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
I. Shell structure from mean field picture(a) Nuclear masses (ground-states)
(b) Ground-states in cold gas of Fermionic atoms: supershell structure
II. 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
III. 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
Shell Structure of Nuclei and Cold Atomic Gases in Traps
Collaborators: Stephanie Reimann, Massimo Rontani, Patricio LeboeufHenrik Olofsson/Urenholdt, Jeremi Armstrong, Matthias Brack, Jonas Christensson, Christian Forssén, Magnus Ögren, Marc Puig von Friesen, Yongle Yu,
I. Shell structure from mean field picture
Shell energy
I.a Shell structure in nuclear mass
Shell energy = Total energy (=mass) – Smoothly varying energy
P. Möller et al, At. Data and Nucl. Data Tables 59 (1995) 185
I.b Ground states of cold quantum gases
Trapped quantum gases of bosonic or fermionic atoms:
T0
Bose condensate Degenerate fermi gas
Fermionic atoms in a 3D H.O. confinement
N
jiji
N
ii
i rrm
ar
m
m
pH )(4
223
2
1
222
a = s-wave scattering length
Un-polarized two-component system with two spin-states:
)(2)()()( rnrnrnrn
Hartree-Fock approximation:
2/
1
2)()(
N
ii rrn
)()()(2
1
222
2
rerrngrmm iii
Where:
mag /4 2 > 0 (repulsive int.)
Shell energy vs particle number for pure H.O.
Fourier transform
Shell energy: Eosc = Etot - Eav
N Fermionic atoms in harmonic trap – Repulsive int.
No interaction
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
42
41
21 rrV effeff Effective potential:
[1] Y. Yu, M. Ögren, S. Åberg, S.M. Reimann, M. Brack, PRA 72, 051602 (2005)
II. Shell structure of BCS pairing gap [1]
[1] S. Åberg, H. Olofsson and P. Leboeuf, AIP Conf Proc Vol. 995 (2008) 173.
odd N
even N
)1()1(5.0)()(3 NBNBNBN
I. Odd-even mass difference Extraction of pairing contribution from masses:
)(
1)(2
2
2
3
gNN
BN
wherede
dNeg )( is s.p. level density
If no pairing:
23(N) = 0
N=odd
e...
N=even
e
.
.
.
23(N) = e
W. Satula, J. Dobaczewski and W. Nazarewicz, PRL 81 (1998) 3599
(N even) = + e/2(N odd) =
Odd-even mass difference from data
odd
even
odd+even
12/A1/2
2.7/A1/4
(
MeV
)
Single-particle distance from masses
50/A MeV
Pairing delta eliminated in the difference: (3)(even N) - (3)(odd N) = 0.5(en+1 – en) = d/2
Fermi-gas model: MeV/A 473/45.0)(5.0 1 Neee Fnn
See e.g.: WA Friedman, GF Bertsch, EPJ A41 (2009) 109
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).
Pairing gap from different mass models
Average behavior in agreement with exp. but very different fluctuations
Fluctuations of the pairing gap
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)
Periodic orbit description of pairing
pr
rpprp rSAe
,orbits periodic 1
,, )/cos()(~
p.o of period :/
index Maslov :
porbit periodic ofaction :S
amplitudestability :
p
rp,
p
,
ES
pdq
A
p
rp
Level density Insert semiclassical expression
L
L e
dee22
)(
G
2 Pairing gap equation:
dtt
xtxK
020
1
)cos()(
where
2
his ”pairing time”
Divide pairing gap in smooth and fluctuating parts: ~
rp
rppprp erSrKA,
,0, /)(cos)/(2~
Expansion in fluctuating parts gives:
G
L1
exp2
1 x,/)exp( xx
.~
Fluctuations of pairing gap
Fluctuations of pairing gap become
)()/( 2~
0
22
22
KKd o
H
where K is the spectral form factor (Fourier transform of 2-point corr. function):
is shortest periodic orbit,min /hhH is Heisenberg time
If regular: )(4 0
2 DFreg
If chaotic: )(2
112
2 DFch
single-particle mean level spacing) /~2RMS 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→∞
Universal/non-universal fluctuations
g
D2min
min Hg ”dimensionless conductance”
Non-universal spectrum fluctuationsfor energy distances larger than g:
universal
non-universal
g=Lmax
3 statistics
Random matrix limit: g (i.e. D = 0) corresponding to pure GOE spectrum (chaotic)or pure Poisson spectrum (regular)
If regular: )(4 0
2 DFreg
If chaotic: )(2
112
2 DFch
single-particle mean level spacing) /~2RMS pairing fluctuations:
Exp.Theory(regular)
Nuclei Metallic grainsIrregular shape of grain chaotic dynamics
Universal pairing fluctuations
22
2
1
ch
D very small (GOE-limit)
50 000 6Li atoms and kF|a| = 0.2
Fermionic atom gas
24.01
, if regular
02.01
, 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→∞
Fluctuations of nuclear pairing gap from mass models
Shell structure in nuclear pairing gap
Shell structure in nuclear pairing gap
Average over proton-numbers
Shell structure in nuclear pairing gap
P.O. description
Average over Z
III. Cold atomic gases in 2D traps - Exact diagonalizations
III.a Cold Fermionic Atoms in 2D Traps [1]
N atoms of spin ½ and equal masses m confined in 2D harmonic trap,interacting through a contact potential:
[1] M. Rontani, JR Armstrong, Y, Yu, S. Åberg, SM Reimann, PRL 102 (2009) 060401.
Solve many-body S.E. by full diagonalizationGround-state energy and excitated states obtained for all angular momenta
Energy scale: 0Length scale: 0/ mDimensionless coupling const.: )/(' 2
0gg
Contact force regularized by energy cut-off [2].Energy (and w.f.) of 2-body state relates strength g to scattering length a.
[2] M. Rontani, S. Åberg, SM Reimann, arXiv:0810.4305
attractiverepulsive
Non-int
Ground-state energyE(N,g) in units of
g=-0.3
g=-3.0(g=0, pure HO)
2 10 18
Interaction energy:Eint(N,g) = E(N,g) – E(N,g=0)
g=-0.30
-1
g=-0.3
2 10 18
Scaled interaction energy:Eint(N,g)/N3/2
-0.019
-0.015
-0.017
Attractive interaction
Cold Fermionic Atoms in 2D Traps – Pairing versus Hund’s Rule
Interaction energy versus particle number
Negative g (attractive interaction): odd-even staggering (pairing)
Positive g (repulsive interaction): Eint max at closed shells,
min at mid-shell (Hund’s rule)
attractive
repulsive
Repulsive interaction
1 2 3 4 5 6 7
g=0.31.03.05.0
N
2
No interactionRepulsive interaction
1 2 3 4 5 6 7
g=0.31.03.05.0
N
2
Coulomb blockade – interaction blockade
Coulomb blockade:Extra (electric) energy, EC, for a single electron to tunnel to a quantum dot with N electrons
Difference between conductance peaks: CEeNENENEN )1()(2)1()(2
where e is energy distance between s.p. states N and N+1 and E(N) total energy
Interaction (or van der Waals) blockade [1]: Add an atom to a cold atomic gas in a trap
Cheinet et al,PRL 101 (2008) 090404
[1] C. Capelle et al PRL 99 (2007) 010402
Attractive interaction
1 2 3 4 5 6 7
2
N
g=-0.3-1.0-3.0-5.0
Pairing gap:
)(*5.0)( 23 NN
-2 0 1 2-1 m
/E
1
3
2
Non-int. picture, N=2
-2 0 1 2-1 m
/E
1
3
2
Non-int. picture, N=8
M=0M=1M=2 M=0M=1M=2
Angular momentum dependence – yrast line
Angular momentum dependence – 4 and 6 atoms
Yrast line – higher M-values, excited states
Pairing decreases with angular momentum and excitation energy: Gap to excited states decreases ”Moment of inertia” increases
Cold Fermionic Atoms in 2D Traps – 8 atomsN=8 particlesExcitation spectra (6 lowest states for each M)Attractive and repulsive interaction
Ground-stateattractive int.
Ground-staterepulsive int.
Onset of inter-shell pairing
Excited statesalmost deg.with g.s. (cf strongly corr. q. dot)
-g/4 (pert. result)
1st exc. stateN=4, N=8
3(3), 3(7)
Extracted pairing gaps
Two fermions
measure probabilityto find ↑ fermion in xy plane
fix ↓ fermion
g = 0
Structure of w.f. from Conditional probability
g = - 0.1
Two fermions
g = - 0.3
Two fermions
g = - 0.6
Two fermions
g = - 1
Two fermions
g = - 1.5
Two fermions
g = - 2
Two fermions
g = - 2.5
Two fermions
g = - 3
Two fermions
g = - 3.5
Two fermions
g = - 4
Two fermions
g = - 4.5
Two fermions
g = - 5
Two fermions
g = - 7
Two fermions
evolution of “Cooper pair”formation in real space
Conditional probability distr.
Repulsiveinteraction
Attractiveinteraction
N
ij
jiN
ii
irr
grmm
pH
2
2
21
222
2exp
2
1
2
1
2
1
2
N spin-less bosons confined in quasi-2D Harmonic-oscillatorInteract via (short-ranged) Gaussian interactionRange: Strength: g
g → 0 implies interaction becomes -function
Energy of non-interacting ground-state: NE
Form all properly symmetrized many-body wave-functions (permanents) with energy: maxN NE
III.b Effective interaction approach to the many-boson problem
:maxN maximal energy of included states
Effective interaction derived from Lee-Suzuki methodcompared toExact diagonalization with same cut-off energy
J. Christensson, Ch. Forssén, S. Åberg and S.M. Reimann, Phys Rev A 79, 012707 (2009)
•Method works well for strong correlations•Ground-state AND excited states•All angular momenta
Effective interaction approach to the many-boson problem
L=0 L=9
Exact diagonalization
Effective interaction
g=1 g=10 g=10
N=9
maxN
Not so useful for long-ranged interactions:
)/(0.1 m
Effective interaction approach to the many-boson problem
)/(1.0 m
N=9 particlesL=0g=10
maxN maxN
En
ergy
En
ergy
SUMMARY
II. Fluctuations and shell structure of BCS gaps in nuclei well described by periodic orbit theory. Non-universal corrections to BCS fluctuations important (beyond RMT).
III. Cold Fermi-gas in 2D traps - Detailed shell structure: Hund’s rule for repulsive int.; Pairing type for attractive int.
Pairing from: Odd-even energy difference, 1st excited state in even-N system, Cond. prob. function
Interaction blockade. Yrast line spectrum
I. Cold Fermionic gases show supershell structure in harmonic confinement.
VI. Effective interaction scheme (Lee-Suzuki) works well for many-body boson system (short-ranged force)
