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
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