Shell Structure of Nuclei and Cold Atomic Gases in Traps

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. Shell Structure of Nuclei and Cold Atomic Gases in Traps • I. Shell structure from mean field picture • (a) Nuclear masses (ground-states) • (b) Ground-states in cold gas of Fermionic atoms: supershell structure • 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 • 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 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. I.a Shell structure in nuclear mass Shell energy 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 T0 Bose condensate Degenerate fermi gas Trapped quantum gases of bosonic or fermionic atoms:

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

7. N Fermionic atoms in harmonic trap – Repulsive int. Shell energy vs particle number for pure H.O. Fourier transform Shell energy: Eosc = Etot- Eav 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. I. Odd-even mass difference is s.p. level density where If no pairing: odd N N=odd N=even de de even N l l . . . . . . 23(N) = 0 23(N) = de Extraction of pairing contribution from masses: D3(N even) = D + de/2 D3(N odd) = D W. Satula, J. Dobaczewski and W. Nazarewicz, PRL 81 (1998) 3599

11. Odd-even mass difference from data 12/A1/2 even D3 (MeV) odd+even odd 2.7/A1/4

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

13. Pairing gap 3odd 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 Divide pairing gap in smooth and fluctuating parts: where Expansion in fluctuating parts gives: is ”pairing time” Pairing gap equation:

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

19. RMS pairing fluctuations: (d: single-particle mean level spacing) 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=vF/2 RMT-limit: D=0 Bulk-limit: D→∞

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

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

22. Fluctuations of nuclear pairing gap from mass models

23. Shell structure in nuclear pairing gap

24. Shell structure in nuclear pairing gap Average over proton-numbers

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

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

27. III.a Cold Fermionic Atoms in 2D Traps [1] attractive repulsive N atoms of spin ½ and equal masses m confined in 2D harmonic trap, interacting through a contact potential: 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. • Solve many-body S.E. by full diagonalization • Ground-state energy and excitated states obtained for all angular momenta [1] M. Rontani, JR Armstrong, Y, Yu, S. Åberg, SM Reimann, PRL 102 (2009) 060401. [2] M. Rontani, S. Åberg, SM Reimann, arXiv:0810.4305

28. g=-0.3 Non-int (g=0, pure HO) g=-3.0 Ground-state energy E(N,g) in units of 0 g=-0.3 Interaction energy: Eint(N,g) = E(N,g) – E(N,g=0) -1 2 2 10 10 18 18 Scaled interaction energy: Eint(N,g)/N3/2 g=-0.3 -0.015 -0.017 -0.019 Attractive interaction

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

30. Coulomb blockade – interaction blockade g=0.3 g=0.3 1.0 1.0 3.0 3.0 Repulsive interaction Repulsive interaction Coulomb blockade: Extra (electric) energy, EC, for a single electron to tunnel to a quantum dot with N electrons Difference between conductance peaks: 5.0 5.0 D2 D2 where de is energy distance between s.p. states N and N+1 and E(N) total energy N N Pairing gap: Interaction (or van der Waals) blockade [1]: Add an atom to a cold atomic gas in a trap 6 6 6 3 3 3 2 2 2 5 5 5 7 7 7 4 4 4 1 1 1 No interaction g=-0.3 Attractive interaction D2 -1.0 Cheinet et al, PRL 101 (2008) 090404 -3.0 -5.0 N [1] C. Capelle et al PRL 99 (2007) 010402

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

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

35. Extracted pairing gaps 1st exc. state N=4, N=8 D3(3),D3(7) -g/4p (pert. result)

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

37. Two fermions g = - 0.1

38. Two fermions g = - 0.3

39. Two fermions g = - 0.6

40. Two fermions g = - 1

41. Two fermions g = - 1.5

42. Two fermions g = - 2

43. Two fermions g = - 2.5

44. Two fermions g = - 3

45. Two fermions g = - 3.5

46. Two fermions g = - 4

47. Two fermions g = - 4.5

48. Two fermions g = - 5

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

50. Conditional probability distr. Repulsive interaction Attractive interaction