Upper quantum Lyapunov exponent and parametric oscillators

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 45, NUMBER NOVEMBER 2004 Upper quantum Lyapunov exponent and parametric oscillators H. R. Jauslin, O. Sapin, and S. Guérin Laboraoire de Physique CNRS - UMR 5027, Université de Bourgogne, BP 47870, F-2078 Dijon, France W. F. Wreszinski Instituto de Física, Universidade de São Paulo, Caixa Postal 6638, 0535-970 São Paulo, Brazil (Received 2 November 2003; accepted 28 July 2004; published 25 October 2004) We introduce a definition of upper Lyapunov exponent for quantum systems in the Heisenberg representation, and apply it to parametric quantum oscillators. We provide a simple proof that the upper quantum Lyapunov exponent ranges from zero to a positive value, as the parameters range from the classical system s region of stability to the instability region. It is also proved that in the instability region the parametric quantum oscillator satisfies the discrete quantum Anosov relations defined by Emch, Narnhofer, Sewell, and Thirring. 2004 American Institute of Physics. [DOI: 0.063/.803926] I. INTRODUCTION Several definitions of quantum Lyapunov exponents have been proposed (see Refs. 6 and references therein). Due to the fact that there are few examples on which these definitions can be explicitly tested in detail, their range of applicability is not well established. Emch and co-workers,2 formulated a definition of quantum Anosov systems, and the associated quantum Lyapunov exponents. This definition of the Lyapunov exponents is however too rigid, and it is not applicable to systems that do not have a global structure with a constant hyperbolicity property. Majewski and Kuna 5 extracted from Ref. a more general definition, that is still too restrictive, since it assumes the existence of certain limits that one cannot expect to be well defined in general. In a quite different approach, Vilela Mendes 6 introduced a definition of the quantum Lyapunov exponents based on the probability density x,t = x,t 2 in the position representation. The limitation of this approach is that it singles out the position representation, and thus it is not easily related to the dependence on initial conditions in phase space (x, and p). In the present paper we propose a definition of the upper quantum Lyapunov exponent that is close in spirit to the ones of Refs. and 5, but is more general and with a wider range of applicability. It is formulated in terms of the evolution of observables in an algebraic setting, and based by analogy on the classical upper Lyapunov exponent, as defined in the general context of cocycles. 7,8 An essential ingredient is the behavior with respect to changes in the initial conditions. In Refs., 3, and 5 these are taken in a very general framework, by considering the variations generated by all the derivations on the algebra of observables. For our definition we restrict the class of variations of initial conditions to those derivations that correspond to translations in phase space, i.e., in x and in p. We illustrate this definition with the example of a parametric quantum oscillator, which shows the utility of the extension. II. DEFINITION OF THE UPPER QUANTUM LYAPUNOV EXPONENT Consider a quantum mechanical particle and let xˆ and pˆ denote its coordinate and momentum operators, on the Hilbert space H=L 2 R,dx, with xˆ, pˆ =i. We choose the units such that =. We define the self-adjoint operator 0022-2488/2004/45()/4377/9/$22.00 4377 2004 American Institute of Physics

4378 J. Math. Phys., Vol. 45, No., November 2004 Jauslin et al. L ª p pˆ + x xˆ, with p, x R 2. It defines a derivation, acting on operators A A, where A denotes the algebra of observables, by A = L,A, A Dom, 2 where Dom A A such that L,A A. 3 This derivation may be interpreted as a derivation in the direction of phase space determined by p, x and is thus naturally suggested by classical mechanics. Let U t,t 0 denote the unitary propagator which defines the dynamics, with initial time t 0.In order to proceed we must also specify the algebra of observables. Experience with examples 2 suggests the choice of the Weyl algebra W, which consists of finite linear combinations of the operators This is particularly natural because W Dom, since W, = exp i xˆ + pˆ,, R 2. 4 L,W, = p x W,. 5 Equivalently, 2 one can consider polynomials in W=exp i xˆ + pˆ with the multiplication law W z W z =exp i z,z W z+z, with zª,, z ª,, and the symplectic form z,z = /2. Note that once the equations (20) and (2) of Theorem 2 have been established, the C* character of the algebra is inessential: a -algebra is sufficient (see Ref., Remark 3.7, No. 6). We also assume that the dynamics defines an automorphism of W, U t,t 0 AU t,t 0 W, A W, t,t 0 R. 6 Under the above assumptions, we can formulate the following. Definition: The upper quantum Lyapunov exponent is defined as = sup R 2, 7 where U,L,A,t 0 ªlim sup t t ln L,A t,t 0 8 and A t,t 0 ªU t,t 0 AU t,t 0 9 and A W. The norm is chosen as A =sup H A /. Remarks: () This definition is adapted from the general formulation for cocycles described, e.g., in Ref. 8. is expected to be independent of t 0 and of the choice of the observable A, under suitable conditions, e.g., L,A t,t 0 0. 0 (2) If the limit in (8) exists, then it is called the Lyapunov exponent U,L,A,t 0. (3) Because of the unitarity of the time evolution, the exponent can also be expressed as

J. Math. Phys., Vol. 45, No., November 2004 Upper quantum Lyapunov exponent 4379 with U,L,A,t 0 = lim sup t ln L t 0,t,A t L t 0,t ªU t 0,t L U t 0,t. (4) The main differences with the earlier definitions of Refs., 2, and 5 are the lim sup in Eq. (8) and the restriction to the derivations () corresponding to directions in phase space. 2 III. APPLICATION TO PARAMETRICALLY DRIVEN QUANTUM OSCILLATORS In order to show the usefulness of the above definition, we consider the parametric quantum oscillator, 9 one of the simplest paradigms of the transition from regular to unstable behavior in classical mechanics. 2,3 The Hamiltonian (we take the mass=) is where f is a periodic function of period T, H t = 2 pˆ 2 + 2 f t xˆ2, 3 f t + T = f t. It is convenient to decompose f as f t =E+ f za with E= /T 0 T dtf t [and thus /T 0 T dtf za t =0]. We will analyze the one parameter family of systems defined by varying the average E R. The classical equation corresponding to (3) is Hill s equation, 4,5 ẍ + f t x =0, which is well known (Ref. 4, Chap. 4, and Ref. 5) to have bands of stability regions S, and instability regions I ( gaps ), when the parameter E is varied. For the quantum parametric oscillators we will prove the following results: Theorem : For any observable A=W, of the form (4), in the stability region E S one has 4 5 U,L,A,t 0 =0,, t 0. 6 In the instability region E I, there is a stable direction s, which depends on t 0, for which s U,L s,a,t 0 = r 0, 7 whereas for all other directions, U,L,A,t 0 = r 0, 8 where r is the absolute value of the real part of the Floquet exponent of the corresponding classical oscillator defined below in Eq. (32). Thus the upper Lyapunov exponent is = sup = r 0. 9 There is thus a transition in the upper quantum Lyapunov exponent as the parameter E ranges from the classical system s region of stability to the instability region. Theorem 2: If we consider the time evolution in the instability region E I at discrete times t n =n2t given by even integer multiples of the period T, there is an unstable eigendirection + p+, x+ such that the corresponding derivation L + satisfies for all t n,

4380 J. Math. Phys., Vol. 45, No., November 2004 Jauslin et al. e il + se r t n U t n,0 = U t n,0 e il + s, and a stable eigendirection p, x such that 20 e il se r t n U t n,0 = U t n,0 e il s, where r 0 is the real part of the classical Floquet exponent. Thus the system satisfies the discrete quantum Anosov relations defined by Emch, Narnhofer, Sewell, and Thirring. 3 Remarks: This system satisfies the quantum Anosov relations but it lacks two other properties that enter in the definition of quantum Anosov systems as formulated in Refs. 3: There is no state that is invariant with respect to U t n,0, nor with respect to e il ± s [U t n,0 has no eigenvalues for E I and there are no translation invariant states in H=L 2 R,dx ]. However, as it was remarked in Ref. (Remark 3.7, No. 7) the requirement of the invariant state may be dispensed with, depending on the intended application. The relations (20) and (2) can be written equivalently, by changing t n t n as 2 e il + se r t n U t n,0 = U t n,0 e il + s, 22 e il se r t n U t n,0 = U t n,0 e il s. With this representation (22) one can give an intuitive interpretation for the Lyapunov exponent in the Schrödinger picture: We compare the forward time evolution t n+ t n of two initial states and + that are related to each other by a translation in phase space in direction + of size s. The time evolution U t n,0 + to time t n will yield a state +, that can be related to =U t n,0 by a translation in phase space in the same direction + but of a size s that is exponentially amplified s =s e rt n. This can be visualized by the following diagram: e il + s + U t n,0 U t n,0 e il + se r t n + For Anosov systems this property is satisfied globally for any size s of the translation. The definition of the Lyapunov exponent () can be interpreted on the basis of this picture but taking infinitesimally small translations. In order to determine the upper quantum Lyapunov exponent according to Eq. () we first need to calculate L t 0,t. Lemma : 23 24 L t ;t 2 = p t ;t 2 pˆ + x t ;t 2 xˆ 25 with p t ;t 2 = e t t 2 +h p+ t ;t 2 + e t t 2 h p t ;t 2, 26 x t ;t 2 = e t t 2 +h x+ t ;t 2 + e t t 2 h x t ;t 2, and h p± t ;t 2, h x± t ;t 2 are periodic functions of the same period T of the function f, and ± are the Floquet exponents associated to the classical dynamics [defined below in Eq. (32)]. In the stability bands E S, ± are purely imaginary, whereas in the instability regions E I, ± has a nonvanishing real part, that we will denote ± r. Proof of Lemma : The classical equation (5) may be written 27

J. Math. Phys., Vol. 45, No., November 2004 Upper quantum Lyapunov exponent 438 d dt x = p 0 f t p 28 0 x. The propagator F t,t 0 of the classical equation, defined by x t p t = F t,t 0 p t 0 F t,t = t, 29 x t 0, may be written, by Floquet s theorem 5 as the following. Lemma 2: where F t,t 0 = G t e t t 0 B G t 0, 30 G t + T = G t is an invertible differentiable matrix, and B a constant traceless matrix. Thus B is of one of the following three forms: () B has two complex eigenvalues 3 ± =± r + i i 0 32 we choose the notation such that r 0 : Thus B is diagonalizable and there are two cases, a r =0 stable case, band, b r 0 unstable case, gap. In these two cases one can write F t,t 0 = G t Se t t 0 + 0 0 S G t 0, 33 where S is some invertible matrix. 2 ± =0. In this case B is not diagonalizable, and has Jordan canonical form 0 0 band edge, which we shall not discuss further in this paper. 0. It corresponds to a Using the fact that the Heisenberg equations of motion for the operators xˆ t and pˆ t have the same form as the classical equations for p t and x t, we can write U t,t 0 pˆ U t,t 0 U = F t,t t,t 0 xˆ U t,t 0 0 g t pˆxˆ = e t t0 + 0 0 e t t 0 g t 0 pˆxˆ, 34 with g t G t S. Writing explicitly the matrix elements g t g ij t,g t 2 g ij t 2, we find, by (26), (27), and (34) that the functions in Eq. (26) are given by h p+ t ;t 2 = p g t + x g 2 t g t 2, h p t ;t 2 = p g 2 t + x g 22 t g 2 t 2, h x+ t ;t 2 = p g t + x g 2 t g 2 t 2, 35 36 37 h x t ;t 2 = p g 2 t + x g 22 t g 22 t 2, 38 which are periodic in t and in t 2 since g t is periodic. Proof of Lemma 2: The time evolution is symplectic, thus the Jacobi matrix F t,t 0 of the map (2) satisfies 2

4382 J. Math. Phys., Vol. 45, No., November 2004 Jauslin et al. FJF T = J, J = 0 0 39 and hence det F=. Thus we may define e BT ªF T,0, i.e., Bª /T ln F T,0, and define G t ªF t,0 e Bt. Then G t + T = F t + T,0 e B t+t = F t + T,T F T,0 e B t+t = F t,0 e Bt = G t, which is Floquet s theorem. 5 It follows that F t,t 0 = F t,0 F 0,t 0 = F t,0 F t 0,0 = G t e B t t 0 G t 0, which is (30). Now F T,0 =G T e BT G 0 and =det F T,0 =det G T det e BT det G 0 =det e BT because G T =G 0, hence Tr B=0. Proof of Theorem : The Weyl algebra of observables is in the domain of the family of derivations defined as t 0,t Aª L t 0,t,A 40 parametrized by the time variable, Indeed, by (5) and (25), ifa=e i xˆ+ pˆ then t W 0 Dom,t, t 0,t R. L t 0,t,A = p t 0,t x t 0,t A W. 4 In order to determine the Lyapunov exponent we calculate L t 0,t,A = p t 0,t x t 0,t, 42 where we have used A =. By (26), (27), and (35) (38) we may write, for t,t 0 R +, L t 0,t,A = e t t 0 + p g 2 t 0 + x g 22 t 0 g 2 t g 22 t + e t t 0 p g t 0 + x g 2 t 0 g t g 2 t, 43 where we have used = +. The stable direction s is determined by the condition that the curly bracket in the first term vanishes, which leads to p g 2 t 0 + x g 22 t 0 =0, 44 s ps, xs = s g 22 t 0,g 2 t 0, where s is an arbitrary constant. We remark that the second term in (43) is not identically zero since 45 p g t 0 + x g 2 t 0 = s g 22 t 0 g t 0 + g 2 t 0 g 2 t 0 = s det g 0. In this case we can write 46 L s t 0,t,A = e t rs t, 47 where r is the absolute value of the real part of ±, and S t is a periodic function of t. There is thus a sequence of times t k, such that S t k 0, for some constant, and therefore

J. Math. Phys., Vol. 45, No., November 2004 Upper quantum Lyapunov exponent 4383 s U,L s,a,t 0 = r + lim sup t ln S t = r. t For all other directions we can write L t 0,t,A = e t rr t, 48 49 where R t is the sum of a periodic function and one that is either exponentially decreasing or identically zero. Thus there is a sequence of times t k, such that R t k 0, for some constant, and therefore U,L,A,t 0 = r + lim sup t t ln R t = r. 50 In the stability bands E S, ± are imaginary, and thus =0 for all directions. Remark: A simple extension of this proof shows that Theorem is also true for any observable A W provided that L,A 0. Proof of Theorem 2: We remark first that (20) and (2) are equivalent, by deriving with respect to s at s=0 to U t n,0 L + U t n,0 = e r t nl +, 5 U t n,0 L U t n,0 = e r t nl. 52 The eigendirections are thus determined by the conditions L ± t n,0 = e ± t nl ± 0,0. 53 According to Eqs. (25) (27), since h p± t,0 and h x± t,0 are T periodic and since = +, these conditions will be satisfied. For +,ifh p+ t n,0 =h p+ 0,0 =0 and h x+ t n,0 =h x+ 0,0, i.e., according to (35) and (37), if p+ g 0 + x+ g 2 0 g 2 0 =0, 54 p+ g 0 + x+ g 2 0 g 22 0 =0. 55 For,ifh p t n,0 =h p 0,0 =0 and h x t n,0 =h x 0,0, i.e., according to (36) and (38), if p g 2 0 + x g 22 0 g 22 0 =0, 56 p g 2 0 + x g 22 0 g 2 0 =0. 57 Since g is nonsingular, g 2 and g 22 cannot both be zero, and therefore for we obtain the condition which has the solution p g 2 0 + x g 22 0 =0, 58 p, x = g 22 0,g 2 0, where is an arbitrary constant. Since g is nonsingular, g 2 and g 22 cannot both be zero, and thus (59) determines the unique stable eigendirection. It coincides for t 0 with the stable s of Theorem, Eq. (45). By the same type of argument we obtain the unstable eigendirection as 59

4384 J. Math. Phys., Vol. 45, No., November 2004 Jauslin et al. We remark that + p+, x+ = + g 2 0,g 0. 60 e 2T ± = e ±2T re ±i2t i = e ±2T r, since in the gaps E I the imaginary part i of the Floquet eigenvalues takes constant values of the form 6 i = k 2, k Z. 62 2T This can be shown as follows: ± e ±T are the eigenvalues of the propagator matrix F T,0 defined in Eq. (30). Since it is a real matrix with det F T,0 =, its eigenvalues are either real or complex conjugate to each other = * +, in which case ± =. Therefore, in the instability region E I the eigenvalues ± =e ± r+i i are necessarily real, and thus i T=k, k Z. This completes the proof of the Anosov property (5) and (52). Remark : The factor /2 in Eq. (62) explains the need to take time intervals that are multiples of 2T in order to have the Anosov property. For the gaps with even k one has the Anosov property also in the discrete times t n =nt, n Z. Remark 2: We have set t 0 =0 above, but, although the eigenvalues ± do not vary upon variation of t 0, the eigendirections do. This is the reason why we have an Anosov system in discretized time but not in continuous time. Remark 3: The above example of a quantum Anosov structure is interesting because it is not global in the space of parameters E, i.e., if E S it is not realized. For E S, the upper quantum Lyapunov exponent equals zero, a case which is included in our definition, but not by Refs. 3. Remark 4: The parametric quantum oscillator (3) is, of course, very special, being quadratic in xˆ and pˆ. We have used the fact that the Heisenberg equations of motion for xˆ, pˆ have the same form as the classical Hamiltonian equations. The dynamics are, however, rich and nontrivial and not explicitly soluble both classically and quantum mechanically: the complexity manifests itself in the fact that the symmetry group of (3) is SU (,) rather than the Heisenberg group. 6 The transition in the theorem has a physical interpretation in a model of quadrupole radio-frequency traps (Paul Penning traps), see Refs. 9, 0, and 7 and references given there. Remark 5: The case of a quasiperiodic time dependence has a very interesting structure (Refs. 8 and 9), and our approach is, in principle, applicable to this case. ACKNOWLEDGMENTS W.F.W. thanks H. Narnhofer for arranging his stay at ESI, where he learned of Ref. 2. We acknowledge support of the Conseil Régional de Bourgogne. G. G. Emch, H. Narnhofer, G. L. Sewell, and W. Thirring, J. Math. Phys. 35, 5582 (994). 2 W. Thirring, What are the Quantum Mechanical Lyapunov Exponents, Proceedings of the 34 Internationale Universitätswoche für Kern und Teilchen Physik, Schladming, March 995 (Springer, Berlin, 996), pp. 223 237. 3 H. Narnhofer, Infinite Dimen. Anal., Quantum Probab., Relat. Top. 4, 85(200). 4 I. J. Peter and G. G. Emch, J. Math. Phys. 35, 5582 (994). 5 W. A. Majewski and M. Kuna, J. Math. Phys. 34, 5007 (993). 6 R. Vilela Mendes, Phys. Lett. A 7, 253(992); 87, 299 (994). 7 V. Oseledec, Trans. Mosc. Math. Soc. 9, 97(968). 8 A. Katok and B. Hasselblatt, Introduction to Modern Theory of Dynamical Systems (Cambridge University Press, Cambridge, 997). 9 M. Combescure, Ann. I.H.P. Phys. Theor. 47, 63(987); 47, 45 (987); Ann. Phys. (N.Y.) 85, 86(988). 0 M. Combescure, Ann. I.H.P. Phys. Theor. 44, 293 (986). S. Weigert, J. Phys. A 35, 469 (2002). 2 V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, Berlin, 978). 3 V. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Addison-Wesley, New York, 989).

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