Phonon laser in a cavity magnomechanical system

Using phonons to simulate an optical two-level laser action has been the focus of research. We theoretically study phonon laser in a cavity magnomechanical system, which consist of a microwave cavity, a sphere of magnetic material and a uniform external bias magnetic field. This system can realize the phonon-magnon coupling and the cavity photon-magnon coupling via magnetostrictive interaction and magnetic dipole interaction respectively, the magnons are driven directly by a strong microwave field simultaneously. Frist, the intensity of driving magnetic field which can reach the threshold condition of phonon laser is given. Then, we demonstrate that the adjustable external magnetic field can be used as a good control method to the phonon laser. Compared with phonon laser in optomechanical systems, our scheme brings a new degree of freedom of manipulation. Finally, with the experimentally feasible parameters, threshold power in our scheme is close to the case of optomechanical systems. Our study may inspire the field of magnetically controlled phonon lasers.

In this work, we study a cavity magnomechanical system, where a YIG sphere is placed in a microwave cavity, there is an uniform external bias magnetic field H in the vertical direction simultaneously. The magnetostrictive (radiation pressure like) leads to the magnon-phonon coupling, and the photons and magnons are coupled via magnetic dipole interaction. It is worth noting that unlike optical pump in the traditional cavity optomechanical system, we introduce a magnetic drive field to to realize phonon laser. Furthermore, the magnomechanical interaction which is quite weak in experiments can be enhanced by the gain of magnon mode. We found that the adjusted applied magnetic field H can be regarded as a good control method to the phonon laser. Compared with phonon laser in optomechanical systems, our scheme brings an additional degree of freedom of manipulation. The threshold conditions of drive magnetic field intensity for phonon laser is obtained. We can make our system reach the threshold condition by enhancing drive magnetic field. And the threshold power required can be below µ 10 W within the experimental allowable range of parameters. According to recent work, the threshold power in cavity optomechanical system is generally about µ 7 W 38-40 .

Results
Model and dynamical equations. In Fig. 1, a highly polished single-crystal YIG sphere (a sphere of mm 1 in diameter is considered in 30 ) is placed in a microwave cavity. At the same time, an uniform external bias magnetic field H is introduced in the vertical direction, which establish the magnon-photon coupling 27,30 , and the rate of coupling can be tuned by the position of the sphere. There are three modes in this system: cavity photon mode, magnon mode and phonon mode. In addition, the external bias magnetic field H is created by a high precision tunable electromagnet, and the adjusting range of bias magnetic field H is between 0 and T 1 30 . Here, the coupling between magnons and phonons is generated by magnetostrictive interaction (the derivation of the relevant Hamiltonian can be found in 22 ). The magnon excitation causes magnetization to change, which leads to micro deformation of YIG sphere slightly. Based on it, we can consider the sphere as an excellent mechanical resonator, so the phonon mode of the sphere can be introduced. Here, a microwave source is utilized to directly drive the magnon mode and it can enhance the magnomechanical coupling 21,30 . It is worth mentioning that the directions of the applied magnetic field H, the drive magnetic field, and the magnetic field of the cavity mode are perpendicular to each other. So we can adjust only one of them without worrying about the impact on the rest. Furthermore, we assume that the size of YIG sphere is much smaller than the wavelength of the cavity. Accordingly, the interaction between cavity photons and phonons can be neglected. The total Hamiltonian of our system reads (  and ω † m m m denote the cavity photon mode and magnon mode, respectively. The third term describes the mechanical mode. ω a , ω m and ω b denote the resonance frequencies of the cavity, magnon, and mechanical modes. The frequency of uniform magnon mode in the YIG sphere is g is the gyromagnetic ratio). The annihilation (creation) operators of these modes are † a a ( ), † m m ( ) and † b b ( ), respectively. Schematic diagram of our system. The YIG sphere is placed in the maximum magnetic field of the microwave cavity mode. Applying an external magnetic field H along z-direction makes YIG sphere produce uniform magnon mode. The enlarged YIG sphere on the right shows the magnetization of magnon (black down arrows), which leds to the micro-deformations on the surface of spheres (y-direction). Conversely, the deformation can also lead to changes in magnetization of magnetons.
H int is the interaction Hamiltonian of our system, which consists of the Hamiltonian of photon-magnon coupling and the phonon-magnon interaction. g ma and g mb are the coupling rates of the magnon-photon interaction and the magnon-phonon interaction, respectively. And g ma can be tuned by external magnetic field H or the position of the YIG sphere inside the cavity. Finally, H d is the driving field of magnon mode, as shown in 30  By making a frame rotating at the frequency ω d , the total Hamiltonian of the system (under a rotating-wave approximation) is given as The Heisenberg-Langevin equations of the system are written as where γ b is the loss of mechanical mode, a m , int i nt and ξ no are input noise operators of cavity, magnon and mechanical modes, respectively. κ m and κ a are the decay rates of magnon and microwave cavity modes. Like other work 38,40 , under the strong driven of the magnon mode, we have  m 1 at the steady-state. The cavity-magnon beam splitter interaction can lead to the large amplitude of cavity field, i.e.,  a 1. So we can safely ignore the quantum noise terms if the mean-number behaviors are only interested. Here, the semi-classical Langevin equations of motion are valid. In other words, we can rewrite all operators as their respective expectation values. Then making the left-hand side equal to 0, the steady-state mean values of the system read According to the feasible experimental parameters ( < g Hz 1 mb ), the distribution of steady-state magnon number. Here, we give the specific values of the parameters used in this paper 22 . ω π ω π = = . GHz , and the loss of mechanical modes is time average of energy per unit volume, c is the velocity of an electromagnetic wave in vacuum, A is the maximum cross-sectional area of YIG sphere. Fig. 2 shows a plot of the steady-state magnon number m s 2 versus the drive magnetic field B 0 . The number of magnons increases exponentially with the increase of B 0 , which represents significant nonlinearity. The corresponding B 0 is much weak relative to the external magnetic field H. Furthermore, the results under different losses of supermode γ are also given. It can be seen that the smaller the dissipation, the faster m s 2 increases.
Magnetic field-based control of phonon laser action. Here, we study the influence of a magnetic field (including the external bias magnetic field H and the drive magnetic field B 0 ) on the phonon laser. Figure 3  , the frequencies of the magnon mode and drive magnetic field can almost reache the condition of resonance (ω ω  m d ). Therefore, the influence of drive magnetic field is the strongest, and the magnetostrictive effect is strengthened effectively 21 , the vibration of the YIG sphere is enhanced due to the magnetostrictive effect simultaneously, which eventually leads to the increase of emitted phonons. Figure 4 shows an equivalent coupling model, it illustrates the coupling between three modes and the effects of magnetic fields.
www.nature.com/scientificreports www.nature.com/scientificreports/ We find that the distribution of the stimulated emitted phonons are similar to Lorentzian shape, so H can be an additional degree of freedom to control phonon laser. And that phenomenon similar to a switch of a phonon laser can be obtained by adjusting H without changing other parameters. Especially, the width of the windows www.nature.com/scientificreports www.nature.com/scientificreports/ almost does not change with the loss of magnon mode κ m . Moreover, it can be seen that the number of phonons increase with decreasing κ m .
A plot of the phonon number N b versus the drive magnetic field B 0 is shown in Fig. 5, the stimulated emitted phonons are enhanced by input drive magnetic field. Here, threshold condition of phonon laser is Physically, the increase of drive magnetic field B 0 leads to the enhancement of magnon mode. Thus the magnetostrictive effect is also strengthened effectively 21 , and the number of phonons is eventually increased.
In addition, we find the loss of supermode has significant impact on the threshold power P th . For γ = MHz 1 and γ = MHz 4 , we have µ . P 0 57 th W and µ . P 9 79 th W, respectively. It is worth mentioning that P th is calculated in the range of parameters that can be achieved by experiments, and it is not much different from the P th obtained in the cavity optomechanics system, and the threshold power in cavity optomechanical system is generally about µ 7 W in [38][39][40][41]56 .

Discussion
In summary, we have studied theoretically phonon laser in a cavity magnomechanical system, the interaction between magnon mode and mechanical mode can be achieved by magnetostrictive force (radiation pressure like).
Our results have shown that a window with a large number of phonons can be obtained by adjusting the intensity of external magnetic field H. And the width of the window is about . mT 0 5 . Compared with phonon laser in optomechanical systems, our scheme brings a new degree of freedom of manipulation.
We first use the way of resonantly driving the YIG sphere, which can be realized by a microwave source. A threshold condition of drive magnetic field intensity for phonon laser is given. Then, we find the system can reach the threshold power and produce phonon laser by increasing the intensity of drive magnetic field B 0 . With the experimentally feasible parameters, threshold power P th in our system is close to the threshold power of phonon laser in optomechanical systems which are mature in theory and experiments. Finally, with the advance in quantum theory and technology, we hope that the phonon laser in cavity magnomechanical systems will be accessible.
Methods the mechanical gain of system. A phonon laser can be obtained theoretically with two coupled whispering-gallery-mode microtoroid resonators via inversion of the two supermodes 38,57 , where the threshold power of phonon laser is μ P 7 th W. Similarly, our system has two supermodes corresponding to the ground and excited states, respectively, the phonon mode can realize energy level transition. Using magnetic pumping of the upper level, the phenomenon of stimulated emission of phonon appears, then coherent phonon lasing can be realized. Therefore, we introduce supermode operators  = ± ± † R a m ( ) 2 to rewrite the Hamiltonian H 0 and H d of the system, i.e.,   . Applying the rotating-wave approximation, H int is rewritten as int sm mb , is ladder operator. Equation (7) represents the absorption and emission of phonons. In general, the introduction of supermode operators ± R means that the magnon mode and the optical mode have the same resonant frequency. After changing the Hamiltonian into the supermode picture, the equations of motion read  where ∆ = ∆ + ∆ m a , then Eq. (9) is substituted into Eq. (8), the result about b can be obtained , Because this paper mainly studies the phonon laser generated by the system, we are only interested in mechanical gain G, which indicates the mechanical gain of the system. Therefore, only the specific expression of G is obtained. The non-negative G leads to the decrease of γ eff , where γ γ = − G eff b . That makes the instability of the mechanical oscillator at γ < 0 eff . Here, γ eff is effective damping rate of the mechanical mode. This problem has been analyzed and discussed in 38,40,58 from both theoretical and experimental perspectives. the phonon number of system. The stimulated emitted phonon number can be calculated 39,40 , i.e., . B th 0, is the drive magnetic field required to achieve the threshold condition of the phonon laser in our system. Finally, according to the expression given earlier  µ = P B Ac