amplifiers (OPAs) in a coupled cavity optomechanical system, which is controlled by the phase difference from the two OPAs [30].
In this work, we propose a simple and feasible scheme for the generation and manipulation of phonon lasering based on a generic three-mode cavity magnomechanical system containing a YIG sphere in a microwave cavity, in which a magnon mode couples simultaneously with a microwave cavity mode and a phonon mode. In sharp contrast to all previous cavity optomechanical or cavity magnomechanical phonon lasering schemes with only a single drive, the input pump field for the system in the proposed scheme is split into two microwave driving fields to drive the microwave cavity mode and the magnon mode, respectively. The impact of changing relative phase and relative amplitude ratio of the two microwave drives on mechanical gain, stimulated emitted phonon number, threshold power, and phonon emission line shape are theoretically and numerically investigated. It is found that the phonon laser action can be effectively controlled simply by adjusting the relative phase and relative amplitude ratio, so the relative phase and relative amplitude ratio can be regarded as additional and tunable degrees of freedom to control the phonon laser. In comparison with the previous phase-controlled phonon laser scheme [30], our scheme exploits the cavity magnomechanical platform with high tunability and flexibility, without driven nonlinear optical media for OPAs. Considering the experimental feasibility of the generic cavity magnomechanical system and the two-drive approach, the present scheme provides a potentially practical route for the development of tunable phonon lasering devices with low-threshold, high-gain, and narrow-linewidth properties based on the platform of cavity magnomechanics. Furthermore, the approach and mechanism utilized in this scheme may also be applied to other platforms such as cavity optomechanical or cavity electromechanical systems.
The rest of this paper is organized as follows. In section 2, we describe in detail the model and scheme, present analytical expressions for mechanical gain of the system as well as threshold pump power. The results of numerical analysis with experimentally feasible parameters are presented in section 3. Finally, we summarize this work in section 4. In addition, appendix A provides the derivation process of effective Hamiltonian in rotating framework and steady-state solutions of Heisenberg-Langevin equations of motion. Appendix B gives the derivation of mechanical gain and threshold power in supermode representation.

Model and method
The model considered in our scheme is schematically shown in figure 1(a). We consider a highly polished single-crystal YIG sphere placed in a three-dimensional microwave resonator [45]. The YIG sphere is supposed to be placed at the maximum magnetic field of a microwave cavity mode with resonance frequency ω a . A uniform external magnitic field H ext biased along the z direction is applied onto the YIG sphere, leading to a uniform magnon mode resonates in the YIG sphere at frequency ω m = γH ext , where γ is the gyromagnetic ratio. Moreover, the YIG sphere is also a mechanical resonator which supports various phonon modes. The varying magnetization induced by the magnon excitation inside the YIG sphere causes deformation of its spherical geometry (and vice versa), leading to the coupling between the magnon mode and the phonon modes (see figure 1(b)). We consider one of the lowest order spheroidal phonon modes with resonance frequency ω b , which shows the effective coupling with the magnon mode and has a relatively long lifetime compared to other phonon modes [45]. So a generic three-mode cavity magnomechanical system is obtained, in which a magnon mode couples simultaneously with a microwave cavity mode and a phonon mode through magnetic-dipole interaction and magnetostrictive interaction, respectively (see figure 1(c)). The magnon mode can be tuned close to the resonance with the cavity photon mode by adjusting the external bias magnetic field. Then the hybridization between the photon and magnon modes leads to the generation of two nondegenerate hybridized supermodes, which act as the gain medium for a two-level phonon laser. Transitions between the corresponding energy levels of the two supermodes are induced by the phonon mode, which can be well understood by the energy level diagram in figure 1(d).
In our scheme, the system is pumped by a coherent microwave input with power P in and frequency ω d . The input pump field is then divided into two microwave driving fields by a power splitter (see figure 1(a)) [79][80][81]. The first microwave drive with amplitude ε a and phase φ cavity is used to drive the microwave cavity mode through the cavity port, which corresponds to the single drive employed in previous cavity optomechanical or cavity magnomechanical phonon lasering schemes [16,77]. The second microwave drive with amplitude ε m and phase φ magnon is used to drive the magnon mode through the magnon port. Experimentally, the magnon port can be realized by a metallic loop around the YIG sphere. The second drive acts only on the magnon and does not couple to the cavity mode. We define relative amplitude ratio and relative phase of the two microwave drives as δ = ε m /ε a and φ = |φ magnon − φ cavity |, respectively. The value of δ can be controlled by the power splitter. A tunable phase shifter added in the path of the magnon port can be used to control the value of φ [80]. Therefore, both the relative amplitude ratio δ and relative phase φ can be controlled independently, and the impact of changing the values of φ and δ on the phonon laser action should be investigated. We note that the two-drive approach with good tunability has been demonstrated in recent experiments to realize steering between level repulsion and attraction [79], control the coupling strength and linewidth [80], and study the phase-controlled transmission properties [81] in cavity-magnon polariton systems. Many interesting phase-controlled effects with two driving fields also have been theoretically [82,83] and experimentally [84] investigated in the field of optomechanicas.
The total Hamiltonian of the coupled three-mode cavity magnomechanical system can be written as [45] where Here, the first term H 0 is the free Hamiltonian, which describes the three modes in the system.h is the reduced Planck constant. a (a † ), m (m † ), and b (b † ) represent the annihilation (creation) operators of the cavity photon, magnon, and phonon modes, respectively. The second item H I is the interaction Hamiltonian, which describes the coupling between the cavity photon and magnon modes with coupling strength g ma , and the coupling between the magnon and phonon modes with coupling strength g mb (see figure 1(c)). The coupling termhg ma (a + a † )(m + m † ) is obtained from the quantization under the magnetic-dipole approximation, which has the same structure as that of the quantum Rabi mode under the electric-dipole approximation [85]. It can further be reduced tohg ma (am † + a † m) by applying the rotating-wave approximation, which is valid for ω a , ω m ≫ g ma , κ a , κ m in our scheme. The radiation pressure-like coupling termhg mb m † m(b + b † ) results from the fact that the frequency of acoustic phonon (megahertz) is much lower than that of the magnon (gigahertz), so only the nonlinear dispersion magnon-phonon interaction is concerned about in the derivation of the interaction Hamiltonian [45]. The last term H d describes the interactions between the microwave driving fields and the system. κ a and κ m are the decay rates of the cavity photon and magnon modes, respectively. It is assumed that the decay rates are mainly dominated by the external coupling rates of the microwave cavity and the YIG sphere [77]. We define the power of the first (second) microwave drive at the cavity port (magnon port) as P in,a (P in,m ), so the amplitudes ε a = √ P in,a /hω d and ε m = √ P in,m /hω d . Then δ = ε m /ε a = √ P in,m /P in,a , and the total input pump power for the system is P in = P in,a + P in,m = (1 + δ 2 )P in,a .
In the rotating reference frame with respect to the driving field frequency ω d , the quantum Heisenberg-Langevin equations of motion can be derived as (see appendix A) Here, is the frequency detuning between the driving field and the cavity photon (magnon) mode. κ b is the dissipation rate of the phonon mode. a in , m in , and b in are the input noise operators affecting the cavity photon, magnon, and phonon modes, respectively. As we are only interested in the mean response of the system, the operator equations can be reduced to the mean-value equations (see appendix A). Then the steady-state solutions of the equations can be obtained as Assuming that the magnon is resonant with the cavity photon mode, i.e. ω m = ω a , then we set ∆ a = ∆ m = ∆. By bringing in the supermode operators Ψ ± = (a ± m)/ √ 2 [16,77], the total Hamiltonian of the system can be rewritten in the supermode representation as after adopting the rotating-wave approximation (see appendix B). Here, the first (second) term describes the Hamiltonian of the supermode Ψ + (Ψ − ) with the eigenfrequency ω + = −∆ + g ma (ω − = −∆ − g ma ). The third term describes the phonon mode. The fourth term describes the transitions between the two nondegenerate supermodes Ψ ± through the absorption and emission of phonons, which corresponds to the energy level diagram in figure 1(d). The last two terms represent the interactions between the two driving fields and the two supermodes.
is the phonon number operator. G > 1 denotes that the mechanical gain overcomes dissipation, so that the phonon mode is coherently amplified. The enhanced mechanical gain is hence a necessary prerequisite of producing phonon lasering. The stimulated emitted phonon number is expressed as N b = exp [2(G − 1)]. The threshold condition for realizing phonon lasering is G = 1, and the corresponding threshold power P th which reveals the critical value of the input pump power can be expressed as Both the mechanical gain G and threshold power P th depend on the relative phase φ as well as the relative amplitude ratio δ, which is quite different from the results in previous phonon lasering schemes [16,77] without introducing the second drive. The phonon laser action can therefore be controlled by changing the values of φ and δ.

Result and discussion
In this section, we numerically analyze the influence of changing the relative phase φ and relative amplitude ratio δ of the two microwave driving fields on the phonon laser action based on the two-drive cavity magnomechanical system. We choose a set of experimentally feasible system parameters in our calculations according to recent experiments, including ω m = ω a = 2π × 7.86 GHz, 2κ a = 2π × 3.35 MHz, 2κ m = 2π × 1.12 MHz, ω b = 2π × 11.42 MHz, 2κ b = 2π × 300 Hz, and g mb = 2π × 4.1 mHz [45]. Moreover, we set g ma = 0.5ω b , so that the splitting of the two supermodes is resonant with the phonon frequency, i.e. ω + − ω − = 2g ma = ω b . This can be called the matching condition for a two-level phonon laser. Experimentally, the photon-magnon coupling strength g ma can be engineered by changing, e.g. the cavity mode volume, the magnetic dipole moment, or the position of the YIG sphere in the microwave cavity. The proposed model and scheme can hence be used to selectively amplify different phonon modes by designing the system to satisfy different matching conditions in practice.
In figures 2(a) and (b), we plot the calculated steady-state photon number |a s | 2 and steady-state magnon number |m s | 2 versus the normalized detuning ∆/ω b and the relative phase φ with the relative amplitude ratio δ = 0.5, respectively. The calculated |a s | 2 and |m s | 2 versus ∆/ω b and δ with φ = π are plotted in figures 2(c) and (d), respectively. The input pump power is fixed at P in = 10 mW in these calculations. It is obvious that the steady-state populations of the photon and magnon modes can be tuned by changing φ and δ, even though the input pump power is unchanged. The variation of the magnon number in the YIG sphere can modify the radiation-pressure-like magnetostrictive force [77]. As a result, the magnon-phonon coupling would be modified. The stronger magnon-phonon coupling can improve the phonon lasering generation, as the mechanical gain is proportional to the square of the magnon-phonon coupling strength as shown in equation (6). So we can expect that the phonon laser action can be effectively controlled by using the two-drive approach and adjusting φ and δ. On the other hand, it is found that both the steady-state photon and magnon numbers get the maximum at ∆ = 0.5ω b , which corresponds to the case that the input pump field is resonant with the supermode Ψ + [i.e. pumping the up energy level in figure 1(d)]. We can expect that the strongest phonon lasering can be achieved in this case.
In figure 3(a), the calculated mechanical gain G is plotted versus the normalized detuning ∆/ω b and the relative phase φ with the input pump power P in = 10 mW and the relative amplitude ratio δ = 0.5. The solid white lines represent the threshold condition G = 1. The area between the solid white lines is the parameter regime where the mechanical gain G > 1 and the phonon mode is coherently amplified. It can be seen that the mechanical gain gets the maximum at ∆ = 0.5ω b , i.e. when the input pump field is resonant with the  supermode Ψ + . This is consistent with the results in figure 2 and the physical mechanism of the two-level phonon laser (pumping the up energy level) [16]. Moreover, the mechanical gain is periodically varied when changing φ from 0 to 2π, and reaches its maximum at φ = π. This implies that the mechanical gain can be effectively tuned by simply adjusting the value of φ. Specifically, the corresponding mechanical gain spectrums with three different values of φ are plotted in figure 3(b). It is shown that when selecting appropriate detuning ∆, one can change the mechanical gain from G < 1 to G > 1 (and vice versa) by simply adjusting the relative phase, i.e. realize the switching of phonon lasering. For example, when we choose ∆ = 0.67ω b , G = 1 is achieved at φ = 0.5π. Then the mechanical gain is enhanced when increasing the relative phase from φ = 0.5π to φ = π, while it is suppressed when decreasing the relative phase from φ = 0.5π to φ = 0. This phase-dependent and switchable mechanical gain can also be phenomenologically understood by the interference effect [81], as two driving fields splitted from a common coherent source are utilized to excite the equivalent two-level system in our scheme. One of the driving fields induces the population inversion through exciting the cavity mode, and the other driving field achieves this through exciting the magnon mode. Despite both the two fields are applied to pump the same up energy level, different excitation pathways exist due to the presence of photon-magnon coupling and coherently transfers. Enhanced or suppressed mechanical gain can hence be achieved when the constructive or destructive pathway interference happens. The relative phase plays a key role in the controllable interference, which can be engineered to effectively control the phonon laser action. Similar to figures 3(a) and (b), figures 3(c) and (d) show the impact of changing the relative amplitude ratio δ on the mechanical gain G with P in = 10 mW and φ = π. The results indicate that the mechanical gain can also be controlled and switched by adjusting the value of δ. From figure 3(c), one can see that there exists a nonzero relative amplitude ratio which results in the maximal mechanical gain and hence the optimized phonon lasering. For the system parameters adopted in our calculations, the mechanical gain reaches its maximum when δ ≃ 0.58. Although the type of interference (constructive interference or destructive interference) is dependent on the relative phase φ, the degree of interference can also be affected and optimized by the relative amplitude ratio δ. By comparing figures 3(c) and (d) with figures 3(a) and (b), one can also see that the mechanical gain is relatively robust to the relative amplitude ratio δ when φ = π, but it is more sensitive to the relative phase φ when δ = 0.5. A possible interpretation of this effect is that the population inversion has basically the same dependence on the driving amplitudes ε a and ε m due to the interference effect when φ = π. Therefore the mechanical gain is relatively robust to the relative magnitudes between ε m and ε a in this case. Based on this feature, we may achieve a control strategy implementing different control regulations (coarse and fine tuning) for the phonon laser by synthetically adjusting φ and δ. For more clear, we plot the mechanical gain G varies with φ and δ in figure 4 with P in = 10 mW and ∆ = 0.5ω b . It can be seen that the mechanical gain can be effectively tuned by synthetically adjusting φ and δ, and the maximum mechanical gain G ≃ 6.7 can be obtained when δ = 0.58 and φ = π. For comparation, the mechanical gain is about 5.1 and unchanged when δ = 0, which corresponds to the traditional single-drive case and no second drive is applied [16,77]. The relative phase and the relative amplitude ratio can hence be regarded as additional and tunable degrees of freedom to control the phonon laser in addition to the input pump power.
The input pump power for the system is fixed at P in = 10 mW in the above calculations. In the following, we study the effect of tuning the relative phase and relative amplitude ratio on the phonon laser action by varying the input pump power. In figure 5(a) (figure 5(c)), the calculated mechanical gain G is plotted versus the relative phase φ (relative amplitude ratio δ) and the input pump power P in with δ = 0.58 (φ = π). The mechanical gain G as a function of φ (δ) with four different values of P in is shown in figure 5(b) (figure 5(d)) when δ = 0.58 (φ = π). Here, we set ∆ = 0.5ω b for obtaining the large mechanical gain. The solid white line represents the threshold condition G = 1, and the corresponding pump power is the threshold power P th . The area above the solid white line is the parameter regime with the enhanced mechanical gain G > 1. It can  be seen that the mechanical gain G increases monotonously when increasing the pump power P in , which is consistent with the theoretical expectation for a phonon laser. Moreover, the relative phase has a significant impact on the threshold power when δ = 0.58, but the threshold power is relatively robust to the relative amplitude ratio when φ = π. The threshold power is significantly reduced when the maximally enhanced mechanical gain is achieved at φ = π and δ = 0.58. These results are consistent with the results in figures 3 and 4.
The stimulated emitted phonon number N b varies with the input pump power P in is plotted in figure 6(a) with three different values of φ. Here, we set ∆ = 0.5ω b and δ = 0.58. The horizontal dot-dashed line represents N b = 1, which corresponds to the threshold condition G = 1. The corresponding pump power is the threshold power P th . One can see that the stimulated emitted phonon number N b increases very fast with the input pump power P in after reaching the threshold value P th , which indicates that the phonon lasering is well realized. Moreover, it is clearly seen that the threshold power highly depends on the relative phase φ. For example, the threshold power is about 2.51 mW when φ = 0.5π. By adjusting the relative phase to φ = π, the threshold power can be reduced to about 1.47 mW as the mechanical gain is enhanced. In  contrast, a much larger threshold power about 5.84 mW is required to realize the phonon lasering when adjusting the relative phase to φ = 0 due to the suppressed mechanical gain. On the other hand, the effect of changing the relative amplitude ratio δ on the stimulated emitted phonon number and the threshold power is shown in figure 6(b) with ∆ = 0.5ω b and φ = π. The result indicates that the relative amplitude ratio also has an impact on the threshold power as well as the stimulated emitted phonon number. It is shown that the threshold power can be reduced with a nonzero δ compared with the single-drive case (δ = 0). The minimum threshold power is only about 1.47 mW when the maximum mechanical gain is achieved at δ = 0.58 and φ = π. We can therefore reduce the threshold power of the phonon laser by using the two-drive approach, and tune the threshold power in a range by changing the values of δ and φ. To show this effect more clearly, we plot the calculated threshold power changes with δ and φ in figure 7. The result is very consistent with the above analysis. Figures 8(a) and (b) show the phonon emission line shape characterised by N b (ω) with center frequency ω b = 2π × 11.42 MHz below and above the threshold value, respectively. Here, we set g ma = 0.5 × 2π × 11.42 MHz (i.e. the splitting of the two supermodes is 2g ma = 2π × 11.42 MHz) and ∆ = g ma (i.e. pumping the up energy level), and then replace ω b with the independent variable ω in equation (6). Then we can numerically plot N b (ω) = exp {2[G(ω) − 1]} as a function of ω with different P in , δ, and φ [86,87]. According to the above analysis, the threshold power P th is about 1.47 mW when δ = 0.58 and φ = π. It can be seen that the linewidth above the threshold value is much narrower than that below the threshold value by comparing figures 8(a) and (b), which indicates the property of linewidth narrowing in the phonon lasering regime [16,86,87]. The narrower linewidth results in a better coherence for the phonon lasering in practical applications. In addition, we also show the phonon emission line shape for the δ = 0 case in figures 8(c) and (d) for comparation. When the input pump power is P in = 5 mW which exceeds the threshold values for both cases, the stimulated emitted phonon number is significantly enhanced and the linewidth of the phonon lasering is slightly reduced when δ = 0.58 and φ = π in contrast to the δ = 0 case. With these results and the above discussions, a tunable phonon laser featuring low-threshold, high-gain, and narrow-linewidth properties can be achieved based on the cavity magnomechanical system and the two-drive approach.

Summary
In summary, we have presented the scheme for generating and manipulating phonon lasering based on the generic three-mode cavity magnomechanical system and the two-drive approach. In the presented scheme, the input pump field for the system is split into two microwave driving fields to drive the microwave cavity mode and the magnon mode, respectively. The impact of changing the relative phase φ and relative amplitude ratio δ of the two microwave drives on the phonon laser action has been investigated in detail. It is found that the phonon laser action can be effectively controlled simply by tuning the values of φ and δ, so the relative phase and relative amplitude ratio can be introduced as additional and tunable degrees of freedom to control the phonon laser. Considering the experimental feasibility of the generic cavity magnomechanical system and the two-drive approach, the present scheme provides a potentially practical route for the development of tunable phonon lasers featuring low-threshold, high-gain, and narrow-linewidth properties based on the platform of cavity magnomechanics. The approach and mechanism utilized in this scheme may also be applied to other platforms such as cavity optomechanical or cavity electromechanical systems.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).
Here, the quantum noise terms have been safely dropped as all the noise operators have zero mean values, i.e. ⟨a in ⟩ = 0, ⟨m in ⟩ = 0, and ⟨b in ⟩ = 0. In addition, the mean-field approximation, i.e. ⟨ m † m ⟩ = m * m, is used to deal with the nonlinear term ig m m † m. According to equation (A.3), the steady-state solutions of the equations can be obtained as in equation (4) by settingȯ = 0 (o = a, m, b).

Appendix B. Derivation of the mechanical gain and the threshold pump power
By introducing the supermode operators Ψ ± = (a ± m)/ √ 2, the radiation pressure-like magnon-phonon interaction Hamiltonianhg mb m † m(b + b † ) can be expressed as After applying the rotating-wave approximation, the Hamiltonian becomes Therefore, the total Hamiltonian of the system can be rewritten as that in equation (5). Based on the Hamiltonian in equation (5), the dynamical equations for the supermodes Ψ ± and the phonon mode b can be obtained asΨ where γ = (κ a + κ m )/2. With the introduction of the ladder operator p = Ψ † − Ψ + and the inversion operator δn = Ψ † + Ψ + − Ψ † − Ψ − , we can rewrite the Hamiltonian in equation (5) as We can solve the steady-state solutions of Ψ ± and p from equations (B.3) and (B.5) as β − 2i γ∆ , where β = g 2 mb b † b/4 − ∆ 2 + g 2 ma + γ 2 . By substituting the steady-state solutions into equation (B.6), we can obtain the equationḃ and (B.14) The mechanical gain G presented in equation (6) is defined as G = G ′ /κ b , i.e. the ratio between G ′ and the dissipation rate κ b of the phonon mode [16]. G > 1 denotes that the mechanical gain overcomes the dissipation and the phonon mode is coherently amplified. The threshold condition for phonon lasing is G = 1, and the threshold pumping power P th can be calculated using the threshold condition G = 1 as in equation (7).