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Resolution is one of the key indicators in the cavity optomechanical mass sensing. The bound states in the continuum (BIC) enable extremely narrow linewidths, which have great potential for enhancing the resolution of cavity optomechanical mass sensors. In order to enhance the resolution of cavity optomechanical mass sensing, we propose a simple double-cavity optomechanical system under the blue-detuning condition to realize the BIC singularity, and present an ultrahigh-resolution mass sensing scheme based on BIC in this paper. By solving the linearized Heisenberg-Langevin equations, the expressions for the susceptibility and transmission rate of the system are derived. Based on the system’s susceptibility, we study the absorption characteristics of the probe field under the blue-detuning condition. The absorption spectrum of the system exhibits three peaks, among which the central narrow peak exhibits optical gain characteristics, collectively forming a phenomenon analogous to double optomechanically induced transparency. Then, analysis of the dressed-state energy-level structure reveals that the formation of the central narrow peak stems from quantum interference effects in a double-Λ-type dark-state resonance. The linewidth evolution of the quasi-BIC central narrow peak is investigated by analyzing the dependence of the real part and imaginary part of the corresponding eigenvalue on the optomechanical coupling strength. It can be found that the imaginary part of the eigenvalue for the central narrow peak becomes zero when the optomechanical cooperativity coefficient equals the double-cavity cooperativity coefficient plus one, enabling the realization of BIC. The linewidth of the central peak is ultrasmall under this BIC condition, and the shift of the transmission peak in the transmission spectrum is linearly related to the adsorbed mass. Based on these characteristics, the system under the BIC condition can achieve mass sensing with an ultrahigh resolution, with a resolution of approximately 1 ag. Meanwhile, the linewidth of the transmission peak can be suppressed below 1 Hz, which is superior to the traditional optomechanical mass sensing schemes based on four-wave mixing, photonic molecules, and plasmon polaritons. Systematic investigation of eigenvalue variations and the corresponding sensitivity enhancement factors under mechanical resonator frequency shift reveals that the real part and the imaginary part of the eigenvalue associated with the central peak exhibit negligible variations under such perturbations. This indicates that the mass sensing scheme based on BIC in the double-cavity optomechanical system can maintain ultrahigh resolution and precise mass measurement under mechanical resonator frequency shift. Our scheme provides an approach for realizing the BIC singularity in optomechanical systems, and presents a new route to improving the resolution of mass sensors based on cavity optomechanical systems.
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Keywords:
- optomechanics /
- optomechanically induced transparency /
- mass sensing /
- bound-states in the continuum
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图 1 双腔耦合的光力系统示意图, 光力腔a由泵浦场驱动, 其本征频率和衰减率分别为$ {\omega _{\text{a}}} $和$ \kappa $, 通过光力作用与本征频率和衰减率分别为$ {\omega _{\text{m}}} $和$ {\gamma _{\text{m}}} $的机械振子耦合, 而辅助腔c与光力腔a耦合, 耦合强度为J, 其本征频率和衰减率分别为$ {\omega _{\text{c}}} $和$ {\kappa _{\text{c}}} $, 探测场输入辅助腔c
Figure 1. Schematic diagram of a dual-cavity coupled optomechanical system, where the optomechanical cavity with the frequency $ {\omega _{\text{a}}} $ and dissipation rate $ \kappa $ is driven by the pumping field and couples to the mechanical resonator with the frequency $ {\omega _{\text{m}}} $ and dissipation rate $ {\gamma _{\text{m}}} $ by the optomechanical interaction. The auxiliary cavity c with the frequency $ {\omega _{\text{c}}} $ and dissipation rate $ {\kappa _{\text{c}}} $ is coupled with the optomechanical cavity a with the coupling strength J, the probe field is input into the auxiliary cavity.
图 2 ${\varepsilon _{\text{T}}}$的实部${Re} ({\varepsilon _{\text{T}}})$随探测场失谐量变化曲线, 其中$ \kappa = {\kappa _{\text{c}}} = 6.4 \, {\text{MHz}} $, $ {\omega _{\text{m}}} = 2{\pi} \times 10.3\, {\text{MHz}} $, $ {\gamma _{\text{m}}} = 2{\pi} \times $$ 0.83\, {\text{kHz}} $, $ m = 10\, {\text{ng}} $, $ {G_{\text{m}}} = 0.003\, {\omega _{\text{m}}} $和$ {g_1} = 0.1\, {\omega _{\text{m}}} $ [49,61]
Figure 2. Real part ${Re} ({\varepsilon _{\text{T}}})$ of ${\varepsilon _{\text{T}}}$ as a function of the probe field detuning, where $ \kappa = {\kappa _{\text{c}}} = 6.4 \, {\text{MHz}} $, $ {\omega _{\text{m}}} = 2{\pi} \times $$ 10.3\, {\text{MHz}} $, $ {\gamma _{\text{m}}} = 2{\pi} \times 0.83\, {\text{kHz}} $, $ m = 10\, {\text{ng}} $, $ {G_{\text{m}}} = $ $ 0.003\, {\omega _{\text{m}}} $, and $ {g_1} = 0.1\, {\omega _{\text{m}}} $[49,61].
图 3 (a)双腔光力学系统的能级结构示意图及其(b)缀饰态解释示意图, 能级$ \left| 0 \right\rangle $, $ \left| 1 \right\rangle $, $ \left| 2 \right\rangle $和$ \left| 3 \right\rangle $分别表示$ \left| {{n_{\text{a}}}, \, {n_{\text{c}}}, \, m} \right\rangle $, $ \left| {{n_{\text{a}}}, \, {n_{\text{c}}}, \, m + 1} \right\rangle $, $ \left| {{n_{\text{a}}}, \, {n_{\text{c}}} + 1, \, m} \right\rangle $和$ \left| {{n_{\text{a}}} + 1, \, {n_{\text{c}}}, \, m} \right\rangle $, 缀饰态$ \left| \pm \right\rangle $为$ {{(\left| 2 \right\rangle \pm \left| 3 \right\rangle )} {/ } {\sqrt 2 }} $, 其中$ {n_{\text{a}}} $和$ {n_{\text{c}}} $分别表示光力学腔中光模a和辅助腔中光模c的光子数, m表示机械模b的声子数
Figure 3. (a) Energy level structure diagram of the double-cavity optomechanical system and (b) the corresponding dressed-states explanation diagram, the energy level $ \left| 0 \right\rangle $, $ \left| 1 \right\rangle $, $ \left| 2 \right\rangle $, and $ \left| 3 \right\rangle $ represent $ \left| {{n_{\text{a}}}, \, {n_{\text{c}}}, \, m} \right\rangle $, $ \left| {{n_{\text{a}}}, \, {n_{\text{c}}}, \, m + 1} \right\rangle $, $ \left| {{n_{\text{a}}}, \, {n_{\text{c}}} + 1, \, m} \right\rangle $, and $ \left| {{n_{\text{a}}} + 1, \, {n_{\text{c}}}, \, m} \right\rangle $, respectively. The dressed-states $ \left| \pm \right\rangle $ are expressed as $ {{(\left| 2 \right\rangle \pm \left| 3 \right\rangle )} {/ } {\sqrt 2 }} $. Here, $ {n_{\text{a}}} $ and $ {n_{\text{c}}} $ represent the photon numbers of optical mode a in the optomechanical cavity and optical mode c in the auxiliary cavity, and m represents the phonon number of mechanical mode b.
图 4 本征值$ {\lambda _{0, \, \pm }} $的(a)实部和(b)虚部随光力耦合强度${G_{\text{m}}}$变化, 其他参数与图2一致
Figure 4. (a) Real and (b) imaginary parts of the eigenvalues $ {\lambda _{0, \, \pm }} $ as function of the optomechanical coupling strength ${G_{\text{m}}}$, and other parameters are the same as in Fig. 2.
图 5 (a)在吸附物的质量为0 pg, 10 pg, 15 pg和20 pg时, 系统的透射光谱图, 参数取$ G_{\text{m}}^{} = 0.001{\omega _{\text{m}}} $; (b)吸附物质量为10 pg时, 在不同光力耦合强度下的透射光谱, 其他参数与图2相同
Figure 5. (a) Transmission spectrum of the system when the mass of the adsorbate is 0 pg, 10 pg, 15 pg, and 20 pg, with $ G_{\text{m}}^{} = 0.001{\omega _{\text{m}}} $; (b) transmission spectrum for an adsorbate mass of 10 pg under different optomechanical coupling strengths, other parameters are the same as in Fig. 2.
图 6 (a) ${Re} ({\lambda '_0})$和(b) ${Im} ({\lambda '_0})$随频移$\delta {\omega _{\text{m}}}$的变化曲线, 参数取$ G_{\text{m}}^{} = \sqrt {{{(4 J_{}^2{\gamma _{\text{m}}} - \kappa {\kappa _{\text{c}}}{\gamma _{\text{m}}})} {/ } {4{\kappa _{\text{c}}}}}} $, 其余参数与图2一致
Figure 6. Curves of (a) ${Re} ({\lambda '_0})$ and (b) ${Im} ({\lambda '_0})$ as function of $\delta {\omega _{\text{m}}}$ with $ G_{\text{m}}^{} = \sqrt {{{(4 J_{}^2{\gamma _{\text{m}}} - \kappa {\kappa _{\text{c}}}{\gamma _{\text{m}}})} {/ } {4{\kappa _{\text{c}}}}}} $, other parameters are the same as in Fig. 2.
图 7 在不同的腔衰减率下灵敏度增加因子(a) ${\eta _{{\text{real}}}}$和(b) ${\eta _{{\text{imag}}}}$以及不同的腔耦合强度下灵敏度增加因子(c) ${\eta _{{\text{real}}}}$和(d) ${\eta _{{\text{imag}}}}$随频移$\delta {\omega _m}$的变化曲线, 参数取$ G_{\text{m}}^{} = \sqrt {{{(4 J_{}^2{\gamma _{\text{m}}} - \kappa {\kappa _{\text{c}}}{\gamma _{\text{m}}})} {/ } {4{\kappa _{\text{c}}}}}} $, 其余参数与图2一致
Figure 7. Curves of sensitivity enhancement factor (a) ${\eta _{{\text{real}}}}$ and (b) ${\eta _{{\text{imag}}}}$ as a function of $\delta {\omega _{\text{m}}}$ under different cavity decay rates, and curves of sensitivity enhancement factor (c) ${\eta _{{\text{real}}}}$ and (d) ${\eta _{{\text{imag}}}}$ as a function of $\delta {\omega _{\text{m}}}$ under different cavity coupling strengths with $ G_{\text{m}}^{} = \sqrt {{{(4 J_{}^2{\gamma _{\text{m}}} - \kappa {\kappa _{\text{c}}}{\gamma _{\text{m}}})} {/ } {4{\kappa _{\text{c}}}}}} $. Other parameters are the same as in Fig. 2.
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