Please wait a minute...
Chin. Phys. B, 2020, Vol. 29(12): 120302    DOI: 10.1088/1674-1056/abc0dc
GENERAL Prev   Next  

Chaotic dynamics of complex trajectory and its quantum signature

Wen-Lei Zhao(赵文垒)1,†, Pengkai Gong(巩膨恺)1, Jiaozi Wang(王骄子)2, and Qian Wang(王骞)3,
1 School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China; 2 Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China; 3 Department of Physics, Zhejiang Normal University, Jinhua 321004, China
Abstract  We investigate both the quantum and classical dynamics of a non-Hermitian system via a kicked rotor model with $\mathcalPT$ symmetry. For the quantum dynamics, both the mean momentum and mean square of momentum exhibit the staircase growth with time when the system parameter is in the neighborhood of the $\mathcalPT$ symmetry breaking point. If the system parameter is much larger than the $\mathcalPT$ symmetry breaking point, the accelerator mode results in the directed spreading of the wavepackets as well as the ballistic diffusion in momentum space. For the classical dynamics, the non-Hermitian kicking potential leads to the exponentially-fast increase of classical complex trajectories. As a consequence, the imaginary part of the trajectories exponentially diffuses with time, while the real part exhibits the normal diffusion. Our analytical prediction of the exponential diffusion of imaginary momentum and its breakdown time is in good agreement with numerical results. The quantum signature of the chaotic diffusion of the complex trajectories is reflected by the dynamics of the out-of-time-order correlators (OTOC). In the semiclassical regime, the rate of the exponential increase of the OTOC is equal to that of the exponential diffusion of the complex trajectories.
Keywords:  $\calPT$ symmetry      quantum-classical correspondence      quantum chaos  
Received:  11 July 2020      Revised:  28 August 2020      Published:  13 November 2020
PACS:  03.65.-w (Quantum mechanics)  
  03.65.Yz (Decoherence; open systems; quantum statistical methods)  
  05.45.-a (Nonlinear dynamics and chaos)  
  05.45.Mt (Quantum chaos; semiclassical methods)  
Fund: Project partially supported by the National Natural Science Foundation of China (Grant Nos. 12065009, 11804130, and 11805165) and Zhejiang Provincial Nature Science Foundation, China (Grant No. LY20A050001).
Corresponding Authors:  Corresponding author. E-mail: Corresponding author. E-mail:   

Cite this article: 

Wen-Lei Zhao(赵文垒), Pengkai Gong(巩膨恺), Jiaozi Wang(王骄子), and Qian Wang(王骞) Chaotic dynamics of complex trajectory and its quantum signature 2020 Chin. Phys. B 29 120302

[1] Bender C M and Boettcher S Phys. Rev. Lett. 80 5243 DOI: 10.1103/PhysRevLett.80.52431998
[2] Bender C M, Brody D C and Jones H F Phys. Rev. Lett. 89 270401 DOI: 10.1103/PhysRevLett.89.2704012002
[3] Mostafazadeh A 2002 J. Math. Phys. (N. Y.) 43 2814 DOI: 10.1063/1.1461427
[4] Bender C M Rep. Prog. Phys. 70 947 DOI: 10.1088/0034-4885/70/6/R032007
[5] Mostafazadeh A Int. J. Geom. Meth. Mod. Phys. 07 1191 DOI: 10.1142/S02198878100048162010
[6] Jones-Smith K and Mathur H Phys. Rev. A 82 042101 DOI: 10.1103/PhysRevA.82.0421012010
[7] Moiseyev N Non-Hermitian quantum mechanics (Cambridge: Cambridge University Press) p. 211 DOI: 10.1017/CBO97805119761862011
[8] Jones-Smith K and Mathur H Phys. Rev. D 89 125014 DOI: 10.1103/PhysRevD.89.1250142014
[9] Cao H and Wiersig J Rev. Mod. Phys. 87 61 DOI: 10.1103/RevModPhys.87.612015
[10] El-Ganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S and Christodoulides D N 2018 Nat. Phys. 14 11 DOI: 10.1038/nphys4323
[11] Berry M Czech. J. Phys. 54 1039 DOI: 10.1023/B:CJOP.0000044002.05657.042004
[12] Klaiman S, Günther U and Moiseyev N Phys. Rev. Lett. 101 080402 DOI: 10.1103/PhysRevLett.101.0804022008
[13] Graefe E M, Korsch H J and Niederle A E Phys. Rev. Lett. 101 150408 DOI: 10.1103/PhysRevLett.101.1504082008
[14] Mostafazadeh A Phys. Rev. Lett. 102 220402 DOI: 10.1103/PhysRevLett.102.2204022009
[15] Graefe E M and Schubert R Phys. Rev. A 83 060101 DOI: 10.1103/PhysRevA.83.0601012011
[16] Hou T J Phys. Rev. A 95 013824 DOI: 10.1103/PhysRevA.95.0138242017
[17] Joshi S and Galbraith I Phys. Rev. A 98 042117 DOI: 10.1103/PhysRevA.98.0421172018
[18] Rotter I J. Phys. A 42 153001 DOI: 10.1088/1751-8113/42/15/1530012009
[19] Zloshchastiev K G and Sergi A J. Mod. Opt. 61 1298 DOI: 10.1080/09500340.2014.9305282014
[20] Choi Y, Kang S, Lim S, Kim W, Kim J R, Lee J H and An K Phys. Rev. Lett. 104 153601 DOI: 10.1103/PhysRevLett.104.1536012010
[21] Barontini G, Labouvie R, Stubenrauch F, Vogler A, Guarrera V and Ott H Phys. Rev. Lett. 110 035302 DOI: 10.1103/PhysRevLett.110.0353022013
[22] Rüter C E, Makris K G, El-Ganainy R, Christodoulides D N, Segev M and Kip D Nat. Phys. 6 192 DOI: 10.1038/nphys15152010
[23] Longhi S Phys. Rev. Lett. 105 013903 DOI: 10.1103/PhysRevLett.105.0139032010
[24] Buslaev V and Grecchi V J. Phys. A 26 5541 DOI: 10.1088/0305-4470/26/20/0351993
[25] Bender C M Contemp. Phys. 46 277 DOI: 10.1080/001075000726322005
[26] Ruschhaupt A, Delgado F and Muga J G J. Phys. A: Math. Gen. 38 L171 DOI: 10.1088/0305-4470/38/9/L032005
[27] El-Ganainy R, Makris K G, Christodoulides D N and Musslimani Z H Opt. Lett. 32 2632 DOI: 10.1364/OL.32.0026322007
[28] Longhi S Phys. Rev. Lett. 103 123601 DOI: 10.1103/PhysRevLett.103.1236012009
[29] Guo A, Salamo G J, Duchesne D, Morandotti R, Volatier-Ravat M, Aimez V, Siviloglou G A and Christodoulides D N Phys. Rev. Lett. 103 093902 DOI: 10.1103/PhysRevLett.103.0939022009
[30] Feng L, Ayache M, Huang J, Xu Y L, Lu M H, Chen Y F, Fainman Y and Scherer A Science 333 729 DOI: 10.1126/science.12060382011
[31] Alexeeva N V, Barashenkov I V, Sukhorukov A A and Kivshar Y S Phys. Rev. A 85 063837 DOI: 10.1103/PhysRevA.85.0638372012
[32] Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N and Peschel U Nature 488 167 DOI: 10.1038/nature112982012
[33] Nandkishore R and Huse D A Annu. Rev. Condens. Matter Phys. 6 15 DOI: 10.1146/annurev-conmatphys-031214-0147262015
[34] Ponte P, Papić Z, Huveneers F and Abanin D A Phys. Rev. Lett. 114 140401 DOI: 10.1103/PhysRevLett.114.1404012015
[35] Lazarides A, Das A and Moessner R Phys. Rev. Lett. 115 030402 DOI: 10.1103/PhysRevLett.115.0304022015
[36] Zhang J, Hess P, Kyprianidis A, Becker P, Lee A, Smith J, Pagano G, Potirniche I D, Potter A C and Vishwanath A Nature 543 217 DOI: 10.1038/nature214132017
[37] Yao N Y, Potter A C, Potirniche I D and Vishwanath A Phys. Rev. Lett. 118 030401 DOI: 10.1103/PhysRevLett.118.0304012017
[38] Yao N Y, Nayak C, Balents L and Zaletel M P Nat. Phys. 16 438 DOI: 10.1038/s41567-019-0782-32020
[39] Yao N Y and Nayak C2018 Phys. Today 71 40
[40] Lindner N H, Refael G and Galitski V Nat. Phys. 7 490 DOI: 10.1038/nphys19262011
[41] Titum P, Lindner N H, Rechtsman M C and Refael G Phys. Rev. Lett. 114 056801 DOI: 10.1103/PhysRevLett.114.0568012015
[42] Zhou L, Chen C and Gong J Phys. Rev. B 94 075443 DOI: 10.1103/PhysRevB.94.0754432016
[43] Roy R and Harper F Phys. Rev. B 96 155118 DOI: 10.1103/PhysRevB.96.1551182017
[44] Grifoni M and Hanggi P Phys. Rep. 304 229 DOI: 10.1016/S0370-1573(98)00022-21998
[45] Valle G Della, Ornigotti M, Cianci E, Foglietti V, Laporta P and Longhi S Phys. Rev. Lett. 98 263601 DOI: 10.1103/PhysRevLett.98.2636012007
[46] Valle G Della and Longhi S Phys. Rev. A 87 022119 DOI: 10.1103/PhysRevA.87.0221192013
[47] Huang L and Lai Y C Commun. Theor. Phys. 72 047601 DOI: 10.1088/1572-9494/ab69092020
[48] Li X L, Chen X Z, Liu C R and Huang L2020 Acta Phys. Sin. 69 080506 (in Chinese)
[49] Wu J and Xie X T Phys. Rev. A 86 032112 DOI: 10.1103/PhysRevA.86.0321122012
[50] El-Ganainy R, Makris K G and Christodoulides D N Phys. Rev. A 86 033813 DOI: 10.1103/PhysRevA.86.0338132012
[51] Moiseyev N Phys. Rev. A 83 052125 DOI: 10.1103/PhysRevA.83.0521252011
[52] Gong J and Wang Q H J. Phys. A 46 485302 DOI: 10.1088/1751-8113/46/48/4853022013
[53] Zhou L and Gong J Phys. Rev. B 98 205417 DOI: 10.1103/PhysRevB.98.2054172018
[54] West C T, Kottos T and Prosen T Phys. Rev. Lett. 104 054102 DOI: 10.1103/PhysRevLett.104.0541022010
[55] Longhi S2017 J. Phys. A 95 012125
[56] Zhao W L, Wang J, Wang X and Tong P Phys. Rev. E 99 042201 DOI: 10.1103/PhysRevE.99.0422012019
[57] Haake F Quantum Signatures of Chaos (3rd edn) (Berlin Heidelberg: Springer-Verlag) p. 247 DOI: 10.1007/978-1-4899-3698-1382010
[58] Casati G, Chirikov B V, Izraelev F M and Ford J Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Lecture Notes in Physics (Berlin: Springer) p. 334 DOI: 10.1007/BFb00217321979
[59] Casati G, Chirikov B V, Shepelyansky D L and Guarneri I Phys. Rep. 154 77 DOI: 10.1016/0370-1573(87)90009-31987
[60] Izrailev F M Phys. Rep. 196 299 DOI: 10.1016/0370-1573(90)90067-C1990
[61] Wang J, Guarneri I, Casati G and Gong J Phys. Rev. Lett. 107 234104 DOI: 10.1103/PhysRevLett.107.2341042011
[62] Liu J, Zhang C, Raizen M G and Niu Q Phys. Rev. A 73 013601 DOI: 10.1103/PhysRevA.73.0136012006
[63] Fishman S, Grempel D R and Prange R E Phys. Rev. Lett. 49 509 DOI: 10.1103/PhysRevLett.49.5091982
[64] Podolskiy V A, Narimanov E, Fang W and Cao H Proc. Natl. Acad. Sci. USA 101 10498 DOI: 10.1073/pnas.04028051012004
[65] Raizen M G and Steck D L Scholarpedia 6 10468 DOI: 10.4249/scholarpedia.104682011
[66] Rosen A, Fischer B, Bekker A and Fishman S J. Opt. Soc. Am. B 17 1579 DOI: 10.1364/JOSAB.17.0015792000
[67] Chaudhury S, Smith A, Anderson B, Ghose S and Jessen P Nature 461 768 DOI: 10.1038/nature083962009
[68] Zhao W L and Jie Q L Chin. Phys. B 29 080302 DOI: 10.1088/1674-1056/ab8a3a2020
[69] Zhao W L, Jie Q L and Zhou B Commun. Theor. Phys. 54 247 DOI: 10.1088/0253-6102/54/2/092010
[70] Zhao W L and Jie Q L Commun. Theor. Phys. 51 465 DOI: 10.1088/0253-6102/51/3/172009
[71] Yang Y B and Wang W G Chin. Phys. Lett. 32 030301 DOI: 10.1088/0256-307X/32/3/0303012015
[72] Graefe E M, Korsch H J, Rush A and Schubert R J. Phys. A 48 055301 DOI: 10.1088/1751-8113/48/5/0553012015
[73] Bender C M, Feinberg J, Hook D W and Weir D J Pramana J. Phys. 73 453 DOI: 10.1007/s12043-009-0099-32009
[74] Bender C M and Brody D C Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians, Lecture Notes in Physics (Berlin Heidelberg: Springer) 789 341 https://dx.doi.122009/org/10.1007/978-3-642-03174-8122009
[75] Bender C M, Boettcher S and Meisinger P N J. Math. Phys. 40 2201 DOI: 10.1063/1.5328601999
[76] Bender C M and Hook D W J. Phys. A 44 372001 DOI: 10.1088/1751-8113/44/37/3720012011
[77] Larkin A and Ovchinnikov Y N Sov. Phys. JETP 281200
[78] Maldacena J, Shenker S H and Stanford D J. High Energ. Phys. 2016 106 DOI: 10.1007/JHEP08(2016)1062016
[79] Hashimoto K, Murata K and Yoshii R2017 J. High Energ. Phys. 2017 138
[80] Dóra B and Moessner R Phys. Rev. Lett. 119 026802 DOI: 10.1103/PhysRevLett.119.0268022017
[81] Heyl M, Pollmann F and Dóra B Phys. Rev. Lett. 121 016801 DOI: 10.1103/PhysRevLett.121.0168012018
[82] Garcìa-Mata I, Saraceno M, Jalabert R A, Roncaglia A J and Wisniacki D A Phys. Rev. Lett. 121 210601 DOI: 10.1103/PhysRevLett.121.2106012018
[83] Jalabert R A, Garcìa-Mata I and Wisniacki D A Phys. Rev. E 98 062218 DOI: 10.1103/PhysRevE.98.0622182018
[84] Fortes E M, Garcìa-Mata I, Jalabert R A and Wisniacki D A Phys. Rev. E 100 042201 DOI: 10.1103/PhysRevE.100.0422012019
[85] Yan H, Wang J Z and Wang W G Commun. Theor. Phys. 71 1359 DOI: 10.1088/0253-6102/71/11/13592019
[87] Swingle B, Bentsen G, Schleier-Smith M and Hayden P Phys. Rev. A 94 040302(R) DOI: 10.1103/PhysRevA.94.0403022016
[88] Zhu G, Hafezi M and Grover T Phys. Rev. A 94 062329 DOI: 10.1103/PhysRevA.94.0623292016
[89] Li J, Fan R, Wang H, Ye B, Zeng B, Zhai H, Peng X and Du J Phys. Rev. X 7 031011 DOI: 10.1103/PhysRevX.7.0310112017
[90] Gärttner M, Bohnet J G, Safavi-Naini A, Wall M L, Bollinger J J and Rey A M Nat. Phys. 13 781 DOI: 10.1038/nphys41192017
[91] Chirikov B V Phys. Rep. 52 263 DOI: 10.1016/0370-1573(79)90023-11979
[92] Rozenbaum E B, Ganeshan S and Galitski V Phys. Rev. Lett. 118 086801 DOI: 10.1103/PhysRevLett.118.0868012017
[1] Quantum to classical transition induced by a classically small influence
Wen-Lei Zhao(赵文垒), Quanlin Jie(揭泉林). Chin. Phys. B, 2020, 29(8): 080302.
[2] Dynamical stable-jump-stable-jump picture in a non-periodically driven quantum relativistic kicked rotor system
Hsincheng Yu(于心澄), Zhongzhou Ren(任中洲), Xin Zhang(张欣). Chin. Phys. B, 2019, 28(2): 020504.
[3] Identifying the closeness of eigenstates in quantum many-body systems
Hai-bin Li(李海彬), Yang Yang(杨扬), Pei Wang(王沛), Xiao-guang Wang(王晓光). Chin. Phys. B, 2017, 26(8): 080502.
[4] Level spacing statistics for two-dimensional massless Dirac billiards
Huang Liang, Xu Hong-Ya, Lai Ying-Cheng, Celso Grebogi. Chin. Phys. B, 2014, 23(7): 070507.
[5] Rotational symmetry of classical orbits, arbitrary quantization of angular momentum and the role of the gauge field in two-dimensional space
Xin Jun-Li,Liang Jiu-Qing. Chin. Phys. B, 2012, 21(4): 040303.
[6] Chaos and quantum Fisher information in the quantum kicked top
Wang Xiao-Qian, Ma Jian, Zhang Xi-He, Wang Xiao-Guang. Chin. Phys. B, 2011, 20(5): 050510.
[1] Guo Ru-Hai, Shi Hong-Yan, Sun Xiu-Dong. Theoretical study of quantum confined Stark shift in InAs/GaAs quantum dots[J]. Chin. Phys., 2004, 13(12): 2141 -2146 .
[2] Luo Zheng-Ming, Deng Bai-Quan, Peng Li-Lin, Yan Jian-Cheng, Chen Zhi. Damaging impacts of energetic charge particles on materials in plasma energy explosive events[J]. Chin. Phys., 2006, 15(7): 1486 -1491 .
[3] Ding Gang, Zhong Shi-Sheng, Li Yang. Time series prediction using wavelet process neural network[J]. Chin. Phys. B, 2008, 17(6): 1998 -2003 .
[4] Li Dong-Mei, Liu Xiao-Jing, Li Yuan, Li Hai-Hong, Hu Gui-chao, Gao Kun, Liu De-Sheng, Xie Shi-Jie. Dynamical study on charge injection and transport in a metal/polythiophene/metal structure[J]. Chin. Phys. B, 2008, 17(8): 3067 -3076 .
[5] Li Meng-Meng, Long Yun-Ze, Tan Jin-Shan, Yin Hong-Xing, Sui Wan-Mei, Zhang Zhi-Ming. Dielectric properties of electrospun titanium compound/polymer composite nanofibres[J]. Chin. Phys. B, 2010, 19(2): 28102 -028102 .
[6] Wang Shao-Yi, Hong Wei-Yi, Zhang Qing-Bin, Li Qian-Guang, Lu Pei-Xiang. Attosecond manipulation of ionization and efficient broadband supercontinuum generation in stretched molecules[J]. Chin. Phys. B, 2010, 19(8): 83203 -083203 .
[7] Li Hui-Rong, Yin Jian-Ping. Propagation properties of electromagnetic fields in elliptic dielectric hollow fibres and their applications[J]. Chin. Phys. B, 2010, 19(8): 83204 -083204 .
[8] Hu Wei-Xuan, Cheng Bu-Wen, Xue Chun-Lai, Su Shao-Jian, Wang Qi-Ming. Formation of rippled surface morphology during Si/Si (100) epitaxy by ultrahigh vacuum chemical vapour deposition[J]. Chin. Phys. B, 2011, 20(12): 126801 .
[9] Cui Xin, Liang Xi-Xia, Wang Jian-Tao, Zhao Guo-Zhong. Ab initio studies on the mechanic and magnetic properties of PdHx[J]. Chin. Phys. B, 2011, 20(2): 26201 -026201 .
[10] Ma Xiao-Hua, Pan Cai-Yuan, Yang Li-Yuan, Yu Hui-You, Yang Ling, Quan Si, Wang Hao, Zhang Jin-Cheng, Hao Yue. Characterization of Al2O3/GaN/AlGaN/GaN metal–insulator–semiconductor high electron mobility transistors with different gate recess depths[J]. Chin. Phys. B, 2011, 20(2): 27304 -027304 .