Lecture - FUSTIPEN

as an Open System:
Continuum Shell Model and
New Challenges
Vladimir Zelevinsky
NSCL/ Michigan State University
Supported by NSF
Caen, GANIL May 30, 2014
1. From closed to open many-body systems
2. Effective non – Hermitian Hamiltonian
3. Doorways and phenomenon of superradiance
4. Continuum shell model
5. Statistics of complex energies
6. Cross sections, resonances, correlations and
7. Quantum signal transmission
Naftali Auerbach (Tel Aviv University)
Luca Celardo (University of Breschia)
Felix Izrailev (University of Puebla)
Lev Kaplan (Tulane University)
Gavriil Shchedrin (MSU, TAMU)
Valentin Sokolov (Budker Instutute)
Suren Sorathia (University of Puebla)
Alexander Volya (Florida State University)
NSCL and FRIB Laboratory
543 employees, incl. 38 faculty, 59 graduate and 82 undergraduate students
as of April 21, 2014
• NSCL is funded by the U.S. National Science Foundation to operate a flagship user
facility for rare isotope research and education in nuclear science, nuclear
astrophysics, accelerator physics, and societal applications
• FRIB will be a national user facility for the U.S. Department of Energy Office of Science
– when FRIB becomes operational, NSCL will transition into FRIB
User group of over 1300 members with approx. 20 working groups
C.K. Gelbke, 5/5/2014, Slide 4
The Evolution of Nuclear Science at MSU
C.K. Gelbke, 5/5/2014, Slide 5
NSCL Science Is Aligned with National Priorities
Articulated by National Research Council RISAC Report (2006), NSAC LRP (2007),
NRC Decadal Survey of Nuclear Physics (2012), “Tribble Report” (2013)
Properties of nuclei – UNEDF SciDAC, FRIB Theory Center (?)
• Develop a predictive model of nuclei and their interactions
• Many-body quantum problem: intellectual overlap to mesoscopic
science, quantum dots, atomic clusters, etc. – Mesoscopic Theory
Astrophysical processes – JINA
• Origin of the elements in the cosmos
• Explosive environments: novae,
supernovae, X-ray bursts …
• Properties of neutron stars
Tests of fundamental symmetries
• Effects of symmetry violations are
amplified in certain nuclei
Societal applications and benefits
• Bio-medicine, energy, material
sciences – Varian, isotope harvesting, … Reaping benefits from recent
• National security – NNSA
investments while creating future
C.K. Gelbke, 5/5/2014, Slide 6
FRIB Science is Transformational
• FRIB physics is at the core of nuclear science:
“To understand, predict, and use” (David Dean)
• FRIB provides access to a vast unexplored
terrain in the chart of nuclides
FRIB science answers big questions
C.K. Gelbke, 5/5/2014, Slide 7
Examples for Cross-Disciplinary and
Applied Research Topics
• Medical research
• Examples: 47Sc, 62Zn, 64Cu, 67Cu, 68Ge, 149Tb, 153Gd, 168Ho, 177Lu, 188Re, 211At, 212Bi, 213Bi, 223Ra (DOE expert
• MSU Radiology Dept. interested in 60,61Cu
• -emitters 149Tb, 211At: potential treatment of metastatic cancer
Plant biology: role of metals in plant metabolism
Environmental and geosciences: ground water, role of metals as catalysts
Engineering: advanced materials, radiation damage, diffusion studies
Toxicology: toxicology of metals
Biochemistry: role of metals in biological process and correlations to disease
Fisheries and Wildlife Sciences: movement of pollutants through environmental and
biological systems
• Reaction rates important for stockpile stewardship – non-classified research
• Determination of extremely high neutron fluxes by activation analysis
• Rare-isotope samples for (n,g), (n,n’), (n,2n), (n,f) e.g. 88,89Zr
• Same technique important for astrophysics
• Far from stability: surrogate reactions (d,p), (3He, xn) …
Vision: Up to 10 Faculty Positions for Cross-Disciplinary and Applied Research
From closed to open (or marginally stable)
many-body system
Open systems:
Bound states
Mean field, quasiparticles
Residual interactions
Pairing, superfluidity
Collective modes
Quantum many-body chaos (GOE type)
Continuum energy spectrum
Unstable states, lifetimes
Decay channels (E,c)
Energy thresholds
Cross sections
Resonances, isolated or overlapping
Statistics of resonances and cross sections
Unified approach?
From giant resonances to superradiance
The doorway state is connected directly
to external world, other states (next
level) only through the doorway.
Examples: IAS, single-particle resonance, giant resonances
at high excitation energy, intermediate structures.
Feshbach resonance in traps, superradiance
Superradiance, collectivization by
decay Analog in a complex system
Dicke coherent state
N identical two-level atoms
coupled via common radiation
Volume ¿ 3
Interaction via continuum
Trapped states ) self-organization
 ~ D and few channels
• Nuclei far from stability
• High level density (states of
same symmetry)
• Channel thresholds
Single-particle decay in many-body system
Evolution of complex energies
•8 s.p. levels, 3 particles
•One s.p. level in continuum
Total states 8!/(3! 5!)=56; states that decay fast 7!/(2! 5!)=21 – superradiant doorways
Examples of superradiance
Mechanism of superradiance
Interaction via continuum
Trapped states - self-organization
W - m12C [MeV]
Narrow resonances and broad
superradiant state in 12C
in the region of Delta
Bartsch et.al. Eur. Phys. J. A 4, 209 (1999)
N. Auerbach, V.Z.. Phys. Lett. B590, 45 (2004)
• Optics
• Molecules
• Microwave cavities
• Nuclei
• Hadrons
• Quantum computing
• Measurement theory
Physics and mathematics of coupling to continuum
New part of Hamiltonian: coupling through continuum
[1] C. Mahaux and H. Weidenmüller, Shell-model approach to nuclear reactions,
North-Holland Publishing, Amsterdam 1969
Two parts of coupling to continuum
Integration region involves no poles
State embedded in the continuum
Form of the wave function and probability
(Eigenchannels in P-space)
(+) means + i0
Factorization (unitarity), energy dependence
(kinematic thresholds) , coupling through continuum
Self energy, interaction with continuum
Gamow shell model
Correction to Harmonic
Oscillator Wave
s,p, and d waves (red,
blue, black)
A. Volya, EPJ Web of Conf. 38, 03003 (2012).
N Michel, J. Phys. G: Nucl. Part. Phys.
36 (2009) 013101
•Wave functions are not HO
•Phenomenological SM is adjusted to
•No corrections for properly solved mean
The nuclear many-body problem
Single-particles state (particle in the well)
Many-body states (slater determinants)
Hamiltonian and Hamiltonian matrix
Matrix diagonalization
Continuum physics
Effective non-Hermitian and
energy-dependent Hamiltonian
Channels (parent-daughter structure)
Bound states and resonances
Matrix inversion at all energies (time
dependent approach)
Formally exact approach
Limit of the traditional shell model
Unitarity of the scattering matrix
• Intrinsic states: Q-space
• States of fixed symmetry
• Unperturbed energies e1; some e1>0
• Hermitian interaction V
• Continuum states: P-space
• Channels and their thresholds Ecth
• Scattering matrix Sab(E)
• Coupling with continuum
• Decay amplitudes Ac1(E) - thresholds
• Typical partial width g=|A|2
• Resonance overlaps: level spacing vs. width
No approximations until now parameter
One open
Interaction between resonances
• Real V
• Energy repulsion
• Width attraction
• Imaginary W
• Energy attraction
• Width repulsion
Dynamics of states coupled to a common decay channel
• Model
• Mechanism of binding
Dynamics of two states coupled to a common decay channel
• Model H
A1 and A2
opposite signs
A. Volya and V. Zelevinsky,
Phys. Rev. Lett. 94, 052501 (2005);
Phys. Rev. C 67, 054322 (2003);
Phys. Rev. C 74, 064314 (2006).
Oxygen Isotopes
Continuum Shell Model Calculation
• sd space, HBUSD interaction
• single-nucleon reactions
Predictive power of theory
Continuum Shell Model prediction 2003-2006
Measured 2009-2013
[1] C. R. Hoffman et al., Phys. Lett. B 672, 17 (2009); Phys.Rev.Lett.102,152501(2009); Phys.Rev.C 83,031303(R)(2011); E. Lunderberg et al.,
Phys. Rev. Lett. 108, 142503 (2012).
[2] A. Volya and V. Zelevinsky, Phys. Rev. Lett. 94, 052501 (2005); Phys. Rev. C 67, 054322 (2003); 74, 064314 (2006).
[3] G. Hagen et.al Phys. Rev. Lett. 108, 242501 (2012)
[2] A. Volya and V.Z. Phys. Rev. C 74 (2006) 064314, [3] G. Hagen et al. Phys. Rev. Lett. 108 (2012) 242501
Continuum shell model:
Detailed predictions
For Oxygen isotopes;
Color code - for widths
[A. Volya]
VirVirtual excitations into continuum
Figure: 23O(n,n)23O Effect of self-energy term (red
curve). Shaded areas show experimental values
with uncertainties.
experiment 2+
Experimental data from:
C. Hoffman, et.al. Phys. Lett. B672, 17 (2009)
Two-neutron sequential decay of 26O
A. Volya and V. Zelevinsky, Continuum shell model, Phys. Rev. C 74, 064314 (2006).
Predicted Q-value: 21 keV
Z. Kohley, et.al PRL 110, 152501 (2013) (experiment)
CSM calculation of 18O
States marked with longer lines correspond to sd-shell model; only l=0,2 partial waves
included in theoretical results.
Continuum Shell Model
He isotopes
• Cross section and structure
within the same formalism
• Reaction l=1 polarized elastic
[1] A. Volya and V. Zelevinsky
Phys. Rev. C 74 (2006) 064314
[2] A. Volya and V. Zelevinsky
Phys. Rev. Lett. 94 (2005) 052501
[3] A. Volya and V. Zelevinsky
Phys. Rev. C 67 (2003) 054322
Specific features of the
continuum shell model
• Remnants of traditional shell model
• Non-Hermitian Hamiltonian
• Energy-dependent Hamiltonian
• Decay chains
• New effective interaction – unknown…
(self – made recipes) …
Energy-dependent Hamiltonian
• Form of energy-dependence
• Consistency with thresholds
• Appropriate near-threshold behavior
• How to solve energy-dependent H
• Consistency in solution
• Determination of energies
• Determination of open channels
Interpretation of complex energies
• For isolated narrow resonances all
definitions agree
• Real Situation
• Many-body complexity
• High density of states
• Large decay widths
• Result:
• Overlapping, interference, width redistribution
• Resonance and width are definition dependent
• Non-exponential decay
• Solution: Cross section is a true observable
(S-matrix )
Calculation Details,
Time – Dependent
•Scale Hamiltonian so that eigenvalues are in [-1 1]
•Expand evolution operator in Chebyshev polynomials
•Use iterative relation and matrix-vector multiplication to generate
•Use FFT to find return to energy representation
*W.Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in C++
the art of scientific computing, Cambrige University Press, 2002
T. Ikegami and S. Iwata, J. of Comp. Chem. 23 (2002) 310-318
Green’s function calculation
• Advantages of the
• -No need for full
diagonalization or
inversion at different E
• -Only matrix-vector
• -Numerical stability
Interplay of collectivities
n - labels particle-hole state
en – excitation energy of state n
dn - dipole operator
An – decay amplitude of n
Model Hamiltonian
• Two doorway states of
different nature
• Real energy: multipole
• Imaginary energy:
super-radiant state
Driving GDR externally
(doing scattering)
Everything depends on
angle between multi-dimensional vectors
A and d
Interplay of collectivities
n - labels particle-hole state
en – excitation energy of state n
dn - dipole operator
An – decay amplitude of n
Model Hamiltonian
Driving GDR externally
(doing scattering)
Everything depends on
angle between multi dimensional vectors
A and d
Pygmy resonance
GDR not seen
Parallel case:
and particle-hole
states with pion
quantum numbers
A model of 20 equally
distant levels is used
Most effective excitation
of GDR from continuum
At angle:
excitation of GDR
and pigmy
Loosely stated, the PTD is based on the assumptions that
s-wave neutron scattering is a single-channel process, the
widths are statistical, and time-reversal invariance holds;
hence, an observed departure from the PTD implies that
one or more of these assumptions is violated P.E. Koehler et al.
PRL 105, 072502
Time-reversal invariance holds
Single-channel process
Widths are statistical (whatever it means…)
Intrinsic “chaotic” states are uncorrelated
Energy dependence of widths is uniform
No doorway states
No structure pecularities
(b) and (d) are wrong; (c), (e), (f), (g) depend on the nucleus
Resonance width distribution
(chaotic closed system, single open channel)
G. Shchedrin, V.Z., PRC (2012)
Adding many “gamma” - channels
Level spacing distribution
in an open system with
a single decay channel:
No level repulsion at short distances!
(Energy of an unstable state is not well defined)
No level repulsion in
the intermediate region
Super-radiant transition
in Random Matrix Ensemble
N= 1000, m=M/N=0.25
Particle in Many-Well Potential
Hamiltonian Matrix:
•No continuum coupling - analytic solution
•Weak decay - perturbative treatment of decay
•Strong decay – localization of decaying states at the edges
Typical Example
e=0 and v=1
Critical decay strength g about 2
Decay width as a function of energy
Location of particle
Disordered problem
Disordered problem
of a particle
(or signal
Star graph
Ziletti et al. Phys. Rev. B 85, 052201 (2012)
Many-branch (M) junction coupled at the origin
Long-lived quasibound states at the junction
Average width of all widths or of (all-M) widths, M=4
Universal “phase transition”
SIMILAR SYSTEMS: inserted qubit
sequence of two-level atoms
coupled oscillators
heat-bath environment
realistic reservoirs
biological molecules
Transmission picture T(12) for M=4;
Blue dashed lines – very strong continuum coupling;
All equal branches
Non-equal branches
Critical disorder parameter
EPL 88 (2009) 27003
Cross section (conductance) fluctuations
in a system of randomly interacting
fermions, similarly to the shell model,
as a function of the intrinsic interaction
strength. Transition (lambda =1) –
onset of chaos, exactly as in the theory
of universal conductance fluctuations
in quantum wires
7 particles, 14 orbitals,
3432 many-body states, 20 open channels
Cross section (conductance) fluctuations
as a function of openness.
No dependence on the character of chaos,
one-body (disorder) or
many-body (interactions).
Transition to superadiance: kappa=1
(‘’perfect coupling”)
Many – Body
1.C. Mahaux and H.A. Weidenmueller, Shell Model Approach to Nuclear Reactions (1969)
Formalism of effective Hamiltonian
2. R.H. Dicke, Phys. Rev. 93, 99 (1954)
Super-radiance in quantum optics
3. V.V. Sokolov and V.G. Zelevinsky, Nucl. Phys. A504, 562 (1989); Ann. Phys. 216, 323 (1992).
Super-radiance in open many-body systems
4. A. Volya and V. Zelevinsky, Phys. Rev. Lett. 94, 052501(2005); Phys. Rev. C 74, 064314 (2006).
Continuum shell model (CSM)
5. N. Michel, W. Nazarewicz, M. Ploszajczak, and T. Vertse, J. Phys. G 36, 013101 (2009).
Alternative approach: Gamow shell model
6. G.L. Celardo et al. Phys. Rev. E 76, 031119 (2007); Phys. Lett. B 659, 170 (2008);
EPL 88, 27003 (2009); A. Ziletti et al. Phys. Rev. B 851, 052201 (2012).; Y. Greenberg et al. EPJ
Quantum signal transmission
B86, 368 (2013).
7. C.W.J. Beenakker, Rev. Mod. Phys. 69, 731 (1997). Universal conductance fluctuations
8. T. Ericson and T. Mayer-Kuckuk, Ann. Rev. Nucl. Sci. 16, 183 (1966). ”Ericson fluctuations”
9. N. Auerbach and V.Z. Phys. Rev. C 65, 034601 (2002). Pions and Delta-resonance
10. A. Volya, Phys. Rev. C 79, 044308 (2009).
Modern development of CSM
11. N. Auerbach and V.Z. Rep. Prog. Phys. 74, 106301 (2011). Review - Effective Hamiltonian
12. A. Volya. EPJ Web of Conf. 38, 03003 (2012). From structure to sequential decays.
13. A. Volya and V.Z. Phys. At. Nucl. 77, 1 (2014). Nuclear physics at the edge of stability.
• No harmonic oscillator
• Correlated decays
• Cluster decays
• Transfer reactions
• Microscopic derivation of
the Hamiltonian
• Collectivity in continuum
• New applications
• >>>>>>
Quantum Decay: exponential versus non-exponential
* [Kubo] - exponential decay corresponds to the condition for
a physical process to be approximated as a Markovian process
* [Silverman] - indeed a random process, no “cosmic force”
* [Merzbacher] - result of “delicate” approximations
Three stages: short-time
main (exponential)
Why and when decay cannot be exponential
Initial state “memory” time
Internal motion in quasi-bound state
Remote power-law
There are “free” slow-moving non-resonant particles, they escape slowly
Example 14C decay: E0=0.157 MeV t2=10-21 s
Time dependence of decay, Winter’s model
Winter, Phys. Rev., 123,1503 1961.
Winter’s model:
Dynamics at remote times
Internal dynamics in decaying system
Winter’s model
Is it possible to have oscillatory decay?
Decay oscillations are possible
•Kinetic energy - mass eigenstates
•Interaction (barrier)- flavor eigenstates
•Fast and slow decaying modes
Survival probability
[1] A Volya, M. Peshkin, and V. Zelevinsky, work in progress
Oxygen Isotopes
Continuum Shell Model Calculation
• sd space, HBUSD interaction
• single-nucleon reactions
Time-dependent approach
• Reflects time-dependent physics of unstable
• Direct relation to observables
• Linearity of QM equations maintained
• No matrix diagonalization
• New many-body numerical techniques
• Stability for broad and narrow resonances
• Ability to work with experimental data
Time evolution of several SM states
in 24O. The absolute value of the
survival overlap is shown
A. Volya, Time-dependent approach to the continuum shell model, Phys. Rev. C 79, 044308 (2009).
EPL 88 (2009) 27003
Variance of cross section fluctuations
for a system of randomly interacting
fermions similarly to the nuclear shell
model as a function of the strength
of internal chaotic interaction:
In the transition to chaos (lambda=1),
we see precisely the same evolution
from 2/15 to 1/8 as predicted by
theory of universal conductance
fluctuations in quantum wires.
Identical results for many-body
chaos (coming from interactions)
and one-body disorder
as a function of degree of
openness (coupling to continuum);
Kappa=1 is “perfect coupling”
(phase transition to super-radiance)
Many – Body
as an Open System:
Continuum Shell Model and
New Challenges
Vladimir Zelevinsky
NSCL/ Michigan State University
Supported by NSF
Bruyères-le-Châtel, May 2014

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