Quantum integrability and nonintegrability in the spin-boson model

Quan tum in tegrabilit y and nonin tegrabilit y in the spin-boson mo del Vyac heslav V. Stepanov, 1 Gerhard M ¨ uller, 1 and J oac him Stolze 2 1 Dep artment of Physics, University of R ho de I sland, Kingston RI 02881, USA 2 Institut f ¨ ur Phy sik, T e chnische Uni ve rsit¨ at Dortmund, 44221 Dortmund, Germany (Dated: Nov emb er 2, 2018) W e s tu dy the sp ectral prop erties of a spin- boson H a miltonian that dep ends on t wo contin u o u s parameters 0 ≤ Λ < ∞ (interac tion strength) and 0 ≤ α ≤ π / 2 (integrabil ity switch). In the classical limit this system has t wo distinct integrable regimes, α = 0 and α = π / 2. F or eac h in tegrable regime w e can ex press the quan tu m Hamiltonian as a function of tw o acti on op erators. Their eigen v alues (multiples of ~ ) are th e natural quantum n u m b ers for the complete leve l sp ectrum. This functional dep endence cannot b e ext ended in t o the nonintegrable regime (0 < α < π / 2). Here level crossings are prohibited and the level sp ectrum is n atu ral ly describ ed b y a single (energy sorting) quantum num b er. In consequence, the trac kin g of ind iv i d ual eigenstates along closed p ath s th rough both regimes leads to conflicting assignments of quan tum numbers. This effect is a useful and reliable indicator of quantum chaos – a diagnostic to ol that is indep enden t of any lev el-statistical analysis. P ACS num b ers: 05.45.-a; 05.45.Mt I. INTRO DUC TION Classical int eg rabilit y of a sy s tem with tw o degrees of freedom guara n tees that the Hamiltonian can be e x - pressed as a piecewise s mo oth function of tw o actio n co ordinates: H ( p 1 , q 1 ; p 2 , q 2 ) = H C ( J 1 , J 2 ). No such functional relation exists if the system is nonin teg rable [1, 2, 3, 4 ]. Geometrically sp eaking for a parametr ic sys- tem with parameter s sub ject to a n integrability condi- tion, there exist complete foliations of in v aria n t tori in phase s pace for a ll pa rameter po in ts in the integrable regime. Throughout the nonintegrable reg ime the foli- ation is par tially destroy ed. Some tori are replaced b y chaotic tra jector ies, ca n tori, and unstable p erio dic tra- jectories. The surviving tori in the nonint eg rable re gime are no longer dense anywhere in phase space, but eac h one of them can still b e c ha r acterized by t wo lo cal action co ordinates J 1 , J 2 via line integrals H p i dq i along pairs of top ologically indep enden t closed paths. In the integrable regime, the natur al lab el of an in v ar i- ant torus is its set of action co ordinates ( J 1 , J 2 ). T racking an in v ar ian t torus a long a path through the in tegr able regime of pa r ameter space means that we o bserv e ho w the tor us with this specification c hanges its pos ition and shap e in phas e space. In the nonin teg rable r egime, where all int a ct tor i a r e separ a ted b y c haotic phase flow, a n in- dividual to rus can no longer b e characterized by fixed v alues of J 1 , J 2 . T racking a surviving in v ar ia n t torus along a path through the nonintegrable reg ime of par am- eter space now means that we o bserv e it in iso lation from other tori. The lo cal a ction co ordinates v ary smoo thly as the torus changes its lo cation and shape in phase space. Now let us attempt to trac k one torus along a closed path that lies pa r tly in the in tegrable regime and pa r tly in the nonin teg r able regime, assuming that it does sur- vive the presenc e of chaos. Inside the in teg rable regime the iden tity of the to rus is determined by consta n t v al- ues of the action coor dina tes, while outside that regime the action co ordinates v ar y with the sha p e o f the iso- lated torus. The v alues of J 1 , J 2 at the end of the closed path will, in ge neral, b e different from the starting v al- ues, implying that the individuality of a torus cannot b e maintained. No lo ss of individua lit y is suffered b y tori along closed paths em b edded in the integrable r egime or for sur viving tori along closed paths in the no nin tegrable regime [5]. There exis ts a qua n tum co unterpart to this ’cr isis of ident ity’ as will be demonstrated. It ca n b e employ ed to discriminate b et ween reg imes o f integrability and nonin- tegrability on pur ely qua n tum mec ha nical gro unds. Here we s how the w ork ings of this diag nostic to ol in the con- text of the spin-bos on mo del [6, 7, 8 , 9], H = ~ ω B a † a + ~ ω S S z + Λ co s α  S + a + S − a †  + Λ sin α  S + a † + S − a  , (1) one of the s implest nontrivial mo dels describing nonrel- ativistically the in ter a ction betw een an atom and a ra- diation field [10]. This mo del has also been used to de- scrib e the in ter a ction b et ween electronic and vibrational degrees of freedo m in molecules and solids. The relation betw ee n classical a nd quan tum integrability of (1) has bee n the ob ject of previo us inv es tigations [8, 11]. The first tw o terms in (1) describ e one mode o f the electromagne tic field and a (2 σ + 1)-level atom, resp ec- tively . The coupling b et ween the t wo deg r ees of freedom has str e ng th Λ and dep ends on a contin uous parameter α that connects t wo regimes for whic h this mo del is in- tegrable in the classica l limit. T he class ic a l integrability for α = 0 and α = π / 2 is established b y a second integral of the motion. The cas e α = 0 is known as the rotating wa ve approximation in quantum optics. Ea rly studies in one or the other classica l limit of the spin-b oson model revealed c haotic phase space flow turning regular in the rotating w av e a ppro ximation [6, 7, 12, 1 3 ]. In the t wo-dimensional para meter space spanned by the (p olar) co ordinates (Λ , α ), the tw o integrable regimes 2 are lo cated on tw o perp endicular straight lines that in- tersect each other a t the p oin t of zer o coupling stre ngth. Each quadrant of this pa rameter plane r epresen ts a non- int eg rable reg ime. Henceforth we consider the parameter range 0 ≤ Λ < ∞ , 0 ≤ α ≤ π / 2. In prepar ation of our main theme we first discuss the classical in tegrability condition of the s pin-boson model (Sec. I I) and then the classifica tion of its quantum energy levels (Sec. I II ) a nd ce rtain q ua n tum in v ar ian ts (Sec. IV) by distinct sets o f quantum num b ers in the int eg rable and nonintegrable regimes. This distinction has a deeper meaning, which we will further discuss in Sec. V, and which we will emplo y in Sec. VI for the iden tification of the t wo regimes in purely quantum mechanical terms. II. INTEGRABILITY CONDITION In taking the classical limit ~ → 0, σ → ∞ of the spin-b oson mo del, we reno rmalize the co upling constant, Λ = ( ~ / 2) 3 / 2 ¯ Λ, substitute a = r M ω B 2 ~ x + r 1 2 ~ M ω B ıp (2) for the b oson o perator s and, via ~ p σ ( σ + 1 ) = s , co n vert the spin- σ op erator in to a class ic al 3- component vector of fixed length: ( S x , S y , S z ) = s (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ ) . (3) The s pin-boson Hamiltonian (1) thus turns in to the en- ergy function o f t wo linea r one-degr ee-of-freedom systems – a har monic oscillato r and a cla ssical spin in a constant magnetic field – with a nonlinear coupling: H = p 2 2 M + 1 2 M ω 2 B x 2 + ω S S z + 1 2 ¯ Λ cos α  p M ω B xS x − 1 √ M ω B pS y  + 1 2 ¯ Λ sin α  p M ω B xS x + 1 √ M ω B pS y  . (4) A set of canonical co ordinates is ( p, x ; s cos ϑ, ϕ ). The equations of motion for the ph ys ical v a riables ( x, p, S x , S y , S z ) inferred from (4) via dx/dt = ∂ H/∂ p , dp/dt = − ∂ H /∂ x , and d S /dt = − S × ∂ H /∂ S read ˙ x = p M + ¯ Λ 2 √ M ω B S y (sin α − cos α ) , (5a) ˙ p = − M ω 2 B x − ¯ Λ √ M ω B 2 S x (cos α + sin α ) , (5b) ˙ S x = − ω S S y − ¯ Λ p 2 √ M ω B S z (cos α − sin α ) , (5c) ˙ S y = ω S S x − ¯ Λ x √ M ω B 2 S z (cos α + sin α ) , (5d) ˙ S z = ¯ Λ x √ M ω B 2 S y (cos α + sin α ) + ¯ Λ p 2 √ M ω B S x (cos α − sin α ) . (5e) The phase flow gener ated b y these equations is, in g en- eral, chaotic. Chaos gives way to a fully intact to rus structure at α = 0 , π/ 2. The integrability o f these ca ses is established by the fact that one or the other of the t wo functions, I = p 2 2 M ω B + 1 2 M ω B x 2 + S z , (6a) K = p 2 2 M ω B + 1 2 M ω B x 2 − S z , (6b) whose time evolution is determined b y [14] ˙ I = { H , I } = ¯ Λ sin α  pS x √ M ω B − p M ω B xS y  , (7a) ˙ K = { H , K } = ¯ Λ cos α  pS x √ M ω B + p M ω B xS y  , (7b) bec omes a second integral of the motion. The cas e α = 0 is k no wn as the Jaynes-Cummings mo del [15]. The im- pact of the class ical integrability conditions on the quan- tum system is the main theme of this study . II I. ENERGY LEVELS F or the a nalytic or numerical s olution of the spin- bo son mo del (1), it is conv enient to use the pro d- uct vectors of the noninteracting system, | m, n i , m = 0 , 1 , 2 , . . . , 2 σ , n = 0 , 1 , 2 , . . . , a s a bas is. The relev ant op erators act on this basis as follows: ( σ − S z ) | m, n i = m | m, n i , (8a) S + | m, n i = p m (2 σ − m + 1) | m − 1 , n i , (8b) S − | m, n i = p (2 σ − m )( m + 1) | m + 1 , n i , (8c) a † | m, n i = √ n + 1 | m, n + 1 i , (8d) a | m, n i = √ n | m, n − 1 i . (8e) The Hamiltonian matrix can thus b e assembled from the diagonal elemen ts h m, n | S z | m, n i = σ − m, h m, n | a † a | m, n i = n, (9) 3 0 1 2 ... 2σ ... 2 1 0 m n FIG. 1: Basis ve ctors | m, n i with p ositiv e parity (full squares) and negative parit y (op en squares) as coupled by matri x el - ements of op erators S + a, S − a † (solid lines) and S + a † , S − a (dashed lines) of Hamiltonian (1) and fr om the off-diagonal elements h m, n | S + a | m + 1 , n + 1 i = p (2 σ − m )( m + 1 )( n + 1) , h m, n | S − a † | m − 1 , n − 1 i = p (2 σ + 1 − m ) mn, h m, n | S + a † | m + 1 , n − 1 i = p (2 σ − m )( m + 1 ) n, h m, n | S − a | m − 1 , n + 1 i = p (2 σ + 1 − m ) m ( n + 1 ) . The str ucture of this matrix is illustr a ted in Fig. 1. The solid lines repr esen t matrix elements g enerated by the first interaction term in (1), and the da s hed lines repre- sent ma trix elemen ts which arise in the second int er ac- tion term. The tw o- sublattice str ucture is a r e flection o f parity conser v ation. The parity op erator, P = ( − 1) a † a + σ − S z , (10) commutes with H for arbitra ry Λ , α . It divides the eigen- states into t wo symmetry classes . States with P = +1 ( P = − 1) inv olve basis v ecto r s with even m + n (o dd m + n ) only . If α = 0 only the s o lid bo nds are present and if α = π / 2 only the dashed b onds. In either case the Hamiltonian matrix is r educed to in v aria n t blocks of size 2 σ + 1. If 0 < α < π / 2 we mu s t deal with an infinite matrix. In this study we set ω S = ω B ≡ ω exce pt wher e indicated o ther- wise. In the following we analyze the level spectrum for v ar ious cases at α = 0 , π/ 2: systems with σ = 1 2 , σ = 1 for arbitrary n , a nd systems with arbitrary σ asymptot- ically for larg e n . A. Spin σ = 1 2 The integrable spin-b oson mo del with σ = 1 2 inv olves only 2 × 2 matrices. If α = 0, the eigen vectors happ en to be indep enden t of the interaction strength, | ψ 1 , 0 i = | 1 , 0 i (11a) | ψ 1 ,n i = 1 √ 2 {| 1 , n i + | 0 , n − 1 i} , n = 1 , 2 , . . . , (11b) | ψ 0 ,n i = 1 √ 2 {| 1 , n + 1 i − | 0 , n i} , n = 0 , 1 , . . . , (1 1c) and the energy eigenv alues (for n = 0 , 1 , 2 , . . . ) are E 1 ,n = ~ ω ( n − 1 / 2) + Λ √ n, (12a) E 0 ,n = ~ ω ( n + 1 / 2) − Λ √ n + 1 . (12b) If α = π / 2 the eig en vectors do dep end on Λ: | ψ 1 , 0 i = | 0 , 0 i (13a) | ψ 1 ,n i = a 0 ,n | 0 , n i + b 0 ,n | 1 , n − 1 i , n = 1 , 2 , . . . , (1 3 b) | ψ 0 ,n i = a 1 ,n | 0 , n + 1 i − b 1 ,n +1 | 1 , n i , n = 0 , 1 , . . . (13c) with a 0 ,n = √ λ n − 1 p 2( λ 1 − √ λ n ) , a 1 ,n − 1 = √ λ n − 1 p 2( λ 1 + √ λ n ) , b 0 ,n = √ λ n − 1 p 2( λ 1 − √ λ n ) , b 1 ,n − 1 = √ λ n + 1 p 2( λ 1 + √ λ n ) , where λ . = (Λ / ~ ω ) 2 , λ n . = 1 + nλ . The asso ciated ener gy eigenv alues (for n = 0 , 1 , 2 , . . . ) are E 1 ,n = ~ ω ( n − 1 / 2) + ~ ω p λ n , ( 1 4a) E 0 ,n = ~ ω ( n + 1 / 2) − ~ ω p λ n +1 . (14b) B. Spin σ = 1 The case σ = 1 at in tegr abilit y in volves the so lut io n of cubic equations. Here we lis t the (Λ- indep endent ) eigen- vectors and the asso ciated ener gy eig en v a lues for α = 0 . W e have | ψ 1 , 0 i = | 0 , 0 i , | ψ 1 , 1 i = ( | 1 , 0 i + | 0 , 1 i ) / √ 2, | ψ 2 , 1 i = ( | 1 , 0 i − | 0 , 1 i ) / √ 2, with energ ies E 1 , 0 = − ~ ω , E 1 , 1 = √ 2Λ, E 2 , 1 = − √ 2Λ, respectively , and for n ≥ 2 the results ar e | ψ 1 ,n i = r n − 1 4 n − 2 | 0 , n − 2 i + 1 √ 2 | 1 , n − 1 i + r n 4 n − 2 | 2 , n i , (15a) | ψ 2 ,n i = r n 2 n − 1 | 0 , n − 2 i + r n − 1 2 n − 1 | 2 , n i , (15b) | ψ 3 ,n i = r n − 1 4 n − 2 | 0 , n − 2 i − 1 √ 2 | 1 , n − 1 i + r n 4 n − 2 | 2 , n i , (15c) 4 with energies E 1 ,n = ~ ω ( n − 1 ) + Λ √ 4 n − 2 , (16a ) E 2 ,n = ~ ω ( n − 1 ) , (16b) E 3 ,n = ~ ω ( n − 1 ) − Λ √ 4 n − 2 . (16c) C. Arbitrary Spin σ A simple analytic solution exists for arbitrary σ in the asymptotic reg ime of lar g e n . Consider the (2 σ + 1)- dimensional in v aria n t blo c k of (1) a t α = 0 for med by the basis vectors | 2 σ − m, n − m i , m = 0 , 1 , . . . , 2 σ . It is tridiagona l with elements h 2 σ − m , n − m | H | 2 σ − m, n − m i = ~ ω ( n − σ ) , h 2 σ − m , n − m | H | 2 σ − m, n − m − 1 i = Λ p 2 σ ( n − m ) . F or n ≫ σ we ca n write H = ~ ω ( n − σ ) E + 2Λ √ nS x + O  σ √ n  , (17) where E is the (2 σ + 1)-dimensio na l unit matrix and S x is the irreducible representation o f the spin op erator with the same dimensiona lit y . The asymptotic eigen v alues of this matrix ar e E m,n ≃ ~ ω ( n − σ ) + 2Λ √ n ( σ − m ) . (18) for m = 0 , . . . , 2 σ . The corres p onding ana lysis car ried out for α = π / 2 yields the matrix H = ~ ω ( n − σ ) E + 2 ~ ω S z + 2Λ √ nS x + O  σ √ n  (19) with asymptotic energ y eigenv alues (for m = 0 , . . . , 2 σ ) E m,n ≃ ~ ω ( n − σ ) + 2 ~ ω p λ n ( σ − m ) . (20) Note that in all cases p ertaining to the integrable regimes α = 0 o r α = π / 2 the energy levels are na turally lab elled b y the tw o qua n tum num be r s m, n . The par - it y b ecomes P = ( − 1) m + n . In the nonin teg rable r egime 0 < α < π / 2, by contrast, the nu mer ical a nalysis suggests the use o f a single (energy so rting) qua n tum num b er k for all levels of given parity . IV. QUANTUM INV ARIANTS The quantum coun terpa rts of the t wo analytic in v ari- ants (6) are the oper ators I = ~ ( a † a + S z ) , K = ~ ( a † a − S z ) , (21) which indeed comm ute with (1 ) under exactly the sa me conditions as in the classical limit. W e ha ve [ H, I ] = 2Λ sin α ( S − a − S + a † ) , (22a) [ H, K ] = 2Λ cos α ( S + a − S − a † ) . (22b) How ever, quantum integrability cannot b e infer red fro m quantum inv a rian ts as simply as classical integrabilit y can b e inferred fr o m integrals of the motion (ana lytic inv a rian ts). Co mm uting op erators ca n always be con- structed irres p ective of whether the mo del is (classically) int eg rable or not [16, 1 7 ]. The par it y op erator (10 ), for example, which can b e expressed as a function of either inv a rian t I or K , P = e iπ ( I / ~ − σ ) = e iπ ( K/ ~ + σ ) , (23) commutes with H fo r ar bitrary α . More generally , any op erator A that is not already an inv aria n t, [ H , A ] 6 = 0, can be turned into an inv ar ian t via time a verage. In the energy repres e ntation, the time av er age s trips A of a ll its off-diagona l elemen ts. The resulting operato r I A = h A i th us commut es with H by construction [18, 1 9]. The fact is that in the cla ssical limit neither the parity op erator nor an y of the artificially constructed qua n tum inv a rian ts will tur n in to analytic inv ariants (in tegr a ls of the motio n) if the phase flo w is chaotic. Such quantum inv a rian ts either lose their meaning altogether or turn int o nonanalytic in v ariants [19, 20]. The distinctiv e attr ibutes of q ua n tum in v ar ia n ts in the int eg rable and nonin tegrable r egimes of a quantum sys- tem are subtle but not ambiguous. Here we use I A = h A i , A = a † ( S − + S + ) . (24) F or σ = 1 2 , its eigenv alues at α = 0 can b e calculated from the eigenv ecto rs (11) , h A i 1 ,n = 1 2 √ n, h A i 0 ,n = − 1 2 √ n + 1 , (25) and its e igen v alues at α = π / 2 from the eig en vectors (13): h A i 1 ,n = ( λ n − 1)( √ λ n − 1) 2( λ n − √ λ n ) , (26a ) h A i 0 ,n = − ( λ n +1 − 1)( p λ n +1 + 1) 2( λ n +1 + p λ n +1 ) . (2 6b) F or σ = 1 and α = 0, we obtain the following eigenv alues from the eigenv ecto rs (15): h A i 1 ,n = −h A i 3 ,n = p n − 1 / 2 , h A i 2 ,n = 0 , (27) Asymptotically fo r large n , we can ev alua te the eigenv al- ues fo r arbitra r y σ . The results for α = 0 read h A i m,n = ( σ − m ) √ n, (28) 5 and fo r α = π / 2 w e ha ve h A i m,n = ( σ − m ) √ n p 1 + 1 /nλ . (29) Numerical results of h A i k for 0 < α < π / 2 of systems with σ = 1 2 , 1 , 3 2 were rep orted pr eviously [9]. The pat- terns of p oin ts ( E m,n , h A i m,n ) for int eg rable cases w ere found to b e strik ing ly different from the pa tter n of p oin ts ( E k , h A i k ) for nonintegrable cases. Here this difference will b e used as a demarc a tion to ol for regimes of integra- bilit y and nonin tegr a bilit y . V. QUANTUM ACTIONS One hallmark of integrability in a quant um s y stem with tw o degr ees of fre edom is that the Hamiltonian can be express ed as a function of tw o actio n op erators J 1 , J 2 , i.e. o f tw o quan tum in v ariants whos e sp ectra consis t of equidistant levels. A. F rom Λ = 0 to Λ > 0 In the abs e nce of th e spin-b oson in teraction (Λ = 0), the t wo action oper ators a re J 1 = ~ ( σ − S z ) , J 2 = ~ a † a (30) with in teger eigenv a lues (in units of ~ ) J 1 = m ~ , m = 0 , 1 , . . . , 2 σ , (31a ) J 2 = n ~ , n = 0 , 1 , . . . , (31b) as in (9). The Hamiltonia n, H 0 = ~ ω B a † a + ~ ω S S z , and the tw o quantum inv ar ian ts (21) are expr essible as linear combinations of J 1 , J 2 . Classically , the cont r ibution of each degr ee of freedom to H 0 = p 2 / 2 M + 1 2 M ω 2 B x 2 + ω S S z is tra nsformed int o a function of one action co ordinate by a sepa r ate canonica l transformatio n: ( S z , ϕ ) → ( J 1 , θ 1 ) with S z = s − J 1 , ϕ = − θ 1 ; and ( p, x ) → ( J 2 , θ 2 ) with p = √ 2 J 2 M ω B cos θ 2 , x = p 2 J 2 / M ω B sin θ 2 . The transformed Hamiltonian and the tw o class ical inv aria n ts (6) are linear functions of J 1 , J 2 just as in quantum mechanics. The exact qua n- tum sp ectra of H 0 , I , K can then b e recov er ed exactly via semiclassical quantization, i.e. by subs tit uting the ac- tions quant ize d acco rding to (31) into the classica l Hamil- tonian. Classically , the interaction renders the equations of motion, Eqs. (5), nonlinear. How ever, the effects of an- harmonicity in the time e v olution dep end sensitively on whether int eg rabilit y is susta ined or destroy ed by the in- teraction. Integrability for α = 0 , π/ 2 dictates that the phase flo w is exclusively tor oidal. F or 0 < α < π/ 2 chaotic pha s e flo w is omnipresent alb eit constra ined by surviving tori. Quantum mechanically , the in ter action distorts the eigenv alue sp ectrum and mo difies the selection rules of transition rates. Qua n tum proper ties that are as sensi- tive to the in tegr abilit y status a s their classica l coun ter- parts do e x ist and have pr e viously been explored in the context of a different mo del s y stem [21, 22, 23]. These prop erties are directly related to the existence of actio n op erators as constituent ele ments of the Hamil- tonian such as discussed in Sec. V A for the noninter- acting system. In the in tera cting cases, the existence of action op erators can again b e demonstrated direc tly for α = 0 , π / 2, and their no nexistence for 0 < α < π / 2 can be demonstra ted indirectly . B. σ = 1 2 , α = 0 The unitar y tra nsformation which diagona liz e s the Hamiltonian (1) for σ = 1 2 and α = 0 , expressed in terms of s pin and boson ope rators, rea ds U A = P A 0 + 1 √ 2  − 2 S z + 1 √ a † a a † S − + aS + 1 √ a † a Q A 1  , (32) where P A 0 = | 1 , 0 ih 1 , 0 | , Q A 1 = 1 − | 0 , 0 ih 0 , 0 | − | 1 , 0 ih 1 , 0 | . The o perator s T z = U A S z U − 1 A = P A 0 S z − 1 2 G A 1 (33a) b † b = U A a † aU − 1 A = a † a − S z P A 0 + 1 2 G A 1 (33b) with G A 1 = aS + 1 √ a † a Q A 1 + 1 √ a † a a † ˆ S − (34) are dia gonal in the ener g y representation: T z | ψ m,n i = ( σ − m ) | ψ m,n i , (35a) b † b | ψ m,n i = n | ψ m,n i . (35b) Hence the quantum actions with eigen v alues (3 1) ar e J 1 = ~ ( σ − T z ) , J 2 = ~ b † b. (36) Applying U A to the Hamiltonian yields U A H U − 1 A = ~ ω ( b † b + T z ) + Λ  1 − 2 T z 2 √ b † b − 1 + 2 T z 2 p b † b + 1  , (37) which, together with (36), describes the functional rela- tion b et ween H and J 1 , J 2 . C. σ = 1 2 , α = π / 2 The same metho d also pro duces the quantum actions for the int eg rable case s = 1 2 , α = π / 2 of the spin-b oson 6 Hamiltonian (1). Here the block-diagonal unitar y trans- formation, U B to b e used can also b e expressed in terms of s pin and b oson op erators but has a more complica ted structure than U A . The operato rs T z = U B S z U − 1 B = 1 2 P B 0 + ( G B )) Q B 0 , ( 3 8a) b † b = U B a † aU B =  a † a − S z + G B  Q B 0 , (38b) with G B = G B 1 + G B 2 + G B 3 , G B 1 = 1 p 16 (1 + λa † a ) − 1 p 16 (1 + λ ( a † a + 1)) , G B 2 = S z p 4 (1 + λa † a ) + S z p 4 (1 + λ ( a † a + 1)) , G B 3 = 1 + 2 S z 4 √ 64 λ + a † a a † S + + aS − 1 + 2 S z 4 √ 64 λ + a † a , and P B 0 = | 0 , 0 ih 0 , 0 | , Q B 0 = 1 − | 0 , 0 ih 0 , 0 | , aga in satisfy (35) and ar e related to quantum actions via (36). The functional dep endence of the tra nsformed Hamiltonian on the actions is differen t from (37): U B H U − 1 B = ~ ω ( b † b − T z ) + 1 + 2 T z 2 p 1 + λb † b − 1 − 2 T z 2 q 1 + λ ( b † b + 1) . (39) D. σ > 1 2 The r esults of Secs. V B and V C are gene r alizable to arbitrar y σ , alb eit for the price o f a higher and higher calculational effort. The ca s e σ = 1 , α = 0 can still be presented compactly . The unitary transformatio n U C to be us ed in this case is now determined b y the eig e nvectors (15) and yields T z = U C S z U − 1 C = P C 0 − G C 1 Q C 2 + 1 2 G C 2 P C 1 , (40a) b † b = U C a † aU − 1 C =  a † a + S z + G C 1  Q C 1 + 1 2 G C 2 P C 1 , ( 4 0b) where G C 1 = S 2 z − S z 2 √ 4 a † a − 2 a † S − + 1 − S 2 z √ 4 a † a + 2 aS + + 1 − S 2 z √ 4 a † a + 2 a † S − + S z + S 2 z √ 4 a † a + 6 a † S + , G C 2 = 1 + 1 − S 2 z √ 2 aS + + S 2 z − S z 2 √ 2 a † S − , and P C 0 = | 0 , 0 ih 0 , 0 | , P C 1 = | 1 , 0 ih 1 , 0 | + | 0 , 1 ih 0 , 1 | , Q C 2 = 1 − P C 0 − P C 1 . The transformed Hamiltonia n bec omes U C H U − 1 C = ~ ω  b † b + T z  + Λ √ 2  3 T 2 z − S z − 2  P C 1 +  ( T 2 z − T z ) p 2 b † b − 1 − ( T 2 z + T z ) p 2 b † b + 3  Q C 2 i . (41) U A and U C are sp ecial cases for σ = 1 2 and σ = 1 , resp ectiv ely , of a unitary transfo r mation U 1 ( σ , Λ) that diagonalizes (1) at α = 0 for arbitrary v alues of σ . This transformatio n turns out to b e Λ-indep enden t for the tw o cases we ha ve worked out. It may well b e Λ-indep enden t for arbitra ry σ . Likewise, U B is the s pecial case for σ = 1 2 of a unitary transformation U 2 ( σ , Λ) that diagonalizes (1) at α = π / 2 for arbitrary σ . That transformation is manifestly Λ-dependent. The end-pro duct of these unitary transfor ma tions are t wo functions ¯ H (1) Q ( T z , b † b ; Λ) = H (1) Q ( J 1 , J 2 ; Λ) and ¯ H (2) Q ( T z , b † b ; Λ) = H (2) Q ( J 1 , J 2 ; Λ), which express the functional dep e ndence of the Hamiltonian on actio n o p- erators in the t wo integrable r e gimes α = 0 a nd α = π / 2, resp ectiv ely . The leading term of an asymptotic expa n- sion at high b oson o ccupancy and unrestricted spin s ta te of these functions can b e inferred from (17) and (19): ¯ H Q = ~ ω  b † b ± T z  + Λ √ b † b + O(1) , ( 4 2) where the oper a tors T z , b † b ag ain sa tisfy (35) and the upper (low er ) sign per ta ins to α = 0 ( α = π / 2). W e exp ect a semicla ssical regime to exist at large spin and/o r b oson quantu m n umber s where the functions H (1) Q ( J 1 , J 2 ; Λ) a nd H (2) Q ( J 1 , J 2 ; Λ) connect with func- tions H (1) C ( J 1 , J 2 ; Λ) a nd H (2) C ( J 1 , J 2 ; Λ) of cla ssical ac- tions. Ho wev er, the iden tification of the s emiclassical regime requires a complete solution o f the cla ssical equa- tions of motion (5), a task still o utsta nding. The connectio ns b et ween the quantum and c la ssical functional dep endences of Hamiltonian o n a ctions was inv estigated in a prev io us study for an int eg rable tw o- spin mo del a nd for the (integrable) cir cular billiard mo de l [23]. There we found subtle quantum effects that restrict the r ange of the semicla ssical regime in unexp ected w ays. That may als o b e the case in the spin-b oson model. How- ever, the point w e wish to emphasize in this study is a different one. VI. TRACKING EIGENST A TES The goa l is to demonstr ate that the functions H (1) Q ( J 1 , J 2 ; Λ) and H (2) Q ( J 1 , J 2 ; Λ) canno t b e extended in an y consistent wa y in to the region of no nin tegrability in the (Λ , α )-pla ne. The fu nctio ns H (1) Q and H (2) Q make it pos sible to label all eige ns tates of (1) by the t wo ac- tion quantum num b ers m, n as defined in (31) and to track them with no ambiguit y through each one of the t wo integrable re g imes α = 0 and α = π/ 2. The non- extendability of the tw o fu nctio ns H (1) Q and H (2) Q int o a function H Q ( J 1 , J 2 ; Λ , α ) translates into the imp ossibilit y of co nsisten tly assigning action quant um num b ers m, n to the eigenstates in the entire parameter ra nge 0 ≤ Λ < ∞ , 0 ≤ α ≤ π / 2. One w ay of keeping track of eigensta tes | ν i of (1) is to determine how the eigenv alues of quantum in v a rian ts 7 v ar y alo ng some path in the (Λ , α )-plane. F or the pur- po se of this demonstr ation, we fo cus o n the eigen v alues h H i ν = E ν of the Hamiltonia n (1) with spin quan tum nu mber σ = 1 2 and the eigen v alues h A i ν of the qua ntum inv a rian t I A as defined in (24). Within each of the tw o integrable regimes, both sets of eigenv alues hav e a n e xplicitly known (discrete) dep en- dence on the action quantum num b ers ν = ( m, n ) and an explicitly known (contin uous) dependence on the in- teraction strength Λ. The fu nctio nal relations a re stated in Eqs. (12), (25 ) for α = 0 and (14), (26) fo r α = π / 2. A. Level Cro ss ings In panels (a) a nd (b) of Fig. 2 w e hav e plotted one quantum inv ariant versus the other for all states with po sitiv e parity up to a certain ener gy . In bo th panels w e observe tw o v ertica lly displaced rows of states . Sta tes in the top and bottom rows hav e action quantum num b ers (1 , n ) a nd (0 , n ), respectively . The observed arrangement o f states is due to the fact that h A i m,n ∼ √ n but E m,n ∼ n in le a ding order . No- tice that the spacings b et ween success iv e energy levels in each row v ary slowly , and at different rates in the top and b ottom r o ws. T o enhance the visibilit y o f this effect we have connected successive energy lev els in each panel by dashed lines. The spacing s are somewhat la r ger in the top row co mpared to the bottom row, causing instances in b oth panels where t wo consecutive states of the bo ttom row fit in to the space b et ween tw o states of the top row. These instances where the alter nating (top/b o ttom) se- quence is broken mark loca tions where ener gy levels from opp osite rows can fall a rbitrarily close to ea c h o ther. When we incr e ase the in tera ction strength Λ gradually , the states in the top row of Fig. 2(a) move toward the right and the states in the bottom row tow a rd the left. The same observ a tion can b e made in Fig . 2(b). Here the shift also contains a small v er tical component. W e hav e sing led out one pair of nearly degenerate states in Fig. 2(a) and another pair in Fig. 2(b). E ac h pair is marked by full circle s . In panels (a) a nd (b) of Fig. 3 we hav e plotted the traces o f these states in the plane of in v aria n ts as the interaction strength is increased by a certain amoun t. The gradual change of Λ causes a casca de of level c ross- ings b et ween states from opp o site rows. F o r the tw o pairs of tagged states, the cross ings o ccur at the p oint marked by an asterisk on their tra ces. States from opp osite rows undergo level crossings even thoug h they ha ve the same parity . What ma tt er s are the functional relations H (1) Q and H (2) Q established pr eviously . They remove a ny p ossi- ble cause for level collisions (av oided crossings) betw een states from opp osite r o ws as they mov e (energetically) in opp osite directions when Λ is increased. B. Level Collisons A very differ en t scenar io unfurls when we plot the tw o quantum inv ariants for a nonin tegrable ca se. What hap- pens when we change the integrability parameter from α = 0 [Fig. 2(a)] or from α = π/ 2 [Fig. 2(b)] to α = π / 4 is illustrated in Fig. 2(c). Here the states that used to live in different worlds (top ro w with action quantum num- ber m = 1 a nd bo ttom ro w w ith m = 0) now suddenly get into each other’s wa y . Since they are pro hibit ed from undergoing an y lev el cross ings, it is no w appropriate to lab el them b y the e nergy sorting quant um n umber k . In those parts of the sp ectrum where the energ y level spacings ar e lar ge, the loss of in tegr abilit y has no visible effect on the quantum inv ariants. That is the case near the left and right b order ar eas of Fig. 2(c). Her e the t wo rows of states remain la r gely in tact. Howev er, near the cen ter of the pa nel, where small ener gy level spac- ings occur , the eigenv ectors of nearly degenerate lev els affect each other strongly . The most conspicuous effect is a strong v ertica l displacement of the tw o states from the row p ositions tow ar d each other. Less cons picuous in Fig. 2(c) but of even greater imp ortance is the small hor - izontal displac emen t of the tw o nearly degenerate states aw ay from each other. The effect o f nonintegrability is that ener gy le v els exer t a shor t-distance repulsion o n e ac h other. At the same time, expectatio n v alues in general and the quantum inv aria n t h A i k in particula r tend to b e- come less differentiated than they w ere in the in tegr able case. When we ag ain increa se the in tera ction streng th Λ, now at fixed α = π / 4 in the nonin teg r able regime, we find that no levels with equal parit y e ver underg o a crossing. As in the integrable cases, the states with h A i k > 0 hav e a tendency to move tow ard the right a nd the states with h A i k < 0 toward the left. Inevitably , these trends put states on o pposite sides of h A i k = 0 o n a collision course. When tw o such states approach one another, the sta te starting out with h A i k > 0 swings down as it mo ves to the r igh t and the state with h A i k < 0 swings up as it mov es to the le f t. The t wo states reach their clos est energetic approach when their v er tical p ositions are about the same. After that, the sta te coming fro m below contin ues its upswing, but now it is moving to the right to join the r igh t-moving upper row of states. Mea n while, the state coming from ab o ve co n tinues its do wnswing to join the left-mo ving low er row of sta tes. One s uc h level collisio n, b et ween the tagged s tates in Fig. 2(c), is sho wn in Fig. 3(c). C. Quan tum Numbe r s in Conflict Lo oking at the sp ectrum of the spin-b oson model (1) in the plane of inv ariants ( E ν , h A i ν ) as the int er action strength Λ increases gr a dually , reveals striking ly different patterns o f co ordinated motion of all states with given parity , depending on whether the pa rameter α is set to 8 -4 -3 -2 -1 0 1 2 3 4 0 5 10 15 20 25 30 35 40 〈 A 〉 mn E mn (a) -4 -3 -2 -1 0 1 2 3 4 0 5 10 15 20 25 30 35 40 〈 A 〉 mn E mn (a) -4 -3 -2 -1 0 1 2 3 4 0 5 10 15 20 25 30 35 40 〈 A 〉 mn E mn (a) -4 -3 -2 -1 0 1 2 3 4 0 5 10 15 20 25 30 35 40 〈 A 〉 mn E mn (a) -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 0 5 10 15 20 25 30 35 40 〈 A 〉 mn E mn (b) -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 0 5 10 15 20 25 30 35 40 〈 A 〉 mn E mn (b) -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 0 5 10 15 20 25 30 35 40 〈 A 〉 mn E mn (b) -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 0 5 10 15 20 25 30 35 40 〈 A 〉 mn E mn (b) -2 -1 0 1 2 0 5 10 15 20 25 30 35 40 〈 A 〉 k E k (c) -2 -1 0 1 2 0 5 10 15 20 25 30 35 40 〈 A 〉 k E k (c) -2 -1 0 1 2 0 5 10 15 20 25 30 35 40 〈 A 〉 k E k (c) -2 -1 0 1 2 0 5 10 15 20 25 30 35 40 〈 A 〉 k E k (c) FIG. 2: Quantum inv ariant h A i ν = h ν | a † ( S − + S + ) | ν i versus quantum inv arian t E ν = h ν | H | ν i o ver some energy range for the eigenstates | ν i with parity P = +1 of th e spin - boson mod el (1) with σ = 1 2 , ~ ω = 1, λ . = (Λ / ~ ω ) 2 = 0 . 09, and (a) α = 0, (b ) α = π / 2, (c) α = π/ 4. In th e integrable regimes we u se th e action quantum numbers ν = ( m, n ) and in the nonintegrable regime w e u se the en erg y sorting quantum num b er ν = k . One p ai r of states in eac h panel (full circles) is tagged for further u se in Fig. 3. an integrable reg ime ( α = 0 , π / 2) o r fixed within the nonintegrable regime (0 < α < π / 2). -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 10 10.5 11 11.5 12 12.5 13 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (a) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 10 10.5 11 11.5 12 12.5 13 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (a) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 10 10.5 11 11.5 12 12.5 13 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (a) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 10 10.5 11 11.5 12 12.5 13 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (a) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 10 10.5 11 11.5 12 12.5 13 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (a) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 10 10.5 11 11.5 12 12.5 13 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (a) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 10 10.5 11 11.5 12 12.5 13 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (a) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 10 10.5 11 11.5 12 12.5 13 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (a) -4 -2 0 2 4 32 32.5 33 33.5 34 34.5 35 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (b) -4 -2 0 2 4 32 32.5 33 33.5 34 34.5 35 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (b) -4 -2 0 2 4 32 32.5 33 33.5 34 34.5 35 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (b) -4 -2 0 2 4 32 32.5 33 33.5 34 34.5 35 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (b) -4 -2 0 2 4 32 32.5 33 33.5 34 34.5 35 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (b) -4 -2 0 2 4 32 32.5 33 33.5 34 34.5 35 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (b) -4 -2 0 2 4 32 32.5 33 33.5 34 34.5 35 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (b) -4 -2 0 2 4 32 32.5 33 33.5 34 34.5 35 〈 A 〉 mn E mn λ =0.01 λ =0.25 λ =0.01 λ =0.25 (b) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 16.5 17 17.5 18 18.5 〈 A 〉 k E k λ =0.01 λ =0.25 λ =0.01 λ =0.25 λ =0.09 λ =0.09 (c) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 16.5 17 17.5 18 18.5 〈 A 〉 k E k λ =0.01 λ =0.25 λ =0.01 λ =0.25 λ =0.09 λ =0.09 (c) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 16.5 17 17.5 18 18.5 〈 A 〉 k E k λ =0.01 λ =0.25 λ =0.01 λ =0.25 λ =0.09 λ =0.09 (c) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 16.5 17 17.5 18 18.5 〈 A 〉 k E k λ =0.01 λ =0.25 λ =0.01 λ =0.25 λ =0.09 λ =0.09 (c) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 16.5 17 17.5 18 18.5 〈 A 〉 k E k λ =0.01 λ =0.25 λ =0.01 λ =0.25 λ =0.09 λ =0.09 (c) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 16.5 17 17.5 18 18.5 〈 A 〉 k E k λ =0.01 λ =0.25 λ =0.01 λ =0.25 λ =0.09 λ =0.09 (c) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 16.5 17 17.5 18 18.5 〈 A 〉 k E k λ =0.01 λ =0.25 λ =0.01 λ =0.25 λ =0.09 λ =0.09 (c) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 16.5 17 17.5 18 18.5 〈 A 〉 k E k λ =0.01 λ =0.25 λ =0.01 λ =0.25 λ =0.09 λ =0.09 (c) FIG. 3: T race of one pair of eigenstates | ν i with parity P = +1 (identified by full circles in Fig. 2) i n the plane of quantum inv ariants ( E ν , h A i ν as the interaction p a rameter λ is increased a specified a mount at constant val u e (a) α = 0, (b) α = π / 2, (c) α = π/ 4 of the integ rability parameter. In the integ rable regimes we use ν = ( m, n ) and in the non inte- grable regime ν = k . F or α = 0 or α = π / 2 [panels (a) and (b), res pectively , of Figs. 2 and 3], the t wo rows of states march pa st each other in an orderly fashion, undergo ing a sequence of level cro ssings in co mplete o blivion o f ea ch other’s pres- 9 ence. F or α = π / 4 [panel (c) of Figs. 2 and 3 ], on the other ha nd, all sta tes are part of a co ordinated clo c kwise lo oping motion. While every indiv idua l state main tains the same po sition in the level seque nc e , the wa ve-shap ed top row o f states has the app earance of moving stea dily to the righ t a nd the bottom row to the left. The path of an individual state in the plane of in v a rian ts is not un- like that of an H 2 O molecule in a traveling sur face water wa ve. This qualitativ e c hange in pattern caused by differ- ent settings of the para meter α requir es the assignment of mut ua lly exclusive sets of quantum num b ers to the same set of eigenstates in different para meter r egimes. The a ction quantum n umber s m, n a re the trademark of quantum in tegr abilit y . Their very existence accommo- dates lev el crossings betw een states of equal pa rit y . The level sorting quantum n umber k , on the o ther hand, is applicable when level cr ossings be tw ee n states of equal parity are pro hibited. It is the trademark of quant um nonintegrabilit y . This conclusion brings us full cir cle to the thought exp erimen t on in v ariant tori describ ed in Sec. I. If we track a n eigenstate along a clo sed path in (Λ , α )-plane, sp ecifically a path that lies par tly inside the in tegra ble regime and partly outside, its individuality cannot be maintained through a unique and consistent a ssignmen t of quan tum num b ers. On a path that fir st leads a certain stretch through the integrable reg ime and then returns through the nonin tegra ble regime, the tagged eigenstate may undergo s ev eral cr ossings on the first leg of this path and will then, o n the s econd leg, be una ble to cro ss back to its initial p osition in the level sequenc e . Bar ring a mi- nor caveat (see App endix A) this conflict in the as s ign- men t of qua n tum n umbers to eigens tates is a dep endable detecting device for the demarcation of regimes of in te- grability and nonintegrability in quantum sy s tems with few degrees of freedom. APPENDIX A: POINT OF HIG HER SYMMETR Y Conflicts in the a ssignmen t of quantum n umber s to eigenstates may ar ise for rea s ons unrelated to noninte- grability . In a study o f a tw o -spin system [22] t wo such causes were ident ified: (i) the presence o f p oint s of higher symmetry inside the int eg rable reg ime; (ii) a multiple connectedness of the in tegrable regime in the parameter space. Both causes a re rea dily ident ified as extraneous. In the context of the spin- boson model (1) o nly the first cause comes into play . In the following we describ e one scenario wher e t wo eigenstates swap p ositions in the lev el sp ectrum when track ed along a clo sed path in para meter space, a path that do es not lea ve the integrable regime. F or this pur- po se we co nsider (1) with σ = 1 2 in the extended pa ram- eter space (Λ , ω S , ω B ) at α = 0 . The energy eigen v alues E ± =  n + 1 2  ~ ω B ± 1 2 p 4Λ 2 ( n + 1 ) + ( ~ ω S − ~ ω B ) 2 , (A1) and the eigenv ecto r s | + i = cos φ | 0 , n i + sin φ | 1 , n i , (A 2 a) |−i = − sin φ | 0 , n i + cos φ | 1 , n i , (A2b) depe nd on the ang ular v ar iable φ = arctan E + − n ~ ω B − 1 2 ~ ω S Λ √ n + 1 . (A3) The p oint of higher symmetry is at Λ = 0 , ω B = ω S . Here the ener gy eigenv alues beco me doubly degenerate (for ω B > 0). W e consider the quan tum in v a rian t h S z i ± = ± 1 2 cos 2 φ (A4) defined b y expectation v alues in the eig enstates (A2). The lo op in parameter space is par ametrized as fo llo ws: ~ ω S = ~ ω B (1 + s in β ) , (A5a) Λ = ~ ω B (1 − c os β ) , (A5b) where 0 ≤ β ≤ 2 π . It cuts through the p oin t of higher symmetry a t β = 0. T he crucial p oin t is that one com- plete lo op along this path a dv a nces the angle (A3) by ∆ φ = π / 2, whic h in ter c hanges the t wo states (A2) and do es not bring bo th inv ariants (A1) and (A4) back to the same po sition. It takes t wo lo ops to return the states |±i to their original iden tity and the po in ts ( E ± , h S z i ± ) to their o riginal p osition. [1] M. T ab or, Chaos and Inte gr ability in Nonline ar Dynam- ics ( Wil ey , New Y ork, 1989). [2] M. C. Gutzwiller, Chaos in Cl a ssic al and Quantum Me- chanics (Springer-V erlag, New Y ork, 1990). [3] L. E. R ei chl, The T r ansition to Chaos in Conservat i ve Classic al Systems: Quantum Manifestations (Springer- V erlag, New Y ork, 1992). [4] R. M. Hilb orn, Chaos and Nonline ar Dynamics , (Oxford Universit y Press, 2000). 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