Pavel Stránský

ECT* Seminar*

13 January 2012

Institute of Particle and Nuclear Phycics, Faculty of Mathematics and Physics,Charles University in Prague, Czech Republic

CHAOTIC DYNAMICS AND QUANTUM STATE PATTERNS IN COLLECTIVE

MODELS OF NUCLEI

Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México

Collaborators:

Michal Macek, Pavel Cejnar

Alejandro Frank, Emmanuel Landa, Irving Morales

Jan DobešNuclear Research Institute, Řež, Czech Republic

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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

CHAOTIC DYNAMICS AND QUANTUM STATE PATTERNS IN COLLECTIVE

MODELS OF NUCLEI

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1. Classical Chaos(analysis of trajectories)

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Hamiltonian systemsState of a system: a point in the 4D phase space

Conservative system:

Trajectory restricted to 3D hypersurface

Connected with additional symetries

Integrals of motion:

Integrable system: Number of independent

integrals of motion

number of degrees of freedom

=Canonical transformation to action-angle variables

Quasiperiodic motion on a toroid

J1

J2

1. Classical chaos

Hamiltonian systemsState of a system: a point in the 4D phase space

Conservative system:

Trajectory restricted to 3D hypersurface

Connected with additional symetries

Integrals of motion:

Integrable system: Number of independent

integrals of motion

number of degrees of freedom

=Canonical transformation to action-angle variables

Quasiperiodic motion on a toroid

J1

J2

Chaotic behaviour: property of nonintegrable

systems

1. Classical chaos

y

x

px

px

Section at

y = 0

x

ordered case – “circles”

chaotic case – “fog”

We plot a point every time when the trajectory crosses the plane y = 0

Poincaré sections

1. Classical chaos

Generic conservative system of 2 degrees of freedom

Different initial conditions at the same energy

REGULAR area

CHAOTIC area

freg=0.611 x

px

Fraction of regularity

Measure of classical chaos

Surface of the section covered with regular trajectories

Total kinematically accessible surface of the section

1. Classical chaos

1. Lyapunov exponent

Divergence of two neighboring trajectories

2. SALI (Smaller Alignment Index)

• fast convergence towards zero for chaotic trajectories

• two divergencies

Quasiperiodic X unstable trajectories

Regular: at most polynomial divergence

Chaotic: exponential divergence

Classical chaos –Hypersensitivity to

the initial conditions

Ch. Skokos, J. Phys. A: Math. Gen 34, 10029 (2001); 37 (2004), 6269

1. Classical chaos

2. Quantum Chaos(analysis of energy spectra)

Semiclassical theory of chaos

2. Quantum chaos

Spectral density:

smooth part

given by the volume of the classical phase space

oscillating part

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. E 84, 016203 (2011)

E

Quantum chaos: Spectral statistics

P(s)

Gaussian Orthogona

lEnsemble

CHAOTIC systems

REGULAR system

Transformation

H T invariance

Angular momentum

R invariance

GOE Orthogonal SymmetricYESYES

nn/2 YES

GUE Unitary Hermitian NO

GSE Symplectic YES n/2 NO

Nearest-neighbor spacing distribution

level repulsion

no level interaction

Gaussian Unitary

Ensemble

Gaussian Symplectic

Ensemble

Ensembles of random matrices

O. Bohigas, M. J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984), 1M.V. Berry, M.Tabor, Proc. Roy. Soc. A 356, 375 (1977)

2. Quantum chaos

Wigner

P(s)

s

Poisson

CHAOTIC systemREGULAR system

Brodydistributionparameter

- Tool to test classical-quantum correspondence

- Measure of chaoticity of quantum systems- Artificial interpolation between Poisson and GOE distribution

Spectral statistics

Nearest-neighbor spacing distribution

2. Quantum chaos

Schrödinger equation:(for wave function)

Helmholtz equation:(for intensity of el. field)

Quantum chaos - examples

2. Quantum chaos

They are also extensively studied

experimentally

Billiards

Riemann function:

Prime numbers

Riemann hypothesis:All points z(s)=0 in the complex plane lie on the line s=½+iy (except trivial zeros on the real exis s=–2,–4,–6,…)

GUE

Zeros of function

Quantum chaos - applications

2. Quantum chaos

GOE

Correlation matrix of the human EEG signal

P. Šeba, Phys. Rev. Lett. 91 (2003), 198104

Quantum chaos - applications

2. Quantum chaos

1/f noise

Power spectrum

A. Relaño et al., Phys. Rev. Lett. 89, 244102 (2002)E. Faleiro et al., Phys. Rev. Lett. 93, 244101 (2004)

CHAOTIC system = 1 = 2

Direct comparison of 3 measures of chaos

REGULAR system

= 2

= 1

1 = 0

2

3

4

n = 0k

k

- Fourier transformation of the time series constructed from energy levels fluctuations

J. M. G. Gómez et al., Phys. Rev. Lett. 94, 084101 (2005)

Ubiquitous in the nature (many time signals or space characteristics of complex systems have 1/f power spectrum)

2. Quantum chaos

Peres lattices Quantum system:

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

Infinite number of of integrals of motion can be constructed (time-averaged operators P):

nonintegrable

E

<P>

regular

E

Integrable

<P>

chaoticregular

B = 0 B = 0.445

Lattice: energy Ei versus value of

lattice always ordered for any operator P

partly ordered, partly disordered

2. Quantum chaos

3. Application to the collective models of nuclei

Surface of homogeneous nuclear matter:

Monopole deformations = 0

Geometric collective model

(even-even nuclei – collective character of the lowest excitations)

- Does not contribute due to the incompressibility of the nuclear matter

Dipole deformations = 1- Related to the motion of the center of mass- Zero due to momentum conservation

- “breathing” mode

3a. Geometric collective model

T…Kinetic term V…Potential

Neglect higher order terms

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

Corresponding tensor of momenta

Surface of homogeneous nuclear matter:

Quadrupole deformations = 2

G. Gneuss, U. Mosel, W. Greiner, Phys. Lett. 30B, 397 (1969)

4 external parameters

Geometric collective model

neglect

3a. Geometric collective model

T…Kinetic term V…Potential

Neglect higher order terms

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

Corresponding tensor of momenta

Surface of homogeneous nuclear matter:

Quadrupole deformations = 2

Scaling properties

4 external parametersAdjusting 3 independent scalesenergy

(Hamiltonian)

1 “shape” parameter

size (deformation)

time

1 “classicality” parametersets absolute density of quantum spectrum (irrelevant in classical case)

P. Stránský, M. Kurian, P. Cejnar, Phys. Rev. C 74, 014306 (2006)

Geometric collective model

neglect

3a. Geometric collective model

(order parameter)

Principal Axes System (PAS)

Shape variables:

Shape-phase structure

Deformed shape Spherical shape

VV

B

A

C=1

Phase separatrix

3a. Geometric collective model

Nonrotating case J = 0!

(a) 5D system restricted to 2D (true geometric model

of nuclei)

(b) 2D system

2 physically important quantization options(with the same classical limit):

Classical dynamics– Hamilton equations of motion

• An opportunity to test the Bohigas conjecture in different quantization schemes

Quantization– Diagonalization in the oscillator basis

Dynamics of the GCM

3a. Geometric collective model

(a) 5D system restricted to 2D (true geometric model

of nuclei)

(b) 2D system

IndependentPeres operators in

GCM

H’

L22DL2

5D

Peres operators

Nonrotating case J = 0!

P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79, 046202 (2009); 066201 (2009)

3a. Geometric collective model

Complete map of classical chaos in GCM IntegrabilityIntegrability

Veins ofVeins of regularityregularity

chaotichaoticc

regularegularr

control parameter

““ Arc

of

Arc

of

regula

rity

”re

gula

rity

”

Global minimum and saddle pointRegion of phase transition

Sh

ap

e-p

hase

Sh

ap

e-p

hase

tr

ansi

tion

transi

tion

HO approximationHO approximation

3a. Geometric collective model

Increasing perturbation

E

Peres lattices in GCM

<L2>

B = 0 B = 0.005

<H’>

Integrable Empire of chaos

Small perturbation affects only a localized part of the lattice

B = 0.05 B = 0.24

Remnants ofregularity

3a. Geometric collective model

Peres lattices for two different operators

(The place of strong level interaction)

“Arc of regularity” B = 0.62

<L2>

2D

<VB>

5D

(different quantizations)

E

• – vibrations resonance

Connection with IBM: M. Macek et al., Phys. Rev. C 75, 064318 (2007)

3a. Geometric collective model

Zoom into the sea of levels

Dependence on the classicality parameter

E

<L2>

Dependence of the Brody parameter on energy

3a. Geometric collective model

Selected squared wave functions:

E

Peres operators & Wavefunctions

<L2>

<VB>

2D

Peres invariant classically

Poincaré sectionE = 0.2

3a. Geometric collective model

Classical and quantum measures - comparison Classical

measure

Quantum measure (Brody)

B = 0.24 B = 1.09

3a. Geometric collective model

Integrable case: = 2 expected

3.0 - 1.92x

6.0 - 1.93x

Shortest periodic classical orbit

Universal region

(averaged over 4 successive sets of 8192 levels, starting from level 8000)

(512 successive sets of 64 levels)

2.0 - 1.94x

log<S>

log f

1/f noise

Correlations we are interested in

Averaging of smaller intervals

3a. Geometric collective model

Mixed dynamics A = 0.25

reg

ula

rity

freg

- 11 -

E

Calculation of :Each point –

averaging over 32 successive sets of

64 levels in an energy window

1/f noise

3a. Geometric collective model

Interacting Boson Model

3b. Interacting boson model

IBM Hamiltonian

s-bosons (l=0) d-bosons (l=2)

- Valence nucleon pairs with l = 0, 2- quanta of quadrupole collective excitations

Symmetry

U(6) with 36 generatorstotal number of bosons is conserved

Dynamical symmetries (group chains)

The most general Hamiltonian (constructed from Casimir invariants of the subgoups)

vibrational

-unstable

rotational

nuclei

3b. Interacting boson model

SO(3) – total angular momentum L is conserved

U(5)SU(3)

SO(6)

0 0

1Arc of regularity

a – scaling parameter

Invariant of SO(5) (seniority)

Consistent-Q Hamiltonian

integrable casesClassical limit

via coherent states

F. Iachello, A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987)

quadrupole operator

d-boson number operator

3b. Interacting boson model

Shape phase transition

U(5)SU(3)

SO(6)

0 0

1

Casten triangle

a – scaling parameter

Invariant of SO(5) (seniority)

Consistent-Q Hamiltonian

integrable cases

3 independent

Peres operators

quadrupole operator

d-boson number operator

3b. Interacting boson model

Regular lattices in integrable case

3ˆ.ˆ SUQQ

dn̂v

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

40

-40

-2020

10

30 -10

-30

0

-40

-20

-10

-30

0

0

3ˆ.ˆ SUQQ

6ˆ.ˆ OQQ

dn̂

L = 0

commuting non-commuting

U(5)

limit

N = 40

3b. Interacting boson model

Different invariants

= 0.5

N = 40

U(5)

SU(3)

O(5)

Arc of regularityArc of regularity

classical regularity

M. Macek, J. Dobeš, P. Cejnar, Phys. Rev. C 80, 014319 (2009)

3b. Interacting boson model

Different invariants

= 0.5

N = 40

U(5)

SU(3)

O(5)

Arc of regularityArc of regularity

classical regularity

Correspondence with GCM

<L2>

M. Macek, J. Dobeš, P. Cejnar, Phys. Rev. C 80, 014319 (2009)

3b. Interacting boson model

High-lying rotational bands

dn̂

N = 30L = 0

η = 0.5, χ= -1.04 (arc of regularity)

3ˆ.ˆ SUQQdn̂

3b. Interacting boson model

E

η = 0.5, χ= -1.04 (arc of regularity)

3ˆ.ˆ SUQQdn̂

High-lying rotational bands

3b. Interacting boson model

N = 30L = 0,2

E

η = 0.5, χ= -1.04 (arc of regularity)

3ˆ.ˆ SUQQdn̂

High-lying rotational bands

3b. Interacting boson model

N = 30L = 0,2,4

E

3ˆ.ˆ SUQQ

η = 0.5, χ= -1.04 (arc of regularity)

dn̂

High-lying rotational bands

3b. Interacting boson model

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

Regular areas:Adiabatic separation of the intrinsic and collective motion

E

Numerical evidence of the rotational bands

Pearson correlation coefficient

=10/3 for rotational band

Classical fraction of regularity

M. Macek, J. Dobeš, P. Cejnar, Phys. Rev. C 81, 014318 (2010)

M. Macek, J. Dobeš, P. Stránský, P. Cejnar, Phys. Rev. Lett. 105, 072503 (2010)

3b. Interacting boson model

Components of eigenvectors in SU(3) basis

RB Appears naturally in the SU(3) basis

indices labeling the intrinsic , excitations (SU(3) basis states)

low-lying band highly excited band

Non-rotational sequence of states

3b. Interacting boson model

Quasidynamical symmetryThe characteristic features of a dynamical symmetry (the existence of the rotational bands here) survive despite the dynamical symmetry is broken

li – i-th eigenstate with angular momentum l

Summary1. Peres lattices

• Allow visualising quantum chaos• Capable of distinguishing between chaotic and

regular parts of the spectra• Freedom in choosing Peres operator

2. 1/f Noise• Effective method to introduce a measure of

chaos using long-range correlations in quantum spectra

3. 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

4. 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

Thank you for your attention

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

~stransky

Enjoy the last slide!