Bosonization, Pairing, and Superconductivity of the Fermionic Tonks-Girardeau Gas

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by  M. D. Girardeau

Bosonization, Pairing, and Superconductivity of the Fermionic Tonks-Girardeau Gas
M. D. Girardeau1,∗ and A. Minguzzi2,3,†
1College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA
2Laboratoire de Physique et Mod´elisation des Mileux Condens´es, C.N.R.S., B.P. 166, 38042 Grenoble, France
3Laboratoire de Physique Th´eorique et Mod`eles Statistiques,
6 Universit´e Paris-Sud, Bˆat. 100, F-91405 Orsay, France
0
0 Wedeterminesomeexactstaticandtime-dependentpropertiesofthefermionicTonks-Girardeau
2 (FTG) gas, a spin-aligned one-dimensional Fermi gas with infinitely strongly attractive zero-range
odd-wave interactions. We show that the two-particle reduced density matrix exhibits maximal
n
superconductive off-diagonal long-range order, and on a ring an FTG gas with an even number of
a
atoms has a highly degenerate ground state with quantization of Coriolis rotational flux and high
J
sensitivity to rotation and to external fields and accelerations. For a gas initially under harmonic
9
confinementweshowthatduringanexpansionthemomentumdistributionundergoesa“dynamical

bosonization”,approachingthatofanidealBosegaswithoutviolatingthePauliexclusionprinciple.
]
n
o PACSnumbers: 03.75.-b,05.30.Jp
c
-
r
p IfanultracoldatomicvaporisconfinedinadeBroglie of an even number of atoms on a ring with quantiza-
u wave guide with transverse trapping so tight and tem- tion of Coriolis rotational flux and high sensitivity to
s perature so low that the transverse vibrational excita- rotation and to external fields and accelerations, and a
t. tion quantum ~ω is larger than available longitudinal “dynamicalbosonization”ofthemomentumdistribution
a
m zero point and thermal energies, the effective dynam- following sudden relaxation of the trap frequency.
ics becomes one-dimensional (1D) [1, 2], a regime cur- Untrapped FTG gas: The Hamiltonian is Hˆ =
-
d rently under intense experimental investigation [3, 4]. N ~2 ∂2 + vF (x x ) where vF is
n Confinement-induced 1D Feshbach resonances (CIRs) Pj=1h−2m∂x2ji P1≤j<ℓ≤N int j − ℓ int
the two-body interaction. Since the spatial wave func-
o
reachablebytuningthe1Dcouplingconstantvia3DFes-
c tion is antisymmetric due to spin polarization, there
hbachscatteringresonancesoccurforbothBosegases[1]
[ is no s-wave interaction, but it has been shown [5, 7]
and spin-aligned Fermi gases [5]. Near a CIR the 1D in-
that a strong, attractive, short-range odd-wave interac-
3
teractionisverystrong,leadingtostrongshort-rangecor-
v tion (1D analog of 3D p-wave interactions) occurs near
relations,breakdownofeffective-fieldtheories,andemer-
3 the CIR. This can be modeled by a narrow and deep
6 gence of highly-correlatedN-body ground states. In the squarewellofdepthV andwidth2x . Thecontactcon-
0 bosonic case with very strong repulsion (1D hard-core 0 0
dition at the edges of the well is [7] ψ (x = x ) =
08 GBoirsaerdgaeasuwi(tThGc)ougpalsin),gtchoensetxanacttg1BND-b→od+y∞g,rothuendTosntaktse- −ψF(xjℓ = −x0) = −aF1DψF′ (xjℓ = ±x0F) wjhℓere aF10D is
5 the 1D scattering length and the prime denotes differen-
wasdeterminedsome45yearsagobyaFermi-Bose(FB)
0 tiation. Consider first the relative wave function ψ (x)
t/ mapping to an ideal Fermi gas [6], leading to “fermion- in the case N = 2. The FTG limit is aF F, a
a ization” of many properties of this Bose system, as re- 1D → −∞
zero-energy scattering resonance. The exterior solution
m centlyconfirmedexperimentally[4]. The“fermionicTG”
isψ (x)=sgn(x)= 1(+1forx>0and 1forx<0)
- (FTG) gas[7], a spin-alignedFermi gaswith verystrong F ± −
d attractive 1D odd-wave interactions, can be realized by and the interior solution fitting smoothly onto this is
n sin(κx) with κ = mV /~2 = π/2x . In the zero-range
3DFeshbachresonancemediatedtuningtotheattractive p 0 0
o limit x 0+ the well area 2x V =(π~)2/2mx ,
c sideoftheCIRwith1Dcouplingconstantg1FD →−∞. It stronge0r→thana negativedelta f0un0ction. Inthis li0m→it t∞he
: hasbeenpointedout[5,7]thatthe generalizedFBmap-
v wave function is discontinuous at contact x = 0 , al-
Xi pmianpg[t5h,is7,s8y]stceamnbteoetxhpelotirtaepdpiendthideeoaplpBoossietegdaisr,eclteiaodnintgo lowinganinfinitely strongzero-rangeinteract0ionin±spite
of the antisymmetry of ψ [8]. This generalizes immedi-
r to determination of the exact N-body ground state and F
a ately to arbitraryN: the exact FTG gas ground state is
“bosonization” of many properties of this Fermi system.
We recently examined the equilibrium one-body density
matrix and exact dynamics following sudden turnoff of N
the interactions by detuning from the CIR [9]. Here we ψF(x1,··· ,xN)=A(x1,··· ,xN)Yφ0(xj) (1)
determine some otherexactproperties of the untrapped, j=1
ring-trapped, and harmonically trapped fermionic TG
with A(x , ,x ) = sgn(x x ) the “unit
gas,themoststrikingofwhicharepairing,superconduc- 1 ··· N Q1≤j<ℓ≤N ℓ − j
antisymmetricfunction”employedintheoriginaldiscov-
tive off-diagonal long-range order (ODLRO) of the two-
ery of fermionization [6] and φ = 1/√L the ideal Bose
body density matrix, a highly degenerate ground state 0
gas ground orbital, L being the periodicity length. Its
2
energyiszero[10]anditsatisfiesperiodicboundarycon- FTG gas is maximally superconductive in the sense of
ditions for odd N and antiperiodic boundary conditions Yang’s ODLRO criterion.
for even N [11]. FTG gas on a ring: If the FTG gas is trapped on a
The exact single-particle density matrix ρ (x,x′) = circular loop of radius R, with particle coordinates x
1 j
NR ψF(x,x2,··· ,xN)ψF∗(x′,x2,··· ,xN)dx2···dxN measured around the circumference L = 2πR, the FTG
is [9, 12] ρ1(x,x′) = Nφ0(x)φ∗0(x′)[F(x,x′)]N−1 with gas must satisfy periodic boundary conditions for both
F(x,x′) = L/2 sgn(x y)sgn(x′ y)φ (y)2dy = odd and even N because of single-valuedness of its wave
R−L/2 − − | 0 |
1 2x x′ /L. In the thermodynamic limit N , function. Since the mapping function A(x1, ,xN) =
···
− | − | → ∞ sgn(x x )is periodic (antiperiodic)forodd
L , N/L = n this gives an exponential decay Q1≤j<ℓ≤N ℓ− j
[12]→: ρ∞(x,x′) = ne−2n|x−x′|. Its Fourier transform n , (even) N as a result of its definition, it follows that the
1 k
mapped ideal Bose gas used to solve the FTG problem
normalized to n = N (allowed momenta ν2π/L
Pk k must satisfy periodic (antiperiodic) boundary conditions
with ν = 0, 1, 2, ), is the momentum distribution
function n =±[1±+(k··/·2n)2]−1. It satisfies the exclusion for odd (even) N. The ground state of a FTG gas on
k
a ring is then different depending on the particle num-
principle limitation n 1, but nevertheless, for n 0
k
≤ → ber parity. For odd N the FTG ground state in Eq. (1)
the continuous momentum density n(k) = (L/2π)n
k
is built from the zero-momentum orbital φ = 1/√L
reduces to N times a representation of the Dirac delta 0
and correspondsto mapping the FTG gas onto the ideal
function, simulating the ideal Bose gas distribution:
Bose gasgroundstate, the usual complete Bose-Einstein
n(k) Nδ(k) [12].
Thn−e→→0two-particle density matrix ρ (x ,x ;x′,x′) = condensate (BEC), and is nondegenerate. On the other
2 1 2 1 2
N(N 1) ψ (x ,..,x )ψ∗(x′,x′,x ,..,x )dx ...dx hand,for evenN,whichwe henceforthassume,antiperi-
− R F 1 N F 1 2 3 N 3 N odicityrequiresthattheonlyplane-waveorbitalsallowed
also has a simple closed form:
are eikxj/√L with k = π/L, 3π/L, . The ground
ρ (x ,x ;x′,x′)=N(N 1)sgn(x x )φ (x )φ (x ) state of this fictitious id±eal Bo±se gas,··a·nd hence that
2 1 2 1 2 − 1− 2 0 1 0 2
sgn(x′ x′)φ∗(x′)φ∗(x′)[G(x ,x ;x′,x′)]N−2 (2) of the mapped FTG gas, is then (N + 1)-fold degen-
× 1− 2 0 1 0 2 1 2 1 2 erate, with energy eigenvalue N(~2/2m)(π/L)2. These
where [G(x ,x ;x′,x′)]N−2 = [ L/2 sgn(x degenerategroundstatesarefragmentedBECswithwN
xe2)nsg(yn1(−xy22+−y3x−1)ys4g)n2(ixn′11 −t2hxe)sgtnh(exr′2m−odxy)n|aφmR0−|i2cL(/x2)ldimx]iNt−12an−=d awtiothms0in≤thwe o≤rb1it,alanediπxajr/eL caonndve(n1ie−ntwly)Nlabinellee−diπbxyj/La
yin1 a≤sceyn2di≤ngyo3rd≤ery.4ρareisthoef oarrdgeurmne2ntisn(xth1e,xf2o;lxlo′1w,xin′2g) qreulaantetdumtontuhmebeiegrenℓzva=lue(wP−of12c)irNcu=mfe0r,e±n1c,ia±l2li,n·e·a·r,±moN2-
2
cases: (a) x x′ O(1/n), x x′ O(1/n); mentumandthatLz ofangularmomentumz-component
(b) x x|′1 − 1O|(1≤/n), x |x2′ − 2|O≤(1/n); (c) by P = ℓz~/R and Lz = ℓz~. The angular momentum
x | x1 − O2|(1/≤n), x′ x′| 2 −O(1/1|n).≤These are just per particle is half-integral due to antiperiodicity of the
| 1− 2| ≤ | 1− 2| ≤ orbitals, and the degenerate ground states are in one-
Yang’s criteria [14] for superconductive ODLRO of ρ in
2
one correspondence with the eigenstates of spin angular
the absence of ODLRO of ρ . In case (c) ρ remains of
1 2
order n2 for arbitrarily large separation of the centers momentum z-component of N spin-1/2 fermions.
of mass X = (x + x )/2 and X′ = (x′ + x′)/2, the The groundstate degeneracymakesthe FTGgasona
1 2 1 2
ring a good candidate for detecting small external fields
hallmark of ODLRO. On the other hand, in cases (a)
and (b) ρ decays exponentially with X X′ . In the and linear accelerations. Suppose that there is a poten-
2
| − | tial gradient parallel to a diameter of the ring, or an
thermodynamic limit only configurations (c) contribute
acceleration leading to a gradient in the inertial poten-
to the largest eigenvalue of ρ , and ρ separates apart
2 2
tial arising from Einstein’s principle of equivalence, with
from negligible contributions (a) and (b) [13]:
the circumferential minimum of this potential occurring
ρ (x ,x ;x′,x′)=n2sgn(x x )e−2n|x1−x2| at a point x . Then the degeneracy is lifted and to low-
2 1 2 1 2 1− 2 0
sgn(x′ x′)e−2n|x′1−x′2|+terms negligible for λ .(3) est order in degenerate perturbation theory all N atoms
× 1− 2 1 occupythe orbitalφ (x)= 2/Lcos[π(x x )/L],lead-
0 p − 0
ByYang’sargument[14]thelargesteigenvalueisλ =N, ing to an observable asymmetric density profile n(x) =
1
and this is confirmed by comparison with the λ1 contri- 2ncos2[π(x−x0)/L].
bution λ u (x ,x )u (x′,x′) to the spectral representa- Due to its quantum coherence the FTG gas is also a
1 1 1 2 1 1 2
tion of ρ , implying that the corresponding eigenfunc- goodcandidateforasensitiverotationdetector. Suppose
2
tion is u (x ,x ) = sgn(x x )e−2n|x1−x2| with [15] thattheringtrapisrotatingwithangularvelocity~ω per-
1 1 2 1 2
C −
= n/L, confirming the value λ = n2/ 2 = N. pendicular to the plane of the ring. In the rotating coor-
C p 1 C
The range 1/2n of u is in the region of onset of a dinate system each atom sees an effective Coriolis force
1
BEC-BCS crossover between tightly bound bosons and F~ =2m~v ~ω. Comparingthiswiththeusualmagnetic
Cor
loosely bound Cooper pairs. There is an upper bound force F~ =×(e/c)~v B~, one sees that the kinetic energy
mag
×
[14] λ N on the largest eigenvalue, so the untrapped operators in the Hamiltonian in the rotating system are
1
≤
3
exist. InthecaseofevenN boththegroundstateandthe
excitationbranchesare(N+1)-folddegenerate,butitis
2R] sufficientheretoconsidertheℓz =0groundstateandthe
m excitations arising fromit by promoting atoms to higher
/20,25 k-values. Generalizing Bloch’s analysis, we note that for
2
~ −5N~ −3N~ −N~ N~ 3N~ 5N~ 0 < ν N/2 the lowest branch corresponds to excita-
N 2 2 2 2 2 2 ≤
[ tion of ν atoms from k = π/L to k = 3π/L, yielding
E a state with angular mome−ntum z-component ℓ ~ with
z
ℓ = 2ν, and with excitation energy ǫ(ℓ ) = ℓ ~2/mR2.
z z z
At ν = N/2 one has reached a state differing from the
ground state by translation of all atoms by an amount
0 2π/L in k-space, and one can repeat this process, pro-
-2 0 2
Φ/Φ0 moting atoms from k = π/L to 5π/L, yielding another
straight-line segment connecting the points ℓ = N and
FIG.1: DependenceofenergiesEonrotationalfluxΦ. Heavy z
line: Ground state energy E0(Φ). Lighter lines: Lowest en- ℓz = 2N on a parabolic curve (ℓz~)2/2NmR2, etc. To-
ergy for each valueof total angular momentum. gether with symmetry ǫ(ℓz) = ǫ( ℓz) this yields an ex-
−
citationenergycurvecomposedofstraight-linesegments
asinthe dashedcurveofBloch’sFig. 2[16]withthe no-
tationP =ℓ ~/R. HenceforbothoddandevenN there
[pˆ h Φ ]2/2m where pˆ = (~/i)∂/∂x , Φ = πR2ω is z
j − LΦ0 j j arenoenergybarriers,andtheFTGgasonanonrotating
the Coriolis flux through the loop, and Φ = h/2m is
0 ring does not exhibit flow metastability.
the Coriolis flux quantum. The energy of each state ℓ
z
then becomes E = E (Φ = 0)+ N~2 [( Φ )2 2ℓ |Φi] Expansion from a longitudinal harmonic trap: We
0 2mR2 Φ0 − zΦ0 focus finally on a 1D expansion, as could be achievedby
which is minimized when ℓ = N if Φ>0 and ℓ = N
z 2 z −2 keepingonthetransverseconfinement. Ifthe1Dinterac-
if Φ < 0, i.e., even a very small angular velocity leads
tions are suddenly turned off before the gas is let free to
to a nondegenerate ground state with all N atoms at
expand from a longitudinal harmonic trap, the density
either k = π/L or k = π/L. Generalizing to states
profileatlongtimesreflectstheinitialmomentumdistri-
−
differing from the Φ = 0 ground states by displacement
bution [9]. If instead the interactions are kept on during
in k-space by integral multiples of 2π/L one obtains the
the expansion we find that the density profile expands
Φ-dependent ground state energy E (Φ) shown by the
0 self-similarly, while the momentum distribution evolves
heavy line in Fig. 1, in which the lighter lines show the
fromaninitialoverallLorentzianshape[12]tothatofan
lowestenergiesforℓ = N, 3N, .Thegroundstate
z ±2 ± 2 ··· ideal Bose gas. These properties can be demonstrated
energy is a periodic function of Φ with period Φ in ac-
0 with the aid of an exact scaling transformation as
cord with a general theorem [14], but unlike the usual
we outline below. Since the FB mapping holds also
situation for a superconductor, (a) there is no smaller
for time-dependent phenomena induced by one-body
period Φ /2, and (b) for even N, Φ = 0 is a relative
0 0 external fields [17], the exact many-body wavefunc-
maximum of E0 rather than a minimum (as is the case tion ψ (x , ,x ;t) = A(x , ,x ) N φ (x ;t)
of odd N), the first minima occuring at Φ = Φ /2. F 1 ··· N 1 ··· N Qj=1 0 j
± 0 during the dynamics is fully determined by the
Thebarrierheightsofthe energylandscapeinFig.1van-
solution of the single-particle Schr¨odinger equa-
ish like 1/N for N , so flux quantization will not
→ ∞ tion for the orbital φ0(xj;t). For the case of
be observable for a macroscopic ring. However, it may
an external potential V (x,t) = mω(t)2x2/2
ext
be observable for mesoscopic rings using BEC-on-a-chip
with ω(0) = ω the solution is known [18] to be
0
technology. For example, assuming a ring radius R =5 φ (x;t) = φ (x/b(t);0)eimx2b˙/2b~−iE0τ(t)/~ where
µm, one finds that for 6Li, ∆E >k T for T <50 nK. 0 0
B
b(t) is the solution of the differential equation
Flow properties on a nonrotating ring: According to ¨b + ω2(t)b = ω2/b3 with b(0) = 1 and b˙(0) = 0,
0
theFBmappingtheexcitationspectrumoftheFTGgas τ(t) = tdt′1/b2 and E = ~ω /2. Since the unit anti-
is the same as that of an ideal Bose gas, and hence it is R0 0 0
symmetric wavefunctionA is invariantunder the scaling
sufficient to analyze the latter. Since the excitation en-
transformation, we immediately obtain the expres-
ergy of the ideal Bose gas is quadratic in the excitation
sion for the many-body wavefunction, ψ (x ,..,x ;t)=
Bmoogmoelinutbuomv~cqr,ittehreioFnTfGorgasuspdeoreflsunidotitsya.tisWfyethinevLeasntidgaaute- b−N/2ψF(x1/b,..,xN/b;0)ei(b˙/bω0)PNj=1x2jF/2x2o1sce−iNNE0τ(t)/~,
and for the one-body density matrix, ρ (x,x′;t) =
here the possibility of flow metastability associated with 1
barriers in the excitation energy landscape as a function 1bρ1(cid:0)xb,xb;0(cid:1)exph−ibb˙ (x22x−2oxsc′2)i. This yields the mo-
of the transferredmomentum. It was shownby F. Bloch mentum distribution as a function of time. While
[16] that for the usual ideal Bose gas, which corresponds the intermediate-time dynamics has to be determined
to the case of odd N in our treatment, no such barriers numerically, the stationary-phase method determines
4
for the opportunity to participate, and to S. Giorgini,
30 ideal Bose gas −→
R. Seiringer, F. Zhou, E. Zaremba, and G. Shlyapnikov
for helpful comments. The Aspen Center for Physics is
25
c t=40 −→ supported by the U.S. National Science Foundation, re-
os t=30 −→
k20 searchofM.D.G.attheUniversityofArizonabyU.S.Of-
k,t) t=20 −→ ficeofNavalResearchgrantN00014-03-1-0427througha
n(15 subcontract from the University of Southern California,
t=10 −→ andthatofA.M.bytheCentreNationaldelaRecherche
10 t=5 −→ Scientifique (CNRS).
5 t=0 −→
-6 -4 -2 0 2 4 6
k/kosc
∗ Electronic address: [email protected]
FIG. 2: Momentum distributions of a FTG gas (solid lines) † Electronic address: [email protected]
with N = 9 particles as functions of the wavevector k at [1] M. Olshanii, Phys. Rev.Lett. 81, 938 (1998).
subsequenttimest (in unitsof 1/ω0) duringa 1D expansion, [2] D.S. Petrov, G.V. Shlyapnikov, and J.T.M. Walraven,
and asymptotic long-time expression (4) (dashed line). Phys. Rev.Lett. 85, 3745 (2000).
[3] B.L. Tolra et al.,Phys.Rev.Lett.92,190401 (2004); T.
St¨oferle,H.Moritz,C.Schori,M.K¨ohl,andT.Esslinger,
the long-time evolution of the momentum distribution ibid. 92, 130403 (2004).
[4] B.Paredes,et al.,Nature429,277(2004); T.Kinoshita,
in the same way as for the bosonic TG gas [19]. For
T. Wenger, and D.S. Weiss, Science 305, 1125 (2004).
the case of a 1D expansion the scaling parameter is
[5] B.E.GrangerandD.Blume,Phys.Rev.Lett.92,133202
b(t)= 1+ω2t2 and the momentum distribution tends
p 0 (2004).
tothatofanidealBosegasunderharmonicconfinement, [6] M. Girardeau, J. Math. Phys. 1, 516 (1960); M.D. Gi-
rardeau, Phys.Rev.139, B500 (1965), Secs. 2,3, and 6.
[7] M.D. Girardeau and M. Olshanii, cond-mat/0309396;
n(k,t ) ω /b˙ n (kω /b˙), (4) M.D. Girardeau, Hieu Nguyen, and M. Olshanii, Optics
0 B 0
→∞ ≃| | Communications 243, 3 (2004).
where n (k) = 2πN φ˜ (k)2, with φ˜ (k) = [8] T. Cheon and T. Shigehara, Phys. Lett. A 243, 111
B 0 0
π−1/4ko−s1c/2e−k2/2ko2sc and |kosc =| 1/xosc. This be- [9] M(19.D98.)GainradrdPehayus.aRndevE..LMet.tW. 8r2ig,h2t5,3P6h(y1s9.9R9)e.v. Lett. 95,
havior is illustrated in Fig. 2. Quite noticeably, the
010406 (2005).
“bosonization”time appears to be muchlonger than the [10] Theinfinitelylargenegativeinteractionenergyisexactly
“fermionization” time of the momentum distribution cancelled by the infinitely large positive kinetic energy,
of the bosonic TG gas [19]. Note that the “dynamical as one sees by passing to the zero-range limit x0 → 0+
bosonization” described above does not violate the of thefinitesquare well.
[11] Thisisthemostconvenientchoicetopasstothethether-
Pauli exclusion principle: by using the above scaling
modynamiclimitofaninfinitelylonglineartrapcontain-
solution for the one-body density matrix and fixing
ing an FTG gas of uniform density.
unit normalization of the natural orbitals at all times
[12] S.A. Bender, K.D. Erker, and B.E. Granger, Phys. Rev.
it follows that the eigenvalues αj of ρ1(x,x′;t) are Lett. 95, 230404 (2005).
invariant during the expansion and hence always satisfy [13] The omitted terms determine the smaller eigenvalues,
the condition α 1. whichareoforderunityandsmaller,butmakethedomi-
j
In conclusion,≤we have found that (a) the un- nantcontributiontoTrρ2=N(N−1);see[14].Theyare
trapped system exhibits superconductive ODLRO of the importantforthediagonalelementsofρ2.Fromtheexact
two-body density matrix ρ2 associated with its maxi- eyx3p=rexss2i,oannfdory4G=oxn′2e,steheas,toρn2(xse1t,txin2;gxy11,x=2)x=1,ny22. = x′1,
mal eigenvalue N and pair eigenfunction C sgn(x1 − [14] C.N.Yang,Rev.Mod.Phys.34,694(1962), Sec.18and
x2)e−2n|x1−x2|; (b) on a ring it has a highly degenerate AppendixA.
groundstateforevenatomnumber,anditexhibitsquan- [15] To correctly calculate C onemust periodically extendρ2
tizationofrotationalCoriolisflux andhighsensitivity to and u1 in accordance with Sec. 37 of [14].
rotation and to accelerations, making it a good candi- [16] F. Bloch, Phys. Rev.A 7, 2187 (1973).
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5691 (2000).
trappedsystemundergoesa“dynamicalbosonization”of
[18] A.M. Perelomov and V.S. Popov, Sov. Phys. JETP 30,
its momentum distribution during a 1D expansion.
910 (1970); A.M.Perelomov andY.B.Zel’dovich,Quan-
This work was initiated at the Aspen Center for tum Mechanics, (World Scientific,Singapore, 1998).
Physics during the summer 2005 workshop “Ultracold [19] A. Minguzzi and D.M. Gangardt, Phys. Rev. Lett. 94,
Trapped Atomic Gases”. We are grateful to the orga- 240404 (2005).
nizers, G Baym, R. Hulet, E. Mueller, and F. Zhou,