1. Super - Radiance, Collectivity and Chaos in the Continuum Vladimir Zelevinsky NSCL/ Michigan State University Supported by NSF Workshop on Level Density and Gamma Strength in Continuum Oslo, May 22, 2007

2. OUTLINE 1.Super-radiance 2.Analogs in open many-body systems 3.Theory - how to handle 4.Statistics of resonances 5.Statistics of cross sections 6.Towards continuum shell model 7.Collectivity in continuum 8.Mesoscopic example 9.Future challenge

3. THANKS Valentin Sokolov (Novosibirsk) Alexander Volya (Florida State) Naftali Auerbach (Tel Aviv) Felix Izrailev (Puebla) Luca Celardo (Puebla) Gennady Berman (Los Alamos)

7. Examples of superradiance Mechanism of superradiance Interaction via continuum Trapped states ) self-organization Optics Molecules Microwave cavities Nuclei Hadrons Quantum computing Measurement theory Narrow resonances and broad superradiant state in 12 C D Bartsch et.al. Eur. Phys. J. A 4, 209 (1999)

9. Basic Theory [1] C. Mahaux and H. Weidenmüller, Shell-model approach to nuclear reactions, North-Holland Publishing, Amsterdam 1969

12. Ingredients Intrinsic states: P-space –States of fixed symmetry –Unperturbed energies  1 ; some  1 >0 –Hermitian interaction V Continuum states: Q-space –Channels and their thresholds E c th –Scattering matrix S ab ( E ) Coupling with continuum –Decay amplitudes A c 1 ( E ) –Typical partial width  =|A| 2 –Resonance overlaps: level spacing vs. width

13. EFFECTIVE HAMILTONIAN

17. Specific features of the continuum shell model Remnants of traditional shell model Non-Hermitian Hamiltonian Energy-dependent Hamiltonian Decay chains New effective interaction – unknown…

18. 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 )

19. Complex Energy eigenvalue problem Eigenvalue problem has only complex (E>threshold) roots but E is real? Definitions of resonance Gamow : poles of scattering matrix Eigenvalue proglem with regular inside outgoing outside boundary condition ! discrete resonant states + complex energies Breit-Wigner: Find roots on real axis Cross section peaks and lifetimes

24. Probability P(s) for s<0.04 as a function of the overlap

26. 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

27. Parameterization of energy dependence

28. 11 Li model Dynamics of two states coupled to a common decay channel Model H Mechanism of binding by Hermitian interaction

29. Solutions with energy-dependent widths Energy-independent width is not consistent with definitions of threshold Squeezing of phase-space volume in s and p waves, Threshold E c =0 Model parameters:  1 =100,  2 =200, A 1 =7.1 A 2 =3.1 (red);  =1,  =0.05 (blue) (in units based on keV) Upper panel: Energies with A 1 =A 2 =0 (black)

30. Model parameters:  1 =100,  2 =200, A 1 =8.1 A 2 =12.8 (red);  =15,  =0.05 (blue) (in units based on keV) Upper panel: Energies with A 1 =A 2 =0 (black) and case b (blue) Bound state in the continuum:  =0 above threshold

31. Interaction between resonances Real V –Energy repulsion –Width attraction Imaginary W –Energy attraction –Width repulsion

32. Scattering and cross section near threshold Scattering Matrix Cross section Solution in two-level model

33. Cross section near threshold No direct interaction v=0 Breit-Wigner resonance Below critical v (both states in continuum)- sharp resonances Above critical v One state is bound- “attraction” to sub-threshold region fig (a) Second state –resonance, fig (b) Model parameters:  1 =100,  2 =200, v=180 (keV) A 1 2 =0.05 (E) 3/2, A 2 2 =15 (E) 1/2

34. 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

35. Realistic model - Oxygen isotopes sd-shell, USD interaction Exact pairing+monopole; truncation of space s=0,1 0d 3/2 in the continuum, L=2  (2) 3/2 = 2  (1) 3/2 =Γ( 17 O)/e 5/2 Fully self consistent treatment

36. Oxygen Isotopes Continuum Shell Model Calculation sd space, HBUSD interaction single-nucleon reactions

37. Interplay of collectivities Definitions n - labels particle-hole state  n – excitation energy of state n d n - dipole operator A n – decay amplitude of n Model Hamiltonian Everything depends on angle between multi dimensional vectors A and d Driving GDR externally (doing scattering)

40. Pigmy resonance Orthogonal: GDR not seen Parallel: Most effective excitation of GDR from continuum At angle: excitation of GDR and pigmy A model of 20 equally distant levels is used

41. Model Example

42. 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

43. Typical Example N=1000  =0 and v=1 Critical decay strength  about 2 Decay width as a function of energy Location of particle

44. Disordered problem

45. Example: disorder + localization N=1000  =random number and v=1 Critical decay strength  about 2 Location of a particle

46. Distribution of widths as a function of decay strength Weak decay: Random Distribution Transitional region: Formation of superradiant states Strong decay: Superradiance

47. The role of internal degrees of freedom in scattering and tunneling m2m2 m1m1 x1x1 x2x2 The composite object The External Potential: Intrinsic Potential: Incident wave Transmission Reflection

49. Effective potential for slowly moving “deuteron”

50. Transmission probability through the effective barrier

51. Questions and Future Interactions Pairing and continuum Interplay of collective modes and particle decay Channels with internal degrees of freedom Applications beyond nuclear physics

52. PAIR CORRELATOR (b) Only pairing (d) Non-pairing interactions (f) All interactions