双模光机械腔中冷原子的量子相变和超辐射相塌缩

赵秀琴; 张文慧

doi:10.7498/aps.73.20241103

摘要

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.

关键词:

Abstract

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.

Keywords:

作者及机构信息

赵秀琴^1,2,3,
张文慧^1,2,3

1.
太原师范学院物理系, 晋中　030619

2.
太原师范学院计算与应用物理研究所, 晋中　030619

3.
太原师范学院, 智能优化计算与区块链技术山西省重点实验室, 晋中　030619

通信作者: 张文慧, zhangwh@tynu.edu.cn

基金项目: 国家自然科学基金(批准号: 12304404)和山西省基础研究计划(批准号: 202203021222236)资助的课题.

Authors and contacts

1.
Department of Physics, Taiyuan Normal University, Jinzhong 030619, China

2.
Institute of Computational and Applied Physics, Taiyuan Normal University, Jinzhong 030619, China

3.
Shanxi Key Laboratory for Intelligent Optimization Computing and Blockchain Technology, Taiyuan Normal University, Jinzhong 030619, China

Corresponding author: Zhang Wen-Hui, zhangwh@tynu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 12304404) and the Basic Research Project of Shanxi Province, China (Grant No. 202203021222236).

文章全文 : translate this paragraph

参考文献

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施引文献

图 1 双模光机械腔的示意图. 在水平方向上, 机械谐振子的频率是$ {\omega _{\text{b}}} $, 腔模的频率是$ {\omega _1} $; 在垂直方向上, 腔模的频率是$ {\omega _2} $

Fig. 1. Schematic diagram of a two-mode opto-mechanical cavity. $ {\omega _{\text{b}}} $ is the frequency of the mechanical harmonic oscillator in the horizontal direction, $ {\omega _1} $ is the frequency of the cavity mode, and $ {\omega _2} $ is the frequency of the cavity mode in the vertical direction.

下载: 全尺寸图片幻灯片

图 2 $ {g_1}/{\omega _{\mathrm{a}}} $ -$ {g_2}/{\omega _{\text{a}}} $的平面被分成五个区域, 其中$ {g_1}/ {\omega _{\text{a}}} $$ \lt 1, $ $ g_1^2/\omega _{\text{a}}^2 + g_2^2/\omega _{\text{a}}^2 \lt 1 $在区域I; $ {g_1}/{\omega _{\text{a}}} \leqslant 1, {\text{ }}{g_2}/{\omega _{\text{a}}} \leqslant 1, $$ g_1^2/\omega _{\text{a}}^2 + g_2^2/\omega _{\text{a}}^2 \geqslant 1 $在区域II; $ {g_1}/{\omega _{\text{a}}} \lt 1, {\text{ }}{g_2}/{\omega _{\text{a}}} \gt 1 $在区域III; $ {g_1}/{\omega _{\text{a}}} \gt 1, {\text{ }}{g_2}/{\omega _{\text{a}}} \gt 1 $在区域IV; $ {g_1}/{\omega _{\text{a}}} \gt 1, ~~{g_2}/{\omega _{\text{a}}} $$ \lt 1 $在区域V

Fig. 2. Plane $ {g_1}/{\omega _{\text{a}}} {\text{-}} {g_2}/{\omega _{\text{a}}} $ is divided into five regions, $ {g_1}/{\omega _{\text{a}}} \lt 1 $, $ g_1^2/\omega _{\text{a}}^2 + g_2^2/\omega _{\text{a}}^2 \lt 1 $ in region I; $ {g_1}/{\omega _{\text{a}}} \leqslant 1, $$ {\text{ }}{g_2}/{\omega _{\text{a}}} \leqslant 1,~~ g_1^2/\omega _{\text{a}}^2 + g_2^2/\omega _{\text{a}}^2 \geqslant 1 $ in region II; $ {g_1}/{\omega _{\text{a}}} \lt 1 $, $ {g_2}/{\omega _{\text{a}}} \gt 1 $ in region III; $ {g_1}/{\omega _{\text{a}}} \gt 1 $, $ {g_2}/{\omega _{\text{a}}} \gt 1 $ in region IV; $ {g_1}/{\omega _{\text{a}}} \gt 1 $, $ {g_2}/{\omega _{\text{a}}} \lt 1 $ in region V.

下载: 全尺寸图片幻灯片

图 3 平均光子数${n_{{\text{p}}1}}$ (a), ${n_{{\text{p}}2}}$ (b), 原子布居数差分布$\Delta {n_{\text{a}}}$ (c)和平均基态能量$\varepsilon $ (d)

Fig. 3. Average photon number ${n_{{\text{p}}1}}$ (a), ${n_{{\text{p}}2}}$ (b), atomic population difference $\Delta {n_{\text{a}}}$ (c) and average energy $\varepsilon $ (d).

下载: 全尺寸图片幻灯片

图 4 $ g/{\omega _{\text{a}}}{\text{ - }}\zeta /{\omega _{\text{a}}} $平面中的相图　$ \left( {\text{a}} \right){\text{ }}\delta = - {1}. {\text{0;}} $ $ \left(\text{b}\right)\text{ }\delta =-0.5; $ $ \left( {\text{c}} \right){\text{ }}\delta = 0.0; $ $ \left( {\text{d}} \right){\text{ }}\delta = 0.5 $

Fig. 4. Phase diagram in a plane $ g/{\omega _{\text{a}}}{\text{ - }}\zeta /{\omega _{\text{a}}} $: $ \left( {\text{a}} \right){\text{ }}\delta = - {1}. {\text{0;}} $ $ \left( {\text{b}} \right){\text{ }}\delta = - 0.5; $ $ \left( {\text{c}} \right){\text{ }}\delta = 0.0; $ $ \left( {\text{d}} \right){\text{ }}\delta = 0.5 $

下载: 全尺寸图片幻灯片

表 1 光-声子的非线性参量$ \zeta = 0 $、平均基态能量为$ {\varepsilon _ - } $时, 相应点的计算值

Table 1. Calculated values of the corresponding point when nonlinear parameters of the light-phonon $ \zeta = 0 $ and the average ground state energy is $ {\varepsilon _ - } $

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)

下载: 导出CSV

表 2 光-声子的非线性参量$ \zeta /{\omega _{\text{a}}} = 1.0 $、平均基态能量为$ {\varepsilon _ - } $时, 相应点的计算值

Table 2. Calculated values of the corresponding point when nonlinear parameters of the light-phonon $ \zeta /{\omega _{\text{a}}} = 1.0 $ and the average ground state energy is $ {\varepsilon _ - } $.

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. $

下载: 导出CSV

表 3 模式2光场$ {\gamma _2} = 0 $、光-声子的非线性参量$ \zeta /{\omega _{\text{a}}} = 1.0 $、基态能量取$ {\varepsilon _ + } $时, 相应点的计算值

Table 3. Calculated values of the corresponding point when the light field of mode 2 $ {\gamma _2} = 0 $, nonlinear parameters of the light-phonon $ \zeta /{\omega _{\text{a}}} = 1.0 $ and the average ground state energy is $ {\varepsilon _ + } $.

$ {{{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

下载: 导出CSV

[1]	Dicke R H 1954 Phys. Rev. 93 99 Google Scholar
[2]	Hepp K, Lieb E H 1973 Ann. Phys. 76 360 Google Scholar
[3]	Wang Y K, Hioe F T 1973 Phys. Rev. A 7 831 Google Scholar
[4]	Hioe F T 1973 Phys. Rev. A 8 1440 Google Scholar
[5]	Baumann K, Guerlin C, Brennecke F, Esslinger T 2010 Nature 464 1301 Google Scholar
[6]	Baumann K, Mottl R, Brennecke F, Esslinger T 2011 Phys. Rev. Lett. 107 140402 Google Scholar
[7]	Emary C, Brandes T 2003 Phys. Rev. Lett. 90 044101 Google Scholar
[8]	Chen G, Li J Q, Liang J Q 2006 Phys. Rev. A 74 054101 Google Scholar
[9]	Aspelmeyer M, Kippenberg T J, Marquardt F 2014 Rev. Mod. Phys. 86 1391 Google Scholar
[10]	Brennecke F, Ritter S, Donner T, Esslinger T 2008 Science 322 235 Google Scholar
[11]	Gröblacher S, Hammerer K, Vanner M R, Aspelmeyer M 2009 Nature 460 724 Google Scholar
[12]	Anetsberger G, Arcizet O, Unterreithmeier Q P, Rivière R, Schliesser A, Weig E M, Kotthaus J P, Kippenberg T J 2009 Nat. Phys. 5 909 Google Scholar
[13]	Chan J, Alegre T P M, Safavi-Naeini A H, Hill J T, Krause A, Gröblacher S, Aspelmeyer M, Painter O 2011 Nature 478 89 Google Scholar
[14]	陈华俊, 米贤武 2011 物理学报 60 124206 Google Scholar Chen H J, Mi X W 2011 Acta Phys. Sin. 60 124206 Google Scholar
[15]	严晓波, 杨柳, 田雪冬, 刘一谋, 张岩 2014 物理学报 63 204201 Google Scholar Yan X B, Yang L, Tian X D, Liu Y M, Zhang Y 2014 Acta Phys. Sin. 63 204201 Google Scholar
[16]	韩明, 谷开慧, 刘一谋, 张岩, 王晓畅, 田雪冬, 付长宝, 崔淬砺 2014 物理学报 63 094206 Google Scholar Han M, Gu K H, Liu Y M, Zhang Y, Wang X C, Tian X D, Fu C B, Cui C L 2014 Acta Phys. Sin. 63 094206 Google Scholar
[17]	Brooks D W C, Botter T, Schreppler S, Purdy T P, Brahms N, Stamper-Kurn D M 2012 Nature 488 476 Google Scholar
[18]	Ian H, Gong Z R, Liu Y X, Sun C P, Nori F 2008 Phys. Rev. A 78 013824 Google Scholar
[19]	Jiang C, Bian X T, Cui Y S, Chen G B 2016 J. Opt. Soc. Am. B: Opt. Phys. 33 2099 Google Scholar
[20]	Wang Z M, Lian J L, Liang J Q, Yu Y M, Liu W M 2016 Phys. Rev. A 93 033630 Google Scholar
[21]	Lian J L, Liu N, Liang J Q, Chen G, Jia S T 2013 Phys. Rev. A 88 043820 Google Scholar
[22]	Zhao X Q, Liu N, Bai X M, Liang J Q 2017 Ann. Phys. 378 448 Google Scholar
[23]	Santos J P, Semião F L, Furuya K 2010 Phys. Rev. A 82 063801 Google Scholar
[24]	Sankey J C, Yang C, Zwickl B M, Jayich A M, Harris J G E 2010 Nat. Phys. 6 707 Google Scholar
[25]	Clerk A A, Marquardt F, Harris J G E 2010 Phys. Rev. Lett. 104 213603 Google Scholar
[26]	Purdy T P, Brooks D W C, Botter T, Brahms N, Ma Z Y, Stamper-Kurn D M 2010 Phys. Rev. Lett. 105 133602 Google Scholar
[27]	Wang B, Nori F, Xiang Z X 2024 Phys. Rev. Lett. 132 053601 Google Scholar
[28]	Léonard J, Morales A, Zupancic P, Esslinger T, Donner T 2017 Nature 543 87 Google Scholar
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