supershell structure in gases of fermionic atoms

Supershell Structure in Gases of Fermionic AtomsMagnus 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

gives possibilities to study new phenomena in physics of finite many-body systems

T0

Bose condensate Degenerate fermi gas

Neutral atoms: # electrons = # protons # neutrons determines quantum statistics

e.g. : 6Li3 fermionic7Li4 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)

a=scattering length (s-wave) (Total cross section: )2

0 8 a

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 .................

)(4)( 21)3(

2

21 rrm

arrV

Approximate int. with:

C.A. Regal, D.S. Jin, PRL 90 (2003) 230404

40K

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):

)(42

1

2)3(

2

1

222

jiji

N

ii

i rrm

arm

m

pH

s-wave interaction interaction only between spin-up and spin-down particlesin relative S=0, =0 states.

2

1

Equal density of spin-up and spin-down particles:

Assume total S=0, i.e. N/2 particles spin-up and N/2 particles spin-down :

The interaction term is replaced by a mean field for spin-down particles:

Solved numerically on a grid

)(2

1)()(

2/

1

2rrr

N

ii

iii ergrm

m )(

2

1

222

2

Where:

Theoretical Treatment

m

ag

2

2 1 mConstants:

Gross-Pitaevskii like single particle equation. (Skyrme)

Ground state by filling lowest N/2 levels

g=0 (H.O)g>0

4/)3( 3/4.. NE OH

Total energy of ground state:

rdgeNEN

ii

322/

1 2

12)(

Total energy

Microscopically calc.energy similar toThomas-Fermi expr.(in this resolution)

(r)

g=0 (H.O)

g>0

(N=10 000)

2/322

.. 2

1~)(

rmrOH

Density profile of the cloud

Shell structure

Total energy:

rdgeNEN

ii

)3(22/

1 2

12)(

Shell energy: )(~

)()( NENENEosc

is a smoothly varying function of N.)(~NE

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.

Pertubativeresult

N=6552

Nosc=26

N=5850

Nosc=24

25,23...

...3,1

N=6928

Shell structure, single particle spectra

)1(~, geFN

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!

Two close-lying frequencies give rise to the beating pattern

(ArXiv:cond-mat/0502096)

Eshell/Etot 10-5

Shell energy for different interaction strengths, g

Supershell structure

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

• For the circle term there is a sin function in the denominator responsibly for bifurcations where (3-fold-) orbits of tori type are born.

ToriNM ):(

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

I. Supershell structure found in gases of Fermionic atoms confined in H.O. potential, with repulsive -interaction

Summary

II. 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.