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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
Instituto de Ciencias Nucleares,
Universidad Nacional Autónoma de México
ECT* Seminar*
13 January 2012
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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
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1. Classical Chaos
(analysis of trajectories)
1. Classical chaos
Hamiltonian systems
State of a system:
a point in the 4D
phase space
Conservative system:
Trajectory restricted to 3D hypersurface
Integrals of motion:
Connected with
additional symetries
Integrable system:
Canonical transformation
to action-angle variables
Number of independent
integrals of motion
=
number of degrees
of freedom
J2
Quasiperiodic
motion on a toroid
J1
1. Classical chaos
Hamiltonian systems
State of a system:
a point in the 4D
phase space
Conservative system:
Trajectory restricted to 3D hypersurface
Integrals of motion:
Connected with
additional symetries
Integrable system:
Canonical transformation
to action-angle variables
Number of independent
integrals of motion
=
number of degrees
of freedom
J2
Quasiperiodic
behaviour: motion on a toroid
Chaotic
J1 systems
property of nonintegrable
1. Classical chaos
Poincaré sections
Generic conservative
system of 2 degrees of
freedom
We plot a point every time
when the trajectory crosses
the plane y = 0
px
y
x
chaotic case
– “fog”
px
Section at
y=0
x
ordered case – “circles”
Different initial conditions at the same energy
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
px
REGULAR area
CHAOTIC area
freg=0.611
x
1. Classical chaos
Quasiperiodic X unstable trajectories
1. Lyapunov exponent
Classical chaos –
Hypersensitivity to
the initial conditions
Divergence of two neighboring trajectories
Regular: at most
polynomial divergence
2. SALI (Smaller Alignment Index)
Chaotic: exponential
divergence
• two divergencies
• fast convergence towards zero for chaotic trajectories
Ch. Skokos, J. Phys. A: Math. Gen 34, 10029 (2001); 37 (2004), 6269
2. Quantum Chaos
(analysis of energy spectra)
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. E 84, 016203 (2011)
2. Quantum chaos
Quantum chaos: Spectral statistics
E
level repulsion
no level interaction
Nearestneighbor
spacing
distribution
Gaussian
Orthogonal
P(s)
Ensemble
REGULAR system
Gaussian
Unitary
Ensemble
Gaussian
Symplectic
Ensemble
CHAOTIC systems
Ensembles of random matrices
Transformation
H
T invariance
Angular momentum
R invariance
GOE
Orthogonal
Symmetric
YES
YES
n
n/2
YES
GUE
Unitary
Hermitian
NO
GSE
Symplectic
n/2
NO
YES
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
Spectral statistics
Nearestneighbor
spacing
distribution
P(s)
Poisson
Wigner
s
REGULAR system CHAOTIC system
Brody
distribution
parameter w
- Artificial interpolation between Poisson and GOE distribution
- Measure of chaoticity of quantum systems
- Tool to test classical-quantum correspondence
2. Quantum chaos
Quantum chaos - examples
Billiards
They are also
extensively studied
experimentally
Schrödinger equation:
(for wave function)
Helmholtz equation:
(for intensity of el. field)
2. Quantum chaos
Quantum chaos - applications
Riemann z
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 z function
2. Quantum chaos
Quantum chaos - applications
GOE
Correlation matrix
of the human EEG signal
P. Šeba, Phys. Rev. Lett. 91 (2003), 198104
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
Direct comparison of
3 measures of chaos
A. Relaño et al., Phys. Rev. Lett. 89, 244102 (2002)
E. Faleiro et al., Phys. Rev. Lett. 93, 244101 (2004)
a=1
J. M. G. Gómez et al., Phys. Rev. Lett. 94, 084101 (2005)
2. Quantum chaos
Peres lattices
Quantum system:
Infinite number of of integrals of
motion can be constructed
(time-averaged operators P):
Lattice: energy Ei versus value of
Integrable
lattice always ordered
for any operator P
nonintegrable
B=0
partly ordered,
partly disordered
B = 0.445
<P>
<P>
regular
E
E
regular
chaotic
A. Peres, Phys. Rev. Lett. 53, 1711 (1984)
3. Application to the collective
models of nuclei
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
3a. Geometric collective model
Geometric collective model
Surface of homogeneous nuclear matter:
Quadrupole deformations l = 2
Corresponding tensor of momenta
Quadrupole tensor of collective coordinates
(2 shape parameters, 3 Euler angles)
T…Kinetic term
V…Potential
Neglect higher
order terms
neglect
4 external parameters
G. Gneuss, U. Mosel, W. Greiner, Phys. Lett. 30B, 397 (1969)
3a. Geometric collective model
Geometric collective model
Surface of homogeneous nuclear matter:
Quadrupole deformations l = 2
Corresponding tensor of momenta
Quadrupole tensor of collective coordinates
(2 shape parameters, 3 Euler angles)
T…Kinetic term
V…Potential
Neglect higher
order terms
neglect
4 external parameters
Scaling
properties
1 “shape” parameter
(order parameter)
Adjusting 3 independent scales
energy (Hamiltonian)
size (deformation)
time
1 “classicality” parameter
sets absolute density of quantum
spectrum (irrelevant in classical case)
P. Stránský, M. Kurian, P. Cejnar, Phys. Rev. C 74, 014306 (2006)
3a. Geometric collective model
Principal Axes System (PAS)
g
Shape
variables:
b
Shape-phase structure
B
Phase
separatrix
V
V
A
b
C=1
Deformed shape
Spherical shape
b
3a. Geometric collective model
Dynamics of the GCM
Nonrotating case J = 0!
Classical dynamics
– Hamilton equations of motion
Quantization
– Diagonalization in the oscillator basis
2 physically important quantization 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
3a. Geometric collective model
Peres operators
Nonrotating case J = 0!
H’
Independent
Peres operators in GCM
L22D
L25D
(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)
3a. Geometric collective model
Complete map of classical chaos in GCM
chaotic
Shape-phase transition
regularity”
Integrability
Veins of
regularity
regular
“Arc of
control
parameter
Global minimum
and saddle point
HO approximation
Region of phase
transition
3a. Geometric collective model
Peres lattices in GCM
Small perturbation affects 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
3a. Geometric collective model
“Arc of regularity” B = 0.62
• b – g vibrations resonance
<L2>
<VB>
2D
(different quantizations)
5D
E
Connection with IBM: M. Macek et al., Phys. Rev. C 75, 064318 (2007)
3a. Geometric collective model
Dependence on the classicality parameter
<L2>
Zoom into the sea of levels
E
Dependence of the Brody
parameter on energy
3a. Geometric collective model
Peres operators & Wavefunctions
2D
Selected squared wave functions:
<L2>
<VB>
E
Poincaré section
E = 0.2
Peres invariant
classically
3a. Geometric collective model
Classical and quantum
measures - comparison
B = 0.24 Classical measure
B = 1.09
Quantum measure (Brody)
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>
3.0 - 1.92x
Correlations we
are interested in
2.0 - 1.94x
6.0 - 1.93x
Averaging of
smaller intervals
Universal region
Shortest periodic classical orbit
log f
3a. Geometric collective model
1/f noise
Mixed dynamics A = 0.25
regularity
a-1
Calculation of a:
Each point –
averaging over 32
successive sets of 64
levels in an energy
window
1-w
freg
E
3b. Interacting boson model
Interacting Boson Model
3b. Interacting boson model
IBM Hamiltonian
- Valence nucleon pairs with l = 0, 2
s-bosons (l=0)
Symmetry
d-bosons (l=2)
- quanta of quadrupole collective excitations
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)
3b. Interacting boson model
Consistent-Q Hamiltonian
d-boson number
operator
quadrupole operator
a – scaling parameter
SO(6)
Classical limit via
coherent states
integrable cases
0
0
Arc of regularity
SU(3)
Shape phase transition
Invariant of SO(5)
(seniority)
1
U(5)
F. Iachello, A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987)
3b. Interacting boson model
Consistent-Q Hamiltonian
d-boson number
operator
quadrupole operator
a – scaling parameter
3 independent
Peres operators
SO(6)
integrable cases
0
0
Invariant of SO(5)
(seniority)
1
Casten triangle
SU(3)
U(5)
3b. Interacting boson model
Regular lattices in integrable case
- even the operators non-commuting with
Casimirs of U(5) create regular lattices !
commuting
40
30
non-commuting
0
n̂d
20
-10
U(5)
10
SU
SU33
ˆ
ˆ
Q
.
Q
ˆ
ˆ
Q.Q
-20
-30
limit
0
-40
0
v
n̂d
-10
N = 40
L=0
-20
-30
-40
Qˆ .Qˆ O 6
3b. Interacting boson model
Different invariants
classical
regularity
Arc of regularity
h = 0.5
N = 40
U(5)
SU(3)
O(5)
M. Macek, J. Dobeš, P. Cejnar, Phys. Rev. C 80, 014319 (2009)
3b. Interacting boson model
Different invariants
Arc of regularity
<L2>
classical
regularity
Correspondence
with GCM
h = 0.5
N = 40
U(5)
SU(3)
O(5)
M. Macek, J. Dobeš, P. Cejnar, Phys. Rev. C 80, 014319 (2009)
3b. Interacting boson model
High-lying rotational bands
η = 0.5, χ= -1.04 (arc of regularity)
N = 30
L=0
Qˆ .Qˆ SU 3
n̂d
n̂d
E
3b. Interacting boson model
High-lying rotational bands
η = 0.5, χ= -1.04 (arc of regularity)
N = 30
L = 0,2
Qˆ .Qˆ SU 3
n̂d
E
3b. Interacting boson model
High-lying rotational bands
η = 0.5, χ= -1.04 (arc of regularity)
N = 30
L = 0,2,4
Qˆ .Qˆ SU 3
n̂d
E
3b. Interacting boson model
High-lying rotational bands
η = 0.5, χ= -1.04 (arc of regularity)
N = 30
L = 0,2,4,6
Qˆ .Qˆ SU 3
n̂d
Regular areas:
Adiabatic separation
of the intrinsic and
collective motion
E
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)
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)
Enjoy the last slide!
Summary
1. Peres lattices
•
•
•
Thank you for
your attention
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
http://www-ucjf.troja.mff.cuni.cz/~geometric
~stransky