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Phases of Wave Functions and Level Repulsion W.D. Heiss Centre for Nonlinear Studies and Department of Physics University of the Witwatersrand, PO Wits 2050, Johannesburg, South Africa PACS: 03.65.Bz, 02.30.Dk, 84.40.-x occur[7],but,dependingontheparticularsituation,they E mayaccumulateatonepointinthefiniteplane[8];there are2s.p2ecial cases, where no exceptional point occurs in Avoided level crossings are associated with exceptional the finite plane as canbe seen fromthe analytic solution 2 points which are thesingularities of the spectrum and eigen- for the single particle motion in the Hulthen potential functions, when they are considered as functions of a cou- [9].1.8 9 pling parameter. It is shown that the wave function of one 9 state changes sign but not theother, if theexceptional point 1.6 9 is encircled in the complex plane. An experimental setup is 1 1.4 suggestedwherethispeculiarphasechangecouldbeobserved. n 1.2 a J λ 1 The phase of the wave function changes in a charac- 0.1 0.2 0.3 0.4 0.5 0.6 1 teristic way, if a selfadjoint Hamiltonian has an energy FIG. 1. Level repulsion using in Eq.(3) ǫ1 = 1, ǫ2 = 2, degeneracy at specific values in some parameter space ω1=2, ω2 =−1and φ=π/25. 1 andifalooparoundthe pointofdegeneracyis described v in parameter space [1]. In the simple case of a real sym- All essentialaspects of exceptionalpoints canbe illus- 3 2 metric Hamiltonian two parameters are needed to get a trated in a two level model on an elementary level. In 0 diabolic point. Inthis case Berry’sphaseis π whenloop- fact, for finite or infinite dimensional problems an iso- 1 ingaroundthedegeneracyintwodimensionalparameter latedexceptionalpointcanbe describedlocallyby atwo 0 space. dimensional problem [10]. In other words, even though 9 If a selfadjoint Hamiltonian depends only on one pa- a high or infinite dimensional problem is globally more 9 / rameter, its variation will in generalgive rise to level re- complex than the two dimensional problem, we do not h pulsion [2]. Associated with a level repulsion is a pair of loosegeneralityforourspecificpurposewhentherestric- p exceptionalpoints[3]whicharethepointswherethetwo tion to a two dimensional problem is made. For easy - t levels actually coalesce when continued analytically into illustration we therefore confine ourselves to the discus- n a the complex plane of the parameter [4]. In the present sion of u letterwediscussthebehaviourofthewavefunctionsand q their phases when an exceptional point is encircled. We H = ǫ1 0 +λU ω1 0 U† (1) : (cid:18) 0 ǫ2(cid:19) (cid:18) 0 ω2(cid:19) v suggest an experimental situation where such behaviour i could be measured. It is distinctly different from a dia- X with bolic pointwhereBerry’sphaseoccurs. Inparticular,an r a exceptionalpointmustnotbeconfusedwithacoinciden- cosφ −sinφ U = . (2) taldegeneracyofresonancestates, whichwasconsidered (cid:18)sinφ cosφ (cid:19) in [5] as a generalisation of Berry’s phase. This is, up to a similarity transformation, the most gen- The difference between a diabolic point and an excep- eral form of a real two dimensional Hamilton matrix of tional point is due to the selfadjointness of the Hamilto- nian in the former and the lack of it in the latter case. the type H0 +λH1. The particular dependence on the parameter λ has been chosen as it is of a nature widely Also,whencontinuing intothe complexparameterplane used in physical applications. We emphasise that our we are faced with analytic functions of a complex vari- aimisnotinparticulardirectedataphysicalmodelthat able which implies a more rigid mathematical structure. is describable by a two dimensional problem although In the references quoted above a thorough discussion is there may exist interesting problems in our special con- given of the spectrum and eigenfunctions, when one pa- text. The restriction has been chosen for illustration, rameter on which the Hamiltonian depends is continued whilethephysicalapplicationthatwehaveinmindisan intothecomplexplane. Theparameterchosencanbean interactionstrengthasconsideredinthispaper,butother choices are possible [6]. Generically, an N-dimensional matrix problem yields N(N −1) exceptional points. For an infinite dimensional problem, an infinite number can 1 infinite dimensional situation. λ =− ǫ1−ǫ2 exp(±2iφ). (5) c ω1−ω2 Im E Atthese points,the twolevelsE (λ) areconnectedby k a square rootbranch point, in fact the two levels are the 0.06 values ofone analyticfunction on twodifferent Riemann sheets. In Fig.2a we display contours in the complex 0.04 energyplaneforeachlevel,whichareobtainedifaloopis describedinthecomplexλ-plane,whichdoesnotencircle theexceptionalpoint. Correspondingly,Fig.2bshowsthe 0.02 contours,iftheexceptionalpointisencircled. Inthiscase only a double loop in the λ-plane yields a closed loop in Re E theenergyplane. Obviously,thisconnectionisnotofthe 1.55 1.65 1.7 1.75 type encountered at a genuine diabolic point. Im E Thedifferencehasabearingalsoonthescatteringma- trix [11] and on the wave functions ψ1(λ) and ψ2(λ). In 0.08 [11], although the notion exceptional points is not used, thepertinentdistinctionbetweenagenuinedegeneracyof 0.06 tworesonancesandthe analyticcoalescence(exceptional point) of two (complex) eigenvalues is nicely discussed. 0.04 A usual degeneracyof two resonancesstill givesrise to a simple pole in the scattering matrix or Green’s function, 0.02 whileanexceptionalpointproducesadouble pole. With regardtotheeigenfunctions,werecallthatforcomplexλ Re E theHamiltonianisnolongerselfadjoint. Thismeansthat 1.55 1.65 1.7 1.75 the eigenfunctions are no longer orthogonal. We rather FIG.2. Contours in the complex energy plane. The top have a biorthogonal system which can be normalized as illustratescontoursforeachlevel,ifaclosedloopisdescribed ibnottthoemλi-lplulasntreatweisththoeuteneenrcgiyrccloinngtotuhreperxocdeupcteiodnbayltpwooin(te.qTuahle) hψ˜1(λ)|ψ2(λ)i=δ1,2 (6) closed loops in the λ-plane, which encircle the exceptional where |ψ i and hψ˜ | are the right hand and left hand point. The solid line corresponds to the first closed loop in k k eigenvectors of H, respectively. Note that Eq.(6) causes the λ-plane and the dotted line to the subsequent loop. The problems at the exceptional point, since it is exactly at soliddotisthepositionwherethecontourswouldmeet,ifthe loops would cross theexceptional point in the λ-plane. this point where two linearly independent eigenfuctions no longer exist. This is in contrast to a genuine degen- The eigenvalues of H are given by eracyofa selfadjointoperatorwhere ak-folddegeneracy always gives rise to a k-dimensional eigenspace. At the E1,2(λ)= ǫ1+ǫ2+λ(ω1+ω2) ±R (3) exceptional point, not only the eigenvalues but also the 2 eigenfunctions coalesce. As a consequence, the orthog- where onality conflicts with the normalization, in other words, if Eq.(6) is enforced globally, that is also at λ = λ , the ǫ1−ǫ2 2 c R= ( ) components of the wave function have to blow up. This (cid:26) 2 canbemadeexplicitbyparametrizingthewavefunctions 1/2 λ(ω1−ω2) 2 1 by the complex angle θ, viz. +( ) + λ(ǫ1−ǫ2)(ω1−ω2)cos2φ . (4) 2 2 (cid:27) cosθ −sinθ Clearly, when φ = 0 the spectrum is given by the two ψ1(λ)=(cid:18)sinθ(cid:19), ψ2(λ)=(cid:18) cosθ (cid:19) (7) lines with 0 E (λ)=ǫ +λω , k =1,2 k k k 2 E1(λ)−E2(λ)−(ǫ1−ǫ2)−λ(ω1−ω2)cos2φ which intersect at the point of degeneracy λ = −(ǫ1 − tan θ(λ)= . E1(λ)−E2(λ)+(ǫ1−ǫ2)+λ(ω1−ω2)cos2φ ǫ2)/(ω1−ω2). When the coupling between the two levels is turned on by switching on φ, the degeneracy is (8) lifted and avoided level crossing occurs as is illustrated in Fig.1. Now the two levels coalesce in the complex λ- At λ = λc it is E1 = E2 and hence tan2θ = −1 which implies |cosθ| = |sinθ| = ∞, that is the components of plane where R vanishes which happens at the complex 2 the wavefunctions blowup. (Note thattan θ ≡0inthe conjugate points 2 trivialcaseφ=0). Theincreaseofthecomponentsofthe Eq.(1)istraditionallyreplacedby−iGwithrealabsorp- wavefunctions while approachingexceptionalpoints has tionparameterG. With the replacementthe eigenvalues been used in similar contextas a theoreticalsignatureof acquire imaginary parts which are related to the inverse a phase transition [8], but we do not believe that it has life times of the states of the open system. The excep- observational consequences. tional points appear now in the complex G-plane at We now study the behaviour of the wave functions in ǫ1−ǫ2 more detail for two contours in the λ-plane which start, Gc =− exp(±2iφ+iπ/2). (12) ω1−ω2 say,atλ=0andendatlargerealvaluesofλ,butenclose If the coupling of the two channels is equal, that is if an exceptional point between the two contours. For the φ = π/4, the two exceptional points lie on the real G- complex angle θ we choose an expression which is more axis at convenient for this purpose, viz. ǫ1−ǫ2 G =± . tanθ(λ)= λ(ω1−ω2)sin2φ . c ω1−ω2 E1(λ)−E2(λ)+ǫ1−ǫ2+λ(ω1−ω2)cos2φ If the coupling is nearly equal, they lie just above or be- (9) lowtherealaxisdependingonthecouplingbeingslightly weaker or stronger (φ < π/4 or φ > π/4). Controlling The first path can be taken along the real λ-axis. From theabsorptionparameterGandtherelativecouplingen- Eq.(9) we read off the expected result that θ(0)=0 and ables one to pass the exceptional point on a path below θ(λ) → φ for λ ≫ |(ǫ1 −ǫ2)/(ω1 −ω2)|. In obtaining and above, which is the situation described above. A this result use is made of E1 −E2 = 2R → λ(ω1 −ω2) setup similar to the one proposed by Berry [1] and im- for λ ≫ |(ǫ1 −ǫ2)/(ω1 −ω2)|. For the second path we plemented in [13] should make feasible a distinction of move into the upper λ-plane in order to pass above the the two cases. In a two dimensionalelectromagnetic res- exceptional point before returning down to the real axis onatortherelativecouplingoftwosuitablelevels,thatis again. Using again Eq.(9) we now have to observe that levels that do coalesce, can be controlled by a judicious we crossed into the other sheet which means E1−E2 = choice of the geometry of the resonator [13]. The global −2R → −λ(ω1 − ω2). As a consequence we find this absorptioncanbecontrolledbyradiationlossesregulated time tanθ = −cotφ = tan(φ+π/2). Surely we would bysuitableantennas. Inthiswaythechangeofthephase expect the wave functions to interchange just like the of one but not the other wavefunction should be observ- energies, if an exceptional point is encircled. But our able. Note that, for large values of G, one state will be finding indicates that one wave function has changed its much broader than the other [8]. sign. In fact, we obtain along the second path We stress again that the occurrence of exceptional points is a generic mathematical feature associated with ψ1 →ψ2, ψ2 →−ψ1. (10) any system that has avoided level crossing. While this presentation used a two dimensional illustration, the re- Note that this result is equally obtained, if a closed con- sults can be immediately generalised. In fact, if more tour which encloses the exceptional point is described in than one exceptional point is encircled, the resulting the λ-plane. Of interest is the result of a double loop in phase change is simply a combination of several two– the λ-plane. It yields dimensional cases. The findings presented in this paper ψ1 →−ψ1, ψ2 →−ψ2 (11) have therefore universal significance. Whether the particular phase changes of the respective wave functions which is, in accordance with the corresponding single aredetectable,issubjecttoexperimentation. Duetothe loop in the energy plane, just Berry’s phase retrieved. short life time involved for one of the states, scattering We mention that this implies that the wave functions, if experiments, where a steady incoming flux can be main- parametrized as in Eq.(7), have an algebraic singularity tained, are expected to be particularly suitable. Fortu- that is determined by a fourth root at the exceptional itously, in the spirit of the experiment performed by [13] point; only a four-fold loop in the λ-plane restores com- the adiabatic change of the parameters is not essential, pletely the original situation. since it is the movement of the nodes of the wave func- Next we address the question concerning the physi- tions that are being observed under parameter variation cal significance of these findings. It boils down to the [14]. Possiblejumps ofthe phasesdo thereforenotaffect problem of varying a complex interaction parameter in the result of interest. the laboratory. Problems of a similar nature have been discussed in connection with Berry’s phase for dissipa- tive systems [12], but not as yet for exceptional points. We are guided by the phenomenological description of open quantum systems [8] and submit as one suggestion a strongly absorptive system, where the parameter λ in 3 [1] M.V.Berry,Quantum Chaos, ed.byG.Casati (London: Plenum) 1985; Proc. R. Soc. A 239, 45 (1983) [2] J. von Neumann and E. Wigner, Z.Phys. 30, 467 (1929) [3] T.Kato,PerturbationtheoryoflinearoperatorsSpringer, Berlin 1966 [4] W.D.HeissandA.L.Sannino,J.Phys.A231167(1990); W.D. Heiss and A.L. Sannino, Phys. Rev. A 43 4159 (1991) [5] C. Mimiatura, C. Sire, J. Baudon and J. Bellissard, Eur.Lett. 13, 199 (1990) [6] C. M. Bender and S. Boettcher, Phys.Rev.Lett. 80 5243 (1998); S.C. Creagh and N.D. 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