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纳米机械谐振器耦合量子比特非厄米哈密顿量诱导的声子阻塞

廖庆洪 邓伟灿 文健 周南润 刘念华

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纳米机械谐振器耦合量子比特非厄米哈密顿量诱导的声子阻塞

廖庆洪, 邓伟灿, 文健, 周南润, 刘念华

Phonon blockade induced by a non-Hermitian Hamiltonian in a nanomechanical resonator coupled with a qubit

Liao Qing-Hong, Deng Wei-Can, Wen Jian, Zhou Nan-Run, Liu Nian-Hua
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  • 研究了纳米机械谐振器耦合量子比特系统中的声子阻塞现象. 发现在弱驱动的条件下, 非厄米哈密顿量的所有本征值都等于零时, 出现了强声子反聚束现象. 于是提出了零本征值方法来诱导声子阻塞. 本文详细分析了此方法对声子阻塞的影响, 给出了最佳条件并解释了背后的物理机理. 不同于传统的声子阻塞和非传统的声子阻塞, 不管是强非线性还是弱非线性, 都可以实现声子的反聚束效应.
    Nanomechanical resonator has important applications in the field of high-precision detection because it has a high-Q factor, high vibration frequency, small size, and other excellent characteristics. Superconducting qubit has very large magnetic dipole moments, so it can be easily combined with nanomechanical resonator. Furthermore, the system parameters including frequency and coupling strength can be designed according to requirements beforehand, which makes a superconducting qubit an ideal artificial atom. Compared with natural atom, superconducting qubit has abundant energy levels. For these reasons, nanomechanical system has aroused wide interest in the engineering, electron, physical and other fields of science and technology. According to the recent research, a new approach to the zero eigenvalues of non-Hermitian Hamiltonian is applied to the optomechanical system. It was found that the scheme is superior to conventional photon blockade (CPB) and unconventional photon blockade (UPB) in the cavity quantum electrodynamics (QED) system. So we propose a scheme to induce phonon blockade in order to explore a new avenue to the research about phonon blockades in the quantum open system. We study the phonon blockade in an optomechanical system that a qubit is coupled with nanomechanical resonator (NAMR) driven by two external weakly driving fields respectively in this way. In this paper, the Hamiltonian of such a system can be treated by the non-Hermitian Hamiltonian and it can be described in the form of matrix. Then the phenomenon of phonon blockade occurs when all the eigenvalues in the form of matrix are equal to zero. It is found that strong phonon antibunching can be triggered in both strong and weak nonlinearity when we use the method which has been already used in a gain optical cavity system. The distinct result reflects the advantage of our approach which possesses some outstanding characters between the ordinary methods (conventional phonon blockade and unconventional phonon blockade). In addition, the effect of our avenue on phonon blockade is analyzed and also the distinction between the conventional phonon blockade (CPNB) and unconventional phonon blockade (UPNB) is compared with each other in detail. By analytical calculation, the optimal conditions are given and the underlying physical mechanism is explained. In the comparison between CPNB and UPNB, we show the superiority of our scheme through some graphs. Finally, we describe briefly the measurements of phonon blockade in the NAMR-qubit system via a superconducting cavity. The proposal may provide a theoretical way to guide the manufacture of phonon devices in the future. The results obtained here may have a great significance and application in the field of quantum information processing and precision measurement.
      通信作者: 廖庆洪, nculqh@163.com
      Corresponding author: Liao Qing-Hong, nculqh@163.com
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    Zhou Y H, Shen H Z, Shao X Q, Yi X X 2016 Opt. Express 24 17332Google Scholar

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    Wang X, Miranowicz A, Li H R, Nori F 2016 Phys. Rev. A 93 063861Google Scholar

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    Ramos T, Sudhir V, Stannigel K, Zoller P, Kippenberg T J 2013 Phys. Rev. Lett. 110 193602Google Scholar

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    Xu X W, Chen A X, Liu Y X 2016 Phys. Rev. A 94 063853Google Scholar

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    Gerace D, Savona V 2014 Phys. Rev. A 89 031803Google Scholar

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    Lemonde M A, Didier N, Clerk A A 2014 Phys. Rev. A 90 063824Google Scholar

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    Bamba M, Imamoğlu A, Carusotto I, Ciuti C 2011 Phys. Rev. A 83 021802Google Scholar

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    Qian Z, Zhao M M, Hou B P, Zhao Y H 2017 Opt. Express 25 33097Google Scholar

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    Kong C, Li S, You C, Xiong H, Wu Y 2018 Sci. Rep. 8 1060Google Scholar

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    Li G, Xiao X, Li Y, Wang X 2018 Phys. Rev. A 97 023801Google Scholar

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    Guan S, Bowen W P, Liu C, Duan Z 2017 EPL 119 58001Google Scholar

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    Xie H, Liao C G, Shang X, Ye M Y, Lin X M 2017 Phys. Rev. A 96 013861Google Scholar

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    Yin T S, Lu X Y, Wan L L, Bin S W, Wu Y 2018 Opt. Lett. 43 2050Google Scholar

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    Debnath S, Linke N M, Wang S T, Figgatt C, Landsman K A, Duan L M, Monroe C 2018 Phys. Rev. Lett. 120 073001Google Scholar

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    Li S S 2018 Int. J. Theor. Phys. 57 2359Google Scholar

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    Shen H, Zhen B, Fu L 2018 Phys. Rev. Lett. 120 146402Google Scholar

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    Longhi S 2017 J. Phys. A: Math. Theor. 50 505201Google Scholar

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    Park K W, Kim J, Jeong K 2016 Opt. Commun. 368 190Google Scholar

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    Khantoul B, Bounames A, Maamache M 2017 Eur. Phys. J. Plus 132 258Google Scholar

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    Baradaran M, Panahi H 2017 Chin. Phys. B 26 060301Google Scholar

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    Maamache M, Lamri S, Cherbal O 2017 Ann. Phys. 378 150Google Scholar

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    Sergi A, Giaquinta P 2016 Entropy 18 451Google Scholar

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    Sergi A, Zloshchastiev K G 2016 J. Stat. Mech: Theory Exp. 2016 033102Google Scholar

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    Chen J H, Fan H Y 2012 Front. Phys. 7 632Google Scholar

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    Greenberg Y S, Shtygashev A A 2015 Phys. Rev. A 92 063835Google Scholar

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    Zloshchastiev K G 2015 Eur. Phys. J. D 69 253Google Scholar

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    Bagarello F, Lattuca M, Passante R, Rizzuto L, Spagnolo S 2015 Phys. Rev. A 91 042134Google Scholar

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    Giusteri G G, Mattiotti F, Celardo G L 2015 Phys. Rev. B 91 094301Google Scholar

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    Zhou Y H, Shen H Z, Zhang X Y, Yi X X 2018 Phys. Rev. A 97 043819Google Scholar

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    Li J, Yu R, Wu Y 2015 Phys. Rev. A 92 053837Google Scholar

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    Xing Y, Qi L, Cao J, Wang D Y, Bai C H, Wang H F, Zhu A D, Zhang S 2017 Phys. Rev. A 96 043810Google Scholar

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    Peng B, Ozdemir S K, Rotter S, Yilmaz H, Liertzer M, Monifi F, Bender C M, Nori F, Yang L 2014 Science 346 328Google Scholar

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

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

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

    [51]

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

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    Fan L R, Fong K Y, Poot M, Tang H X 2015 Nat. Commun. 6 5850

    [53]

    Xiong C, Fan L R, Sun X K, Tang H X 2013 Appl. Phys. Lett 102 021110

    [54]

    Winger M, Blasius T D, Alegre T P M, Safavi Naeini A H, Meenehan S, Cohen J, Stobbe S, Painter O 2011 Opt. Express 19 24905Google Scholar

    [55]

    Zhang J, Peng B, Ozdemir S K, Liu Y X, Jing H, Lu X Y, Liu Y L, Yang L, Nori F 2015 Phys. Rev. B 92 115407

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    Bahl G, Tomes M, Marquardt F, Carmon T 2012 Nat. Phys. 8 203Google Scholar

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    Dong C H, Shen Z, Zou C L, Zhang Y L, Fu W, Guo G C 2015 Nat. Commun. 6 6193

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    Kepesidis K V, Milburn T J, Huber J, Makris K G, Rotter S, Rabl P 2016 New J. Phys. 18 095003

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    Xu X W, Liu Y X, Sun C P, Li Y 2015 Phys. Rev. A 92 013852

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    刘玉龙2017 博士学位论文 (北京: 清华大学)

    Liu Y L 2017 Ph. D. Dissertation (Beijing: Tsinghua University) (in Chinese)

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    Didier N, Pugnetti S, Blanter Y M, Fazio R 2011 Phys. Rev. B 84 054503Google Scholar

  • 图 1  NAMR-量子比特系统的物理模型图

    Fig. 1.  Schematic diagram of the nanomechanical resonator coupled with a qubit.

    图 2  外加驱动场共振的情况下声子阻塞现象的能级图 (a)满足(8)式时的能级图, 其中$\omega _ \pm ^{\left( 1 \right)} = {\omega _{\rm{m}}}$$\omega _ \pm ^{\left( 2 \right)} = 2{\omega _{\rm{m}}} \pm $ $2g + {\rm{i}}\dfrac{\gamma }{2}$; (b)满足(9)式时的能级图, 其中$\omega _ \pm ^{\left( 1 \right)} = {\omega _{\rm{m}}} \pm g$$\omega _ \pm ^{\left( 2 \right)} = 2{\omega _{\rm{m}}} \pm \sqrt 2 g + {\rm{i}}\dfrac{\gamma }{2}$

    Fig. 2.  Schematic energy-level diagram explaining the occurrence of the phonon blockade by the driving field satisfying the resonance condition: (a) Energy-level diagram when Eqs. (8) are satisfied, where $\omega _ \pm ^{\left( 1 \right)} = {\omega _{\rm{m}}}$ and $\omega _ \pm ^{\left( 2 \right)} = 2{\omega _{\rm{m}}} \pm$ $ 2g + {\rm{i}}\dfrac{\gamma }{2}$; (b) energy-level diagram when Eqs. (9) are satisfied, where $\omega _ \pm ^{\left( 1 \right)} = {\omega _{\rm{m}}} \pm g$ and $\omega _ \pm ^{\left( 2 \right)} = 2{\omega _{\rm{m}}} \pm \sqrt 2 g + {\rm{i}}\dfrac{\gamma }{2}$.

    图 3  在不同的$g/\gamma $关系下, 二阶关联函数${g^{\left( 2 \right)}}\left( 0 \right)$$\varDelta /\gamma $的关系图 (a) $\kappa /\gamma = - 1$, $\varepsilon /\gamma = 0.1$, $\varOmega = 0$, $\varepsilon \ne 0$; (b) $\kappa /\gamma = - 1$, $\varOmega /\gamma = 0.1$, $\varepsilon = 0$, $\phi = 0$

    Fig. 3.  Logarithmic plot (of base e) of the zero-delay-time second-order correlation functions ${g^{\left( 2 \right)}}\left( 0 \right)$ as a function of $\varDelta /\gamma $ for different $g/\gamma $ with $\kappa /\gamma = - 1$: (a) $\varepsilon /\gamma = 0.1$, $\varOmega = 0$, $\varepsilon \ne 0$; (b) $\varOmega /\gamma = 0.1$, $\varepsilon = 0$, $\phi = 0$.

    图 4  UPNB能级图

    Fig. 4.  Schematic energy-level diagram of unconventional phonon blockade.

    图 5  本文方案与CPNB和UPNB的区别

    Fig. 5.  Different parameter range for the CPNB, the UPNB, and the present scheme.

    图 6  二阶关联函数${g^{\left( 2 \right)}}\left( 0 \right)$ 和平均声子数$N = \left\langle {{a^ + }a} \right\rangle = \left[ {{\rm{Tr}}\left( {{\rho _s}{a^ + }a} \right)} \right]$的对数图像 (a), (b) $\varepsilon /\gamma = 0.1$, $\varOmega = 0$; (c), (d) $\varOmega /\gamma = 0.1$, $\varepsilon = 0$, $\phi = 0$. 其他参数为$\kappa /\gamma = - 1$.

    Fig. 6.  The logarithmic plot (of base e) of the zero-delay-time second-order correlation functions ${g^{\left( 2 \right)}}\left( 0 \right)$ and average photon numberN: (a) and (b) $\varepsilon /\gamma = 0.1$, $\varOmega = 0$; (c) and (d) $\varOmega /\gamma = 0.1$, $\varepsilon = 0$, $\phi = 0$. The shared parameters: $\kappa /\gamma = - 1$.

    图 7  二阶关联函数${g^{\left( 2 \right)}}\left( 0 \right)$(对数)与$\varDelta /\gamma $的关系图, 其中$\varepsilon /\gamma = 0.1$, $\varOmega = 0$ (a) $g/\gamma = 0.5$, $\kappa /\gamma = - 1$(本文方案), $\kappa /\gamma = 1$(UPNB); (b) $g/\gamma = 5$, $\kappa /\gamma = - 1$(本文方案), $\kappa /\gamma = 1$(CPNB)

    Fig. 7.  The logarithmic plot (of base e) of the zero-delay-time second-order correlation functions ${g^{\left( 2 \right)}}\left( 0 \right)$ as a function of $\varDelta /\gamma $. The shared parameters: $\varepsilon /\gamma = 0.1$, $\varOmega = 0$. Other parameters: (a) $g/\gamma = 0.5$, $\kappa /\gamma = - 1$(our scheme), $\kappa /\gamma = 1$ (UPNB); (b) $g/\gamma = 5$, $\kappa /\gamma = - 1$(our scheme), $\kappa /\gamma = 1$(CPNB).

    图 8  二阶关联函数${g^{\left( 2 \right)}}\left( 0 \right)$(对数)与$\varDelta /\gamma $的关系图, $\varOmega \ne 0$, $\varepsilon = 0$, $\phi = 0$ (a) $\varOmega /\gamma = 0.1$, $g/\gamma = 0.5$, $\kappa /\gamma = - 1$(本文方案), $\kappa /\gamma = 1$(UPNB); (b) $\varOmega /\gamma = 2$, $g/\gamma = 10$, $\kappa /\gamma = - 1$(本文方案), $\kappa /\gamma = 1$(CPNB)

    Fig. 8.  The logarithmic plot (of base e) of the zero-delay-time second-order correlation functions ${g^{\left( 2 \right)}}\left( 0 \right)$ as a function of $\varDelta /\gamma $: $\varOmega \ne 0$, $\varepsilon = 0$, $\phi = 0$. Other parameters: (a) $\varOmega /\gamma = 0.1$, $g/\gamma = 0.5$, $\kappa /\gamma = - 1$(our scheme), $\kappa /\gamma = 1$ (UPNB); (b) $\varOmega /\gamma = 2$, $g/\gamma = 10$, $\kappa /\gamma = - 1$(our scheme), $\kappa /\gamma = 1$(CPNB).

    图 9  声子统计特性探测模型

    Fig. 9.  The detecting model of the statistical properties of the phonons.

  • [1]

    You J Q, Nori F 2011 Nature 474 589Google Scholar

    [2]

    Armour A D, Blencowe M P, Schwab K C 2002 Phys. Rev. Lett. 88 148301Google Scholar

    [3]

    Blencowe M 2004 Phys. Rep. 395 159Google Scholar

    [4]

    Cleland A N, Geller M R 2004 Phys. Rev. Lett. 93 070501Google Scholar

    [5]

    O'Connell A D, Hofheinz M, Ansmann M, Bialczak R C, Lenander M, Lucero E, Neeley M, Sank D, Wang H, Weides M, Wenner J, Martinis J M, Cleland A N 2010 Nature 464 697Google Scholar

    [6]

    Liu Y X, Miranowicz A, Gao Y B, Bajer J, Sun C P, Nori F 2010 Phys. Rev. A 82 032101Google Scholar

    [7]

    Imamoglu A, Schmidt H, Woods G, Deutsch M 1997 Phys. Rev. Lett. 79 1467Google Scholar

    [8]

    Barzanjeh S, Vitali D 2016 Phys. Rev. A 93 033846Google Scholar

    [9]

    Zhou Y H, Shen H Z, Yi X X 2015 Phys. Rev. A 92 023838Google Scholar

    [10]

    Zhou Y H, Shen H Z, Shao X Q, Yi X X 2016 Opt. Express 24 17332Google Scholar

    [11]

    石海泉, 谢智强, 徐勋卫, 刘念华 2018 物理学报 67 044203Google Scholar

    Shi H Q, Xie Z Q, Xu X W, Liu N H 2018 Acta Phys. Sin. 67 044203Google Scholar

    [12]

    Cai K, Pan Z W, Wang R X, Ruan D, Yin Z Q, Long G L 2018 Opt. Lett. 43 1163Google Scholar

    [13]

    Miranowicz A, Bajer J, Lambert N, Liu Y X, Nori F 2016 Phys. Rev. A 93 013808Google Scholar

    [14]

    Wang X, Miranowicz A, Li H R, Nori F 2016 Phys. Rev. A 93 063861Google Scholar

    [15]

    Ramos T, Sudhir V, Stannigel K, Zoller P, Kippenberg T J 2013 Phys. Rev. Lett. 110 193602Google Scholar

    [16]

    Xu X W, Chen A X, Liu Y X 2016 Phys. Rev. A 94 063853Google Scholar

    [17]

    Gerace D, Savona V 2014 Phys. Rev. A 89 031803Google Scholar

    [18]

    Lemonde M A, Didier N, Clerk A A 2014 Phys. Rev. A 90 063824Google Scholar

    [19]

    Bamba M, Imamoğlu A, Carusotto I, Ciuti C 2011 Phys. Rev. A 83 021802Google Scholar

    [20]

    Qian Z, Zhao M M, Hou B P, Zhao Y H 2017 Opt. Express 25 33097Google Scholar

    [21]

    Shi H Q, Zhou X T, Xu X W, Liu N H 2018 Sci. Rep. 2018 2212

    [22]

    Kong C, Li S, You C, Xiong H, Wu Y 2018 Sci. Rep. 8 1060Google Scholar

    [23]

    Li G, Xiao X, Li Y, Wang X 2018 Phys. Rev. A 97 023801Google Scholar

    [24]

    Guan S, Bowen W P, Liu C, Duan Z 2017 EPL 119 58001Google Scholar

    [25]

    Xie H, Liao C G, Shang X, Ye M Y, Lin X M 2017 Phys. Rev. A 96 013861Google Scholar

    [26]

    Yin T S, Lu X Y, Wan L L, Bin S W, Wu Y 2018 Opt. Lett. 43 2050Google Scholar

    [27]

    Debnath S, Linke N M, Wang S T, Figgatt C, Landsman K A, Duan L M, Monroe C 2018 Phys. Rev. Lett. 120 073001Google Scholar

    [28]

    Li S S 2018 Int. J. Theor. Phys. 57 2359Google Scholar

    [29]

    Shen H, Zhen B, Fu L 2018 Phys. Rev. Lett. 120 146402Google Scholar

    [30]

    Longhi S 2017 J. Phys. A: Math. Theor. 50 505201Google Scholar

    [31]

    Park K W, Kim J, Jeong K 2016 Opt. Commun. 368 190Google Scholar

    [32]

    Khantoul B, Bounames A, Maamache M 2017 Eur. Phys. J. Plus 132 258Google Scholar

    [33]

    Baradaran M, Panahi H 2017 Chin. Phys. B 26 060301Google Scholar

    [34]

    Maamache M, Lamri S, Cherbal O 2017 Ann. Phys. 378 150Google Scholar

    [35]

    Sergi A, Giaquinta P 2016 Entropy 18 451Google Scholar

    [36]

    Miao Y G, Xu Z M 2016 Phys. Lett. A 380 1805Google Scholar

    [37]

    Ferry D K, Akis R, Burke A M, Knezevic I, Brunner R, Bird J P, Meisels R, Kuchar F, Bird J P 2013 Fortschr. Phys. 61 291Google Scholar

    [38]

    Sergi A, Zloshchastiev K G 2016 J. Stat. Mech: Theory Exp. 2016 033102Google Scholar

    [39]

    Chen J H, Fan H Y 2012 Front. Phys. 7 632Google Scholar

    [40]

    Greenberg Y S, Shtygashev A A 2015 Phys. Rev. A 92 063835Google Scholar

    [41]

    Zloshchastiev K G 2015 Eur. Phys. J. D 69 253Google Scholar

    [42]

    Bagarello F, Lattuca M, Passante R, Rizzuto L, Spagnolo S 2015 Phys. Rev. A 91 042134Google Scholar

    [43]

    Giusteri G G, Mattiotti F, Celardo G L 2015 Phys. Rev. B 91 094301Google Scholar

    [44]

    Zhou Y H, Shen H Z, Zhang X Y, Yi X X 2018 Phys. Rev. A 97 043819Google Scholar

    [45]

    Li J, Yu R, Wu Y 2015 Phys. Rev. A 92 053837Google Scholar

    [46]

    Xing Y, Qi L, Cao J, Wang D Y, Bai C H, Wang H F, Zhu A D, Zhang S 2017 Phys. Rev. A 96 043810Google Scholar

    [47]

    Peng B, Ozdemir S K, Rotter S, Yilmaz H, Liertzer M, Monifi F, Bender C M, Nori F, Yang L 2014 Science 346 328Google Scholar

    [48]

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

    [49]

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

    [50]

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

    [51]

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

    [52]

    Fan L R, Fong K Y, Poot M, Tang H X 2015 Nat. Commun. 6 5850

    [53]

    Xiong C, Fan L R, Sun X K, Tang H X 2013 Appl. Phys. Lett 102 021110

    [54]

    Winger M, Blasius T D, Alegre T P M, Safavi Naeini A H, Meenehan S, Cohen J, Stobbe S, Painter O 2011 Opt. Express 19 24905Google Scholar

    [55]

    Zhang J, Peng B, Ozdemir S K, Liu Y X, Jing H, Lu X Y, Liu Y L, Yang L, Nori F 2015 Phys. Rev. B 92 115407

    [56]

    Bahl G, Tomes M, Marquardt F, Carmon T 2012 Nat. Phys. 8 203Google Scholar

    [57]

    Dong C H, Shen Z, Zou C L, Zhang Y L, Fu W, Guo G C 2015 Nat. Commun. 6 6193

    [58]

    Kepesidis K V, Milburn T J, Huber J, Makris K G, Rotter S, Rabl P 2016 New J. Phys. 18 095003

    [59]

    Xu X W, Liu Y X, Sun C P, Li Y 2015 Phys. Rev. A 92 013852

    [60]

    刘玉龙2017 博士学位论文 (北京: 清华大学)

    Liu Y L 2017 Ph. D. Dissertation (Beijing: Tsinghua University) (in Chinese)

    [61]

    Didier N, Pugnetti S, Blanter Y M, Fazio R 2011 Phys. Rev. B 84 054503Google Scholar

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出版历程
  • 收稿日期:  2018-12-25
  • 修回日期:  2019-02-27
  • 上网日期:  2019-06-01
  • 刊出日期:  2019-06-05

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