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Supershell Structure in Gases of Fermionic Atoms

Supershell Structure in Gases of Fermionic Atoms Magnus Ögren , Lund Institute of Technology, Lund, Sweden. Nilsson conference, June, 2005. Collaborators: Yongle Yu, Lund Sven Åberg, Lund Stephanie Reimann, Lund Matthias Brack, Regensburg. Dilute gases of Atoms. Bose condensate.

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Supershell Structure in Gases of Fermionic Atoms

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  1. Supershell Structure in Gases of Fermionic Atoms Magnus Ögren, Lund Institute of Technology, Lund, Sweden Nilsson conference, June, 2005 • Collaborators: • Yongle Yu, Lund • Sven Åberg, Lund • Stephanie Reimann, Lund • Matthias Brack, Regensburg

  2. Dilute gases of Atoms Bose condensate Degenerate fermi gas Trapped quantum gases of bosons or fermions T0 gives possibilities to study new phenomena in physics of finite many-body systems Neutral atoms: # electrons = # protons  # neutrons determines quantum statistics e.g. : 6Li3 fermionic 7Li4 bosonic

  3. Dilute gases of Fermionic Atoms  Approximate int. with: Repulsive interaction (a>0): This study ................. Atom-atom interaction is short-ranged (1-10 Å) and much smaller than interparticle range (~ 10-6 m) (dilute gas) a=scattering length (s-wave) (Total cross section: ) 40K Via Feshbach resonance one can experimentally control size and sign of interaction (via external magnetic field): C.A. Regal, D.S. Jin, PRL 90 (2003) 230404 Attractive interaction (a<0): Pairing, Bose-Einstein condensate, collective modes, .... Many studies!

  4. Theoretical Treatment Assume total S=0, i.e. N/2 particles spin-up and N/2 particles spin-down : Equal density of spin-up and spin-down particles: N s=1/2 fermions at temperature T=0 are trapped in a harmonic oscillator potential and interact via a two-body interaction with repulsive s-wave (=0) scattering length, a (a>0): s-wave interaction  interaction only between spin-up and spin-down particles in relative S=0, =0 states.

  5. Theoretical Treatment Gross-Pitaevskii like single particle equation. (Skyrme) Where: Ground state by filling lowest N/2 levels Constants: The interaction term is replaced by a mean field for spin-down particles: Solved numerically on a grid

  6. Total energy Total energy of ground state: g>0 g=0 (H.O) Microscopically calc. energy similar to Thomas-Fermi expr. (in this resolution)

  7. Density profile of the cloud (N=10 000) g=0 (H.O) g>0 (r)

  8. Shell structure Total energy: Shell energy: is a smoothly varying function of N. • Calculational procedure: • Fix the interaction strength, g. • Solve self-consistently the Gross-Pitaevskii like s.p. equation for systems with N varying from 2 to 106. • Find a smoothing function and deduce the shell energy. • Plot the shell energy vs N1/3.

  9. Shell structure, single particle spectra Nosc=26 N=6928 N=6552 N=5850 Nosc=24 H. Heiselberg and B. Mottelson, PRL 88 (2002) 190401 Pertubative result Spherical symmetry: Each  state has 2 +1 degenerate m-states

  10. Shell energy - non-interacting system Shell energy vs particle number for pure H.O. Fourier transform

  11. Shell energy – interacting system Eshell/Etot  10-5 Two close-lying frequencies give rise to the beating pattern (ArXiv:cond-mat/0502096) Supershell structure!

  12. Supershell structure Shell energy for different interaction strengths, g

  13. Semiclassical analysis • Major contribution to the U(3) symmetry breaking • in our problem can be modeled by a quartic term!? • Study the following model potential (m=1) for • non-interacting particles (no selfconcistent meanfield). • For small  we have used a perturbative approach* to • derive a traceformula for the U(3)→SO(3) transition. • Further on we have derived a uniform traceformula • for the diameter and circle orbits, valid for • all values of .* Creagh, Ann. Phys. (N.Y.) 248, 60 (1996) • (ArXiv:nlin.SI/0505060)

  14. EBK + Poisson sum. (B.-T.) → Uniform traceformula • The diameter orbit, which has no angular momentum , • comes from the lower integration limit in l (scaled • angular momentum). • The circle orbit, which has maximal angular • momentum, comes from the upper integration limit in l • For the circle term there is a sin function in the • denominator responsibly for bifurcations • where (3-fold-) orbits of tori type are born.

  15. Uniform trace formula vs QM To test our uniform trace formula (including only diameter and circle contributions) we have calculated the oscillating part of the quantum mechanical spectra for a few values of  (e.g.  =0.01).

  16. Supershell Structure in Gases of Fermionic Atoms Summary • Supershell structure found in gases of Fermionic atoms confined in H.O. potential, with repulsive -interaction II. H.O. magic numbers – not square well numbers like in e.g. metall clusters. • Semiclassical understanding: • Spherical perturbed H.O. is dominated by diameter • and circle orbits.

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