1 / 47

Pavel Str ánský

1 / 674. C HAOTIC DYNAMICS AND QUANTUM STATE PATTERNS IN COLLECTIVE MODELS OF NUCLEI. Pavel Str ánský. Collaborators:. Michal Macek, Pavel Cejnar. Institute of Particle and Nuclear Phycics , Faculty of Mathematics and Physics , Charles University in Prague, Czech Republic. Jan Dobe š.

aspen-sanders
Download Presentation

Pavel Str ánský

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 1 / 674 CHAOTIC DYNAMICS AND QUANTUM STATE PATTERNS IN COLLECTIVE MODELS OF NUCLEI Pavel Stránský Collaborators: Michal Macek, Pavel Cejnar Institute of Particle and Nuclear Phycics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic Jan Dobeš Nuclear Research Institute, Řež, Czech Republic Alejandro Frank, Emmanuel Landa, Irving Morales Institutode Ciencias Nucleares, Universidad Nacional Autónoma de México 13 January 2012 ECT* Seminar*

  2. 2 / 674 CHAOTIC DYNAMICS AND QUANTUM STATE PATTERNS IN COLLECTIVE MODELS OF NUCLEI 1. Classical chaos - Stable x unstable trajectories - Poincaré sections: a manner of visualization - Fraction of regularity: a measure of chaos 2. Quantum chaos - Statistics of the quantum spectra, spectral correlations - 1/f noise: long-range correlations - Peres lattices: ordering of quantum states 3. Applications in the nuclear physics - Geometric collective model and Interacting boson model - Quantum – classical correspondence - Adiabatic separation of the collective and intrinsic motion

  3. 3 / 674 • Classical Chaos (analysis of trajectories)

  4. 1. Classical chaos Hamiltonian systems State of a system: a point in the 4D phase space Conservativesystem: Trajectory restricted to 3D hypersurface Integrals of motion: Connected with additional symetries Integrablesystem: Number of independent integrals of motion number of degrees of freedom = Canonical transformation to action-angle variables J2 Quasiperiodic motion on a toroid J1

  5. 1. Classical chaos Hamiltonian systems State of a system: a point in the 4D phase space Conservativesystem: Trajectory restricted to 3D hypersurface Integrals of motion: Connected with additional symetries Integrablesystem: Number of independent integrals of motion number of degrees of freedom = Canonical transformation to action-angle variables J2 Quasiperiodic motion on a toroid Chaotic behaviour: property of nonintegrable systems J1

  6. chaotic case – “fog” Section at y = 0 px ordered case – “circles” x 1. Classical chaos Generic conservative system of 2 degrees of freedom We plot a point every time when the trajectory crosses the plane y = 0 Poincaré sections px y x Different initial conditions at the same energy

  7. px x 1. Classical chaos Fraction of regularity Measure of classical chaos Surface of the section covered with regular trajectories Total kinematically accessible surface of the section REGULARarea CHAOTICarea freg=0.611

  8. 1. Lyapunov exponent Classical chaos –Hypersensitivity to the initial conditions 1. Classical chaos Quasiperiodic X unstable trajectories Divergence of two neighboring trajectories Regular: at most polynomial divergence Chaotic: exponential divergence 2. SALI (Smaller Alignment Index) • twodivergencies • fast convergence towardszero for chaotic trajectories Ch. Skokos, J. Phys. A: Math. Gen 34, 10029 (2001); 37 (2004), 6269

  9. Quantum Chaos (analysis of energy spectra)

  10. 2. Quantum chaos Semiclassical theory of chaos Spectral density: smooth part oscillating part given by the volume of the classical phase space Gutzwiller formula (given by the sum of all classical periodic orbits and their repetitions) The oscillating part of the spectral density can give relevant information about quantum chaos (related to the classical trajectories) Unfolding: A transformation of the spectrum that removes the smooth part of the level density Note: Improved unfolding procedure using the Empirical Mode Decomposition method in: I. Morales et al., Phys. Rev.E84, 016203 (2011)

  11. M.V. Berry, M.Tabor, Proc. Roy. Soc.A 356, 375 (1977) O. Bohigas, M. J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984), 1 2. Quantum chaos Quantum chaos: Spectral statistics level repulsion no level interaction E Nearest-neighbor spacing distribution GaussianOrthogonalEnsemble GaussianUnitary Ensemble GaussianSymplectic Ensemble P(s) REGULAR system CHAOTIC systems Ensembles of random matrices

  12. 2. Quantum chaos Spectral statistics Nearest-neighbor spacing distribution Wigner Poisson P(s) s REGULAR system CHAOTIC system distribution parameter w Brody - Artificial interpolationbetween Poisson andGOEdistribution - Measure of chaoticity of quantum systems - Tool to test classical-quantum correspondence

  13. 2. Quantum chaos Quantum chaos - examples Billiards They are also extensively studied experimentally Schrödingerequation: (for wave function) Helmholtz equation: (for intensity of el. field)

  14. 2. Quantum chaos Quantum chaos - applications Riemann zfunction: Prime numbers Riemann hypothesis: All points z(s)=0in the complex plane lieon the line s=½+iy (except trivial zeros on the real exis s=–2,–4,–6,…) GUE Zeros ofzfunction

  15. 2. Quantum chaos Quantum chaos - applications GOE Correlation matrix of the human EEG signal P. Šeba, Phys. Rev. Lett. 91 (2003), 198104

  16. 2. Quantum chaos Ubiquitous in the nature (many time signals or space characteristics of complex systems have 1/f power spectrum) 1/f noise - Fourier transformation of the time series constructed from energy levels fluctuations dn = 0 dk d4 Power spectrum d3 k d2 d1 = 0 a= 2 a= 2 a= 1 CHAOTIC system REGULAR system a= 1 Direct comparison of 3 measures of chaos A. Relañoet al., Phys. Rev. Lett. 89, 244102 (2002) E. Faleiro et al., Phys. Rev. Lett. 93, 244101 (2004) J. M. G. Gómez et al., Phys. Rev. Lett. 94, 084101 (2005)

  17. regular regular chaotic 2. Quantum chaos Peres lattices Quantumsystem: Infinite number of of integrals of motion can be constructed (time-averaged operators P): Lattice: energy Eiversus value of partly ordered, partly disordered lattice always ordered for any operator P Integrable nonintegrable B = 0 B = 0.445 <P> <P> E E A. Peres, Phys. Rev. Lett.53, 1711 (1984)

  18. 3. Application to the collective models of nuclei

  19. 3a. Geometric collective model Geometric collective model Surface of homogeneous nuclear matter: (even-even nuclei – collective character of the lowest excitations) Monopole deformations l = 0 - “breathing” mode - Does not contribute due to the incompressibility of the nuclear matter Dipole deformations l = 1 • Related to the motion of the center of mass • - Zero due to momentum conservation

  20. 3a. Geometric collective model Geometric collective model Surface of homogeneous nuclear matter: Quadrupole deformations l = 2 Quadrupole tensor ofcollective coordinates (2 shape parameters, 3 Euler angles) Corresponding tensor of momenta T…Kinetic term V…Potential Neglect higher order terms neglect 4 external parameters G. Gneuss, U. Mosel, W. Greiner, Phys. Lett. 30B, 397 (1969)

  21. 3a. Geometric collective model Geometric collective model Surface of homogeneous nuclear matter: Quadrupole deformations l = 2 Quadrupole tensor ofcollective coordinates (2 shape parameters, 3 Euler angles) Corresponding tensor of momenta T…Kinetic term V…Potential Neglect higher order terms neglect 4 external parameters Adjusting 3 independent scales energy (Hamiltonian) Scaling properties size (deformation) time 1 “classicality” parameter 1 “shape” parameter sets absolute density of quantum spectrum (irrelevant in classical case) (order parameter) P. Stránský, M. Kurian, P. Cejnar, Phys. Rev. C 74, 014306 (2006)

  22. 3a. Geometric collective model Principal Axes System (PAS) g Shape variables: b Shape-phase structure Phase separatrix B V V A b C=1 b Deformedshape Spherical shape

  23. 3a. Geometric collective model Dynamics of the GCM Classical dynamics – Hamilton equations of motion Nonrotating case J = 0! Quantization – Diagonalizationinthe oscillator basis 2 physically importantquantization options (with the same classical limit): • An opportunity to test the Bohigas conjecture in different quantization schemes (a) 5D system restricted to 2D (true geometric model of nuclei) (b) 2D system

  24. H’ Independent Peresoperators in GCM L2 5D L2 2D 3a. Geometric collective model Peres operators Nonrotating case J = 0! (a) 5D system restricted to 2D (true geometric model of nuclei) (b) 2D system P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79, 046202 (2009); 066201 (2009)

  25. 3a. Geometric collective model Complete map of classical chaos in GCM Integrability Veins of regularity chaotic Shape-phase transition regular “Arc of regularity” control parameter Global minimum and saddle point HO approximation Region of phase transition

  26. 3a. Geometric collective model Peres lattices in GCM Small perturbationaffects only a localized part of the lattice (The place of strong level interaction) B = 0 B = 0.005 B = 0.05 B = 0.24 <L2> Peres lattices for two different operators Remnants of regularity <H’> E Integrable Increasing perturbation Empire of chaos

  27. 5D 3a. Geometric collective model “Arc of regularity”B= 0.62 • b – g vibrations resonance <L2> <VB> 2D (different quantizations) E Connection with IBM: M. Macek et al., Phys. Rev. C 75, 064318 (2007)

  28. <L2> Zoom into the sea of levels E 3a. Geometric collective model Dependenceon the classicality parameter Dependence of the Brody parameter on energy

  29. Peres invariant classically Poincaré section E = 0.2 3a. Geometric collective model Peres operators & Wavefunctions 2D Selected squared wave functions: <L2> <VB> E

  30. 3a. Geometric collective model Classical and quantum measures - comparison B= 1.09 B= 0.24 Classical measure Quantum measure (Brody)

  31. 3a. Geometric collective model 1/f noise Integrable case: a = 2 expected (averaged over 4 successive sets of 8192 levels, starting from level 8000) (512 successive sets of 64 levels) log<S> Correlations we are interested in 3.0 - 1.92x 2.0 - 1.94x 6.0 - 1.93x Averaging of smaller intervals log f Universal region Shortest periodic classical orbit

  32. 3a. Geometric collective model 1/f noise Mixed dynamics A = 0.25 Calculation of a: a - 1 Each point –averaging over 32 successive sets of 64 levels in an energy window 1 - w regularity freg E

  33. 3b. Interacting boson model Interacting Boson Model

  34. 3b. Interacting boson model IBMHamiltonian - Valence nucleon pairs with l = 0, 2 - quanta of quadrupole collective excitations s-bosons (l=0) d-bosons (l=2) Symmetry U(6) with 36 generators total number of bosons is conserved SO(3) – total angular momentum L is conserved Dynamical symmetries (group chains) vibrational g-unstable nuclei rotational The most general Hamiltonian (constructed from Casimir invariants of the subgoups)

  35. SO(6) 0 0 Invariant of SO(5) (seniority) 1 Arc of regularity U(5) SU(3) 3b. Interacting boson model Consistent-QHamiltonian d-boson number operator quadrupole operator a – scaling parameter Classical limit via coherent states integrable cases Shape phase transition F. Iachello, A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987)

  36. 3 independent Peres operators SO(6) 0 0 Invariant of SO(5) (seniority) 1 Casten triangle U(5) SU(3) 3b. Interacting boson model Consistent-QHamiltonian d-boson number operator quadrupole operator a – scaling parameter integrable cases

  37. 3b. Interacting boson model Regular lattices in integrable case - even the operators non-commuting with Casimirs of U(5) create regular lattices ! commuting non-commuting 0 40 -10 30 U(5) limit 20 -20 10 -30 0 -40 0 -10 N = 40 -20 L = 0 -30 -40

  38. Arc of regularity 3b. Interacting boson model Different invariants classical regularity h = 0.5 N = 40 U(5) SU(3) O(5) M. Macek, J. Dobeš, P. Cejnar, Phys. Rev. C 80, 014319(2009)

  39. Arc of regularity <L2> Correspondence with GCM 3b. Interacting boson model Different invariants classical regularity h = 0.5 N = 40 U(5) SU(3) O(5) M. Macek, J. Dobeš, P. Cejnar, Phys. Rev. C 80, 014319(2009)

  40. 3b. Interacting boson model High-lying rotational bands N = 30 L = 0 η = 0.5, χ= -1.04 (arc of regularity) E

  41. 3b. Interacting boson model High-lying rotational bands N = 30 L = 0,2 η = 0.5, χ= -1.04 (arc of regularity) E

  42. 3b. Interacting boson model High-lying rotational bands N = 30 L = 0,2,4 η = 0.5, χ= -1.04 (arc of regularity) E

  43. 3b. Interacting boson model High-lying rotational bands N = 30 L = 0,2,4,6 η = 0.5, χ= -1.04 (arc of regularity) Regular areas: Adiabatic separation of the intrinsic and collective motion E

  44. 3b. Interacting boson model Numerical evidence of the rotational bands Pearson correlation coefficient =10/3 for rotational band Classical fraction of regularity M. Macek, J. Dobeš, P. Stránský, P. Cejnar, Phys. Rev. Lett. 105, 072503 (2010) M. Macek, J. Dobeš, P. Cejnar, Phys. Rev. C 81, 014318 (2010)

  45. 3b. Interacting boson model Components of eigenvectors in SU(3) basis RB Appears naturally in the SU(3) basis li – i-th eigenstate with angular momentum l low-lying band highly excited band Quasidynamical symmetry The characteristic features of a dynamical symmetry (the existence of the rotational bands here) survive despite the dynamical symmetry is broken Non-rotational sequence of states indices labeling the intrinsic b, g excitations (SU(3) basis states)

  46. Enjoy the last slide! Thank you for your attention Summary • Peres lattices • Allow visualising quantum chaos • Capable of distinguishing between chaotic and regular parts of the spectra • Freedom in choosing Peres operator • 1/f Noise • Effective method to introduce a measure of chaos using long-range correlations in quantum spectra • Geometrical Collective Model • Complex behavior encoded in simple equations • (order-chaos-order transition) • Possibility of studying manifestations of both classical and quantum chaos and their relation • Good classical-quantum correspondence found even in the mixed dynamics regime • Interacting boson model • Peres operators come naturally from the Casimirs of the dynamical symmetries groups • Evidence of connection between chaoticity and separation of collective and intrinsic motions http://www-ucjf.troja.mff.cuni.cz/~geometric ~stransky

More Related