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周期驱动的二能级系统中的准宇称-时间对称动力学

胡洲 曾招云 唐佳 罗小兵

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周期驱动的二能级系统中的准宇称-时间对称动力学

胡洲, 曾招云, 唐佳, 罗小兵

Quasi-parity-time symmetric dynamics in periodically driven two-level non-Hermitian system

Hu Zhou, Zeng Zhao-Yun, Tang Jia, Luo Xiao-Bing
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  • 本文研究了一个周期驱动的非宇称-时间对称二能级量子系统的非厄米动力学. 通过经典相空间分析方法,解出了该非厄米系统的Floquet态和准能谱, 并解析构造了由该非厄米哈密顿量支配下的量子态的非幺正时间演化算符, 给出了不同参数区域的量子态演化. 本文数值和分析证明, 该非宇称-时间对称二能级Floquet系统, 类似于宇称-时间对称系统, 存在一个准能谱从实数谱到复数谱的相变. 本文还揭示了在量子态的动态演化中存在一种准宇称-时间对称动力学, 即, 该系统的粒子布居概率演化完全满足时间空间对称(宇称-时间对称), 但是由于相位演化违反了宇称-时间对称性的要求, 因此包含相位信息的量子态演化不满足时间空间对称(宇称-时间对称). 这些结果加深了对非厄米物理的理解, 拓展和推广了传统的宇称-时间对称概念.
    In recent years, there have been intensive studies of non-Hermitian physics and parity–time (PT) symmetry due to their fundamental importance in theory and outstanding applications. A distinctive character in PT-symmetric system is phase transition (spontaneous PT-symmetry breaking), i.e. an all-real energy spectrum changes into an all-complex one when the non-Hermitian parameter exceeds a certain threshold. However, the conditions for PT-symmetric system with real energy spectrum to occur are rather restrictive. The generalization of PT-symmetric potentials to wider classes of non-PT-symmetric complex potentials with all-real energy spectra is a currently important endeavor. A simple PT-symmetric two-level Floquet quantum system is now being actively explored, because it holds potential for the realization of non-unitary single-qubit quantum gate. However, studies of the evolution dynamics of non-PT-symmetric two-level non-Hermitian Floquet quantum system are still relatively rare.In this paper, we investigate the non-Hermitian physics of a periodically driven non-PT-symmetric two-level quantum system. By phase-space analysis, we find that there exist so-called pseudo fixed points in phase space representing the Floquet solutions with fixed population difference and a time-dependent relative phase between the two levels. According to these pseudo fixed points, we analytically construct a non-unitary evolution operator and then explore the dynamic behaviors of the non-PT-symmetric two-level quantum system in different parameter regions. We confirm both analytically and numerically that the two-level non-Hermitian Floquet quantum system, although it is non-parity-time-symmetric, still features a phase transition with the quasienergy spectrum changing from all-real to all-complex energy spectrum, just like the PT symmetric system. Furthermore, we reveal that a novel phenomenon called quasi-PT symmetric dynamics occurs in the time evolution process. The quasi-PT symmetric dynamics is so named in our paper, in the sense that the time-evolution of population probabilities in the non-PT-symmetric two-level system satisfies fully the time-space symmetry (PT symmetry), while time-evolution of the quantum state (containing the phase) does not meet such a PT symmetry, due to the fact that time-evolution of the phases of the probability amplitudes on the two levels violates the requirement for the PT symmetry.
      通信作者: 罗小兵, xiaobingluo2013@aliyun.com
    • 基金项目: 浙江省自然科学基金(批准号: LY21A050002)、国家自然科学基金(批准号: 11975110, 12164022)、江西省教育厅科技项目(批准号: GJJ211026)和浙江理工大学科研启动基金(批准号: 20062318-Y)资助的课题.
      Corresponding author: Luo Xiao-Bing, xiaobingluo2013@aliyun.com
    • Funds: Project supported by the Natural Science Foundation of Zhejiang Province, China (Grant No. LY21A050002), the National Natural Science Foundation of China (Grant Nos. 11975110, 12164022), the Scientific and Technological Research Fund of Jiangxi Provincial Education Department (Grant No. GJJ211026), and Zhejiang Sci-Tech University Scientific Research Start-up Fund (Grant No. 20062318-Y).
    [1]

    Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243Google Scholar

    [2]

    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 093902Google Scholar

    [3]

    Rüter C E, Makris K G, El-Ganainy R, Christodoulides D N, Segev M, Kip D 2010 Nat. Phys. 6 192Google Scholar

    [4]

    Doppler J, Mailybaev A A, Böhm J, Kuhl U, Girschik A, Libisch F, Milburn T J, Rabl P, Moiseyev N, Rotter S 2016 Nature 537 76Google Scholar

    [5]

    Peng B, Özdemir Ş K, Lei F, Monifi F, Gianfreda M, Long G Lu, Fan S, Nori F, Bender C M, Yang L 2014 Nat. Phys. 10 394Google Scholar

    [6]

    Chang L, Jiang X, Hua S, Yang C, Wen J, Jiang L, Li G, Wang G, Xiao M 2014 Nat. Photonics 8 524Google Scholar

    [7]

    Bender C M, Berntson B K, Parker D, Samuel E 2013 Am. J. Phys. 81 173Google Scholar

    [8]

    Schindler J, Li A, Zheng M C, Ellis F M, Kottos T 2011 Phys. Rev. A 84 040101(R)

    [9]

    Fleury R, Sounas D, Alù A 2015 Nat. Com. 6 5905

    [10]

    Liu T, Zhu X, Chen F, Liang S, Zhu J 2018 Phys. Rev. Lett. 120 124502Google Scholar

    [11]

    Tang J S, Wang Y T, Yu S, He D Y, Xu J S, Liu B H, Chen G, Sun Y N, Sun K, Han Y J, Li C F, Guo G C 2016 Nat. Photonics 10 642Google Scholar

    [12]

    Xiao L, Zhan X, Bian Z H, Wang K K, Zhang X, Wang X P, Li J, Mochizuki K, Kim D, Kawakami N, Yi W, Obuse H, Sanders B C, Xue P 2017 Nat. Phys. 13 1117Google Scholar

    [13]

    Gao W C, Zheng C, Liu L, Wang T J, Wang C 2021 Optics Express 29 517Google Scholar

    [14]

    Li J, Harter A K, Liu J, Melo L de, Joglekar Y N, Luo L 2019 Nat. Com. 10 855Google Scholar

    [15]

    Zhang D K, Luo X Q, Wang Y P, Li T F, You J Q 2017 Nat. Com. 8 1368Google Scholar

    [16]

    沈瑞昌, 张国强, 王逸璞, 游建强 2019 物理学报 68 230305Google Scholar

    Shen R C, Zhang G Q, Wang Y P, You J Q 2019 Acta Phys. Sin. 68 230305Google Scholar

    [17]

    Wu Y, Liu W, Geng J, Song X, Ye X, Duan C K, Rong X, Dun J 2019 Science 364 878Google Scholar

    [18]

    Zheng C, Hao L, Long G L 2013 Philos. Trans. R. Soc. A 371 20120053Google Scholar

    [19]

    Wen J, Zheng C, Kong X, Wei S, Xin T, Long G 2019 Phys. Rev. A 99 062122Google Scholar

    [20]

    Wang W C, Zhou Y L, Zhang H L, Zhang J, Zhang M C, Xie Y, Wu C W, Chen T, Ou B Q, Wu W, Jing H, Chen P X 2021 Phys. Rev. A 103 L020201Google Scholar

    [21]

    Ding L, Shi K, Zhang Q, Shen D, Zhang X, Zhang W 2021 Phys. Rev. Lett. 126 083604Google Scholar

    [22]

    Lin Z, Ramezani H, Eichelkraut T, Kottos T, Cao H, Christodoulides D N 2011 Phys. Rev. Lett. 106 213901Google Scholar

    [23]

    Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N, Peschel U 2012 Nature 488 167Google Scholar

    [24]

    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 108Google Scholar

    [25]

    Sun Y, Tan W, Li H Q, Li J, Chen H 2014 Phys. Rev. Lett. 112 143903Google Scholar

    [26]

    Jin L, Song Z 2018 Phys. Rev. Lett. 121 073901Google Scholar

    [27]

    Hodaei H, Hassan A U, Wittek S, Garcia-Gracia H, El-Ganainy R, Christodoulides D N, Khajavikhan M 2017 Nature 548 187Google Scholar

    [28]

    Yu S, Meng Y, Tang J, Xu X, Wang Y, Yin P, Ke Z, Liu W, Li Z, Yang Y, Chen G, Han Y, Li C, Guo G 2020 Phys. Rev. Lett. 125 240506Google Scholar

    [29]

    Feng L, Wong Z J, Ma R M, Wang Y, Zhang X 2014 Science 346 972Google Scholar

    [30]

    Hodaei H, Miri M A, Heinrich M, Christodoulides D N, Khajavikhan M 2014 Science 346 975Google Scholar

    [31]

    Assawaworrarit S, Yu X, Fan S 2017 Nature 546 387Google Scholar

    [32]

    Xu H, Mason D, Jiang L, Harris J G E 2016 Nature 537 80Google Scholar

    [33]

    Bender C M, Brody D C, Jones H F 2002 Phys. Rev. Lett. 89 270401

    [34]

    Bender C M 2007 Rep. Prog. Phys. 70 947Google Scholar

    [35]

    Mostafazadeh A 2002 J. Math. Phys. 43 205Google Scholar

    [36]

    Mostafazadeh A 2002 J. Math. Phys. 43 2814Google Scholar

    [37]

    黄永峰, 曹怀信, 王文华 2020 物理学报 69 030301Google Scholar

    Huang Y F, Cao H X, Wang W H 2020 Acta Phys. Sin. 69 030301Google Scholar

    [38]

    Nixon S, Yang J 2016 Phys. Rev. A 93 031802(R)

    [39]

    Hang C, Gabadadze G, Huang G 2017 Phys. Rev. A 95 023833Google Scholar

    [40]

    Pan J, Zhou L 2020 Phys. Rev. B 102 094305Google Scholar

    [41]

    Luo X B, Huang J H, Zhong H H, Qin X Z, Xie Q T, Kivshar Y S, Lee C H 2013 Phys. Rev. Lett. 110 243902Google Scholar

    [42]

    Chitsazi M, Li H, Ellis F M, Kottos T 2017 Phys. Rev. Lett. 119 093901Google Scholar

    [43]

    Duan L, Wang Y, Chen Q 2020 Chin. Phys. Lett. 37 081101Google Scholar

    [44]

    Xie Q, Rong S, Liu X 2018 Phys. Rev. A 98 052122Google Scholar

    [45]

    Koutserimpas T T, Alù A, Fleury R 2018 Phys. Rev. A 97 013839Google Scholar

    [46]

    Luo X B, Wu D, Luo S, Guo Y, Yu X, Hu Q 2014 J. Phys. A:Math. Theor 47 345301Google Scholar

    [47]

    Yang B, Luo X B, Hu Q, Yu X 2016 Phys. Rev. A 94 043828Google Scholar

    [48]

    Luo X B, Yang B, Zhang X F, Li L, Yu X 2017 Phys. Rev. A 95 052128Google Scholar

    [49]

    Cui B, Wang L C, Yi X X 2010 Phys. Rev. A 82 062105

    [50]

    Liu Z P, Zhang J, Özdemir Ş K, Peng B, Jing H, Lu X Y, Li C W, Yang L, Nori F, Liu Y 2016 Phys. Rev. Lett. 117 110802Google Scholar

    [51]

    Bender C M, Brody D C, Jones H F, Meister B K 2007 Phys. Rev. Lett. 98 040403Google Scholar

  • 图 1  不同系统参数下的相空间轨道($ \omega = 2, \nu = 1 $) (a) ${\gamma _0} = $$ 1,\; {\gamma _1} = 0.5$; (b) ${\gamma _0} = 1,\; {\gamma _1} = 1$; (c) ${\gamma _0} = 1, \;{\gamma _1} = 2$; (d) ${\gamma _0} = 2,\; {\gamma _1} = 0.5$

    Fig. 1.  Phase-space trajectories with different system parameters($ \omega = 2, \nu = 1 $): (a) ${\gamma _0} = 1,\; {\gamma _1} = 0.5$; (b) ${\gamma _0} = 1, $$ \;{\gamma _1} = 1$; (c) ${\gamma _0} = 1, \;{\gamma _1} = 2$; (d) ${\gamma _0} = 2,\; {\gamma _1} = 0.5$.

    图 2  非厄米系统(1)在周期驱动(7)作用下共振($2{\gamma _0} = $$ \omega$)情况时的准能谱的实部(a)和虚部(b)随参数${\gamma _1}/\nu $的关系. 红线和蓝线代表直接对角化一个驱动周期的时间演化算符的数值结果, 圆圈代表经典相图中赝定点对应的准能量解析结果. 参数取为${\gamma _0} = 1,\; \omega = 2$

    Fig. 2.  Real (a) and imaginary (b) parts of the quasienergies as a function of ${\gamma _1}/\nu $ for the non-Hermitian system (1) subject to a periodic modulation (7) in the resonant ($2{\gamma _0} = $$ \omega$) case. The red and blue lines denote the numerical results of quasienergies computed through direct diagonalization of the time-evolution operator over one period of the driving, while the circles denote exact analytical results of quasienergies corresponding to the pseudo fixed points in phase space. The system parameters are set as ${\gamma _0} = 1, \;\omega = 2$

    图 3  在共振情况下的系统动力学($\tfrac{{{\gamma _1}}}{\nu } < 1$) (a) 两能级上的占有概率${\left| {{\psi _n}} \right|^2}$($n = 1, 2$)随时间演化; (b)相位${\theta _n}$($n = $$ 1, 2$)随时间演化; (c)以$ {t_0} = - 0.605 $为时间反演点的$ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $演化. 系统参数取为${\gamma _0} = 1, \;{\gamma _1} = 0.5,\; $$ \omega = 2,\; \nu = 1$ 初态为${\psi _1}(0) = 1, {\psi _2}(0) = 0$

    Fig. 3.  System dynamics for the resonance case with the non-Hermitian parameters $\tfrac{{{\gamma _1}}}{\nu } < 1$, starting the system with the state ${\psi _1}(0) = 1,\; {\psi _2}(0) = 0$: (a) Time evolutions of the occupation probabilities ${\left| {{\psi _1}} \right|^2}$ and ${\left| {{\psi _2}} \right|^2}$; (b) time evolutions of phases ${\theta _1}(t)$ and ${\theta _2}(t)$; (c) time evolution of the sum of phases, $ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $. Here we choose the time-inversion point $ {t_0} = - 0.605 $. The system parameters are ${\gamma _0} = 1,\; {\gamma _1} = 0.5,\; \omega = 2, \;\nu = 1$.

    图 4  在共振情况下的系统动力学(${{{\gamma _1}}}/{\nu } = 1$) (a) 两能级上的占有概率${\left| {{\psi _n}} \right|^2}$($n = 1, 2$)随时间演化;(b)相位${\theta _n}$($n = $$ 1, 2$)随时间演化; (c)以$ {t_0} = - 0.5 $为时间反演点的$ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $演化. 系统参数取为${\gamma _0} = 1,\; {\gamma _1} = 1,\; $$ \omega = 2,\; \nu = 1$ 初态为${\psi _1}(0) = 1,\; {\psi _2}(0) = 0$

    Fig. 4.  System dynamics for the resonance case with the non-Hermitian parameters ${{{\gamma _1}}}/{\nu } = 1$, starting the system with the state ${\psi _1}\left( 0 \right) = 1,\; {\psi _2}\left( 0 \right) = 0$: (a) Time evolutions of the occupation probabilities ${\left| {{\psi _1}} \right|^2}$ and ${\left| {{\psi _2}} \right|^2}$; (b) time evolutions of phases ${\theta _1}(t)$ and ${\theta _2}(t)$; (c) time evolution of the sum of phases, $ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $. Here the time-inversion point is given by $ {t_0} = - 0.5 $. The system parameters are ${\gamma _0} = 1,\; {\gamma _1} = 1,\; \omega = 2, \;\nu = 1$.

    图 5  在共振情况下的系统动力学. ${{{\gamma _1}}}/{\nu } > 1$. (a) 两能级上的占有概率${\left| {{\psi _n}} \right|^2}$($n = 1, 2$)随时间演化. (b)相位${\theta _n}$($n = $$ 1, 2$)随时间演化. (c)以$ {t_0} = - 0.380 $为时间反演点的$ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $演化. 系统参数取为${\gamma _0} = 1,\;{\gamma _1} = 2, \; $$ \omega = 2,\; \nu = 1$ 初态为${\psi _1}(0) = 1,\; {\psi _2}(0) = 0$

    Fig. 5.  System dynamics for the resonance case with the non-Hermitian parameters ${{{\gamma _1}}}/{\nu } > 1$, starting the system with the state ${\psi _1}(0) = 1, {\psi _2}(0) = 0$. (a) Time evolutions of the occupation probabilities ${\left| {{\psi _1}} \right|^2}$ and ${\left| {{\psi _2}} \right|^2}$. (b) Time evolutions of phases ${\theta _1}(t)$ and ${\theta _2}(t)$. (c) Time evolution of the sum of phases, $ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $. Here the time-inversion point is given by $ {t_0} = - 0.380 $. The system parameters are ${\gamma _0} = 1,\; {\gamma _1} = 2,\; \omega = 2,\;\nu = 1$.

  • [1]

    Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243Google Scholar

    [2]

    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 093902Google Scholar

    [3]

    Rüter C E, Makris K G, El-Ganainy R, Christodoulides D N, Segev M, Kip D 2010 Nat. Phys. 6 192Google Scholar

    [4]

    Doppler J, Mailybaev A A, Böhm J, Kuhl U, Girschik A, Libisch F, Milburn T J, Rabl P, Moiseyev N, Rotter S 2016 Nature 537 76Google Scholar

    [5]

    Peng B, Özdemir Ş K, Lei F, Monifi F, Gianfreda M, Long G Lu, Fan S, Nori F, Bender C M, Yang L 2014 Nat. Phys. 10 394Google Scholar

    [6]

    Chang L, Jiang X, Hua S, Yang C, Wen J, Jiang L, Li G, Wang G, Xiao M 2014 Nat. Photonics 8 524Google Scholar

    [7]

    Bender C M, Berntson B K, Parker D, Samuel E 2013 Am. J. Phys. 81 173Google Scholar

    [8]

    Schindler J, Li A, Zheng M C, Ellis F M, Kottos T 2011 Phys. Rev. A 84 040101(R)

    [9]

    Fleury R, Sounas D, Alù A 2015 Nat. Com. 6 5905

    [10]

    Liu T, Zhu X, Chen F, Liang S, Zhu J 2018 Phys. Rev. Lett. 120 124502Google Scholar

    [11]

    Tang J S, Wang Y T, Yu S, He D Y, Xu J S, Liu B H, Chen G, Sun Y N, Sun K, Han Y J, Li C F, Guo G C 2016 Nat. Photonics 10 642Google Scholar

    [12]

    Xiao L, Zhan X, Bian Z H, Wang K K, Zhang X, Wang X P, Li J, Mochizuki K, Kim D, Kawakami N, Yi W, Obuse H, Sanders B C, Xue P 2017 Nat. Phys. 13 1117Google Scholar

    [13]

    Gao W C, Zheng C, Liu L, Wang T J, Wang C 2021 Optics Express 29 517Google Scholar

    [14]

    Li J, Harter A K, Liu J, Melo L de, Joglekar Y N, Luo L 2019 Nat. Com. 10 855Google Scholar

    [15]

    Zhang D K, Luo X Q, Wang Y P, Li T F, You J Q 2017 Nat. Com. 8 1368Google Scholar

    [16]

    沈瑞昌, 张国强, 王逸璞, 游建强 2019 物理学报 68 230305Google Scholar

    Shen R C, Zhang G Q, Wang Y P, You J Q 2019 Acta Phys. Sin. 68 230305Google Scholar

    [17]

    Wu Y, Liu W, Geng J, Song X, Ye X, Duan C K, Rong X, Dun J 2019 Science 364 878Google Scholar

    [18]

    Zheng C, Hao L, Long G L 2013 Philos. Trans. R. Soc. A 371 20120053Google Scholar

    [19]

    Wen J, Zheng C, Kong X, Wei S, Xin T, Long G 2019 Phys. Rev. A 99 062122Google Scholar

    [20]

    Wang W C, Zhou Y L, Zhang H L, Zhang J, Zhang M C, Xie Y, Wu C W, Chen T, Ou B Q, Wu W, Jing H, Chen P X 2021 Phys. Rev. A 103 L020201Google Scholar

    [21]

    Ding L, Shi K, Zhang Q, Shen D, Zhang X, Zhang W 2021 Phys. Rev. Lett. 126 083604Google Scholar

    [22]

    Lin Z, Ramezani H, Eichelkraut T, Kottos T, Cao H, Christodoulides D N 2011 Phys. Rev. Lett. 106 213901Google Scholar

    [23]

    Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N, Peschel U 2012 Nature 488 167Google Scholar

    [24]

    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 108Google Scholar

    [25]

    Sun Y, Tan W, Li H Q, Li J, Chen H 2014 Phys. Rev. Lett. 112 143903Google Scholar

    [26]

    Jin L, Song Z 2018 Phys. Rev. Lett. 121 073901Google Scholar

    [27]

    Hodaei H, Hassan A U, Wittek S, Garcia-Gracia H, El-Ganainy R, Christodoulides D N, Khajavikhan M 2017 Nature 548 187Google Scholar

    [28]

    Yu S, Meng Y, Tang J, Xu X, Wang Y, Yin P, Ke Z, Liu W, Li Z, Yang Y, Chen G, Han Y, Li C, Guo G 2020 Phys. Rev. Lett. 125 240506Google Scholar

    [29]

    Feng L, Wong Z J, Ma R M, Wang Y, Zhang X 2014 Science 346 972Google Scholar

    [30]

    Hodaei H, Miri M A, Heinrich M, Christodoulides D N, Khajavikhan M 2014 Science 346 975Google Scholar

    [31]

    Assawaworrarit S, Yu X, Fan S 2017 Nature 546 387Google Scholar

    [32]

    Xu H, Mason D, Jiang L, Harris J G E 2016 Nature 537 80Google Scholar

    [33]

    Bender C M, Brody D C, Jones H F 2002 Phys. Rev. Lett. 89 270401

    [34]

    Bender C M 2007 Rep. Prog. Phys. 70 947Google Scholar

    [35]

    Mostafazadeh A 2002 J. Math. Phys. 43 205Google Scholar

    [36]

    Mostafazadeh A 2002 J. Math. Phys. 43 2814Google Scholar

    [37]

    黄永峰, 曹怀信, 王文华 2020 物理学报 69 030301Google Scholar

    Huang Y F, Cao H X, Wang W H 2020 Acta Phys. Sin. 69 030301Google Scholar

    [38]

    Nixon S, Yang J 2016 Phys. Rev. A 93 031802(R)

    [39]

    Hang C, Gabadadze G, Huang G 2017 Phys. Rev. A 95 023833Google Scholar

    [40]

    Pan J, Zhou L 2020 Phys. Rev. B 102 094305Google Scholar

    [41]

    Luo X B, Huang J H, Zhong H H, Qin X Z, Xie Q T, Kivshar Y S, Lee C H 2013 Phys. Rev. Lett. 110 243902Google Scholar

    [42]

    Chitsazi M, Li H, Ellis F M, Kottos T 2017 Phys. Rev. Lett. 119 093901Google Scholar

    [43]

    Duan L, Wang Y, Chen Q 2020 Chin. Phys. Lett. 37 081101Google Scholar

    [44]

    Xie Q, Rong S, Liu X 2018 Phys. Rev. A 98 052122Google Scholar

    [45]

    Koutserimpas T T, Alù A, Fleury R 2018 Phys. Rev. A 97 013839Google Scholar

    [46]

    Luo X B, Wu D, Luo S, Guo Y, Yu X, Hu Q 2014 J. Phys. A:Math. Theor 47 345301Google Scholar

    [47]

    Yang B, Luo X B, Hu Q, Yu X 2016 Phys. Rev. A 94 043828Google Scholar

    [48]

    Luo X B, Yang B, Zhang X F, Li L, Yu X 2017 Phys. Rev. A 95 052128Google Scholar

    [49]

    Cui B, Wang L C, Yi X X 2010 Phys. Rev. A 82 062105

    [50]

    Liu Z P, Zhang J, Özdemir Ş K, Peng B, Jing H, Lu X Y, Li C W, Yang L, Nori F, Liu Y 2016 Phys. Rev. Lett. 117 110802Google Scholar

    [51]

    Bender C M, Brody D C, Jones H F, Meister B K 2007 Phys. Rev. Lett. 98 040403Google Scholar

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出版历程
  • 收稿日期:  2022-02-14
  • 修回日期:  2022-03-10
  • 上网日期:  2022-04-06
  • 刊出日期:  2022-04-05

周期驱动的二能级系统中的准宇称-时间对称动力学

  • 1. 浙江理工大学物理系, 杭州 310018
  • 2. 井冈山大学数理学院, 吉安 343009
  • 通信作者: 罗小兵, xiaobingluo2013@aliyun.com
    基金项目: 浙江省自然科学基金(批准号: LY21A050002)、国家自然科学基金(批准号: 11975110, 12164022)、江西省教育厅科技项目(批准号: GJJ211026)和浙江理工大学科研启动基金(批准号: 20062318-Y)资助的课题.

摘要: 本文研究了一个周期驱动的非宇称-时间对称二能级量子系统的非厄米动力学. 通过经典相空间分析方法,解出了该非厄米系统的Floquet态和准能谱, 并解析构造了由该非厄米哈密顿量支配下的量子态的非幺正时间演化算符, 给出了不同参数区域的量子态演化. 本文数值和分析证明, 该非宇称-时间对称二能级Floquet系统, 类似于宇称-时间对称系统, 存在一个准能谱从实数谱到复数谱的相变. 本文还揭示了在量子态的动态演化中存在一种准宇称-时间对称动力学, 即, 该系统的粒子布居概率演化完全满足时间空间对称(宇称-时间对称), 但是由于相位演化违反了宇称-时间对称性的要求, 因此包含相位信息的量子态演化不满足时间空间对称(宇称-时间对称). 这些结果加深了对非厄米物理的理解, 拓展和推广了传统的宇称-时间对称概念.

English Abstract

    • 众所周知, 量子体系的状态演化由哈密顿量确定并服从薛定谔方程. 传统量子力学认为, 描述一个量子力学系统的哈密顿量必须具有厄米性, 才具有实数的能量本征值, 从而保证能量的可观测性和体系概率守恒. 1998 年, Bender和 Boettcher[1]发现, 满足宇称-时间(parity-time, PT)对称性的非厄米哈密顿量的本征值也可以为实数. 当非厄米参数强度增大时, PT对称系统的能量本征值存在一个从实数谱到复数谱的对称破缺相变. 尽管PT对称量子理论源于量子力学, 但随后被类比到经典物理系统中, 在光学[2-6]、力学[7]、电学[8]、声学[9,10]等多种经典系统中被实验验证. 但受制于量子力学的退相干性及量子实验系统的复杂性, 真正的量子体系中实现宇称时间对称哈密顿量的演化具有巨大挑战. 最近, 人们在光子[11-13]、冷原子[14]、腔自旋波混合系统[15,16]、金刚石氮-空位色心[17]、核磁共振[18,19]、离子阱[20,21]等量子体系实验平台中成功构造了等效PT对称量子哈密顿量. PT对称物理的发展不但加深了人们对量子力学基本问题的理解, 而且为单向隐身[10,22-24]、相干完美吸收[25]、非互易性传输[5,6,26]、超灵敏传感[27,28]、单模激光[29,30]、无线传能[31]、拓扑相的控制[4,32]等一系列应用技术带来了新的前景.

      同时, PT对称的非厄米量子系统相应的数学结构也吸引了人们的关注[33-37]. 实际上, 算符厄米性只是本征值为实数的充分条件, 而非必要条件. 2002年, Mostafazadeh[35,36]在数学上拓展了PT 对称理论: 对于一个任意哈密顿量$ H $, 若在线性厄米算符$ \eta$的作用下, 可以将该哈密顿量变成它的厄米共轭, 即满足$\eta H{\eta ^{ - 1}} = {H^\dagger }$, 则这个哈密顿量是赝厄米的, 其本征值也可以为实数. 这种赝厄米哈密顿量更为广泛, 它包含了PT对称哈密顿量. 由于非厄米物理带来了丰富的新奇效应, 构造和理解具有实本征谱的非PT对称的非厄米哈密顿量也引起了人们的兴趣. 例如, Nixon 和Yang[38]最近在数学上构造了一类具有任意增益和耗散分布的复数势函数, 证明了这类非厄米哈密顿量不仅具有实数谱, 而且和PT对称体系一样, 能量本征谱存在一个从实数谱到复数谱的相变. 这类非PT对称势函数简洁而且容易调控, 有望在多能级原子系统[39]等实验平台中实现.

      对非厄米哈密顿量量子体系加以周期性调控, 可以实现许多平衡态系统中无法达到的新奇的物理学现象, 如非厄米Floquet拓扑物态[12,40]、Floquet调控下的PT相变[41-44]、Floquet调控实现增益和双向隐身[45]等. 周期调制的宇称-时间对称性物理已经在经典电路[42]、冷原子[14]、光子[12]等体系中实验实现. 二能级系统(称为量子比特)是量子信息和量子计算的基本单元, PT对称二能级非厄米量子系统自然引起了理论和实验的关注和兴趣. 人们可以将宇称-时间对称哈密顿量扩展成更大空间中的厄米哈密顿量, 从而实现量子比特子系统在非厄米哈密顿量支配下的演化. 在前期工作中研究了周期性驱动对PT对称二能级系统的量子调控[41,46-48], 并获得了一些精确解[47,48], 有助于对非厄米哈密顿量的精确操控. 但是, 目前对具有实能谱的非PT对称二能级非厄米量子系统的Floquet调控的研究较少.

      本文研究了一个周期驱动的非PT对称二能级量子系统的非厄米动力学. 通过经典相空间分析方法, 首先得到了该非厄米系统的Floquet态和准能谱, 从而解析构造了非幺正时间演化算符, 分析了不同参数区域量子态演化行为. 通过解析和数值证明, 该非PT对称二能级Floquet系统, 类似于PT对称系统, 存在一个准能谱从实数谱到复数谱的相变. 而且, 在量子态的动态演化过程中, 存在一种准PT对称量子动力学: 该系统的粒子布居概率演化满足时间空间对称(PT对称), 但是量子态演化(由于相位演化违反PT对称性要求)不满足时间空间对称(PT对称). 这种准PT对称量子动力学推广了传统的PT对称概念, 加深了对非厄米物理和非传统量子力学的理解.

    • 考虑一个一般的周期驱动的非厄米二能级哈密顿量系统, 其动力学演化满足的薛定谔方程可写成:

      $ {\text{i}}\frac{{\text{d}}}{{{\text{d}}t}}\left| \psi \right\rangle = {\boldsymbol{H}}\left| \psi \right\rangle , $

      其中, 态矢量$ \left| \psi \right\rangle = {\left( {{\psi _1}, {\psi _2}} \right)^{\text{T}}} $(文中上标T表示矩阵转置), 哈密顿量$ H $具有如下形式

      $ {\boldsymbol{H}}(t) = \left( {\begin{array}{*{20}{c}} {{\gamma _0} + {\text{i}}{\gamma _1}}&{V(t)} \\ {{V^*}(t)}&{ - {\gamma _0} - {\text{i}}{\gamma _1}} \end{array}} \right). $

      这里, ${\gamma _0}$为两能级的静态失谐, ${\gamma _1}$为非厄米参数, $V(t)$为两能级间的耦合强度, 假定两能级的耦合强度具有时间周期性, 即$V(t + T') = V(t)$.

      首先介绍PT对称系统的Floquet本征模和准能量的基本概念和性质. 若哈密顿量${\boldsymbol{H}}(t)$${\boldsymbol{PT}}$算符满足对易关系, 即$\left[ {{\boldsymbol{H}}, {\boldsymbol{PT}}} \right] = 0$, 则该哈密顿量被认为具有PT不变性, 这里的${\boldsymbol{P}} = \left( {\begin{array}{*{20}{c}} 0&1 \\ 1&0 \end{array}} \right)$为宇称算符, ${\boldsymbol{ T}}$是时间反演算符, 具有如下效应${\text{i}} \to $$ - {\text{i}}, {t_0} + t \to {t_0} - t$这里, ${t_0}$为某个时间反演点. 可以证明: 当${\gamma _0} = 0$, 且$V({t_0} + t) = V({t_0} - t)$时, 哈密顿量(2)系统变成了标准的具有PT对称性的二能级Floquet系统, 此时$\left[ {{\boldsymbol{H}}, {\boldsymbol{PT}}} \right] = 0$. 与厄米Floquet系统一样, 周期驱动的非厄米(不限于PT)系统(1)同样具有Floquet态解, 即$\left| \psi \right\rangle {\text{ = }}\left| \varphi \right\rangle {{\text{e}}^{{{ - {\rm{i}}}}\varepsilon t}}$, $\left| {\varphi (t + T')} \right\rangle = $$ \left| {\varphi (t)} \right\rangle$ 在把时间作为另外一个维度的广义希尔伯特空间中, 态矢量 $ \left| \varphi \right\rangle = {\left( {{\varphi _1} , {\varphi _2}} \right)^{\text{T}}} $满足如下本征值方程:

      $ {\boldsymbol{F}}\left| \varphi \right\rangle = \varepsilon \left| \varphi \right\rangle , $

      这里的${\boldsymbol{F}}$称为Floquet哈密顿量, 其定义为${\boldsymbol{F}}: = $$ {\boldsymbol{H}} - i\partial /\partial t$, $\varepsilon $称为准能量, $\left| \varphi \right\rangle $称为Floquet态本征模(或简称Floquet态). 若$\left[ {{\boldsymbol{H}}, {\boldsymbol{PT}}} \right] = 0$, 则$[ {{\boldsymbol{F}}, {\boldsymbol{PT}}}] = $$ 0$也同样成立, 该Floquet系统具有PT对称性. 对于具有PT对称性的Floquet系统, 如果Floquet态$\left| \varphi \right\rangle $满足本征值方程(3), 那么${\boldsymbol{PT}}\left| \varphi \right\rangle$也是Floquet哈密顿量的本征模, 具有复共轭的准能量${\varepsilon ^*}$, 即${\boldsymbol{FPT}}\left| \varphi \right\rangle = {\varepsilon ^*}{\boldsymbol{PT}}\left| \varphi \right\rangle$. 当准能量为实数时, 即$\varepsilon = {\varepsilon ^*}$, ${\boldsymbol{PT}}$算符和${\boldsymbol{F}}$算符具有共同的本征态, ${\boldsymbol{PT}}\left| \varphi \right\rangle = $$ {e^{{\text{i}}\beta }}\left| \varphi \right\rangle$, $\beta $为任意相位(可以通过一个简单的规范变换去掉), 此时系统具有稳定的本征动力学模, 处于PT对称未破缺相. 当准能量为复数时, ${\boldsymbol{PT}}$算符和${\boldsymbol{F}}$算符不再拥有共同的本征态, 系统处于PT对称破缺相. 在PT对称系统中, 准能谱从实能谱到复能谱的转变, 称为PT对称破缺相变.

      其次, 简单讨论具有PT对称性的二能级系统的一般量子态演化的性质. 定义${t_{0 \pm }} = {t_0} \pm t$, 如果$\left| {\psi ({t_{0 \pm }})} \right\rangle = {[{\psi _1}({t_{0 + }}), {\psi _2}({t_{0 + }})]^{\text{T}}}$是薛定谔方程(1)的解, 那么${\boldsymbol{ PT}}\left| {\psi ({t_{0 + }})} \right\rangle = {[{\psi _2}^*({t_{0 - }}), {\psi _1}^*({t_{0 - }})]^{\text{T}}}$${[{\psi _1}({t_{0 + }}){{\rm{e}}^{ - {\text{i}}\xi }}, {\psi _2}({t_{0 + }}){{\rm{e}}^{ - {\text{i}}\xi }}]^{\text{T}}}$($\xi $为任意相位)也是薛定谔方程(1)的解. 假如在${t_0}$时刻, ${\psi _1}({t_0}) = {\psi _2}^*({t_0}){{\rm{e}}^{{\text{i}}\xi }}$, 则${\boldsymbol{PT}}\left| {\psi ({t_{0 + }})} \right\rangle$${{\rm{e}}^{ - {\text{i}}\xi }}\left| {\psi ({t_{0 + }})} \right\rangle$是薛定谔方程(1)的同一个解, 即:

      $ \left( {\begin{array}{*{20}{c}} {{\psi _1}({t_0} + t)} \\ {{\psi _2}({t_0} + t)} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{\psi _2^*}({t_0} - t)} \\ {{\psi _1^*}({t_0} - t)} \end{array}} \right){{\rm{e}}^{{\text{i}}\xi }}. $

      方程(4)等价于

      $ \left| {{\psi _1}({t_0} + t)} \right| = \left| {{\psi _2}({t_0} - t)} \right|, $

      $ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) = \xi . $

      其物理含义是, 在PT对称二能级系统中, 从${t_0}$时刻开始演化的量子动力学, 沿着$ + t$方向演化和沿着$ - t$方向演化在宇称算符(交换两个基矢, 空间反演)操作下是完全相同的.

      一个有趣的问题自然提出: 是否存在非PT对称的非厄米Floquet系统, 它具有类似PT对称的性质, 其准能谱也存在从实数谱到复数谱的相变? 如果存在, 那么其一般量子态的动力学演化是否有可能也和PT对称系统一样, 仍然存在某种对称性呢?

      本文以一个具体的Floquet模型为例尝试去回答上述问题. 在哈密顿模型(2)中, 选择简单的驱动[49]:

      $ V(t) = \nu \cos \omega t - {\text{i}}\nu {\sin}\omega t. $

      这样的哈密顿量可以描述一个被外磁场${\boldsymbol{B}} = $$ ({B_x}, {B_y}, {B_z})$ 驱动的二能级非厄米系统. 在实验方面, 以双势阱系统为例, 这样一个依赖于时间的复数隧穿系数, 可以同时调制两个势阱间的势垒高度和两个势阱的量子态的相角来实现. 容易证明, 对于任意时刻${t_0}$, 等式$V({t_0} + t) = V({t_0} - t)$都不成立, 即这个二能级非厄米Floquet系统本质上不具有PT对称性. 在下一节, 本文将求解该二能级非厄米系统的Floquet态本征模和任意量子态的非幺正演化.

    • 由于哈密顿量的非厄米性, 该系统不再保持概率守恒, 即量子态的模(范数)$n(t) = \left\langle {{\psi (t)}} \mathrel{\left | {\vphantom {{\psi (t)} {\psi (t)}}} \right. } {{\psi (t)}} \right\rangle = $$ {\left| {{\psi _1}} \right|^2} + {\left| {{\psi _2}} \right|^2}$不再为常数, 它随时间的演化可由非厄米薛定谔方程(1)得到:

      $ \frac{{{\text{d}}n}}{{{\text{d}}t}} = 2{\gamma _1}\left( {{{\left| {{\psi _1}} \right|}^2} - {{\left| {{\psi _2}} \right|}^2}} \right) . $

      因此, 引入归一化的量子态:

      $ \left| {\psi '(t)} \right\rangle = \frac{{|\psi (t)\rangle }}{{\sqrt {n(t)} }}. $

      该归一化的量子态满足新的薛定谔方程:

      $ {\text{i}}\frac{{\text{d}}}{{{\text{d}}t}}\left| {\psi '(t)} \right\rangle = {\boldsymbol{H}}'(t)\left| {\psi 't)} \right\rangle , $

      约化后的有效哈密顿量的形式为:

      $ \begin{split} {\boldsymbol{H}}'(t) =\;& {\text{i}}{\gamma _{\text{1}}}{\text{(}}{\left| {{\psi _1'}} \right|^{\text{2}}}-{\left| {{\psi _2'}} \right|^{\text{2}}}{\text{) }}+\nu {\text{cos(}}\omega {\text{t)}}{\sigma _x} \\ &+ \nu \sin (\omega {\text{t)}}{\sigma _y} + \left( {{\gamma _0} + {\text{i}}{\gamma _{\text{1}}}} \right){\sigma _z}. \end{split} $

      其中, ${\sigma _x}, {\sigma _y}, {\sigma _z}$为泡利矩阵的3个分量. 值得强调的是, 虽然有效哈密顿量${\boldsymbol{H}}'$是非厄米的, 但此时新的量子态的范数保持守恒, 即$\dfrac{{\text{d}}}{{{\text{d}}t}}\left\langle {\psi '(t)} \right.\left| {\psi '(t)} \right\rangle = 0$.

    • 由于$ {\psi' _1} = \left| {{\psi _1'}} \right|{{\rm{e}}^{{\text{i}}{\theta _1}}},\; {\psi' _2} = \left| {{\psi _2'}} \right|{{\rm{e}}^{{\text{i}}{\theta _2}}} $, 定义两能级的布居数差$Z(t) = {\left| {{\psi _1'}} \right|^2} - {\left| {{\psi _2'}} \right|^2}$和相对相位$\theta ' = $$ {\theta _2} - {\theta _1}$, 归一化后的新量子态可以表述为:

      $ \left( {\begin{array}{*{20}{l}} {{\psi _1'}} \\ {{\psi _2'}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\sqrt {\dfrac{{1 + Z(t)}}{2}} {{\rm{e}}^{{\text{i}}{\theta _1}}}} \\ {\sqrt {\dfrac{{1 - Z(t)}}{2}} {{\rm{e}}^{{\text{i}}{\theta _2}}}} \end{array}} \right). $

      以布居数差$ Z $和相对相位$\theta '$为两个独立变量, 薛定谔方程(10)变成了如下的两个耦合方程:

      $ \begin{split} & \frac{{{\text{d}}Z}}{{{\text{d}}t}} = 2{\gamma _1}\left( {1 - {Z^2}} \right) + 2\nu \sqrt {1 - {Z^2}} \sin (\theta ' - \omega t), \hfill \\ &\frac{{{\text{d}}\theta '}}{{{\text{d}}t}} = 2{\gamma _0} - 2\nu \frac{Z}{{\sqrt {1 - {Z^2}} }}\cos (\theta ' - \omega t). \end{split} $

      现在我们可以在$(Z, \theta ')$的相空间研究系统的动力学演化, 在相空间定义一种“赝定点”,

      $ \frac{{{\text{d}}Z}}{{{\text{d}}t}} = 0,\;\frac{{{\text{d}}\theta '}}{{{\text{d}}t}} = \omega . $

      这样一种“赝定点”可以认为是非含时哈密顿量系统的传统定点在Floquet系统的一种拓展. 它描述量子态在两能级的布居数差不随时间变化, 但相对相位线性增长(每隔一个驱动周期相对相位增长2π, 等价于每隔一个驱动周期$T' = 2{\text{π }}/\omega $, 相对相位不变). 实际上, 这样一种“赝定点”对应周期驱动量子系统的Floquet态. 为了方便研究这种“赝定点”, 我们引入一个新的变量, $\theta = \theta ' - \omega t$, 方程(13)变成:

      $ \begin{split} &\frac{{{\text{d}}Z}}{{{\text{d}}t}} = 2{\gamma _1}\left( {1 - {Z^2}} \right){\text{ + }}2\nu \sqrt {1 - {Z^2}} \sin \theta ,\\ &\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 2{\gamma _0} - \omega - 2\nu \frac{Z}{{\sqrt {1 - {Z^2}} }}\cos \theta .\end{split} $

      这样, 寻找“赝定点”就转化为寻找方程(15)的定点, $\dfrac{{{\text{d}}Z}}{{{\text{d}}t}} = 0,\; \dfrac{{{\text{d}}\theta }}{{{\text{d}}t}} = 0$.

      首先, 考虑共振(即两能级的失谐等于外场驱动频率, $2{\gamma _0} = \omega $)情况的定点. 为了方便计算, 若无特殊说明, 本文设定: 当$ \dfrac{{{\gamma _1}}}{\nu } < 1 $时, $\sin \alpha = \dfrac{{{\gamma _1}}}{\nu }$; 当$\dfrac{{{\gamma _1}}}{\nu } > 1$时, $\cosh \alpha = \dfrac{{{\gamma _1}}}{\nu }$. 在共振情况下, 方程(15)的定点可以分成以下三类情况: 1)当$\dfrac{{{\gamma _1}}}{\nu } < 1$时, 存在两个定点$\left( {{Z_f}, {\theta _f}} \right) = (0, 2{\text{π }} - \alpha )$, $\left( {{Z_f}, {\theta _f}} \right) = ( 0, \alpha + $$ {\text{π }} )$, 这里$\sin \alpha = \dfrac{{{\gamma _1}}}{\nu }$; 2)当$\dfrac{{{\gamma _1}}}{\nu } = 1$时, 只存在一个定点$ \left( {{Z_f}, {\theta _f}} \right) = \left( {0, \dfrac{{3{\text{π }}}}{2}} \right) $; 3)当$\dfrac{{{\gamma _1}}}{\nu } > 1$时, 存在两个定点$ \left( {{Z_f}, {\theta _f}} \right) = \left( { \pm \dfrac{{\sinh \alpha }}{{\cosh \alpha }}, \dfrac{{3{\text{π }}}}{2}} \right) $, 这里$\cosh \alpha = \dfrac{{{\gamma _1}}}{\nu }$.

      其次, 考虑非共振($2{\gamma _0} \ne \omega $)情况. 由方程(15)可知, 非共振时, ${Z_f} \ne 0$. 令$\dfrac{{{\text{d}}Z}}{{{\text{d}}t}} = 0,\; \dfrac{{{\text{d}}\theta }}{{{\text{d}}t}} = 0$, 可以得到关于${Z_f}$的四次方程,

      $ \begin{split} &- \gamma _1^2Z_f^4 - \left[ {{{\left( {{\gamma _0} - \frac{1}{2}\omega } \right)}^2} + {\nu ^2} - \gamma _1^2} \right]Z_f^2\\ &+ {\left( {{\gamma _0} - \frac{1}{2}\omega } \right)^2} = 0 .\end{split} $

      求解方程(16), 得到定点的布居数差:

      $ {Z_f} = \pm \sqrt{\frac{1}{{ - 2\gamma _1^2}} \Bigg\{{\left[ { - \gamma _1^2 + {{\left( {{\gamma _0} - \dfrac{1}{2}\omega } \right)}^2} + {\nu ^2}} \right] - \sqrt {{{\left[ { - \gamma _1^2 + {{\left( {{\gamma _0} - \dfrac{1}{2}\omega } \right)}^2} + {\nu ^2}} \right]}^2} + 4\gamma _1^2{{\left( {{\gamma _0} - \dfrac{1}{2}\omega } \right)}^2}} }\Bigg\}}. $

      这些定点解的稳定性可以通过计算雅可比矩阵:

      $ \left( {\begin{array}{*{20}{c}} { - 4{\gamma _1}{Z_f} - \dfrac{{2\nu {Z_f}\sin {\theta _f}}}{{\sqrt {1 - Z_f^2} }}}&{2\nu \sqrt {1 - Z_f^2} \cos {\theta _f}} \\ { - 2\nu \dfrac{{\cos {\theta _f}}}{{{{\left( {1 - Z_f^2} \right)}^{\tfrac{3}{2}}}}}}&{\dfrac{{2\nu {Z_f}\sin {\theta _f}}}{{\sqrt {1 - Z_f^2} }}} \end{array}} \right). $

      的本征值${\lambda _{1, 2}}$来分析. 如果雅可比矩阵两个本征值的实部都小于或等于0, 则这个定点是稳定的, 否则, 定点是不稳定的. 在共振($2{\gamma _0} = \omega $)情况下, 当$\dfrac{{{\gamma _1}}}{\nu } < 1$时, 两个定点$ \left( {{Z_f}, {\theta _f}} \right) = \left( {0, 2{\text{π }} - \alpha } \right) $, $\left( {{Z_f}, {\theta _f}} \right) = $$ \left( {0, \alpha + {\text{π }}} \right)$对应的雅可比矩阵的本征值为${\lambda _{1, 2}} = $$ \pm 2{\text{i}}\nu \sqrt {{{1}} - \dfrac{{\gamma _{{1}}^{{2}}}}{{{\nu ^{{2}}}}}}$. 雅可比矩阵的本征值为纯虚数, 因此, 这两个定点是稳定的. 当$\dfrac{{{\gamma _1}}}{\nu } = 1$时, 定点$\left( {{Z_f}, {\theta _f}} \right) = $$ \left( {0, \dfrac{{3{\text{π }}}}{2}} \right)$对应的雅可比矩阵的本征值$\lambda = 0$, 对应临界稳定. 当$\dfrac{{{\gamma _1}}}{\nu } > 1$时, 两个定点$( {{Z_f}, {\theta _f}} )= \left( { \pm \dfrac{{\sinh \alpha }}{{\cosh \alpha }}, \dfrac{{3{\text{π }}}}{2}} \right)$对应的雅可比矩阵的本征值为${\lambda _1} = - 2{\gamma _1}{Z_f}, \;{\lambda _2} = $$ - \dfrac{{2\nu {Z_f}}}{{\sqrt {1 - Z_f^2} }}$. 其中, 定点${Z_f} = \dfrac{{\sinh \alpha }}{{\cosh \alpha }}$对应的雅可比矩阵的两个本征值都为负的实数, 为稳定的结点; 另外一个定点对应的雅可比矩阵的两个本征值都为正的实数, 为不稳定的结点. 同理, 非共振($2{\gamma _0} \ne \omega $)情况的雅可比矩阵分析可知, 对于任意大小的非厄米参数${\gamma _1}$, 一个定点是稳定的, 一个定点是非稳定的.

      图1给出了$\left( {Z, \theta } \right)$的经典相图. 图1(a)(c)给出了共振($2{\gamma _0} = \omega $)的3种情况, 图1(d)给出了非共振($2{\gamma _0} \ne \omega $)情况. 图中的红点表示定点. 文中所有参数都已经无量纲化.

      图  1  不同系统参数下的相空间轨道($ \omega = 2, \nu = 1 $) (a) ${\gamma _0} = $$ 1,\; {\gamma _1} = 0.5$; (b) ${\gamma _0} = 1,\; {\gamma _1} = 1$; (c) ${\gamma _0} = 1, \;{\gamma _1} = 2$; (d) ${\gamma _0} = 2,\; {\gamma _1} = 0.5$

      Figure 1.  Phase-space trajectories with different system parameters($ \omega = 2, \nu = 1 $): (a) ${\gamma _0} = 1,\; {\gamma _1} = 0.5$; (b) ${\gamma _0} = 1, $$ \;{\gamma _1} = 1$; (c) ${\gamma _0} = 1, \;{\gamma _1} = 2$; (d) ${\gamma _0} = 2,\; {\gamma _1} = 0.5$.

      图1(a)所示, 当$\dfrac{{{\gamma _1}}}{\nu } < 1$时, 两个定点都位于${Z_f} = 0$处, 它们各自被一层层封闭轨道包围, 两者都是稳定的定点(中心). 如果增大非厄米参数${\gamma _1}$的强度, 两个定点相互趋近, 向$ \left( {{Z_f}, {\theta _f}} \right) = \left( {0, \dfrac{{3{\text{π }}}}{2}} \right) $处移动, 保持${Z_f} = 0$. 当$\dfrac{{{\gamma _1}}}{\nu } = 1$时, 两个定点合并成了一个定点$ \left( {{Z_f}, {\theta _f}} \right) = \Big( {0, \dfrac{{3{\text{π }}}}{2}} \Big) $, 如图1(b)所示. 当$\dfrac{{{\gamma _1}}}{\nu } > 1$时, 如图1(c)所示, 两个定点都居于$ {\theta _f} = $$ \dfrac{3}{2}{\text{π }} $处, 但${Z_f} \ne 0$, 意味着两能级上的布居数不对称(一个能级上占有的粒子数相对于另外一能级的粒子数占优势), 这是由于非厄米导致的自囚禁(布居数不对称)量子态. 这时, 其中一个定点(${Z_f} > 0$)为收点(sink), 邻域的所有轨线最终都要落在其上, 另外一个定点(${Z_f} < 0$)为源点或发点(source), 邻域中的所有解都要远离它. 如果继续增大非厄米参数${\gamma _1}$的强度, 这两个自囚禁(布居数不对称)的定点会沿着$ {\theta _f} = \dfrac{3}{2}{\text{π }} $移动, 一个趋向${Z_f} = 1$, 另一个趋向${Z_f} = - 1$, 量子态的自囚禁效应增强. 图1(d)给出了非共振($2{\gamma _0} \ne \omega $)情况, 这时, 取任意小的非厄米参数${\gamma _1}$, 两能级上的布居数都不相等, ${Z_f} \ne 0$, 系统也存在一个稳定的收点(${Z_f} > 0$), 一个不稳定的源点(${Z_f} < 0$), 所有轨线都发自源点流向收点.

    • 本节将从相空间的赝定点构造系统的Floquet态和准能量. 约化后的薛定谔方程(10)存在归一化的Floquet态解$\left| {\psi '} \right\rangle = \left| {\varphi '} \right\rangle {{\text{e}}^{{{ - {\rm{i}}}}\varepsilon 't}}$, $\left| {\varphi '(t + T')} \right\rangle = $$ \left| {\varphi '(t)} \right\rangle$, 这里, $T' = 2{\text{π }}/\omega $, 量子态的归一化条件要求$\varepsilon '$为实数. Floquet态$\left| {\varphi '} \right\rangle $满足本征值方程$( {\boldsymbol{H}}' - $$ {\text{i}}{\partial _t} )\left| {\varphi '} \right\rangle = \varepsilon '\left| {\varphi '} \right\rangle$. 由定点$\left( {{Z_f}, {\theta _f}} \right)$可以构造出归一化的Floquet态解$\left| {\psi '} \right\rangle $:

      $ \begin{split} \;& \left( {\begin{array}{*{20}{l}} {{\psi _1'}} \\ {{\psi _2'}} \end{array}} \right) =\left( {\begin{array}{*{20}{c}} {{\varphi _1'}} \\ {{\varphi _2'}} \end{array}} \right){{\rm{e}}^{ - {\text{i}}\varepsilon 't}} \\ =\;& \left( {\begin{array}{*{20}{l}} {\sqrt {\dfrac{{1 + {Z_f}}}{2}} {{\rm{e}}^{ - {\text{i}}\omega t}}} \\ {\sqrt {\dfrac{{1 - {Z_f}}}{2}} {{\rm{e}}^{{\text{i}}{\theta _f}}}} \end{array}} \right){{\rm{e}}^{ - {\text{i}}\varepsilon 't - {\text{i}}m\omega t}}.\end{split} $

      显然, 态向量中两组分的布居数差为$ {Z_f} $, 相位差为$ {\theta _2} - {\theta _1} = {\theta _f} + \omega t $, 对应经典相空间的赝定点. 式中的相位$ - m\omega t$反映了准能谱的类似布里渊区的周期性结构. 本文中如无特殊说明取$m = 0$, 把准能量限制在一个布里渊区内.

      由(9)式可以得到原始哈密顿量的Floquet态解:

      $ |\psi \rangle = \sqrt n \left| {\psi '} \right\rangle = \sqrt {\left\langle {\varphi } \mathrel{\left | {\vphantom {\varphi \varphi }} \right. } {\varphi } \right\rangle } \left| {\varphi '} \right\rangle {{\rm{e}}^{ - {\text{i}}\varepsilon 't +\text {Im} (\varepsilon )t}}. $

      此处用到关系式$ n = \left\langle {\varphi } \mathrel{\left | {\vphantom {\varphi \varphi }} \right. } {\varphi } \right\rangle {{\text{e}}^{2{\text{Im}}\left( \varepsilon \right)t}} $, $\varepsilon = {\text{Re(}}\varepsilon ) + $$ {\rm{i\,Im}}(\varepsilon {\text{)}}$. (20)式和原始Floquet态解$\left| \psi \right\rangle = \left| \varphi \right\rangle {{\text{e}}^{{{ - {\rm{i}}}}\varepsilon t}},$相比较, 可得到具有时间周期性的Floquet态本征模$ |\varphi \rangle = \sqrt {\left\langle {\varphi } \mathrel{\left | {\vphantom {\varphi \varphi }} \right. } {\varphi } \right\rangle } \left| {\varphi '} \right\rangle $, 以及$\varepsilon ' = {\rm Re} (\varepsilon )$, 也就是说, 归一化的Floquet态解的准能量是原始Floquet态解的准能量的实部, 其值求解如下:

      $ \begin{split} & \varepsilon ' = \left\langle {\varphi '} \right|{\boldsymbol{H}}'\left| {\varphi '} \right\rangle - {\text{i}}\left\langle {\varphi {'}} \right|\frac{{\text{d}}}{{{\text{d}}t}}\left| {\varphi '} \right\rangle \hfill \\ =\;& {\gamma _0}{Z_f} + \sqrt {1 - Z_f^2} \nu \cos {\theta _f} - \omega /2 - {Z_f}\omega /2 + m\omega . \end{split} $

      由(8)式可知, 原始哈密顿量Floquet态解的模(范数)随时间的演化方程满足,

      $ n(t) = n(0){{\text{e}}^{{\text{2}}{\gamma _{\text{1}}}{Z_f}t}}, $

      式中的$n(0)$表示初始时刻$t = 0$的原始哈密顿量Floquet态解的模(范数). (22)式与$ n = \left\langle {\varphi } \mathrel{\left | {\vphantom {\varphi \varphi }} \right. } {\varphi } \right\rangle {{\text{e}}^{{\text{2Im}}\left( \varepsilon \right)t}} $比较, 可以得到原始Floquet态解的准能量的虚部,

      $ \text{Im} (\varepsilon ) = {\gamma _1}{Z_f} , $

      以及,

      $ \left\langle {\varphi } \mathrel{\left | {\vphantom {\varphi \varphi }} \right. } {\varphi } \right\rangle = n(0) . $

      假如设定$ n(0)=1 $, 则原始哈密顿量的Floquet态解(20)可以简化为:

      $ \left| \psi \right\rangle = \left| {\varphi '} \right\rangle {{\rm{e}}^{ - {\text{i}}\varepsilon t}} \text{, } $

      准能量为:

      $ \begin{split} \varepsilon =\;& \varepsilon ' + {{\rm i\,Im}}\left( \varepsilon \right) \\ = \;&\left( {{\gamma _0} + {\text{i}}{\gamma _{\text{1}}}} \right){Z_f} + \sqrt {1 - Z_f^2} \nu \cos {\theta _f} \\ &- \frac{1}{2}\omega - \frac{1}{2}{Z_f}\omega + m\omega . \end{split} $

      由(26)式可知, 如果Floquet态在两能级上的占有概率(布居数)相等, ${Z_f} = 0$, 那么准能谱是全实数谱, 如果两能级上的占有概率(布居数)不相等, ${Z_f} \ne 0$, 准能谱就会变成复数谱.

      本文所考虑的周期驱动非厄米系统, 在归一化的相空间中具有特殊的“赝定点”, 它对应不依赖时间的布居数差和依赖于时间的相对相位. 但是对于更一般的周期驱动非厄米系统, 在归一化的经典相空间中, Floquet态本征模应该对应庞加莱截面(每隔驱动周期间隔采样)的定点. 即, 布居数差一般是时间的周期函数, $ Z = {Z_d} + {Z_a}\left( t \right) $, 这里, $ {Z_d} $是常数(${Z_d} = \dfrac{1}{{T'}}\displaystyle\int_0^{T'} {Z{\text{d}}t}$ ), 代表庞加莱截面的定点位置, $ {Z_a}\left( t \right) $是平均值为0的时间周期函数. 这时, Floquet态解的模(范数)随时间的演化方程满足$ n(t) = n(0){{\text{e}}^{{\text{2}}\gamma \int_0^t {{Z_a}{\text{d}}t} }}{{\text{e}}^{{\text{2}}{\gamma _1}{Z_d}t}}, $$ n = \left\langle {\varphi } \mathrel{\left | {\vphantom {\varphi \varphi }} \right. } {\varphi } \right\rangle {{\text{e}}^{{\text{2Im}}\left( \varepsilon \right)t}} $ 比较, 可以得到, 准能量的虚部为, $\text {Im} (\varepsilon ) = {\gamma _1}{Z_d} $, 以及$ \left\langle {\varphi } \mathrel{\left | {\vphantom {\varphi \varphi }} \right. } {\varphi } \right\rangle = n(0){{\text{e}}^{{\text{2}}\gamma \int_0^t {{Z_a}{\text{d}}t} }} $ , 这样, 原始哈密顿量的概率不守恒的Floquet态解可以构造出来, 即: $\left| \psi \right\rangle = $$ \sqrt n \left| {\psi '} \right\rangle {\text{ = }}\sqrt {\left\langle {\varphi } \mathrel{\left | {\vphantom {\varphi \varphi }} \right. } {\varphi } \right\rangle } \left| {\varphi {'}} \right\rangle {{\text{e}}^{{{ - {\rm{i}}}}\varepsilon 't + {\text{Im(}}\varepsilon {\text{)}}t}}$ . 可以知道, 原始哈密顿量Floquet态解的准能量的虚部由${Z_d}$决定.

      结合(19), (25), (26)式, 可以确定薛定谔方程(1)的Floquet态和准能量. 考虑非共振($2{\gamma _0} \ne \omega $)情况, 其定点解(17)${Z_f} \ne 0$, 结合$v\sqrt {1 - Z_f^2} \cos {\theta _f} = $$ \dfrac{{2{\gamma _0} - \omega }}{{2{Z_f}}}\left( {1 - Z_f^2} \right)$, 给出准能量($m = 0$)为:

      $ \varepsilon = \frac{{2{\gamma _0} - \omega }}{{2{Z_f}}} + {\text{i}}{\gamma _1}{Z_f} - \frac{1}{2}\omega . $

      显然, 在非共振情况下, 由于${Z_f} \ne 0$, 对于任意非零的非厄米强度${\gamma _1}$, 其准能量都是复数.

      接下来, 本文侧重于研究共振($2{\gamma _0} = \omega $)情况的本征模和量子动力学演化, 将分别给出共振情况三类定点对应的Floquet态和准能量.

      1)$\dfrac{{{\gamma _1}}}{\nu } < 1$. 这时, 方程(15)有两个布居数对称的定点. 定点$ \left( {{Z_f}, {\theta _f}} \right) = \left( {0, 2{\text{π }} - \alpha } \right) $对应的Floquet态解为:

      $ \left( \begin{gathered} {\psi _1} \hfill \\ {\psi _2} \end{gathered} \right) = \left( \begin{gathered} {\varphi _1} \hfill \\ {\varphi _2} \end{gathered} \right){{\text{e}}^{ - {\text{i}}\varepsilon t}} = \left( \begin{gathered} \sqrt {\dfrac{1}{2}} {{\text{e}}^{ - {\text{i}}\omega t}} \hfill \\ \sqrt {\dfrac{1}{2}} {{\text{e}}^{ - {\text{i}}\alpha }} \end{gathered} \right){{\text{e}}^{ - {\text{i}}\varepsilon t}}, $

      准能量为:

      $ \varepsilon = - \left( {{\gamma _0} - \nu \cos \alpha } \right). $

      另外一个定点$ \left( {{Z_f}, {\theta _f}} \right) = \left( {0, \alpha + {\text{π }}} \right) $对应的Floquet态解为:

      $ \left( \begin{gathered} {\psi _1} \hfill \\ {\psi _2} \end{gathered} \right) = \left( \begin{gathered} {\varphi _1} \hfill \\ {\varphi _2} \end{gathered} \right){{\text{e}}^{ - {\text{i}}\varepsilon t}} = \left( \begin{gathered} \sqrt {\frac{1}{2}} {{\text{e}}^{ - {\text{i}}\omega t}} \hfill \\ - \sqrt {\frac{1}{2}} {{\text{e}}^{{\text{i}}\alpha }} \end{gathered} \right){{\text{e}}^{ - {\text{i}}\varepsilon t}}, $

      准能量为:

      $ \varepsilon = - \left( {{\gamma _0} + \nu \cos \alpha } \right). $

      2)$\dfrac{{{\gamma _1}}}{\nu } = 1$. 这时, 两个定点合并成一个定点$\left( {{Z_f}, \;{\theta _f}} \right) = \left( {0,\; \dfrac{{3{\text{π }}}}{2}} \right)$, 其对应的Floquet态解为:

      $ \left( \begin{gathered} {\psi _1} \hfill \\ {\psi _2} \end{gathered} \right) = \frac{1}{{\sqrt 2 }}\left( \begin{gathered} {\text{i}}{{\text{e}}^{{{ - {\rm{i}}}}\omega t}} \hfill \\ 1 \end{gathered} \right){{\text{e}}^{{{ - {\rm{i}}}}\varepsilon t}}, $

      准能量为$\varepsilon = - {\gamma _0}$. 这时, 两个Floquet态本征模塌缩成一个, 这个参数点被称为奇异点(exceptional point). 系统能谱在奇异点附近有独特的拓扑结构, 将带来许多新奇的物理现象和应用, 如增强传感灵敏度[27,28,50], 非对称模式转化和手性传输[4]等.

      3)$\dfrac{{{\gamma _1}}}{\nu } > 1$. 这时, 两个布居数不对称的定点出现. 其中一个定点$\left( {{Z_f},\; {\theta _f}} \right) = \left( {\dfrac{{\sinh \alpha }}{{\cosh \alpha }}, \;\dfrac{{3{\text{π }}}}{2}} \right)$对应的Floquet态解为:

      $ \left( \begin{gathered} {\psi _1} \hfill \\ {\psi _2} \end{gathered} \right) = \frac{1}{{\sqrt {1 + {{\text{e}}^{{\text{ - 2}}\alpha }}} }}\left( \begin{gathered} {\text{i}}{{\text{e}}^{{{ - {\rm{i}}}}\omega t}} \hfill \\ {{\text{e}}^{-\alpha }} \end{gathered} \right){{\text{e}}^{{{ - {\rm{i}}}}\varepsilon t}}, $

      准能量为:

      $ \varepsilon = {\text{i}}\nu \sinh \alpha - {\gamma _0}. $

      另一个定点$\left( {{Z_f}, {\theta _f}} \right) = \left( { - \dfrac{{\sinh \alpha }}{{\cosh \alpha }},\; \dfrac{{3{\text{π }}}}{2}} \right)$对应的Floquet态解为:

      $ \left( \begin{gathered} {\psi _1} \hfill \\ {\psi _2} \hfill \\ \end{gathered} \right) = \frac{1}{{\sqrt {1 + {{\text{e}}^{{\text{2}}\alpha }}} }}\left( \begin{gathered} {\text{i}}{{\text{e}}^{{{ - i}}\omega t}} \hfill \\ {{\text{e}}^\alpha } \hfill \\ \end{gathered} \right){{\text{e}}^{{{ - {\rm{i}}}}\varepsilon t}}, $

      准能量为:

      $ \varepsilon = - {\text{i}}\nu \sinh \alpha - {\gamma _0}. $

      图2给出了非厄米系统(1)在周期驱动(7)作用下共振($2{\gamma _0} = \omega $)情况时的准能量的实部(见图2(a))和虚部(见图2(b))随参数${\gamma _1}/\nu $的关系. 红线和蓝线代表直接对角化一个驱动周期的时间演化算符的数值结果, 圆圈代表经典相图中赝定点对应的准能量解析结果. 如图所示, 数值结果和解析结果完全符合. 尽管在周期驱动(7)作用下, $V({t_0} + t) \ne V({t_0} - t)$, 从而非厄米系统(1)不具有PT对称性, $\left[ {{\boldsymbol{H}}, {\boldsymbol{PT}}} \right] \ne 0$, 但可发现, 它和PT对称系统一样, 存在一个准能谱从实数谱到复数谱的相变. 当非厄米系统(1)的准能谱为实数时, 将${\boldsymbol{PT}}$算符作用在实数准能谱$\varepsilon = - \left( {{\gamma _0} - \nu \cos \alpha } \right)$对应的Floquet态本征模$\left| {\varphi \left( {{t_0} + t} \right)} \right\rangle = \dfrac{1}{{\sqrt 2 }}( {{\text{e}}^{{{ - {\rm{i}}}}\omega ({t_0} + t)}}, - {{\text{e}}^{{\text{i}}\alpha }} )^{\text{T}}$上, 可以验证, 对于任意的时刻点${t_0}$,

      图  2  非厄米系统(1)在周期驱动(7)作用下共振($2{\gamma _0} = $$ \omega$)情况时的准能谱的实部(a)和虚部(b)随参数${\gamma _1}/\nu $的关系. 红线和蓝线代表直接对角化一个驱动周期的时间演化算符的数值结果, 圆圈代表经典相图中赝定点对应的准能量解析结果. 参数取为${\gamma _0} = 1,\; \omega = 2$

      Figure 2.  Real (a) and imaginary (b) parts of the quasienergies as a function of ${\gamma _1}/\nu $ for the non-Hermitian system (1) subject to a periodic modulation (7) in the resonant ($2{\gamma _0} = $$ \omega$) case. The red and blue lines denote the numerical results of quasienergies computed through direct diagonalization of the time-evolution operator over one period of the driving, while the circles denote exact analytical results of quasienergies corresponding to the pseudo fixed points in phase space. The system parameters are set as ${\gamma _0} = 1, \;\omega = 2$

      $\begin{split} & {\boldsymbol{PT}}\left( \begin{gathered} {\varphi _1}\left( {{t_0} + t} \right) \hfill \\ {\varphi _2}\left( {{t_0} + t} \right) \end{gathered} \right) \\ =\;& \frac{1}{{\sqrt 2 }}\left( \begin{gathered} - {{\text{e}}^{{{ - {\rm{i}}}}\alpha }} \hfill \\ {{\text{e}}^{{\text{i}}\omega ({t_0} - t)}} \hfill \\ \end{gathered} \right) \ne C\left( \begin{gathered} {\varphi _1}\left( {{t_0} + t} \right) \hfill \\ {\varphi _2}\left( {{t_0} + t} \right) \end{gathered} \right), \end{split}$

      式中, $ \boldsymbol{C} $为任意常数. 这说明, 该实数准能谱对应的Floquet态$ \left| {\varphi \left( {{t_0} + t} \right)} \right\rangle $不是${\boldsymbol{PT}}$算符的本征模. 同理可证, 另外一支实数准能谱$\varepsilon = - \left( {{\gamma _0} + \nu \cos \alpha } \right)$对应的Floquet态也不是${\boldsymbol{PT}}$算符的本征模. 这进一步说明, 在周期驱动(7)作用下的非厄米Floquet系统(1)具有实数准能谱, 但该系统不具有PT对称性.

    • 在第3节中, 已经通过相空间中的定点方法解析构造了系统的Floquet态本征模和准能量. 由这些Floquet态和准能量, 我们可以求解任意量子态的时间演化. 在共振情况下, 即使本文所研究的Floquet系统不具有PT对称性, 也存在能谱从实数谱到复数谱的相变. 那么, 一般量子态的动力学演化又会具有什么独特的量子性质呢? 分3种情况求解在共振($ 2 \gamma_{0}=\omega $)情况下的一般量子态的动力学演化.

      1)$\dfrac{{{\gamma _1}}}{\nu } < 1$. 由两个基本的Floquet态解(28)和(30), 构造通解:

      $\begin{split} & {\psi _1}\left( t \right) = {C_1}{{\text{e}}^{{\text{i}}\left( {{\gamma _0} - \nu \cos \alpha } \right)t}}{{\text{e}}^{ - {\text{i}}\omega t}} + {C_2}{{\text{e}}^{{\text{i}}\left( {{\gamma _0} + \nu \cos \alpha } \right)t}}{{\text{e}}^{ - {\text{i}}\omega t}}, \\ &{\psi _2}\left( t \right) = {C_1}{{\text{e}}^{{\text{i}}\left( {{\gamma _0} - \nu \cos \alpha } \right)t}}{{\text{e}}^{ - {\text{i}}\alpha }} - {C_2}{{\text{e}}^{{\text{i}}\left( {{\gamma _0} + \nu \cos \alpha } \right)t}}{{\text{e}}^{{\text{i}}\alpha }},\end{split} $

      式中叠加系数${C_1}, {C_2}$由初态决定.

      假设给定$t = 0$时刻的态矢量, $\left| {\psi (0)} \right\rangle = [ {\psi _1}(0), $$ {\psi _2}(0) ]^{\text{T}}$, 时间演化算符作用在该态矢量上, 可得:

      $ \begin{split} &{\psi _1}\left( t \right) = {U_{11}}\left( {t,0} \right){\psi _1}\left( 0 \right) + {U_{12}}\left( {t,0} \right){\psi _2}\left( 0 \right), \hfill \\ & {\psi _2}\left( t \right) = {U_{21}}\left( {t,0} \right){\psi _1}\left( 0 \right) + {U_{22}}\left( {t,0} \right){\psi _2}\left( 0 \right). \end{split} $

      由初态$ \left| {\psi (0)} \right\rangle $决定(38)式的叠加系数${C_1}, {C_2}$后, 比较(38)和(39)式, 可以写出从$t = 0$开始的非幺正时间演化算符的具体的矩阵形式:

      $ \begin{split} & U\left(t,0\right)=\left(\begin{array}{cc} {U}_{11}(t,0)&{U}_{12}(t,0) \\ {U}_{21}(t,0)&{U}_{22}(t,0)\end{array}\right)=\frac{1}{1+{\text{e}}^{\text{2i}\alpha }} \\ & \times \left(\begin{array}{cc} {\text{e}}^{\text{2i}\alpha }{a}^{*}\left(t\right)+b\left(t\right) & {\text{e}}^{\text{i}\alpha }\left[{a}^{*}\left(t\right)- b\left(t\right)\right]\\ {\text{e}}^{\text{i}\alpha }\left[{b}^{*}\left(t\right)-a\left(t\right)\right] & {b}^{*}\left(t\right)-{\text{e}}^{\text{2i}\alpha }a\left(t\right)\end{array}\right). \end{split} $

      式中, $a\left( t \right) = {{\text{e}}^{{\text{i(}}\nu {\cos}\alpha + \tfrac{\omega }{2})t}}$ , $b\left( t \right) = {{\text{e}}^{{\text{i(}}\nu {\cos}\alpha - \tfrac{\omega }{2})t}}$.

      以初始条件$ {\psi _1}\left( 0 \right) = 1, {\psi _2}\left( 0 \right) = 0 $为例, 由(39)和(40)式可知:

      $ {\psi _1}\left( t \right) = \frac{1}{{1 + {{\text{e}}^{{\text{2i}}\alpha }}}}\left[ {{{\text{e}}^{{\text{2i}}\alpha - {\text{i}}\left( {{\gamma _0} + \nu {\cos}\alpha } \right)t}} + {{\text{e}}^{{\text{i(}}\nu {\cos}\alpha - \tfrac{\omega }{2})t}}} \right], $

      $ {\psi _2}\left( t \right) = \frac{{{{\text{e}}^{{\text{i}}\alpha }}}}{{1 + {{\text{e}}^{{\text{2i}}\alpha }}}}\left[ {{{\text{e}}^{ - {\text{i(}}\nu {\cos}\alpha - \tfrac{\omega }{2})t}} - {{\text{e}}^{{\text{i(}}\nu {\cos}\alpha + \tfrac{\omega }{2})t}}} \right]. $

      因此有, ${| {{\psi _1}(t)}|^2} = \dfrac{{{{\cos }^2}(\alpha - \nu t\cos \alpha )}}{{{{\cos }^2}\alpha }}$, ${| {{\psi _2}(t)} |^2} = $$ \dfrac{{{{\sin }^2}(\nu t\cos \alpha )}}{{{{\cos }^2}\alpha }}$, ${\theta _1}(t) = - \dfrac{\omega }{2}t + \arg [\cos (\alpha - \nu t\cos \alpha )]$, ${\theta _2}(t) = \dfrac{\omega }{2}t + \dfrac{3}{2}{\text{π }} + \arg [\sin (\nu t\cos \alpha )]$. 其中, $\arg $代表某函数的相角, 当函数取值为正实数时, 该相角为0, 当函数取值为负实数时, 该相角为$ {\text{π }} $. 当${t_0} = $$ \dfrac{{\alpha - ( {k + {1}/{2}} ){\text{π }}}}{{{\text{2}}\nu {\cos}\alpha }}$($k$为整数)时, 可以验证, ${| {{\psi _1}({t_0} + t)} |^2} = $$ {\left| {{\psi _2}({t_0} - t)} \right|^2},$$ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) = \xi - \omega t $, 其中, $\xi = {\theta _1}\left( {{t_0}} \right) + {\theta _2}\left( {{t_0}} \right) = \dfrac{3}{2}{\text{π }} + \arg \left[ {\cos \left( {\alpha - \nu {t_0}\cos \alpha } \right)} \right] +$ $\arg \left[ {\sin \left( {\nu {t_0}\cos \alpha } \right)} \right]$.

      ${\psi _1}(0) = 1, {\psi _2}(0) = 0$为初态直接数值模拟原始薛定谔方程(1), 图3给出了在共振情况下, 当$\dfrac{{{\gamma _1}}}{\nu } < 1$时的系统动力学. 从图3可以看出, 两能级上的占有概率${\left| {{\psi _1}} \right|^2}$${\left| {{\psi _2}} \right|^2}$随时间周期振荡, 当${t_0} = \dfrac{{\alpha - \left( {k + {1}/{2}} \right){\text{π }}}}{{{\text{2}}\nu {\cos}\alpha }}$($k$为整数)时, 在两能级上的占有概率相等, ${\left| {{\psi _1}} \right|^2} = {\left| {{\psi _2}} \right|^2}$. 图中以$ {t_0} = - 0.605 $为例说明了这个特性. 图3显示, 从$ {t_0} $时刻开始, 沿着正向演化($ + t $方向)和沿着反向演化($ - t $方向)在宇称算符(交换两个基矢$\left| 1 \right\rangle $$\left| 2 \right\rangle $, 空间反演)操作下, 概率是完全相同的, 即${\left| {{\psi _1}({t_0} + t)} \right|^2} = $$ {\left| {{\psi _2}({t_0} - t)} \right|^2}$. 但相位之和, $ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $, 却不等于$ {t_0} $时刻的值$ \xi = {\theta _1}({t_0}) + {\theta _2}({t_0}) $, 反而随时间$t$线性变化. 注意, 在数值中, 相位每隔2π有个跳跃. 图中的数值结果完全符合理论预言.

      图  3  在共振情况下的系统动力学($\tfrac{{{\gamma _1}}}{\nu } < 1$) (a) 两能级上的占有概率${\left| {{\psi _n}} \right|^2}$($n = 1, 2$)随时间演化; (b)相位${\theta _n}$($n = $$ 1, 2$)随时间演化; (c)以$ {t_0} = - 0.605 $为时间反演点的$ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $演化. 系统参数取为${\gamma _0} = 1, \;{\gamma _1} = 0.5,\; $$ \omega = 2,\; \nu = 1$ 初态为${\psi _1}(0) = 1, {\psi _2}(0) = 0$

      Figure 3.  System dynamics for the resonance case with the non-Hermitian parameters $\tfrac{{{\gamma _1}}}{\nu } < 1$, starting the system with the state ${\psi _1}(0) = 1,\; {\psi _2}(0) = 0$: (a) Time evolutions of the occupation probabilities ${\left| {{\psi _1}} \right|^2}$ and ${\left| {{\psi _2}} \right|^2}$; (b) time evolutions of phases ${\theta _1}(t)$ and ${\theta _2}(t)$; (c) time evolution of the sum of phases, $ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $. Here we choose the time-inversion point $ {t_0} = - 0.605 $. The system parameters are ${\gamma _0} = 1,\; {\gamma _1} = 0.5,\; \omega = 2, \;\nu = 1$.

      2)${{{\gamma _1}}}/{\nu } = 1$. 这时, 系统处于奇异点, 只有一个Floquet态解(32), 但可以构造出另外一个线性无关的基本解:

      $ \left( \begin{gathered} {\psi _1} \hfill \\ {\psi _2} \end{gathered} \right) = {{\text{e}}^{{\text{i}}{\gamma _0}t}}\left( \begin{gathered} {\text{i}}t{{\text{e}}^{{{ - {\rm{i}}}}\omega t}} \hfill \\ t - \frac{1}{\nu } \end{gathered} \right). $

      因此, 由这两个线性无关的基本解, 可以写出薛定谔方程(1)的通解:

      $ {\psi _1}\left( t \right) = \left( {{C_1} + {C_2}t} \right){\text{i}}{{\text{e}}^{{\text{i}}\left( {{\gamma _{\text{0}}} - \omega } \right)t}}, $

      $ {\psi _2}\left( t \right) = \left[ {{C_1} + {C_2}\left( {t - \frac{1}{\nu }} \right)} \right]{{\text{e}}^{{\text{i}}{\gamma _{\text{0}}}t}}, $

      式中叠加系数${C_1}, {C_2}$由初态决定. 由(44)和(45)式, 可以得到, 从$t = 0$开始的非幺正时间演化算符为

      $ U\left(t,0\right)=\left(\begin{array}{ccc} &\left(1+\nu t\right){\text{e}}^{-\text{i}\tfrac{\omega }{\text{2}}t}&-\text{i}\nu t{\text{e}}^{-\text{i}\tfrac{\omega }{\text{2}}t}\\ &-\text{i}\nu t{\text{e}}^{\text{i}\tfrac{\omega }{\text{2}}t}&\left(1-\nu t\right){\text{e}}^{\text{i}\tfrac{\omega }{\text{2}}t}\end{array}\right). $

      $t = 0$时刻的态制备为$ {\psi _1}\left( 0 \right) = 1, {\psi _2}\left( 0 \right) = 0 $时, 把时间演化算符(46)作用在$t = 0$的态矢量上, 可得:

      $ {\psi _1}\left( t \right) = \left( {1 + \nu t} \right){{\text{e}}^{ - {\text{i}}\tfrac{\omega }{{\text{2}}}t}}, $

      $ {\psi _2}\left( t \right) = - {\text{i}}\nu t{{\text{e}}^{{\text{i}}\tfrac{\omega }{{\text{2}}}t}}. $

      可以证明, 当${t_0} = - \dfrac{1}{{{\text{2}}\nu }}$时, $| {{\psi _1}({t_0} + t)}| = | {{\psi _2}({t_0} - t)} |,$ ${\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) = {{\text{π }}}/{2} - \omega t$.

      $ {\psi _1}\left( 0 \right) = 1, {\psi _2}\left( 0 \right) = 0 $为初态, 通过直接数值积分模拟原始薛定谔方程(1), 图4给出了在共振情况下, 当${{{\gamma _1}}}/{\nu } = 1$时的系统动力学演化. 从图4可以看出, 两能级上的占有概率从$ {t_0} = - 0.5 $开始沿着正向演化($ + t $方向), 呈时间$t$的平方增长. 如理论预期一样, 数值结果显示, ${\left| {{\psi _1}({t_0} + t)} \right|^2} = $$ {\left| {{\psi _2}({t_0} - t)} \right|^2}$, 相位之和, $ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $, 却不恒等于$ {t_0} $时刻的相位之和, 而是随时间$t$线性减小. 注意, 在数值中, 相位每隔$2{\text{π }}$有个跳跃.

      图  4  在共振情况下的系统动力学(${{{\gamma _1}}}/{\nu } = 1$) (a) 两能级上的占有概率${\left| {{\psi _n}} \right|^2}$($n = 1, 2$)随时间演化;(b)相位${\theta _n}$($n = $$ 1, 2$)随时间演化; (c)以$ {t_0} = - 0.5 $为时间反演点的$ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $演化. 系统参数取为${\gamma _0} = 1,\; {\gamma _1} = 1,\; $$ \omega = 2,\; \nu = 1$ 初态为${\psi _1}(0) = 1,\; {\psi _2}(0) = 0$

      Figure 4.  System dynamics for the resonance case with the non-Hermitian parameters ${{{\gamma _1}}}/{\nu } = 1$, starting the system with the state ${\psi _1}\left( 0 \right) = 1,\; {\psi _2}\left( 0 \right) = 0$: (a) Time evolutions of the occupation probabilities ${\left| {{\psi _1}} \right|^2}$ and ${\left| {{\psi _2}} \right|^2}$; (b) time evolutions of phases ${\theta _1}(t)$ and ${\theta _2}(t)$; (c) time evolution of the sum of phases, $ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $. Here the time-inversion point is given by $ {t_0} = - 0.5 $. The system parameters are ${\gamma _0} = 1,\; {\gamma _1} = 1,\; \omega = 2, \;\nu = 1$.

      3)${{{\gamma _1}}}/{\nu } > 1$. 由系统的两个基本的Floquet态解(33)和(35), 可以构造在这种情况下的通解

      $ {\psi _1}\left( t \right) = {C_1}{\text{i}}{{\text{e}}^{{\text{i}}\left( {{\gamma _{\text{0}}} - \omega } \right)t + \nu \sinh \alpha t}} + {C_2}{\text{i}}{{\text{e}}^{{\text{i}}\left( {{\gamma _{\text{0}}} - \omega } \right)t - \nu \sinh \alpha t}}, $

      $ {\psi _2}\left( t \right) = {C_1}{{\text{e}}^{ - \alpha }}{{\text{e}}^{{\text{i}}{\gamma _{\text{0}}}t + \nu \sinh \alpha t}} + {C_2}{{\text{e}}^\alpha }{{\text{e}}^{{\text{i}}{\gamma _{\text{0}}}t - \nu \sinh \alpha t}}. $

      式中叠加系数${C_1}, {C_2}$由初态决定. 同理, 由(49)和(50)式, 可以得到, 从$t = 0$开始的非幺正时间演化算符为

      $ U\left(t,0\right)=\left(\begin{array}{ccc} &\dfrac{{\text{e}}^{-\text{i}\tfrac{\omega }{2}t+\nu t\mathrm{sinh}\alpha }}{1-{\text{e}}^{-2\alpha }}+\dfrac{{\text{e}}^{-\text{i}\tfrac{\omega }{2}t-\nu t\mathrm{sinh}\alpha }}{1-{\text{e}}^{2\alpha }}&-\text{i}\left(\dfrac{{\text{e}}^{-\alpha }{\text{e}}^{-\text{i}\tfrac{\omega }{2}t+\nu t\mathrm{sinh}\alpha }}{1-{\text{e}}^{-2\alpha }}+\dfrac{{\text{e}}^{\alpha }{\text{e}}^{-\text{i}\tfrac{\omega }{2}t-\nu t\mathrm{sinh}\alpha }}{1-{\text{e}}^{2\alpha }}\right) \\ & -\text{i}\left(\dfrac{{\text{e}}^{-\alpha }{\text{e}}^{\text{i}\tfrac{\omega }{2}t+\nu t\mathrm{sinh}\alpha }}{1-{\text{e}}^{-2\alpha }}+\dfrac{{\text{e}}^{\alpha }{\text{e}}^{\text{i}\tfrac{\omega }{2}t-\nu t\mathrm{sinh}\alpha }}{1-{\text{e}}^{2\alpha }}\right)&-\dfrac{{\text{e}}^{-2\alpha }{\text{e}}^{\text{i}\tfrac{\omega }{2}t+\nu t\mathrm{sinh}\alpha }}{1-{\text{e}}^{-2\alpha }}-\dfrac{{\text{e}}^{2\alpha }{\text{e}}^{\text{i}\tfrac{\omega }{2}t-\nu t\mathrm{sinh}\alpha }}{1-{\text{e}}^{2\alpha }}  \end{array}\right). $

      $t = 0$时刻的态制备为$ {\psi _1}\left( 0 \right) = 1, {\psi _2}\left( 0 \right) = 0 $时, 把时间演化算符(51)作用在该态矢量上, 可得

      $ {\psi _1}\left( t \right) = \frac{{{{\text{e}}^{ - {\text{i}}\tfrac{\omega }{2}t{\text{ + }}\nu t\sinh \alpha }}}}{{1 - {{\text{e}}^{ - 2\alpha }}}} + \frac{{{{\text{e}}^{ - {\text{i}}\tfrac{\omega }{2}t - \nu t\sinh \alpha }}}}{{1 - {{\text{e}}^{2\alpha }}}}, $

      $ {\psi _2}\left( t \right) = - {\text{i}}\left( {\frac{{{{\text{e}}^{ - \alpha }}{{\text{e}}^{{\text{i}}\tfrac{\omega }{2}t{\text{ + }}\nu t\sinh \alpha }}}}{{1 - {{\text{e}}^{ - 2\alpha }}}} + \frac{{{{\text{e}}^\alpha }{{\text{e}}^{{\text{i}}\tfrac{\omega }{2}t - \nu t\sinh \alpha }}}}{{1 - {{\text{e}}^{2\alpha }}}}} \right). $

      类似地, 可以证明, 当${t_0} = - \dfrac{\alpha }{{2\nu \sinh \alpha }}$时, $ \left| {{\psi _1}({t_0} + t)} \right| = \left| {{\psi _2}({t_0} - t)} \right|, $ ${\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) = $$ \xi - \omega t$, 这里$\xi = {\theta _1}\left( {{t_0}} \right) + {\theta _2}\left( {{t_0}} \right) = {{\text{π }}}/{2}$.

      图5给出了在共振情况下, 当${{{\gamma _1}}}/{\nu } > 1$时的系统动力学演化. 同样地, 这里也是以${\psi _1}(0) = 1,\; {\psi _2}(0) = $$ 0$为初态直接数值模拟了原始薛定谔方程(1). 如图5所示, 两能级上的占有概率从$ {t_0} = - 0.380 $开始沿着正向演化($ + t $方向), 呈指数形式增长. 类似地, 如理论预期的一样, 数值结果显示, ${\left| {{\psi _1}({t_0} + t)} \right|^2} = $$ {\left| {{\psi _2}({t_0} - t)} \right|^2},$ 相位之和, $ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $, 随时间$t$线性减小.

      图  5  在共振情况下的系统动力学. ${{{\gamma _1}}}/{\nu } > 1$. (a) 两能级上的占有概率${\left| {{\psi _n}} \right|^2}$($n = 1, 2$)随时间演化. (b)相位${\theta _n}$($n = $$ 1, 2$)随时间演化. (c)以$ {t_0} = - 0.380 $为时间反演点的$ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $演化. 系统参数取为${\gamma _0} = 1,\;{\gamma _1} = 2, \; $$ \omega = 2,\; \nu = 1$ 初态为${\psi _1}(0) = 1,\; {\psi _2}(0) = 0$

      Figure 5.  System dynamics for the resonance case with the non-Hermitian parameters ${{{\gamma _1}}}/{\nu } > 1$, starting the system with the state ${\psi _1}(0) = 1, {\psi _2}(0) = 0$. (a) Time evolutions of the occupation probabilities ${\left| {{\psi _1}} \right|^2}$ and ${\left| {{\psi _2}} \right|^2}$. (b) Time evolutions of phases ${\theta _1}(t)$ and ${\theta _2}(t)$. (c) Time evolution of the sum of phases, $ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) $. Here the time-inversion point is given by $ {t_0} = - 0.380 $. The system parameters are ${\gamma _0} = 1,\; {\gamma _1} = 2,\; \omega = 2,\;\nu = 1$.

      以上分析和数值证明, 具有周期驱动形式(7)的系统(1)虽然不具有PT对称性, 但在共振情况下, 其量子态演化具有类似PT对称的性质. 当非厄米参数强度小于某个临界值时, 两能级系统布居概率呈周期振荡; 当非厄米参数强度大于该临界值时, 布居概率呈指数增强. 而且, 在共振情况下, 从${t_0}$时刻开始演化的量子态, 沿着正方向($ + t $方向)演化和沿着反方向($ - t $方向)演化在宇称算符(交换两个基矢, 空间反演)操作下, 概率是完全相同的. 也就是说, 从系统粒子布居概率来看, 该非厄米系统的动力学和PT对称系统完全一样. 但实际上, 两能级系统的概率幅的相位不满足PT对称性的要求. 从数学上讲, 该系统只满足PT对称的其中一个条件: $ \left| {{\psi _1}({t_0} + t)} \right| = \left| {{\psi _2}({t_0} - t)} \right|, $ 但不满足另外一个条件$ {\theta _1}({t_0} + t) + {\theta _2}({t_0} - t) = \xi $, 这里$\xi $是个常数, 由${t_0}$时刻的概率幅的相位决定. 我们把这种类似PT对称性的量子动力学称为准PT对称量子动力学.

    • 采用经典相空间分析方法研究了一个周期驱动的非PT对称二能级量子系统的非厄米物理特性. 由于哈密顿量的非厄米性, 体系概率不守恒, 引入一个归一化的量子态, 并把薛定谔方程转化成了一个以两能级布居数差和相对相位为两个独立变量的经典动力学方程. 研究发现, 该量子系统的Floquet态对应经典相空间的“赝定点”, 即不随时间变化的布居数差和依赖于时间的相对相位. 在经典相空间“赝定点”的基础上, 我们解析构造了由该非厄米哈密顿量支配下的量子态的非幺正时间演化算符, 给出了不同参数区域的量子态演化. 研究结果表明, 该非PT对称二能级量子系统也存在类似PT对称系统一样的从对称未破缺区到破缺区的过渡, 而且在该系统量子态的动力学演化中, 存在一种准PT对称量子动力学, 即系统粒子布居概率演化完全满足时间空间对称(PT对称), 但由于相位演化不满足PT对称性的要求, 因此, 该量子态演化不满足时间空间对称(PT对称). 这些研究结果有利于进一步加深对非厄米量子系统, 特别是对PT对称概念的理解.

      对于更一般的周期驱动非厄米系统, 在归一化的经典相空间中, Floquet态本征模对应庞加莱截面(每隔驱动周期间隔采样)的定点. 本文建立的经典相空间定点同Floquet态和准能量的数学联系, 可以推广到非线性的非厄米量子系统中, 有助于研究开放量子多体系统的经典和量子对应, 有助于构造新的超越标准量子极限的快速演化非幺正量子门[18,51], 这些都是未来进一步的研究工作的重点.

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