Exotic quantum holonomy in Hamiltonian systems
EExotic quantum holonomy in Hamiltonian systems
Taksu Cheon ∗ ,a , Atushi Tanaka b , Sang Wook Kim c a Laboratory of Physics, Kochi University of Technology Tosa Yamada, Kochi 782-8502, Japan b Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan c Department of Physics Education, Pusan National University, Busan 609-735, South Korea
Abstract
We study the evolution of quantum eigenstates in the presence of level crossing under adiabatic cyclic change of environmentalparameters. We find that exotic holonomies, indicated by exchange of the eigenstates after a single cyclic evolution, can arise fromnon-Abelian gauge potentials among non-degenerate levels. We illustrate our arguments with solvable two and three level models.
Key words: geometric phase, non-Abelian gauge potential, adiabatic quantum control
PACS:
1. Introduction
Berry phase phenomena are known to be wide-spread, in-triguing, and useful in controlling quantum systems [1, 2]. TheWilczek-Zee variation, in which di ff erent eigenstates sharinga degeneracy are turned into each other after the cyclic varia-tion of environmental parameter [3], is found particularly use-ful, since it supplies the basis for so-called holonomic quantumcomputing [4]. Whether such state transformations require theexistence of degeneracy throughout the parameter variation, isa matter in need of further analysis, although it has been thewide-spread assumption.The exotic holonomies are defined as exchange of eigen-states after one period of cyclic parametric evolution withoutany relevant degeneracy. They have been found in time-periodicsystems [5, 6] and in singular systems [7, 8], but not in fi-nite Hamiltonian systems up to now. One obvious reason isthat the two real numbers in a real axis cannot be continuouslyexchanged without colliding with each other, i.e. degeneracy.Note that in a time periodic system described by a unitary ma-trix, its eigenvalues are complex number on the unit circle in thecomplex plane so that the eigenvalues smoothly exchange theirposition without crossing over each other. Once level cross-ing is allowed during the variation of environmental parameter,even in Hamiltonian system there is a possibility of the eigen-levels being exchanged after the cyclic parameter variation evenamong non-degenerate levels, which would be instrumental inenlarging the resource for holonomic quantum control.It appears that this is related to a myth that adiabatic param-eter variation excludes level crossing, and therefore, there isno possibility for exotic quantum holonomies for non-singularHamiltonian system. An argument often raised against systems ∗ corresponding author Email addresses: [email protected] (Taksu Cheon ), [email protected] (Atushi Tanaka), [email protected] (Sang Wook Kim) with level crossing is that it is not generic, and it represents a setof measure zero in parameter space of all systems. However,systems of our interest often lives in a world with some sym-metry, exact or approximate. Under certain circumstances, asystem can always pass through the point of symmetry when anenvironmental parameter is varied along a circular path. Then,the level crossing become a common feature rather than an ex-ception.Surprisingly, it has been known for quite sometime that theadiabatic theorem is extendable to the case of crossing levels[9]. The eigenstates change smoothly as functions of environ-mental parameter even when the level crossing takes place, andthis opens up the possibility for eigenvalue holonomy for non-singular Hamiltonian systems. In this article, we explicitly con-struct such models, which seem to have immediate extension to N level cases. The solvability of our model allows us to ex-amine the analytic structure of the gauge potentials, which isknown to be the mathematical origin behind the existence ofexotic holonomy [10, 11].
2. Adiabatic level crossing and exotic holonomy
Consider a Hamiltonian system with an environmental pa-rameter, which we call θ . We assume, for the moment, that theeigenvalues E n ( θ ) and eigenstates Ψ n ( θ ) are non-degenerate forall possible values of the parameter. Let us suppose two levels qDD Figure 1: Schematic diagram showing adiabatic level-crossing. ∆ and D repre-sent energy gaps. Preprint from
Kochi University of Technology
November 4, 2018 a r X i v : . [ qu a n t - ph ] S e p ery closely approach each other at a certain value of the pa-rameter, which can be thought of as an avoided crossing withthe closest energy gap ∆ . Let us further assume that the two lev-els are separated apart from the next closest level (See Fig. 1)by D . Consider a smooth cyclic variation of θ ( t ) with a period τ , namely, θ ( τ ) = θ (0). Specifically, we require that θ ( t ) startssmoothly at the beginning t = t = τ [12] toensure the applicability of the Landau-Zener formula [13, 14].Let us assume that we have inequalities,1 D (cid:28) τ (cid:28) ∆ . (1)The period τ is small enough compared with the inverse of theenergy gap ∆ so that the levels completely cross over the gapduring the parametric variation, while it is large enough to ig-nore any transition among levels except the interacting two lev-els considered here, which is the reason why we call this pro-cess adiabatic. The situation remains intact even when the lev-els do cross according to some exact symmetry, ∆ =
0, insteadof showing avoided crossing. In fact, although a tiny avoidedcrossing caused by a slight symmetry-breaking takes place, theafore mentioned adiabatic level cross-over is robust irrespectiveof small parametric perturbation. This is the physics behind theso-called adiabatic level crossing [9].Once such an adiabatic level crossing occurs, there is noreason to assume that each quantum eigenstate should comeback to the corresponding initial state after a cyclic parametricvariation. Only requirement is that the entire set of eigenstatesshould be the same as before, since the solutions of eigenvalueequation with a given parameter are uniquely determined. Itis thus allowed that the two levels are exchanged after the para-metric variation. With a parameter θ , which describes the cyclicparametric variation along the path C from θ i to θ f satisfying H ( θ i ) = H ( θ f ), such a transition is described by the holonomymatrix M as [15, 16] Ψ n ( θ f ) = (cid:88) m M n , m Ψ m ( θ i ) e − i φ m (2)with M = T ∗ e − i (cid:82) C d θ A ( θ ) T e i (cid:82) C d θ A D ( θ ) (3)where T and T ∗ represents the path-ordering and anti-path or-dering of operator integrals, the A ( θ ) is the non-Abelian gaugepotential A n , m ( θ ) = (cid:104) Ψ n ( θ ) | i ∂ θ Ψ m ( θ ) (cid:105) , (4)and A D ( θ ) its diagonal reduction A Dn , m ( θ ) = A n , n ( θ ) δ n , m . (5)The dynamical phase φ n depends on the precise history of theparameter variation, while the holonomy matrix M is solely de-termined by the geometry of the path C in the parameter space.Non-zero o ff -diagonal component of M , if any, signifies the ex-istence of exotic holonomy. Physical requirement that an eigen-state does not split with adiabatic parameter variation limits theform of M to be permutation matrix supplemented by possible Manini-Pistolesi o ff -diagonal phases [17, 18] for each non-zeroelements. Namely, there is only a single non-zero entry to eachraw and each column, and the absolute value of this entry isone.
3. Two-level model with exotic holonomy
Consider a two level quantum system described by a para-metric Hamiltonian H ( θ ) = R ( θ ) (cid:20) cos θ Z (2) + v sin θ F (2) (cid:21) (6)with a real number v , Z (2) = σ z = (cid:32) − (cid:33) , F (2) = I (2) + σ x = (cid:32) (cid:33) , (7)and anti-periodic function R ( θ ) with period 2 π , R ( θ + π ) = − R ( θ ) . (8)A convenient choice we adopt in the numerical examples is R ( θ ) = cos θ . The system thus becomes 2 π periodic; H ( θ + π ) = H ( θ ) , (9)and the parameter θ ∈ [0 , π ) forms a ring, S . Note that if (6)is written in the form H ( x , y ) = x Z (2) + y vF (2) , the parametricevolution θ : 0 → π represents a circle on ( x , y ) plane with itscenter shifted by the radius into x axis, namely x = + cos θ and y = sin θ , so that it touches the origin. If we were to vary R ∈ ( −∞ , ∞ ) and θ ∈ [0 , π ) independently, the entire plane( x , y ) is covered.The eigenvalue equation H ( θ ) Ψ n ( θ ) = E n ( θ ) Ψ n ( θ ) (10)is analytically solvable with eigenvalues given by E n ( θ ) = R ( θ ) cos θ P n ( θ ) , (11)and eigenstates Ψ n ( θ ) = (cid:112) P n ( θ ) + (cid:32) P n ( θ ) + P n ( θ ) − (cid:33) , (12)for n = ,
2, with P n ( θ ) = v tan θ + ( − ) n sgn (cid:20) v cos θ (cid:21) (cid:114) + v tan θ . (13)Energy eigenvalues as a function of θ are shown in Fig.2, inwhich the most notable feature is the occurrence of degeneracyat θ = π and the related exchange of eigenvalues. It guaranteesthat the set of eigenvalues at θ = θ = π ,which is a direct result of H (0) being identical to H (2 π ). Thedegeneracy of eigenvalues at θ = π is a direct consequence ofvanishing Hamiltonian, H ( π ) = P n ( θ ) is 4 π -periodic. Moreover, we have P ( θ + π ) = P ( θ ) and P ( θ + π ) = P ( θ ), which leads to the appear-ance of exotic holonomy, i.e., E ( θ + π ) = E ( θ ) , E ( θ + π ) = E ( θ ) , (14)2 E q v=0.577 Figure 2: Energy eigenvalues E n ( θ ) of the model (6) as a function of θ with v = √ . and also, Ψ ( θ + π ) ∝ Ψ ( θ ) and Ψ ( θ + π ) ∝ Ψ ( θ ).The structure of the eigenstates becomes clearer with there-parameterization of P n ( θ ) with new angle variable χ = χ ( θ ),which we define as P n ( χ ) = tan χ + (2 n − π , (15)namely, P ( χ ) = tan χ − π and P ( χ ) = tan χ + π . The monotonouslyincreasing function χ ( θ ) maps θ ∈ [0 , π ) to χ ∈ [0 , π ). Theeigenstates is written, with the new angle parameter χ , as Ψ n ( χ ) = (cid:32) sin χ − (2 n − π − cos χ − (2 n − π (cid:33) , (16)namely Ψ ( χ ) = (cid:32) − sin χ cos χ (cid:33) , Ψ ( χ ) = (cid:32) cos χ sin χ (cid:33) . (17)Note that Ψ n ( θ ) is 8 π -periodic , with anti-periodicity of period4 π . This is not immediately evident from the expression (12)which is in fact discontinuous at several values of θ = n π , andhad to be amended with factor − sign(sin θ − ( − ) n π ) to turn into(16). A numerical example of eigenstates as functions of θ isdepicted in Fig. 3. Y -1.0-0.50.00.51.0 Y q v=0.577 Figure 3: Eigenstates Ψ ( θ ) (bottom) and Ψ ( θ ) (top) of the model (6) with R ( θ ) = cos θ , and v = √ . The solid and the dashed lines represent the up-per and the lower component of the eigenvectors, respectively, both of whichare chosen to be real. The range indicated by thick lines represents a singleperiod θ ∈ [0 , π ]. The values outside of this range are shown to display theperiodicities and mutual relations of Ψ ( θ ) and Ψ ( θ ) Adiabatic change of eigenstates are determined by the gaugepotential A nm ( θ ) given by A ( θ ) = (cid:34) − ii (cid:35) f ( θ ) (18) where f ( θ ) = ∂χ ( θ ) ∂θ . (19)In Fig. 4, we depicts two examples of function f ( θ ). Obviously,we have (cid:82) π d θ f ( θ ) = π , and we obtain the holonomy matrix M = (cid:34) − (cid:35) (20)showing the exotic holonomy with Manini-Pistolesi phase ( − n . f q v=0.577 Figure 4: Functional form of gauge potential f ( θ ) of the model (6) with v = √ . pπ pπ (cid:15459) - (cid:15459) Re qθ Im qθ pπ pπ (cid:15459) - (cid:15459) Re qθ Im qθ Figure 5: Exceptional points on the Mercator projection of the complex θ planeof system described by (6) with v > v < A ( θ ), whilethe unfilled crosses are the points of eigenvalue degeneracy which has no e ff ecton the singular behavior of A ( θ ). The solid lines are the branch cuts on whichRe( E − E ) =
0. In the limit v →
1, the two complex exceptional points (thefilled crosses) move to ± i ∞ . Nontrivial holonomy is known to be related to the analyticstructure of the gauge potential A ( θ ) in the complex θ plain [10],specifically, its singularities. These singularities can arise at theexceptional points θ (cid:63) , which is defined as the point where twocomplex energy coincide; E m ( θ (cid:63) ) − E n ( θ (cid:63) ) =
0. In our example, R ( θ (cid:63) ) cos θ (cid:63) (cid:16) P ( θ (cid:63) ) − P ( θ (cid:63) ) (cid:17) = θ (cid:63) = π, θ (cid:63) ± = ∓ iv ) , (22)where the first one coming from R ( θ (cid:63) ) cos θ (cid:63) =
0, and the θ (cid:63) ± from P ( θ (cid:63) ) − P ( θ (cid:63) ) =
0. The exceptional points coming from R ( θ (cid:63) ) cos θ (cid:63) = A ( θ ), while at θ (cid:63) ± we have the poles of A ( θ ) in the form A ( θ ) = − A ( θ ) ≈ ∓ i θ − θ (cid:63) ± ( θ → θ (cid:63) ± ) . (23)The existence of the poles explains the non-vanishing valuesof (cid:72) d θ A ( θ ) and (cid:72) d θ A ( θ ) around the real axis, i.e. M ,3 (cid:44) E n ( θ ) in Mercator representation of complex parameterplane θ .
4. Three-level model with exotic holonomy
Let us now consider a three level quantum system describedby a parametric Hamiltonian H ( θ ) = R ( θ ) (cid:20) cos θ Z (3) + v sin θ F (3) (cid:21) (24)where v is real, Z (3) = − , F (3) = , (25)and R ( θ ) is anti-periodic with period 2 π , i.e. R ( θ + π ) = − R ( θ ) . (26)As before, in the numerical examples, we adopt R ( θ ) = cos θ .The system then becomes 2 π periodic; H ( θ + π ) = H ( θ ) , (27)and the parameter θ ∈ [0 , π ) forms a ring, S . -2-1012 E q v=1 Figure 6: Energy eigenvalues E n ( θ ) of the model (24) with v =
1. Exoticeigenvalue holonomy is clearly observed.
The eigenvalue equation H ( θ ) Ψ n ( θ ) = E n ( θ ) Ψ n ( θ ) (28)is analytically solvable with eigenvalues given by E n ( θ ) = R ( θ ) cos θ Q n ( θ ) , (29)and eigenstates Ψ n ( θ ) = (cid:112) Q n ( θ ) + Q n ( θ )( Q n ( θ ) + Q n ( θ ) − Q n ( θ )( Q n ( θ ) − , (30)for n = , ,
3, with Q n ( θ ) = sgn (cid:16) cos η ( θ )2 (cid:17)(cid:113) − sin η ( θ )2 ) (cid:34) sin η ( θ )2 − η ( θ ) − η n (cid:35) , (31) in which the angle η ( θ ) is defined by η ( θ ) = v tan θ + v tan θ , (32)and the state dependent shift η n = (2 n − π . The function η ( θ )is monotonously increasing and maps θ ∈ [0 , π ) to η ∈ [0 , π ).One example of the energy eigenvalues as function of environ-mental parameter θ is shown in Fig. 6. All eigenvalues are de-generate at θ = π , as a consequence of vanishing Hamiltonian H ( π ) = Q n ( θ ) is 6 π -periodic. Moreover, we have Q n ( θ + π ) = Q n + ( θ ), where the subscripts are to be understood in thesense of modulo three, which clearly signifies the existence ofexotic holonomy, E n ( θ + π ) = E n + ( θ ) , ( n = , , . (33) Y -1.0-0.50.00.51.0 Y -1.0-0.50.00.51.0 Y q v=1 Figure 7: Eigenstates Ψ ( θ ) (bottom), Ψ ( θ ) (middle) and Ψ ( θ ) (top) of themodel (24) with R ( θ ) = cos θ , and v =
1. The solid, the dotted, and thedashed lines represent the upper, the middle, and the bottom components ofthe eigenvectors, respectively, all of which are chosen to be real. See also thecaption of Fig. 3.
The structure of the eigenstates becomes clearer with there-parameterization of Q n ( θ ) with a new angle variable ξ = ξ ( θ ); Q n ( ξ ) = Rt ξ + η n , (34)namely, Q ( ξ ) = Rt [( ξ − π ) / Q ( ξ ) = Rt [ ξ/
6] and Q ( ξ ) = Rt [( ξ + π ) / π -periodic function Rt is defined byRt ξ = sgn(tan ξ ) (cid:112) | tan ξ | = Rs ξ Rc ξ , (35)along with 2 π -periodic functions Rs and Rc defined asRs ξ = sgn(sin ξ ) (cid:112) | sin ξ | , Rc ξ = sgn(cos ξ ) (cid:112) | cos ξ | . (36)The functions Rt, Rs and Rc are analytic on a single sheetcomplex θ plane in contrast to √ tan ξ , (cid:112) sin ξ , and √ cos ξ , re-spectively, which are analytic on a double sheet θ plane. The4onotonously increasing function ξ ( θ ) maps θ ∈ [0 , π ) to ξ ∈ [0 , π ). The eigenstates is written, with the new angle parame-ter ξ , as Ψ n ( ξ ) = Rs ξ + η n (cid:16) Rs ξ + η n − Rc ξ + η n (cid:17) Rs ξ + η n − Rc ξ + η n Rs ξ + η n (cid:16) Rs ξ + η n i − Rc ξ + η n (cid:17) . (37)From this from, we see that Ψ n ( θ ) is 6 π -periodic .The gauge potential A nm ( θ ), which determines the adiabaticvariation of eigenstates, is given by A ( θ ) = − i i g ( θ + π ) (38) − − i i g ( θ ) + − ii g ( θ − π ) , where g ( θ ) is defined by g ( θ ) = ∂ξ ( θ ) ∂θ (cid:104) Ψ ( ξ ( θ )) | i ∂ ξ Ψ ( ξ ( θ )) (cid:105) . (39)The calculation of the holonomy matrix involves fully orderedmatrix integral, thus no simple analytical calculation can be per-formed. However, we can deduce from (37), that it is given by M = (40)showing the spiral type exotic holonomy with Manini-Pistolesiphase 1 for all states. g q v=1 Figure 8: Functional form of gauge potential g ( θ ) (solid line), g ( θ − π ) (dashedline) and g ( θ + π ) (dotted line) of the model (24) with v = As in the two level case, we examine exceptional points θ (cid:63) where two complex energies coalesce, E m ( θ (cid:63) ) − E n ( θ (cid:63) ) = θ (cid:63) = π, θ (cid:63) (12) ± , θ (cid:63) (23) ± , (41)where the first solution comes from cos θ (cid:63) =
0, while θ (cid:63) (12) ± and θ (cid:63) (23) ± are obtained from P ( θ (cid:63) ) − P ( θ (cid:63) ) = P ( θ (cid:63) ) − P ( θ (cid:63) ) =
0, respectively. We then immediately obtain θ (cid:63) (12) ± = − (cid:32) (cid:113) e ± i π − v (cid:33) ,θ (cid:63) (23) ± = (cid:32) (cid:113) e ∓ i π − v (cid:33) . (42)Near the exceptional points, A ( θ )s are approximated as A jk ( θ ) = − A jk ( θ ) ≈ ∓ i θ − θ (cid:63) ( jk ) ± ( θ → θ (cid:63) ( jk ) ± ) . (43) The existence of the poles explain the non-vanishing values of M , M , and M , implying the existence of exotic holonomy.Fig. 9 shows the locations of poles and branching structure ofenergy surface E n ( θ ) in Mercator representation of the complexparameter plane θ . (cid:15459) - (cid:15459) Re qθ Im qθ pπ pπ Figure 9: Exceptional points on the Mercator projection of the complex θ planeof system described by (24). The filled crosses represent the exceptional pointsthat are the poles of gauge potential A ( θ ), while the unfilled cross is the point ofeigenvalue degeneracy having no e ff ect on the singular behavior of A ( θ ). Thesolid and the dashed lines represent the branch cuts satisfying Re( E − E ) = E − E ) =
0, respectively.
All the results obtained here do not depend on the specificchoice of the matrices (25). In fact, we can enlarge our modelby replacing the two matrices Z (3) and F (3) by Z (3) = Σ + c Σ , F (3) = I (3) + c Σ + c Σ + c Σ , (44)where c j are real numbers, I (3) three-dimensional unit matrixand Σ j given by Σ = , Σ = , Σ = , Σ = − , Σ = √ − . (45)If we were to make independent choice of six parameters, R ∈ ( −∞ , ∞ ), θ ∈ [0 , π ), c j ∈ ( −∞ , ∞ ), ( j = , , , c j s and binding R and θ by R ( θ ) = cos θ ,we go back to the same game of considering the system as afunction of a single parameter θ which forms a ring S . It canbe checked numerically, that the exotic holonomy characterizedby (40), or equivalently, the eigenvalue flow { , , } → { , , } ,in obvious notation, is a common characteristics of system with c = c = c . Here, we have made an assumption that the unper-turbed spectrum is not much di ff erent from the original model, | c | (cid:28)
1, All possible patterns of eigenvalue flow are obtainedwith suitable choice of c j s. Specifically, c (cid:44) c = c = { , , } → { , , } , c (cid:44) c = c = { , , } →{ , , } , and c (cid:44) c = c =
0, in { , , } → { , , } .Generic case c (cid:44) c (cid:44) c also produces the second pattern, { , , } → { , , } . It is now clear, that there is finite subset ofparameter space, in which exotic holonomies of various typesarise after cyclic variation of the parameter θ .5 . Outlook Our results obtained in the two and three level cases canbe extended to N levels ( N ≤
4) in a straightforward way. Itis possible to prove the existence of the exotic holonomy forsystems described by the Hamiltonian H ( θ ) = cos θ (cid:20) cos θ Z ( N ) + v sin θ | w ( N ) (cid:105)(cid:104) w ( N ) | (cid:21) , (46)where | w ( N ) (cid:105) is a normalized N -dimensional vector and Z ( N ) isan N × N Hermitian matrix, as long as all eigenvectors of Z ( N ) have non-zero overlap with | w ( N ) (cid:105) .The Hamiltonian exotic holonomy shares a common featurewith the Wilczek-Zee holonomy of having SU ( N ) non-Abeliangauge potential at their base. However, they are distinct in that,in the former, N eigenstates are exchanged among themselveswith their internal dynamics, while, in the latter, involvementof another eigenstate, or a set of degenerate eigenstates [3] isrequired.The exotic holonomy also seems to have resemblance tothe o ff -diagonal holonomy of Manini and Pistolesi. It is im-portant to point out that the eigenstates are obtained indepen-dently from the choice of envelope function R ( θ ). If we makethe choice R ( θ ) =
1, the Hamiltonian becomes anti-periodic, H ( θ + π ) = − H ( θ ), so that new period is now 4 π . The eigen-state holonomy, occurring now at the midpoint of the new fullcycle θ ∈ [0 , π ), is nothing but the o ff -diagonal holonomy dis-cussed by Manini and Pistolesi. In the o ff -diagonal holonomy,the set of eigenstates at the starting value of environmental pa-rameter “accidentally” coincides with that at another value ofparameter in a midpoint of cyclic evolution. In general, suchcoincidence is highly unlikely, and it is often a result of thesame Hamiltonian multiplied by di ff erent numbers appearing atdi ff erent value of environmental parameter. In such a case, withthe introduction of new envelope function and reinterpretationof the period of parameter variation, a system with o ff -diagonalholonomy can be mapped to another one with exotic holonomy.In this instance, the Manini-Pistolesi holonomy is the exoticholonomy in disguise. Figure 10: Two types of Hamiltonian holonomy in the presence of a diabolicalpoint (cross surrounded by small circle) on energy surface standing on a para-metric plane. The picture on the left represents circular parameter variation thatresults in the Berry and Wilczek-Zee holonomies, while the picture on the rightdepicts the one leading to the exotic holonomy.
Our findings can be placed in context by considering a stan-dard double-cone structure of energy surface standing on the parameter space, whose connected apices of two cones repre-sent Berry’s diabolical point. When the parameters are variedalong a circle that surrounds the diabolical point, Berry phasearises (Fig. 10, left). We can ask a question: what will hap-pen when the circle touches the diabolical point. Obviously,the trajectory on the energy surface should be “smooth”, andit wanders both cones. With a cyclic variation of parameters,the trajectory moves from one cone to the other. This is exactlythe Hamiltonian exotic holonomy (Fig. 10, right). This processis fully described by holonomy matrix given in terms of thepath-ordered integral of the gauge potential just as in the caseof Berry phase. The gauge potential now has singularities incomplexified parameter space, not on the diabolical point itself.The Hamiltonian exotic holonomy can be viewed as an ex-tension of, and a natural complement to the Berry phase, and itforms an integral part of physics of adiabatic quantum control.The general equation for quantum holonomy (3) is just a verynatural expression of the basic requirement that the entire set ofeigenstates is to be mapped to itself after the cyclic variation ofenvironmental parameter.
Acknowledgements
We acknowledge the financial support by the Grant-in-Aidfor Scientific Research of Ministry of Education, Culture, Sports,Science and Technology, Japan (Grant number 21540402), andby Korea Research Foundation Grant (KRF-2008-314-C00144).
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