We study the geometric phase factors underlying the classical and the
corresponding quantum dynamics of a driven nonlinear oscillator exhibiting
chaotic dynamics. For the classical problem, we compute the geometric phase
factors associated with the phase space trajectories using Frenet-Serret
formulation. For the corresponding quantum problem, the geometric phase
associated with the time evolution of the wave function is computed. Our
studies suggest that the classical geometric phase may be related to the the
difference in the quantum geometric phases between two neighboring eigenstates.Comment: Copy with higher resolution figures can be obtained from
  http://physics.gmu.edu/~isatija by clicking on publications. to appear in the
  Yukawa Institute conference proceedings, {\it Quantum Mechanics and Chaos:
  From Fundamental Problems through Nano-Science} (2003

Balakrishnan, Radha

Satija, Indubala I.

English

arXiv

この論文は国立情報学研究所の電子図書館事業により電子化されました。We study the geometric phase factors underlying the classical and the corresponding quantum dynamics of a driven nonlinear oscillator exhibiting chaotic dynamics. For the classical problem, we compute the geometric phase factors associated with the phase space trajectories using Frenet-Serret formulation. For the corresponding quantum problem, the geometric phase associated with the time evolution of the wave function is computed. Our studies suggest that the classical geometric phase may be related to the the difference in the quantum geometric phases between two neighboring eigenstates

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TitleGeometric Phase and Classical-Quantum CorrespondenceAuthor(s)Satija, Indubala I.; Balakrishnan, RadhaCitation物性研究 (2004), 82(5): 782-785Issue Date2004-08-20URL http://hdl.handle.net/2433/97845RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityGeometric Phase and Classical-Quantum CorrespondenceIndubala 1. SatijaDepartment of Physics, George Mason University, Fairfax, VA 22030Radha BalakrishnanThe Institute of Mathematical Sciences, Chennai 600 113, IndiaWe study the geometric phase factors underlying the classical and the correspondingquantum dynamics of a driven nonlinear oscillator exhibiting chaotic dynamics. For theclassical problem, we compute the geometric phase factors associated with the phase spacetrajectories using Frenet-Serret formulation. For the corresponding quantum problem, thegeometric phase associated with the time evolution of the wave function is computed. Ourstudies suggest that the classical geometric phase may be related to the the difference in thequantum geometric phases between two neighboring eigenstates.PACS numbers: 02.40-k, 05-45.AcSince the discovery of Berry phase [1], the study of geo-metric phases based on the common mathematical themeof anholonomy, has emerged in a variety of fields. [2] TheBerry phase is a path-dependent geometric phase associ-ated with the adiabatic time evolution of the wave func-tion, associated with circuits in parameter space. Thisconcept has been extended [3-5] to non-adiabatic casesand also to non-cyclic circuits, since the phase acquiredby the wave function in any type of time evolution, mayhave a component that is of purely geometric origin. Thisphase is a gauge invariant quantity and is equal to the dif-ference between the total phase and the dynamical phaseacquired by the wave function.One of the open questions has been the classical limitof the Berry phase or its generalization describing thegeometric part of the phase of the quantum wave func-tion. For the special case of integrable Hamiltonian sys-tems, described in terms of action-angle variables, theso-called Hannay angle fh [6] represents the semiclassicallimit of the Berry phase. Berry gave an explicit formularelating the classical angle, called the Hannay angle andthe nth quantum eigenstate ¢n as, (h = -8n¢n. Therehave been some attempts to describe the classical limitof Berry phase for chaotic systems where the effort hasbeen focused on finding a generalization of the 2-formwithin Wigner-Weyl formalism. [7]Viewing the Berry phase as an anholonomy effect un-derlying dynamical evolution described by Schrodingerequation, we seek a classical analog of anholonomy, un-derlying the corresponding classical evolution describedby the Newton's equations of motion. Here, we are con-cerned with the geometric phases associated with pe-riodic, quasiperiodic and chaotic dynamics of a drivennonlinear oscillator. We compute the geometric com-ponent of the phase of the wave function using a kine-matic formulation [4] of the Berry phase [4] as givenby Mukunda and Simon. Tn the corresponding classicalproblem, we find the anholonomy underlying nonplanarperiodic phase space trajectories and then extend thisformulation to quasiperiodic and chaotic trajectories. Itshould be emphasized that unlike in Berry phase, our cir-cuits are not in parameter space but are in phase space.By treating a classical phase space trajectory as a spacecurve, we show that a geometric phase can be associ-ated with every trajectory, by using a Frenet-Serret (FS)formulation [8]. The classical geometric phase is the in-tegrated torsion of the phase trajectory, and it bears astrong analogy to the geometric phase factor associatedwith the wave function.To illustrate this correspondence, we begin with theFrenet-Serret equations, describing the time evolution ofthe orthonormal FS triad, consisting of the tangent T,the normal N, and the binormal B to the space curveret),T = VI\;N, N = -v I\; T + V T B, B = -v TN, (1)where I\; and T are respectively the curvature and tor-sion of the curve and v = Irl. Here, the overdot denotesderivative with respect to time. The above equations canbe written as, F = ~ x F, where F = T, N, or B, and~ = -v I\;B + vTT. That is, the FS triad rotates aroundT and B. One way to quantify this rotation is to work ina frame in which T is parallel transported, and measurethe angle of rotation around the tangent T. This will begiven by the angle ¢c(t) = f~ TvdtJ • Thus ¢c(t) is the ge-ometric phase characterizing the anholonomy associatedwith the corresponding phase space orbit.If we define a complex vector M = (N + iB) /.../2, theclassical geometric phase can be written ast •¢c(t) = i fo M* . M vdtJ•This expression has exactly the same form as the quan-tum geometric phase found by Berry [1], when the clas-sical vector M is replaced by a quantum eigenstate 'l/Jn.Additionally, by mapping the closed phase space trajec-tory to a circuit on a unit sphere traced by the tip of the- 782-1.051.051.04Xo1.040.8c0.6><:0.40.201.030.5c 00--0.51.03solution for the quantum wave function, we can computethe classical and the quantum geometric phases with ex-treme precision. These will be presented below.Figures 1 shows richness and complexity underlyingthe classical dynamics of the oscillator as we vary theinitial energy (initial conditions) of the oscillator. Wesee periodic orbits and quasiperiodic tori, in addition tochaotic trajectories.All results are for a fixed Wo = 1.6 which corresponds tothe oscillator frequency of 3.2, in units of w, the frequencyof the driver, putting our analysis is close to the adiabaticregime.Furthermore, we believe that the probability of induc-ing a transition (due to driving) is rather small becausethe classical energies of the particle under considerationhere are far below the threshold for the transition be-tween two quantum states nl and n2 in units of 1iwo.(2)tangent vector ( tangent indicatrix), ¢c can be shown tobe the solid angle subtended by this tangent indicatrixat the center of the sphere. [10] These results are validalso for a non-periodic trajectory, since it can always beclosed using a geodesic on the sphere. [4]Motivated by this close analogy between the geometricphase in a quantum system and the FS geometric phase,we investigate any possible relationship between the two.The underlying key question is whether the classical an-holonomy is related to the corresponding quantum one.Here we calculate the classical and the quantum geomet-ric phases for a periodically driven nonlinear oscillatorexhibiting complex dynamics. The system under inves-tigation is an "impact oscillator" [9], the oscillator thatrebounds elastically whenever its displacement x dropsto zero. For x > 0, the system is described by,The system is piecewise linear, and the analytic solutionscan be obtained for x > O. The discontinuity at the originmakes it essentially nonlinear. Without loss of generality,we choose the units of t and x such that w = 1 andf = 1. The phase space trajectory of the dynamicalsystem can be viewed as a space curve generated by thethree-dimensional vector r(t) = (x, x, x) parameterizedby the time t.As we discuss below, the impact oscillator exhibits veryrich and complex dynamics, where periodic, quasiperi-odic and chaotic dynamics coexist. Our choice of thisexample was motivated by the fact that in addition tothe simplicity underlying the classical analysis of the os-cillator, the quantum wave functions for the driven os-cillator are known in terms of the classical solution as,[11],(3)Here x' = x - xc(t), with xc(t) being the solution ofthe classical equation of motion and L is the Lagrangianof the driven system. X(x, t) is the wave function of theoscillator in the absence of driving. Note that the wavefunctions of the driven oscillator are centered on the po-sition of the classical forced oscillator.If we take X to be an eigenstate of the undriven os-cillator, Xn(x, t) = un(x) exp -i(Ent/h), the eigenfunc-tion can be written in terms of Hermite polynomials asun(x) = exp -[wo/(21i)x2]Hn(.Jwo/1ix). It should benoted that corresponding to the eigenstate, 1'IjJ(x, tW =IXn(x', tWo Therefore, the center of the wave packetxc(t) obeys classical equation of motion, and the shapeof the wave packet ( the density distribution with respectto the center xc) is unaffected by the driving force.In view of the fact that we have the analytic solutionfor the classical system for x > 0, and also a closed formFIG. 1. For po = 0, the figure shows the possible x and pvalues (once every period of the driver), as a function of Xo,the initial position of the oscillator. For Xo ~ 1.0416, there isa period-ll orbit. For Xo > 1.046, we get invariant quasiperi-odic tori or cantori as seen in the lower figure. For Xo < 1.035,almost al1 initial conditions result in chaotic dynamics.Using the classical solution, the FS geometric phase¢c can be computed as the integral of the torsion, givenby r = r· (i x f)/Ir x i1 2 . For the corresponding quan-tum problem, we can obtain the geometric phase, usingkinematic formulation by Mukunda and Simon [4]. Fora given wave function 'IjJ(x, t), the quantum geometricphase ¢qis a gauge invariant quantity is given by ¢q(t) =arg J['IjJ*(x, O)'/;'J(x, t)]dx - 1m f~ 't/J*(x, e)8t 'IjJ(x, t')dt'. Itis clear that the first term describes total phase while thesecond term describes the dynamical phase accumulatedby the wave function in time t. Note that this formu-lation does not require any circuits to define geometric- 783-should be contrasted with the the driven and dampedoscillator phases [12J where the classical phase averagedover the period of the driver was finite.phase.For the driven impact oscillator, if we substitute thesolution for the wave function, given by equation (3). ,the geometric phase can be written in terms of the ex-pectation values of < x > and the classical solution Xc-(UPq,n = it[L(t')- < x> iAt')]de + G(t) (4)where G(t) is given by, G(t) = (xe(t)xe(t) -xe(O)xe(O)) +arg[f[(un(x'(O)un(x'(t) exp W(xe(t) - xe(O))Jdx. For pe-riodic evolution, G(t) = O. Also, G(t) = 0 if we consider tvalues where the classical particle is at the turning points.In view of this, we will confine our calculations of geo-metric phases to only such values of t.Since the classical impact oscillator is confined to x ~0, the eigenstates of the corresponding quantum systemare restricted to odd n values only. Substituting the ex-plicit expression for the Hermite polynomials, < x > canbe expressed in terms of parabolic cylindrical functionswhich are functions of xc.o 5 10t15 20 25Lf~l A4n- 1(Ye)D-1 (5)< x >2n-l = 4n-lLl=l B4n- I(Ye)D-1where D-m(Ye) is a parabolic cylinder functions of orderm. Here Ye = aXe where a = J2wo/h. AI(Ye) andBI(Ye) are polynomials of Ye of degree 1that are uniquelydetermined by n. For example, for n = 1, we haveFor higher values of n, the expressions are very com-plicated. Our analysis has been confined to the groundstate n = I, and the first excited state n = 3. For all val-ues of n, h<pq,n reduces to the classical action as h -t O.We factor out this part of the phase factor <Po and write<Pq,n = <Po + <Pn· As we show below, it is the difference inthe phases between the two neighboring eigenstates thatexhibit quantum fingerprints of classical dynamics.Figures 2 and 3 show geometric phases for a fixed timeinterval T , equal to the driving period, for a periodic anda chaotic trajectory. These results as well as other similaranalysis suggest that ¢;e may be related to ¢;n-l - ¢;n.For dynamical evolution involving arbitrary time t,our detailed analysis shows that the integrated quantumphase factors can be expressed as a sum of linear andoscillatory functions of t and hence can be written as ,<Pn(t) = vnt + fn(t). Here Vn are constants, independentof t and fn(t) are oscillatory functions which are foundto retain the fingerprints of the corresponding classicaldynamics for all times.In contrast to quantum phases, the classical <Pe(t)is found to be an oscillatory function for periodic ,quasiperiodic as well as for the chaotic trajectories. ThisFIG. 2. For a period-ll trajectory, the four figures (fromtop to bottom) show the quantum geometric phases for theground state( n = 1), the first excited state ( n = 3), classicalgeometric phase <Pc and d4 = <PI - <P3' respectively."l~60 70 80 90 100<>1~60 70 80 90 100~:;~60 70 80 90 1000.2~~ -o.~-0.4-0.660 70 80 90 100tFIG. 3. For a chaotic trajectory, same plots as in Fig. 2In the third figure, we superimpose d¢ with ¢c.The results for arbirary time evolution are shown infigures 4 and 5. Here we factor out the linearly increas-ing parts of the phases and show the fluctuating partsalong with the classical phase. For the parameter valuescorresponding to the figures 2 to 5, vt/(21r) = .4218 andv3/(21r) = .3956 for the periodic orbit and vt/(21r) =- 784-.3723 and va/(27f) = .4021 for the chaotic orbits. We no-tice that cPc is of the same order of magnitude as f1 - fawith ¢c exhibiting intermittent fluctuations. In view ofthe fact that VI ~ Va, it is conceivable that Vn approachesa constant: independent of n as n -+ 00 and therefore ¢cmay be related to the drj; for arbitrary time t. This isanalogous to Berry's relation en = -on'l./;n which wastrue for integrable systems in semiclassical limit.o 5tIn summary, our results as depicted above describe pre-liminary studies of classical and quantum phases under-lying a driven nonlinear oscillator. An interesting resultis the possible relationship between the classical phasesand the difference in the quantum phases between thetwo neighboring states, reminiscent of the Berry relationBit = -on¢n' As a consequence, the smalless of the clas-sical phase will have its origin in the relative stability ofthe quantum phase with respect to the quantum num-ber n. It is rather intriguing that the fluctuations in thequantum geometric phases retain the finger prints of thecorresponding classical dynamics. Further detailed stud-ies involving higher excited states are needed to confirmthese speculative views.In physical applications such as atom optics, Hamil-tonians of the form, H(x, t) = Ho + "\xsin(wt) are rele-vant where He is the time-independent Hamiltonian andthe time-dependent term describes the interaction witha single mode radiation field in dipole approximation.For nonlinear He, nontrivial dynamics may lead to manysurprises. The impact oscillators exhibit dynamics withmany features that are typical of a nonlinear oscillator.This suggests that the driven impact oscillator may pro-vide an interesting theoretical model to explore variousissues relevant to quantum chaos.The research of lIS is supported by National ScienceFoundation Grant No. DMR 0072813.FIG. 5. For a chaotic trajectory, the same plots as in Fig.4.FIG. 4. For a period-ll trajectory, the top two figures showthe oscillatory components it and h of the net accumulatedphases for the ground state and the excited state. The thirdfigure shows the classical phase and the differnce d/ = it - h.The lowest figure shows the position of the oscillator.-:!~60 70 80 90 100_o:!~~:f~O60 70 80 90 100x o~60 70 80t90 100[1] M. V. Berry, Proc. Roy. Soc. London A 392, 45 (1984).[2] See, for example, Geometrical Pha.ses in Physics: editedby A. Shapere and F. Wilczek ( World Scientific, Singa-pore, 1989), and references therein.[3] Y. Aharanov and J. Anandan, Phys. Rev. Lett. 58, 1593(1987);[4] N. Mukunda and R. Simon, Ann. Phys. (N.Y.) 228, 205(1993).[5] J. Samuel and R. Bhandari, Phys. Rev. Lett. 60, 2339(1988).[6] J. H. Hannay, J. Phys. A: Math. Gen. 18, 221 (1985).[7] J. M. Robbins and M. V. Berry, Proc. R. Soc. Lond. A(1992) 436, 631.[8] D.J. Struik, Lectures on Classical Differential Geometry(Addison-Wesley, Reading, Mass., 1961).[9] See, for example, J.M.T. Thompson and H. B. Stewart,Nonlinear Dynamics and Chaos (Wiley, New York, 1986),Chapter 14, and references therein.[10] Radha Balakrishnan, A. R. Bishop, and R. Dandoloff,Phys. Rev. Lett. 64, 2107 (1990).[11] See Y. Nogamin, Am. J. Phys, 59, 64 (1991).[12] "Anholonomy and Geometrical localization in Nonlin-ear oscillator", Radha Balakrishnan and Indubala Satija,(preprint)nlin.CD/0303071.- 785-

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