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

Fig. 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]

Fig. 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

Fig. 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

Fig. 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

Fig. 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

Fig. 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

Fig. 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.