Exceptional-point-engineered phonon laser in a cavity magnomechanical system

We propose a scheme to engineer phonon laser in a non-Hermitian cavity magnomechanical (CMM) system with dissipative magnon-photon coupling. The exceptional point (EP) (the analog of the -symmetric regime), emerging in the system and changing the properties of photons, magnons, and phonons, can be observed with a tunable dissipative magnon-photon coupling caused by the cavity Lenz’s law. At the EP, we find that a strong nonlinear relation appears between the mechanical amplification factor and the detuning parameter, which results in a dramatic enhancement of magnetostrictive force and mechanical gain, and leading to the highly efficient phonon laser and the ultralow threshold power. Furthermore, EP induced by dissipative coupling is flexible and tunable compared to the -symmetric regime, and the ultralow threshold power phonon laser is immune to the loss rates of the photon and magnon modes. Our scheme provides a theoretical basis for phonon laser in non-Hermitian systems and presents potential applications ranging from preparing coherent phonon sources to operating on-chip functional acoustic devices.


Introduction
Hybrid magnonic devices involving collective excitation of magnetization in ferromagnetic and antiferromagnetic crystals possess excellent nonlinearity and quantum properties and become a vital information carrier for modern information technology and constructing quantum networks based on its powerful compatibility [1][2][3][4][5]. Recent experiments have focused on a yttrium-iron-garnet (YIG) sphere with a large spin density and a high Curie temperature, which ensures that magnons can be coherently coupled to photons, phonons, and superconducting qubits [6][7][8]. Moreover, the magnon mode in the YIG sphere excited by an external magnetic field has a frequency that can be easily tuned. Based on magnetic dipole interaction, the coherent strong and even ultrastrong magnon-photon coupling can be generated with the mode anticrossing effect and the emergence of magnon polaritons [9][10][11][12][13][14][15]. The radio-frequency magnetic field used in the magneto-optical coupling process significantly reduces the quality factor of the optical mode. Then this problem is resolved by using a three-dimensional cavity [3,4]. The intrinsically excellent tunability and long coherence time give rise to scientists' interest in magnon-based hybrid systems [16,17].
Magnon systems offer us a powerful tool to explore the unique effects of quantum mechanics [18,19]. The relatively small external dissipation and inherent damping of the coupling modes make the dissipative coupling have no obvious correlation. Thus coherent coupling is dominating in such system [20,21]. Although the effect of coherent coupling is extensive, it is only a small part of magnon-photon hybridization [22]. In 2018, the dissipative magnon-photon coupling is discovered [23], and is implemented in numerous hybrid systems [24][25][26][27]. The pure dissipative coupling with obvious coupling characteristics has been revealed recently via the damping-like Lenz's law [28]. The dissipative magnon-photon coupling makes that the non-Hermitian physical system possess a real eigenvalue, which is analogous to PT -symmetric regime [29,30]. In parallel, a cooperative radiative damping picture rooted in the theory of reservoir engineering has been demonstrated, and the dissipative coupling is observed experimentally [31].
The EP has generated a series of fascinating phenomena to research various quantum effects in the non-Hermitian systems and the (anti) PT -symmetric cavity magnonic systems [32][33][34].
The magnon and phonon modes can be coupled by a nonlinear magnetostrictive interaction, hence the YIG sphere can also serve as an excellent phonon mode [35]. Phonons, quantum particles that represent sound like photons, similar to self-organizing synchronization, which is the basis for the theory of phonon lasers [36]. Similar to the optical laser realized by Maiman in 1960 [37], an efficient scheme was proposed to prove the feasibility of phonon lasers in 2003 [38]. The phonon laser, coherent sound oscillations (mechanical vibrations) induced by optical pumping, has potential applications in audio filtering, acoustic imaging, highly precise sensing, imaging or switching, and topological sound control [39][40][41][42]. Recently, a large number of theoretical and experimental researches related to phonon laser have attracted the attention of researchers, including low threshold phonon laser, PT -symmetric phonon laser, non-reciprocal phonon laser, and polarization controlled phonon laser [43][44][45][46][47][48][49]. Phonon laser under coherent coupling regime has been widely considered [50], and has been extended to dissipative coupled systems [51].
In this paper, we demonstrate the exceptional-point-engineered phonon laser in the non-Hermitian CMM system composed of a cavity and a YIG sphere. Benefits from the cavity Lenz's law, the tunable dissipative magnon-photon coupling can be obtained. The phenomenon of energy level attraction caused by non-Hermitian dissipative coupling can lead to the existence of pure real eigenvalue of the non-Hermitian Hamiltonian, which indicates that it is possible to find out the EP. Similar to the EP caused by PT -symmetry, the EP caused by dissipative magnon-photon coupling can be utilized to obtain a strong nonlinear relationship between the mechanical amplification factor and the detuning parameter, which leads to a dramatic enhancement of magnetostrictive force and mechanical gain. Therefore, a highly efficient phonon laser can be realized and ultralow threshold power can be ensured around the EP. Compared to the traditional PT -symmetric system, the EP is realized without the gain of the subsystem and the restrictive conditions of strict gain-loss balance are avoided, which can be applied to a wide range of system parameters. The results provide a promising approach for engineering EP induced by dissipative coupling, and the EP is flexible and tunable compared to the PT -symmetric regime, resulting in that the ultralow threshold power phonon laser is immune to the loss rates of the photon and magnon modes.

Model and Hamiltonian
As shown in figure 1(a), the three-mode coupling photon-magnon-phonon hybrid system we consider consists of a microwave cavity and a YIG sphere supporting a magnon and mechanical breathing modes interacting via magnetostrictive force. In particular, there is a dissipative magnon-photon coupling due to the effect of Lenz's law (Φ = π). Therefore, the voltage caused by Faraday's law creates an induced current. In the frame rotating at the cavity pump frequency and with the rotating-wave approximation, the system Hamiltonian can be written as where a (m and b) is the annihilation operator of the photon (magnon and phonon) mode with resonance frequency ω a (ω m and ω b ).
is the detuning between the pump frequency and the photon (magnon) mode. g ma (g mb ) is the magnon-photon (magnon-phonon) coupling strength.
The pump field applied to the photon mode is considered by the last term with the amplitude is the loss rate of the photon (magnon, phonon) mode, and P in is pump power. The model can also be implemented by constructing a magnon-circult-QED system, as shown in figure 1(b). The coherent or dissipative magnon-photon coupling can be realized by RLC-circuit structure, which corresponds to the existence of resistance-dominated or inductance-dominated coupling [52]. The noise terms can be safely ignored if the mean-number behaviors are only considered. The steady-state mean values of the system can be written as To describe the enhanced steady-state mechanical displacement in the dissipative coupling regime compared to the coherent coupling, we define the mechanical displacement amplification factor as The absorption and emission of phonons can be achieved via inverting both supermodes. With the supermode operators (Ψ ± = (a ± m)/ √ 2) and the rotating-wave approximation (2g ma + ω b , ω b ≫ |2g ma − ω b |), the Hamiltonian can be rewritten as where ω ± = [−∆ a − ∆ m ± g ma (1 + e iΦ )]/2 and Ω ± = [−∆ a + ∆ m ± g ma (1 − e iΦ )]/2. The first two terms describe the Hamiltonian of two supermodes Ψ ± , the third term describes the free energy of the phonon mode, and the fourth term denotes the absorption and emission of phonons between two nondegenerate supermodes Ψ ± . Similar to the optical laser produced by a two-level system, the energy level difference between two supermodes matches the phonon frequency [53]. The fifth term includes a dissipative (or coherent) magnon-photon coupling and the last term describes the interaction between supermodes Ψ ± and the pump field. Using the standard procedures, mechanical gain can be obtained as where The first term is proportional to the population − inversion of two supermodes, while the second term is derived from the optical pump applied to the supermodes. The stimulated emitted phonon number can be defined as [36] We consider a set typical accessible parameters: Hz, P in = 1 mW, and g mb /2π = 9.9 mHz [23, 28].

EP-like in model
Benefits from the cavity Lenz's law, the tunable dissipative magnon-photon coupling can be obtained. The phenomenon of energy level attraction caused by non-Hermitian dissipative coupling can lead to the existence of pure real eigenvalue of the non-Hermitian Hamiltonian, which means that it is possible to find out the EP. The expression of the eigenvalues the supermode Ψ ± can be written as  where and the real and imaginary parts of the eigenvalues represent the frequencies and linewidths of the coupling mode respectively. As shown in figure 2, the coupled modes frequencies ∆ ± and linewidths γ ± are plotted as a function of the detuning parameter ∆. For coherent coupling Φ = 0, the real parts of two eigenvalues of the coupling modes ∆ ± are separated, while the imaginary parts γ ± coalesce in figure 2(a). By considering dissipative coupling Φ = π, the EP between the photon and magnon modes can be observed in figure 2 The real parts of the eigenvalues of two coupled supermodes are coalescent, while the imaginary parts are separated when ∆ < 0.5ω b . It is worth noting that the real and imaginary parts of the eigenvalues can simultaneously coalesce at ∆ = 0.5ω b . By increasing the loss rate κ/2π = 5 MHz, the imaginary parts γ ± are far away from 0 for the coherent coupling regime in figure 2(c). For the larger κ, the real parts of the eigenvalues remain constant while their imaginary parts shift downward for the dissipative coupling regime in figure 2(d). According to the eigenfrequencies of supermodes, the EP can be obtained when ∆ = √ g 2 ma − κ 2 (please see appendix C). Compared with EP in PT -symmetric regime, the dissipative magnon-photon coupling induced EP does not require the introduction of a gain medium and avoids the strict restriction of gain loss balance, which can be applied to a wide range of system parameters.
In figure 3(a), both peak values of the steady-state mean photon and magnon numbers are at ∆ = 0.5ω b in coherent magnon-photon coupling regime for ∆ a = ∆ m = −∆. In contrast, the peak values vanish in the dissipative magnon-photon coupling regime in figure 3(b). To illustrate the superiority of the dissipative magnon-photon coupling, the steady-state mean photon and magnon numbers is shown in figure 3(c). Compared with coherent magnon-photon coupling, the mean photon and magnon numbers are considerably raised around EP for the dissipative coupling regime. The enhancing magnon number enhances the magnetostrictive force, hence the steady-state mechanical displacement can be effectively changed. The mechanical displacement amplification factor is shown in figure 3(d) for different κ. The nonlinear phenomenon at EP induced by dissipative coupling implies that the capacity of preparing phonon laser can be considerably enhanced. Compared with the coherent coupling regime, the dissipative coupling can radically change the properties of photons, magnons, and phonons. Meanwhile, a significant nonlinear relation appears between the mechanical amplification factor and the detuning parameter.

Mechanical gain and threshold power
The greater the mechanical gain, the more stimulated emitted phonons, thus the engineering phonon laser is quantized by the mechanical gain G. The phonon laser can be generated when G > 1. The supermodes of the two-level structure formed by photon and magnon modes lead to the emergence of phonon laser with the coherent coupling when ∆ a = ∆ m = −∆. In the dissipative coupling regime, the merging of the intrinsic energy levels of the supermodes leads to the disappearance of the phonon laser. Figure 4(a) shows the mechanical gain that arises from the energy transition from a higher energy level to a lower one accompanied by coherent amplification of stimulated emission phonons for coherent coupling. However, the supermodes composed of photon and magnon modes merge for dissipative coupling, thus the two-level structure generating the phonon laser is destroyed, which leads to the mechanical gain G < 0. Figure 4(b) shows the variation of mechanical gain with detuning parameter in different coupling regimes. The phonon laser can appear near EP induced by dissipative coupling, and there is a slight offset caused by κ. In the dissipative coupling regime, the original two-level structure is destroyed, that is, ω + − ω − = 0, but the coefficients Ω + and Ω − produce a difference value of mechanical resonator frequency, that is, Ω + − Ω − = ω b . Therefore, phonon lasers can still be prepared in dissipative coupled systems. Moreover, the EP can be modulated by g ma in figure 4(c). Then we observe that the position of achieving mechanical gain varies with the detuning parameter for different g ma . According to figure 4(d), the EP can be found in the vicinity of g ma = 0.5ω b , and there is a slight offset caused by κ. For a larger loss rate, the offset of EP corresponding to excellent mechanical gain is more remarkable, which is consistent with figure 2. Furthermore, the area where the phonon laser is generated has also significantly increased.
The considerable steady-state mechanical displacement implies that the ultrastrong phonon laser can be prepared at EP. Then we focus on EP to discuss the realization of phonon laser. In figures 5(a) and (b), we show that the logarithm of mechanical gain lg(G) varies with the detuning parameter and the magnon-photon coupling for different κ. As shown in figure 5(a), the phonon laser can be obtained in the area between the two black contour lines. The peak represented by the red area is near ∆ = g ma , where there is a slight offset caused by κ and ∆ m . For the larger κ, the area generating phonon laser becomes larger significantly, and the peak represented by the red area also shifts upward in figure 5(b). Figure 5(c) shows that the area generating phonon laser can be not only widened but also the position of the peak value is shifted to the left by increasing κ. The phonon laser is immune to the loss rates of the photon and magnon modes even if there is a large κ. In figure 5(d), the peak value of the mechanical gain occurs around ∆∆ m = 0.25ω 2 b , and there is a small offset induced by κ. It is worth noting that the EP appears at ∆ = ∆ m ≈ 0.495ω b , namely, ∆ = √ g 2 ma − κ 2 . For increasing P in , the mechanical gain near the EP increases significantly. In figure 6(a), we show the variation of mechanical gain with P in for different κ a(m) . The increasing κ a results in the position of EP being further away from ∆ = 0.5ω b , thus the threshold power increases with the increase of κ a . Then we explore the threshold power generating phonon laser around the EP. Figure 6(b) shows the variation of mechanical gain with P in for different g ma . Combined with EP in figure 2, the excellent mechanical gain appears at g ma ≈ 0.504ω b , where the ultralow threshold power is close to 1.5 µW. Thus, extremely strong mechanical gain can be prepared, which possesses ultralow threshold power. The threshold power of the phonon laser under these conditions is simplified as . It is worth mentioning that our simplification is effective, that is, the above simplified threshold power is consistent with figures 6(a) and (b). |β| is close to 0 at EP, so the threshold power of the phonon laser also approaches zero. The EP induced by dissipative coupling is flexibly tunable compared to PT -symmetric regime, and the ultralow threshold power required to achieve the phonon laser can still be guaranteed in the appropriate parameters. To verify that the ultralow threshold phonon laser can be achieved even in the case of larger loss rates κ a and κ m , we investigate the influence of the loss rates κ a(m) on the threshold power as shown in figure 6(c). By numerical simulating computing (setting G = 1 in equation (5)), the ultralow threshold power can appear in the parameter range of the lower loss rates for g ma = 0.5ω b , and the threshold power increases exponentially with the increase of the loss rates. For larger loss rates, ultralow threshold phonon laser can still be achieved by appropriately adjusting the coupling strength g ma . For g ma = √ (0.5ω b ) 2 + nκ 2 (n ⩾ 0), the   optimal phonon laser will appear at κ a (m) = √ nκ according to the offset EP. Similarly, different loss rates will result in changes in the optimal ultralow threshold phonon laser. Figure 6(d) indicates the influence of the threshold power of the phonon laser with coupling strength g ma for different loss rates κ a(m) . For lower loss rates, the optimal phonon laser exists around g ma = 0.5ω b . As the loss rate increases, the position of the optimal phonon laser shifts to the right as the EP is offset. Our numerical solutions are in agreement with the analytical solutions for both the optimal phonon laser and the threshold power.

Conclusion
In conclusion, we have investigated how to engineer a phonon laser in a non-Hermitian CMM system with the dissipative coupling regime caused by the cavity Lenz's law. We find that the EP between the damping photon and magnon modes changes the properties of photons, magnons, and phonons. The loss rates of the photon and magnon modes can offset the EP with pure real eigenvalues in the non-Hermitian system. Thus the dissipative-induced EP is flexibly tunable compared to PT -symmetric regime of the gain-loss balance. At the EP, a significant nonlinear relation appears between the mechanical amplification factor and the detuning parameter compared to coherent coupling, which leads to a dramatic enhancement of magnetostrictive force and mechanical gain. Therefore, a highly efficient phonon laser can be achieved and the ultralow threshold phonon laser can be ensured around the EP. The results show a more flexible way for the generation of EP induced by dissipative coupling, and the ultralow threshold phonon laser is immune to the loss rates of the photon and magnon modes. These concepts can be straightforwardly extended to the integrated platforms and unlock new potentials to engineer ultralow threshold acoustic (magnetic) devices.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grants Nos. 12074330 and 62201493.

Appendix A. Dissipative coupling
According to the Landau-Lifshitz-Gilbert (LLG) equation, the magnetization dynamic evolution of YIG can be described as where M t = M x (t)x + M y (t)ŷ + M 0ẑ and H t = h x (t)x + h y (t)ŷ +Ĥ 0ẑ , in which M x,y (t) and h x,y (t) are the magnetization and the radio-frequency magnetic field, respectively (M 0 are the saturation magnetization). γ is the gyromagnetic ratio, and α is assumed to be scalar for uniaxial symmetry and small oscillations, which is a damping parameter that is characteristic of the material. The damping term is an added 'damping field' that can reduce the effective magnetic field and change the torque exerted on the magnetization field, resulting in magnetic damping and subsequent energy transfer [54]. In practice, the magnetization damping can be manipulated by spin pumping, eddy currents, and incoherent scattering of magnons. For the combined case of level repulsion (K A ) and level attraction (K L ), we can obtain the relation between the microwave current j and the magnetization m as (ω − ω m + iκ m )m − iω r (K A − K L )j = 0, where κ m = αω, ω r = γM 0 (neglecting anisotropies), and h = h x + ih y = −i(K A − K L )j. According to Faraday's law, magnetization precession generates a voltage along with the RLC circuit. Adding this effect to the cavity equation (ω 2 − ω 2 a + iκ a )j + iK F ω 2 m = iωV 0 /L, where the applied (induced) voltage is V 0(F) = ∓K F Ldm/dt. Therefore, we arrive at the combined LLG and RLC equations [23,28] ( where the term K F represents the impact of dynamic magnetization m on induced microwave current j, as described by Faraday's law. Meanwhile, the term K A is derived from Ampère's law, which demonstrates that the rf magnetic field produced by microwave current j drives magnetization via a field torque. By utilizing these laws, the system achieves coherent magnon-photon coupling. The voltage V induced by Faraday's law generates an induced current, which in turn produces an additional magnetic field due to the effect of Lenz's law. The cavity Lenz effect is manifested in the K L term, which has an opposite sign to the K A term because the backaction from the induced microwave current impedes magnetization dynamics instead of driving it [23]. Therefore, dissipative magnon-photon coupling can arise from Faraday's law, Ampère's law, and Lenz's law. The competition between the K L and K A terms will lead to a net coherent or dissipative coupling. Following the standard quantization procedures and combined with the magnons (photons) annihilation operator m (a), the magnon-photon coupling for can be written as in which where K n (n = A, F, or L) is real positive, thus e iΦ = ±1. For K A > K L or K A < K L , there is coherent (Φ = 0) or dissipative (Φ = π) coupling between magnons and photons.

Appendix B. Phonon laser
The microwave cavity is aligned along the x axis, and an external magnetic field H is applied along the z axis. Using Ampère and Faraday effects, coherent magnon-photon coupling in the cavity has been extensively studied, while dissipative magnon-photon coupling based on the Lenz' effect has recently been experimentally achieved by placing the moving YIG sphere with the processional magnetization in a disk near the inner edge of the guide. By adjusting the angular position of the YIG sphere, dissipative (or coherent) magnon-photon coupling can be achieved when θ ∈ (65 . Following the standard quantization procedures and combined with the magnon (photon) annihilation operator m (a), the magnon-photon coupling can be written as H ma =hg ma (a † m + e iΦ am † ) with coherent (Φ = 0) or dissipative (Φ = π) coupling between magnons and photons [23]. In the frame rotating at the cavity drive frequency ω d and with the rotating-wave approximation, the Hamiltonian of the CMM system can be written as (h = 1) where a (m and b) is the annihilation operator of the photon (magnon and phonon) mode with resonance frequency ω a (ω m and ω b ). ∆ a = ω d − ω a (∆ m = ω d − ω m ) is the detuning between the driving laser frequencies and the photon (magnon) mode. The first (second and third) term of the Hamiltonian describes the free Hamiltonian of the photon (magnon and phonon) mode. The fourth term describes linear interaction Hamiltonian between magnon and photon modes with interaction strength g ma . The fifth term denotes the interaction between the magnon and phonon modes via nonlinear magnetostrictive force. The microwave control field applied to the photon mode is considered by the last term with the amplitude E d = √ 2κ a P in /hω d , in which κ a is the loss rate of the photon mode, and P in is driving power of control field. Heisenberg Langevin equations of motion in the CMM system are obtained from the coherent evolution of total Hamiltonian (under the rotating-wave approximation) after including the system dissipations aṡ Applying the semiclassical approximation, all operators are reduced to their expectation values. The noise has been ignored due to the fact that we consider the mean value of all the operators [55,56]. κ a (κ m and κ b ) is the loss rate of photon (magnon and phonon) mode. Under strong driving of the photon mode, the noise terms can be safely ignored if the mean-number behaviors are only interested. The steady-state mean values of the system can be written as The steady-state mechanical displacement x s is proportional to By considering the supermode operators Ψ ± = (a ± m)/ √ 2, the Hamiltonian of system can be written as By diagonalizing the matrix in equation (B5), the eigenfrequencies of supermodes Ψ ± are where which clearly depends on the coupling phase Φ. To describe the enhanced steady-state mechanical displacement in the dissipative coupling regime compared to the coherent coupling, we define the mechanical displacement amplification factor as In close analogy to an optical laser, the absorption and emission of phonons can be achieved via inverting both supermodes in the hybrid system. With the supermode operators Ψ ± , the Hamiltonian of the system can be rewritten as where , under the rotating-wave approximation (2g ma + ω b , ω b ≫ |2g ma − ω b |). The first two terms describe the Hamiltonian of two supermodes Ψ ± , respectively. The third term describes the free energy of the phonon mode. The forth term denotes the absorption and emission of phonons between two nondegenerate supermodes Ψ ± . Similar to the optical laser produced by a two-level atomic ensemble, the energy level difference between two supermodes matches the frequency of the phonon mode. The fifth term includes a dissipative (or coherent) magnon-photon coupling. The last term describes the interaction between supermodes Ψ ± and the driving field.
In the supermode picture, the equations of motion of the system can be given aṡ where κ ′ = (κ a + κ m )/2. The ladder operator, and population − inversion operator of two supermodes is defined as Using the standard procedures, we can obtain where β = (g 2 ma cosΦ − ∆ m ∆ a + κ ′2 ) + i[g 2 ma sinΦ − (∆ m + ∆ a )κ ′ ] in the case of g mb ≪ g ma . The mechanical gain is defined as the ratio between G ′ and the mechanical loss rate κ b (i.e. G = G ′ /κ b ). In equation (B13), the first term is proportional to the population − inversion of two supermodes, while the second term is derived from the optical drive applied to the supermodes.

Appendix C. EP and threshold power
With dissipative coupling, a real eigenvalue in the non-Hermitian system ensures the appearance of EP, which can be adjusted by loss rate κ. According to equation (B6), we can obtain the condition of the real eigenvalue as follows In order to more clearly show the relationship between EP position and system parameters, we consider the conditions ∆ a − ∆ m = 2∆ ≫ κ a − κ m and κ a + κ m ≫ κ a − κ m . The expressions for the EP position can be written as where Φ = π. In the previous results, the introduction of loss rates of photon and magnon modes causes the displacement of EP. Thus EP is located at ∆ = g ma in the absence of loss rate κ. The larger the loss rate, the larger the displacement of EP, as shown in figure 7. The EP is usually accompanied by strong nonlinear phenomena. Benefits to this nonlinearity, the coherent amplification effect of stimulated radiation in mechanical breathing modes may be enhanced. In consideration of g mb ≪ ∆, we have δn = 2|E d | 2 g ma ∆ m |β| −2 . In the dissipative magnon-photon coupling regime, we consider the parameters ∆ a = −∆ m = −∆. In this case, the second term in the expression of the mechanical gain tends to zero, so the mechanical gain can be simplified as G ′ = g 2 mb κ ′ δn/(8κ ′2 + 2ω 2 b ) (for Φ = π). To obtain the optimal phonon laser, we can consider the condition of ∆ m ∆ a = κ ′2 − g 2 ma . At this point, the population − inversion operator has a fairly large value, which ensures the emergence of ultralow threshold phonon laser. Therefore, the threshold power of the phonon laser under these conditions is defined as 2g ma ∆ m κ ′ κ a g 2 mb . (C3) From the formula, we can observe that the ultralow threshold phonon laser can be obtained by setting |β| → 0, which is consistent with the previous analysis of the EP.