Continuum Shell Model and New Challenges 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

OUTLINE From 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)

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From closed to open (or marginally stable) many-body system CLOSED SYSTEMS: Bound states Mean field, quasiparticles Symmetries Residual interactions Pairing, superfluidity Collective modes Quantum many-body chaos (GOE type) Open systems: 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? (Many…)

Examples: IAS, single-particle resonance, giant resonances DooRWAY STATES 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

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 Narrow resonances and broad superradiant state in 12C in the region of Delta Optics Molecules Microwave cavities Nuclei Hadrons Quantum computing Measurement theory Bartsch 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 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

(+) means + i0 (Eigenchannels in P-space) (off-shell) (on-shell) Factorization (unitarity), energy dependence (kinematic thresholds) , coupling through continuum

Self energy, interaction with continuum Correction to Harmonic Oscillator Wave Function s,p, and d waves (red, blue, black) 17O momentum Gamow shell model N Michel, J. Phys. G: Nucl. Part. Phys. 36 (2009) 013101 Notes: Wave functions are not HO Phenomenological SM is adjusted to observation No corrections for properly solved mean field A. Volya, EPJ Web of Conf. 38, 03003 (2012).

The nuclear many-body problem Single-particles state (particle in the well) Many-body states (slater determinants) Hamiltonian and Hamiltonian matrix Matrix diagonalization Traditional shell-model Effective non-Hermitian and energy-dependent Hamiltonian Channels (parent-daughter structure) Bound states and resonances Matrix inversion at all energies (time dependent approach) Continuum physics Formally exact approach Limit of the traditional shell model Unitarity of the scattering matrix

Ingredients Intrinsic states: Q-space Continuum states: P-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 =|A|2 Resonance overlaps: level spacing vs. width “kappa” parameter No approximations until now

EFFECTIVE HAMILTONIAN One open channel

Interaction between resonances Real V Energy repulsion Width attraction Imaginary W Energy attraction Width repulsion

Dynamics of states coupled to a common decay channel 11Li model Dynamics of states coupled to a common decay channel Model Mechanism of binding

11Li model Dynamics of two states coupled to a common decay channel A1 and A2 opposite signs Model H

Continuum Shell Model Calculation sd space, HBUSD interaction 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) http://www.nscl.msu.edu/general-public/news/2012/O26

[2] A. Volya and V. Z. Phys. Rev. C 74 (2006) 064314, [3] G [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+ 1+ 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 channel References [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 method -No need for full diagonalization or inversion at different E -Only matrix-vector multiplications -Numerical stability

Interplay of collectivities Definitions n - labels particle-hole state n – excitation energy of state n dn - dipole operator An – decay amplitude of n Two doorway states of different nature Real energy: multipole resonance Imaginary energy: super-radiant state Model Hamiltonian Driving GDR externally (doing scattering) Everything depends on angle between multi-dimensional vectors A and d

Interplay of collectivities Definitions n - labels particle-hole state n – 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

Most effective excitation Pygmy resonance Orthogonal: GDR not seen Parallel: Most effective excitation of GDR from continuum At angle: excitation of GDR and pigmy Parallel case: Delta-resonance and particle-hole states with pion quantum numbers A model of 20 equally distant levels is used

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 (2010) - 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

0.1 0.5 1.0 5.0 No level repulsion at short distances! Level spacing distribution in an open system with a single decay channel: No level repulsion in the intermediate region 0.1 0.5 1.0 5.0 No level repulsion at short distances! (Energy of an unstable state is not well defined)

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 solution Weak decay - perturbative treatment of decay Strong decay – localization of decaying states at the edges

Typical Example N=1000 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 Localization of a particle (or signal transmission)

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 Many – Body One-Body 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 One-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 B86, 368 (2013). Quantum signal transmission 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 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

CHALLENGES: 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) Oscillations? long-time

Quantum mechanics of decay Why exponential decay? Time evolution and decay in quantum mechanics Survival amplitude and probability Resonance wave function E

Discussion continues: Is radioactive decay exponential? The 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). Period = 7 sec ! Half life 5,730 ± 40 years mean-life time 8,033 years Carbon 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).

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 =73

Time dependence of decay, Winter’s model Winter, Phys. Rev., 123,1503 1961.

Dynamics at remote times Winter’s model: Dynamics at remote times background resonance

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 Current oscillations 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

CSM 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 Decay and nuclear mean field At low energies amplitudes are defined by penetrability which is given by channel radius R-Matrix expressions e(MeV) γ(keV) r(fm) 5He 0.895 648 4.5* 17O 0.941 98 3.8 19O 1.540 310 3.9

Time-dependent approach Reflects time-dependent physics of unstable systems 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 One-Body

Continuum Shell Model and New Challenges Nucleus as an Open System: Continuum Shell Model and New Challenges Vladimir Zelevinsky NSCL/ Michigan State University Supported by NSF Bruyères-le-Châtel, May 2014