宇称-时间对称与反对称研究进展

唐原江; 梁超; 刘永椿

doi:10.7498/aps.71.20221323

摘要

在标准量子力学中, 描述物理系统的哈密顿量一般是厄米的,以保证系统具有实能谱及系统演化的幺正性. 近些年来, 研究发现具有宇称-时间(parity-time, ${\cal {PT}} $)对称特性的非厄米哈密顿量也具有实能谱, 并且在${\cal {PT}} $对称相和${\cal {PT}} $对称破缺相之间存在一个新奇的非厄米奇异点, 这是厄米系统所不具有的. 最近, 人们在各种各样的物理系统中实现了${\cal {PT}} $对称和${\cal {PT}} $反对称的非厄米哈密顿量, 并演示了新奇的量子现象, 这不仅加深了对基本量子物理规律的理解, 也促进了应用技术的突破. 本综述将介绍${\cal {PT}} $对称和${\cal {PT}} $反对称的基本物理原理, 总结在光学系统和原子系统中实现${\cal {PT}} $对称和${\cal {PT}} $反对称的方案, 并回顾利用${\cal {PT}} $对称系统非厄米奇异点进行精密传感的研究.

关键词:

Abstract

In standard quantum mechanics, the Hamiltonian describing the physical system is generally Hermitian, so as to ensure that the system has real energy spectra and that the system’s evolution is unitary. In recent years, it has been found that non-Hermitian Hamiltonians with parity-time (${\cal {PT}}$) symmetry also have real energy spectra, and there is a novel non-Hermitian exceptional point between ${\cal {PT}}$-symmetric phase and ${\cal {PT}} $-symmetry-broken phase, which is unique to non-Hermitian systems. Recently, people have realized ${\cal {PT}} $ symmetric and anti-${\cal {PT}}$ symmetric non-Hermitian Hamiltonians in various physical systems and demonstrated novel quantum phenomena, which not only deepened our understanding of the basic laws of quantum physics, but also promoted the breakthrough of application technology. This review will introduce the basic physical principles of ${\cal {PT}} $ symmetry and anti-${\cal {PT}}$ symmetry, summarize the schemes to realize ${\cal {PT}} $ symmetry and anti-${\cal {PT}} $ symmetry in optical and atomic systems systematically, including the observation of ${\cal {PT}} $-symmetry transitions by engineering time-periodic dissipation and coupling in ultracold atoms and single trapped ion, the realization of anti-${\cal {PT}} $ symmetry in dissipative optical system by indirect coupling, and realizing anti-${\cal {PT}} $-symmetry through fast atomic coherent transmission in flying atoms. Finally, we review the research on precision sensing using non-Hermitian exceptional points of ${\cal {PT}} $-symmetric systems. Near the exceptional points, the eigenfrequency splitting follows an ${\varepsilon }^{\tfrac{1}{N}}$-dependence, where the $\varepsilon$ is the perturbation and $ N $ is the order of the exceptional point. We review the ${\cal {PT}}$-symmetric system composed of three equidistant micro-ring cavities and enhanced sensitivity at third-order exceptional points. In addition, we also review the debate on whether exceptional-point sensors can improve the signal-to-noise ratio when considering noise, and the current development of exceptional-point sensors, which is still an open and challenging question.

Keywords:

作者及机构信息

1.
清华大学物理系, 低维量子物理国家重点实验室, 北京　100084

2.
教育部量子信息前沿科学中心, 北京　100084

通信作者: 刘永椿, ycliu@tsinghua.edu.cn

基金项目: 广东省重点领域研发计划(批准号: 2019B030330001)、国家自然科学基金(批准号: 92050110, 91736106, 11674390, 91836302)和国家重点研发计划(批准号: 2018YFA0306504)资助的课题.

Authors and contacts

1.
State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China

2.
Frontier Science Center for Quantum Information, Beijing 100084, China

Corresponding author: Liu Yong-Chun, ycliu@tsinghua.edu.cn

Funds: Project supported by the Key-Area Research and Development Program of Guangdong Province, China (Grant No. 2019B030330001), the National Natural Science Foundation of China (NSFC) (Grant Nos. 92050110, 91736106, 11674390, 91836302), and the National Key R&D Program of China (Grant No. 2018YFA0306504).

文章全文 : translate this paragraph

参考文献

[1]	Bender C M 2007 Rep. Prog. Phys. 70 947 Google Scholar
[2]	Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243 Google Scholar
[3]	Heiss W D 2004 J. Phys. A. Math. Gen. 37 2455 Google Scholar
[4]	Bender C M, Brody D C, Jones H F 2003 Am. J. Phys. 71 1095 Google Scholar
[5]	Bagchi B, Quesne C 2000 Phys. Lett. A 273 285 Google Scholar
[6]	Makris K G, El-Ganainy R, Christodoulides D N, Musslimani Z H 2008 Phys. Rev. Lett. 100 103904 Google Scholar
[7]	Longhi S 2009 Phys. Rev. Lett. 103 123601 Google Scholar
[8]	Klaiman S, Günther U, Moiseyev N 2008 Phys. Rev. Lett. 101 080402 Google Scholar
[9]	Rüter C E, Makris K G, El-Ganainy R, Christodoulides D N, Segev M, Kip D 2010 Nat. Phys. 6 192 Google Scholar
[10]	Ramezani H, Kottos T, El-Ganainy R, Christodoulides D N 2010 Phys. Rev. A 82 043803 Google Scholar
[11]	Chang L, Jiang X, Hua S, Yang C, Wen J, Jiang L, Li G, Wang G, Xiao M 2014 Nat. Photonics 8 524 Google Scholar
[12]	Peng B, Özdemir S K, Lei F, Monifi F, Gianfreda M, Long G L, Fan S, Nori F, Bender C M, Yang L 2014 Nat. Phys. 10 394 Google Scholar
[13]	Lin Z, Ramezani H, Eichelkraut T, Kottos T, Cao H, Christodoulides D N 2011 Phys. Rev. Lett. 106 213901 Google Scholar
[14]	Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N, Peschel U 2012 Nature 488 167 Google Scholar
[15]	Feng L, Xu Y L, Fegadolli W S, Lu M H, Oliveira J E B, Almeida V R, Chen Y F, Scherer A 2013 Nat. Mater. 12 108 Google Scholar
[16]	Feng L, Wong Z J, Ma R M, Wang Y, Zhang X 2014 Science 346 972 Google Scholar
[17]	Hodaei H, Miri M A, Heinrich M, Christodoulides D N, Khajavikhan M 2014 Science 346 975 Google Scholar
[18]	Miao P, Zhang Z, Sun J, Walasik W, Longhi S, Litchinitser N M, Feng L 2016 Science 353 464 Google Scholar
[19]	Zhang Z, Qiao X, Midya B, Liu K, Sun J, Wu T, Liu W, Agarwal R, Jornet J M, Longhi S, Litchinitser N M, Feng L 2020 Science 368 760 Google Scholar
[20]	Hang C, Huang G, Konotop V V 2013 Phys. Rev. Lett. 110 083604 Google Scholar
[21]	Zhang Z, Zhang Y, Sheng J, Yang L, Miri M A, Christodoulides D N, He B, Zhang Y, Xiao M 2016 Phys. Rev. Lett. 117 123601 Google Scholar
[22]	Sun Y, Tan W, Li H, Li J, Chen H 2014 Phys. Rev. Lett. 112 143903 Google Scholar
[23]	Yang X, Li J, Ding Y, Xu M, Zhu X F, Zhu J 2022 Phys. Rev. Lett. 128 065701 Google Scholar
[24]	Schindler J, Li A, Zheng M C, Ellis F M, Kottos T 2011 Phys. Rev. A 84 040101 Google Scholar
[25]	Wu Y, Liu W, Geng J, Song X, Ye X, Duan C K, Rong X, Du J 2019 Science 364 878 Google Scholar
[26]	Jing H, Özdemir S K, Geng Z, Zhang J, Lü X Y, Peng B, Yang L, Nori F 2015 Sci. Rep. 5 9663 Google Scholar
[27]	Schönleber D W, Eisfeld A, El-Ganainy R 2016 New J. Phys. 18 045014 Google Scholar
[28]	Zhu X, Ramezani H, Shi C, Zhu J, Zhang X 2014 Phys. Rev. X 4 031042
[29]	Fleury R, Sounas D, Alù A 2015 Nat. Commun. 6 5905 Google Scholar
[30]	Bittner S, Dietz B, Günther U, Harney H L, Miski-Oglu M, Richter A, Schäfer F 2012 Phys. Rev. Lett. 108 024101 Google Scholar
[31]	Yang F, Liu Y C, You L 2017 Phys. Rev. A 96 053845 Google Scholar
[32]	Antonosyan D A, Solntsev A S, Sukhorukov A A 2015 Opt. Lett. 40 4575 Google Scholar
[33]	Wu J H, Artoni M, La Rocca G C 2014 Phys. Rev. Lett. 113 123004 Google Scholar
[34]	Ge L, Türeci H E 2013 Phys. Rev. A 88 53810 Google Scholar
[35]	Zhao J, Liu Y, Wu L, Duan C K, Liu Y, Du J 2020 Phys. Rev. Appl. 13 014053 Google Scholar
[36]	Bergman A, Duggan R, Sharma K, Tur M, Zadok A, Alù A 2021 Nat. Commun. 12 486 Google Scholar
[37]	Zhang X L, Jiang T, Chan C T 2019 Light Sci. Appl. 8 88 Google Scholar
[38]	Peng P, Cao W, Shen C, Qu W, Wen J, Jiang L, Xiao Y 2016 Nat. Phys. 12 1139 Google Scholar
[39]	Wu H C, Jin L, Song Z 2021 Phys. Rev. B 103 235110 Google Scholar
[40]	Xu H S, Jin L 2021 Phys. Rev. A 104 012218 Google Scholar
[41]	Jin L 2018 Phys. Rev. A 98 022117 Google Scholar
[42]	Makris K G, El-Ganainy R, Christodoulides D N, Musslimani Z H 2011 Int. J. Theor. Phys. 50 1019 Google Scholar
[43]	Zyablovsky A A, Vinogradov A P, Pukhov A A, Dorofeenko A V, Lisyansky A A 2014 Physics-Uspekhi 57 1063 Google Scholar
[44]	Feng L, El-Ganainy R, Ge L 2017 Nat. Photonics 11 752 Google Scholar
[45]	El-Ganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S, Christodoulides D N 2018 Nat. Phys. 14 11 Google Scholar
[46]	Özdemir S K, Rotter S, Nori F, Yang L 2019 Nat. Mater. 18 783 Google Scholar
[47]	Krasnok A, Nefedkin N, Alu A 2021 IEEE Antennas Propag. Mag. 63 110 Google Scholar
[48]	Hang C, Huang G 2017 Adv. Phys. X 2 737
[49]	Qi B, Chen H, Ge L, Berini P, Ma R 2019 Adv. Opt. Mater. 7 1900694 Google Scholar
[50]	Konotop V V, Yang J, Zezyulin D A 2016 Rev. Mod. Phys. 88 035002 Google Scholar
[51]	Suchkov S V, Sukhorukov A A, Huang J, Dmitriev S V, Lee C, Kivshar Y S 2016 Laser Photon. Rev. 10 177 Google Scholar
[52]	Shankar R 1994 Principles of Quantum Mechanics (New York: Springer US) pp145–147
[53]	Guo A, Salamo G J, Duchesne D, Morandotti R, Volatier-Ravat M, Aimez V, Siviloglou G A, Christodoulides D N 2009 Phys. Rev. Lett. 103 093902 Google Scholar
[54]	Li J, Harter A K, Liu J, de Melo L, Joglekar Y N, Luo L 2019 Nat. Commun. 10 855 Google Scholar
[55]	Ding L Y, Shi K Y, Zhang Q X, Shen D N, Zhang X, Zhang W 2021 Phys. Rev. Lett. 126 083604 Google Scholar
[56]	Wiersig J 2014 Phys. Rev. Lett. 112 203901 Google Scholar
[57]	Chen W, Özdemir S K, Zhao G, Wiersig J, Yang L 2017 Nature 548 192 Google Scholar
[58]	Hodaei H, Hassan A U, Wittek S, Garcia-Gracia H, El- Ganainy R, Christodoulides D N, Khajavikhan M 2017 Nature 548 187
[59]	Lau H K, Clerk A A 2018 Nat. Commun. 9 4320 Google Scholar
[60]	Zhang M, Sweeney W, Hsu C W, Yang L, Stone A D, Jiang L 2019 Phys. Rev. Lett. 123 180501 Google Scholar
[61]	Lai Y H, Lu Y K, Suh M G, Yuan Z, Vahala K 2019 Nature 576 65 Google Scholar
[62]	Xiao Z, Li H, Kottos T, Alù A 2019 Phys. Rev. Lett. 123 213901 Google Scholar
[63]	Kononchuk R, Cai J, Ellis F, Thevamaran R, Kottos T 2022 Nature 607 697 Google Scholar
[64]	Kepesidis K V, Milburn T J, Huber J, Makris K G, Rotter S, Rabl P 2016 New J. Phys. 18 095003 Google Scholar
[65]	Schomerus H 2010 Phys. Rev. Lett. 104 233601 Google Scholar
[66]	Ge L, Stone A D 2014 Phys. Rev. X 4 031011
[67]	Ge L, Makris K G, Christodoulides D N, Feng L 2015 Phys. Rev. A 92 062135 Google Scholar
[68]	Malzard S, Poli C, Schomerus H 2015 Phys. Rev. Lett. 115 200402 Google Scholar
[69]	Miri M A, Heinrich M, El-Ganainy R, Christodoulides D N 2013 Phys. Rev. Lett. 110 233902 Google Scholar
[70]	Heinrich M, Miri M A, Stützer S, El-Ganainy R, Nolte S, Szameit A, Christodoulides D N 2014 Nat. Commun. 5 3698 Google Scholar
[71]	Christodoulides D, Yang J 2018 Parity-Time Symmetry and Its Applications (Singapore: Springer Singapore) pp513–534
[72]	Yang J 2017 Opt. Lett. 42 4067 Google Scholar
[73]	Nixon S, Yang J 2016 Phys. Rev. A 93 031802 Google Scholar
[74]	He Z, Li L, Cui W, Wang Y, Xue W, Xu H, Yi Z, Li C, Li Z 2021 New J. Phys. 23 053015 Google Scholar
[75]	Zhu X, Peng X, Qiu Y, Wang H, He Y 2020 New J. Phys. 22 033035 Google Scholar
[76]	Jing H, Özdemir S K, Lü X Y, Zhang J, Yang L, Nori F 2014 Phys. Rev. Lett. 113 53604 Google Scholar

施引文献

图 1 $\cal {PT} $对称系统(a)与$\cal {PT} $反对称系统(b)示意图

Fig. 1. Schematic diagram of $\cal {PT} $-symmetric system (a) and anti-$\cal {PT} $symmetric system (b).

下载: 全尺寸图片幻灯片

图 2 传统和$\cal {PT} $对称耦合光学系统　(a) 复折射率的实部($ {\mathit{n}}_{\mathrm{R}} $, 红线)和虚部($ {\mathit{n}}_{\mathrm{I}} $, 绿线)分布; (b) 传统和PT对称系统的叠加态; (c) 对于传统和$\cal {PT} $对称系统, 当系统在通道1或通道2处被激发时的光波传播情况^[9]

Fig. 2. Conventional and $\cal {PT} $-symmetric optical systems: (a) The distribution of real part ($ {n}_{\mathrm{R}} $, red line) and imaginary part ($ {n}_{\mathrm{I}} $ green line) of the complex refractive index; (b) superposition state of conventional and PT-symmetric systems; (c) light wave propagation when the system is excited at channel 1 or channel 2 ^[9].

下载: 全尺寸图片幻灯片

图 3 (a) 在冷原子系统中实现$\cal {PT}$对称的示意图^[54]; (b) 在单个囚禁离子系统中实现$\cal {PT} $对称的镱离子$ {}_{}{}^{171}{\mathrm{Y}\mathrm{b}}^{+} $的能级示意图^[55]; (c) 系统密度矩阵测量图^[55]; (d) 系统的相图^[55], 红色和黄色区域对应$\cal {PT} $对称相, 蓝色区域对应$\cal {PT} $对称破缺相

Fig. 3. (a) Schematic diagram of realizing $\cal {PT} $ symmetry in cold atom system^[54]; (b) schematic diagram of energy levels of ytterbium ion $ {}_{}{}^{171}{\mathrm{Y}\mathrm{b}}^{+} $ for realizing $\cal {PT} $ symmetry in a single trapped ion system^[55]; (c) system density matrix measurement diagram^[55]; (d) the phase diagram of the system^[55]. The red and yellow areas correspond to the $\cal {PT} $-symmetric phase, and the blue area corresponds to the $\cal {PT} $-symmetry-broken phase.

下载: 全尺寸图片幻灯片

图 4 (a) 耦合波导示意图; (b) 耦合波导的截面示意图, 波导$ c $红色部分表示存在较大耗散; (c), (d) 波导本征模式的特性; (e) 波导场强的特性; 数据点是有限元模拟结果, 实线是理论计算结果^[31]

Fig. 4. (a) Schematic diagram of coupled waveguide; (b) cross section diagram of coupled waveguide, the red part of waveguide c indicates large dissipation; (c), (d) characteristics of waveguide eigenmodes; (e) property of waveguide field strength. Data points are finite element simulation results, and solid lines are theoretical calculation results^[31].

下载: 全尺寸图片幻灯片

图 5 波导内的光场演化图^[31]　(a), (b) $\cal {PT} $对称相和$\cal {PT} $对称破缺相的传播特性, 数据点是有限元模拟结果, 实线是理论计算结果; (c) 传统厄米系统和$\cal {PT} $反对称系统的光场分布对比图; (d) 分束比例对波长的依赖特性, 红色线是$ \cal {PT} $反对称系统, 蓝色线是传统厄米系统

Fig. 5. Evolution diagram of light field in the waveguides^[31]: (a) (b) The propagation characteristics of $\cal {PT} $-symmetric phase and $\cal {PT} $-symmetry-broken phase, respectively, the data points are the result of finite element simulation, and the solid lines are the result of theoretical calculation; (c) comparison diagram of light field distribution between traditional Hermitian system and anti-PT-symmetric system; (d) the dependence of beam splitting ratio on wavelength, the red line is the anti-$\cal {PT} $-symmetric system, and the blue line is the traditional Hermitian system.

下载: 全尺寸图片幻灯片

图 6 光学微腔构型I (a)和构型II (b)及相应本征频率在复平面上的演化(c)(d), 数据点是有限元模拟结果, 实线是理论计算结果^[31]

Fig. 6. Optical microcavity configuration I (a) and configuration II (b) and the corresponding eigenfrequencies on the complex plane. Data points are finite element simulation results, and solid lines are theoretical calculation results^[31].

下载: 全尺寸图片幻灯片

图 7 (a) 通过热$ {}_{}{}^{87}\mathrm{R}\mathrm{b} $蒸汽池中的快速原子相干传输, 实现$\cal {PT} $反对称性的示意图; (b) 两个通道中的三能级${\Lambda }$型EIT构型^[38]

Fig. 7. (a) Schematic diagram of realizing anti-$ \cal {PT} $-symmetry through fast atomic coherent transmission in hot ⁸⁷Rb vapor cell; (b) three level Λ-type EIT configuration in two channels^[38].

下载: 全尺寸图片幻灯片

图 8 (a) 3个等距微环腔构成的$\cal {PT} $对称系统示意图, 两侧的谐振腔具有平衡的增益和损耗, 而中间的谐振腔是中性的; (b) 系统处于三阶非厄米奇异点的激光模式的强度分布; (c) 相邻激光谱线之间的分裂随微扰强度$\varepsilon $的变化, 数据点是实验测量结果, 实线是理论计算结果^[58]

Fig. 8. (a) Schematic diagram of $\cal {PT} $-symmetric system composed of three equidistant micro-ring cavities, the resonators on both sides have balanced gain and loss, while the resonators in the middle are neutral; (b) the intensity distribution of the laser mode with the system at the third-order non-Hermitian exception point; (c) splitting between adjacent laser spectral lines with perturbation intensity $\varepsilon$. Data points are experimental measurement results, and solid lines are theoretical calculation results^[58].

下载: 全尺寸图片幻灯片

map

[1]	Bender C M 2007 Rep. Prog. Phys. 70 947 Google Scholar
[2]	Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243 Google Scholar
[3]	Heiss W D 2004 J. Phys. A. Math. Gen. 37 2455 Google Scholar
[4]	Bender C M, Brody D C, Jones H F 2003 Am. J. Phys. 71 1095 Google Scholar
[5]	Bagchi B, Quesne C 2000 Phys. Lett. A 273 285 Google Scholar
[6]	Makris K G, El-Ganainy R, Christodoulides D N, Musslimani Z H 2008 Phys. Rev. Lett. 100 103904 Google Scholar
[7]	Longhi S 2009 Phys. Rev. Lett. 103 123601 Google Scholar
[8]	Klaiman S, Günther U, Moiseyev N 2008 Phys. Rev. Lett. 101 080402 Google Scholar
[9]	Rüter C E, Makris K G, El-Ganainy R, Christodoulides D N, Segev M, Kip D 2010 Nat. Phys. 6 192 Google Scholar
[10]	Ramezani H, Kottos T, El-Ganainy R, Christodoulides D N 2010 Phys. Rev. A 82 043803 Google Scholar
[11]	Chang L, Jiang X, Hua S, Yang C, Wen J, Jiang L, Li G, Wang G, Xiao M 2014 Nat. Photonics 8 524 Google Scholar
[12]	Peng B, Özdemir S K, Lei F, Monifi F, Gianfreda M, Long G L, Fan S, Nori F, Bender C M, Yang L 2014 Nat. Phys. 10 394 Google Scholar
[13]	Lin Z, Ramezani H, Eichelkraut T, Kottos T, Cao H, Christodoulides D N 2011 Phys. Rev. Lett. 106 213901 Google Scholar
[14]	Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N, Peschel U 2012 Nature 488 167 Google Scholar
[15]	Feng L, Xu Y L, Fegadolli W S, Lu M H, Oliveira J E B, Almeida V R, Chen Y F, Scherer A 2013 Nat. Mater. 12 108 Google Scholar
[16]	Feng L, Wong Z J, Ma R M, Wang Y, Zhang X 2014 Science 346 972 Google Scholar
[17]	Hodaei H, Miri M A, Heinrich M, Christodoulides D N, Khajavikhan M 2014 Science 346 975 Google Scholar
[18]	Miao P, Zhang Z, Sun J, Walasik W, Longhi S, Litchinitser N M, Feng L 2016 Science 353 464 Google Scholar
[19]	Zhang Z, Qiao X, Midya B, Liu K, Sun J, Wu T, Liu W, Agarwal R, Jornet J M, Longhi S, Litchinitser N M, Feng L 2020 Science 368 760 Google Scholar
[20]	Hang C, Huang G, Konotop V V 2013 Phys. Rev. Lett. 110 083604 Google Scholar
[21]	Zhang Z, Zhang Y, Sheng J, Yang L, Miri M A, Christodoulides D N, He B, Zhang Y, Xiao M 2016 Phys. Rev. Lett. 117 123601 Google Scholar
[22]	Sun Y, Tan W, Li H, Li J, Chen H 2014 Phys. Rev. Lett. 112 143903 Google Scholar
[23]	Yang X, Li J, Ding Y, Xu M, Zhu X F, Zhu J 2022 Phys. Rev. Lett. 128 065701 Google Scholar
[24]	Schindler J, Li A, Zheng M C, Ellis F M, Kottos T 2011 Phys. Rev. A 84 040101 Google Scholar
[25]	Wu Y, Liu W, Geng J, Song X, Ye X, Duan C K, Rong X, Du J 2019 Science 364 878 Google Scholar
[26]	Jing H, Özdemir S K, Geng Z, Zhang J, Lü X Y, Peng B, Yang L, Nori F 2015 Sci. Rep. 5 9663 Google Scholar
[27]	Schönleber D W, Eisfeld A, El-Ganainy R 2016 New J. Phys. 18 045014 Google Scholar
[28]	Zhu X, Ramezani H, Shi C, Zhu J, Zhang X 2014 Phys. Rev. X 4 031042
[29]	Fleury R, Sounas D, Alù A 2015 Nat. Commun. 6 5905 Google Scholar
[30]	Bittner S, Dietz B, Günther U, Harney H L, Miski-Oglu M, Richter A, Schäfer F 2012 Phys. Rev. Lett. 108 024101 Google Scholar
[31]	Yang F, Liu Y C, You L 2017 Phys. Rev. A 96 053845 Google Scholar
[32]	Antonosyan D A, Solntsev A S, Sukhorukov A A 2015 Opt. Lett. 40 4575 Google Scholar
[33]	Wu J H, Artoni M, La Rocca G C 2014 Phys. Rev. Lett. 113 123004 Google Scholar
[34]	Ge L, Türeci H E 2013 Phys. Rev. A 88 53810 Google Scholar
[35]	Zhao J, Liu Y, Wu L, Duan C K, Liu Y, Du J 2020 Phys. Rev. Appl. 13 014053 Google Scholar
[36]	Bergman A, Duggan R, Sharma K, Tur M, Zadok A, Alù A 2021 Nat. Commun. 12 486 Google Scholar
[37]	Zhang X L, Jiang T, Chan C T 2019 Light Sci. Appl. 8 88 Google Scholar
[38]	Peng P, Cao W, Shen C, Qu W, Wen J, Jiang L, Xiao Y 2016 Nat. Phys. 12 1139 Google Scholar
[39]	Wu H C, Jin L, Song Z 2021 Phys. Rev. B 103 235110 Google Scholar
[40]	Xu H S, Jin L 2021 Phys. Rev. A 104 012218 Google Scholar
[41]	Jin L 2018 Phys. Rev. A 98 022117 Google Scholar
[42]	Makris K G, El-Ganainy R, Christodoulides D N, Musslimani Z H 2011 Int. J. Theor. Phys. 50 1019 Google Scholar
[43]	Zyablovsky A A, Vinogradov A P, Pukhov A A, Dorofeenko A V, Lisyansky A A 2014 Physics-Uspekhi 57 1063 Google Scholar
[44]	Feng L, El-Ganainy R, Ge L 2017 Nat. Photonics 11 752 Google Scholar
[45]	El-Ganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S, Christodoulides D N 2018 Nat. Phys. 14 11 Google Scholar
[46]	Özdemir S K, Rotter S, Nori F, Yang L 2019 Nat. Mater. 18 783 Google Scholar
[47]	Krasnok A, Nefedkin N, Alu A 2021 IEEE Antennas Propag. Mag. 63 110 Google Scholar
[48]	Hang C, Huang G 2017 Adv. Phys. X 2 737
[49]	Qi B, Chen H, Ge L, Berini P, Ma R 2019 Adv. Opt. Mater. 7 1900694 Google Scholar
[50]	Konotop V V, Yang J, Zezyulin D A 2016 Rev. Mod. Phys. 88 035002 Google Scholar
[51]	Suchkov S V, Sukhorukov A A, Huang J, Dmitriev S V, Lee C, Kivshar Y S 2016 Laser Photon. Rev. 10 177 Google Scholar
[52]	Shankar R 1994 Principles of Quantum Mechanics (New York: Springer US) pp145–147
[53]	Guo A, Salamo G J, Duchesne D, Morandotti R, Volatier-Ravat M, Aimez V, Siviloglou G A, Christodoulides D N 2009 Phys. Rev. Lett. 103 093902 Google Scholar
[54]	Li J, Harter A K, Liu J, de Melo L, Joglekar Y N, Luo L 2019 Nat. Commun. 10 855 Google Scholar
[55]	Ding L Y, Shi K Y, Zhang Q X, Shen D N, Zhang X, Zhang W 2021 Phys. Rev. Lett. 126 083604 Google Scholar
[56]	Wiersig J 2014 Phys. Rev. Lett. 112 203901 Google Scholar
[57]	Chen W, Özdemir S K, Zhao G, Wiersig J, Yang L 2017 Nature 548 192 Google Scholar
[58]	Hodaei H, Hassan A U, Wittek S, Garcia-Gracia H, El- Ganainy R, Christodoulides D N, Khajavikhan M 2017 Nature 548 187
[59]	Lau H K, Clerk A A 2018 Nat. Commun. 9 4320 Google Scholar
[60]	Zhang M, Sweeney W, Hsu C W, Yang L, Stone A D, Jiang L 2019 Phys. Rev. Lett. 123 180501 Google Scholar
[61]	Lai Y H, Lu Y K, Suh M G, Yuan Z, Vahala K 2019 Nature 576 65 Google Scholar
[62]	Xiao Z, Li H, Kottos T, Alù A 2019 Phys. Rev. Lett. 123 213901 Google Scholar
[63]	Kononchuk R, Cai J, Ellis F, Thevamaran R, Kottos T 2022 Nature 607 697 Google Scholar
[64]	Kepesidis K V, Milburn T J, Huber J, Makris K G, Rotter S, Rabl P 2016 New J. Phys. 18 095003 Google Scholar
[65]	Schomerus H 2010 Phys. Rev. Lett. 104 233601 Google Scholar
[66]	Ge L, Stone A D 2014 Phys. Rev. X 4 031011
[67]	Ge L, Makris K G, Christodoulides D N, Feng L 2015 Phys. Rev. A 92 062135 Google Scholar
[68]	Malzard S, Poli C, Schomerus H 2015 Phys. Rev. Lett. 115 200402 Google Scholar
[69]	Miri M A, Heinrich M, El-Ganainy R, Christodoulides D N 2013 Phys. Rev. Lett. 110 233902 Google Scholar
[70]	Heinrich M, Miri M A, Stützer S, El-Ganainy R, Nolte S, Szameit A, Christodoulides D N 2014 Nat. Commun. 5 3698 Google Scholar
[71]	Christodoulides D, Yang J 2018 Parity-Time Symmetry and Its Applications (Singapore: Springer Singapore) pp513–534
[72]	Yang J 2017 Opt. Lett. 42 4067 Google Scholar
[73]	Nixon S, Yang J 2016 Phys. Rev. A 93 031802 Google Scholar
[74]	He Z, Li L, Cui W, Wang Y, Xue W, Xu H, Yi Z, Li C, Li Z 2021 New J. Phys. 23 053015 Google Scholar
[75]	Zhu X, Peng X, Qiu Y, Wang H, He Y 2020 New J. Phys. 22 033035 Google Scholar
[76]	Jing H, Özdemir S K, Lü X Y, Zhang J, Yang L, Nori F 2014 Phys. Rev. Lett. 113 53604 Google Scholar

[1]	张慧洁, 贺衎. 哈密顿量宇称-时间对称性的刻画. , 2024, 73(4): 040302. doi: 10.7498/aps.73.20230458
[2]	古燕, 陆展鹏. 非厄米耦合链中的局域化转变. , 2024, 73(19): 197101. doi: 10.7498/aps.73.20240976
[3]	刘敬鹄, 徐志浩. 随机两体耗散诱导的非厄米多体局域化. , 2024, 73(7): 077202. doi: 10.7498/aps.73.20231987
[4]	王利凯, 王宇倩, 郭志伟, 江海涛, 李云辉, 羊亚平, 陈鸿. 基于高阶非厄密物理的磁共振无线电能传输研究进展. , 2024, 73(20): 201101. doi: 10.7498/aps.73.20241079
[5]	张光成, 孙武, 周志鹏, 全秀娥, 叶伏秋. 周期调制四通道光学波导的宇称-时间对称特性调控和动力学研究. , 2024, 73(16): 164201. doi: 10.7498/aps.73.20240690
[6]	江翠, 李家锐, 亓迪, 张莲莲. 具有宇称-时间反演对称性的虚势能对T-型石墨烯结构能谱和边缘态的影响. , 2024, 73(20): 207301. doi: 10.7498/aps.73.20240871
[7]	蒋宏帆, 林机, 胡贝贝, 张肖. 非宇称时间对称耦合器中的非局域孤子. , 2023, 72(10): 104205. doi: 10.7498/aps.72.20230082
[8]	潘磊. 非厄米线性响应理论及其应用. , 2022, 71(17): 170305. doi: 10.7498/aps.71.20220862
[9]	施婷婷, 张露丹, 张帅宁, 张威. 两量子比特系统中相互作用对高阶奇异点的影响. , 2022, 0(0): 0-0. doi: 10.7498/aps.71.20220716
[10]	施婷婷, 张露丹, 张帅宁, 张威. 两量子比特系统中相互作用对高阶奇异点的影响. , 2022, 71(13): 130303. doi: 10.7498/aps.70.20220716
[11]	范辉颖, 罗杰. 非厄密电磁超表面研究进展. , 2022, 71(24): 247802. doi: 10.7498/aps.71.20221706
[12]	高洁, 杭超. 里德伯原子中非厄米电磁诱导光栅引起的弱光孤子偏折及其操控. , 2022, 71(13): 133202. doi: 10.7498/aps.71.20220456
[13]	祝可嘉, 郭志伟, 陈鸿. 实验观测非厄米系统奇异点的手性翻转现象. , 2022, 71(13): 131101. doi: 10.7498/aps.71.20220842
[14]	曲登科, 范毅, 薛鹏. 高维宇称-时间对称系统中的信息恢复与临界性. , 2022, 0(0): . doi: 10.7498/aps.71.20220511
[15]	曲登科, 范毅, 薛鹏. 高维宇称-时间对称系统中的信息恢复与临界性. , 2022, 71(13): 130301. doi: 10.7498/aps.70.20220511
[16]	胡洲, 曾招云, 唐佳, 罗小兵. 周期驱动的二能级系统中的准宇称-时间对称动力学. , 2022, 71(7): 074207. doi: 10.7498/aps.70.20220270
[17]	胡洲, 曾招云, 唐佳, 罗小兵. 周期驱动的二能级系统中的准宇称-时间对称动力学. , 2022, 0(0): 0-0. doi: 10.7498/aps.71.20220270
[18]	张翔宇, 刘会刚, 康明, 刘波, 刘海涛. 金属-介质-金属多层结构可调谐Fabry-Perot共振及高灵敏折射率传感. , 2021, 70(14): 140702. doi: 10.7498/aps.70.20202058
[19]	张高见, 王逸璞. 腔光子-自旋波量子耦合系统中各向异性奇异点的实验研究. , 2020, 69(4): 047103. doi: 10.7498/aps.69.20191632
[20]	党婷婷, 王娟芬, 安亚东, 刘香莲, 张朝霞, 杨玲珍. 亮孤子在宇称时间对称波导中的传输和控制. , 2015, 64(6): 064211. doi: 10.7498/aps.64.064211

计量

文章访问数: 11204
PDF下载量: 1087
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板

宇称-时间对称与反对称研究进展