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在双模Dicke模型的基础上, 研究了光场(模式1)与机械振子有非线性耦合的双模光机械腔中冷原子的量子相变. 利用自旋相干态及变分法得到了系统基态能量的泛函. 通过求解和判定稳定性, 得到了相变点和基态相图. 发现存在正常相和反转正常相的双稳态, 超辐射相和反转正常相的共存态, 以及单独存在的反转正常相. 原子与两模光场相互作用强度的不同对相变点的值有较大影响, 存在正常相经过相变点到超辐射相的量子相变. 光-声子非线性耦合对相变点没有影响, 但诱导了超辐射相的塌缩, 存在一个转折点, 经过转折点可以实现超辐射相到反转的正常相的量子相变. 超辐射相的区域随着光子-声子耦合的增加而减小, 在耦合的临界值处收缩为零, 即转折点和相变点重合, 并且有可能出现两个正常相之间的原子布居数的反转; 光-声子的非线性耦合还产生了不稳定的非零光子态, 它与超辐射态相对应. 在不含机械振子时, 回到双模Dicke模型的结果.In this paper, the quantum phase transition of cold atoms in a two-mode photomechanical cavity with nonlinear coupling between the optical field (mode 1) and the mechanical oscillator is studied on the basis of the two-mode Dicke model. The functional of the ground state energy of the system is obtained by spin coherent states and variational method. By solving and judging the stability, the phase transformation point and ground state phase diagram are obtained. It is found that there are bistable state of normal phase and reverse normal phase, coexistent state of superradiation phase and reversed normal phase, and reversed normal phase that exists alone. The different interaction strengths between atoms and two-mode light fields greatly affect the value of the phase transition point. There is a quantum phase transition from a normal phase through a phase transition point to a superradiant phase. The light-phonon nonlinear coupling has no effect on the phase transition point, but induces the collapse of the superradiant phase. There is a turning point through which the quantum phase transition from the superradiant phase to the reversed normal phase can be realized. The region of the superradiation phase decreases with the increase of the photon-phonon coupling, and it shrinks to zero at the critical value of the coupling, that is, the turning point and the phase transition point coincide, and there may be a reversal of the atomic population between the two normal phases. The nonlinear coupling of the light-phonon also produces an unstable non-zero photon state, which corresponds to the superradiation state. In the absence of mechanical oscillators, the results of the two-mode Dicke model are returned.
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Keywords:
- two model optomechanical cavity /
- light-phonon nonlinear coupling /
- quantum phase transition /
- superradiation phase collapse
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区域 $ \left( {\dfrac{{{g_1}}}{{{\omega _{\text{a}}}}}, \dfrac{{{g_2}}}{{{\omega _{\text{a}}}}}} \right) $ $ (\overline{{\gamma }_{1}^{2}}, \text{ }\overline{{\gamma }_{2}^{2}}) $ $ {\varepsilon _ - } $ $ {{\boldsymbol{H}}_ - } = \left[ {\begin{array}{*{20}{c}} {\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial \gamma _1^2}}}&{\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial {\gamma _1}\partial {\gamma _2}}}} \\ {\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial {\gamma _2}\partial {\gamma _1}}}}&{\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial \gamma _2^2}}} \end{array}} \right] $ 特征值$ \left[ {{H_ - }} \right] $ I (0.3, 0.4) (0, 0) –0.5 $ \left[ {\begin{array}{*{20}{c}} {1.82}&{ - 0.24} \\ { - 0.24}&{1.68} \end{array}} \right] $ (2, 1.5) II (0.8, 0.7) (0.0347, 0.0266) –0.5037 $ \left[ {\begin{array}{*{20}{c}} {1.1129}&{ - 0.7762} \\ { - 0.7762}&{1.3208} \end{array}} \right] $ (2, 0.4337) III (0.6, 1.6) (0.0794, 0.5649) –0.8156 $ \left[ {\begin{array}{*{20}{c}} {1.9711}&{ - 0.0711} \\ { - 0.0711}&{1.7943} \end{array}} \right] $ (2, 1.7654) IV (1.5, 1.5) (0.5347, 0.5347) –1.1806 $ \left[ {\begin{array}{*{20}{c}} {1.9506}&{ - 0.0494} \\ { - 0.0494}&{1.9506} \end{array}} \right] $ (2, 1.9012) V (1.5, 0.5) (0.4725, 0.0525) –0.725 $ \left[ {\begin{array}{*{20}{c}} {1.7119}&{ - 0.0960} \\ { - 0.0960}&{1.9680} \end{array}} \right] $ (2, 1.6800) 
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区域 $ \left( {\dfrac{{{g_1}}}{{{\omega _{\text{a}}}}}, \dfrac{{{g_2}}}{{{\omega _{\text{a}}}}}} \right) $ $ (\overline {\gamma _1^2} , {\text{ }}\overline {\gamma _2^2} ) $ $ {\varepsilon _ - } $ $ {{\boldsymbol{H}}_ - } = \left[ \begin{array}{*{20}{c}} {\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial \gamma _1^2}}}&{\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial {\gamma _1}\partial {\gamma _2}}}} \\ {\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial {\gamma _2}\partial {\gamma _1}}}}&{\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial \gamma _2^2}}} \end{array} \right] $ 特征值$ \left[ {{H_ - }} \right] $ I (0.4, 0.3) (4.5925, 0.01716) 1.2585 $ \left[ \begin{array}{*{20}{c}} { - 3.5436}&{ - 0.0326} \\ { - 0.0326}&{1.9674} \end{array} \right] $ (–3.5438, 1.9676) II (0.7, 0.8) $ \left\{ {\begin{aligned} &{\left( {4.1771, 0.1478} \right)} \\ &{\left( {0.0273, 0.0353} \right)} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{0.7714} \\ &{ - 0.5038} \end{aligned}} \right. $ $ \left\{ \begin{aligned} {\left[ \begin{array}{*{20}{c}} { - 3.033}&{ - 0.0237} \\ { - 0.0237}&{1.9730} \end{array} \right]} \\ {\left[ \begin{array}{*{20}{c}} {1.2929}&{ - 0.7707} \\ { - 0.7707}&{1.1192} \end{array} \right]} \end{aligned} \right. $ $ \left\{ {\begin{aligned} &{\left( { - 0.0334, 1.9731} \right)} \\ &{\left( {1.9816, 0.4305} \right)} \end{aligned}} \right. $ III (0.8, 1.2) $ \left\{ {\begin{aligned} &{\left( {4.0258, 0.3439} \right)} \\ &{\left( {0.1303, 0.2781} \right)} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{0.3866} \\ & { - 0.6418} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} {\left[ \begin{array}{*{20}{c}} { - 2.8431}&{ - 0.0182} \\ { - 0.0182}&{1.9727} \end{array} \right]} \\ {\left[ \begin{array}{*{20}{c}} {1.7048}&{ - 0.2082} \\ { - 0.2082}&{1.6878} \end{array} \right]} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{\left( { - 2.8432, 1.9728} \right)} \\ &{\left( {1.9046, 1.4880} \right)} \end{aligned}} \right. $ IV (1.5, 1.5) $ \left\{ {\begin{aligned} &{\left( {2.7669, 0.5519} \right)} \\ &{\left( {0.7439, 0.5390} \right)} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{ - 1.0907} \\ & { - 1.2190} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} {\left[ \begin{array}{*{20}{c}} { - 1.3319}&{ - 0.0116} \\ { - 0.0116}&{1.9884} \end{array} \right]} \\ {\left[ \begin{array}{*{20}{c}} {1.7048}&{ - 0.2082} \\ { - 0.2082}&{1.6878} \end{array} \right]} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{\left( { - 1.3220, 1.9884} \right)} \\ & {\left( {1.9632, 1.0673} \right)} \end{aligned}} \right. $ V (1.2, 0.8) $ \left\{ {\begin{aligned} &{\left( {3.4015, 0.1540} \right)} \\ &{\left( {0.3252, 0.1263} \right)} \end{aligned}} \right. $ $ \left\{ {\begin{aligned}& { - 0.1776} \\ &{ - 0.6491} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{\left[ \begin{array}{*{20}{c}} { - 2.1072}&{ - 0.0141} \\ { - 0.0141}&{1.9907} \end{array} \right]} \\ &{\left[ \begin{array}{*{20}{c}} {1.3319}&{ - 0.1853} \\ { - 0.1853}&{1.8765} \end{array} \right]} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{\left( { - 2.1073, 1.9907} \right)} \\ &{\left( {1.9336, 1.2748} \right)} \end{aligned}} \right. $
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$ {{{g_1}}}/{{{\omega _{\text{a}}}}} $ $ \overline {\gamma _1^2} $ $ {\varepsilon _ + } $ $ {{{\partial ^2}{\varepsilon _ + }}}/{{\partial \gamma _1^2}} $ 0.4 5.0910 3.531 –4.0728 0.8 5.0614 4.3676 –4.0491 1.2 5.0437 5.2408 –4.0350 1.6 5.0336 6.1243 –4.0269
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