Super-shell stru
tures and pairing in ultra
old trapped Fermi gases
Magnus Ögren
and Henning Heiselberg
Mathemati
al Physi
s, Lund Institute of Te
hnology, P.O. Box 118, SE-22100 Lund, Sweden
University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
(Dated: April 3, 2007)
We 
al
ulate level densities and pairing gaps for an ultra
old dilute gas of fermioni
 atoms in
harmoni
 traps under the in�uen
e of mean �eld and anharmoni
 quarti
 trap potentials. Super-
shell stru
tures, whi
h were found in Hartree-Fo
k 
al
ulations, are 
al
ulated analyti
ally within
periodi
 orbit theory as well as from WKB 
al
ulations. For attra
tive intera
tions, the underlying
level densities are 
ru
ial for pairing and super-shell stru
tures in gaps are predi
ted.
PACS numbers: 03.75.Ss, 05.30.Fk
Ultra
old atomi
 gases have re
ently been used to 
re-
ate novel quantum many-body systems su
h as strongly
intera
ting high temperature super�uids of fermions,
Bose-Einstein 
ondensates, Mott insulators in opti
al lat-
ti
es, et
. These lab phenomena have a strong over-
lap with 
ondensed matter [1℄, nu
lear [2℄ and neutron
star physi
s [3℄. Finite fermion systems su
h as atoms in
traps, nu
lei, helium and metal 
lusters, semi
ondu
tor
quantum dots, super
ondu
ting grains, et
., have addi-
tional interesting quantum stru
tures su
h as level spe
-
tra, densities and pairing. These will be observable as
temperatures are further lowered in atomi
 trap experi-
ments. The high degree of 
ontrol over physi
al param-
eters, in
luding intera
tion strength and density, makes
the atomi
 traps marvelous model systems for general
quantum phenomena.
The purpose here is to 
al
ulate the level spe
tra,
densities and pairing for zero-temperature Fermi gases
in harmoni
 os
illator (HO) traps with anharmoni
 and
mean �eld perturbations, and to show that novel super-
shell stru
tures appear in both level densities and pairing.
In 
al
ulating level spe
tra by analyti
al periodi
 orbit
theory and WKB as well as numeri
al Hartree-Fo
k, we
also relate these di�erent theoreti
al approa
hes to one
another.
We treat a gas ofN fermioni
 atoms of massm in a HO
potential at zero temperature, intera
ting via a two-body
intera
tion with s-wave s
attering length a. We shall
mainly dis
uss a spheri
ally symmetri
 trap and a dilute
gas (i.e. where the density ρ obeys the 
ondition ρ|a|3 ≪
1) of parti
les with two spin states of equal population.
The Hamiltonian is then given by
mω2r2i + U(ri)
, (1)
We will 
onsider both external anharmoni
 potentials of
the form U = εr4 and parti
le intera
tions: U(ri) =
(2π~2a/m)
j 6=i δ
3(ri−rj). When intera
tions are weak,
the latter 
an be approximated by the mean �eld poten-
U(r) =
2π~2a
ρ(r) . (2)
For a large number of parti
les and U = 0 the Fermi
energy is EF = ñF~ω where nF = ñF − 3/2 ≃ (3N)1/3
is the HO quantum number at the Fermi surfa
e. The
HO shells are highly degenerate with states having an-
gular momenta l = nF , nF −2, ...,mod(nF , 2), due to the
U(3) symmetry of the 3D spheri
ally symmetri
 HO po-
tential. However, intera
tions split this degenera
y. In
the Thomas-Fermi (TF) approximation (see, e.g., [4℄) the
Fermi energy is
2k2F (r)
mω2r2 + U(r) . (3)
The density ρ(r) = k3F (r)/3π
ρ(r) = ρ0
1− r2/R2TF
, (4)
inside the 
loud r ≤ RTF = aosc
2ñF , where ρ0 =
(2ñF )
3/2/3π2a3osc is the 
entral density [5℄. For 
onve-
nien
e we set the os
illator length aosc =
~/mω = 1 in
the following.
Taylor expanding the density and thereby also the
mean �eld of Eq. (2) around the 
enter gives
ρ(r) ≃ ρ0
r2/R2TF +
r4/R4TF + ...
, (5)
the �rst term will simply in
orporate a 
onstant shift
in energies whereas the term quadrati
 in radius renor-
malizes the HO frequen
y as ωeff = ω
1− 6πaρ0/R2TF .
The third term is quarti
 in radius and is therefore also
of the same form as the external potential
U(r) ≃ εr4 , (6)
with ε = (3π~2a/4m)ρ0/R
TF . Both the pure quarti
potential and the mean �eld potential of Eq. (2) are an-
harmoni
 and 
hange the level density by splitting the l
degenera
y of the HO shell nF at the Fermi surfa
e.
We will now 
al
ulate analyti
ally the level spe
tra
from perturbative periodi
 orbit theory for the quarti
potential and subsequently within semi
lassi
al WKB
wavefun
tions for both the quarti
 and the mean �eld
potential of Eq. (2). We will start with repulsive inter-
a
tions where pairing is not present.
http://arxiv.org/abs/0704.0385v1
In periodi
 orbit theory [6℄, the level density 
an be
written (to leading order in ~
) in terms of a perturba-
tive HO tra
e formula [7, 8℄
g(E) =
1 +Re
(−1)k M ei2πkE/~ω
. (7)
For the unperturbed HO (U = 0) the modulation fa
tor is
M = 1. For a quarti
 perturbed potential, as in Eq. (6),
the modulation fa
tor was 
al
ulated in [8℄
e−i2kσ/~−iπ/2 + e−i3kσ/~+iπ/2
, (8)
with σ = επE2/~2ω3, being a small 
lassi
al a
tion. The
two terms arise from the 
hange in a
tions for the 
ir
le
and diameter orbits respe
tively due to the quarti
 po-
tential [8℄. The resulting level density 
an be written in
the fa
torised form [9℄
g(E) =
(~ω)3
(−1)k
. (9)
Here, the �rst term is the average level density, the

osine fa
tor gives the rapid HO shell os
illations (mod-
i�ed by the perturbation) whi
h, however, are slowly
modulated by the sine fa
tor resulting in a beating pat-
tern. Moreover, the non-perturbed HO limit, equivalent
to M = 1 in Eq. (7), is re
overed in the limit of |ε| → 0,
where the U (3) symmetry is restored. The k = 1 term
in Eq. (9) gives the major os
illations in the level density
and is shown in Fig. 1 (a). The beating pattern or super-
shells is 
learly observed. The shell os
illations vanish
when the argument of the sine in Eq. (9) is an integer
S = 1, 2, 3, ... times π, i.e. |ε|E2/2(~ω)3 = S. This gives
the supernode 
ondition
nF = E/~ω =
2S~ω/|ε| . (10)
We now turn to an alternative 
al
ulation of the level
density with WKB. The splitting of the HO shells degen-
erate levels l = nF , nF −2, ...,mod (nF , 2) in the shell nF
by the mean-�eld potential 
an be 
al
ulated perturba-
tively in the dilute limit. An ex
ellent approximation for
the radial HO wave fun
tion with angular momentum l
and (nF − l)/2 radial nodes in the HO shell when nF ≫ 1
is the WKB one [10, 11℄:
RnF l(r) ≃
sin(kl(r)r + θ)
l (r)r
, (11)
between turning points r2± = ñF ±
ñ2F − l(l + 1). Here,
ñF = nF + 3/2 and the WKB wave number kl(r) is
k2l (r) = 2ñF − r2 − l(l+ 1)/r2 . (12)
When nF ≫ 1 the wave fun
tion has many nodes 1 ≪
l ≪ nF and the os
illations in R2nl(r) 
an be averaged
〈sin2(kl(r)r)〉 = 1/2 [10℄. The phase θ is then unim-
portant. The single-parti
le energies for the anharmoni
potential of Eq. (6) are simply
EnF ,l − ñF~ω =
U(r)|RnF l(r)|2r2dr (13)
πkl(r)
3ñ2F − l(l + 1)
.(14)
It is spe
ial for the quarti
 perturbation that the level
energies are linear in l(l+1). The resulting level spa
ing
in
reases as (2l+1) just as the level degenera
y for SO(3)
symmetry. Therefore the level density is 
onstant within
the bandwidth
D ≡ |EnF ,l≡0 − EnF ,l=nF | = εn2F /2 (15)
on energy s
ales larger than 2D/nF but smaller than D.
The level density vanishes between the bandwidths of
two neighbouring n shells and therefore it generally has
a strong os
illatory behavior as shown in Fig. 1 (a). Its
amplitude is largest when D ∼ ~ω/2. However, when
D ≃ ~ω the level density is 
onstant and the os
illa-
tory behavior vanishes. This phenomenon repeats when
D = S~ω sin
e the level spe
tra then overlap S times.
With the bandwidth of Eq. (15) under this 
ondition, we
obtain exa
tly the same supernode 
ondition as for pe-
riodi
 orbit theory, Eq. (10). We 
on
lude that Craig's
perturbative periodi
 orbit theory [7℄ is in exa
t agree-
ment with perturbative WKB for a quarti
ally perturbed
spheri
al symmetri
 HO in three dimensions.
We now turn to the slightly more 
ompli
ated mean
�eld potential of Eq. (2). Its level spe
trum 
an also be

al
ulated from the WKB wave fun
tions of Eq. (11).
Inserting them in Eq. (13), we obtain
EnF ,l − ñF~ω = 2/
F ~ω I . (16)
Here, the integral I is
1− l(l+ 1)/ñ2F
1− x2
where x = (r2−ñF )/
ñ2F − l(l + 1). This integral is I =
π for l ≃ nF and I = 8
2/3 for l = 0. The bandwidth is
therefore
D = 2/
2/3− π
. (18)
Inserting this bandwidth in the supernode 
ondition D =
S~ω gives
2/3− π
F = S . (19)
For example in the 
ase 2πa = 1 the supernodes S =
1, 2, 3, .. should o

ur when nF ∼ 28, 44, 58, et
. The
Hartree-Fo
k (HF) 
al
ulations of the os
illating part of
the total energy, whi
h is proportional to the level den-
sity at the Fermi level [6℄, result in slightly higher su-
pernodes, as in Fig 1 (b). The di�eren
es arise be
ause
the WKB 
al
ulations are perturbative in the intera
tion
strength, whereas in the HF 
al
ulation the MF poten-
tial U in
ludes a large s
attering length whi
h, e.g., leads
to 
orre
tions for the e�e
tive os
illator frequen
y. Also
for the purely quarti
 term the perturbative approa
h
underestimates the exa
t supernodes (see Fig. 3 of [8℄).
For weaker intera
tions 2πa = 0.1, the �rst supernode
S = 1 should o

ur at nF = 130 a

ording to the 
ondi-
tion of Eq. (19), in 
loser agreement with the HF result
of Fig. 1 (
).
For 
omparison, the Taylor expansion of the mean �eld
potential leads to the supernode 
ondition of Eq. (10)
with ε = (3π~2a/4m)ρ0/R
TF . It di�ers from Eq. (19)
by the prefa
tor, whi
h is ∼ 34% smaller. It is a better
approximation to expand e.g. around r = RTF /2
nF /2, where the 
orresponding prefa
tor is only ∼ 8%
smaller, su
h that the supernode in Fig 1 (
) is predi
ted
to nF = 137. Now expanding I of Eq. (17) for small
l ≪ nF , one �nds
I = (8/3)
2− l2/
2n2F , (20)
resulting in the level spe
trum [10℄
EnF ,l − ñ~ω =
− l(l+ 1)
. (21)
This level density is 
onstant at low l as for the potential
in Eq. (14). However, near l ∼ nF the density of lev-
els is slightly smaller as 
an be seen from the bandwidth

orresponding to Eq. (21), whi
h is ∼ 12% larger, for a
given nF , than the bandwidth of Eq. (18). That the level
density is not 
ompletely 
onstant within the bandwidth
has the e�e
t that a small periodi
ity remains even at
the super-shell 
ondition D = S~ω. Therefore the shell
os
illations do not disappear 
ompletely at the supern-
odes, as 
an be seen in Fig. 1 (b,
), whereas for the purely
quarti
 
ase (a) the os
illations disappear 
ompletely at
the supernodes.
Most atomi
 traps are not spheri
al but 
igar shaped
(prolate) with ωz
<∼ω⊥. The unperturbed HO energies
E = nz~ωz + n⊥~ω⊥ will generally lead to a 
onstant
level density for energy s
ales larger than ~ωz but smaller
than ~ω⊥. When the os
illator frequen
y ratio ω⊥/ωz is a
rational number, level degenera
ies and larger os
illations
will o

ur on the s
ale ~ωz. Intera
tions will, however,
smear this level density. In any 
ase, super-shell stru
ture
is not expe
ted as in the spheri
al symmetri
 
ase. In
very oblate traps ωz ≫ ω⊥ the mean �eld potential is
e�e
tively two-dimensional and quadrati
, i.e. it does
not split the HO shells [10, 13℄. Thus we may expe
t
strong os
illations in the level density on the s
ale ~ω⊥,
but again no super-shell stru
ture.
0 20 40 60 80 100
−1000
1000 (a)
0  20 40 60 80 100 120 140
200 (c)
0 20 40 60 80 100
40 (b)
Figure 1: (
olor online) The upper �gure (a) shows the leading
term (k = 1) of the os
illating part of the perturbative level
density of Eq. (9) as a fun
tion of nF = E/~ω, for the 
ase of
an external potential V = VHO+εr
with ε = 2/402 ≈ 0.0013.
The middle and lower �gures (b,
) show the os
illating part
of the total energy a

ording to a numeri
al HF 
al
ulation
[12℄, with intera
tion strength 2πa = 1 and 2πa = 0.1, as a
fun
tion of the HO shell number (~ = ω = 1). This illustrates
qualitatively that a supernode, e.g. at nF = 40, 
an be due
to intera
tion (b) and/or an additional quarti
 term to the
HO potential (a).
Attra
tive intera
tions lead to pairing by an amount
that is exponentially sensitive to the underlying level den-
sity near the Fermi surfa
e [2, 10, 11, 14℄. The level den-
sity is the same for repulsive and attra
tive intera
tions
ex
ept that the levels are reversed when the sign of ε
(Eqs. (9)) and (13)) or a is 
hanged (Eq. (16)). Therefore
we 
an use the level densities and bandwidths 
al
ulated
above for pairing 
al
ulations. Pairing in �nite systems
is des
ribed by the Bogoliubov-de Gennes (BdG) equa-
tions [15℄ and take pla
e between time-reversed states.
As shown in [14℄ these states 
an be approximated by HO
wave fun
tions in dilute HO traps as long as the gap does
not ex
eed the os
illator energy, ∆<∼~ω. Solving BdG for
su
h �nite systems is numeri
ally 
ompli
ated and we
shall therefore apply further simplifying approximations,
namely that the pairing gap ∆nl and the wavefun
tion
overlap matrix elements vary slowly with level l in a shell
n. Both approximations are fair for the trapped atoms
as argued in [11℄ and deviations 
an be understood. As
result we arrive at a mu
h simpli�ed gap equation
∫ ∼2nF
g(E) dE
(E − µ)2 +∆(µ)2
. (22)
Here, the supergap G = 32
2nF |a|~ω/15π2 was 
al-

ulated in [10℄ as the pairing gap when all states in
a shell 
an pair; this is the 
ase for a region of in-
tera
tion strengths and parti
le number where the gap
is large as 
ompared to the level splitting, yet small
0 10 20 30 40 50 60
=(3N)1/3
38 40 42 44
Figure 2: (
olor online) Multi-shell pairing gaps for a HO trap
with an additional quarti
 term in the potential with ε =
2/402, i.e. for the level density of Fig. 1 (a) with supernodes
at nF ≃ 40, 40
2 ≈ 57, etc. The intera
tion strength is a =
−0.05 (top red 
urve), a = −0.03 (middle blue 
urve, with the
inset �gure around the �rst supernode) and a = −0.01 (lower
green 
urve). In the inset plot it is 
learly seen that the lo
al
minima for l ∼ nF and l ∼ 0 before the supernode turns
into lo
al maxima after the supernode, as a 
onsequen
e of
overlapping shells. The dashed (red) line is the multi-shell gap
∆ = G/(1−2G ln(nF )/~ω) for a = −0.05 and the upper/lower
thin solid line (bla
k) are the single mid-/end-shell pairing for
a = −0.01 (see text).

ompared to the shell splitting ~ω. ∆(µ) = ∆nl is
the gap at the Fermi surfa
e. g(E) = n2F /D is
the level density within ea
h bandgap D around ev-
ery shell n = 0, 1, ...,∼2nF but vanishes between
the bandgaps. The gap equation thus redu
es to
1 = (G/D)
∑∼2nF
(E + n~ω − µ)2 +∆2. The

hemi
al potential µ 
an be determined from the level
spe
trum; as we gradually �ll parti
les into the shell nF
at the Fermi surfa
e, µ in
reases from nF~ω to nF~ω+D.
The 
ut-o� n<∼2nF in the sum of the gap equation models
as a �rst approximation the more rigorous regularization
pro
edure des
ribed in Ref. [16℄ that is required for a
delta-fun
tion pseudo-potential.
By solving this simpli�ed gap equation of Eq. (22),
we �nd that it still 
ontains and displays the essential
interplay between the variation in level density and pair-
ing. To illustrate the super-shell stru
ture in pairing,
we take the strongly anharmoni
 trap potential used for
the level spe
tra in Fig. 1 (a), and 
al
ulate the pairing
arising from a weak attra
tive s
attering length a < 0.
For su�
iently weak intera
tions su
h that pairing only
takes pla
e in the shell at the Fermi surfa
e, we obtain
the expe
ted result from the gap equation: ∆ = G when
D ≪ ∆, whereas for D ≫ ∆ we get ∆ = D exp(−D/2G)
midshell (µ = nF~ω + D/2) and ∆ = 2D exp(−D/G)
endshell (µ = nF~ω or µ = nF~ω +D). Pairing is thus
stronger at midshell than at endshell, where there are
fewer states to pair [11℄, and strong shell os
illations fol-
low as shown in Fig. 2. For stronger intera
tions, pairing
also takes pla
e between states in shells around the Fermi
shell and Eq. (22) gives: ∆ = G/ (1− 2G ln (nF ) /~ω) for
small bandwidth [14℄. In Fig. 2 this 
urve is 
ompared
with the �nite bandwidth result, whi
h has strong os
il-
lations ex
ept at the supernodes where the level density
is 
ontinuous. At a supernode D = ~ω and the gap equa-
tion (22) leads to a gap ∆ = 2nF~ω exp(−~ω/2G) [11℄.
In summary, level densities, shell-os
illations and
super-shell stru
tures in anharmoni
 traps 
al
ulated
from numeri
al Hartree-Fo
k and analyti
al periodi
 or-
bit theory as well as WKB were found to mat
h to lead-
ing order. Analogous super-shell stru
tures were found
in pairing from an approximated BdG 
al
ulation. The
mean �eld in atomi
 nu
lei also have a large anharmoni
potential and the HO shells start to overlap (the �rst
supernode) already for heavy nu
lei with nF ∼ 5 − 6.
The interplay of level spe
tra and multishell pairing is,
however, di�
ult to disentangle in nu
lear pairing due to
strong spin-orbit e�e
t and small parti
le number. Ul-
tra
old atomi
 traps, however, provide ideal systems for
observing the ri
h quantum stru
tures su
h as level den-
sities and pairing.
Dis
ussions with Matthias Bra
k on periodi
 orbit the-
ory, Ben Mottelson on (nu
lear) shell theory and pairing,
and proof reading by Joel Corney, are gratefully a
knowl-
edged.
[1℄ J. Bardeen, L. N. Cooper, J. R. S
hrie�er, Phys. Rev.
108, 1175 (1957).
[2℄ A. Bohr and B. R. Mottelson, Nu
lear Stru
ture Vols.
I+II, Benjamin, New York 1969.
[3℄ A. Bohr, B. R. Mottelson, D. Pines, Phys. Rev. 110, 936
(1958).
[4℄ C. J. Pethi
k and H. Smith, Bose-Einstein Condensation
in Dilute Gases, Cambridge Univ. Press 2002.
[5℄ For a �nite number of parti
les the fa
tor ñ = nF + 3/2
in
ludes a 
orre
tion to nF , whi
h has been 
he
ked nu-
meri
ally to improve the TF approximation and slightly

hange the predi
tion of supernodes.
[6℄ M. Bra
k and R. K. Bhaduri, Semi
lassi
al Physi
s, re-
vised edn (Boulder, CO: Westview) (2003).
[7℄ S. C. Creagh, Ann. Phys., NY 248 60 (1996).
[8℄ M. Bra
k et al., J. Phys. A 38, 9941 (2005).
[9℄ M. Ögren, unpublished (2006):
www.magnus.ogren.se/notes/pot/derivationofgpert.pdf
[10℄ H. Heiselberg and B. R. Mottelson, Phys. Rev. Lett. 88,
190401 (2002).
[11℄ H. Heiselberg, Phys. Rev. A 68, 053616 (2003). Note that
the square root of kl was missing in Eq. (6) of this Ref.
as 
ompared to Eq. (11).
[12℄ Y. Yu et al., Phys. Rev A 72, 051602(R) (2005).
[13℄ B. P. van Zyl et al., Phys. Rev. A 67, 023609 (2003).
[14℄ G. M. Bruun and H. Heiselberg, Phys. Rev. A 65, 053407
(2002).
[15℄ P. G. de Gennes, Super
ondu
tivity of Metals and Alloys
(Addison-Wesley, New York, 1989).
[16℄ G. M. Bruun et al., Eur. Phys. J. D9, 433 (1999).

  We calculate level densities and pairing gaps for an ultracold dilute gas of
fermionic atoms in harmonic traps under the influence of mean field and
anharmonic quartic trap potentials. Super-shell structures, which were found in
Hartree-Fock calculations, are calculated analytically within periodic orbit
theory as well as from WKB calculations. For attractive interactions, the
underlying level densities are crucial for pairing and super-shell structures
in gaps are predicted.

Super-shell stru
tures and pairing in ultra
old trapped Fermi gases
Magnus Ögren
and Henning Heiselberg
Mathemati
al Physi
s, Lund Institute of Te
hnology, P.O. Box 118, SE-22100 Lund, Sweden
University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
(Dated: April 3, 2007)
We 
al
ulate level densities and pairing gaps for an ultra
old dilute gas of fermioni
 atoms in
harmoni
 traps under the in�uen
e of mean �eld and anharmoni
 quarti
 trap potentials. Super-
shell stru
tures, whi
h were found in Hartree-Fo
k 
al
ulations, are 
al
ulated analyti
ally within
periodi
 orbit theory as well as from WKB 
al
ulations. For attra
tive intera
tions, the underlying
level densities are 
ru
ial for pairing and super-shell stru
tures in gaps are predi
ted.
PACS numbers: 03.75.Ss, 05.30.Fk
Ultra
old atomi
 gases have re
ently been used to 
re-
ate novel quantum many-body systems su
h as strongly
intera
ting high temperature super�uids of fermions,
Bose-Einstein 
ondensates, Mott insulators in opti
al lat-
ti
es, et
. These lab phenomena have a strong over-
lap with 
ondensed matter [1℄, nu
lear [2℄ and neutron
star physi
s [3℄. Finite fermion systems su
h as atoms in
traps, nu
lei, helium and metal 
lusters, semi
ondu
tor
quantum dots, super
ondu
ting grains, et
., have addi-
tional interesting quantum stru
tures su
h as level spe
-
tra, densities and pairing. These will be observable as
temperatures are further lowered in atomi
 trap experi-
ments. The high degree of 
ontrol over physi
al param-
eters, in
luding intera
tion strength and density, makes
the atomi
 traps marvelous model systems for general
quantum phenomena.
The purpose here is to 
al
ulate the level spe
tra,
densities and pairing for zero-temperature Fermi gases
in harmoni
 os
illator (HO) traps with anharmoni
 and
mean �eld perturbations, and to show that novel super-
shell stru
tures appear in both level densities and pairing.
In 
al
ulating level spe
tra by analyti
al periodi
 orbit
theory and WKB as well as numeri
al Hartree-Fo
k, we
also relate these di�erent theoreti
al approa
hes to one
another.
We treat a gas ofN fermioni
 atoms of massm in a HO
potential at zero temperature, intera
ting via a two-body
intera
tion with s-wave s
attering length a. We shall
mainly dis
uss a spheri
ally symmetri
 trap and a dilute
gas (i.e. where the density ρ obeys the 
ondition ρ|a|3 ≪
1) of parti
les with two spin states of equal population.
The Hamiltonian is then given by
mω2r2i + U(ri)
, (1)
We will 
onsider both external anharmoni
 potentials of
the form U = εr4 and parti
le intera
tions: U(ri) =
(2π~2a/m)
j 6=i δ
3(ri−rj). When intera
tions are weak,
the latter 
an be approximated by the mean �eld poten-
U(r) =
2π~2a
ρ(r) . (2)
For a large number of parti
les and U = 0 the Fermi
energy is EF = ñF~ω where nF = ñF − 3/2 ≃ (3N)1/3
is the HO quantum number at the Fermi surfa
e. The
HO shells are highly degenerate with states having an-
gular momenta l = nF , nF −2, ...,mod(nF , 2), due to the
U(3) symmetry of the 3D spheri
ally symmetri
 HO po-
tential. However, intera
tions split this degenera
y. In
the Thomas-Fermi (TF) approximation (see, e.g., [4℄) the
Fermi energy is
2k2F (r)
mω2r2 + U(r) . (3)
The density ρ(r) = k3F (r)/3π
ρ(r) = ρ0
1− r2/R2TF
, (4)
inside the 
loud r ≤ RTF = aosc
2ñF , where ρ0 =
(2ñF )
3/2/3π2a3osc is the 
entral density [5℄. For 
onve-
nien
e we set the os
illator length aosc =
~/mω = 1 in
the following.
Taylor expanding the density and thereby also the
mean �eld of Eq. (2) around the 
enter gives
ρ(r) ≃ ρ0
r2/R2TF +
r4/R4TF + ...
, (5)
the �rst term will simply in
orporate a 
onstant shift
in energies whereas the term quadrati
 in radius renor-
malizes the HO frequen
y as ωeff = ω
1− 6πaρ0/R2TF .
The third term is quarti
 in radius and is therefore also
of the same form as the external potential
U(r) ≃ εr4 , (6)
with ε = (3π~2a/4m)ρ0/R
TF . Both the pure quarti
potential and the mean �eld potential of Eq. (2) are an-
harmoni
 and 
hange the level density by splitting the l
degenera
y of the HO shell nF at the Fermi surfa
e.
We will now 
al
ulate analyti
ally the level spe
tra
from perturbative periodi
 orbit theory for the quarti
potential and subsequently within semi
lassi
al WKB
wavefun
tions for both the quarti
 and the mean �eld
potential of Eq. (2). We will start with repulsive inter-
a
tions where pairing is not present.
http://arxiv.org/abs/0704.0385v1
In periodi
 orbit theory [6℄, the level density 
an be
written (to leading order in ~
) in terms of a perturba-
tive HO tra
e formula [7, 8℄
g(E) =
1 +Re
(−1)k M ei2πkE/~ω
. (7)
For the unperturbed HO (U = 0) the modulation fa
tor is
M = 1. For a quarti
 perturbed potential, as in Eq. (6),
the modulation fa
tor was 
al
ulated in [8℄
e−i2kσ/~−iπ/2 + e−i3kσ/~+iπ/2
, (8)
with σ = επE2/~2ω3, being a small 
lassi
al a
tion. The
two terms arise from the 
hange in a
tions for the 
ir
le
and diameter orbits respe
tively due to the quarti
 po-
tential [8℄. The resulting level density 
an be written in
the fa
torised form [9℄
g(E) =
(~ω)3
(−1)k
. (9)
Here, the �rst term is the average level density, the

osine fa
tor gives the rapid HO shell os
illations (mod-
i�ed by the perturbation) whi
h, however, are slowly
modulated by the sine fa
tor resulting in a beating pat-
tern. Moreover, the non-perturbed HO limit, equivalent
to M = 1 in Eq. (7), is re
overed in the limit of |ε| → 0,
where the U (3) symmetry is restored. The k = 1 term
in Eq. (9) gives the major os
illations in the level density
and is shown in Fig. 1 (a). The beating pattern or super-
shells is 
learly observed. The shell os
illations vanish
when the argument of the sine in Eq. (9) is an integer
S = 1, 2, 3, ... times π, i.e. |ε|E2/2(~ω)3 = S. This gives
the supernode 
ondition
nF = E/~ω =
2S~ω/|ε| . (10)
We now turn to an alternative 
al
ulation of the level
density with WKB. The splitting of the HO shells degen-
erate levels l = nF , nF −2, ...,mod (nF , 2) in the shell nF
by the mean-�eld potential 
an be 
al
ulated perturba-
tively in the dilute limit. An ex
ellent approximation for
the radial HO wave fun
tion with angular momentum l
and (nF − l)/2 radial nodes in the HO shell when nF ≫ 1
is the WKB one [10, 11℄:
RnF l(r) ≃
sin(kl(r)r + θ)
l (r)r
, (11)
between turning points r2± = ñF ±
ñ2F − l(l + 1). Here,
ñF = nF + 3/2 and the WKB wave number kl(r) is
k2l (r) = 2ñF − r2 − l(l+ 1)/r2 . (12)
When nF ≫ 1 the wave fun
tion has many nodes 1 ≪
l ≪ nF and the os
illations in R2nl(r) 
an be averaged
〈sin2(kl(r)r)〉 = 1/2 [10℄. The phase θ is then unim-
portant. The single-parti
le energies for the anharmoni
potential of Eq. (6) are simply
EnF ,l − ñF~ω =
U(r)|RnF l(r)|2r2dr (13)
πkl(r)
3ñ2F − l(l + 1)
.(14)
It is spe
ial for the quarti
 perturbation that the level
energies are linear in l(l+1). The resulting level spa
ing
in
reases as (2l+1) just as the level degenera
y for SO(3)
symmetry. Therefore the level density is 
onstant within
the bandwidth
D ≡ |EnF ,l≡0 − EnF ,l=nF | = εn2F /2 (15)
on energy s
ales larger than 2D/nF but smaller than D.
The level density vanishes between the bandwidths of
two neighbouring n shells and therefore it generally has
a strong os
illatory behavior as shown in Fig. 1 (a). Its
amplitude is largest when D ∼ ~ω/2. However, when
D ≃ ~ω the level density is 
onstant and the os
illa-
tory behavior vanishes. This phenomenon repeats when
D = S~ω sin
e the level spe
tra then overlap S times.
With the bandwidth of Eq. (15) under this 
ondition, we
obtain exa
tly the same supernode 
ondition as for pe-
riodi
 orbit theory, Eq. (10). We 
on
lude that Craig's
perturbative periodi
 orbit theory [7℄ is in exa
t agree-
ment with perturbative WKB for a quarti
ally perturbed
spheri
al symmetri
 HO in three dimensions.
We now turn to the slightly more 
ompli
ated mean
�eld potential of Eq. (2). Its level spe
trum 
an also be

al
ulated from the WKB wave fun
tions of Eq. (11).
Inserting them in Eq. (13), we obtain
EnF ,l − ñF~ω = 2/
F ~ω I . (16)
Here, the integral I is
1− l(l+ 1)/ñ2F
1− x2
where x = (r2−ñF )/
ñ2F − l(l + 1). This integral is I =
π for l ≃ nF and I = 8
2/3 for l = 0. The bandwidth is
therefore
D = 2/
2/3− π
. (18)
Inserting this bandwidth in the supernode 
ondition D =
S~ω gives
2/3− π
F = S . (19)
For example in the 
ase 2πa = 1 the supernodes S =
1, 2, 3, .. should o

ur when nF ∼ 28, 44, 58, et
. The
Hartree-Fo
k (HF) 
al
ulations of the os
illating part of
the total energy, whi
h is proportional to the level den-
sity at the Fermi level [6℄, result in slightly higher su-
pernodes, as in Fig 1 (b). The di�eren
es arise be
ause
the WKB 
al
ulations are perturbative in the intera
tion
strength, whereas in the HF 
al
ulation the MF poten-
tial U in
ludes a large s
attering length whi
h, e.g., leads
to 
orre
tions for the e�e
tive os
illator frequen
y. Also
for the purely quarti
 term the perturbative approa
h
underestimates the exa
t supernodes (see Fig. 3 of [8℄).
For weaker intera
tions 2πa = 0.1, the �rst supernode
S = 1 should o

ur at nF = 130 a

ording to the 
ondi-
tion of Eq. (19), in 
loser agreement with the HF result
of Fig. 1 (
).
For 
omparison, the Taylor expansion of the mean �eld
potential leads to the supernode 
ondition of Eq. (10)
with ε = (3π~2a/4m)ρ0/R
TF . It di�ers from Eq. (19)
by the prefa
tor, whi
h is ∼ 34% smaller. It is a better
approximation to expand e.g. around r = RTF /2
nF /2, where the 
orresponding prefa
tor is only ∼ 8%
smaller, su
h that the supernode in Fig 1 (
) is predi
ted
to nF = 137. Now expanding I of Eq. (17) for small
l ≪ nF , one �nds
I = (8/3)
2− l2/
2n2F , (20)
resulting in the level spe
trum [10℄
EnF ,l − ñ~ω =
− l(l+ 1)
. (21)
This level density is 
onstant at low l as for the potential
in Eq. (14). However, near l ∼ nF the density of lev-
els is slightly smaller as 
an be seen from the bandwidth

orresponding to Eq. (21), whi
h is ∼ 12% larger, for a
given nF , than the bandwidth of Eq. (18). That the level
density is not 
ompletely 
onstant within the bandwidth
has the e�e
t that a small periodi
ity remains even at
the super-shell 
ondition D = S~ω. Therefore the shell
os
illations do not disappear 
ompletely at the supern-
odes, as 
an be seen in Fig. 1 (b,
), whereas for the purely
quarti
 
ase (a) the os
illations disappear 
ompletely at
the supernodes.
Most atomi
 traps are not spheri
al but 
igar shaped
(prolate) with ωz
<∼ω⊥. The unperturbed HO energies
E = nz~ωz + n⊥~ω⊥ will generally lead to a 
onstant
level density for energy s
ales larger than ~ωz but smaller
than ~ω⊥. When the os
illator frequen
y ratio ω⊥/ωz is a
rational number, level degenera
ies and larger os
illations
will o

ur on the s
ale ~ωz. Intera
tions will, however,
smear this level density. In any 
ase, super-shell stru
ture
is not expe
ted as in the spheri
al symmetri
 
ase. In
very oblate traps ωz ≫ ω⊥ the mean �eld potential is
e�e
tively two-dimensional and quadrati
, i.e. it does
not split the HO shells [10, 13℄. Thus we may expe
t
strong os
illations in the level density on the s
ale ~ω⊥,
but again no super-shell stru
ture.
0 20 40 60 80 100
−1000
1000 (a)
0  20 40 60 80 100 120 140
200 (c)
0 20 40 60 80 100
40 (b)
Figure 1: (
olor online) The upper �gure (a) shows the leading
term (k = 1) of the os
illating part of the perturbative level
density of Eq. (9) as a fun
tion of nF = E/~ω, for the 
ase of
an external potential V = VHO+εr
with ε = 2/402 ≈ 0.0013.
The middle and lower �gures (b,
) show the os
illating part
of the total energy a

ording to a numeri
al HF 
al
ulation
[12℄, with intera
tion strength 2πa = 1 and 2πa = 0.1, as a
fun
tion of the HO shell number (~ = ω = 1). This illustrates
qualitatively that a supernode, e.g. at nF = 40, 
an be due
to intera
tion (b) and/or an additional quarti
 term to the
HO potential (a).
Attra
tive intera
tions lead to pairing by an amount
that is exponentially sensitive to the underlying level den-
sity near the Fermi surfa
e [2, 10, 11, 14℄. The level den-
sity is the same for repulsive and attra
tive intera
tions
ex
ept that the levels are reversed when the sign of ε
(Eqs. (9)) and (13)) or a is 
hanged (Eq. (16)). Therefore
we 
an use the level densities and bandwidths 
al
ulated
above for pairing 
al
ulations. Pairing in �nite systems
is des
ribed by the Bogoliubov-de Gennes (BdG) equa-
tions [15℄ and take pla
e between time-reversed states.
As shown in [14℄ these states 
an be approximated by HO
wave fun
tions in dilute HO traps as long as the gap does
not ex
eed the os
illator energy, ∆<∼~ω. Solving BdG for
su
h �nite systems is numeri
ally 
ompli
ated and we
shall therefore apply further simplifying approximations,
namely that the pairing gap ∆nl and the wavefun
tion
overlap matrix elements vary slowly with level l in a shell
n. Both approximations are fair for the trapped atoms
as argued in [11℄ and deviations 
an be understood. As
result we arrive at a mu
h simpli�ed gap equation
∫ ∼2nF
g(E) dE
(E − µ)2 +∆(µ)2
. (22)
Here, the supergap G = 32
2nF |a|~ω/15π2 was 
al-

ulated in [10℄ as the pairing gap when all states in
a shell 
an pair; this is the 
ase for a region of in-
tera
tion strengths and parti
le number where the gap
is large as 
ompared to the level splitting, yet small
0 10 20 30 40 50 60
=(3N)1/3
38 40 42 44
Figure 2: (
olor online) Multi-shell pairing gaps for a HO trap
with an additional quarti
 term in the potential with ε =
2/402, i.e. for the level density of Fig. 1 (a) with supernodes
at nF ≃ 40, 40
2 ≈ 57, etc. The intera
tion strength is a =
−0.05 (top red 
urve), a = −0.03 (middle blue 
urve, with the
inset �gure around the �rst supernode) and a = −0.01 (lower
green 
urve). In the inset plot it is 
learly seen that the lo
al
minima for l ∼ nF and l ∼ 0 before the supernode turns
into lo
al maxima after the supernode, as a 
onsequen
e of
overlapping shells. The dashed (red) line is the multi-shell gap
∆ = G/(1−2G ln(nF )/~ω) for a = −0.05 and the upper/lower
thin solid line (bla
k) are the single mid-/end-shell pairing for
a = −0.01 (see text).

ompared to the shell splitting ~ω. ∆(µ) = ∆nl is
the gap at the Fermi surfa
e. g(E) = n2F /D is
the level density within ea
h bandgap D around ev-
ery shell n = 0, 1, ...,∼2nF but vanishes between
the bandgaps. The gap equation thus redu
es to
1 = (G/D)
∑∼2nF
(E + n~ω − µ)2 +∆2. The

hemi
al potential µ 
an be determined from the level
spe
trum; as we gradually �ll parti
les into the shell nF
at the Fermi surfa
e, µ in
reases from nF~ω to nF~ω+D.
The 
ut-o� n<∼2nF in the sum of the gap equation models
as a �rst approximation the more rigorous regularization
pro
edure des
ribed in Ref. [16℄ that is required for a
delta-fun
tion pseudo-potential.
By solving this simpli�ed gap equation of Eq. (22),
we �nd that it still 
ontains and displays the essential
interplay between the variation in level density and pair-
ing. To illustrate the super-shell stru
ture in pairing,
we take the strongly anharmoni
 trap potential used for
the level spe
tra in Fig. 1 (a), and 
al
ulate the pairing
arising from a weak attra
tive s
attering length a < 0.
For su�
iently weak intera
tions su
h that pairing only
takes pla
e in the shell at the Fermi surfa
e, we obtain
the expe
ted result from the gap equation: ∆ = G when
D ≪ ∆, whereas for D ≫ ∆ we get ∆ = D exp(−D/2G)
midshell (µ = nF~ω + D/2) and ∆ = 2D exp(−D/G)
endshell (µ = nF~ω or µ = nF~ω +D). Pairing is thus
stronger at midshell than at endshell, where there are
fewer states to pair [11℄, and strong shell os
illations fol-
low as shown in Fig. 2. For stronger intera
tions, pairing
also takes pla
e between states in shells around the Fermi
shell and Eq. (22) gives: ∆ = G/ (1− 2G ln (nF ) /~ω) for
small bandwidth [14℄. In Fig. 2 this 
urve is 
ompared
with the �nite bandwidth result, whi
h has strong os
il-
lations ex
ept at the supernodes where the level density
is 
ontinuous. At a supernode D = ~ω and the gap equa-
tion (22) leads to a gap ∆ = 2nF~ω exp(−~ω/2G) [11℄.
In summary, level densities, shell-os
illations and
super-shell stru
tures in anharmoni
 traps 
al
ulated
from numeri
al Hartree-Fo
k and analyti
al periodi
 or-
bit theory as well as WKB were found to mat
h to lead-
ing order. Analogous super-shell stru
tures were found
in pairing from an approximated BdG 
al
ulation. The
mean �eld in atomi
 nu
lei also have a large anharmoni
potential and the HO shells start to overlap (the �rst
supernode) already for heavy nu
lei with nF ∼ 5 − 6.
The interplay of level spe
tra and multishell pairing is,
however, di�
ult to disentangle in nu
lear pairing due to
strong spin-orbit e�e
t and small parti
le number. Ul-
tra
old atomi
 traps, however, provide ideal systems for
observing the ri
h quantum stru
tures su
h as level den-
sities and pairing.
Dis
ussions with Matthias Bra
k on periodi
 orbit the-
ory, Ben Mottelson on (nu
lear) shell theory and pairing,
and proof reading by Joel Corney, are gratefully a
knowl-
edged.
[1℄ J. Bardeen, L. N. Cooper, J. R. S
hrie�er, Phys. Rev.
108, 1175 (1957).
[2℄ A. Bohr and B. R. Mottelson, Nu
lear Stru
ture Vols.
I+II, Benjamin, New York 1969.
[3℄ A. Bohr, B. R. Mottelson, D. Pines, Phys. Rev. 110, 936
(1958).
[4℄ C. J. Pethi
k and H. Smith, Bose-Einstein Condensation
in Dilute Gases, Cambridge Univ. Press 2002.
[5℄ For a �nite number of parti
les the fa
tor ñ = nF + 3/2
in
ludes a 
orre
tion to nF , whi
h has been 
he
ked nu-
meri
ally to improve the TF approximation and slightly

hange the predi
tion of supernodes.
[6℄ M. Bra
k and R. K. Bhaduri, Semi
lassi
al Physi
s, re-
vised edn (Boulder, CO: Westview) (2003).
[7℄ S. C. Creagh, Ann. Phys., NY 248 60 (1996).
[8℄ M. Bra
k et al., J. Phys. A 38, 9941 (2005).
[9℄ M. Ögren, unpublished (2006):
www.magnus.ogren.se/notes/pot/derivationofgpert.pdf
[10℄ H. Heiselberg and B. R. Mottelson, Phys. Rev. Lett. 88,
190401 (2002).
[11℄ H. Heiselberg, Phys. Rev. A 68, 053616 (2003). Note that
the square root of kl was missing in Eq. (6) of this Ref.
as 
ompared to Eq. (11).
[12℄ Y. Yu et al., Phys. Rev A 72, 051602(R) (2005).
[13℄ B. P. van Zyl et al., Phys. Rev. A 67, 023609 (2003).
[14℄ G. M. Bruun and H. Heiselberg, Phys. Rev. A 65, 053407
(2002).
[15℄ P. G. de Gennes, Super
ondu
tivity of Metals and Alloys
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