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As one of the essential features in non-Hermitian systems coupled with environment, the exceptional point has attracted much attention in many physical fields. The phenomena that eigenvalues and eigenvectors of the system simultaneously coalesce at the exceptional point are also one of the important properties to distinguish from Hermitian systems. In non-Hermitian systems with parity-time reversal symmetry, the eigenvalues can be continuously adjusted in parameter space from all real spectra to pairs of complex-conjugate values by crossing the phase transition from the parity-time reversal symmetry preserving phase to the broken phase. The phase transition point is called an exceptional point of the system, which occurs in company with the spontaneous symmetry broken and many novel physical phenomena, such as sensitivity-enhanced measurement and loss induced transparency or lasing. Here, we focus on a two-qubit quantum system with parity-time reversal symmetry and construct an experimental scheme, prove and verify the features at its third-order exceptional point, including high-order energy response induced by perturbation and the coalescence of eigenvectors. We first theoretically study a two-qubit non-Hermitian system with parity-time reversal symmetry, calculate the properties of eigenvalues and eigenvectors, and prove the existence of a third-order exceptional point. Then, in order to study the energy response of the system induced by perturbation, we introduce an Ising-type interaction as perturbation and quantitatively demonstrate the response of eigenvalues. In logarithmic coordinates, three of the eigenvalues are indeed in the cubic root relationship with perturbation strength, while the fourth one is a linear function. Moreover, we study the eigenvectors around exceptional point and show the coalescence phenomenon as the perturbation strength becomes smaller. The characterization of the response of eigenvalues at high-order exceptional points is a quite difficult task as it is in general difficult to directly measure eigenenergies in a quantum system composed of a few qubits. In practice, the time evolution of occupation on a particular state is used to indirectly fit the eigenvalues. In order to make the fitting of experimental data more reliable, we want to determine an accurate enough expressions for the eigenvalues and eigenstates. To this aim, we employ a perturbation treatment and show good agreement with the numerical results of states occupation obtained by direct evolution. Moreover, we find that after the system evolves for a long enough time, it will end up to one of the eigenstates, which gives us a way to demonstrate eigenvector coalescence by measuring the density matrix via tomography and parity-time reversal transformation. To show our scheme is experimentally applicable, we propose an implementation using trapped $ ^{171} {\rm{Yb}}^{+}$ ions. We can map the parity-time reversal symmetric Hamiltonian to a purely dissipative two-ion system: use microwave to achieve spin state inversion, shine a 370 nm laser to realize dissipation of spin-up state, and apply Raman operation for Mølmer-Sørensen gates to implement Ising interaction. By adjusting the corresponding microwave and laser intensity, the spin coupling strength, the dissipation rate and the perturbation strength can be well controlled. We can record the probability distribution of the four product states of the two ions and measure the density matrix by detecting the fluorescence of each ion on different Pauli basis.-
Keywords:
- non-Hermitian system /
- exceptional point /
- parity-time reversal symmetry /
- ion trap
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图 1 微扰作用下高阶EP点处本征值和本征态的特征 (a), (b)在高阶EP点处存在微扰时, 体系本征值的实部((a))和虚部((b))随微扰
$J_x/J$ 的变化曲线. 其中, 实线对应本征值的数值结果, 虚线代表对$J_x/J$ 做微扰处理后, 仅保留最低阶的结果. 同一种颜色的曲线对应同一个本征值. (c)本征态$\left| {{\psi _1}} \right\rangle $ 分别和$\left| {{\psi _2}} \right\rangle $ (红色),$\left| {{\psi _3}} \right\rangle $ (黄色),$\left| {{\psi _4}} \right\rangle $ (紫色)的区分度$D_{1, n}$ Figure 1. Eigenvalues and eigenstates near a high-order exceptional point: (a) The real part and (b) imaginary part of the eigenvalues of Hamiltonian
$\hat{H}$ versus the perturbation$J_x/J$ . Here, solid lines are numerical results obtained by direct diagonalization, and dashed lines are those from perturbation. (c) Trace distance between$\left| {{\psi _1}} \right\rangle $ and$\left| {{\psi _2}} \right\rangle $ (red),$\left| {{\psi _3}} \right\rangle $ (yellow),$\left| {{\psi _4}} \right\rangle $ (purple)
图 2 系统从初态
$\left| {{ \downarrow _1}{ \downarrow _2}} \right\rangle $ 演化的量子态布据数结果 (a)从上到下红色和黑色的实线分别代表$J_x/J=0.1$ 和$J_x/J=0.01$ 时, 初态$\left| {{ \downarrow _1}{ \downarrow _2}} \right\rangle $ 的占据数随时间演化的结果. 两条几乎重合的灰色虚线为相应参数下的微扰近似值. (b)主图和子图分别展示了4个直积态$\left| {{\sigma _1}{\sigma _2}} \right\rangle $ 的归一化占据数和未归一化的占据数随时间的变化. 蓝色点划线和紫色点线分别代表$\left| {{ \uparrow _1}{ \uparrow _2}} \right\rangle $ 和$\left| {{ \downarrow _1}{ \downarrow _2}} \right\rangle $ 的演化, 红色实线和黄色虚线分别对应$\left| {{ \uparrow _1}{ \downarrow _2}} \right\rangle $ 和$\left| {{ \downarrow _1}{ \uparrow _2}} \right\rangle $ . (c), (d)演化时间为$tJ=10$ 时密度矩阵的实部((c))和虚部((d)). 这里选取$J_x/J=0.1$ Figure 2. (a) The evolution of
$\left| {{ \downarrow _1}{ \downarrow _2}} \right\rangle $ state population with$J_x/J=0.1$ (red, top) and$J_x/J=0.01$ (black, bottom). The gray lines on the top are the approximate results up to the second order. (b) The evolution of normalized spin product state population$\left| {{\sigma _1}{\sigma _2}} \right\rangle $ and the corresponding unnormalized populations (inset) with initial state$\left| {{ \downarrow _1}{ \downarrow _2}} \right\rangle $ . From top to bottom, the four lines represent results for states$\left| {{ \downarrow _1}{ \downarrow _2}} \right\rangle $ (purple dotted),$\left| {{ \uparrow _1}{ \downarrow _2}} \right\rangle $ (red solid),$\left| {{ \downarrow _1}{ \uparrow _2}} \right\rangle $ (yellow dashed), and$\left| {{ \uparrow _1}{ \uparrow _2}} \right\rangle $ (blue dot-dashed), respectively. (c) The real part and (d) the imaginary part of all density matrix elements at$tJ=10$ . For panels (a)–(d), the coupling strength of Ising interaction is$J_x/J=0.1$ 
图 3 实验方案设计 (a)
${^{171}{\rm{Yb}}}^+$ 的能级结构及耗散过程; (b) 伊辛相互作用的实现过程; (c)两离子系统中PT对称的哈密顿量与伊辛相互作用的实现. 可以利用微波(黄色)实现自旋态的翻转; 利用370 nm激光(蓝色)实现自旋上态的耗散; 利用基于拉曼激光(紫色)操作的MS门实现伊辛相互作用Figure 3. Experimental scheme: (a) The energy level of
${^{171}{\rm{Yb}}}^+$ and dissipation process; (b) realization of Ising interaction; (c) realization of${\cal{PT}}$ symmetric Hamiltonian and Ising interaction in a two-ion system: we can use microwave (yellow color) to achieve spin state inversion, shine a 370 nm laser (blue color) to realize dissipation of the spin-up state, and apply Raman laser (purple color) operation for MS gates to implement Ising interaction
图 A1 无相互作用两量子比特系统的本征值的(a)实部和(b)虚部随耦合强度
$J/\varGamma$ 的变化Figure A1. (a) The real and (b) imaginary parts of eigenvalues of a non-interacting two-qubit system as functions of coupling strength
$J/\varGamma$ 
图 A2 对数坐标系下具有伊辛相互作用的两量子比特体系本征值的(a)实部和(b)虚部的绝对值随相互作用强度
$J_x/J$ 的变化. 以及在更大的相互作用强度范围内, 该体系的本征值的(c)实部和(d)虚部随相互作用强度的变化Figure A2. The absolute values of (a) the real and (b) imaginary parts of eigenvalues of a two-qubit system with Ising interaction versus the interaction strength
$J_x/J$ . (c) The real and (d) the imaginary parts of eigenvalues of such a system versus$J_x/J$ in an extended range of the interaction strength -
[1] Kato T 1966 Perturbation Theory for Linear Operators (Berlin: Springer) pp62–86
[2] Teller E 1937 J. Phys. Chem. 41 109
Google Scholar
[3] Heiss W D 2012 J. Phys. A: Math. Theor. 45 444016
Google Scholar
[4] Ashida Y, Gong Z, Ueda M 2020 Adv. Phys. 69 249
[5] Berry M V 2004 Czech. J. Phys. 54 1039
Google Scholar
[6] Chen W, Kaya Özdemir S K, Zhao G, Wiersig J, Yang L 2017 Nature 548 192
Google Scholar
[7] Hodaei H, Hassan A U, Wittek S, Garcia-Gracia H, El-Ganainy R, Christodoulides D N, Khajavikhan M 2017 Nature 548 187
Google Scholar
[8] Lai Y H, Lu Y K, Suh M G, Yuan Z, Vahala K 2019 Nature 576 65
Google Scholar
[9] Chu Y, Liu Y, Liu H, Cai J 2020 Phys. Rev. Lett 124 020501
Google Scholar
[10] Ding L, Shi K, Zhang Q, Shen D, Zhang X, Zhang W 2021 Phys. Rev. Lett. 126 083604
Google Scholar
[11] Guo A, Salamo G J, Duchesne D, Morandotti R, Volatier-Ravat M, Aimez V, Siviloglou G A, Christodoulides D N 2009 Phys. Rev. Lett. 103 093902
Google Scholar
[12] Peng B, Özdemir S K, Rotter S, Yilmaz H, Liertzer M, Monifi F, Bender C M, Nori F, Yang L 2014 Science 346 328
Google Scholar
[13] Gao T, Estrecho E, Bliokh K Y, Liew T C H, Fraser M D, Brodbeck S, Kamp M, Schneider C, Höfling S, Yamamoto Y, Nori F, Kivshar Y S, Truscott A G, Dall R G, Ostrovskaya E A 2015 Nature 526 554
Google Scholar
[14] Kanki K, Garmon S, Tanaka S, Petrosky T 2017 J. Math. Phys. 58 092101
Google Scholar
[15] Wu Y, Zhou P, Li T, Wan W, Zou Y 2021 Opt. Express 29 6080
Google Scholar
[16] Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243
Google Scholar
[17] El-Ganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S, Christodoulides D N 2018 Nat. Phys. 14 11
Google Scholar
[18] Dembowski C, Gräf H D, Harney H L, Heine A, Heiss W D, Rehfeld H, Richter A 2001 Phys. Rev. Lett. 86 787
Google Scholar
[19] Peng B, Özdemir S K, Lei F, Monifi F, Gianfreda M, Long G L, Fan S, Nori F, Bender C M, Yang L 2014 Nat. Phys. 10 394
Google Scholar
[20] Rüter C E, Makris K G, El-Ganainy R, Christodoulides D N, Segev M, Kip D 2010 Nat. Phys. 6 192
Google Scholar
[21] 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 76
Google Scholar
[22] Schindler J, Li A, Zheng M C, Ellis F M, Kottos T 2011 Phys. Rev. A 84 040101(R
Google Scholar
[23] Bender N, Factor S, Bodyfelt J D, Ramezani H, Christodoulides D N, Ellis F M, Kottos T 2013 Phys. Rev. Lett. 110 234101
Google Scholar
[24] Naghiloo M, Abbasi M, Joglekar Y N, Murch K W 2019 Nat. Phys. 15 1232
Google Scholar
[25] 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 1117
Google Scholar
[26] Li J, Harter A K, Liu J, de Melo L, Joglekar Y N, Luo L 2019 Nat. Commun. 10 855
Google Scholar
[27] 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 L020201
Google Scholar
[28] Nielsen M A, Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) p399-416
[29] Gilchrist A, Langford N K, Nielsen M A 2005 Phys. Rev. A 71 062310
Google Scholar
[30] Fuchs C A, van de Graaf J 1999 IEEE Trans. Inf. Theory 45 1216
Google Scholar
[31] Kawabata K, Ashida Y, Ueda M 2017 Phys. Rev. Lett. 119 190401
Google Scholar
[32] Wang Y T, Li Z P, Yu S, Ke Z J, Liu W, Meng Y, Yang Y Z, Tang J S, Li C F, Guo G C 2020 Phys. Rev. Lett. 124 230402
Google Scholar
[33] Brody D C, Graefe E M 2012 Phys. Rev. Lett. 109 230405
Google Scholar
[34] Ding L, Shi K, Wang Y, Zhang Q, Zhu C, Zhang L, Yi J, Zhang S, Zhang X, Zhang W 2022 Phys. Rev. A 105 L010204
Google Scholar
[35] Sørensen A, Mølmer K 1999 Phys. Rev. Lett. 82 1971
Google Scholar
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