Abstract
Realizing a convenient way to control the phonon laser action is of great importance and may find applications in phonon laser devices engineering. Here we propose a vector parity time (PT)symmetric optomechanical system to study the phonon laser action. We find that there is a specific region for the optimum mechanical gain appearing in parameter areas of the cavity gain and loss. The intensity of phonon laser action can be well controlled by adjusting the polarization of the pump field. The threshold value of phonon laser action manifests as a function relationship of the polarization direction θ. Furthermore, an ultralow threshold (even if threshold less) can be obtained around the exceptional point with the gain–loss balance. Our results indicate that the intensity and threshold of the phonon laser action can be continuously adjusted by only tuning the photon polarization, which provides a new degree of freedom to realize phonon laser regulation.
Introduction
Just like optical lasers, recent research into phonon lasers has been increasingly rapid. With the development of onchip devices, the phonon laser has also made spectacular advances by means of microfabricated architectures, these are not only theoretical investigations but numerous experimental proposals have also been put forward to produce a phonon laser via coherent amplification of stimulated emission of phonons in various physical systems. Many years ago, Chen and Khurgin analyzed the feasibility of phonon lasers and pointed out a promising scheme to achieve phonon laser action^{1}. Next, based on prominent advances in experimental realization on the nanoscale, single trapped ions^{2} and quantum dots^{3,4} have been utilized to study phonon lasers. Recent experimental phonon laser platforms including semiconductors^{5}, nanomechanics^{6}, and an electromechanical resonator^{7} have made significant progress. And phononstimulated emission also has been observed experimentally in cryogenic ionic compounds^{8,9,10,11}, semiconductor superlattices^{12}, and electromagnetically trapped ions^{13}, which builds a vital system to produce phonon lasers theoretically and experimentally. Furthermore, phonon lasers also have attracted wide interest in matching the significant utility of its optical cousins such as medical imaging and highprecision measurement devices.
Cavity optomechanics (COM)^{14}, characterized by exploring the radiation–pressure interaction between optical modes and mechanical modes, have received great attention in combining nanomechanics with nanophotonics. The unique optomechanical nonlinearity of the system has revealed abundant intriguing physical phenomena, including numerous interesting quantum^{15,16,17,18,19} and classical^{20,21,22,23,24,25,26,27,28,29} nonlinearity phenomena. Significantly, considering orthogonally polarized modes in a whisperinggallerymode resonator has revealed important physical mechanisms, for example, based on orthogonally polarized modes in a whisperinggallerymode resonators, we can observe couplings between orthogonally polarized modes in a birefringent whisperinggallerymode resonator, which can be used for beneficial effect in sensing and optomechanical experiments^{30}. In addition, due to the effect of the polarization coupling occurring in the ring waveguide, the singlering phase shifter can be realized experimentally^{31}. Recently, the concept of vector cavity optomechanics^{32,33} has been put forward, where the polarization behavior of light is introduced to achieve optomechanical control, which can manipulate the onchip light propagation with nanobeam optomechanical crystal.
Parity time (PT)symmetric optomechanical system has recently undergone extensive research to achieve distinctive optical behavior, which is unattainable with ordinary optomechanical systems^{34,35,36,37}. It is known that the transition from an unbroken PTsymmetric phase (real eigenvalue spectra) to a spontaneously broken PTsymmetric phase (complex eigenvalue spectra) emerges as the system parameters in the Hamiltonian are properly modified^{36,38}. NonHermitian degeneracies known as exceptional points occuring in open physical systems have recently been utilized to enhance sensitivity in an optical microcavity ^{39,40}, which suggests that the sensitivity enhancement conforms to squareroot and cuberoot decay law. Furthermore, observation of subPoissonian phonon lasing in a threemode optomechanical system^{41}, and the discovery of phonon lasing action in a compound whisperinggallerymicrocavities (WGM) system^{42}, have received increasing attention. More strikingly, a significative regime of phonon lasing in a PTsymmetric system has been reported^{43}, which reveals the PTsymmetric system has been proved to be an excellent platform for exploring phonon laser action.
In this work, inspired by the vector cavity optomechanics, we present the vector PTsymmetriccoupled resonators to study phonon laser action. In contrast to other conventional phonon lasing regimes^{42,43}, where the phonon laser action can only be adjusted by the intensity of coherent pump field, the polarized PTsymmetry system proposed here provides a new degree of freedom to realize phonon laser regulation based on the mechanical effect of light. Our regime is pumped by a linear polarized light field, in close analogy to a twolevel laser system, the phonon laser is formed through the coherent amplification of the stimulated emission phonons arising from the interaction between photon and phonon. The theoretical analysis of our results was formulated by Heisenberg–Langevin equations. First of all, we find that there is an optimal region for mechanical gain G in a certain parameter range of the gain and loss of the active and passive cavity. Moreover, our regime provides a novel way to improve the performance of controlling a phonon laser, that is achieved by tuning the polarization state of pump light, instead of only regulating the intensity of the pump field^{42,43}. Furthermore, the strong polarization dependence of the phonon laser action may offer insight into the understanding of nonlinear optomechanical interactions and find applications in photon and phonon manipulation. Specifically, the phonon laser action is a polarizationdependent effect instead of a polarizationindependent effect when the polarizationdependent optomechanical nonlinearity are taken into account and this polarizationcontrolled phonon laser action may reveal nonlinear optomechanical interactions in a more robust way. Recently, onchip manipulation and control of light propagation has made remarkable progress by means of PTsymmetric systems, for instance, nonreciprocal light transmission in the PTsymmetrybreaking phase can be observed in the PTsymmetric whisperinggallery microcavities^{35}. We believe that the polarizationrelated phonon laser action can stimulate further remarkable applications in polarizationcontrolled onchip optical architectures due to the improvement of nanofabrication techniques.
Results
Physical setup and dynamical equation
Figure 1a gives the schematic of the system we propose, a vector parity time (PT)symmetric COM system which is composed by two directly coupled microcavities consisting of orthogonally polarized cavity modes. In this setup, one of cavities is fabricated from silica doped with Er^{3+} ions, and gain in this cavity is derived from optically pumping the doped Er^{3+} ions, whereas another cavity has passive loss without dopants, which was identified experimentally as the PTsymmetry system^{35}. The passive cavity is formed by one fixed mirror which is partially transparent and located to the left, and one movable, totally reflecting mirror placed to the right. The movable mirror is pushed by the radiation pressure, which is produced by the photon’s momentum transfer. Consequently, the movable mirror is modeled as a harmonic oscillator with effective mass m, eigenfrequency ω_{m}, and decay rate Γ_{m}. As Fig. 1b shows, each resonator has two modes of TE and TM, and a set of orthogonal basis vectors of polarization (\(\vec e_ \updownarrow\), \(\vec e_ \leftrightarrow\)) corresponding to TE and TM modes can be introduced^{32}. For entirely arbitrary polarized orientation of input light (so long as it is perpendicular to the direction of propagation), the vector \(\vec e\) can be disintegrated as \(\vec e = \alpha \vec e_ \updownarrow + \beta \vec e_ \leftrightarrow\) with α^{2} + β^{2} = 1 (shown in Fig. 1b), and α and β are complex. In this letter, our system is pumped by linearly polarized optical field, viz. α and β are real. Figure 1c shows that the optical supermodes ω_{±} are coupled by the mechanical mode b.
We consider that a dualmode active cavity couples to a dualmode passive optomechanical cavity via optical tunneling, i.e., the vector PTsymmetric COM system (see Fig. 1a). In view of the generic optomechanical system^{44}, the Hamiltonian of our system pumped by linearly polarized optical field can be described by^{32}
where \(a_{1{\mathrm{j}}}{\kern 1pt} (a_{1{\mathrm{j}}}^\dagger )\), \(a_{2{\mathrm{j}}}{\kern 1pt} (a_{2{\mathrm{j}}}^\dagger )\), and \(b{\kern 1pt} (b^\dagger )\) are the annihilation (creation) operators of the orthogonal cavity modes of the two resonators and the mechanical mode, respectively. The commutation relations are \(\left[ {a_{1{\mathrm{j}}},a_{1{\mathrm{j}}}^\dagger } \right] = 1\), \(\left[ {a_{2{\mathrm{j}}},a_{2{\mathrm{j}}}^\dagger } \right] = 1\), and \(\left[ {b,b^\dagger } \right] = 1\). The cavity mode a_{1j} coupling to a_{2j} (with the same resonance frequency ω_{c}) is pumped by a light field with frequency ω_{d} and amplitude ε_{d}. Considering the case of resonance, i.e., ω_{d} = ω_{c}, \(\varepsilon _{\mathrm{d}} = \sqrt {P_{{\mathrm{in}}}/(\hbar \omega _{\mathrm{d}})}\) where P_{in} is the input power. As depicted in Fig. 1b, the linearly polarized input field can be expressed as \(\varepsilon _{\mathrm{d}}e^{  i\omega _{\mathrm{d}}t}\vec e\), and \(\vec e\) is the unit vector of polarization of the input field, which can be decomposed as \(\vec e = a\vec e_ \updownarrow + b\vec e_ \leftrightarrow\), with a = cosθ and b = sinθ being the projections of \(\vec e\) at the vertical and horizontal modes, respectively, while θ is the included angle between \(\vec e\) and the vertical mode, hence \(\varepsilon _{{\mathrm{d}} \updownarrow } = \varepsilon _{\mathrm{d}}\,{\mathrm{cos}}\theta e^{  i\omega _{\mathrm{d}}t}\), \(\varepsilon _{{\mathrm{d}} \leftrightarrow } = \varepsilon _{\mathrm{d}}\,{\mathrm{sin}}\theta e^{  i\omega _{\mathrm{d}}t}\). The Hamiltonian of the cavity part can take the form \(H_{\mathrm{c}} = \hbar \omega _{\mathrm{c}}\left( {a_{1 \updownarrow }^\dagger a_{1 \updownarrow } + a_{1 \leftrightarrow }^\dagger a_{1 \leftrightarrow } + a_{2 \updownarrow }^\dagger a_{2 \updownarrow } + a_{2 \leftrightarrow }^\dagger a_{2 \leftrightarrow }} \right)\). The Hamiltonian of the interaction part can be given as \(H_{{\mathrm{int}}} =  \mathop {\sum}\nolimits_{j = \updownarrow , \leftrightarrow } \hbar \left[ {J_{\mathrm{j}}\left( {\hat a_{1{\mathrm{j}}}^\dagger a_{2{\mathrm{j}}} + a_{2{\mathrm{j}}}^\dagger a_{1{\mathrm{j}}}} \right) + gx_0\left( {a_{2{\mathrm{j}}}^\dagger a_{2{\mathrm{j}}}} \right)\left( {b + b^\dagger } \right)} \right]\), therein the first term of H_{int} represents the interaction between the active cavity and the passive cavity with coupling strength J_{j}, and the second term expresses optomechanical interaction induced by the radiation pressure of both TE and TM modes in the passive cavity with coupling coefficient g = ω_{c}/R^{44}, and R is the cavity size. The last term in Eq. (1) \(H_{{\mathrm{drive}}} = i\hbar\left[{\sqrt {2\kappa }({a_{1 \updownarrow}^\dagger \varepsilon _{d \updownarrow }  {\mathrm{H}}{\mathrm{.c}}{\mathrm{.}}}) + \sqrt {2\kappa \prime}({a_{1 \leftrightarrow }^\dagger \varepsilon _{{\mathrm{d}} \leftrightarrow}  {\mathrm{H}}{\mathrm{.c}}{\mathrm{.}}})}\right]\) describes the optical driving of the active cavity modes.
To study the temporal evolution of the system, we use the Heisenberg–Langevin equations of motion which give an evolutionary description of the cavity fields and mechanical motion: (in a rotating frame at ω_{c} where the system Hamiltonian is transformed by a unitary transformation \(U(t) = {\mathrm{exp}}\left( {  i\omega _{\mathrm{c}}a^\dagger at} \right)\))
where
\({\mathrm{\Psi }} = ( {a_{1 \updownarrow },{\kern 1pt} a_{2 \updownarrow },{\kern 1pt} a_{1 \leftrightarrow },{\kern 1pt} a_{2 \leftrightarrow },x} ),{\mathrm{\Omega }} = \frac{1}{{m{\mathrm{\Gamma }}_{\mathrm{m}}}}( {  \frac{{d^2}}{{dt^2}}  \omega _{\mathrm{m}}^2} ),\) the noise correlators are defined as \(\langle {a_{{\mathrm{in}}}(t)a_{{\mathrm{in}}}^\dagger (t\prime )} \rangle = \delta \left( {t  t\prime } \right),\langle {a_{{\mathrm{in}}}^\dagger (t)a_{{\mathrm{in}}}(t\prime )} \rangle = 0,\langle {b_{{\mathrm{in}}}(t)b_{{\mathrm{in}}}^\dagger (t\prime )} \rangle = \left( {n_{{\mathrm{th}}} + 1} \right)\delta \left( {t  t\prime } \right),\) and \(\langle {b_{{\mathrm{in}}}^\dagger (t)b_{{\mathrm{in}}}(t\prime )} \rangle = n_{{\mathrm{th}}}\delta ( {t  t\prime } ),\) with the average number of mechanical quanta n_{th} ≈ k_{B}T/ℏω_{m}. In this case, using the semiclassical approximation, e.g., 〈AB〉 = 〈A〉〈B〉 with A and B being optical or mechanical operators, the Heisenberg–Langevin equations can be reduced to a group of nonlinear evolution equations in which the noise has been ignored due to the fact that we consider the mean values of all the operators. The position of the mechanical oscillator is defined as \(x = x_0\left( {b^\dagger + b} \right)\), \(x_0 = \sqrt {\hbar /\left( {2m\omega _{\mathrm{m}}} \right)}\) is the zeropoint fluctuation amplitude of the mirror, and the operator \(\hat b\) denotes the phonon mode. κ(κ′) and γ(γ′) are the optical gain and decay of the vertical and the horizontal modes in cavity 1 and cavity 2, respectively. Γ_{m} is the mechanical damping rate.
The distribution of steadystate photon number in the passive cavity
Recently, PTsymmetric whisperinggallery microcavities have been demonstrated experimentally, where lowpower optical isolation and nonreciprocal light transmission have been observed under the balanced gain and loss^{35,45}. In addition, the experimental observation of PTsymmetry in optically induced atomic lattices with periodical gain and loss profiles^{38}, and an antiPTsymmetric potential has been experimentally produced in a pair of optically induced waveguides coupled by steadystate flying atoms^{46}. These experimental facts indicate that the steady state of PTsymmetric systems can be observed in a short period of time via current experimental technology. First, we analyze the property of the steadystate situation of our system. Setting the time differential in dynamical Eqs. (2) to zero, thus the steadystate solutions can be obtained, i.e., the steadystate intracavity photons a_{1j,s} and a_{2j,s} (j = ↕, ↔). Their analytical expressions are
with
and the displacement of the mechanical resonator \(x_{\mathrm{s}} = \frac{{\hbar g}}{{m\omega _{\mathrm{m}}^2}}\left( {\left {a_{2 \updownarrow ,{\mathrm{s}}}} \right^2 + \left {a_{2 \leftrightarrow ,{\mathrm{s}}}} \right^2} \right)\). It can be inferred that the steadystate value of intracavity photons is a fifthorder polynomial equation, thus, there are at most five real roots. But the pump field is too weak to reach the multiple steady states in our regime, so there is only one real root under the weak pump power. Figure 2a shows how the steadystate populations of the intracavity photons in the cavity 2 vary with input power due to the Eq. (3). It is obvious that the photon number N acquires giant enhancement with the weak pump power \(\left( {P_{{\mathrm{in}}} \le 10\,\mu {\mathrm{W}}} \right)\) increasing. Specifically, the rise of photon number in the case of θ = π/2 (the violet line) features linear responses, while the significant nonlinear enhancement appears in other polarization states, leading to the appearance of the accompanied giant enhancement of COM interactions. Interestingly, the polarization state of the pump light serves as a sensitive controller for the physical phenomenon, namely, the photon number experiences distinct rising tendency with the different polarization state of pump light. In other words, the photontunneling effect is equivalently controlled by the polarization states of the pump field. From this phenomenon, we can work out that the gaintoloss ratio may also affect the behavior of photon tunneling, then we plot how the photon number varies with γ/κ at a fixed input power P_{in} in Fig. 2b. As γ/κ increases, the change of N has a obvious peak value at γ/κ = 1 (the gain–loss balance) and the vicinity of γ/κ = 0.6 for θ = 0 and θ = π/2, which indicates a giant enhancement of the intracavity field intensity at the gain–loss balance and the vicinity of γ/κ = 0.6. The remaining curves show N have two peak values at the two points for 0 < θ < π/2, owing to the projections of linearly polarized pump light onto the vertical and horizontal modes. The two extreme values of N satisfy \(J_ \updownarrow ^2 = \kappa \gamma\) and \(J_ \leftrightarrow ^2 = \kappa \prime \gamma \prime\) resulting from the fact that first derivative of N is equal to 0, and the point of κ/γ = 1 is the exceptional point (EP) for θ = 0. We proceed to study the remarkable influence of the photon polarization on the intracavity photons with the balanced gain and loss (κ/γ = 1), and the intracavity photons in the passive cavity is shown in Fig. 2c. Clearly, the photon number can be significantly enhanced in the vicinity of θ = 0, π, which coincides well with the EP, while the slight enhancement appears around θ = π/2, 3π/2 due to the considerable shift away from the actual EP. Physically, the field localization induces the dynamical accumulations of photon energy, in addition, the unidirectional energy transfer from the active cavity to the passive cavity. Hence, the enormously enhanced photon number can be generated at the EP. The proposed mechanism is especially suited for the condition of the gain–loss balance, where the intensity of intracavity photon can be significantly enhanced. Then leading to a giant enhancement of both optical pressure and mechanical gain. This fact provides a route for achieving a lowpower optomechaical amplifier in the presence of polarizationoptomechaics EP.
Polarizationbased control of phonon laser action
In order to explore the specific relationship between the gain (loss) and the mechanical gain G, we have investigated the gain G of the mechanical mode in a parameter map of the optical gain κ and loss γ in Fig. 3. The system is in the PTsymmetric phase when we choose \(J_ \updownarrow = \omega _{\mathrm{m}}/2\) and \(J_ \leftrightarrow = 1.3J_ \updownarrow\). We find that the optimum mechanical gain can be reached by choosing the proper optical gain κ and loss γ. As shown in Fig. 3, the optimum mechanical gain formation starts at a critical point κ = γ = 15 MHz, namely, the mechanical gain is very small even close to zero until the κ and γ arrive at this point. Obviously, we can get a clear description that the mechanical oscillator cannot possess gain when κ > γ above the critical point, in contrast to the fact that the mechanical gain can be produced in an optimal area of κ < γ below the critical point. This result indicates that optical gain and loss within the optimal area can significantly contribute to the supermode population inversion for magnifying the stimulated phonon number. Such that we find the phonon laser action can achieve its optimal solution in the specific parameter region while can not be acquired in other wide range of gain κ and loss γ. It is worth noting that the optimal parameter area can guide us to design more efficient phonon laser devices.
In Fig. 4, we present the dependence of the phonon number n on the pumped light field P_{in} for different polarizations, which clearly shows that the phonon number n can be controlled by the polarization of the pump field. Based on ref. ^{42}, only above the threshold can the phonon lasering action occur, Fig. 4a shows that the phonon laser is not triggered where γ/κ = 0.8. As γ/κ goes up in Fig. 4b–d, i.e., γ/κ = 4/3, 5/3, 2, the polarization state of pump light can directly influence the intensity of the phonon laser action for different loss–gain ratio γ/κ(γ′/κ′). As shown in Fig. 2, the photon number in cavity 2 can be adjusted by tuning the polarization of the pump field, and this phenomenon can be explained by the fact that the vertical and horizontal modes in cavity 2 have different photon number according to the fact that the two orthogonal components of linearly polarized pump light act as two different intensities of driving field, resulting in different optomechanical nonlinear strength. Moreover, the two orthogonal modes can produce two different intensities of the phonon laser action, which means that the phonon laser action in our regime possesses the polarizationrelated property. As a transversely oscillating electromagnetic wave, polarization is a fundamental property of light, and optical polarization is often a major consideration in the construction of many optical systems, such as optical biomedical imaging, communication, and sensor systems^{47,48}. This demonstrates that it is very important and achievable to utilize the polarization nature of light field to study phonon laser action. Recently, the photon polarization degree of freedom has been introduced to achieve optomechanical control in the vector cavity optomechanics^{32}, which may enable onchip optical control of polarization management with remarkable applications in nanophotonic polarizationrelated devices, such as optomechanical polarizers^{33}. Compared with previous studies of the phonon laser action^{42,43}, as shown in Fig. 4, our scheme provides a new degree of freedom to adjust the intensity of coherent phonon laser action by varying the polarization angle of pump light without changing other system parameters. This fact opens the way for polarizationcontrolled phonon lasing and may thus be applied to engineer onchip phonon laser devices.
In Fig. 5, we plot the phonon number n versus the polarization angle θ in the PTsymmetric phase under different pump power P_{in}, which shows the role of the light field polarization behavior on the control of phonon laser action more clearly. We can see that the phonon number has a Lorentzianlikeshape dependence on the polarization direction θ, which indicates that the intensity of phonon laser action can be continuously adjusted by only tuning the photon polarization behavior without changing other parameters of the device. Compared with the conventional phonon laser regime^{41,42,43}, our scheme provides an additional degree of freedom to control phonon laser action. In addition, the slight enhancement of the pump power leads to the fact that the maximum value of the phonon number increases dramatically arising from the population inversion of the supermodes, corresponding to the exponential growth of phonon number in Fig. 4. Compared with the pumppowercontrolled method, the change of phonon number induced by the light polarization is slower, which can boost the feasibility of controlling the intensity of phonon laser action quite accurately. This result is a manifestation of the intrinsic merit of polarizationdependent optomechanics, and we can utilize the light polarization to realize alloptical switcher of phononlasing devices. On the other hand, our work may offer a useful platform to convert the information carried by optical polarization into the stimulated emission of phonons while preserving their intrinsic coherence, which is of great importance in communication systems and modern communication networks.
We next explore the impact of polarization on the phonon laser threshold value P_{th}, which behaves as a function relationship of the polarization direction θ as shown in Fig. 6. Remarkably, unlike previously reported observations of phonon lasing effects in microcavity structures^{42,43}, in which the phonon laser threshold value can only be adjusted by changing the system parameters, in our scheme the threshold value of phonon laser action can also be controlled by adjusting the polarization direction of the pump field. In this work, the threshold condition and the threshold power of phonon laser action are G = Γ_{m}/2 and P_{th} = N_{+}(γ_{−} + γ_{+})ℏω_{+}^{42,43}, respectively. Therefore, the thresholds of the vertical and horizontal modes yield \(P_{{\mathrm{th}} \updownarrow } = 4{\mathrm{\Gamma }}_m\omega _{ + \updownarrow }\left[ {\left( {{\mathrm{\Delta }}\omega _ \updownarrow  \omega _{\mathrm{m}}} \right)^2 + \left( {\frac{{\kappa  \gamma }}{2}} \right)^2} \right]/\left( {gx_0} \right)^2,P_{{\mathrm{th}} \leftrightarrow } = 4{\mathrm{\Gamma }}_{\mathrm{m}}\omega _{ + \leftrightarrow } \left[ {\left( {{\mathrm{\Delta }}\omega _ \leftrightarrow  \omega _{\mathrm{m}}} \right)^2 + \left( {\frac{{\kappa \prime  \gamma \prime }}{2}} \right)^2} \right]/\left( {gx_0} \right)^2\). Clearly, \(P_{{\mathrm{th}} \updownarrow } \to 0\) and P_{th↔}→0 under the conditions \((a){\kern 1pt} {\mathrm{\Delta }}\omega _ \updownarrow = \omega _{\mathrm{m}},{\mathrm{\Delta }}\omega _ \leftrightarrow = \omega _{\mathrm{m}},(b)\kappa = \gamma ,\kappa \prime = \gamma \prime\), and this implies that an ultralow threshold (even threshold less) phonon laser action can be obtained by tuning the parameter configuration of the system in the vector PTsymmetric regime. As shown in Fig. 5, the threshold values become much smaller as the value of γ/κ approaches 1, as expected, phonon lasing can be enhanced greatly in the vicinity of EP. Interestingly, we can regulate the threshold value of phonon laser action via modifying the polarization of pump light in our regime. This attribute offers a promising approach to engineer polarizationcontrolled onchip phononic devices in practical applications.
Discussion
We propose a feasible scheme to realize a polarizationcontrolled phonon laser in a vector PTsymmetric system based on the mechanical effect of light. Different from the phonon laser action previously discussed, our results have shown that the generation and the intensity regulation of phonon laser action can be well controlled by only tuning the polarization of the pump field while keeping other parameters unchanged. This indicates a novel way to improve the performance of controlling a phonon laser, that is by tuning the polarization state of the pump light, instead of only adjusting the intensity of the pump field. Moreover, the threshold value of phonon laser action behaves as a function relationship of the polarization direction θ. Furthermore, an ultralow threshold (even threshold less) phonon laser action can be obtained around EP with the gain–loss balance. The underlying physical mechanism can be explained as follows: the dynamical accumulations of photon energy result in drastic population transition processes from the upsupermode level to downsupermode level, which accompany the phonon emission. This is reminiscent of randomdefectinduced phonon lasing at EPs^{49} and the lossinduced suppression and revival of lasing at EPs^{50}. Physically, the enormously enhanced photon number can induce the great reinforcement of radiating pressure, consequently resulting in stronger phonon laser action. These attributes may inspire the exploration of engineering new polarizationrelated phonon laser devices featuring lowerthreshold power and convenient regulation ways. In addition, our research which associates the vector regime and PTsymmetric optomechanics may induce more opportunities in both areas and provide new possibilities to develop the corresponding applications.
Finally, we present some discussions on the experimental implementation of our proposal with coupled optical cavity systems. Specifically, it is achievable to implement our proposal experimentally in the PTsymmetric optomechanical system with current accessible technology. First, we introduce a group of orthogonal basis vectors of polarization corresponding to TE and TM modes in the coupled whisperinggallery mode (WGM) optical microcavities^{32}. Considering the system is pumped by linearly polarized light field, whose orthogonal components obey Malus law toward orthogonally polarized directions in the vector regime which reveals a nontrivial phenomenon in analogy to optomechanical polarizer. Second, the cavity gain \(\kappa \sim 10\,{\kern 1pt} {\mathrm{MHz}}\) and coupling strength \(J\sim 20\,{\mathrm{MHz}}\) can be realized in the PTsymmetric system, which was identified experimentally as the experimental arrangement in ref. ^{35}, leading to the system reaching PTsymmetric phase required here. Last, the active cavity is fabricated from silica doped with Er^{3+} ions, and the Er^{3+} ions can emit photons in the 1550 nm band arising from optical pump with a light in the 1460 nm band. This leads to the amplification of weak signal light in the 1550 nm band^{35}, and the microtoroids are ~34.5 μm in radius^{43}. Due to the fact that the two different polarizations have two different photontunneling values which are determined by geometry size and material properties as well as the index contrast between the cavity core and cladding^{51,52}, the tunneling between the two resonators of the two orthogonal polarizations we choose are different, viz. \(J_ \leftrightarrow = 1.3J_ \updownarrow\) where J_{↔} and \(J_ \updownarrow\) denote the tunneling values of horizontal and vertical directions, respectively. Consequently, all parameter values in our scheme are taken from current experiments, implying that our system is well within the phonon laser regime based on the above discussion^{42}. In summary, the experimental implementation of our scheme is feasible under currently existing experimental techniques, i.e., all relevant system parameters and experimental techniques used in our scheme are well within the reach of the current experimental level. With regard to controlling the phonon laser action, we can employ nanophotonic polarizer^{33} to tune the polarization direction of light field, which can be utilized to achieve optomechanical control and regulate the behavior of the phonon laser action.
Methods
Derivation of the polarizationcontrolled phonon laser
The polarizationdependent character possesses important practical application. Here, we integrate the feature of vector cavity optomechanics into PTsymmetry, and introduce the vector PTsymmetric phonon laser. According to ref. ^{43}, PTsymmetriccoupled resonators provide a coherent phonon amplification by phononmediated transitions between optical supermodes in parallel with a twolevel laser by the electronic transitions. On this basis, we propose a controllable phonon laser via tuning the polarization of pump field. Before proceeding further, we illuminate the important influence of the cavity coupling intensity J in the regime. To be specific, the nonHermitian Hamiltonian of general PTsymmetric optical system is \(\hbar \omega _{\mathrm{c}}\left( {a_1^\dagger a_1 + i\kappa } \right) + \hbar \omega _{\mathrm{c}}\left( {a_2^\dagger a_2  i\gamma } \right) + \hbar J\left( {a_1^\dagger a_2 + a_2^\dagger a_1} \right)\), which can be diagonalized as \(\hbar \left( {\omega _ + + i\gamma _ + } \right)o_1^\dagger o_1 + \hbar \left( {\omega _  + i\gamma _  } \right)o_2^\dagger o_2\), with \(\omega _ \pm = \omega _{\mathrm{c}} \pm \sqrt {J^2  \left( {\kappa + \gamma } \right)^2/4} ,\gamma _ \pm = \left( {\kappa  \gamma } \right)/2\). For the case of J ≥ (κ + γ)/2, the system is in the \({\cal P}{\cal T}\)symmetric phase. And only in this regime can the two nondegenerate supermodes exchange energy through the phonon and possess the population inversion, and then generating the phonon lasing^{43}. Note that the situation J < (κ + γ)/2 corresponds to degenerate supermodes \(\omega _ \pm = \omega _{\mathrm{c}},\gamma _ \pm = \left( {\kappa  \gamma } \right)/2 \pm \sqrt {J^2  \left( {\kappa + \gamma } \right)^2/4}\), thus this situation which has no supermode splitting results in no phonon lasing action.
TE and TM modes are independent of each other in our regime, we can use the same method to deal with the two modes, respectively. Derived from Eq. (1), the optical supermodes \(a_{ \pm {\mathrm{j}}} = \left( {a_{1{\mathrm{j}}} \pm a_{2{\mathrm{j}}}} \right)/\sqrt 2\), \(\left( {j = \updownarrow , \leftrightarrow } \right)\) correspond to the energy levels \(\omega _{ \pm \updownarrow } = \omega _{\mathrm{c}} \pm \sqrt {J_ \updownarrow ^2  \left( {\kappa + \gamma } \right)^2/4} ,\omega _{ \pm \leftrightarrow } = \omega _{\mathrm{c}} \pm \sqrt {J_ \leftrightarrow ^2  \left( {\kappa \prime + \gamma \prime } \right)^2/4}\) can be introduced. Looking at one of the two orthogonal modes, the Fig. 1c describes the transition between the upper level ω_{+} and the lower level ω_{−} while absorbing or emitting a phonon simultaneously, that is the eigenfrequencies ω_{+} and ω_{−} can exchange energy through the phonon mode. In analogy to a twolevel laser system, the stimulated emission of phonon can be generated by virtue of optical pumping of the upper level, then leading to the appearance of coherent phonon lasing. Additionally, the stimulated emission linewidth which describing the firstorder coherence above the threshold is much narrower in comparison with that below the threshold^{42,43}. The interaction is provided by radiation pressure within the optomechanical system. Applying the rotatingwave approximation (RWA), the optomechanical interaction of Hamiltonian can be converted to another form^{43}, i.e.,
\(\Re _{\mathrm{j}} = a_{  {\mathrm{j}}}^\dagger a_{ + {\mathrm{j}}}\,\left( {j = \leftrightarrow , \updownarrow } \right)\) is the population inversion of the supermodes. And the partly diagonalizable form of the Hamiltonian H in Eq. (1) is:
where ω_{±j} are the optical frequencies of the supermodes and defined to be blue and red, respectively. The third term of the Hamiltonian \(\tilde H\) describes the absorption of one phonon, which leads to transition from the red supermode to the blue supermode and the reverse process. Using the Hamiltonian in Eq. (5), we can rewrite the evolution equations for the mechanical mode and the operator \(\Re _{\mathrm{j}}\) as follows:
\(\Theta = \left( {\kappa  \gamma } \right)/2,\Theta \prime = \left( {\kappa \prime  \gamma \prime } \right)/2,{\mathrm{\Delta }}\aleph _{\mathrm{j}} = a_{ + {\mathrm{j}}}^\dagger a_{ + {\mathrm{j}}}  a_{  {\mathrm{j}}}^\dagger a_{  {\mathrm{j}}},\) \({\mathrm{\Delta }}\omega _{\mathrm{j}} = \omega _{ + {\mathrm{j}}}  \omega _{  {\mathrm{j}}},\,\Re = \Re _ \updownarrow + \Re _ \leftrightarrow\). We ignore the influence of Ξ_{1}(t), Ξ_{2}(t), which represent fluctuation operators correspond to the supermodes and the mechanical resonator. In order to solve Eq. (6) and obtain the mechanical gain, we introduce the slow varying amplitudes: \(\tilde b = be^{i\omega _{\mathrm{m}}t},\tilde \Re _ \updownarrow = \Re _ \updownarrow e^{i{\mathrm{\Delta }}\omega _ \updownarrow t},\tilde \Re _ \leftrightarrow = \Re _ \leftrightarrow e^{i{\mathrm{\Delta }}\omega _ \leftrightarrow t}\), and substituting these into Eq. (6), we can get
\({\mathrm{\Delta }} =  \frac{{{\mathrm{\Gamma }}_{\mathrm{m}}}}{2} + G + i\eta ,\) where the mechanical gain G is given by
\(G_ \updownarrow = \frac{{\left( {gx_0/2} \right)^2{\mathrm{\Delta }}\aleph _ \updownarrow \left( {  \Theta } \right)}}{{\left( {{\mathrm{\Delta }}\omega _ \updownarrow  \omega _{\mathrm{m}}} \right)^2 + \left( \Theta \right)^2}}\) and \(G_ \leftrightarrow = \frac{{\left( {gx_0/2} \right)^2{\mathrm{\Delta }}\aleph _ \leftrightarrow \left( {  \Theta \prime } \right)}}{{\left( {{\mathrm{\Delta }}\omega _ \leftrightarrow  \omega _{\mathrm{m}}} \right)^2 + \left( {\Theta \prime } \right)^2}}\) indicate that the mechanical gain of orthogonally polarized modes. According to the expression of G, the mechanical gain of the mechanical oscillator can be modified by varying the gain κ or loss γ of cavities 1 and 2. Finally, we can work out that the phonon number n is expressed by n = exp[2(G − Γ_{m}/2)/(Γ_{m}/2)] at t = [Γ_{m}/2]^{−1}. As shown in Fig. 4, we can achieve the polarizationcontrolled phonon laser action in the vector PTsymmetric optomechanical systems.
Data availability
The data that support the findings of this study are available from the corresponding author on request.
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Acknowledgements
The work was supported by the National Basic Research Program of China (Grant No. 2016YFA0301203) and the National Science Foundation (NSF) of China (Grant Nos. 11375067, 11405061, and 11574104).
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B.W. carried out the calculations, wrote the main manuscript text and prepared all figures. Z.X.L., X.J., H.X., and Y.W. participated in the discussions. All authors reviewed the manuscript and contributed to the interpretation of the work and the writing of the manuscript.
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Wang, B., Liu, ZX., Jia, X. et al. Polarizationbased control of phonon laser action in a Parity Timesymmetric optomechanical system. Commun Phys 1, 43 (2018). https://doi.org/10.1038/s4200501800423
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