PowerPoint Presentation

Nucleus as an Open System: Continuum Shell Model and New Challenges

Vladimir Zelevinsky NSCL/ Michigan State University Supported by NSF Caen, GANIL May 30, 2014

OUTLINEFrom closed to open many-body systems

Effective non Hermitian Hamiltonian

Doorways and phenomenon of super-radiance

Continuum shell model

Statistics of complex energies

Cross sections, resonances, correlations and fluctuations

Quantum signal transmission

THANKS 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 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 applicationsFRIB will be a national user facility for the U.S. Department of Energy Office of Science when FRIB becomes operational, NSCL will transition into FRIBNSCL and FRIB Laboratory 543 employees, incl. 38 faculty, 59 graduate and 82 undergraduate studentsas of April 21, 2014200320092011User group of over 1300 members with approx. 20 working groupswww.fribusers.orgC.K. Gelbke, 5/5/2014, Slide #The Evolution of Nuclear Science at MSU

C.K. Gelbke, 5/5/2014, Slide #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 interactionsMany-body quantum problem: intellectual overlap to mesoscopic science, quantum dots, atomic clusters, etc. Mesoscopic Theory Astrophysical processes JINAOrigin of the elements in the cosmosExplosive environments: novae, supernovae, X-ray bursts Properties of neutron starsTests of fundamental symmetriesEffects of symmetry violations are amplified in certain nucleiSocietal applications and benefitsBio-medicine, energy, material sciences Varian, isotope harvesting, National security NNSA

Reaping benefits from recent investments while creating future opportunitiesC.K. Gelbke, 5/5/2014, Slide #UNEDF: Unified Nuclear Energy Density FunctionalSciDAC: Scientific Discovery through Advanced ComputingSC proton-therapy cyclotrons (ACCEL/VARIAN): PSI - Switzerland, St. Petersburg - Russia, RPTC Munich - Germany, Scripps - San Diego, Riyadh - Saudi Arabia

FRIB Science is TransformationalFRIB 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 #Examples for Cross-Disciplinary and Applied Research TopicsMedical researchExamples: 47Sc, 62Zn, 64Cu, 67Cu, 68Ge, 149Tb, 153Gd, 168Ho, 177Lu, 188Re, 211At, 212Bi, 213Bi, 223Ra (DOE expert panel)MSU Radiology Dept. interested in 60,61Cu-emitters 149Tb, 211At: potential treatment of metastatic cancerPlant biology: role of metals in plant metabolismEnvironmental and geosciences: ground water, role of metals as catalystsEngineering: advanced materials, radiation damage, diffusion studiesToxicology: toxicology of metalsBiochemistry: role of metals in biological process and correlations to diseaseFisheries and Wildlife Sciences: movement of pollutants through environmental and biological systemsReaction rates important for stockpile stewardship non-classified researchDetermination of extremely high neutron fluxes by activation analysisRare-isotope samples for (n,g), (n,n), (n,2n), (n,f) e.g. 88,89ZrSame technique important for astrophysics Far from stability: surrogate reactions (d,p), (3He,a xn) Vision: Up to 10 Faculty Positions for Cross-Disciplinary and Applied ResearchFrom closed to open (or marginally stable) many-body system CLOSED SYSTEMS:Bound statesMean field, quasiparticlesSymmetriesResidual interactionsPairing, superfluidityCollective modesQuantum many-body chaos (GOE type)Open systems: Continuum energy spectrumUnstable states, lifetimesDecay channels (E,c)Energy thresholdsCross sections Resonances, isolated or overlappingStatistics of resonances and cross sections

Unified approach? (Many) DooRWAY STATESFrom 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

Single-particle decay in many-body systemTotal states 8!/(3! 5!)=56; states that decay fast 7!/(2! 5!)=21 superradiant doorways8 s.p. levels, 3 particlesOne s.p. level in continuum Evolution of complex energies

Examples of superradiance Mechanism of superradianceInteraction via continuumTrapped states - self-organization

Optics Molecules Microwave cavities Nuclei Hadrons Quantum computing Measurement theory

Narrow resonances and broad superradiant state in 12C in the region of DeltaBartsch et.al. Eur. Phys. J. A 4, 209 (1999) N. Auerbach, V.Z.. Phys. Lett. B590, 45 (2004)

Physics and mathematics of coupling to continuum[1] C. Mahaux and H. Weidenmller, Shell-model approach to nuclear reactions, North-Holland Publishing, Amsterdam 1969

New part of Hamiltonian: coupling through continuum

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)

(off-shell)(on-shell)Factorization (unitarity), energy dependence (kinematic thresholds) , coupling through continuum(+) means + i0Self energy, interaction with continuum

Correction to Harmonic Oscillator Wave Functions,p, and d waves (red, blue, black)

17O

momentum

N Michel, J. Phys. G: Nucl. Part. Phys. 36 (2009) 013101Gamow shell modelA. Volya, EPJ Web of Conf. 38, 03003 (2012).Wave functions are not HOPhenomenological SM is adjusted to observationNo corrections for properly solved mean fieldNotes:

19

The nuclear many-body problemEffective non-Hermitian and energy-dependent HamiltonianChannels (parent-daughter structure)Bound states and resonancesMatrix inversion at all energies (time dependent approach)Continuum physics Single-particles state (particle in the well)Many-body states (slater determinants)Hamiltonian and Hamiltonian matrixMatrix diagonalizationTraditional shell-model Formally exact approachLimit of the traditional shell modelUnitarity of the scattering matrixIngredientsIntrinsic states: Q-space States of fixed symmetryUnperturbed energies e1; some e1>0Hermitian interaction VContinuum states: P-space Channels and their thresholds EcthScattering matrix Sab(E)Coupling with continuumDecay amplitudes Ac1(E) - thresholdsTypical partial width =|A|2Resonance overlaps: level spacing vs. width No approximations until nowkappa parameter

EFFECTIVE HAMILTONIANOne open channel

Interaction between resonancesReal VEnergy repulsionWidth attractionImaginary WEnergy attractionWidth repulsion

11Li modelDynamics of states coupled to a common decay channelModel

Mechanism of binding

11Li modelDynamics of two states coupled to a common decay channelModel H

A1 and A2 opposite signs

Oxygen IsotopesContinuum Shell Model Calculation sd space, HBUSD interaction single-nucleon reactions A. Volya and V. Zelevinsky, Phys. Rev. Lett. 94, 052501 (2005); Phys. Rev. C 67, 054322 (2003); Phys. Rev. C 74, 064314 (2006).2728

[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)

Predictive power of theory

Measured 2009-2013 Continuum Shell Model prediction 2003-2006http://www.nscl.msu.edu/general-public/news/2012/O26

[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. Experimental data from: C. Hoffman, et.al. Phys. Lett. B672, 17 (2009)

experiment 2+ 1+

Two-neutron sequential decay of 26OA. Volya and V. Zelevinsky, Continuum shell model, Phys. Rev. C 74, 064314 (2006).

Predicted Q-value: 21 keVZ. Kohley, et.al PRL 110, 152501 (2013) (experiment)

CSM calculation of 18OStates 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 channelReferences[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

34Specific features of thecontinuum shell model Remnants of traditional shell modelNon-Hermitian HamiltonianEnergy-dependent HamiltonianDecay chainsNew effective interaction unknown

(self made recipes) Energy-dependent HamiltonianForm of energy-dependenceConsistency with thresholdsAppropriate near-threshold behaviorHow to solve energy-dependent HConsistency in solutionDetermination of energiesDetermination of open channelsInterpretation of complex energiesFor isolated narrow resonances all definitions agreeReal SituationMany-body complexityHigh density of statesLarge decay widths Result: Overlapping, interference, width redistributionResonance and width are definition dependentNon-exponential decaySolution: 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 polynomialsUse iterative relation and matrix-vector multiplication to generate

Use FFT to find return to energy representationT. Ikegami and S. Iwata, J. of Comp. Chem. 23 (2002) 310-318*W.Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in C++ the art of scientific computing, Cambrige University Press, 2002 Greens function calculationAdvantages of the method-No need for full diagonalization or inversion at different E-Only matrix-vector multiplications-Numerical stability

Interplay of collectivities

Definitionsn - labels particle-hole staten excitation energy of state ndn - dipole operator An decay amplitude of n Model HamiltonianEverything depends onangle between multi-dimensional vectorsA and d Driving GDR externally(doing scattering)

Two doorway states of different natureReal energy: multipole resonanceImaginary energy: super-radiant state

Interplay of collectivities

Definitionsn - labels particle-hole staten excitation energy of state ndn - dipole operator An decay amplitude of n Model HamiltonianEverything depends onangle between multi dimensional vectorsA and d Driving GDR externally(doing scattering)

Pygmy resonance

Orthogonal:GDR not seenParallel:Most effective excitation of GDR from continuumAt angle: excitation of GDR and pigmyA model of 20 equally distant levels is usedParallel case:Delta-resonanceand particle-holestates with pionquantum numbers Loosely stated, the PTD is based on the assumptions thats-wave neutron scattering is a single-channel process, thewidths are statistical, and time-reversal invariance holds;hence, an observed departure from the PTD implies thatone or more of these assumptions is violatedP.E. Koehler et al.PRL 105, 072502 (2010) -Time-reversal invariance holdsSingle-channel processWidths are statistical (whatever it means)Intrinsic chaotic states are uncorrelatedEnergy dependence of widths is uniformNo doorway statesNo 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

0.1 0.51.05.0No level repulsion at short distances!

(Energy of an unstable state is not well defined)Level spacing distributionin an open system witha single decay channel:No level repulsion inthe intermediate region

Super-radiant transition

in Random Matrix Ensemble

N= 1000, m=M/N=0.25

Particle in Many-Well Potential

Hamiltonian Matrix:Solutions: No continuum coupling - analytic solutionWeak decay - perturbative treatment of decayStrong decay localization of decaying states at the edgesTypical Example

N=1000e=0 and v=1Critical decay strength g about 2Decay width as a function of energyLocation of particleDisordered problem

Disordered problem

Localizationof a particle(or signaltransmission)

Star graphZiletti et al. Phys. Rev. B 85, 052201 (2012)

Many-branch (M) junction coupled at the originAverage width of all widths or of (all-M) widths, M=4Universal phase transitionLong-lived quasibound states at the junctionSIMILAR 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 branchesNon-equal branchesCritical disorder parameter

EPL 88 (2009) 27003

Many Body One-BodyCross 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 theoryof universal conductance fluctuationsin quantum wires Cross section (conductance) fluctuations as a function of openness. No dependence on the character of chaos,one-body (disorder) ormany-body (interactions).Transition to superadiance: kappa=1(perfect coupling)7 particles, 14 orbitals, 3432 many-body states, 20 open channels2. R.H. Dicke, Phys. Rev. 93, 99 (1954)3. V.V. Sokolov and V.G. Zelevinsky, Nucl. Phys. A504, 562 (1989); Ann. Phys. 216, 323 (1992).4. A. Volya and V. Zelevinsky, Phys. Rev. Lett. 94, 052501(2005); Phys. Rev. C 74, 064314 (2006).5. N. Michel, W. Nazarewicz, M. Ploszajczak, and T. Vertse, J. Phys. G 36, 013101 (2009).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 B86, 368 (2013).

Formalism of effective Hamiltonian Super-radiance in quantum opticsSuper-radiance in open many-body systems Continuum shell model (CSM) Alternative approach: Gamow shell model Quantum signal transmission7. C.W.J. Beenakker, Rev. Mod. Phys. 69, 731 (1997). Universal conductance fluctuations8. T. Ericson and T. Mayer-Kuckuk, Ann. Rev. Nucl. Sci. 16, 183 (1966). Ericson fluctuations9. N. Auerbach and V.Z. Phys. Rev. C 65, 034601 (2002). Pions and Delta-resonance

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.

10. A. Volya, Phys. Rev. C 79, 044308 (2009). Modern development of CSM 1.C. Mahaux and H.A. Weidenmueller, Shell Model Approach to Nuclear Reactions (1969) CHALLENGES:No harmonic oscillator Correlated decaysCluster decaysTransfer reactions Microscopic derivation of the HamiltonianCollectivity in continuumNew 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 approximationsThree stages: short-time main (exponential) Oscillations? long-time

Quantum mechanics of decay61

Why exponential decay? Survival amplitude and probabilityResonance wave functionTime evolution and decay in quantum mechanics

EDiscussion continues: Is radioactive decay exponential?62The GSI oscillations Mystery (2008)Periodic modulation of the expected exponential law in EC-decays of different highly charged ions Litvinov et al. Phys. Lett. B 664, 162 (2008);P. Kienle et al. Phys. Lett. B 726, 638 (2013).

Half life 5,730 40 years mean-life time 8,033 yearsCarbon dating and non-exponential decay (2012)If the decay of 14C is indeed non-exponential... this would remove a foundation stone of modern dating methods." Aston EPL 97, 52001 (2012).

Period = 7 sec !Why and when decay cannot be exponential63Initial 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

=73

64

Time dependence of decay, Winters modelWinter, Phys. Rev., 123,1503 1961.

65

Winters model: Dynamics at remote times

backgroundresonanceInternal dynamics in decaying system Winters model66

t2t1

Is it possible to have oscillatory decay?[1] A Volya, M. Peshkin, and V. Zelevinsky, work in progress

Decay oscillations are possibleKinetic energy - mass eigenstatesInteraction (barrier)- flavor eigenstatesFast and slow decaying modesoscillationsCurrentSurvival probability

Oxygen IsotopesContinuum Shell Model Calculation sd space, HBUSD interaction single-nucleon reactions L. V. Grigorenko, et al. Phys. Rev. C 84, 021303 (2011)V. Zelevinsky, A. Volya, Yad. Fiz. 77, issue 7, 1-14 (2014).

CSM

Low-energy phase-space decay laws

e(MeV)(keV)r(fm)5He0.8956484.5*17O0.941983.819O1.5403103.9Decay and nuclear mean field

At low energies amplitudes are defined by penetrability which is given by channel radius

R-Matrix expressions

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).Time-dependent approach24OReflects time-dependent physics of unstable systemsDirect relation to observablesLinearity of QM equations maintainedNo matrix diagonalizationNew many-body numerical techniquesStability for broad and narrow resonancesAbility to work with experimental data

Variance of cross section fluctuationsfor a system of randomly interactingfermions similarly to the nuclear shellmodel as a function of the strengthof internal chaotic interaction:In the transition to chaos (lambda=1),we see precisely the same evolutionfrom 2/15 to 1/8 as predicted by theory of universal conductance fluctuations in quantum wires. EPL 88 (2009) 27003

Many Body One-BodyIdentical results for many-bodychaos (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) Nucleus as an Open System: Continuum Shell Model and New Challenges

Vladimir Zelevinsky NSCL/ Michigan State University Supported by NSF Bruyres-le-Chtel, May 2014

Superradiance, collectivization by decay
Dicke coherent stateN identical two-level atomscoupled via common radiation
Analog in a complex systemInteraction via continuumTrapped states ) self-organization
Volume 3
g ~ D and few channels Nuclei far from stability High level density (states of same symmetry) Channel thresholds