QUANTUM 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. Examples of chaos in:- Geometric Collective Model (GCM)- Interacting Boson Model (IBM)

1. Classical and quantum chaos- visualising (Peres lattices)- measuring

QUANTUM CHAOS IN THE COLLECTIVE DYNAMICS OF

NUCLEI

Classical chaos

Poincaré sections

y

x

vx

vx

Section aty = 0

x

ordered case – “circles”

chaotic case – “fog”

(2D system)

Fraction of regularity

Measure of classical chaos

regular

totalnumber of

trajectories (with random initial conditions)

energy

control parameter

regularegularr

chaotichaoticc

Quantum chaos

Peres lattices Quantum system:

A. Peres, Phys. Rev. Lett. 53 (1984), 1711

E

Integrable

<P>

lattice always ordered for any operator P

Infinite number of of integrals of motion can be constructed:

Lattice: energy Ei versus value of

nonintegrable

E

<P>

partly ordered, partly disordered

chaoticregular

regular

E

GOE

GUE

GSE

P(s)

s

Poisson

CHAOTIC systemREGULAR system

Brody parameter

Nearest Neighbour Spacing

distribution

Brodydistributionparameter

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)

Examples 1. Geometric Collective

Model

T…Kinetic term

V…Potential

GCM Hamiltonian

neglect higher order terms

neglectQuadrupole 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:

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

neglectQuadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles)

Corresponding tensor of momenta

T…Kinetic term

V…Potential

Nonrotating case J = 0!

Principal axes system (PAS)

(a) 2D system

GCM Hamiltonian

neglect higher order terms

neglectQuadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles)

Corresponding tensor of momenta

2 differentPeres

operatorsL2

H’

Mapping classical chaos

Arc of regularity Arc of regularity B B = 0.62= 0.62Empire of chaos

Integrability

Increasing perturbation

E

A=-1, K=C=1Integrability x Onset of chaos

<L2>

B = 0 B = 0.001 B = 0.05 B = 0.24

<H’>

Integrable Empire of chaos

• Connection with the arc of regularity (IBM)• – vibrations resonance

Selected squared wave functions:

Peres invariant classically

Poincaré sectionE = 0.2

<L2>

<H’>

E

Arc of regularity Arc of regularity B B = = 0.620.62

Classical-Quantum correspondence

B = 0.62 B = 1.09

<L2>

<H’>

1-

freg

Classical freg

Brody

good qualitative agreemen

t

Examples2. Interacting Boson Model

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)

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

Regular Lattices in Integrable case N = 40U(5) limit

even the operators non-commuting with Casimirs of U(5) create regular lattices !

Different invariants

= 0.5

N = 40

U(5)

SU(3)

O(5)

Arc of regularityArc of regularity

classical regularity

Application: Rotational bands

dn̂

N = 30L = 0

η = 0.5, χ= -1.04 (arc of regularity)

3ˆ.ˆ SUQQdn̂

Application: Rotational bands

dn̂

N = 30L = 0,2

η = 0.5, χ= -1.04 (arc of regularity)

3ˆ.ˆ SUQQdn̂

Application: Rotational bands N = 30L = 0,2,4

η = 0.5, χ= -1.04 (arc of regularity)

dn̂

3ˆ.ˆ SUQQdn̂

Application: Rotational bands

dn̂

3ˆ.ˆ SUQQ

N = 30L = 0,2,4,6

η = 0.5, χ= -1.04 (arc of regularity)

dn̂

http://www-ucjf.troja.mff.cuni.cz/~geometric

Summary1. 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

Thank you for your attention

E

PT

Zoom into sea of levels

Dependence on the classicality parameter

E

1- Quantum

Classical

freg

Peres lattices and invariant

A. Peres, Phys. Rev. Lett. 53 (1984), 1711

constant of motion

J1

J2

Arbitrary 2D system

constant for each trajectory and more generally for each torus

EBK Quantization quantu

m numbers

Difference between eigenvalues of A

(valid for any constant of motion)

Classical x quantum view (more examples)

(a)

(b)

(c)

(b) B=0.445 (c) B=1.09(a) B=0.24

<P>

freg

E

E

Variance lattices • U(5) invariant

• Phonon calculationn

nexc

(mean-field approximation)

basis:

= -1.32

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