原子-光子相互作用下耗散光腔中的量子相变和奇异点

刘妮; 罗芸青; 梁九卿

doi:10.7498/aps.74.20250075

摘要

基于Dicke模型引入非厄米原子-场耦合, 本文通过调节非线性原子-光子相互作用研究开放Dicke模型的量子相变和奇异点. 通过厄米算符进行相似变换, 然后对系统对角化得到有效的非厄米哈密顿量, 并利用自旋相干态变分法计算宏观量子态的能量泛函. 非厄米Dicke模型主要结果是: 系统超辐射相和相关量子相变完全消失, 出现了不稳定的非零光子数态; 在泵浦场产生的非线性原子-光子相互作用影响下, 系统的量子相变发生了显著变化; 能谱出现两个奇异点, 两个奇异点之间的能谱为复数, 而其他区域为纯实数; 红、蓝腔-泵浦场失谐调控下, 相图显示出丰富的量子相变. 随着原子-场耦合强度的增大, 系统出现从超辐射相到正常相的新奇反向量子相变, 不同于Dicke模型.

关键词:

Abstract

Dicke model, as an important many-body model in quantum optics, describes the interaction between multiple identical two-level atoms and a quantized electromagnetic field. This spin-boson model shows collective phenomena in light-matter interaction systems and can undergo a second-order quantum phase transition from a normal phase to a superradiant phase when the coupling strength between the two-level atoms and the optical field exceeds a critical value. Dicke model embodies unique many-body quantum theories. And it has been widely studied and obtained many significant research results in quantum information, quantum process, and other quantum systems. Meanwhile, Dicke model also has wide applications in quantum optics and condensed matter physics.The extended Dicke model, describing the interaction of a Bose-Einstein condensate in an optical cavity, provides a remarkable platform for studying extraordinary quantum phase transitions in theory and experiment. Based on the recent experiment on non-Hermitian coupling between two long-lived atomic spin waves in an optical cavity, in this work we use spin-coherent-state variational method and present the macroscopic quantum-state energy of the non-Hermitian Dicke model.The spin coherent state variational method has an advantage in the theoretical research of macroscopic quantum states, especially in the normal and the inverted pseudospin states. In the variational method, optical coherent states and atomic extremum spin coherent states are used as the trial wave functions. A Hermitian transformation operator is proposed to diagonalize the non-Hermitian Hamiltonian, which is different from the ordinary quantum mechanics where the transformation operator must be unitary. Herein, the energy function is not necessarily real in the entire coupling region. Beyond an exceptional point, the spectrum becomes complex and introducing biorthogonal sets of atomic extremum states is necessary to evaluate the average quantities.The normal phase (for the zero average photon number) possesses real energy and atomic population. The non-Hermitian interaction destroys the superradiant phase (for the stable nonzero average photon number) and leads to the absence of quantum phase transition. However, the introduced atom-photon interaction, which is induced by the pump field experimentally, can change the situation, dramatically. The pump field can balance the loss by the non-Hermitian atom-photon interaction to achieve the superradiant phase.An interesting double exceptional point are observed in the energy functional. There is the real spectrum below the first exceptional point and beyond the second exceptional point, while there is a complex spectrum between these two exceptional points. The superradiant phase appears only beyond a critical value, which is related to the nonlinear interaction and the pump laser. A new and inverted quantum phase transition from the superradiant phase to the normal phase, is observed by modulating the atom-field coupling strength. The superradiant phase of the population inversion state appears for a negative effective frequency and a large atom-photon interaction. The influence of the dissipative coupling may be observed in cold atom experiment in an optical cavity. All the parameters adopted in this work are the actual experimental parameters.

Keywords:

作者及机构信息

山西大学理论物理研究所, 量子光学与光量子器件国家重点实验室, 太原　030006

通信作者: 刘妮, 317446484@qq.com

基金项目: 国家自然科学基金(批准号: 12374312)和山西省回国留学人员科研经费(批准号: 2022-014)资助的课题.

Authors and contacts

State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China

Corresponding author: LIU Ni, 317446484@qq.com

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 12374312) and the Scholarship Council of Shanxi Province, China (Grant No. 2022-014).

文章全文

参考文献

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施引文献

图 1 (a) 横向泵浦激光器捕获光腔中冷原子的实验原理图; (b) 大失谐$ {\varDelta _{\mathrm{a}}} $下原子能级跃迁图

Fig. 1. (a) Experimental schematic for cold atoms trapped in an optical cavity with a transverse pumping laser; (b) transition diagram of atomic energy level in the large detuning $ {\varDelta _{\mathrm{a}}} $.

下载: 全尺寸图片幻灯片

图 2 能量本征值的虚部$\text{Im} [{\varepsilon _ \pm }]$在${n_{\text{p}}}{\text{-}}g$平面的相图

Fig. 2. Phase diagram of ${n_{\text{p}}}{\text{-}}g$ plane about the imaginary part of the eigenvalues $\text{Im} [{\varepsilon _ \pm }]$.

下载: 全尺寸图片幻灯片

图 3 能量本征值${\varepsilon _ \pm }$的实部和虚部随原子-场耦合强度$g$(a)和平均光子数${n_{\text{p}}}$(b)变化的示意图

Fig. 3. The real part and the imaginary part of the eigenvalue ${\varepsilon _ \pm }$ as a function of the atom-field coupling strength $g$(a) and the mean photon number ${n_{\text{p}}}$(b).

下载: 全尺寸图片幻灯片

图 4 (a) 平均能量${\varepsilon _ \pm }$和(b) 平均光子数${n_{\text{p}}}$随原子-场耦合强度$g$变化的示意图, 腔频率与原子跃迁频率满足关系$ \omega_{\mathrm{f}}=\omega_0 $

Fig. 4. (a) The average energy ${\varepsilon _ \pm }$ and (b) the average photon number ${n_{\text{p}}}$ as a function of the atom-field coupling strength $g$ with the cavity frequency and the atomic transition frequency satisfy the relation $ \omega_{\mathrm{f}}=\omega_0 $.

下载: 全尺寸图片幻灯片

图 5 本征值的虚部$\text{Im} [{\varepsilon _ \pm }]$在U-g平面的相图, 光子数${n_{\text{p}}} = 1$

Fig. 5. Phase diagram of U-g plane about the imaginary part of the eigenvalues $\text{Im} [{\varepsilon _ \pm }]$ for mean photon number ${n_{\text{p}}} = 1$.

下载: 全尺寸图片幻灯片

图 6 本征值${\varepsilon _ \pm }$的实部和虚部随原子-场耦合强度$g$(a)和原子-光子相互作用$U$(b)变化的示意图

Fig. 6. The real part and the imaginary part of the eigenvalue ${\varepsilon _ \pm }$ as a function of the atom-field coupling strength $g$ (a) and the atom-photon interaction $U$(b).

下载: 全尺寸图片幻灯片

图 7 本征值的虚部$\text{Im} [{\varepsilon _ \pm }]$在${n_{\text{p}}}{\text{-}}U$平面的相图, 原子-场耦合强度$g = 5$

Fig. 7. Phase diagram of ${n_{\text{p}}}{\text{-}}U$ plane about the imaginary part of the eigenvalues $\text{Im} [{\varepsilon _ \pm }]$ for atom-field strength $g = 5$.

下载: 全尺寸图片幻灯片

图 8 本征值${\varepsilon _ \pm }$的实部和虚部随光子数${n_{\text{p}}}$ (a)和原子-光子相互作用$U$(b)变化的示意图

Fig. 8. The real part and the imaginary part of the eigenvalue ${\varepsilon _ \pm }$ as functions of the photon number ${n_{\text{p}}}$(a) and the atom-photon interaction $U$(b).

下载: 全尺寸图片幻灯片

图 9 g-U空间(a)红失谐$\varDelta = 40$和(b)蓝失谐$\varDelta = - 40$的相图. 图(b)中标签(1)表示${\text{NP}_{\text{co}}}\left( {{N_ - }, {S_ + }} \right)$

Fig. 9. Phase diagrams in the g-U space for (a) red detuning $\varDelta = 40$ and (b) blue detuning $\varDelta = - 40$. Label (1) is the NP denoted by ${\text{N}}{{\text{P}}_{{\text{co}}}}\left( {{N_ - }, {S_ + }} \right)$.

下载: 全尺寸图片幻灯片

[1]	Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243 Google Scholar
[2]	Bender C M, Boettcher S, Meisinger P N 1999 J. Math. Phys. 40 2201 Google Scholar
[3]	Bender C M, Brody D C, Jones H F 2002 Phys. Rev. Lett. 89 270401 Google Scholar
[4]	Bender C M, Brody D C, Jones H F 2004 Phys. Rev. D. 70 025001 Google Scholar
[5]	Bender C M 2007 Rep. Prog. Phys. 70 947 Google Scholar
[6]	Heiss W D 2004 J. Phys. A. 37 2455 Google Scholar
[7]	Brody D C, Graefe E M 2012 Phys. Rev. Lett. 109 230405 Google Scholar
[8]	Roccati F, Palma G M, Bagarello F, Ciccarello F 2022 Open Syst. Inf. Dyn. 29 2250004 Google Scholar
[9]	Lin Z, Ramezani H, Eichelkraut T, Kottos T, Cao H, Christodoulides D N 2011 Phys. Rev. Lett. 106 213901 Google Scholar
[10]	Sticlet D, Dóra B, Moca C P 2022 Phys. Rev. Lett. 128 016802 Google Scholar
[11]	Peng Y, Jie J W, Yu D P, Wang Y C 2022 Phys. Rev. B. 106 L161402 Google Scholar
[12]	Yao S Y, Wang Z 2018 Phys. Rev. Lett. 121 086803 Google Scholar
[13]	Hou J K, Zhu J F, Ma R X, Xue B Y, Zhu Y C, Lin J T, Jiang X S, Zheng Y L, Chen X F, Cheng Y, Ge L, Wan W J 2024 Phys. Rev. Lett. 132 256902 Google Scholar
[14]	Zhang C, Liang P F, Lambert N, Cirio M 2024 Phys. Rev. Res. 6 023012 Google Scholar
[15]	Masson S J, Covey J P, Will S, Asenjo- Garcia A 2024 Phys. Rev. X Quantum 5 010344 Google Scholar
[16]	黄珊, 刘妮, 梁九卿 2018 物理学报 67 183701 Google Scholar Huang S, Liu N, Liang J Q 2018 Acta Phys. Sin. 67 183701 Google Scholar
[17]	Matsumoto N, Kawabata K, Ashida Y, Furukawa S, Ueda M 2020 Phys. Rev. Lett. 125 260601 Google Scholar
[18]	Turkeshi X, Schiró M 2023 Phys. Rev. B 107 L020403 Google Scholar
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[20]	Chen W J, Özdemir Ş K, Zhao G M, Wiersig J, Yang L 2017 Nature 548 192 Google Scholar
[21]	Dicke R H 1954 Phys. Rev. 93 99 Google Scholar
[22]	Zhao X Q, Liu N, Liang J Q 2014 Phys. Rev. A. 90 023622 Google Scholar
[23]	Bhaseen M J, Mayoh J, Simons B D, Keeling J 2012 Phys. Rev. A 85 013817 Google Scholar
[24]	Keeling J, Bhaseen M J, Simons B D 2010 Phys. Rev. Lett. 105 043001 Google Scholar
[25]	Buijsman W, Gritsev V, Sprik R 2017 Phys. Rev. Lett. 118 080601 Google Scholar
[26]	Das P, Bhakuni D S, Sharma A 2023 Phys. Rev. A. 107 043706 Google Scholar
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[28]	Guerra C A E, Mahecha-Gómez J, Hirsch J G 2020 Eur. Phys. J. D. 74 200 Google Scholar
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原子-光子相互作用下耗散光腔中的量子相变和奇异点