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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
T0
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
 8a 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):
2
 pi2 1


a
2 2
H   
 m ri   4
 (3) (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  eii
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 3r
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.