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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 Trapped quantum gases of bosons or fermions T0 Degenerate fermi gas Bose condensate gives possibilities to study new phenomena in physics of finite many-body systems Neutral atoms: # electrons = # protons # neutrons determines quantum statistics e.g. : 6Li 3 7Li 4 fermionic bosonic Dilute gases of Fermionic Atoms Atom-atom interaction is short-ranged (1-10 Å) and much smaller than interparticle range (~ 10-6 m) (dilute gas) Approximate int. with: 2 a ( 3) V (r1 r2 ) 4 (r1 r2 ) m a=scattering length (s-wave) (Total cross section: 0 8a ) 2 40K Via Feshbach resonance one can experimentally control size and sign of interaction (via external magnetic field): Attractive interaction (a<0): Pairing, Bose-Einstein condensate, collective modes, .... Many studies! Repulsive interaction (a>0): This study ................. C.A. Regal, D.S. Jin, PRL 90 (2003) 230404 Theoretical Treatment 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): pi2 1 2a 2 2 ( 3) H m ri 4 (ri rj ) 2 m i j i 1 2m N s-wave interaction interaction only between spin-up and spin-down particles in relative S=0, =0 states. 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: 1 2 Theoretical Treatment The interaction term is replaced by a mean field for spin-down particles: 2 1 2 2 m r g (r ) i eii 2 2m Where: N /2 1 (r ) (r ) (r ) 2 i 1 i 2 Gross-Pitaevskii like single particle equation. (Skyrme) Ground state by filling lowest N/2 levels Solved numerically on a grid 2 Constants: a g 2 m m 1 Total energy N /2 Total energy of ground state: 1 E ( N ) 2 ei g 2 d 3 r 2 i 1 Microscopically calc. energy similar to Thomas-Fermi expr. (in this resolution) g>0 g=0 (H.O) EH .O. (3N ) 4 / 3 / 4 Density profile of the cloud (N=10 000) g=0 (H.O) g>0 (r) 1 H .O. (r ) ~ m 2 r 2 2 3/ 2 Shell structure N /2 Total energy: 1 E ( N ) 2 ei g 2 d (3) r 2 i 1 Shell energy: ~ Eosc ( N ) E( N ) E ( N ) ~ E ( N ) 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. Shell structure, single particle spectra Nosc=26 N=6928 Pertubative result N=6552 1, 3... N=5850 ...23, 25 Nosc=24 eN F , ~ g ( 1) H. Heiselberg and B. Mottelson, PRL 88 (2002) 190401 Spherical symmetry: Each state has 2 +1 degenerate m-states Shell energy - non-interacting system Shell energy vs particle number for pure H.O. Fourier transform Shell energy – interacting system Supershell structure! Eshell/Etot 10-5 Two close-lying frequencies give rise to the beating pattern (ArXiv:cond-mat/0502096) Supershell structure Shell energy for different interaction strengths, g 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) 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 ( M : N ) Tori For the circle term there is a sin function in the denominator responsibly for bifurcations where (3-fold-) orbits of tori type are born. 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). Supershell Structure in Gases of Fermionic Atoms Summary I. II. Supershell structure found in gases of Fermionic atoms confined in H.O. potential, with repulsive interaction H.O. magic numbers – not square well numbers like in e.g. metall clusters. III. Semiclassical understanding: Spherical perturbed H.O. is dominated by diameter and circle orbits.