Q UANTUM CHAOS IN THE COLLECTIVE DYNAMICS OF NUCLEI

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

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

3. Classical chaos

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

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

6. Quantum chaos

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

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

9. Examples • 1. Geometric Collective Model

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

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

12. Nonrotating case J = 0! H’ 2 different Peres operators L2 GCMHamiltonian T…Kinetic term V…Potential neglect higher order terms neglect Quadrupole tensor ofcollective coordinates (2 shape parameters, 3 Euler angles) Corresponding tensor of momenta Principal axes system (PAS) (a) 2D system

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

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

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

16. Classical-Quantum correspondence B = 0.62 B = 1.09 <L2> <H’> Brody 1-w Classicalfreg good qualitative agreement freg

17. Examples 2. Interacting Boson Model

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

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

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

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

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

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

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

26. Summary • The geometric collective model of nuclei – complex behaviour encoded in simple dynamical equation • 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 • Peres lattices and the nuclear collective models provide excellent tools for studying classical-quantum correspondence More results in clickable form on http://www-ucjf.troja.mff.cuni.cz/~geometric ~stransky

27. Thank you for your attention

29. Zoom into sea of levels PT freg Classical 1-w Quantum E E Dependence on the classicality parameter

30. Peres lattices and invariant J2 EBK Quantization quantum numbers J1 constant of motion Arbitrary 2D system Difference between eigenvalues of A (valid for any constant of motion) constant for each trajectory and more generally for each torus A. Peres, Phys. Rev. Lett.53 (1984), 1711

31. (b) (c) (a) (c) B=1.09 (a) B=0.24 (b) B=0.445 <P> E freg E Classical x quantumview (more examples)

32. Variance lattices • U(5) invariant c = -1.32 • Phonon calculation (mean-field approximation) nb basis: nexc

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