Phase-Controlled Phonon Laser

TL;DR

This paper proposes a phase-controlled phonon laser with ultralow threshold powered by tunable optical amplifiers, enabling enhanced photon-phonon interactions and applications in quantum phononics and sensitive motion detection.

Contribution

It introduces a novel phase-controlled phonon laser using tunable amplifiers in a coupled-cavity system, achieving giant enhancement of interactions and low-power operation.

Findings

Achieved giant enhancement of photon-phonon interactions.

Enabled single-photon optomechanics and low-power phonon lasing.

Potential applications in quantum phononics and ultrasensitive detection.

Abstract

A phase-controlled ultralow-threshold phonon laser is proposed by using tunable optical amplifiers in coupled-cavity-optomechanical system. Giant enhancement of coherent photon-phonon interactions is achieved by engineering the strengths and phases of external parametric driving. This in turn enables single-photon optomechanics and low-power phonon lasing, opening up novel prospects for applications, e.g. quantum phononics and ultrasensitive motion detection.

Equations

$$H=$$

$$H_c=$$

$$H_m=$$

$$a_j$$

$$a_{sj}=\cosh \left(r\right)A_j-e^{-i\Phi }\sinh \left(r\right)A_k^{†},$$

$$H_{int}$$

$$G_1$$

$$G_2$$

$$r$$

$$\frac{J^{\prime }}{2J}$$

$$G_{p12}$$

$$\mathcal{G}=\frac{{\left|G_{p12}\right|}^2\Delta N\kappa }{{\left(W_1-W_2-{\omega }_m\right)}^2+{\left(\kappa /2\right)}^2},$$

$$P_{th}\approx \frac{{\gamma }_m{W}_1\left[{\left(W_1-W_2-{\omega }_m\right)}^2+{\left(\kappa /2\right)}^2\right]}{{\left|G_{p12}\right|}^2},$$

$$H=$$

$$a_j=\cosh \left(r_{dj}\right)a_{sj}-e^{-i{\Phi }_{dj}}\sinh \left(r_{dj}\right)a_{sj}^{†},$$

$$r_{dj}=1/4\ln \left[\left({\Delta }_j+2{\Lambda }_j\right)/\left({\Delta }_j-2{\Lambda }_j\right)\right],$$

$$H$$

$${\omega }_{sj}$$

$$g_{s2}$$

$$g_{p2}$$

$${\lambda }_1$$

$${\lambda }_2$$

$$F$$

$$C$$

$$f_1\equiv \left|\frac{{\lambda }_2\left({\omega }_{s1}-{\omega }_{s2}\right)}{{\lambda }_1\left({\omega }_{s1}+{\omega }_{s2}\right)}\right|\gg 1,$$

$$a_{sj}=\cosh \left(r\right)A_j-e^{-i\Phi }\sinh \left(r\right)A_k^{†}\left(j\ne k\right),$$

$$r=\frac{1}{4}\ln \left[\left({\omega }_{s1}+{\omega }_{s2}+\left|J^{\prime }\right|\right)/\left({\omega }_{s1}+{\omega }_{s2}-\left|J^{\prime }\right|\right)\right],$$

$$f_2\equiv \left|{\omega }_{s1}+{\omega }_{s2}\right|-\left|J^{\prime }\right|>0.$$

$$H$$

$$W_1$$

$$W_2$$

$$G_1$$

$$G_2$$

$$G_{12}$$

$$G_{11}$$

$$G_{22}$$

$$G_{p12}$$

$$F^{\prime }$$

Full text

Introduction.- As a promising platform to study fascinating macroscopic quantum phenomena Chen2013 , cavity optomechanics Schwab2012 ; Vahala2008 has received tremendous attentions in recent years. All kinds of optomechanical couplings and applications Marquardt2014 have been opened up due to remarkable experimental advances in e.g., mechanical ground-state cooling Teufel2011 ; Wang2009 , optomechanical non-reciprocity Dong2015 ; Shen2016 ; Kim2015 , optomechanically induced transparency Weis2010 ; Safavi-Naeini2011 , nonclassical state preparation Purdy2013 ; Naeini2013 , coherent state transfer between light and sound Fiore2011 ; Zhou2013 , and various phonon-mediated hybrid devices Dong2012 . To extend more applications, on the one hand, the unique regime of single-photon quantum optomechanics Rabl2011 ; Nunnenkamp2011 , however, is still pursued in current experimental efforts; on the other hand, we need to realize convenient tuning, especially the switching between different optomechanical couplings.

To realize single-photon coupling, many theoretical schemes have been proposed based on, for examples, undriven two-cavity set-ups Bhattacharya2008 , optomechanical arrays Xuereb2012 , Josephson effect Heikkil=0000E42014 , and the transient scheme Xu2015 . Very recently, the parametric drive has been used to enhance the nonlinear coupling Xin2015 ; Clerk2016 ; Li2016 in optomechanical systems. By exploiting coupled cavities, many applications have been studied, such as single-photon generation Bamba99 , steady-state entanglement shen2011 , thermal phonon squeezing Mahboob2014 , and phonon laser He2016 . In the study of phononic devices Li2012 ; LaHaye2009 ; Hatanaka2013 , the phonon laser Vahala2009 ; Khurgin2012 plays a key role in integrating coherent phonon sources, detectors, and waveguides Eichenfeld2009 . Phonon lasing have been demonstrated in the electromechanical resonator Mahboob2013 , the nanomechanical resonator Cohen2015 , the vertical cavity structure Maryam2013 , and the compound microcavity system Grudinin2010 , and some schemes are also proposed to realize phonon laser in the quantum-dot system Khaetskii2013 ; Kabuss2012 . In particular, the ultralow-threshold phonon laser Jing2014 ; Wang2017 still gives rise to the broad interest and remains largely unplored.

In this paper, we present a scheme for both enhancing optomechanical couplings into the single-photon strong-coupling regime and realizing the switching between different optomechanical interactions using optical parametric amplifiers (OPAs). The key idea is to put two OPAs into both the auxiliary cavity and the optomechanical system, which leads to the squeezing of transformational optical modes. Due to the squeezing, we can obtain exponentially enhanced radiation-pressure, parametric amplification, and three-mode optomechanical couplings, which are controlled by the phase difference from the two OPAs. As one of applications, we study a phase-controlled ultralow-threshold phonon laser in detail. In addition, we consider the noise of the squeezed modes, which can be suppressed greatly via dissipative squeezing or an additional optical mode. With current experimentally accessible parameters, our scheme should be feasible to study quantum optomechanics.

Model.- We consider an optomechanical system with two coupled cavities, and each cavity contains a driven nonlinear optical medium for OPA, as shown in Fig. 1(a), which can be described by the Hamiltonian ($\hbar=1$)

[TABLE]

where $a_{j}$ and $b$ are the annihilation operators for the $j$th ($j=1,~{}2$) cavity mode with frequency $\omega_{j}$ and the mechanical mode with frequency $\omega_{m}$, respectively, and $J$ is the photon-hopping interaction strength between two cavities. $H_{c}$ describes the optical modes containing two different OPAs. $H_{m}$ describes the optomechanical system associated with the 2nd cavity, in which $g_{0}$ is the radiation-pressure optomechanical coupling strength.

Phase-controlled optomechanical systems.- For simplicity, we take the two parametric driving frequencies satisfying $\omega_{d1}=\omega_{d2}=\omega_{d}$; then to diagonalize $H_{c}$, we define the squeezing operator $a_{sj}$ via the transformation Xin2015 :

[TABLE]

where $r_{dj}=\left(1/4\right)\ln\left[\left(\Delta_{j}+2\varLambda_{j}\right)/\left(\Delta_{j}-2\varLambda_{j}\right)\right]$ and $\Delta_{j}=\omega_{j}-\omega_{d}/2$, which requires $\left|\Delta_{j}\right|>\left|2\varLambda_{j}\right|$ to avoid the system instability. Due to the photon-hopping, the transformational optical modes are coupled though the coherent term $\lambda_{1}a_{s1}a_{s2}^{\dagger}+\mathrm{H.c.}$ and squeezing term $\lambda_{2}a_{s1}a_{s2}+\mathrm{H.c.}$, in which $\lambda_{1,2}$ is the effective photon-hopping.

For convenient discussion, we define the effective coupling ratio between the squeezing and coherent terms $f_{1}\equiv\left|\frac{\lambda_{2}\left(\omega_{s1}-\omega_{s2}\right)}{\lambda_{1}\left(\omega_{s1}+\omega_{s2}\right)}\right|$, where $\omega_{s1,2}$ is the frequency of the transformational optical mode $a_{s1,2}$. With the rotating wave approximation, it is obvious that we can reserve the squeezing (coherent) term for $f_{1}\gg 1$ ($f_{1}\ll 1$), which can be used to realize different optomechanical interactions.

When we have $f_{1}\gg 1$, the squeezing term can be used to enhance optomechanical coupling strength Li2016 , and we can further diagonalize the two-mode squeezing terms via the squeezing transformation ($j\neq k$)

[TABLE]

with $A_{j}$ is the annihilation operator for the supermode $j$ with frequency $W_{j\left(k\neq j\right)}=\omega_{sj}\cosh^{2}\left(r\right)+\omega_{sk}\sinh^{2}\left(r\right)-\frac{\left|J^{\prime}\right|\sinh\left(2r\right)}{2}$. The effective interaction Hamiltonian can be rewritten as

[TABLE]

which describes the typical optomechanical forms including the radiation-pressure, parametric amplification, and three-mode optomechanical couplings. Here $G_{j}$ is the effective coupling of optomechanical systems, where

[TABLE]

with

[TABLE]

depending on the phase difference $\Delta\Phi=\Phi_{d1}-\Phi_{d2}$. As illustrated in Fig. 1(b), the phase difference $\Delta\Phi$ determines the effective optomechanical couplings. As a comparison with the previous proposals Xin2015 ; Li2016 , the coupling $G_{j}$ is greatly enhanced as the product of enhancement from the single-mode Xin2015 and two-mode Li2016 squeezing. Here $\Phi=\mathrm{arg}\left(J^{\prime}\right)$ and the explicit expressions for the parameters $\lambda_{1,2}$, $\omega_{sj}$, $G_{jk}$, and $G_{p12}$ can be found in the Supplemental Material Zhang2017 .

In Fig. 2(a), the optomechanical coupling strengths $G_{1,2}$ are plotted with reasonable parameters, which demonstrate the significant enhancement by controlling the phase $\Delta\Phi$ and show the strong-coupling regime is achievable (i.e. $G_{1},~{}G_{2}\sim\omega_{m}>\kappa$) for $\Delta\Phi$ around the optimal $\Delta\Phi=\pi$. Because we choose the parametric pump detuning $\Delta_{1}<0$ and $\Delta_{2}>0$ , which lead to $r_{d1}<0$ and $r_{d2}>0$, the effective $\left|J^{\prime}\right|$ reaches its maximum when $\Delta\Phi=\pi$ and the minimum when $\Delta\Phi=0$. In the Fig. 2(b), we plot the dependence of supermode frequencies $\left|W_{1}\right|/\omega_{m}$, $W_{2}/\omega_{m}$ on $\Delta\Phi$. When $\Delta\Phi$ tends to [math] or $2\pi$, we have the coupling strength $G_{2}\gg G_{1}$, while the other couplings $G_{jk}$ and $G_{p12}$ can be ignored for $\left|W_{j}+W_{k}\pm\omega_{m}\right|\gg G_{jk}$ and $\left|W_{1}-W_{2}\pm\omega_{m}\right|\gg G_{p12}$. To show the enhanced coupling strengths for different driving amplitude $\varLambda_{1}$ and phase $\Delta\Phi$, we plot the equipotential lines of $f_{1}$, $G_{1}/\omega_{m}$, $G_{2}/\omega_{m}$, and $\eta=G_{1}/G_{2}$ in the Fig. 2(c). The inner region surrounded by the blue line $f_{1}=10\gg 1$ means that only the squeezing term dominates and the rotating wave approximation is appropriate. The red line $G_{2}=0.1\omega_{m}>\kappa$ and green line $\eta=\frac{1}{20}$ show that only the second optomechanical coupling reaches the strong-coupling regime. When the parameters $\varLambda_{1}$ and $\Delta\Phi$ tend to the central area, as shown by both the pink and black lines, both $G_{1}$ and $G_{2}$ can reach strong-coupling regime.

With appropriate parameters, the parametric amplification coupling forms in Eq. 6 can also been obtained when $\omega_{m}\gg G_{j}$ and $\left|W_{1}-W_{2}\pm\omega_{m}\right|\gg G_{p12}$, and meanwhile the frequency matching $\left|W_{j}+W_{k}\pm\omega_{m}\right|\approx 0$ is satisfied. The detailed discussion for the parametric amplification can be found in the Supplemental Material Zhang2017 . Compared to previous schemes that also employ the parametric interaction Liu2013 ; Lemonde2013 ; B=0000F8rkje2013 , the coefficient of parametric amplification is further improved by our coupled-cavity configuration, which can be used to generate photon-phonon pairs. Even when only one parametric driving field exists in the cavity, the optomechanical coupling can still be enhanced than no parametric driving Zhang2017 .

Phase-controlled phonon laser.- The laser term in Eq. 6 could be utilized for realizing the phonon laser if $G_{p12}$ is dominated over other coupling strengths. This interaction is a triply-resonant interaction, with the advantage that the pump and idle optical field are resonantly enhanced. When the triply-resonant frequency $W_{1}-W_{2}\approx\omega_{m}$ is matched, we have the parameter $f_{1}\ll 1$. By a similar transformation Zhang2017 with the Eq. 5, we obtain

[TABLE]

with $\Phi=\mathrm{arg}\left(J^{\prime}\right)$ and $\theta=\arctan\left[\left|J^{\prime}\right|/\left(\omega_{s2}-\omega_{s1}\right)\right]$, in which $J^{\prime}=2J\left[\cosh\left(r_{d1}\right)\cosh\left(r_{d2}\right)+\sinh\left(r_{d1}\right)\sinh\left(r_{d2}\right)e^{i\Delta\Phi}\right]$.

In Fig. 3(a), we plot the triply-resonant phonon lasing coupling strength $\left|G_{p12}\right|$ versus the phase difference $\Delta\Phi$, which can reach the strong-coupling regime $\left|G_{p12}\right|\simeq\kappa$, and there is no obvious change with the increasing of the phase difference. When the frequency matches $W_{1}-W_{2}\approx\omega_{m}$, we have $\left|G_{j}\right|/\omega_{m},~{}\left|G_{jk}/\left(W_{j}+W_{k}\pm\omega_{m}\right)\right|\ll 1$, therefore, the other coupling strengthscan be neglected.

If the effective optical cavity decay rate exceeds the mechanical dissipation rate ($\kappa\gg\gamma_{m}$), we find the mechanical gain Grudinin2010

[TABLE]

where $\Delta N=N_{+}-N_{-}\approx N_{+}$ with $N_{+}=A_{1}^{\dagger}A_{1}$ and $N_{-}=A_{2}^{\dagger}A_{2}$. The gain has a spectral bandwidth $\kappa$ and $W_{1}-W_{2}=\omega_{m}$ is corresponding to the maximum gain.

The threshold condition $\mathscr{G}=\gamma_{m}$ determines the emitted phonon number, which is shown in Fig. 3(c). The solid lines are stimulated emitted phonon number $n_{b}\left[\gamma_{m}\right]=\exp\left[2\left(\mathscr{G}-\gamma_{m}\right)/\gamma_{m}\right]$ as a function of the density $N_{+}$ for different $\Delta\Phi$. If there is no any OPA in the cavity, the emitted phonon number $n_{b}$ with the resonance $W_{1}-W_{2}=\omega_{m}$ is shown by the dashed line in Fig. 3(c). Clearly, it indicates an ultralow-threshold phonon laser by tuning the phase difference $\Delta\Phi$.

The black square points denote the threshold density $N_{+}$ for $\mathscr{G}=\gamma_{m}$ in Fig. 3(c). We know the threshold pump power as $P_{th}=N_{+}\kappa W_{1}$, and we obtain

[TABLE]

which is plotted as the function of phase difference $\Delta\Phi$ in Fig. 3(b). There are two dips, which mean an ultralow-threshold power with the near resonance $W_{1}-W_{2}\approx\omega_{m}$. The ultralow-threshold power $P_{th}$ is related to the frequency difference $W_{1}-W_{2}$ controlled by the strengths and phases of parametric driving terms. From the Fig. 3, it is noted that the threshold density $N_{+}\leq 1$ can be obtained by changing the phase difference $\Delta\Phi$. In other words, the phonon lasing is possible with an ultralow-threshold power, as low as single photon.

*Discussion.- *In the presence of a parametric drive, the noise from the optical cavity decay might also be amplified. To circumvent the amplified noise, a possible strategy is to introduce a broadband single-mode or two-mode squeezed vacuum via dissipative squeezing Xin2015 ; Clerk2016 ; Li2016 . This steady-state technique has recently been implemented experimentally Wollman2015 ; Pirkk2015 ; Lecocq2015 , and recently it has been experimentally demonstrated that squeezed light can be used to cool the motion of a macroscopic mechanical object without resolved-sideband condition Clark2016 . One can also take advantage of the tunability of the parametric drive to avoid significant perturbation of the initial photon state Clerk2016 . It is feasible to suppress the cavity noise in the experiment for realizing the optomechanical strong-coupling regime.

Conclusion.- we present a scheme for enhancing phase-controlled optomechanical couplings into the single-photon strong-coupling regime by optical squeezing. With two OPAs in two coupled optical cavities, we obtain the squeezing of transformational optical modes, which leads to exponentially enhanced optomechanical systems. The phase difference between the two driving fields on OPAs can control the enhanced radiation-pressure, parametric amplification, and three-mode optomechanical couplings. In particular, the three-mode optomechanical coupling can be used to realize a low-threshold phonon laser, and the threshold pump power is decreased greatly with the giant enhancement of mechanical gain. With current experimentally accessible parameters, our scheme should be feasible to study quantum optomechanics. This allows us to explore a number of interesting quantum optomechanics applications ranging from single-photon sources to nonclassical quantum states.

Acknowledgments. We thank Yue-Man Kuang, Xin-You Lü, Ya-Feng Jiao, and Jie-Qiao Liao for useful suggestions. This work was funded by the National Key R & D Program (Grants No. 2016YFA0301300 and No. 2016YFA0301700), the National Natural Science Foundation of China (Grants No. 11474271, No. 11674305, and No. 61505195), and the China Postdoctoral Science Foundation (No. 2016M602013). H. J. is supported by the National Natural Science Foundation of China (Grants No. 11474087 and No. 11422437).

S-1 effective Hamiltonian

From the main text, we know that the Hamiltonian of the system can be written as

[TABLE]

For simplicity, we take the two parametric driving frequencies satisfying $\omega_{d1}=\omega_{d2}=\omega_{d}$. In the interaction picture $H_{0}=\frac{\omega_{d}}{2}\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\right)$, the Hamiltonian of the system can be written as

[TABLE]

where the detuning $\Delta_{j}=\omega_{j}-\omega_{d}/2$.

To diagonalize the $H_{c}$, we introduce a squeezing transformation Xin2015

[TABLE]

where

[TABLE]

which requires $\left|\Delta_{j}\right|>\left|2\varLambda_{j}\right|$ to avoid the system instable.

The Hamiltonian of the system can be changed into

[TABLE]

where

[TABLE]

S-2 phase-controlled optomechanical systems with $f_{1}\gg 1$

With the rotating approximation, we can eliminate the term $\lambda_{1}a_{s1}a_{s2}^{\dagger}+\mathrm{H.c.}$ when we have

[TABLE]

which means that the effective interaction from squeezing terms is much larger. It is obvious that only the squeezing terms can be reserved to ehance optomechanical coupling strength Li2016 .

Similar to the above, we can diagonalize the two-mode squeezing via the squeezing transformation Li2016

[TABLE]

where

[TABLE]

in which $J^{\prime}=2J\lambda_{2}$ and $\Phi=\mathrm{arg}\left(J^{\prime}\right)$. To avoid the system instability, we need

[TABLE]

This leads to the following Hamiltonian

[TABLE]

where

[TABLE]

We notice that $F^{\prime}-F$ and $C+C^{\prime}$ are only the displaced term and a constant, respectively, which can be neglected in the optomechanical system.

We have discussed the radiation-pressure optomechanical coupling in the main text. With the appropriate parameters, the parametric amplification coupling forms can also been obtained.

To obtain the effective Hamiltonian , we notice that there are the followling conditions: (a) $f_{1}\gg 1$ (rotating wave approximation); (b) $\left|\Delta_{j}\right|>\left|2\varLambda_{j}\right|$ and $f_{2}>0$ (stable conditions). Naturally, the system parameters are chosen to satisfy $\left|\Delta_{j}\right|>\left|2\varLambda_{j}\right|$. Equipotential lines $f_{1}=10$ (blue line) and $f_{2}=0$ (red line) versus $\varLambda_{1}$ and $\Delta\Phi$ are plotted in Fig. S1(a), and the area between the blue and red lines fully satisfies the above conditions. In Fig. S1(b), we plot the coupling $G_{2}/\omega_{m}$ (red-dashed line) and $G_{12}/\omega_{m}$ (blue-solid line) versus phase difference $\Delta\Phi$, and we can reach the strong-coupling regime when $\Delta\Phi=\pi$, however, which is much smaller than the mechanical frequency $\omega_{m}$. The supermode frequencies $\left|W_{1}\right|/\omega_{m}$ (red-solid line) and $W_{2}/\omega_{m}$ (blue-dashed line) are shown in Fig. S1(c), which means that we have $\omega_{m}\gg G_{j}$, $\left|2W_{j}\pm\omega_{m}\right|\gg\left|G_{jj}\right|$ and $\left|W_{1}-W_{2}\pm\omega_{m}\right|\gg\left|G_{p12}\right|$. While we find the frequency matching (red square points) $\left|W_{1}+W_{2}-\omega_{m}\right|\approx 0$ inFig. S1(d), which leads that only the term $G_{12}$ can be reserved. It is obvious that we can also obtain the parametric amplification coupling forms when $\left|2W_{j}\pm\omega_{m}\right|\approx 0$ with appropriate parameters.

When only one parametric driving field exists in the cavity, we can still realize enhanced optomechanical coupling without phase control. If the parametric driving field exists in the second cavity, it means $\Lambda_{1}=0$ $\left(r_{d1}=0\right)$, and all coupling forms are same to the above. The phase difference does not appear in the expression of effective coupling $\left|J^{\prime}\right|=2J\sinh\left(r_{d2}\right)$, which means the coupling parameters can not be tuned by the phase difference. It needs a stronger photon-hopping interaction $J$ because of no product factor $\sinh\left(r_{d1}\right)$ or $\cosh\left(r_{d1}\right)$ in the effective $\left|J^{\prime}\right|$. We can still realize the controlled optomechanical coupling by tuning $\Lambda_{2}$, $\Delta_{j}$ and $J$. If the parametric driving field exists in the first cavity, we have $\Lambda_{2}=0$ $\left(r_{d2}=0\right)$. The coupling strength becomes $G_{1}=g_{0}\sinh^{2}\left(r\right)$ and $G_{2}=g_{0}\cosh^{2}\left(r\right)$, and the enhanced optomechanical coupling can still be obtained. The OPA is put into the auxiliary cavity, which may be easier to implement in the experiment.

S-3 phase-controlled phonon laser with $f_{1}\ll 1$

When $f_{1}\ll 1$, we can neglect the term $\lambda_{2}a_{s1}^{\dagger}a_{s2}^{\dagger}+\mathrm{H.c.}$ (rotating wave approximation), and the Hamiltonian of the system can be written as

[TABLE]

To diagonalize the interaction term $\lambda_{1}$, we introduce the transformation

[TABLE]

where $\theta=\arctan\left(\frac{\left|J^{\prime}\right|}{\omega_{s2}-\omega_{s1}}\right)$, in which $J^{\prime}=2J\lambda_{1}$ and $\Phi=arg\left(J^{\prime}\right)$.

The Hamiltonian can be written as

[TABLE]

where

[TABLE]

We know that the phonon laser can be realized by the Hamiltonian

[TABLE]

when we have $\omega_{m}\gg G_{j}$, $\left|W_{j}+W_{k}\pm\omega_{m}\right|\gg\left|G_{jk}\right|$ and $\left|W_{1}-W_{2}\pm\omega_{m}\right|\approx 0$.

In Fig. S2(a), we plot the equipotential lines $f_{1}=0.1$ (blue line) versus $\varLambda_{1}$ and $\Delta\Phi$, and the area between the blue and red lines fully satisfies $f_{1}\ll 1$. The frequencies $W_{1}/\omega_{m}$ (red-dashed line), $W_{2}/\omega_{m}$ (blue-dashed line), and $\left(W_{1}-W_{2}\right)/\omega_{m}$ (blue-solid line) are shown in Fig. S2(b). We find the frequency matching (red square points) $\left|W_{1}-W_{2}-\omega_{m}\right|\approx 0$, which leads that only the term $G_{p12}$ can be reserved. While the rotating wave approximation is satisfied, we can realize the phase-controlled phonon laser. In Fig. S2(b), the frequency $\left|2W_{2}-\omega_{m}\right|\approx 0$ can also be matched, which is denoted by the green dots. It is obvious that we can obtain the parametric amplification coupling form $G_{22}$ only by the tuning phase difference.