1. Q UANTUM CHAOS IN THE COLLECTIVE DYNAMICS OF NUCLEI Pavel Cejnar, Pavel Stránský, Michal Macek DPG Frühjahrstagung, Bochum 2009, Germany 18.3.2009 Institute of Particle and Nuclear Phycics Faculty of Mathematics and Physics Charles University in Prague, Czech Republic

2. 2.Examples of chaos in: - Geometric Collective Model (GCM) - Interacting Boson Model (IBM) 1.Classical and quantum chaos - visualising (Peres lattices) - measuring Q UANTUM CHAOS IN THE COLLECTIVE DYNAMICS OF NUCLEI

3. Classical chaos

4. Poincaré sections y x vxvx vxvx Section at y = 0 x ordered case – “circles” chaotic case – “fog” (2D system)

5. Fraction of regularity Measure of classical chaos regular total number of trajectories (with random initial conditions) energy control parameter regular chaotic

6. Quantum chaos

7. Peres lattices Quantum system: A. Peres, Phys. Rev. Lett. 53 (1984), 1711 E Integrable lattice always ordered for any operator P Infinite number of of integrals of motion can be constructed: Lattice: energy E i versus value of nonintegrable E partly ordered, partly disordered chaotic regular

8. E GOE GUEGSE P(s)P(s) s Poisson CHAOTIC system REGULAR system Brody parameter Nearest Neighbour Spacing distribution Brody distribution parameter  Standard way of measuring quantum chaos by means of spectral statistics spectrum Bohigas conjecture (O. Bohigas, M. J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984), 1)

9. Examples 1. Geometric Collective Model

10. T…Kinetic term V…Potential GCM Hamiltonian neglect higher order terms neglect Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles) Corresponding tensor of momenta Principal axes system (PAS) B … strength of nonintegrability ( B = 0 – integrable quartic oscillator) shape variables:

11. T…Kinetic term V…Potential Nonrotating case J = 0 ! Principal axes system (PAS) (b) 5D system restricted to 2D (true geometric model of nuclei) (a) 2D system 2 physically important quantization options (with the same classical limit): GCM Hamiltonian neglect higher order terms neglect Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles) Corresponding tensor of momenta

12. T…Kinetic term V…Potential Nonrotating case J = 0 ! Principal axes system (PAS) (a) 2D system GCM Hamiltonian neglect higher order terms neglect Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles) Corresponding tensor of momenta 2 different Peres operators L2L2 H’

13. Mapping classical chaos Arc of regularity B = 0.62 Empire of chaos Integrability

14. Increasing perturbation E A=-1, K=C=1 Integrability x Onset of chaos <L2><L2> B = 0 B = 0.001 B = 0.05 B = 0.24 Integrable Empire of chaos

15. Connection with the arc of regularity (IBM)  –  vibrations resonance Selected squared wave functions: Peres invariant classically Poincaré section E = 0.2 <L2><L2> E Arc of regularity B = 0.62

16. Classical-Quantum correspondence B = 0.62 B = 1.09 <L2><L2> 1-  f reg Classical f reg Brody good qualitative agreement

17. Examples 2. Interacting Boson Model

18. IBM Hamiltonian 3 different dynamical symmetries U(5) SU(3) O(6) 0 0 1 Casten triangle a – scaling parameter Invariant of O(5) (seniority)

19. 3 different dynamical symmetries U(5) SU(3) O(6) IBM Hamiltonian 0 0 1 Casten triangle Invariant of O(5) (seniority) a – scaling parameter 3 different Peres operators

20. Regular Lattices in Integrable case N = 40 U(5) limit even the operators non-commuting with Casimirs of U(5) create regular lattices !

21. Different invariants  = 0.5 N = 40 U(5) SU(3) O(5) Arc of regularity classical regularity

22. Application: Rotational bands N = 30 L = 0 η = 0.5, χ= -1.04 (arc of regularity)

23. Application: Rotational bands N = 30 L = 0,2 η = 0.5, χ= -1.04 (arc of regularity)

24. Application: Rotational bands N = 30 L = 0,2,4 η = 0.5, χ= -1.04 (arc of regularity)

25. Application: Rotational bands N = 30 L = 0,2,4,6 η = 0.5, χ= -1.04 (arc of regularity)

26. http://www-ucjf.troja.mff.cuni.cz/~geometric Summary 1.The geometric collective model of nuclei – complex behaviour encoded in simple dynamical equation 2.Peres lattices: allow visualising quantum chaos capable of distinguishing between chaotic and regular parts of the spectra freedom in choosing Peres operator independent on the basis in which the system is diagonalized 3.Peres lattices and the nuclear collective models provide excellent tools for studying classical-quantum correspondence More results in clickable form on ~stransky

27. Thank you for your attention

29. E PTPT Zoom into sea of levels Dependence on the classicality parameter E 1-  Quantum Classical f reg

30. Peres lattices and invariant A. Peres, Phys. Rev. Lett. 53 (1984), 1711 constant of motion J1J1 J2J2 Arbitrary 2D system constant for each trajectory and more generally for each torus EBK Quantization quantum numbers Difference between eigenvalues of A (valid for any constant of motion)

31. Classical x quantum view (more examples) (a)(a) (b)(b) (c)(c) (b) B=0.445 (c) B=1.09 (a) B=0.24 <P><P> f reg E E

32. Variance lattices U(5) invariant Phonon calculation nn n exc (mean-field approximation) basis:  = -1.32

33. Wave functions components in SU(3) basis Phonon calculation (mean-field approximation) basis: Quasidynamical symmetry (same amplitude for all low-L states) L = 0,2,4,6,8