1. Shell Structure of Nuclei and Cold Atomic Gases in Traps Sven Åberg, Lund University, Sweden From Femtoscience to Nanoscience: Nuclei, Quantum Dots, and Nanostructures July 20 - August 28, 2009

2. I.Shell structure from mean field picture (a)Nuclear masses (ground-states) (b)Ground-states in cold gas of Fermionic atoms: supershell structure II.Shell structure of BCS pairing gap (a)Nuclear pairing gap from odd-even mass difference (b)Periodic-orbit description of pairing gap fluctuations - role of regular/chaotic dynamics (c)Applied to nuclear pairing gaps and to cold gases of Fermionic atoms III.Cold atomic gases in a trap – Solved by exact diagonalizations (a)Cold Fermionic atoms in 2D traps: Pairing versus Hund’s rule (b)Effective-interaction approach to interacting bosons Shell Structure of Nuclei and Cold Atomic Gases in Traps Collaborators: Stephanie Reimann, Massimo Rontani, Patricio Leboeuf Henrik Olofsson/Urenholdt, Jeremi Armstrong, Matthias Brack, Jonas Christensson, Christian Forssén, Magnus Ögren, Marc Puig von Friesen, Yongle Yu,

3. I. Shell structure from mean field picture

4. Shell energy I.a Shell structure in nuclear mass Shell energy = Total energy (=mass) – Smoothly varying energy P. Möller et al, At. Data and Nucl. Data Tables 59 (1995) 185

5. I.b Ground states of cold quantum gases Trapped quantum gases of bosonic or fermionic atoms: T0T0 Bose condensate Degenerate fermi gas

6. Fermionic atoms in a 3D H.O. confinement a = s-wave scattering length Un-polarized two-component system with two spin-states: Hartree-Fock approximation: Where: > 0 (repulsive int.)

7. Shell energy vs particle number for pure H.O. Fourier transform Shell energy: E osc = E tot - E av N Fermionic atoms in harmonic trap – Repulsive int. No interaction

8. Super-shell structure predicted for repulsive interaction[1] g=0.2 g=0.4 g=2 Two close-lying frequencies give rise to the beating pattern: circle and diameter periodic orbits Effective potential: [1] Y. Yu, M. Ögren, S. Åberg, S.M. Reimann, M. Brack, PRA 72, 051602 (2005)

9. II. Shell structure of BCS pairing gap [1] [1] S. Åberg, H. Olofsson and P. Leboeuf, AIP Conf Proc Vol. 995 (2008) 173.

10. odd N even N I. Odd-even mass difference Extraction of pairing contribution from masses: where is s.p. level density If no pairing: 2  3 (N) = 0 N=odd ee...... N=even ee...... 2  3 (N) =  e W. Satula, J. Dobaczewski and W. Nazarewicz, PRL 81 (1998) 3599   (N even) =  +  e/2   (N odd) = 

11. Odd-even mass difference from data odd even odd+even 12/A 1/2 2.7/A 1/4   (MeV)

12. Single-particle distance from masses 50/A MeV Pairing delta eliminated in the difference:  (3) (even N) -  (3) (odd N) = 0.5(e n+1 – e n ) = d/2 Fermi-gas model: See e.g.: WA Friedman, GF Bertsch, EPJ A41 (2009) 109

13. Pairing gap  3 odd from different mass models Mass models all seem to provide pairing gaps in good agreement with exp. P. Möller et al, At. Data and Nucl. Data Tables 59 (1995) 185. M. Samyn et al, PRC70, 044309 (2004). J. Duflo and A.P. Zuker, PRC52, R23 (1995).

14. Pairing gap from different mass models Average behavior in agreement with exp. but very different fluctuations

15. Fluctuations of the pairing gap

16. II.b Periodic orbit description of BCS pairing - Role of regular and chaotic dynamics [1] H. Olofsson, S. Åberg and P. Leboeuf, Phys. Rev. Lett. 100, 037005 (2008)

17. Periodic orbit description of pairing Level density Insert semiclassical expression Pairing gap equation: where is ”pairing time” Divide pairing gap in smooth and fluctuating parts: Expansion in fluctuating parts gives:

18. Fluctuations of pairing gap Fluctuations of pairing gap become where K is the spectral form factor (Fourier transform of 2-point corr. function): is shortest periodic orbit, is Heisenberg time

19. If regular: If chaotic:  single-particle mean level spacing) RMS pairing fluctuations: Dimensionless ratio: D=2R/  0 Size of system: 2R (Number of Cooper pairs along 2R)Corr. length of Cooper pair:  0 =  v F /2  RMT-limit: D=0Bulk-limit: D→∞

20. Universal/non-universal fluctuations ”dimensionless conductance” Non-universal spectrum fluctuations for energy distances larger than g: universal non-universal g=L max  3 statistics Random matrix limit: g   (i.e. D = 0) corresponding to pure GOE spectrum (chaotic) or pure Poisson spectrum (regular)

21. If regular: If chaotic:  single-particle mean level spacing) RMS pairing fluctuations: Exp. Theory (regular) NucleiMetallic grains Irregular shape of grain  chaotic dynamics Universal pairing fluctuations D very small (GOE-limit) 50 000 6 Li atoms and k F |a| = 0.2 Fermionic atom gas, if regular, if chaotic Dimensionless ratio: D=2R/  0 Size of system: 2R (Number of Cooper pairs along 2R)Corr. length of Cooper pair:  0 =  v F /2  RMT-limit: D=0Bulk-limit: D→∞

22. Fluctuations of nuclear pairing gap from mass models

23. Shell structure in nuclear pairing gap

24. Average over proton-numbers

25. Shell structure in nuclear pairing gap P.O. description Average over Z

26. III. Cold atomic gases in 2D traps - Exact diagonalizations

27. III.a Cold Fermionic Atoms in 2D Traps [1] N atoms of spin ½ and equal masses m confined in 2D harmonic trap, interacting through a contact potential: [1] M. Rontani, JR Armstrong, Y, Yu, S. Åberg, SM Reimann, PRL 102 (2009) 060401. Solve many-body S.E. by full diagonalization  Ground-state energy and excitated states obtained for all angular momenta Energy scale: Length scale: Dimensionless coupling const.: Contact force regularized by energy cut-off [2]. Energy (and w.f.) of 2-body state relates strength g to scattering length a. [2] M. Rontani, S. Åberg, SM Reimann, arXiv:0810.4305 attractive repulsive

28. Non-int Ground-state energy E(N,g) in units of g=-0.3 g=-3.0 (g=0, pure HO) 2 10 18 Interaction energy: E int (N,g) = E(N,g) – E(N,g=0) g=-0.3 0 g=-0.3 2 10 18 Scaled interaction energy: E int (N,g)/N 3/2 -0.019 -0.015 -0.017 Attractive interaction

29. Cold Fermionic Atoms in 2D Traps – Pairing versus Hund’s Rule Interaction energy versus particle number Negative g (attractive interaction): odd-even staggering (pairing) Positive g (repulsive interaction): E int max at closed shells, min at mid-shell (Hund’s rule) attractive repulsive

30. Repulsive interaction 1 2 3 4 5 6 7 g=0.3 1.0 3.0 5.0 N 22 No interaction Repulsive interaction 1 2 3 4 5 6 7 g=0.3 1.0 3.0 5.0 N 22 Coulomb blockade – interaction blockade Coulomb blockade: Extra (electric) energy, E C, for a single electron to tunnel to a quantum dot with N electrons Difference between conductance peaks: where  e is energy distance between s.p. states N and N+1 and E(N) total energy Interaction (or van der Waals) blockade [1]: Add an atom to a cold atomic gas in a trap Cheinet et al, PRL 101 (2008) 090404 [1] C. Capelle et al PRL 99 (2007) 010402 Attractive interaction 1 2 3 4 5 6 7 22 N g=-0.3 -3.0 -5.0 Pairing gap:

31. -2 0 1 2 m 1 3 2 Non-int. picture, N=2 -2 0 1 2 m 1 3 2 Non-int. picture, N=8 M=0 M=1M=2 M=0 M=1M=2 Angular momentum dependence – yrast line

32. Angular momentum dependence – 4 and 6 atoms

33. Yrast line – higher M-values, excited states Pairing decreases with angular momentum and excitation energy:  Gap to excited states decreases  ”Moment of inertia” increases

34. Cold Fermionic Atoms in 2D Traps – 8 atoms N=8 particles Excitation spectra (6 lowest states for each M) Attractive and repulsive interaction Ground-state attractive int. Ground-state repulsive int. Onset of inter- shell pairing Excited states almost deg. with g.s. (cf strongly corr. q. dot)

35. -g/4  (pert. result) 1 st exc. state N=4, N=8  3 (3),  3 (7) Extracted pairing gaps

36. Two fermions measure probability to find ↑ fermion in xy plane fix ↓ fermion g = 0 Structure of w.f. from Conditional probability

37. g = - 0.1 Two fermions

38. g = - 0.3 Two fermions

39. g = - 0.6 Two fermions

40. g = - 1 Two fermions

41. g = - 1.5 Two fermions

42. g = - 2 Two fermions

43. g = - 2.5 Two fermions

44. g = - 3 Two fermions

45. g = - 3.5 Two fermions

46. g = - 4 Two fermions

47. g = - 4.5 Two fermions

48. g = - 5 Two fermions

49. g = - 7 Two fermions evolution of “Cooper pair” formation in real space

50. Conditional probability distr. Repulsive interaction Attractive interaction

51. N spin-less bosons confined in quasi-2D Harmonic-oscillator Interact via (short-ranged) Gaussian interaction Range:  Strength: g g → 0 implies interaction becomes  -function Energy of non-interacting ground-state: Form all properly symmetrized many-body wave-functions (permanents) with energy: III.b Effective interaction approach to the many-boson problem maximal energy of included states

52. Effective interaction derived from Lee-Suzuki method compared to Exact diagonalization with same cut-off energy J. Christensson, Ch. Forssén, S. Åberg and S.M. Reimann, Phys Rev A 79, 012707 (2009) Method works well for strong correlations Ground-state AND excited states All angular momenta Effective interaction approach to the many-boson problem L=0 L=9 Exact diagonalization Effective interaction g=1 g=10 N=9

53. Not so useful for long-ranged interactions: Effective interaction approach to the many-boson problem N=9 particles L=0 g=10 Energy

54. SUMMARY II.Fluctuations and shell structure of BCS gaps in nuclei well described by periodic orbit theory. Non-universal corrections to BCS fluctuations important (beyond RMT). III.Cold Fermi-gas in 2D traps - Detailed shell structure: Hund’s rule for repulsive int.; Pairing type for attractive int. Pairing from: Odd-even energy difference, 1st excited state in even-N system, Cond. prob. function Interaction blockade. Yrast line spectrum I.Cold Fermionic gases show supershell structure in harmonic confinement. VI.Effective interaction scheme (Lee-Suzuki) works well for many-body boson system (short-ranged force)