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本文在包含两模光场、N个原子以及机械振子的耦合光机械腔中, 从理论上探讨了光与原子以及光与机械振子的相互作用引起的量子相变. 采用Holstein-Primakoff变换法, 假设了新的平移玻色算符和4个参量, 给出了系统的基态能量泛函和4个参量之间的关系, 通过两个特例证明了假设的平移玻色算符的正确性. 在共振情况下有正常相到超辐射相的相变, 调控两腔光场的耦合强度可以改变相变点. 当考虑辐射压力产生的非线性光子-声子相互作用时, 系统的相图由原来的2个相区扩展为3个相区, 包括正常相和超辐射相的共存区, 双稳的超辐射相区, 以及不稳定的真空宏观相区. 同时, 还出现了一条转折点曲线, 该曲线与相变点曲线有重叠区域, 表明系统中存在多重量子相变. 这些相变现象可以通过测量平均光子数来检测. 当不考虑两模光场的耦合作用时回到旋波近似的Dicke模型的量子相变.
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关键词:
- 耦合光机械腔 /
- Holstein-Primakoff变换 /
- 光场耦合强度 /
- 双稳超辐射相
In this paper, the quantum phase transitions caused by the interactions between light and atoms, as well as between light and mechanical oscillators, are discussed theoretically in a coupled optomechanical cavity containing two optical field modes (N atoms and mechanical oscillator). By using Holstein-Primak off transformation method, new translational boson operators and four parameters are assumed. The ground state energy functional of the system and a set of equations composed of four parameters are given. The correctness of the assumed translation boson operators is proved by two special cases. In the case of resonance, the characteristics of the obtained solutions are shown by solving the equations, graphical method and Hessian matrix judgment. The stable zero solution is called the normal phase, the unstable zero solution is named the unstable vacuum macroscopic phase, and the stable non-zero solution is referred to as the superradiation phase. The phase can transition from normal phase to superradiation phase, and the point of phase transition can be changed by adjusting the coupling intensity of the two cavity light fields. When the nonlinear photon-phonon interaction caused by radiation pressure is considered, the phase diagram of the system is expanded from the original two phase regions to three phase regions, including the coexistent normal phase and superradiation phase, the bistable superradiation phase, and the unstable vacuum macroscopic phase region, where the bistable superradiation phase is similar to the optical bistable phenomenon. At the same time, there is also a turning point curve, which overlaps with the phase transition point curve, indicating the existence of multiple quantum phase transitions in the system. These predictions can be detected by measuring the average number of photons. The coupled optomechanical cavity that we studied, when considering the coupling of the two-mode optical field and the atomic ensemble but no mechanical oscillator, reflects the interaction between the two-mode optical field and the atom, thus concluding that the transformation point is small and the quantum phase transition is easy to occur. When the coupling between the mechanical oscillator and the two-mode optical field is not considered, the interaction between the single-mode optical field and the atom is reflected, returning to the quantum phase transition of the Dicke model under rotating wave approximation.-
Keywords:
- coupled optomechanical cavity /
- Holstein-Primak off transformation /
- coupling intensity of the light fields /
- bistable superradiation phase
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图 1 耦合光机械腔的示意图[20–22], A腔和C腔通过两模光场耦合强度J耦合, A腔中有二能级原子系综, 由具有强度$ {\varOmega _{\text{p}}} $和频率$ {\omega _{\text{p}}} $的外部单模激光场相干驱动, C腔是光机械腔
Fig. 1. Schematic illustration of the coupled optomechanical cavities [20–22], cavity A and cavity C are coupled by a two-mode optical field coupling strength J. An ensemble of two-level atoms is placed into the cavity A which is coherently driven by an external monochromatic laser field with strength $ {\varOmega _{\text{p}}} $ and frequency $ {\omega _{\text{p}}} $, cavity C is a optomechanical cavity.
图 2 $ g/{\omega _{\text{b}}}{\text{ - }}J/{\omega _{\text{b}}} $的相图
Fig. 2. Phase diagram of $ g/{\omega _{\text{b}}}{\text{ - }}J/{\omega _{\text{b}}} $.
图 3 选择A, B两个点, 参量为$ J = 10{\omega _{\text{b}}} $, $ g/{\omega _{\text{b}}} = 10, 15 $, $ \zeta /{\omega _{\text{b}}} = 1.0 $ (a1), (a2)对应的一阶偏导数p; (b1), (b2)对应的平均基态能量$ \varepsilon $
Fig. 3. Select points A and B, parameters $ J = 10{\omega _{\text{b}}} $, $ g/{\omega _{\text{b}}} = 10, 15 $, $ \zeta /{\omega _{\text{b}}} = 1.0 $: (a1), (a2) The corresponding first partial derivatives p; (b1), (b2) the corresponding average ground state energies $ \varepsilon $.
图 4 选择D点, 参量为$ J = 10{\omega _{\text{b}}} $, $ g/{\omega _{\text{b}}} = 30 $, $ \zeta /{\omega _{\text{b}}} = 0 $, $ 1.0 $, $ 2.0 $ (a1)—(a3) 对应的一阶偏导数p; (b1)—(b3) 对应的平均基态能量$ \varepsilon $
Fig. 4. Select D point, parameters $ J = 10{\omega _{\text{b}}} $, $ g/{\omega _{\text{b}}} = 30 $, $ \zeta /{\omega _{\text{b}}} = 0 $, $ 1.0 $, $ 2.0 $: (a1)–(a3) Corresponding first partial derivatives p; (b1)–(b3) the corresponding average ground state energies $ \varepsilon $.
图 5 取参量$ {\text{ }}g/{\omega _{\text{b}}} = 25 $, $ \zeta /{\omega _{\text{b}}} = 1.0 $, $ {\text{ }}J/{\omega _{\text{b}}} = 0 \to 20{\text{ }} $, A腔和C腔的平均光子数分布$ {n_{{\text{pA}}}} $ (a1)和$ {n_{{\text{pC}}}} $(a3), A腔中原子布居差的分布$ \varDelta {n_{\text{a}}} $(a2)随两模光场的耦合参量J变化曲线, $ {\text{ }}{J_{\text{t}}}/{\omega _{\text{b}}} $是转折点
Fig. 5. Take parameters $ {\text{ }}g/{\omega _{\text{b}}} = 25 $, $ \zeta /{\omega _{\text{b}}} = 1.0 $, $ {\text{ }}J/{\omega _{\text{b}}} = 0 \to 20{\text{ }} $: The changing curves of average photon number distribution $ {n_{{\text{pA}}}} $ (a1) and $ {n_{{\text{pC}}}} $ (a3) in cavity A and cavity C, and the distribution of atomic population difference $ \varDelta {n_{\text{a}}} $ (a2) in cavity A, with the coupling parameter J of the two-mode optical field, $ {\text{ }}{J_{\text{t}}}/{\omega _{\text{b}}} $ is a turning point.
图 6 $ g/{\omega _{\text{b}}}\sim J/{\omega _{\text{b}}} $平面相图 (a1) $ \zeta /{\omega _{\text{b}}} = 0.0 $; (a2) $ \zeta /{\omega _{\text{b}}} = 1.0 $; (a3) $ \zeta /{\omega _{\text{b}}} = 2.0 $
Fig. 6. Phase diagram in a plane $ g/{\omega _{\text{b}}}\sim J/{\omega _{\text{b}}} $: (a1) $ \zeta /{\omega _{\text{b}}} = 0.0 $; (a2) $ \zeta /{\omega _{\text{b}}} = 1.0 $; (a3) $ \zeta /{\omega _{\text{b}}} = 2.0 $.
表 1 旋波近似时Dicke模型基态物理量的分布情况
Table 1. Distribution of physical quantities in the ground state of Dicke model in rotating-wave approximation.
基态物理量 $g \leqslant g_{\text{c}}^{\text{R}}$ $g > g_{\text{c}}^{\text{R}}$ 平均光子数分布${n_{{\text{pA}}}} = \alpha $ 0 $ \dfrac{1}{4}\dfrac{{{g^2}}}{{\varDelta _{\text{a}}^{2}}}\left( {1 - \dfrac{{g_{\text{c}}^{{\text{R4}}}}}{{{g^4}}}} \right) $ 布局数差分布$\varDelta {n_{\text{a}}} = \beta - \dfrac{1}{2}$ $ - \dfrac{1}{2}$ $ - \dfrac{{g_{\text{c}}^{{\text{R2}}}}}{{2{g^2}}} $ 平均基态能量$\varepsilon = {H_0}$ $ - \dfrac{1}{2}$ $ - \dfrac{{{g^2}}}{{4{\varDelta _{\text{a}}}}}\left( {1 + \dfrac{{g_{\text{c}}^{{\text{R4}}}}}{{{g^4}}}} \right) $ 
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表 2 考虑与C腔中光场有相互作用时基态物理量的分布情况
Table 2. Distribution of physical quantities in the ground state interacting with the light field in the cavity C is considered.
基态物理量 $g \leqslant g_{\text{c}}^J$ $g > g_{\text{c}}^J$ 平均光子数分布$\left\{ \begin{aligned} {n_{{{\text{p}}_{\text{a}}}}} = \alpha \\ {n_{{{\text{p}}_{\text{c}}}}} = \rho\end{aligned} \right.$ 0 $ \left\{ \begin{aligned}& \alpha = \dfrac{{{g^2}}}{{4\varDelta _{\text{a}}^{2}}}\left(1 - \dfrac{{g_{\text{c}}^{J4}}}{{{g^4}}}\right) \\ &\rho = \dfrac{{{J^2}}}{{4\varDelta _{\text{c}}^{2}}}\dfrac{{{g^2}}}{{\varDelta _{\text{a}}^{2}}}\left(1 - \dfrac{{g_{\text{c}}^{J4}}}{{{g^4}}}\right) \end{aligned} \right. $ 布局数差分布$\varDelta {n_{\text{a}}} = \beta - \dfrac{1}{2}$ $ - \dfrac{1}{2}$ $ - \dfrac{{g_{\text{c}}^{J2}}}{{2{g^2}}} $ 平均基态能量$\varepsilon = \dfrac{{{H_0}}}{{{\omega _0}}}$ $ - \dfrac{1}{2}$ $ - \dfrac{{{g^2}}}{{4{\varDelta _{\text{a}}}}}\left(1 + \dfrac{{g_{\text{c}}^{J4}}}{{{g^4}}}\right) $
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表 3 4个红点A—D在$ \zeta /{\omega _{\text{b}}}{\text{ = 0, 1}}{\text{.0, 2}}{.0} $时, 3个参量$ \alpha $, $ \beta $, $ \rho $的解的具体数值
Table 3. When the four red dots are at $ \zeta /{\omega _{\text{b}}}{\text{ = 0, 1}}{\text{.0, 2}}{.0} $, the specific values of the solutions of the three parameters $ \alpha $, $ \beta $, $ \rho $.
正常相区点A点 正常相区点B点 $ \zeta /{\omega _{\text{b}}} = 0 $ $ \zeta /{\omega _{\text{b}}} = 1.0 $ $ \zeta /{\omega _{\text{b}}} = 2.0 $ $ \zeta /{\omega _{\text{b}}} = 0 $ $ \zeta /{\omega _{\text{b}}} = 1.0 $ $ \zeta /{\omega _{\text{b}}} = 2.0 $ 0 $ \left\{ \begin{aligned} \alpha = {2}{.380} \\ \beta = {0}{.228} \\ \rho = {7}{.107}\end{aligned} \right.{\text{ }} $ $ \left\{ \begin{aligned} \alpha = {0}{.663} \\ \beta = {0}{.112} \\ \rho = {1}{.725}\end{aligned} \right.{\text{ }} $ $ 0 $ $ \left\{ \begin{aligned} \alpha = {2}{.739} \\ \beta = {0}{.313} \\ \rho = {6}{.835}\end{aligned} \right.{\text{ }} $ $ \left\{ \begin{aligned} \alpha = {0}{.868} \\ \beta = {0}{.209} \\ \rho = {1}{.571}\end{aligned} \right.{\text{ }} $ 超辐射区相点C点 超辐射区相点D点 $ \zeta /{\omega _{\text{b}}} = 0 $ $ \zeta /{\omega _{\text{b}}} = 1.0 $ $ \zeta /{\omega _{\text{b}}} = 2.0 $ $ \zeta /{\omega _{\text{b}}} = 0 $ $ \zeta /{\omega _{\text{b}}} = 1.0 $ $ \zeta /{\omega _{\text{b}}} = 2.0 $ $ \left\{ \begin{aligned} \alpha = {0}{.534} \\ \beta = {0}{.260} \\ \rho = {0}{.134}\end{aligned} \right.{\text{ }} $ $ \left\{ \begin{aligned} {\alpha _1} = {0}{.541} \\ {\beta _1} = {0}{.261} \\ {\rho _1} = {0}{.139}\end{aligned} \right.{\text{ }} $
$ \left\{ \begin{aligned} {\alpha _2} = {3}{.474} \\ {\beta _2} = {0}{.395} \\ {\rho _2} = {6}{.280}\end{aligned} \right.{\text{ }} $$ \left\{ \begin{aligned} {\alpha _1} = {0}{.568} \\ {\beta _1} = {0}{.262} \\ {\rho _1} = {0}{.162}\end{aligned} \right.{\text{ }} $
$ \left\{ \begin{aligned} {\alpha _2} = {1}{.295} \\ {\beta _2} = {0}{.334} \\ {\rho _2} = {1}{.203}\end{aligned} \right.{\text{ }} $$ \left\{ \begin{aligned} \alpha = 0.889 \\ \beta = {0}{.333} \\ \rho = {0}{.222}\end{aligned} \right. $ $ \left\{ \begin{aligned} {\alpha _1} = {0}{.905} \\ {\beta _1} = {0}{.335} \\ {\rho _1} = {0}{.237}\end{aligned} \right. $
$ \left\{ \begin{aligned} {\alpha _2} = {3}{.853} \\ {\beta _2} = {0}{.416} \\ {\rho _2} = {5}{.989}\end{aligned} \right.{\text{ }} $$ \left\{ \begin{aligned} {\alpha _1} = {1}{.000} \\ {\beta _1} = {0}{.342} \\ {\rho _1} = {0}{.332}\end{aligned} \right.{\text{ }} $
$ \left\{ \begin{aligned} {\alpha _2} = {1}{.462} \\ {\beta _2} = {0}{.367} \\ {\rho _2} = {0}{.946}\end{aligned} \right.{\text{ }} $
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