Detecting topological exceptional points in a parity-time symmetric system with cold atoms
aa r X i v : . [ qu a n t - ph ] N ov Detecting topological exceptional points in a parity-time symmetric system with cold atoms
Jian Xu,
1, 2, ∗ Yan-Xiong Du, Wei Huang, and Dan-Wei Zhang † College of Electronics and Information Engineering, Guangdong Ocean University, Zhanjiang 524088, China Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,SPTE, South China Normal University, Guangzhou 510006, China (Dated: November 9, 2016)We reveal a novel topological property of the exceptional points in a two-level parity-time symmetric systemand then propose a scheme to detect the topological exceptional points in the system, which is embedded in alarger Hilbert space constructed by a four-level cold atomic system. We show that a tunable parameter in thepresented system for simulating the non-Hermitian Hamiltonian can be tuned to swept the eigenstates throughthe exceptional points in parameter space. The non-trivial Berry phases of the eigenstates obtained in this loopfrom the exceptional points can be measured by the atomic interferometry. Since the proposed operations anddetection are experimentally feasible, our scheme may pave a promising way to explore the novel properties ofnon-Hermitian systems.
PACS numbers: 67.85.-d, 42.50.Gy, 11.30.Rd, 03.65.Vf
I. INTRODUCTION
Non-Hermitian Hamiltonian used to describe open or dis-sipative systems usually have complex eigenvalues. However,it is recently found that a series of non-Hermitian Hamiltoni-ans have real eigenvalues if they are invariant under the parity ( P ) and time-reversal ( T ) union operation [1–3]. Meanwhile,the eigenstates of these Hamiltonians are also commuted withthe PT-symmetry operation[4]. Due to various intriguingproperties, the PT-symmetric systems have raised broad at-tentions. Some PT-symmetric models have been experimen-tally realized in physical systems, such as active LRC circuits[5, 6], two coupled waveguides [7], photonic lattices [8, 9],microwave billiard [10], transmission line [11], whispering-gallery microcavities [12, 13] and single-mode lasing [14, 15].Recently, the schemes for simulating the PT-symmetry poten-tials with cold atoms in optical systems have been theoreti-cally proposed [16–18] and experimentally realized [19].Exceptional point (EP) is a special point in parameter spaceof non-Hermitian systems where both eigenvalues and eigen-states coalesce into only one value and state [20, 21]. One ofthe important properties of EPs is that the states of EPs are chi-ral [22], which has been experimentally observed [23, 24]. Onthe other hand, the chirality leads to an unique effect that theeigenstates exchange themselves but only one of them obtainsa π Berry phase in a cyclic evolution [25, 26]. This chiral phe-nomenon of EPs has been experimentally demonstrated in mi-crowave cavity for the first time [27, 28]. Very recently, a fulldynamical encircling of EP has been realized [29] and a non-reciprocal topological energy transfer due to dynamical encir-cling of such point has been measurement [30]. In contrast,other theoretical and numerical results suggest in this case theeigenstates change to the other one but both of them obtain a π/ Berry phase due to the linear dependence of eigenstates[31, 32], which has been verified in a recent experiment [33]. ∗ Electronic address: xujian˙[email protected] † Electronic address: [email protected]
In this paper, we demonstrate that there are two differentchiral EPs in parameter space and the non-trivial Berry phaseof EPs emerge different due to the chirality breaking when theeigenstates exactly to sweep through EPs in a cyclic evolu-tion. Then we propose a scheme to realize the PT-symmetricHamiltonian in cold atomic systems that the parameters can beexactly controlled in time. Based on the idea of the Naimark-Dilated operation [34] and the embedding quantum simula-tor [35], we show that a two-level PT-symmetric Hamiltoniancan be constructed through a four-level Hermitian Hamilto-nian in an embedding cold atomic system. Finally, we pro-pose to detect this Berry phase through the observation of thephase difference between atomic levels, which can be mea-sured through atomic interferometry. The proposed schemeprovides a promising approach to realize the PT-symmetricHamiltonian in cold atomic systems and to further explore theexotic properties of the EPs.The paper is organized as follows. Section II describesour two-level PT-Hamiltonian and the topological propertiesof the intrinsic EPs. In Sec. III, we propose an experimentallyfeasible scheme for emulating the non-Hermitian two-levelHamiltonian in a Hermitian four-level cold atomic system. InSec. IV, we show that the topological EPs can be measured bythe atomic interferometry in the cold atom system. Finally, abrief discussion and a short conclusion are given in Sec. V.
II. TOPOLOGICAL EXCEPTIONAL POINTS IN APT-SYMMETRIC SYSTEM
If a Hamiltonian H is non-Hermitian for describing an openor dissipative system, there are gain or loss effects in this sys-tem and the eigenvalues are generally complex values. How-ever, if gain and loss of this system are balanced, this systemremains stable and all eigenvalues are real numbers. This phe-nomenon is described by the PT-symmetric theory. Supposingthat in this case σ i ( i = x, y, z ) are Pauli matrices, the parityoperator P is σ x and the time-reversal operator T is the com-plex conjugation operator, which is an antilinear operator. Asimple PT-symmetric Hamiltonian can be constructed as H P T = S (cid:18) i sin( α ) 11 − i sin( α ) (cid:19) , (1)where S is a general scaling factor of the matrix. The angle α characterizes the non-Hermiticity of the Hamiltonian. When α = 0 , the Hamiltonian H is a Hermitian operator, when α =0 the Hamiltonian H becomes a non-Hermitian operator. Inthe case of α = ± π/ , the eigenvalues and the eigenstatescoalesce into a single value and state, respectively.The eigenvalues of Eq. (1) are E ± ( α ) = ± χ = ± S cos( α ) and the corresponding eigenstates are given by | E + ( α ) i = e iα/ p α ) (cid:18) e − iα (cid:19) , | E − ( α ) i = ie − iα/ p α ) (cid:18) − e iα (cid:19) . (2)In addition, the non-Hermitian Hamiltonian H P T has a bi-orthogonal basis | E ± ( α ) i , | Λ ± ( α ) i [37]: H ∗ P T | Λ ± ( α ) i = E ∗± ( α ) | Λ ± ( α ) i , | Λ ± ( α ) i = −| E ∓ ( α ) i . (3)When α = ± π/ , the two eigenvalues become E ± = 0 andthe corresponding eigenstates coalescence at the same time: | E ( π/ i ∝ (cid:18) − i (cid:19) , | E ( − π/ i ∝ (cid:18) i (cid:19) . (4)The signs before i in Eq. (4) depend on the system and givethe chirality of these specific degeneracy points dubbed as EPs[20, 22, 25, 26], such that the two EPs are different from eachother. The Hamiltonian thus is restricted to purely real eigen-values, and the time evolution operator ˆ U P T ( t ) = e − iH PT t isunitary with the explicit form ˆ U P T ( t ) = 1cos( α ) (cid:18) cos( χt − α ) − i sin( χt ) − i sin( χt ) cos( χt + α )) (cid:19) . (5)With the analytical solution of the PT-symmetric Hamilto-nian, we can study the topological properties of the EPs. Con-sidering the eigenvalues E ± ( α ) , we can find that the eigen-states | E + ( α ) i and | E − ( α ) i respectively represent the higherand lower levels when α ∈ [ − π/ N π, π/
N π ] ,but respectively represent the lower and higher levels when α ∈ [ π/ N π, π/ N π ] , with N being a positive inte-ger. This means that the definition of the domain in the systemis [ − π/ , π/ and the eigenstates exchange themselves when α sweeps through the EPs every time. With the eigenvalues E ± ( α ) = ± S cos( α ) , one can also find that the correspond-ing point of α in the other Riemann surface is the point of ± π − α . Due to the degeneracy of the EPs, the eigenstates ob-tain non-Abelian geometric phases through passing the EPs. For this degenerate non-Hermitian system, the Berry phase inthe cyclic evolution is [38] γ = I C Adα, (6)where A is the non-Abelian Berry connection: A = i (cid:18) h Λ + | d α | E + i h Λ + | d α | E − ih Λ − | d α | E + i h Λ − | d α | E − i (cid:19) , (7)with d α being the α derivative. For the PT-symmetric Hamil-tonian H P T here, we can obtain the non-Abelian geometricphases for two different loops from α to ± π − α (which passthrough two EPs of different chiralities) as γ α →± π − α = (cid:18) ± π ± π (cid:19) . (8)Consequently, when α successively sweeps through two dif-ferent EPs in the same evolutionary direction, the eigenstatesbecome original with an additional π Berry phase. In thiscase, the eigenstates under the whole evolution can be writ-ten as | E ± ( α ) i−→ e i π | E ∓ ( π − α ) i−→ − | E ± ( α ) i , | E ± ( α ) i−→ e − i π | E ∓ ( − π − α ) i−→ − | E ± ( α ) i . (9)Unlike | E ± i→| E ∓ i→ − | E ± i or | E ± i→− | E ∓ i→ − | E ± i inthe general case, here Eq. (9) shows that there is an obviouslydifferent behavior in the intermediate process when one passesthrough different EPs successively.The above results show that in non-Hermitian systems, theeigenvalue surfaces exhibit a complex-square-root topologywith a branch point named by EP, which can also be consid-ered as a critical point where a transition from PT-symmetricphase to broken-PT-symmetric phase. A consequence of thistopology is that encircling an EP once in the parameter spaceresults in the exchange of both eigenvalues and eigenstates.This means that one has to encircle an EP twice to recover theoriginal eigenvalues and eigenstates. On the other hands, oneof the eigenstates acquires a Berry phase of ± π when encir-cling an EP once and the other one acquires the same phasein the second loop. However, because one not longer distin-guishes the clockwise or anticlockwise direction of the stateevolution, the result of passing through the EPs once will bedifferent. In this system, one acquires this phase in each loopis determined by the chirality of the EP and the evolutionarydirection of the eigenstates in the parameter space. III. REALIZATION OF THE TWO-LEVELPT-SYMMETRIC HAMILTONIAN IN A FOUR-LEVELCOLD ATOMIC SYSTEM
The PT-symmetric Hamiltonian may be difficult to achievein a practical non-Hermitian two-level system. In this sec-tion, we propose to use a four-level Hermitian system to sim-ulate the two-level PT-symmetric Hamiltonian (1). In the ba-sis ( | i , | i , | i , | i ) T , the four-level Hermitian Hamiltonian FIG. 1: Schematic representation of the light-atom interaction con-figuration of the four-level Hamiltonian. The relevant atomic levelsare coupled by microwave radios with corresponding Rabi frequen-cies. takes the form H F = χ α ) i sin( α ) 0cos( α ) 0 0 − i sin( α ) − i sin( α ) 0 0 cos( α )0 i sin( α ) cos( α ) 0 . (10)The corresponding time evolution operator is given by ˆ U F = e − iH F t . For an arbitrary state υ = ( a, b ) T , we can find that: ˆ U F (cid:18) υηυ (cid:19) = (cid:18) ˆ U P T ( t ) 00 η ˆ U P T ( t ) η − (cid:19) (cid:18) υηυ (cid:19) , (11)with η = (cid:18) − i sin( α ) i sin( α ) 1 (cid:19) / cos( α ) . So if we take thefour-level states as ( | E ± ( α ) i , η | E ± ( α ) i ) T , we can simulate the evolution of the states in Eq. (2) in the two-level PT-symmetric Hamiltonian.Now we present an experimentally realizable scheme im-plementing with cold atoms. We consider an atomic cloudof Rb with five internal states in the ground-state mani-fold, noted by | m F i = | F = 1 , m F = − , , i and | m F i = | F = 2 , m F = − , i , which are separated by thehyperfine splitting ω HF and the Zeeman splitting ω Z causedby a uniform static magnetic field. One can apply four mi-crowave radios to couple these atomic levels through the an-nulus configuration as shown in Fig. 1. Here Ω i and ω i denotethe Rabi frequencies and the frequencies of the correspondingmicrowave radios, respectively. In particular, we use resonantmicrowaves to couple | i ←→ | i and | i ←→ | i and useradio-frequency fields to couple | i ←→ | i and | i ←→ | i ,respectively. Supposing that ω ei are the energies of | i i and theenergy of | i is the zero of energy, the total Hamiltonian canbe written as H = H + H int with H = X j ( ω ej − ω e ) | j ih j | ,H int =Ω e iω t | ih | + Ω e iω t | ih | +Ω e iω t | ih | + Ω e iω t | ih | + H.c., (12)In the bare-state basis ( | i , | i , | i , | i ) T , one has V = e − ω t e − ω t
00 0 0 e − ( ω + ω ) t (13)and the Hamiltonian in the rotating frame becomes ˜ H = i dV † dt V + V † HV = Ω ∗ − ω − ω e + ω e e − i ( ω + ω − ω − ω ) t Ω Ω ∗ − ω − ω e + ω e Ω e i ( ω + ω − ω − ω ) t Ω ∗ Ω ∗ − ω − ω − ω e + ω e . (14)To ensure the Hamiltonian being time-independent, wechoose ω + ω = ω + ω . On the other hand, consid-ering the resonance condition ∆ = ω e − ω e − ω = 0 , ∆ = ω e − ω e − ω = 0 , ∆ = ω e − ω e − ω = 0 , ∆ = ω e − ω e − ω = 0 , with ω = ω = ω HF , ω = ω = ω Z .In particular, we choose the corresponding Rabi frequencies Ω = − Ω = iS sin( α ) cos( α ) and Ω = Ω = S cos ( α ) .Under these conditions, the Hamiltonian ˜ H can be representedas H F in Eq. (10): ˜ H = Ω ∗ Ω ∗ ∗ Ω ∗ = H F . (15)Up to this step, we have proposed a method to realize the re- quired four-level Hamiltonian for simulating the two-level PT-symmetric Hamiltonian in a cold atomic system. It is notewor-thy that in this system, we can precisely and easily control thenon-Hermitian parameter α by adjusting Ω i . IV. DETECTING THE BERRY PHASE IN THEPT-SYMMETRIC SYSTEM
In the section, we show how to detect the Berry phasesof the mimicked EPs in the four-level cold atomic system.First, we needs an additional atomic level for this measure-ment, which is denoted by | i in the cold atomic system asshown in Fig. 1. We assume the atoms are initially pumpedto | i and the transitions | i → | i i ( i = 1 , , , can be re- FIG. 2: (Color online) The phase difference and atomic popula-tion versus angle. The red, green, yellow and blue line denotethe | E + ( α ) i , | E − ( π − α ) i , | E − ( α ) i and | E + ( π − α ) i , respec-tively. (a,c) The dynamic phase γ d = 0 . (b,d) The dynamic phase γ d = π/ . alized successively through the stimulated-Raman-adiabatic-passage (STIRAP) [41, 42]. It is noted that the microwave ra-dios must be phase-locking between each STIRAP for keep-ing the coherence between the states. On the other hands,only the phase difference between | i and | i i is needed to beconcerned, so it is nonsignificant what the population differ-ences between | i and other levels are. Under this condiction,we make | i → | i + η | ψ ± ( α ) i T from the very beginning,where η is an arbitrary real number, for the preparation of thePT-symmetric initial state. The phase difference between | i and | i is the only distinction between | E ± ( α ) i for a given α , and the phase differences between | i and | i ( | i ) can beused to detect the Berry phases of | E ± ( α ) i . Thus we can de-tected the Berry phases by measuring the phase differencesbetween the corresponding atomic levels.The phase difference between two atomic levels can bemeasured through the atomic interferometry. For an arbitrarystate denoted by | a i + e − iϕ | b i , a π/ -pulse operation takesthe form U π/ ,φ = (cid:18) − ie − iφ − ie iφ (cid:19) , (16)with φ being a controllable phase of the π/ -pulse. After ap-plying a π/ -pulse to the state | a i + e − iϕ | b i , one can findthe relationship between the atomic populations and the phasedifferences as N a,b = 12 [1 ∓ sin( φ + ϕ )] . (17)Considering Eq. (2) and Eq. (17), the atomic populations ofthe different levels are given by N , ( α ) = ( [1 ∓ sin( φ + α )] / for | E + ( α ) i [1 ∓ sin( φ + π − α )] / for | E − ( α ) i . (18) For a given α , the function of φ in Eq. (18) can be used todistinguish | E + ( α ) i and | E − ( α ) i , so we can verify whetherthe eigenstates exchange themselves through measuring thefunction of φ in Eq. (18) when sweeping an EP.After confirming the two states | E ± ( α ) i , we can measurethe phase difference between | i and | i to determine theBerry phases γ ± for | E ± ( α ) i , which is related to the valueof α and the evolution loop. For | i and | i , we can also findthe relationship between atomic populations and total phasedifferences: N , ( α ) = ( (1 ∓ sin ϕ + ) / for | E + ( α ) i (1 ∓ sin ϕ − ) / for | E − ( α ) i , (19)where the total phases between the corresponding atomic lev-els in the case of different eigenstates ϕ + = γ d − α/ − θ ( α ) π/ and ϕ − = γ d − ( π − α ) / − θ ( α ) π/ with γ d = ∓ S cos( α ) t + φ + ω Z t being the dynamic phase due tothe evolution ˆ U F , the π/ -pulse and the Zeeman energy dif-ference, respectively, and θ ( α ) = θ [ − cos( α )] being a Heav-iside unit step function whose value is 0 for − cos( α ) < and1 for − cos( α ) > . It is clear that the topology of the EPis the source of the Heaviside unit step function. Due to thesymmetry of the trigonometric functions, confirming | E + ( α ) i and | E − ( α ) i in Eq. (19) needs two values of γ d .For simplicity, here we choose the dynamic phase as and π/ , and the phases ϕ ± and the atomic populations N , in the different eigenstates are shown in Fig. (2). For agiven eigenstate, one can measure the phases ϕ ± to obtainthe Berry phases of the eigenstates from Eq. (19). In par-ticular, for a given eigenstate and the parameter α , we firstmeasure N , in Eq. (18) to determine which eigenstate isthrough the function of φ . Then one can detect the total phase ϕ ± by measuring N , in Eq. (19), as shown in Fig. 2. Af-ter that, one can control the systemic parameter α to sweepan EP in parameter space that the eigenstate is predicted toobtain a non-Abelian phase in Eq. (8). Again one can suc-cessively measure N , and N , to determine whether theeigenstate has been changed and the variation of total phase.Here we can find that: i) The eigenstates obtain a phase of + π/ or − π/ alternately when α sweeps through the differ-ent EPs successively. ii) The eigenstates exchange themselveswhen α sweeps through an EP and the phase differences from | E ± ( α ) i to | E ∓ ( π − α ) i are always ± π/ . iii) The eigen-states obtain a Berry phase of ± π when α sweeps ± π . Inshort, the measurement of the phase differences between | i i ( i = 0 , , provides a simple way to experimentally verifyEq. (9) and demonstrate the Riemann sheet structure and theintrinsic properties of the EPs. V. DISCUSSION AND SUMMARY
In the above case, there is no dynamical phase contributionbetween the four states | i i ( i = 1 , , , ), but the evolutionof | i i must be still adiabatic in order to avoid non-adiabatictransitions. To be specific, the four Rabi frequencies shouldkeep the adiabatic cyclic evolutions in the parameter spacewith the change rate of α being significantly smaller than thetypical Rabi frequencies. This means that the evolution periodof the system is much larger than the inverses of the energygaps between the atomic levels. On the other hands, with thenon-adiabatic transition between | E ± ( α ) i being avoided, thechange rate of α is also smaller than χ , which is the differ-ence of eigenvalue. Thus, the adiabatic condition takes theform dαdt ≪ Ω i , χ. (20)To fulfill this condition, one should assure the system remainsin the eigenstate of the Hamiltonian ˜ H all the time.In summary, we have proposed an experimental scheme torealize a two-level PT-symmetric system with parameter be-ing controllable through an embedding four-level cold atomicsystem. We further clarify the different properties between en-circling and passing through EPs. And then we demonstratethat the change of the eigenstates and the relevant topologi- cal phase in the case of passing through EPs can be confirmedby measuring the phases of atomic levels and this novel phe-nomenon can be probed through the standard phase measure-ment. Our work proposes a method to realize PT-SymmetricHamiltonian in clod atomic systems and therefore provides apowerful tool to explore the properties of PT-symmetry andPEs further. Acknowledgements
We thank Profs. Hui Yan and Shi-Liang Zhu and Dr. FengMei for helpful discussions. This work was supported by theNKRDP of China (Grant No. 2016YFA0301803), the NSFC(Grants No. 11604103), the NSF of Guangdong province(Grants No. 2015A030310277, No. 2016A030310462, andNo. 2016A030313436), and the SRFYTSCNU (Grants No.15KJ15 and No. 15KJ16). [1] C. M. Bender and S. Boettcher, Real Spectra of Non-HermitianHamiltonian Having P T Symmetry, Phys. Rev. Lett. , 5243(1998).[2] C. M. Bender, D. C. Brody, and H. F. Jones, Complex Extensionof Quantum Mechanics, Phys. Rev. Lett. , 270401 (2002).[3] C. M. Bender, Making sense of non-Hermitian Hamiltonians,Rep. Prog. Phys. , 947 (2007).[4] T. Kottos, Broken symmetry makes light work, Nat. Phys. ,166 (2010).[5] J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos,Experimental study of active LRC circuits with PT symmetries,Phys. Rev. A , 040101(R) (2011).[6] Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, Experimentalobservation of the dual behavior of PT-symmetric scattering,Phys. Rev. A , 050101(R) (2012).[7] C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N.Christodoulides, M. Segev, and D. Kip, Observation of pari-tyCtime symmetry in optics, Nat. Phys. , 192 (2010).[8] A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev,PT-symmetry in honeycomb photonic lattices, Phys. Rev. A ,021806(R) (2011).[9] A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov , D.N. Christodoulides, and U. Peschel, ParityCtime synthetic pho-tonic lattices, Nature (London) , 167 (2012).[10] S. Bittner, B. Dietz, U. G ¨ u nther, H. L. Harney, M. Miski-Oglu,A. Richter, and F. Sch ¨ a fer, PT Symmetry and SpontaneousSymmetry Breaking in a Microwave Billiard, Phys. Rev. Lett. , 024101 (2012).[11] Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, ExperimentalDemonstration of a Coherent Perfect Absorber with PT PhaseTransition, Phys. Rev. Lett. , 143903 (2014).[12] L. Chang, X.-S. Jiang, S.-Y. Hua, C. Yang, J.-M. Wen, L. Jiang,G.-Y. Li, G.-Z. Wang and M. Xiao, ParityCtime symmetry andvariable optical isolation in activeCpassive-coupled microres-onators, Nat. Photonics , 524 (2014).[13] B. Peng, S. K. ¨ O zdemir, F.-C. Lei, F. Monifi, M. Gianfreda,G.-L. Long, S.-H. Fan, F. Nori, Carl M. Bender and L. Yang,ParityCtime-symmetric whispering-gallery microcavities, Nat. Physics , 394 (2014).[14] L. Feng, Z.-J. Wong, R.-M. Ma, Y. Wang, X. Zhang, Single-mode laser by parity-time symmetry breaking, Science , 972(2014).[15] H. Hodaei, M. Miri, M Heinrich, Demetrios N. Christodoulides,M. Khajavikhan, Parity-timeCsymmetric microring lasers, Sci-ence , 975 (2014).[16] C. Hang, G.-X. Huang and Vladimir V. Konotop, PT Symme-try with a System of Three-Level Atoms, Phys. Rev. Lett. ,083604 (2013).[17] J.-H. Wu, M. Artoni, and G.C. La Rocca, Non-Hermitian De-generacies and Unidirectional Reflectionless Atomic Lattices,Phys. Rev. Lett. , 123004 (2014).[18] C. Hang and G.-X. Huang, Weak-light solitons and their activecontrol in a parity-time-symmetric atomic system, Phys. Rev. A , 043833 (2015).[19] P. Peng, W.-x. Cao, C. Shen, W.-z Qu, J.-m. Wen, L. Jiang,and Y.-h Xiao, Anti-Parity-Time Symmetric Optics via FlyingAtoms, Nat. Phys. 3842 (2016).[20] T. Kato, Perturbation Theory of Linear Operators (Berlin:Springer) (1966).[21] H. Cao and J. Wiersig, Dielectric microcavities: Model systemsfor wave chaos and non-Hermitian physics, Rev. Mod. Phys. ,61 (2015).[22] W.D. Heiss and H.L. Harney, The chirality of exceptionalpoints, Eur. Phys. J. D , 149 (2001).[23] C. Dembowski, B. Dietz, H.-D. Gr ¨ a f, H. L. Harney, A. Heine,W. D. Heiss, and A. Richter, Observation of a Chiral State in aMicrowave Cavity, Phys. Rev. Lett. , 034101 (2003).[24] Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K.An, Quasieigenstate Coalescence in an Atom-Cavity QuantumComposite, Phys. Rev. Lett. ,153601 (2010).[25] W.D. Heiss, Phases of wave functions and level repulsion, Eur.Phys. J. D , 1C4 (1999).[26] W. D. Heiss, Repulsion of resonance states and exceptionalpoints, Phys. Rev. E , 929 (2000).[27] C. Dembowski, H.-D. Gr ¨ a f, H. L. Harney, A. Heine, W. D.Heiss, H. Rehfeld, and A. Richter, Experimental Observation of the Topological Structure of Exceptional Points, Phys. Rev.Lett. , 787 (2001).[28] C. Dembowski, B. Dietz, H.-D. Gr ¨ a f, H. L. Harney, A. Heine,W. D. Heiss, and A. Richter, Encircling an exceptional point,Phys. Rev. E , 056216 (2004).[29] J. Doppler, A. A. Mailybaev, J. B?hm, U. Kuhl, A. Girschik, F.Libisch, T. J. Milburn, P. Rabl, N. Moiseyev and S. Rotter, Dy-namically encircling an exceptional point for asymmetric modeswitching, Nature , 76 (2016).[30] H. Xu, D. Mason, L. Jiang and J. G. E. Harris, Topologicalenergy transfer in an optomechanical system with exceptionalpoints, Nature , 80 (2016).[31] I. Rotter, A non-Hermitian Hamilton operator and the physicsof open quantum systems, J. Phys. A , 153001 (2009).[32] H. Eleuch, I. Rotter, Quantum states talk via the environment,Eur. Phys. J. D
229 (2015).[33] B. Wahlstrand, I. I. Yakimenko, and K.-F. Berggren, Wavetransport and statistical properties of an open non-Hermitianquantum dot with parity-time symmetry, Phy. Rev. E ,062910 (2014).[34] Uwe G ¨ u nther, and Boris F. Samsonov, Naimark-Dilated PT-Symmetric Brachistochrone, Phys. Rev. Lett. , 230404(2008).[35] X. Zhang, Yangchao Shen, J.-h. Zhang, J. Casanova, L. Lamata, E. Solano, M.-H. Yung, J.-N. Zhang and K. Kim, Time reversaland charge conjugation in an embedding quantum simulator,Nat. Commun. , 7917 (2015).[36] C. M. Bender, D. C. Brody and H. F. Jones, Must a Hamiltonianbe hermitian?, Am. J. Phys. , 1095 (2003).[37] P. M. Morse and H. Feshbach. Methods of Theoretical Physics.McGraw-Hill, New York, (1953).[38] S. W. Kim, T. Cheon, and A. Tanaka, Exotic quantum holonomyinduced by degeneracy hidden in complex parameter space,Phys. Lett. A , 1958 (2010).[39] S.-L. Zhu, B.-G. Wang, and L.-M. Duan, Simulation and de-tection of Dirac fermions with cold atoms in an optical lattice,Phys. Rev. Lett. , 260402 (2007).[40] D.-W. Zhang, Z. D. Wang, and S.-L. Zhu, Relativistic quantumeffects of Dirac particles simulated by ultracold atoms, Front.Phys. , 31 (2012).[41] P. Kral, I. Thanopulos, and M. Shapiro, Coherently controlledadiabatic passage. Rev. Mod. Phys. , 53 (2007).[42] Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W.Huang, X. Chen, H. Yan, and S.-L. Zhu, Experimental real-ization of stimulated Raman shortcut-to-adiabatic passage withcold atoms, Nat. Commun.7