Brillouin cavity optomechanics sensing with enhanced dynamical backaction

Guo-Qing Qin; Min Wang; Jing-Wei Wen; Dong Ruan; Gui-Lu Long

doi:doi:10.1364/PRJ.7.001440

Photonics Research, 2019, 7 (12): 12001440, Published Online: Nov. 19, 2019

Brillouin cavity optomechanics sensing with enhanced dynamical backaction Download： 581次

论文大纲

Guo-Qing Qin ^1†Min Wang ^1†Jing-Wei Wen ¹Dong Ruan ^1,4,*Gui-Lu Long ^1,2,3,5,*

Author Affiliations

¹ State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China

² Beijing Information Science and Technology National Research Center, Beijing 100084, China

³ Beijing Academy of Quantum Information Sciences, Beijing 100193, China

⁴ e-mail: dongruan@tsinghua.edu.cn

⁵ e-mail: gllong@tsinghua.edu.cn

Abstract

Based on the dispersive interaction between a high quality factor microcavity and nano-objects, whispering-gallery-mode microcavities have been widely used in highly sensitive sensing. Here, we propose a novel method to enhance the sensitivity of the optical frequency shift and reduce the impact of the laser frequency noise on the detection resolution through Brillouin cavity optomechanics in a parity-time symmetric system. The optical spring effect is sensitive to the perturbation of optical modes around the exceptional point. By monitoring the shift of the mechanical frequency, the detection sensitivity for the optical frequency shift is enhanced by 2 orders of magnitude compared with conventional approaches. We find the optical spring effect is robust to the laser frequency noise around the exceptional point, which can reduce the detection limitation caused by the laser frequency instability. Thus, our method can improve the sensing ability for nano-object sensing and other techniques based on the frequency shift of the optical mode.

1 INTRODUCTION

Whispering-gallery-mode (WGM) optical resonators, with their high quality factors and small mode volumes, enable strong light–matter interactions, and have become a significant platform for fundamental studies and technological applications of light [1 –10" target="_self" style="display: inline;">–10]. The microcavity system has been widely applied to ultrasensitive detection [11 –22" target="_self" style="display: inline;">–22], such as biomolecule sensing [23,24] and nanoparticle sensing [25 27" target="_self" style="display: inline;">–27].

In recent years, sensitivity enhancement of microcavity sensing has been theoretically and experimentally demonstrated at non-Hermitian spectral degeneracies known as the exceptional point (EP) [7,28 –37" target="_self" style="display: inline;">–37]. For conventional microcavity sensing, the induced eigenvalue changes of normal modes are proportional to the perturbation strength [25,38 40" target="_self" style="display: inline;">–40]. While the frequency splitting, resulting from the non-degeneracy induced by a small perturbation, scales as the square-root of the perturbation strength around the EPs, thus exceeding the splitting observed in the conventional sensing schemes. To further improve the sensitivity of WGM sensors, the system could be operated at the $N$ th-order exceptional points [31].

Besides directly utilizing the optical spectrum for sensing, optomechanical systems [5,41 –44" target="_self" style="display: inline;">–44] have also been revealed as excellent candidates for sensing [14,45,46]. This method can also enhance the sensing sensitivity and has been used in different fields, such as mass sensing [16], ultrasensitive gas detection [47], and single molecule detection [48]. By monitoring the optical spring effect, the detection resolution of the optical mode change is enhanced by orders of magnitude [48]. The main reason of such sensing enhancement lies in the fact that the effective mechanical linewidth (e.g., 0.1 Hz in Ref. [48]) is much smaller than the linewidth of the optical mode. The external perturbation to the optical mode is easy to detect with the optomechanical interaction. Thus, optomechanical interactions offer an effective and promising way for sensing.

In this paper, we propose a novel method that utilizes the Brillouin scattering interaction for nano-object sensing based on parity-time ( $PT$ ) symmetric microcavities. In this model, a passive optical microcavity with Brillouin optomechanics couples with an active microcavity, as shown in Fig. 1(a). Here, the passive optical sphere could support three modes: the Brillouin mechanical mode, coupled with the anti-Stokes optical mode, and the Stokes optical mode, as shown in Fig. 1(b). For this Brillouin optomechanical interaction, strict constraints of energy conservation and momentum conservation are required for the modes [49 –52" target="_self" style="display: inline;">–52]. Thus the Brillouin scattering interaction is sensitive to the changes of the optical mode or mechanical mode [47,53,54]. Different from previous works using $PT$ optomechanical systems [14,55,56], our system operates in the $PT$ symmetry broken regime. We show that the detection resolution of the optical frequency shift is enhanced when the passive cavity with Brillouin cavity optomechanics couples to an active cavity. Specifically, when the Stokes optical mode couples to the gain cavity, the Stokes scattering process is enhanced, which leads to giant enhancement of the optical spring effect. Benefiting from the optical spring effect, 2 orders of magnitude enhancement has been achieved near the EP. Moreover, the robustness of the optical spring effect to the laser frequency instability around the EP is first revealed in this paper.

(a) Schematic of a Brillouin interaction in the parity-time symmetric system. The modes a2, a3, and b in the lossy cavity represent the anti-Stokes mode, Stokes mode, and mechanical mode, respectively, and the gain microsphere supports the cavity mode a1. The waveguide can couple the light into the cavity and collect the cavity emission power for sensing. (b) The cavity modes and mechanical mode are illustrated in the frequency domain.

Fig. 1. (a) Schematic of a Brillouin interaction in the parity-time symmetric system. The modes $a_{2}$ , $a_{3}$ , and $b$ in the lossy cavity represent the anti-Stokes mode, Stokes mode, and mechanical mode, respectively, and the gain microsphere supports the cavity mode $a_{1}$ . The waveguide can couple the light into the cavity and collect the cavity emission power for sensing. (b) The cavity modes and mechanical mode are illustrated in the frequency domain.

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2 THEORETICAL MODEL

In this model, a lossy optical microcavity with the Brillouin optomechanics couples with a gain cavity, as schematically shown in Fig. 1. The gain optical cavity is described by the annihilation (creation) operator $a_{1}$ ( $a_{1}^{†}$ ) with frequency $ω_{1}$ and gain $κ$ . The lossy cavity supports the Brillouin mechanical mode $b$ with the anti-Stokes optical mode $a_{2}$ and Stokes optical mode $a_{3}$ . The Hamiltonian can be described in the frame rotating at the driving field frequency $ω_{L}$ : $H / ℏ = H_{0} + H_{I} + H_{d}, H_{0} = (- Δ_{1} + i κ) a_{1}^{†} a_{1} - (Δ_{2} + i γ_{2}) a_{2}^{†} a_{2} - (Δ_{3} + i γ_{3}) a_{3}^{†} a_{3} + (ω_{m} - i \frac{Γ_{m}}{2}) b^{†} b, H_{I} = J (a_{1}^{†} a_{3} + a_{3}^{†} a_{1}) + g_{0} (a_{2}^{†} a_{3} b + a_{2} a_{3}^{†} b^{†}), H_{d} = i \sqrt{κ_{ex 2}} α_{L} (a_{2}^{†} - a_{2}),$ (1)where $Δ_{j} = ω_{L} - ω_{j}$ ( $j = 1, 2, 3$ ) are detunings of optical modes with respect to the driving field. $J$ is the optical coupling rate between the optical modes $a_{1}$ and $a_{3}$ . And $g_{0}$ denotes the coupling between the optical modes and mechanical mode, $α_{L}$ is the pump rate for optical mode $a_{2}$ , and $κ_{ex 2}$ indicates the coupling rate between the tapered fiber and optical mode $a_{2}$ . $γ_{2}$ , $γ_{3}$ denote the optical damping rate for the optical modes $a_{2}$ and $a_{3}$ , respectively. $Γ_{m}$ is the mechanical damping rate of the mechanical mode. Under the non-depletion approximation of the non-linear three-wave mixing, following the standard linearization procedure, the effective Hamiltonian in the blue sideband driving regime can be described as $H_{0} = (- Δ_{1} + i κ) a_{1}^{+} a_{1} - (Δ_{3} + i γ_{3}) a_{3}^{+} a_{3} + (ω_{m} - i \frac{Γ_{m}}{2}) b^{+} b, H_{I} = G (a_{3}^{+} b^{+} + a_{3} b) + J (a_{1}^{+} a_{3} + a_{3}^{+} a_{1}),$ (2)with $G = g_{0} \sqrt{\frac{κ_{ex 2} P_{in}}{(Δ_{2}^{2} + γ_{2}^{2}) ℏ ω_{2}}} .$ (3)Here the power of the pump beam is denoted by $P_{in}$ with $α_{L} = \sqrt{P_{in} / (ℏ ω_{L})}$ . The quantum Langevin equations of all fluctuation operators for the linearized Hamiltonian obey $\frac{d X}{d t} = M X + ϵ$ , where $X = {(a_{1}, a_{3}, b^{†})}^{T}$ is the operator representing the system, $ϵ = {(0, 0, b_{in}^{†})}^{T}$ is the input noise vector, $b_{in}$ is the noise operator associated with the mechanical dissipation, and coefficient matrix M is $M = (\begin{matrix} i Δ_{1} + κ & - i J & 0 \\ - i J & i Δ_{3} - γ_{3} & - i G \\ 0 & i G & i ω_{m} - \frac{Γ_{m}}{2} \end{matrix}) .$ (4)Operators $(a_{1}, a_{3}, b^{†})$ and their Hermitian conjugates constitute the coupled operator equations, which could be solved by the method of the Fourier transform. After the calculation, the mechanical frequency $Ω$ , and the linewidth $Γ$ without external perturbation are described as $Ω = Ω_{m} + ω_{opt}$ and $Γ = Γ_{m} + Γ_{opt}$ , where the optical damping rate $Γ_{opt}$ and the spring effect $ω_{opt}$ are then given by $Γ_{opt} = - 2 G^{2} \frac{γ_{3} {(δ_{1}^{2} + κ^{2})}^{2} - J^{2} κ (δ_{1}^{2} + κ^{2})}{{[γ_{3} (δ_{1}^{2} + κ^{2}) - J^{2} κ]}^{2} + {[δ_{3} (δ_{1}^{2} + κ^{2}) - J^{2} δ_{1}]}^{2}},$ (5) $ω_{opt} = - G^{2} \frac{δ_{3} {(δ_{1}^{2} + κ^{2})}^{2} - J^{2} δ_{1} (δ_{1}^{2} + κ^{2})}{{[γ_{3} (δ_{1}^{2} + κ^{2}) - J^{2} κ]}^{2} + {[δ_{3} (δ_{1}^{2} + κ^{2}) - J^{2} δ_{1}]}^{2}},$ (6)where $δ_{1} = ω_{m} - Δ_{1}$ and $δ_{3} = ω_{m} - Δ_{3}$ .

Before we discuss the effects of EPs for sensing, we first study the $PT$ symmetric optical system of optical modes $a_{1}$ and $a_{3}$ . The modal field evolution for modes $a_{1}$ and $a_{3}$ without Brillouin scattering interaction is described by $i d V / d t = H_{13} V$ , where $V = {(a_{1}, a_{3})}^{T}$ represents the modal column vector. The Hamiltonian $H_{13}$ can be written in a $2 \times 2$ matrix form: $H_{13} = (\begin{matrix} ω_{1} + i κ & J \\ J & ω_{3} - i γ_{3} \end{matrix}) .$ (7)By diagonalizing this optical subsystem via the transformation $A = D V$ , where $A = {(A_{+}, A_{-})}^{T}$ and $D$ represent the supermodes and the transform matrix [14], respectively. The corresponding associated eigenvalues are $ω_{\pm} = \frac{ω_{1} + ω_{3} + i κ - i γ_{3} \pm \sqrt{{[ω_{1} - ω_{3} + i (κ + γ_{3})]}^{2} + 4 J^{2}}}{2} .$ (8)When $ω_{1} = ω_{3}$ , the eigenfrequencies are simplified to $ω_{\pm} = ω_{3} + \frac{i}{2} (κ - γ_{3}) \pm \frac{\sqrt{4 J^{2} - {(κ + γ_{3})}^{2}}}{2} .$ (9)In this configuration, the two eigenstates coalesce when $J^{2} = {(κ + γ_{3})}^{2} / 4$ . The system exhibits its second-order exceptional point, also known as the $PT$ -phase transition point. When the mode coupling rate is tuned to satisfy $J^{2} < {(κ + γ_{3})}^{2} / 4$ , the system is in a symmetry broken phase and the two eigenstates have the same eigenfrequency but different loss.

To study the optical damping rate $Γ_{opt}$ and the spring effect $ω_{opt}$ , we plot the mechanical responses around the EP in Fig. 2. Here the parameters are feasible in the experiment [7,49,50]: $κ = 1 MHz$ , $P_{in} = 17.5 μ W$ , $γ_{2} = 2.6 MHz$ , $g_{0} = 4 Hz$ , $Ω_{m} = 50 MHz$ , $Γ_{m} = 1 kHz$ , and $J = κ$ . Figure 2, region A1 indicates that the mechanical frequency shifts more than 7 MHz in the blue sideband driving regime, which means the optical spring effect is enhanced compared with the performance in a single passive cavity [57,58]. We plot the optical damping rate $Γ_{opt}$ and the supermodes spectrum $ω_{\pm}$ in Fig. 2, regions A2 and A3, respectively. The blue dashed line plots the imaginary parts of the eigenfrequencies ( $Im ω_{\pm}$ ) and the red dashed line indicates the real parts of the eigenfrequencies ( $Re ω_{\pm}$ ) as a function of the $γ_{3}$ in Fig. 2, region A3. The optical damping rate $Γ_{opt}$ behaves very differently around the EP. Unlike the conventional Brillouin scattering interaction, where the mechanical mode will only be heated in the blue sideband driving regime, the mechanical mode could even be cooled down in this regime. When the system is in a symmetry broken phase, the mechanical mode is heated, which indicates the Stokes scattering process is enhanced. While in the symmetry unbroken phase, the mechanical mode is cooled by annihilating phonons with $γ_{3} < κ$ , which indicates the anti-Stokes scattering process is enhanced. Figure 2, regions B1 and B2 present the optomechanical backaction around the EP in the red sideband pumping regime. The similar feature is that the mechanical damping rate has the sign change at the $PT$ -phase transition point. $Γ$ is positive in the $PT$ symmetry broken phase, while negative in the $PT$ symmetric phase.

Mechanical responses and the supermodes spectrum as a function of γ3. Frequency shifts are plotted in A1 and B1; A2 and B2 denote the optical damping rate Γopt. Here the real and imaginary parts of complex numbers ω± are denoted by the red and blue dashed lines, respectively. A1–A3 indicate the mechanical responses and supermode spectrum with blue-detuned driving. B1–B3 plot the mechanical responses and supermode spectrum in the red sideband driving regime. The parameters used are κ=1 MHz, Pin=17.5 μW, γ2=2.6 MHz, g0=4 Hz, Ωm=50 MHz, Γm=1 kHz, J=κ, δ3=−1.95 kHz, and δ1=0.99δ3.

Fig. 2. Mechanical responses and the supermodes spectrum as a function of $γ_{3}$ . Frequency shifts are plotted in A1 and B1; A2 and B2 denote the optical damping rate $Γ_{opt}$ . Here the real and imaginary parts of complex numbers $ω_{\pm}$ are denoted by the red and blue dashed lines, respectively. A1–A3 indicate the mechanical responses and supermode spectrum with blue-detuned driving. B1–B3 plot the mechanical responses and supermode spectrum in the red sideband driving regime. The parameters used are $κ = 1 MHz$ , $P_{in} = 17.5 μ W$ , $γ_{2} = 2.6 MHz$ , $g_{0} = 4 Hz$ , $Ω_{m} = 50 MHz$ , $Γ_{m} = 1 kHz$ , $J = κ$ , $δ_{3} = - 1.95 kHz$ , and $δ_{1} = 0.99 δ_{3}$ .

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The reason for the sign change of $Γ_{opt}$ is as follows. For the Brillouin scattering interaction, two optical modes are separated and only the Stokes optical mode couples to the gain cavity. That means the optical frequency $ω_{2}$ and the loss $γ_{2}$ of the pump mode are not affected by the gain $κ$ and the mode coupling strength $J$ . The pump photons are stable and robust to the gain cavity. In Fig. 2, regions A1–A3, it is clear that in the $PT$ symmetric phase regime with $γ_{3} < κ$ , the loss of the $a_{3}$ mode has been compensated by the gain. The Stokes field is much stronger than the anti-Stokes field even when the system is driven by the anti-Stokes pump, and the Stokes scattering is suppressed resulting in the cooling process. While in the $PT$ broken phase regime, the mechanical mode is heated in the blue sideband driving regime. We can obtain that both the optical spring effect and the optical damping rate are enhanced in the coupled gain and loss resonator system, which has been proved in other systems [57,58]. This optical spring effect enhancement makes the coupled system a promising candidate for metrology and sensing applications.

For the Brillouin scattering interaction, the mechanical mode is a traveling wave phonon mode with non-zero momentum $k_{b}$ . Phase matching of these modes in both frequency space and momentum space is essential. The energy and momentum of these three discrete modes must be matched to satisfy $ω_{2} - ω_{3} = ω_{m}$ and $k_{b} = k_{2} - k_{3}$ simultaneously. Here, $k_{b}$ , $k_{2}$ , and $k_{3}$ represent the momentum of the propagating acoustic mode $b$ , anti-Stokes optical mode $a_{2}$ , and Stokes optical mode $a_{3}$ , respectively. In the supermode picture, the mechanical mode $b$ couples to the supermodes $A_{+}$ and $A_{-}$ . Hence the phase matching of the supermodes and the mechanical mode is required, namely $ω_{2} - ω_{\pm} = ω_{m}$ and $k_{b} = k_{2} - k_{\pm}$ , where $k_{\pm}$ indicates the supermodes momentum. Therefore, in the following parts, the parameters are chosen such that the system is in the $PT$ symmetry broken regime. We also set the driving power below the phonon laser threshold to ensure the stability with blue-detuned driving.

We consider the sensing scheme based on dispersive sensing, such as the optical resonant wavelength shift. In general, the resonance wavelength shift is caused by dispersive interactions between the passive cavity and external perturbation. These perturbations can be induced by nanoparticles, molecules, chemical gas, and thin membranes. In our scheme, the passive optical cavity is designed to detect nano-objects. By dispersive coupling, nano-objects perturb the cavity modes $a_{2}$ and $a_{3}$ simultaneously, resulting in a cavity resonant frequency shift $ν$ , where $ω_{2}^{'} = ω_{2} + ν$ and $ω_{3}^{'} = ω_{3} + ν$ . Under the small perturbation $ν$ without the laser noise, the optical spring effect $Ω_{m} (ν)$ and optomechanical damping rate $Γ (ν)$ could be rewritten in $Γ (ν) = Γ_{m} - 2 G^{2} \frac{γ_{eff}}{{(δ + ν)}^{2} + γ_{eff}^{2}},$ (10) $Ω_{m} (ν) = ω_{m} - G^{2} \frac{δ + ν}{{(δ + ν)}^{2} + γ_{eff}^{2}},$ (11)where $δ = δ_{3} - \frac{J^{2}}{δ_{1}^{2} + κ^{2}} δ_{1}$ , $γ_{eff} = γ_{3} - \frac{J^{2}}{δ_{1}^{2} + κ^{2}} κ$ . Then we can obtain the mechanical frequency shift $Δ Ω (ν) = Ω_{m} (ν) - Ω$ and the damping rate change $Δ Γ (ν) = Γ (ν) - Γ$ induced by perturbation: $Δ Γ (ν) = 2 G^{2} \frac{2 γ_{eff} δ ν + γ_{eff} ν^{2}}{[{(δ + ν)}^{2} + γ_{eff}^{2}] (δ^{2} + γ_{eff}^{2})},$ (12) $Δ Ω (ν) = G^{2} \frac{δ^{2} ν - γ_{eff}^{2} ν + δ ν^{2}}{[{(δ + ν)}^{2} + γ_{eff}^{2}] (δ^{2} + γ_{eff}^{2})} .$ (13)To compare the sensitivity of $Δ Γ (ν)$ and $Δ Ω (ν)$ to $ν$ , the absolute value of the mechanical frequency shift $Δ Ω (ν)$ and the effective linewidth $Γ (ν)$ are plotted in Figs. 3(a) and 3(b) as a function of the optical mode shift $ν$ . The nano-object is detectable if the mechanical frequency shift $| Δ Ω (ν) |$ can be resolved from $Γ (ν)$ . $| Δ Ω (ν) |$ (red line) will change linearly with $ν$ when $J = 0.97 κ$ , while $Γ (ν)$ (blue dashed line) is insensitive to the external perturbation in Fig. 3(a). When $| ν | > 800 Hz$ , $| Δ Ω (ν) |$ is larger than $Γ (ν)$ , and thus the optical mode shift $ν$ is recognizable. In Fig. 3(b), both $| Δ Ω (ν) |$ and $Γ (ν)$ are sensitive to the optical mode shift when the system is very close to the EP ( $J = 1.01 κ$ ). As the perturbation $ν$ increases, $| Δ Ω (ν) |$ varies faster than $Γ (ν)$ . While for a large mode shift $ν$ , the mechanical damping rate limits the detection since $Γ (ν)$ increases faster than $| Δ Ω (ν) |$ . We plot the difference value between $| Δ Ω (ν) |$ and $Γ (ν)$ in Figs. 3(c) and 3(d) in different detuning conditions. The red areas indicate $| Δ Ω (ν) | < Γ (ν)$ , which is undetectable. While the blue areas plot the detectable areas with $| Δ Ω (ν) | > Γ (ν)$ . It is clear that approaching the EP, both the spring effect and optical damping rate become more and more sensitive to the external perturbation.

Absolute value of the mechanical frequency shift ΔΩm(ν) and effective linewidth Γ(ν) with the perturbation ν. The red lines show the absolute value of the mechanical frequency shift and the blue dashed lines plot Γ(ν) in (a) and (b) with δ1=δ3=5 kHz. The difference between Abs(ΔΩm(ν)) and Γν are plotted in (c) and (d) with the detuning δ3 and the perturbation ν. The coupling rates J are 0.97κ and 1.01κ in (c) and (d), respectively. Other parameters used here are γ3=1 MHz, κ=0.975γ3, and Γ=5 Hz.

Fig. 3. Absolute value of the mechanical frequency shift $Δ Ω_{m} (ν)$ and effective linewidth $Γ (ν)$ with the perturbation $ν$ . The red lines show the absolute value of the mechanical frequency shift and the blue dashed lines plot $Γ (ν)$ in (a) and (b) with $δ_{1} = δ_{3} = 5 kHz$ . The difference between $A b s (Δ Ω_{m} (ν))$ and $Γ_{ν}$ are plotted in (c) and (d) with the detuning $δ_{3}$ and the perturbation $ν$ . The coupling rates $J$ are $0.97 κ$ and $1.01 κ$ in (c) and (d), respectively. Other parameters used here are $γ_{3} = 1 MHz$ , $κ = 0.975 γ_{3}$ , and $Γ = 5 Hz$ .

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Considering the enhanced dynamical backaction effect, it is necessary to study the effect of the laser frequency instability. When a frequency noise of the pump laser couples into the microcavity, it will induce the frequency noise [59], which could spoil the detection sensitivity. Hence it is necessary to study the effect of the laser frequency instability in this optomechanical system. Without loss of generality, the laser frequency noise $σ$ will affect the pump frequency: $ω_{L}^{'} = ω_{L} + σ$ . Following the same calculation process of Eqs. (12) and (13), we can easily obtain the mechanical changes induced by the pump laser frequency noise. We denote the laser frequency noise-induced mechanical frequency shift and damping rate change as $Δ Ω (σ)$ and $Δ Γ (σ)$ , respectively. To detect the optical frequency shift, we define the symmetric power spectral density of the output mode $S (ω)$ , which is defined as $S (ω) = \int_{- \infty}^{+ \infty} d t e^{- i ω t} κ_{e x 3} ⟨ a_{3}^{⏧} (t) a_{3} (0) ⟩,$ (14)where $κ_{ex 3}$ is the external coupling strength between the Stokes mode and the tapered fiber. Through the cavity emission spectrum $S (ω)$ , we can analyze the cause of the mechanical frequency shift. In Figs. 4(a) and 4(b), we plot the cavity emission spectrum $S (ω)$ with different laser frequency noise $σ$ as the optical mode frequency shifts 300 Hz. The output spectrum is near the EP in Fig. 4(b) while it is away from the EP ( $J = 0.8 κ$ ) in Fig. 4(a). In Fig. 4(a), the spectrum $S (ω)$ shifts according to the frequency noise $σ$ . The inset shows the mechanical frequency shift $Δ Γ (σ)$ as a function of the noise $σ$ . The mechanical frequency shift can be 1.45 Hz (at $σ = 3 kHz$ ) while the effective mechanical linewidth is 1 Hz (at $σ = 0$ ). The laser frequency noise will limit the detection precision in this situation. In Fig. 4(b), the black solid line indicates that $Δ Ω (ν)$ is $- 8.6 Hz$ without the influence of the laser noise. While all the lines overlap making them hard to resolve even the laser frequency shift is 10 times larger than $ν$ . The cavity emission spectrum is immune to the noise $σ$ . Then we define $η = Δ Ω (σ) / Δ Ω (ν)$ to characterize the influence of the laser noise. The relation between $η$ and $σ$ is shown in Figs. 4(c), where the effective mechanical linewidth is 1 Hz. For the green solid line with coupling rate $J = 0.8 κ$ , the laser frequency noise can affect the detection sensitivity; especially when $| η | > 1$ , the laser frequency noise becomes the primary source to induce the mechanical frequency shift. When the system is around the EP with $J = κ$ (red solid line), $η$ approaches zero and the optical spring effect is robust to the laser frequency instability.

Cavity emission spectrum with the effect of laser frequency noise in (a) and (b). The perturbation ν equals 300 Hz and the mechanical linewidth is 1 Hz. The insets show the mechanical frequency shift induced by laser frequency noise σ. The coupling rate J=0.8κ in (a) and J=κ in (b). (c) We plot η as a function of σ with different coupling rates J. The red line indicates the system is robust to laser frequency noise near the exceptional point. The parameters used here are γ3=1.00 MHz, Δ1=Ωm, Δ3=Ωm, κ=0.985γ3, and Γ=1 Hz.

Fig. 4. Cavity emission spectrum with the effect of laser frequency noise in (a) and (b). The perturbation $ν$ equals 300 Hz and the mechanical linewidth is 1 Hz. The insets show the mechanical frequency shift induced by laser frequency noise $σ$ . The coupling rate $J = 0.8 κ$ in (a) and $J = κ$ in (b). (c) We plot $η$ as a function of $σ$ with different coupling rates $J$ . The red line indicates the system is robust to laser frequency noise near the exceptional point. The parameters used here are $γ_{3} = 1.00 MHz$ , $Δ_{1} = Ω_{m}$ , $Δ_{3} = Ω_{m}$ , $κ = 0.985 γ_{3}$ , and $Γ = 1 Hz$ .

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From Fig. 4, we can obtain that the output signal is robust to the laser frequency noise while sensitive to the external perturbation. For further explanation, we rewrite $ω_{\pm}$ in the frame of frequency $ω_{L}$ as a function of $ν$ and $σ$ : $ω_{\pm} = - Δ_{3} - σ + \frac{i}{2} (κ - γ_{3}) \pm \frac{\sqrt{4 J^{2} - {[i ν + (κ + γ_{3})]}^{2}}}{2},$ (15)where $ω_{1} = ω_{3}$ . The frequency splitting between the real components of $ω_{\pm}$ is now expressed by $Re (ω_{+} - ω_{-}) ≅ \sqrt{2 κ ν}$ . When the small frequency change $ν$ is introduced to the optical oscillator, it should be noted that normal modes frequency splitting is enhanced by $\sqrt{2 κ ν}$ in the $PT$ symmetric coupled cavity system. And this frequency splitting is transferred to the mechanical mode by enhanced dynamical backaction. Due to the energy and momentum conservation requirement, the mechanical spring effect is sensitive to the optical frequency shift $ν$ . While the laser noise $σ$ can only move supermodes frequency without any amplification, thus the spring effect is insensitive to the laser noise.

The detection limit of the optical frequency shift depends on the linewidth of the mechanical mode. It is instructive to study the detection sensitivity in a $PT$ symmetric system and a single passive cavity system. Here we define $ν_{P}$ and $ν_{P T}$ to describe the detection resolution in a single passive cavity system and the $PT$ symmetric system, respectively. The relation between $ν_{P T}$ and $ν_{P}$ can be deduced as $ν_{P T} = \frac{γ_{eff}}{γ_{3}} ν_{P}$ . In particular, we can define a sensitivity amplification factor (S.A.) as the ratio of $ν_{P} / ν_{P T}$ : $S.A. = \frac{γ_{3}}{γ_{eff}} .$ (16)The cavity emission spectra corresponding to the optical mode shift $ν$ around the EP are shown in Fig. 5(a). The mechanical oscillator has the effective linewidth $Γ = 1 Hz$ for the unperturbed situation. In Fig. 5(b), we plot the resolution $ν_{P T}$ of the optical frequency shift as a function of the coupling rate $J$ . The dashed lines represent the mechanical linewidth $Γ (ν)$ when $ν = ν_{P T}$ , indicating the mechanical damping rate remains unchanged. In the cavity emission spectra, the optical frequency shift $ν = 80 Hz$ is obviously detectable, plotted by the red solid line in Fig. 5(a), while the detection resolution $ν_{P}$ is around 2 kHz in the single passive cavity system (at $J = 0$ ) in Fig. 5(b) (green solid line). The solid lines plot the detection resolution with different effective mechanical linewidths in Fig. 5(b). Compared with the single passive cavity ( $J = 0$ ), the sensitivity of the frequency shift $ν$ is enhanced for at least 2 orders of magnitude around the EP.

(a) Normalized output spectra with different optical frequency shifts ν. The black solid line represents the cavity emission spectrum without external perturbation when the mechanical linewidth is 1 Hz. And the output spectrum is normalized to 1 separately. (b) The resolution of ν as a function of the coupling rate J. The red, green, and blue solid lines indicate the minimum detectable optical cavity resonance shift with Γ=2, 1, and 0.5 Hz, respectively. The parameters used here are γ3=1.00 MHz, Δ1=Ωm, Δ3=Ωm, and κ=0.985γ3. And the coupling rate J=κ in (a).

Fig. 5. (a) Normalized output spectra with different optical frequency shifts $ν$ . The black solid line represents the cavity emission spectrum without external perturbation when the mechanical linewidth is 1 Hz. And the output spectrum is normalized to 1 separately. (b) The resolution of $ν$ as a function of the coupling rate $J$ . The red, green, and blue solid lines indicate the minimum detectable optical cavity resonance shift with $Γ = 2$ , 1, and 0.5 Hz, respectively. The parameters used here are $γ_{3} = 1.00 MHz$ , $Δ_{1} = Ω_{m}$ , $Δ_{3} = Ω_{m}$ , and $κ = 0.985 γ_{3}$ . And the coupling rate $J = κ$ in (a).

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In summary, we propose a novel method to detect the minute optical frequency shift from the mechanical motion, which relies on the hypersensitized response near the EP. The anti-Stokes mode decouples from the gain cavity; thus the system can operate stably with the blue-detuned driving below the phonon laser threshold. The spring effect is dramatically enhanced around the EP since the loss of the Stokes mode has been compensated. In the supermode picture, the normal modes frequency splitting is enhanced around the EP. By coupling to the Brillouin phonons, the normal modes splitting is transferred to the mechanical mode with enhanced optomechanical dynamical backaction. We observe that the detection sensitivity is increased by 2 orders of magnitude near the phase transition point. In addition, the system is robust to the pump laser frequency noise, suitable for a practical optomechanical sensor. Benefiting from the ultrahigh resolution of the mechanical frequency shift, the system can be applied for nanoparticle sensing, such as the nanoparticle size and position sensing, displacement detection of a thin membrane, and other physical sensing applications.

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1 INTRODUCTION

2 THEORETICAL MODEL

Guo-Qing Qin, Min Wang, Jing-Wei Wen, Dong Ruan, Gui-Lu Long. Brillouin cavity optomechanics sensing with enhanced dynamical backaction[J]. Photonics Research, 2019, 7(12): 12001440.

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