For citation:
Kuznetsov S. P. Arnold’s cat map: quantum chaos and operator dynamics in Heisenberg representation. Izvestiya VUZ. Applied Nonlinear Dynamics, 1998, vol. 6, iss. 3, pp. 3-48.
Arnold’s cat map: quantum chaos and operator dynamics in Heisenberg representation
The quantum model system is considered for which the classic analog is the known Arnold’s саг map. Due to periodicity conditions for the phase space, quantum states are represented by vectors of finite dimension N, and operators by NxN matrices. The integer parameter N characterizes a relative value of quantum effects; classic limit corresponds to N—>∞. Operator тар is suggested which governs discrete time evolution in Heisenberg representation for operators of finite shifts for position and momentum. Explicit form of evolution operator is stated in Schrodinger representation. Solution for non—stationary problem is presented and discussed for initial conditions taken as localized state, two delta—spikes, Gaussian wave packet. Quantum dynamics in terms of Husimi distribution and Wigner function, quasi—cnergy spectrum and eigenvector structure are discussed on а basis of dynamics of Heisenberg operators.
- Cvitanovic P, Percival I, Wirzba A, editors. Quantum Chaos — Quantum Measurement. Dordrecht: Springer; 1992. 332 p. DOI: 10.1007/978-94-015-7979-7.
- Nakamura K. Quantum chaos. A New Paradigm of Nonlinear Dynamics. Cambridge: Cambridge University Press; 1993. 224 p.
- Elyutin PV. The quantum chaos problem. Phys. Usp. 1988;31(7):597-622. DOI: 10.1070/PU1988v031n07ABEH003572.
- Ott Е. Chaos in Dynamical Systems. Cambridge: Cambridge University Press; 1993. 385 p. DOI: 10.1017/CBO9780511803260.
- Cerdeira H, Ramaswamy R, Gutzwiller MC, Casati G, editors. Quantum Chaos. Singapore: World Scientific; 1991. 480 p. DOI: 10.1142/1375.
- Reichl LE. The Transition to Chaos. In Conservative Classical Systems: Quantum manifestation. N.Y.: Springer; 1992. 551 p. DOI: 10.1007/978-1-4757-4352-4.
- Landau LD, Lifshits EM. Quantum Mechanics. Non-relativistic Theory. М.: Nauka; 1974. 752 p. (in Russian).
- Fock VA. Fundamentals of Quantum Mechanics. M.: Mir; 1986. 362 p.
- Feynman RP, Leighton RB, Sands M. The Feynman Lectures on Physics. MA: Addison-Wesley; 1963.
- Dirac P. The Principles of Quantum Mechanics. Oxford: Oxford University Press; 1930. 257 p.
- Einstein А. On the quantum theorem of Sommerfeld and Epstein. In: The Collected Papers of Albert Einstein. The Berlin Years Writings, 1914-1917. Vol. 6. Princeton: Princeton University Press; 1997 P. 434-443.
- Schuster HG. Deterministic Chaos. An Introduction. Weinheim: Physik—Verlag; 1984. 220 p.
- Arnold VI. Ordinary Differential Equation. Berlin: Springer; 1992. 338 p.
- Devaney RL. An Introduction to Chaotic Dynamical Systems. Amsterdam: Addison-Wesley Publ.Comp.; 1986. 320 p.
- Hannay JH, Berry MV. Quantization of linear maps on а torus — Fresnel diffraction by а periodic grating. Physica D. 1980;1(3):267-290. DOI: 10.1016/0167-2789(80)90026-3.
- Zaslavskii GM. Stochasticity of Dynamical Systems. М.: Nauka; 1984. 267 p. (in Russian).
- Ford J, Mantica G, Ristow GH. The Arnold’s cat: Failure оf the correspondence principle. Physica D. 1991;50(3):493-520. DOI: 10.1016/0167-2789(91)90012-X.
- Eckhardt В. Exact eigenfunctions for а quantised mар. J. Phys. A: Math. Gen. 1986;19(10):1823-1831. DOI: 10.1088/0305-4470/19/10/023.
- Weyers J. The quantum groups GLq(n) and Weyl-Heisenberg operators. Phys. Lett. B. 1990;240(3-4):396-400. DOI: 10.1016/0370-2693(90)91118-U.
- Athanasiu GG, Floratos EG. The lightcone SUq(2) quantum algebra аs dynamical symmetry of the Azbel — Hofstadter problem. Phys. Lett. B. 1995;352:105-110. DOI: 10.1016/0370-2693(95)00464-V.
- Shrodinger E. The continuous transition from micro- to macro- mechanics. In: Collected Papers on Wave Mechanics. London: Blackie and Son Limited; 1928. P. 41-44.
- Klauder JR, Sudarshan ECG. Fundamentals of Quantum Optics. N.Y.: W.A. Benjamin; 1968. 279 p.
- Saraceno M. Classical structures in the quantized baker transformation. Аnnals of Physics. 1990;199(1):37-60. DOI: 10.1016/0003-4916(90)90367-W.
- Balescu R. Equilibrium and Nonequilibrium Statistical Mechanics. N.Y.: Wiley; 1975. 742 p. DOI: 10.1002/bbpc.19770810244.
- Ford J, Mantica G. Does quantum mechanics оbеу the correspondence principle? Is it complete? Am. J. Phys. 1992;60(12):1086-1098. DOI: 10.1119/1.16954.
- Kasperkovitz Р, Peev M. Wigner—Weyl Formalisms for toroidal geometrics. Annals of Physics. 1994;230(1):21-51. DOI: 10.1006/aphy.1994.1016.
- Agam O, Brenner N. Semiclassical Wigner functions for quantum maps оn а torus. J. Phys. A: Math. Gen. 1995;28(5):1345-1360. DOI: 10.1088/0305-4470/28/5/020.
- Mehta ML. Eigenvalues and eigenvectors of the finite Fourier transform. J. Math. Phys. 1987;28(4):781-785. DOI: 10.1063/1.527619.
- Gutzwiller МG. Chaos with few degrees оf freedom. Progress оf Theor. Phys. Suppl. 1994;116:1-16. DOI: 10.1143/PTPS.116.1.
- Keating JP. Asymptotic properties of the periodic orbits of the cat maps. Nonlinearity. 1991;4(2):277-308. DOI: 10.1088/0951-7715/4/2/005.
- Keating JP. The cat maps: quantum mechanics and classical motion. Nonlinearity. 1991;4(2):309-342. DOI: 10.1088/0951-7715/4/2/006.
- Ozorio dе Almeida АМ, dа Luz MGE. Path integrals and edge corrections for torus maps. Physica D. 1996;94(1-2):1-18. DOI: 10.1016/0167-2789(95)00308-8.
- Percival I, Vivaldi F. Arithmetical properties оf strongly chaotic motions. Physica D. 1987;25(1-3):105-130. DOI: 10.1016/0167-2789(87)90096-0.
- Percival I, Vivaldi F. A linear code for the sawtooth and cat maps. Physica D. 1987;27(3):373-386. DOI: 10.1016/0167-2789(87)90037-6.
- Bartuccelli M, Vivaldi F. Ideal orbits of toral automorphisms. Physica D. 1989;39(2-3):194-204. DOI: 10.1016/0167-2789(89)90004-3.
- Mantica G, Ford J. On the completeness оf the classical limit of quantum mechanics. In: Cvitanovic P, Percival I, Wirzba A, editors. Quantum Chaos — Quantum Measurement. NATO ASI Series. Vol. 358. Dordrecht: Springer; 1992. P. 241-248. DOI: 10.1007/978-94-015-7979-7_19.
- Eckhardt В. Quantum mechanics of classically non—integrable systems. Phys. Rep. 1988;163(4):205-297. DOI: 10.1016/0370-1573(88)90130-5.
- Toda M, Ikeda K. Quantal Lyapunov exponent. Phys.Lett. A. 1987;124(3):165-169. DOI: 10.1016/0375-9601(87)90245-3.
- Faisal FНМ, Schwengelbeck U. Unified theory оf Lyapunov exponents аnd positive example оf deterministic quantum chaos. Phys. Lett. A. 1994;207(1-2):31-36. DOI: 10.1016/0375-9601(95)00645-J.
- Klimek S, Lesniewski А. Quantized chaotic dynamics and non-commutative КS entropy. Annals of Physics. 1996;248(2):173-198. DOI: 10.1006/aphy.1996.0056.
- Esposti MD, Graffi S, Isola S. Classical limit оf the quantized hyperbolic toral automorphisms. Commun. Math. Phys. 1995;167:471-507. DOI: 10.1007/BF02101532.
- Bouzouina А, De Bievre S. Equipartition оf the eigenfunctions of quantized ergodic maps оn the torus. Commun. Math. Phys. 1996;178:83-105. DOI: 10.1007/BF02104909.
- Tabor М. A semiclassical quantization of area—preserving maps. Physica D. 1983;6(2):195-210. DOI: 10.1016/0167-2789(83)90005-2.
- Dematos MB, Dealmeida AMO. Quantization оf Anosov maps. Annals of Physics. 1995;237(1):46-65. DOI: 10.1006/aphy.1995.1003.
- Lakshminarayan А, Balazs NL. On the quantum cat and sawtooth maps — Return to generic behavior. Chaos, Solitons and Fractals. 1995;5(7):1169-1179. DOI: 10.1016/0960-0779(94)E0060-3.
- Leboeuf P, Voros А. Chaos—revealing multiplicative representation of quantum eigenstates. J. Phys. A: Math. Gen. 1990;23(10):1765-1774. DOI: 10.1088/0305-4470/23/10/017.
- Knabe S. On the quantisation of Arnold’s cat. J. Phys. A: Math. Gen. 1990;23(11):2013-2025. DOI: 10.1088/0305-4470/23/11/025.
- Isola S. 3 —function and distribution оf periodic orbits of toral automorphisms. Europhys. Lett. 1990;11(6):517-522. DOI: 10.1209/0295-5075/11/6/006.
- Weigert St. The configurational quantum саt mар. Z. Physik B - Condensed Matter. 1990;80:3-4. DOI: 10.1007/BF01390645.
- Benatti F, Narnhofer H, Sewell GL. A non—commutative version оf the Arnold cat mар. Lett. Math. Phys. 1991;21:157-172. DOI: 10.1007/BF00401650.
- Kolovsky AR. Condition оf correspondence between quantum and classical dynamics for а chaotic system. Phys. Rev. Lett. 1996;76(3):340-343. DOI: 10.1103/PhysRevLett.76.340.
- Kolovsky AR. Quantum coherence, evolution of the Wigner function, and transition from quantum to classical dynamics for а chaotic system. CHAOS. 1996;6(4):534-542. DOI: 10.1063/1.166201.
- Balazs NL, Voros А. The quantized Baker’s transformation. Europhys. Lett. 1987;4(10):1089-1094. DOI: 10.1209/0295-5075/4/10/001.
- Balazs NL, Voros A. The quantized Baker’s transformation. Annals оf Physics. 1989;190(1):1-31. DOI: 10.1016/0003-4916(89)90259-5.
- O’Connor PW, Tomsovic S. The unusual nature of the quantum Baker’s transformation. Annals оf Physics. 1991;207(1):218-264. DOI: 10.1016/0003-4916(91)90184-A.
- Ozorio de Almeida AM, Saraceno M. Periodic orbit theory for the quantized baker’s mар. Annals of Physics. 1991;210:1.
- Lakshminarayan А, Balazs NL. The classical and quantum mechanics оf lazy baker maps. Annals of Physics. 1993;226(2):350-373. DOI: 10.1006/aphy.1993.1073.
- Saraceno M, Voros А. Towards а semiclassical theory оf the quantum baker’s mар. Physica D. 1994;79(2-4):206-268. DOI: 10.1016/S0167-2789(05)80007-7.
- Dittes FM, Doron E, Smilansky U. Long—time behavior оf the semiclassical baker’s map. Phys. Rev. E. 1994;49(2):R963-R966. DOI: 10.1103/physreve.49.r963.
- Hannay JN, Keating JP, Ozorio de Almeida АМ. Optical realization оf the haker's transformation. Nonlinearity. 1994;7(5):1327-1342. DOI: 10.1088/0951-7715/7/5/003.
- Lakshminarayan А, Balazs NL. On the noncommutativity оf quantization and discerete time evolution. Nuclear Phys. A. 1994;572(1):37-47. DOI: 10.1016/0375-9474(94)90419-7.
- Lakshminarayan А, Balaz NL. Relaxation and localization in interacting quantum maps. J. Stat. Phys. 1994;77(1-2):311-344. DOI: 10.1007/BF02186844.
- Lakshminarayan А. On the quantum baker’s mар and its unusual traces. Annals оf Physics. 1995;239(2):272-295. DOI: 10.1006/aphy.1995.1035.
- Saraceno M, Vallejos RO. The quantized D—fransformation. CHAOS. 1996;6(2):193-199. DOI: 10.1063/1.166164.
- Boasman PA, Smilansky U. Quantization оf monotonic twist maps. J. Phys. А: Math. Gen. 1994;27(4):1373-1386. DOI: 10.1088/0305-4470/27/4/031.
- Lakshminarayan A, Balaz NL. On the quantization оf linear maps. Annals оf Physics. 1991;212(2):220-234. DOI: 10.1016/0003-4916(91)90115-O.
- Lakshminarayan А. Semiclassical theory оf the sawtooth mар. Phys. Lett. A. 1994;192(5-6):345-354. DOI: 10.1016/0375-9601(94)90217-8.
- Nakamura K. Introduction to quantum chaos. Chaos, Solitons and Fractals. 1995;5(7):1035-1048. DOI: 10.1016/0960-0779(94)E0052-Q.
- Bogomolny EB, Georgeot B, Giannoni MJ, Schmit С. Quantum chaos оn constant negative curvature surfaces. Chaos, Solitons and Fractals. 1995;5(7):1311-1323. DOI: 10.1016/0960-0779(94)E0067-Y.
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