Exceptional points enhance sum sideband generation in a mechanical <i>P</i><i>T</i>-symmetric system

Xiao-Hu Lu; Liu-Gang Si; Liu-Gang Si; Xiao-Yun Wang; Ying Wu; Ying Wu

doi:10.1364/OE.417156

1. Introduction

Cavity optomechanical system [1,2], using the radiation pressure of light to explore the interaction between light and matter, has become a very creative platform. Some representative works have been achieved in weak-coupling regime, such as ground state cooling [3], optomechanically induced transparency(OMIT) [4,5], non-reciprocity [6] and optomechanical dark mode [7]. With some experimental breakthroughs [8–12], the intrinsic nonlinearity of the optomechanical interaction has become a new frontier in optomechanical systems. Then, some interesting quantum properties were found in the strong or ultrastrong coupling regime, such as photon blockade effect [13], efficient entangled-photon emitters [14] and spontaneous emission of Schrödinger cats [15].

Additionally, based on the inherent nonlinearity of optomechanical coupling, some higher-order processes and nonlinear optics are analyzed, such as OMIT in nonlinear quantum regime [16,17], higher-order sidebands [18] in OMIT, higher-order sideband comb in the non-perturbative regime [19]. In particular, different from the traditional single probe-field-driven optomechanical systems, the sum and difference sidebands [20,21] realized in a double-probe-field-driven optomechanical systems. Physically, sum sideband generation(SSG) is caused by the nonlinear interaction between the mechanical motion and the multiple driving fields in the optomechanical system. However, due to the weak inherent nonlinearity, the efficiency of SSG obtained in general optomechanical systems is very low. Therefore, the method to enhance the efficiency of SSG by using parametric interaction [22] was proposed. Similarly, enhancing the SSG efficiency was discussed in the quadratically coupled optomechanical system [23].

Recently, the conception of parity-time ($\mathcal {PT}$) symmetry [24] has been extensively studied in optics and photonics, which is achieved by introducing gain and loss balance in optical system [25,26]. The study of $\mathcal {PT}$-symmetric systems has achieved breakout progress whatever in theory and experiment [27–33]. Some interesting phenomena are realized in various systems, such as non-reciprocal light transmission in $\mathcal {PT}$-symmetric resonators [32], optical solitons in $\mathcal {PT}$-symmetric lattices [34], topological edge states in $\mathcal {PT}$-symmetric photonic crystals [35] and ultralow threshold chaos [36] in $\mathcal {PT}$-symmetric optomechanics. Specially, exceptional points(EPs), in which the $\mathcal {PT}$-symmetry start to break, were observed in optical microcavities [37]. In EPs, the eigenvalues of the system is degenerate but the eigenvectors is non-orthogonal. Utilizing the special properties of EPs, some important applications have been realized, such as phonon lasers [38,39], enhancing sensitivity [40,41], stopping light [42] and transferring topological energy [43]. Of course, $\mathcal {PT}$-symmetric nonlinear optics has been discussed already [44,45]. Naturally, mechanical $\mathcal {PT}$-symmetric system [46,47] is also proposed. Basing on the PT-symmetry, some intriguing phenomena were discovered in optomechanical system, such as mechanical EPs-induced transparency and slow light [48], mechanical EPs-enhanced sensitivity of optical gyroscope [49], non-reciprocal transmission of microwave acoustic waves in nonlinear $\mathcal {PT}$-symmetric resonators [50].

Based on the characteristics of low-power drive and high-sensitivity of EPs [40], we theoretically analyze SSG in the mechanical $\mathcal {PT}$-symmetric system by applying $\mathcal {PT}$ symmetry to coupled optomechanical system. It consists of the Fabry-Pérot cavity and two mechanical resonators(MR) and is driven by the strong control field and two weak probe fields. Considering the inherent nonlinearity of optomechanical coupling, the transmission coefficient of the frequency components at the sum sideband can be obtained in the mechanical $\mathcal {PT}$-symmetric system. By analyzing the evolution of the mechanical $\mathcal {PT}$-symmetric system from $\mathcal {PT}$-symmetric phase to broken-$\mathcal {PT}$-symmetric phase, the efficiency of SSG can be significantly enhanced about two orders of magnitude in EPs. Extraordinarily, there is an interesting phenomenon about upper sum sideband generation(USSG) in the vicinity of EPs. By adjusting the appropriate parameters, the efficiency of USSG can be amplified, and even generates a transparent window. Under this circumstance, the efficiency of SSG can be significantly enhanced about three orders of magnitude than the general optomechanical system. Interestingly, the lower sum sideband generation(LSSG) yield two symmetrical transmission peaks. Then, the LSSG can be split into four transmission peaks in the $\mathcal {PT}$-symmetric phase. In addition, the efficiency of SSG can also be enhanced by adjusting the detuning frequency between the dual detection field and the control field. This research is helpful to pave the way for on-chip optical modulation, ultrahigh precision metrology and sensing, and quantum information processing.

2. Theoretical model and formulations

As shown in Fig. 1(a), the mechanical $\mathcal {PT}$-symmetric system consists of a Fabry-Pérot cavity with resonance frequency $\omega _0$ and two directly coupled $\mathcal {PT}$-symmetric MR. When the system is excited by the control field with frequency $\omega _c$ and the two probe fields with frequency $\omega _i(i=1,2)$, the optical mode $\hat {a}$ is coupled to the mechanical mode $\hat {b}_1$ via radiation pressure. And, the mechanical modes $\hat {b}_1$ with the loss frequency $\gamma _1$ is coupled to the mechanical mode $\hat {b}_2$ with the gain frequency $\gamma _2$ through the mechanical coupling parameter $J$. The amplitudes of the control field and the probe fields are $\varepsilon _c=\sqrt {P_c/\hbar \omega _c}$ and $\varepsilon _i=\sqrt {P_i/\hbar \omega _i}(i=1,2)$, respectively. $P_c$ is the pump power of the control field, $P_i$ denotes the power of the ith probe field.

Now, we can write the Hamiltonian formulation of the whole system in the rotating frame at the frequency $\omega _c$ of the control field [46],

(1)$$\begin{aligned} & H=\hbar\Delta\hat{a}^{\dagger}\hat{a}+\hbar\omega_m(\hat{b}^{\dagger}_{1}\hat{b}_1+\hat{b}^{\dagger}_{2}\hat{b}_2)-\hbar{g}\hat{a}^{\dagger}\hat{a}(\hat{b}^{\dagger}_{1}+\hat{b}_1)-\hbar{J}(\hat{b}^{\dagger}_{1}\hat{b}_2+\hat{b}^{\dagger}_{2}\hat{b}_1)\\ &~~~~~~+i\hbar\sqrt{\eta\kappa}\varepsilon_{c}(\hat{a}^{\dagger}+\hat{a})+i\hbar\sqrt{\eta\kappa}[(\varepsilon_{1}\hat{a}^{\dagger}e^{{-}i\delta_1{t}}+\varepsilon_{2}\hat{a}^{\dagger}e^{{-}i\delta_2{t}})-H.c], \end{aligned}$$

where $\Delta =\omega _0-\omega _c$, $\delta _i=\omega _i-\omega _c(i=1,2)$. $g$ is the optomechanical coupling rate, $\omega _m$ is the eigenfrequency of the mechanical resonator, $\kappa$ is the total cavity loss rate, which consists of an intrinsic loss rate $\kappa _0$ and an external loss rate $\kappa _{ex}$, and the coupling parameter $\eta = \kappa _{ex}/\kappa$ is chosen to be the critical coupling value $1/2$ [5]. Similar to the $\mathcal {PT}$-symmetric optical mode [25], the effective Hamiltonian of $\mathcal {PT}$-symmetric $MR$ is

(2)$$H_{eff}=(\omega_m-\gamma_1/2)\hat{b}^{\dagger}_{1}\hat{b}_1+(\omega_m+\gamma_2/2)\hat{b}^{\dagger}_{2}\hat{b}_2-\hbar{J}(\hat{b}^{\dagger}_{1}\hat{b}_2+\hat{b}^{\dagger}_{2}\hat{b}_1)$$

with the eigenvalues

(3)$$\omega_\pm{=}\omega_m+i\frac{(\gamma_2-\gamma_1)}{4}\pm\sqrt{(J^2-\frac{\gamma_1+\gamma_2}{4})^2}.$$

In this scenario, exceptional points(EPs) are the two-state coalescence that exhibits a square-root dependence on the system parameters. Generally, $\mathcal {PT}$ symmetry breaks spontaneously to form broken-$\mathcal {PT}$-symmetric phase, the critical point from the $\mathcal {PT}$-symmetric phase to the broken-$\mathcal {PT}$-symmetric phase is called EPs. At EPs, the eigenvalues are degenerate and the corresponding eigenstates become non-orthogonal [51]. In $\mathcal {PT}$-symmetric phase(as shown in Fig. 2), the real part of the eigenvalue is split cross the EPs, but its imaginary part is identical. Interestingly, in the broken-$\mathcal {PT}$-symmetric phase, the real part of the eigenvalues are degenerate, while its imaginary part is different. The rich physical properties of the $\mathcal {PT}$-symmetric system are conducive to realize a series of interesting phenomena.

Fig. 1. (a). The optical mode is coupled to $MR1$ via radiation pressure, which is driven by a strong control field with frequency $\omega _c$ and two relatively weak probe fields with frequencies $\omega _1$ and $\omega _2$, respectively. The $\mathcal {PT}$-symmetric mechanical resonator(MR) is composed of a passive $MR_1$ and an active $MR_2$ , where the mechanical coupling parameter is $J$. $\kappa$ is the cavity decay rate, $\gamma _1$ and $\gamma _2$ indicate the loss and gain of mechanical modes, respectively. (b). Frequency spectrogram of the mechanical $\mathcal {PT}$-symmetric system. $\Delta =\omega _0-\omega _c$, $\delta _1=\omega _1-\omega _c$, $\delta _2=\omega _2-\omega _c$.

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Fig. 2. The real part and the imaginary part of the eigenvalues of $\mathcal {PT}$-symmetric $MR$ as a function about $J$.

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Here, we are interested in the mean response of the system to the probe field, ignoring quantum and thermal noise terms. Considering the loss of the cavity field and the resonator, the simplified Heisenberg-Langevin equations can be written as

(4)$$\begin{aligned}&\dot{a}=[{-}i\Delta-\kappa/2+ig({b}^{\dagger}_1+{b}_1)]a+\sqrt{\eta\kappa}(\varepsilon_{c}+s_{in}),\\ &\dot{b_1}=({-}i\omega_m-\gamma_1/2)b_1+iJb_2+ig{a}^{\dagger}{a}~,~~~\dot{b_2}=({-}i\omega_m+\gamma_2/2)b_1+iJb_1, \end{aligned}$$

where $s_{in}=\varepsilon _{1}e^{-i\delta _1{t}}+\varepsilon _{2}e^{-i\delta _2{t}}$. Considering that the power of the control field is much greater than the power of the probe field, we will use the linear approximation method to describe the system, via $a=\overline {a}+\delta {a}$, $b_i=\overline {b}_i+\delta {b}_i$. The steady state value of the system can be obtained as follows,

(5)$$\begin{aligned}&\overline{a}=\frac{\sqrt{\eta\kappa}\varepsilon_{c}}{i\Delta+\kappa/2-ig(\overline{b}^\ast_1+\overline{b}_1)},\\ &\overline{b}_1=\frac{-ig\overline{a}^2({-}i\omega_m+\gamma_2/2)}{({-}i\omega_m-\gamma_1/2)({-}i\omega_m+\gamma_2/2)+J^2} ~,~~\overline{b}_2=\frac{-ig\overline{b}_1}{-i\omega_m+\gamma_2/2}. \end{aligned}$$

It can be seen from Eqs. (5) that the number of photons in the cavity field is mainly modulated by the optomechanical coupling. The number of phonons of the $MR_1$ is affected by the gain from the $MR_2$. Considering the influence of the $\mathcal {PT}$-symmetric MR on the cavity field as a whole, it can be known from the output spectrum of the probe field. Now we consider the perturbation made by the probe field. The quantum Langevin quations for the fluctuations are given by

(6)$$\begin{aligned}&\frac{d}{dt}\delta{a}=[{-}i\Delta-\kappa/2+ig(\overline{b}^\ast_1+\overline{b}_1)]\delta{a}+ig[\overline{a}(\delta{b^\ast_1}+\delta{b_1})+\delta{a}(\delta{b^\ast_1}+\delta{b_1})]+\sqrt{\eta\kappa}s_{in},\\ &\frac{d}{dt}\delta{b}_1=({-}i\omega_m-\gamma_1/2)\delta{b}_1+iJ\delta{b}_2+ig(\overline{a}^\ast\delta{a}+\overline{a}\delta{a^\ast}+\delta{a}\delta{a^\ast}),\\ &\frac{d}{dt}\delta{b}_2=({-}i\omega_m+\gamma_2/2)\delta{b}_2+iJ\delta{b}_1. \end{aligned}$$

Regarding Eqs. (6), we reserve some second-order small quantities that can be used to study the nonlinear effects of optomechanics. Particularly, the second-order small amount $\delta {a}(\delta {b^\ast _1}+\delta {b_1})$ and $\delta {a}\delta {a^\ast }$ is very important to the generation of sum sidebands in mechanical $\mathcal {PT}$-symmetric system, similar to the higher-order sidebands generated in OMIT [18]. Generally, Eqs. (6) can be solved analytically with the ansatz

(7)$$\begin{aligned}&\delta{a}=A^{+}_1e^{{-}i\delta_1{t}}+A^{-}_1e^{i\delta_1{t}}+A^{+}_2e^{{-}i\delta_2{t}}+A^{-}_2e^{i\delta_2{t}}+A^{+}_{s}e^{{-}i\Omega{t}}+A^{-}_se^{i\Omega{t}}+{\cdot}{\cdot}\cdot,\\ &\delta{b}_1=B^{+}_1e^{{-}i\delta_1{t}}+B^{-}_1e^{i\delta_1{t}}+B^{+}_2e^{{-}i\delta_2{t}}+B^{-}_2e^{i\delta_2{t}}+B^{+}_se^{{-}i\Omega{t}}+B^{-}_se^{i\Omega{t}}+{\cdot}{\cdot}\cdot,\\ &\delta{b}_2=C^{+}_1e^{{-}i\delta_1{t}}+C^{-}_1e^{i\delta_1{t}}+C^{+}_2e^{{-}i\delta_2{t}}+C^{-}_2e^{i\delta_2{t}}+C^{+}_se^{{-}i\Omega{t}}+C^{-}_se^{i\Omega{t}}+{\cdot}{\cdot}\cdot. \end{aligned}$$

The first group describes the linear response of the probe field, which can be measured via heterodyning [52]. The first-order output amplitude of the optical mode obtaining the power in the signal at ${\omega _p}$ are

(8)$$\begin{aligned}&A^{+}_j=\frac{\sqrt{\eta\kappa}\varepsilon_{j}\beta(\delta_j)}{F_{+}(\delta_j)\beta(\delta_j)+g^2|\overline{a}|^2\alpha(\delta_j)F_{-}(\delta_j)}~,~~ (A^{-}_j)^\ast{=}\frac{g^2(\overline{a}^\ast)^2\alpha(\delta_j)A^{+}_j}{\beta(\delta_j)},\\ &B^{+}_j=\frac{igE_{+}(\delta_j)[\overline{a}(A^{-}_j)^\ast{+}\overline{a}^{{\ast}}A^{+}_j]}{D_{+}(\delta_j)E_{+}(\delta_j)+J^2}~,~~ (B^{-}_j)^\ast{=}\frac{-igE_{-}(\delta_j)[\overline{a}(A^{-}_j)^\ast{+}\overline{a}^{{\ast}}A^{+}_j]}{D_{-}(\delta_j)E_{-}(\delta_j)+J^2}, \end{aligned}$$

where

(9)$$\begin{aligned}&D_\mp(y)={-}iy\mp{i\omega_m}+\gamma_1/2~,~~E_\mp(y)={-}iy\mp{i\omega_m}+\gamma_2/2,\\ &F_\mp(y)={-}iy\mp{[i\Delta-g(\overline{b}^\ast_1+\overline{b}_1)]}+\kappa/2~,~~ L_\mp(y)=D_\mp(y)E_\mp(y)+J^2,\\ &\alpha(y)={-}E_{-}(y)L_{+}(y)+E_{+}(y)L_{-}(y),\\ &\beta(y)=L_{-}(y)L_{+}(y)F_{-}(y)-g^2|\overline{a}|^2\alpha(y). \end{aligned}$$

Generally, the first-order output spectrum can obtain OMIT and slow light phenomena [48]. And, the second group corresponds to the sum sideband process,

(10)$$\begin{aligned}&A^{+}_s=\frac{-F_{-}(\Omega)g^2|\overline{a}|^2\alpha(\Omega)\phi_3+\psi}{\overline{a}^\ast[F_{+}(\Omega)\beta(\Omega)+F_{-}(\Omega)g^2|\overline{a}|^2\alpha(\Omega)]}~,~~B^{+}_s=\frac{igE_{+}(\Omega)(\overline{a}^\ast{A^{+}_s}+\overline{a}(A^{-}_s)^\ast{+}\phi_3)}{D_{+}(\Omega)E_{+}(\Omega)+J^2},\\ &(A^{-}_s)^\ast{=}\frac{g^2(\overline{a}^\ast)^2\alpha(\Omega)(A^{+}_s)+g^2\overline{a}^\ast\alpha(\Omega)\phi_3-ig\sigma\phi_1}{\beta(\Omega)}, \end{aligned}$$

where

(11)$$\begin{aligned}&\sigma=(D_{-}(\Omega)E_{-}(\Omega)+J^2)(D_{+}(\Omega)E_{+}(\Omega)+J^2),\\ &\psi=ig[F_{-}(\Omega)\overline{a}\sigma\phi_1+\beta(\Omega)\overline{a}^\ast\phi_2-\beta(\Omega)\overline{a}\phi_1],\\ &\phi_1=(A^{-}_1)^\ast(B^{+}_2+(B^{-}_2)^\ast)+(A^{-}_2)^\ast(B^{+}_1+(B^{-}_1)^\ast),\\ &\phi_2=A^{+}_1(B^{+}_2+(B^{-}_2)^\ast)+A^{+}_2(B^{+}_1+(B^{-}_1)^\ast)~,~\phi_3=(A^{-}_1)^\ast{A^{+}_2}+(A^{-}_2)^\ast{A^{+}_1}. \end{aligned}$$

Using the input-output relation [5], the output fields(in a rotating frame at $\omega _c$) of mechanical $\mathcal {PT}$-symmetric system can be obtained as follows:

(12)$$\begin{aligned}&S_{out}=\varepsilon_{c}-\sqrt{\eta\kappa}\overline{a}+(\varepsilon_{1}-\sqrt{\eta\kappa}A^{+}_1)e^{{-}i\delta_1{t}}-\sqrt{\eta\kappa}A^{-}_{1}e^{i\delta_1{t}}+(\varepsilon_{2}-\sqrt{\eta\kappa}A^{+}_2)e^{{-}i\delta_2{t}}\\ &~~~~~~~~-\sqrt{\eta\kappa}A^{-}_{2}e^{i\delta_2{t}}-\sqrt{\eta\kappa}A^{+}_{s}e^{{-}i\Omega{t}}-\sqrt{\eta\kappa}A^{-}_{s}e^{i\Omega{t}}. \end{aligned}$$

The term $\varepsilon _{c}-\sqrt {\eta \kappa }\overline {a}$ denotes the frequency spectra of the control field with $\omega _c$. $(\varepsilon _{1}-\sqrt {\eta \kappa }A^{+}_1)$ ($(\varepsilon _{2}-\sqrt {\eta \kappa }A^{+}_2)$)and $-\sqrt {\eta \kappa }A^{-}_{1}$ ($-\sqrt {\eta \kappa }A^{-}_{2}$) responding to the probe field with frequency $\omega _1$($\omega _2$), are the amplitudes of the anti-Stoke field and the Stoke field, respectively. The terms $-\sqrt {\eta \kappa }A^{+}_{s}e^{-i\Omega {t}}$ and $-\sqrt {\eta \kappa }A^{-}_{s}e^{i\Omega {t}}$ describe the upper and lower sum-sideband process, respectively. As shown in Fig. 1(b), the detuning frequency $\delta _i(i=1,2)$ between the probe field $\omega _i$ and the control field $\omega _c$ is $\omega _i-\omega _c$. When the control field and the probe field satisfies certain detuning conditions with the cavity field respectively, that is, $\Delta =\omega _m$ and $\delta _1+\delta _2=\omega _m$, the upper sum sideband can be generated. And, the lower sum sideband is generated at $\delta _1+\delta _2=-\omega _m$. The efficiency of the upper and lower sum sideband generation can be define as $\eta ^{+}_{s}=|-\sqrt {\eta \kappa }A^{+}_{s}/\varepsilon _1|$ and $\eta ^{-}_{s}=|-\sqrt {\eta \kappa }A^{-}_{s}/\varepsilon _1|$, respectively.

3. Results and discussions

The introduction of the $\mathcal {PT}$-symmetric MR into the optomechanical system greatly changes the characteristics of the cavity field. Especially the EPs effect has brought unexpected changes to the system. We mainly discuss the influence of $\mathcal {PT}$ symmetry on the efficiency of SSG by adjusting the mechanical coupling parameter $J$. Next, the enhancement of the efficiency of SSG by optomechanical coupling in the vicinity of EPs is analyzed. Finally, the influence of the detuning frequency $\delta _i$ between the probe field and the control field on SSG is discussed. Here the main parameters we use are from recent works [47,53], as follows: $\lambda =1573$ nm, $g=2\pi$ MHz, $\omega _m/2\pi =3.68$ GHz, $\kappa =0.1~\omega _m$, $\gamma _1=\gamma _2=0.5\times 10^{-2}~\omega _m$. Through comparison and analysis, it can be know that the efficiency of SSG is significantly enhanced in the vicinity of EPs.

Now we discuss the SSG in mechanical $\mathcal {PT}$-symmetric MR in different phases. In Fig. 3(a), in the broken-$\mathcal {PT}$-symmetric phase, the upper sum sideband generation(USSG) can be obtained at $\delta _1=\omega _m$ with $J=0.1(\gamma _1+\gamma _2)$. And, in Fig. 3(d), there are the lower sum sideband generation(LSSG) at $\delta _1=\omega _m$ and $\delta _1=1.05\omega _m$, respectively. Compared with the SSG in the general optomechanical system [20], here its efficiency is about the same in the broken-$\mathcal {PT}$-symmetric phase. In EPs(as shown in Fig. 3(b)), the USSG forms a sharp peak at $\delta _1=\omega _m$ and a absorption peak at $\delta _1=1.05\omega _m$, respectively. Correspondingly, the LSSG produces two peaks as described in Fig. 3(e). Here is a case where the nature of the EPs changes the conditions for optical path interference. In EPs, the eigenvalue of $\mathcal {PT}$-symmetric MR is degenerate, and its eigenfrequency is a complex value. This determines that the interaction between the optical mode and the mechanical mode has a strong sensitivity for perturbation. When the power of the control field drives the cavity field with a certain value, the system has destructive interference resulting in the appearance of the absorption peak. Under this circumstance, the interference condition satisfies $\Delta =\delta _1+\delta _2$. In Fig. 3(c), in the $\mathcal {PT}$-symmetric phase, the USSG splits to form two sharp peaks and two absorption peaks. And, four sharp peaks are formed in the LSSG as shown in Fig. 3(d). This is consistent with the analysis result of Eqs. (3), that is, the eigenfrequency of $\mathcal {PT}$-symmetric MR splits into two supermodes. Specially, an interesting phenomenon is that the USSG have changed from no to splitting into two absorption peaks at $\delta _1=1.05\omega _m$, which highlights the nature of the $\mathcal {PT}$-symmetric mechanical system. Significantly, the efficiency of SSG in EPs is enhanced by about two orders of magnitude compared to the broken $\mathcal {PT}$-symmetric phase.

From the above analysis, the state of $\mathcal {PT}$-symmetric MR has an important influence on SSG. Thereby, the mechanical coupling strength $J$ is an important parameter for adjusting the efficiency of SSG. Then, it is discussed that SSG is affected by the mechanical coupling parameter, and the properties of the $\mathcal {PT}$-symmetric MR are further analyzed. In Fig. 4, with the increase of $J$, the efficiency of SSG has undergone a change from weak to strong, especially near EPs, the efficiency of the SSG has been significantly enhanced. In Fig. 4(a), there are a sharp peak and a transparent window near EPs with $J=0.26(\gamma _1+\gamma _2)$. The physical explanation is that near EPs, the gain of one MR continuously compensates for the dissipation of the other MR, the localized field induces the dynamical accumulations of the acoustical energy in the passive mechanical resonator [36,54]. This corresponds to the increase of the optomechanical nonlinearity, which greatly enhances the control of the up-conversion process field, that is, improves the generation efficiency of the anti-Stokes field. As the mechanical coupling strength $J$ increases, the USSG is split. Likewise, the transparent window will also split into two absorption peaks as describde in Fig. 3(c). In Fig. 4(b), the efficiency of the LSSG still reaches its maximum in the vicinity of EPs , forming two sharp peaks. Furthermore, the output spectrum is split into four by two sharp peaks. A remarked feature prestent in Fig. 4(c) that is the efficiency of USSG is amplified in the vicinity of EPs. Comparing the optomechanical system of the single MR, where $J=0$, the efficiency of SSG is increased by more than three orders of magnitude in the mechanical $\mathcal {PT}$-symmetric system. Similarly, the efficiency of LSSG is significantly enhanced compared to the general optomechanical system, as described in Fig. 4(d).

Fig. 3. Plots the efficiencies (in logarithmic form) of USSG and LSSG vs $\delta _1$ for different values of $J$: (a)(d) $J=0.1(\gamma _1+\gamma _2)$, (b)(e) $J=0.25(\gamma _1+\gamma _2)$, (c)(f) $J=0.5(\gamma _1+\gamma _2)$. We use $\lambda =1573$ nm, $g=2\pi$ MHz, $\omega _m/2\pi =3.68$ GHz, $\kappa =0.1~\omega _m$, and $\gamma _1=\gamma _2=0.5\times 10^{-2}~\omega _m$. The other parameters are $\delta _2=-0.05\omega _m$, $P_c=1 \mathrm {\mu W}$, $P_1=P_2=0.05\mathrm {\mu W}$.

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Fig. 4. Efficiencies (in logarithmic form) of (a) USSG and (b) LSSG as a function of the mechanical coupling strength $J$ and the detuning frequency $\delta _1$. Efficiencies (in logarithmic form) of (c) USSG and (d) LSSG varies with the detuning frequency $\delta _1$. The other parameters are $P_c=10 \mathrm {\mu W}$, $P_1=P_2=0.5\mathrm {\mu W}$, $\delta _2=-0.05\omega _m$.

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In order to analyze the influence of optomechanical coupling on the efficiency of the SSG, it is necessary to discuss how the power of the control field changes SSG near EPs. In Fig. 5(a), the efficiency of USSG increases with the power of the control field and produces two sharp peaks at $P_c=10 \mathrm {\mu W}$. In this scenario, the optomechanical coupling and the coupling between the two oscillators achieve the best matching effect, which makes the efficiency of the SSG reach the maximum. When the power of the control field continues to increase, the efficiency of USSG produces an absorption peak at $\delta _1=1.05\omega _m$, which is similar to the description in Fig. 3(b). In Fig. 5(b), the efficiency of the LSSG increases with the increase of $P_c$ and the line width between the sharp peaks becomes wider. This means that the power of the control field can be used as a means to modulate SSG in the mechanical $\mathcal {PT}$-symmetric system.

Fig. 5. Plots the efficiencies (in logarithmic form) of (a) USSG and (b) LSSG on the pump power of the control field $P_c$ and the detuning frequency $\delta _1$ with $\delta _2=-0.05\omega _m$.

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Furthermore, this discussion is necessary that the impact on the efficiency of SSG is accompanied by changes in $\delta _1$ and $\delta _2$. In Fig. 6(a), the three-dimensional image shows that the efficiency of the USSG has three large peaks. It can be clearly seen in Fig. 6(c) that these three points are $(0,\omega _m)$, $(\omega _m,0)$, $(\omega _m,\omega _m)$, respectively. Obviously, the USSG achieving the maximum when $\delta _1=\omega _m$ or $\delta _2=\omega _m$. It is worth noting that the matching conditions of USSG in the vicinity of EPs are $\delta _1=\pm \omega _m$, $\delta _2=\pm \omega _m$, and $\delta _1+\delta _2=\omega _m$, which is different from previous studies [20]. The newly added matching condition is $\delta _1+\delta _2=\omega _m$, which also explains that the upper sum sideband can be generated at $\delta _1=1.05\omega _m$, as shown in Fig. 4(c). In contrast, in Fig. 6(b), there are ten sharp peaks in LSSG. This shows that LSSG can obviously be enhanced in the vicinity of EPs. In particular, it can be seen from Fig. 6(d) that the matching conditions of the LSSG in the vicinity of EPs are $\delta _1=\pm \omega _m$, $\delta _2=\pm \omega _m$, $\delta _1+\delta _2=\pm \omega _m$, which is consistent with the theoretical calculation results, as shown in Fig. 4(d). This reveals that the detuning frequencies between the probe fields and the control field can be selected well near EPs to enhance the efficiency of SSG.

Fig. 6. Efficiencies (in logarithmic form) of USSG and LSSG vs $\delta _1$ and $\delta _2$. The other parameters are the same as Fig. 4.

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4. Conclusion

In conclusion, we investigate analytically the efficiency of SSG in the mechanical $\mathcal {PT}$-symmetric system through two probe fields. Through numerical calculation and theoretical analysis, the efficiency of SSG is significantly enhanced in the vicinity of EPs. Meanwhile, the efficiency of SSG is also analyzed in broken-$\mathcal {PT}$-symmetric phase and $\mathcal {PT}$-symmetric phase. An interesting phenomenon occurs near the EPs. The efficiency of the USSG can be amplified by selecting appropriate mechanical coupling parameters. And, when the detuning satisfies the condition $\Delta =\delta _1+\delta _2$, a induced transparent window by optomechanical nonlinearity will appear. Additionally, by adjusting the mechanical coupling strength and the power of the control field, the maximum efficiency of SSG is obtained in vicinity of EPs. Significantly, the efficiency of SSG is enhanced by more than three orders of magnitude in the mechanical $\mathcal {PT}$-symmetric than in a general optomechanical system. Finally, the matching conditions of SSG in mechanical $\mathcal {PT}$-symmetric system by using two probe fields are summarized. This can be used for high-precision measurements [55], enhanced mechanical sensing, on-chip optical signal processing. Recently, fabrication of microcavity has made breakthrough progress [56], and some methods have been used to produce single-photon [57], two-photon sources [58] and low-threshold lasers [59,60]. In particular, sum-frequency generation in on-chip lithium niobate microdisk resonators has been observed [61], which will provide experimental possibilities for mechanical $\mathcal {PT}$-symmetric system.

Funding

National Key Research and Development Program of China (2016YFA0301203); National Natural Science Foundation of China (11875029, 11975103); Fundamental Research Funds for the Central Universities (2018KFYYXJJ032).

Acknowledgments

We thank Dr. Bao Wang and Dr. Chang-Sheng Hu for fruitful discussions.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Exceptional points enhance sum sideband generation in a mechanical $\mathcal {PT}$-symmetric system

Abstract

1. Introduction

2. Theoretical model and formulations

3. Results and discussions

4. Conclusion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (6)

Equations (12)

Optics Express