Research progress of parity-time symmetry and anti-symmetry

Tang Yuan-Jiang; Liang Chao; Liu Yong-Chun

doi:10.7498/aps.71.20221323

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:

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).

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References

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Cited By

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

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

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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]

Figure 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].

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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

Figure 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.

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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]

Figure 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].

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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

Figure 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.

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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]

Figure 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].

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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]

Figure 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].

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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]

Figure 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].

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map

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[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
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[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
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[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
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[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
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[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
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Vol. 71, Issue 17 - 2022-09-05

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