Josephson effect in an atomic Fulde-Ferrell-Larkin-Ovchinnikov superfluid

0.36 MB·

by  Hui Hu

Josephson probe of Fulde-Ferrell-Larkin-Ov
hinnikov phases in atomi
Fermi gases
1,2 2
1 Hui Hu and Xia-Ji Liu
2 Department of Physi
s, Renmin University of China, Beijing 100872, China
ARC Centre of Ex
ellen
e for Quantum-Atom Opti
s, Department of Physi
s,
University of Queensland, Brisbane, Queensland 4072, Australia
7
0 (Dated: February 6, 2008)
0
Westudytheoreti
allytwospatiallyseparateone-dimensionalatomi
Fermigasesinadouble-well
2
trap. A Josephson jun
tion between a Fulde-Ferrell-Larkin-Ov
hinnikov (FFLO) super(cid:29)uid and a

b Bardeen-Cooper-S
hrie(cid:27)er super(cid:29)uid
ould be realized by
reating a weak link at a given position
e andbytuningindependentlythespinpolarizationofgasesinea
hwell. Owingtotheproportionality
F betweentheJosephson
urrentandorderparameters,thespatialinhomogeneityoftheFFLOorder
6 parameter
an be dire
tly revealed through the
urrent measurement at di(cid:27)erent positions of the
link. Thus, theJosephson e(cid:27)e
t provides de(cid:28)nitiveeviden
e for theexisten
e of FFLO phases.

]
n PACSnumbers: 03.75.Ss,05.30.Fk,71.10.Pm,74.20.Fg
o
c
-
Strongly attra
tive Fermi gases with imbalan
ed spin low-dimensional systems. Parti
ularly it dominates in
r
p
omponents are ubiquitous systems in diverse (cid:28)elds of the one-dimensional (1D) polarized gases [10, 11℄. We
u
physi
s[1℄. Theyarebuilding-blo
ksofatomi
nu
lei,the therefore
onsidertwospatiallyseparate1Datomi
Fermi
s
. matterinneutronstarsandeventhequark-gluonplasma gases with tunable spin polarizations in a tight double-
t
a that
omprisedtheearlyUniverse. PolarizedFermigases well potential on an atom
hip, where the lateralmotion
m also appear in solid-state super
ondu
tors subje
ted to of fermions is frozen, while axial motion is weakly
on-
x
0
- either an internal ex
hange (cid:28)eld or external magneti
(cid:28)ned. A weak link at a spe
i(cid:28)
position is easily
re-
d
(cid:28)eld. A re
ent example attra
ted intense attentions is ated by superimposing a narrow dipole dimple potential
n
trapped atomi
gases of neutral fermions with unequal to allow tunneling. Fig. 1 presents a s
hemati
view of
o
c spin populations [2℄. Owing to the (cid:29)exibility in the
on- the
on(cid:28)guration, together with the potential and par-
[ trolofthe
onstituentsandintera
tionstrengths,atomi
ti
le density pro(cid:28)les. Its detailed realization will be ad-

Fermi gases thus provide the most promising pla
e for dressed later.
2
v observing many exoti
forms of matter. Theunderlyingphysi
sofourproposaliseasilyunder-
0 stood using Ginsburg-Landau (GL) theory [12℄. Assum-
ThegroundstateofpolarizedFermigasesremainselu-
4 ing that the order parameters or
ondensate wave fun
-
sive [3℄. The mismat
hed Fermi surfa
es in polarized Ψ (x)
7
1 environment
annot guarantee the standard Bardeen- tionsΨinthe(lxe)ftandrightwellsaredes
ribedby BCS
1 Cooper-S
hrie(cid:27)er (BCS) me
hanism, whi
h requires a and FFLO ,respe
tively,theJosephson
urrentbyGL
6 pairing of two fermions on the same Fermi surfa
e with theory is
0 opposite spins. Various exoti
forms of pairing have x0+∆x/2
/ I =Im J dxΨ∗ (x)Ψ (x) ,
at been suggested [4, 5, 6, 7℄, su
h as Fulde-Ferrell-Larkin- J " Zx0−∆x/2 BCS FFLO # (1)
m Ov
hinnikov (FFLO) state with spatially varying order
parameters [4℄, deformed Fermi surfa
e [5℄, interior gap J
- where is the
hara
teristi
tunneling energy and is de-
d or Sarma super(cid:29)uidity [6℄, and phase separation [7℄.
termined by the small overlap of the
ondensate wave
n ∆x
Among these, the FFLO state is of parti
ular inter-
o fun
tions along the lateral dire
tion. is the width
est, sin
e the Cooper pairs may
ondensate into a (cid:28)-
c of the narrow tunneling link. The BCS order parame-
: nite
enter-of-mass momentum state. The request for ter in Eq. (1) is essentially spatially independent, while
v
the FFLO state has lasted for more than four de
ades
Xi inmanybran
hesofphysi
s. Onlyre
entlyexperimental 2thπe~/FqFLO one is oqs
illatory in real spa
e with period
FFLO, where FFLO is∆txhe≪
e2nπte~r/-qof-mass momen-
ar eviden
es of its existen
e have b5een found in the heavy tum. Thus, provided that FFLO, the mea-
fermion super
ondu
tor CeCoIn . The observations in-
surement of the maximum Josephson
urrent results di-
Ψ (x = x )

lude the spe
i(cid:28)
heat, magnetization, and penetration 0
re
tly FFLO . By displa
ing axially the har-
depth measurements [8℄. However, de(cid:28)nitive identi(cid:28)
a-
moni
traps and
onsequently
hanging the position of
x
tionsoftheFFLOphasesaredesirable. Theseshouldbe 0
weak link , a series of measurements therefore reveal
based on phase-sensitivemeasurementsthat
andire
tly
the whole spatial inhomogeneity of the FFLO order pa-
revealthespatialvariationsofthe phaseoftheorderpa-
rameter. As shown in Fig. 2, the simple GL pi
ture is
rameter. One possibility is the Josephson e(cid:27)e
t [9℄.
veri(cid:28)ed by mu
h
ompli
ated mi
ros
opi

al
ulations.
In this Letter we propose an atomi
Josephson e(cid:27)e
t Our results are obtained by solving mean-(cid:28)eld
to dete
t the existen
e of FFLO phases. Theoreti
al Bogoliubov-de Gennes (BdG) equations for ea
h well,
studies suggest that su
h phase is more favorable in the whiletreatingthetunnelingthroughtheweaklinkwithin
2
HHHH axial x- EF1.0 BCS(x)
weak liTnk at x = x0HHHH (FFLOa)xis (x)/
HHHH (BCS) L E), F0.5
(a) R lateral z-direction /(NNx0.0 FFLO(x)
6 6 ma
x=x), n(z, x=x) (cid:1)(cid:2)0,024 (b)BCSd ~dtho e1u bp0lo0emt-e wmmmnetmilal lpF aoFtt exLn =tOi axl0 V(z=0, x) [arb. units]24 x = x0 I-J,0.5 -1.0 -0.5 x / (I0NJ.,m10/a2xa(hxo0)) 0.5 1.0
V(z, 0 0 (c) z=0 DDDDx < 10m mmm m Figure 2: (Color online) Underlying physi
s of the Joseph-
son e(cid:27)e
t, as a probe of FFLO phases. With a BCS and a
-2 -1 0 1 2 -20 0 20
z / a x / a FFLOsuper(cid:29)uidpla
edontheleftandrightsidesofthejun
-
ho ho
tion,respe
tively,themaximumJosephson
urrentisroughly
proportional to the FFLONLor=derNpRar=am1e2t8er. The number of
Figure1: (Coloronline)S
hemati
viewofaproposedatomi
fermions oγnLe=a
γhRw=ell1.i6s , while intera
tion
Josephsonjun
tiononanatom
hip. (a)Thedouble-wellpo- strengths EF.=ThNeLo~rωde/r2parameteris normalized
tential lands
ape. Two needle-like (BCS and FFLO) super- by the Fermi ene0rg.5y0EF , and is shifted upwards
(cid:29)uidsarelo
ated intheleft andrightwells, respe
txiv=elyx.0Re- tbhyeaFnFaLmOousnutpoefr(cid:29)uid is 0f.o2r5.
laσrNityN. Tishtehsep
inonpdoula
rtiazna
teionofina
du
ingthedouble-wellbarrierataspe
i(cid:28)
position by

orresponding normal jun
tion.
superimposing a narrow dipole dimple potential, the Cooper
pairs in super(cid:29)uids
an tunnel ba
kand forth between wells,
leading to a Josephson
urrent. (b) Lateral distributions of ∆x
many-bodyintera
tions. Assuming thatthe width is
the potential (thi
k dot-dashed line) and spin up and down H
T
thesmallestlengths
ale,in weapproximatealloper-
parti
le density pro(cid:28)les (thin solid and dashed lines) away ψ(x) ≈ ψ(x )
from the weak link at x100.0µTmhe inter-well distan
e in experi- aVt0or=s J∆x 0 and introdu
e a∆trxan≪sfer2πp~a/raqmeter
ments would be about . The potential with superpo- . This is valid as far as FFLO.
sition of a dimple potential is shown by the thi
k solid line. Weshalltakeasmall
onstanttransferparameter,whi
h
Then, the parti
le densities and the order parameters
an
orrespondstothesmalloverlapoftwoorderparameters.
ahave=smpa~ll/movωerlaps within40tµhme weak link. The length s
ale The overlap
ould depend weakly on the position of the
ho z=0is around . (
) Axial pro(cid:28)le of the po- weak link. However, the assumption of a (cid:28)xed transfer
tential at .
parameter is su(cid:30)
ient to
apture the qualitative feature
of the Josephson e(cid:27)e
t.
linear response theory. Our
al
ulations are performed In ea
h well the ground state of an attra
tive gas of
N = N + N P = (N −
↑ ↓ ↑
spe
i(cid:28)
ally for atomi
Fermi gases. However, as the fermions with polarization
N )/N
↓
FFLO physi
s is a fundamental issue that is of impor- is
onvenientlydeterminedbyusingtheBdGfor-
tan
e to many resear
h (cid:28)elds, they
an have potential malism [12℄ that des
ribes the quasiparti
le wave fun
-
u (x) v (x) g
η η
impli
ations beyond ultra
old atoms. tions and with a
onta
t intera
tion (the
We assume that the dipole dimple potential is a small well index is suppressed for
larity),
x
0
perturbation and the resulting weak link at does not H↑0−µ↑ ∆(x) uη(x) =E uη(x) ,
distrub
onsiderablythedistributionsoftheorderparam- ∆∗(x) −H0+µ v (x) η v (x)
eterand parti
ledensitypro(cid:28)lein ea
hwell. The atomi
(cid:20) ↓ ↓ (cid:21)(cid:20) η (cid:21) (cid:20) η (cid:21)
Josephson jun
tion in Fig. 1a is then well des
ribed by H0 = −~2∇2/2m + mω2x2/2 + g n (x)(4)
a tunneling Hamiltonian. By integrating out the lateral where ↑,↓ 1d ↓,↑ is
degree of freedoms in a tight-binding approximation, it the single parti
le Hamiltonian under axial harmoni
takes three terms: trap and Hµartr=eeµp±otδeµntial. The
hemi
al potentials are
↑,↓
shiftedas toa

ountfortheunequalpopu-
H = HL+HT +HR, (2) lation N↑,↓. The order parameter ∆(x) and µ↑,↓ are
al-
H = V ψ+ (x )ψ (x )+H.c. , ∆(x)=
T 0 σ Lσ 0 Rσ 0 (3) g
ulateduby(xse)lvf∗-
(oxn)fsi(sEten)
yequationsforthegap,n (x) =
X (cid:2) (cid:3) 1d η η 2 η η , and for the densities:2 ↑
σ =↑,↓ H H |u (x)| f(E ) n (x) = |v (x)| f(−E )
where is the spin index. The terms L and R ηPη η and ↓ η η η ,
f(x) = 1/(exp[x/k T]+1)
B
arerespe
tivelytheHamiltoniansforfermionsontheleft Pwith bePing the Fermi fun
-
dxn (x) =
and right sides of the jun
tion, and
an be expressed tion. These must be
onstrained so that ↑,↓
ψ (x) N
L,R;σ ↑,↓
in terms of operators . They
ontains all the . Wenotethattheunequal
hemi
alpoRtentialsbreak
3
I (t) = exp[i(ϕ −ϕ )+i2(µ −
J R L R
0.4 BdG n (x) tµeLr)tis/~g]iveσn b−ty∞[1d3t1℄exp[−i(µR,−σ −µL,−σ)(t−t1)/~] <
a)ho Gaudin [A˜σ(t),A˜P−σ(Rt1)]>0 +c.c.,whereweintrodu
eanintera
-
1/2 /N0.3 tionrepresentationwithrespe
ttoHL andHR,andrep-
) / (0.2 n (x) resentit by atildϕeLin opeϕrRators. The globalphasesIoJf(otr)-
(x derparameters, and , aremade exp2li(
µitly−inµ )t/~.
, Theirdi(cid:27)eren
e, togetherwith thefa
tor R L ,
n
0.1 drives the d
and/or a
Josephson
urrents even at the
zero
hemi
al potential di(cid:27)eren
e between wells.
(a) BCS (b) FFLO
0.0 With the help of the Wi
k theorem, in the statisti
al
0.0 0.5 1.0 0.0 0.5 1.0 IJ(t)
1/2 1/2 average of we split the four fermioni
(cid:28)eld oper-
x / (N aho) x / (N aho) ators. The integration of the average over time t1
an
then be expressed in terms of the retarded
orrelation
Figure3: (Coloronline)ZerotemperatureNme=an1(cid:28)2e8lddenγsit=y fun
tions [13℄,
1p.r6o(cid:28)les (solid lines) of a polarized gas at and
[f(E )−f(E )]
,
ompared with the results from Gaudin solutions (dot- χ (Ω) = (u v∗) u∗v i j ,
dashedlines)[10℄. In(b)theos
illationsinthespin-updensity ↓↑ ij i i L j j R +Ω+Ei−Ej (5)
outside theFFLO
ore are dueto Friedel os
illations. X (cid:0) (cid:1) [f(E )−f(E )]
χ (Ω) = (u v∗) u∗v i j ,
↑↓ ij i i L j j R −Ω+Ei−Ej (6)
X (cid:0) (cid:1)
the time-reversal symmetry. Thus, the sum over energy i j
wheretheindi
es and referto,respe
tively,theenergy
levels is done for all the eigenstates with both positive L
and negative energies Eη. Generi
ally, the intera
tion levelsRin the left and right wells, and the subs
ripts
and are the indi
es for wells. We have abbreviated
strength γis=pa−rmamg1edt/er(i~z2end0)by a dimne0nsionless
oupling u = u(x0) and v = v(x0). Consequently, the Josephson

onstant , where is the
enter den-

urrent
an be
al
ulated as [13℄,
sityofanidealγga.s.1I0n0theweakorintermediate
oupling I (t)=Imaxsin[(ϕ −ϕ )+2(µ −µ )t/~],
regimes (i.e., ), the mean-(cid:28)eld BdG theory ap- J J R L R L (7)
pears to be very a

urate. Fig. 3 shows the mean-(cid:28)eld
γ =1.6 P =0
results of density pro(cid:28)les of a gas at and , where the maximum
urrent
0.25 2V2
,
ompared with that from exa
t Gaudin solutions Imax = 0 χ (µ −µ )+χ (µ −µ ) .
and lo
al density approximation [10℄. The agreement is J ~ ↓↑ R↓ L↓ ↑↓ R↑ L↑ (8)
ex
ellent. In
ase of a nonzero spin polarization, both (cid:2) (cid:3)
theoriespredi
tatwo-shellstru
turewithapartiallypo- The JosephIsmonax
urrent thus os
illates in phase with a
larizedsuper(cid:29)uidatthetrap
enterandafullypolarized peak value J . It is
lear from the expressions(5) and
normal state at edge. The mean-(cid:28)eld order parameters (6) that
orrelation fun
tions be
ome roughly a produ
t
aregiveninFig. 2. Thespatialvariationidenti(cid:28)es
learly oftwo∆o(xrd)e∝rparamuet(exr)svi∗f(xw)eapproximatethegapequa-
that the partially polarized phase at
enter is indeed a tion η η η . This parti
ular stru
ture
FFLO super(cid:29)uid. emphasizes thPe mi
ros
opi
origin of the GL theory of
The main observable of interest, the rate of trans- Josephson e(cid:27)e
t. Imax
A simple expression for J
an be derived when we
(cid:28)fenrereddbaytoIm(ts) f=ro<m,dNeˆ.Lg.(,t)r/idgtht>w.elIlntoanlaelfotgywetlol, siuspdeer-- aIJsmsauxm=e i(dπe/n2t)iσ
aNlNu∆ni,fowrmherBeCσSNsNup=er((cid:29)Vu02id/sπ)in[2mbo/t(h~3wEeFll)s],

ondu
tors where the (cid:29)ow of ele
trons out of the super-
I may be viewed as the
ondu
tan
e of a normal jun
tion,

ounrσrd(eAun
σtt.o+rBAeys+σtar)ebwwlihrsihetreiensgAantσhe=elet
ψrtL+ariσn
sa(fxle0r
)uHψrraRemσnti(l,xtow0)ne,iat
nhaellHeqTutha=e- owfitghapEF
anbebinegdtehteerFmeirnmedi ebnyermgye.asTurhienrgefIoJmreaxt.heInvatluhee
itPionσ[oAfσm(to)ti−onAl+σea(dt)s].toBdeNˆaLri(ntg)/dint =miin[HdTt(hta)t,NˆtLhe(t)l]in=k ptorerseesno
retotfottrhaepsnuormienrhi
oamlo
gael
nueloautisosnusp.eFr(cid:29)igu.id2s,pornoevihdaess
x0 the most promising s
heme, where a BCS super(cid:29)uid is
atP is weak so that the transfer of atoms
an be
set in the left well as a referen
e system, while the order
treated as a perturbation, we useI(tth)e=lin−eair trespdto′ns<e parameter of a FFLO super(cid:29)uid in the right well is to
theory [13℄, in whi
h the
urrent −∞
A[d+σNˆ(Lt)(,t)A/σd′t(,tH′)T+(t′A)]+σ′(>t′)0]=>0P. σσH′eRr−et∞thdet′ su<bsR
r[Aipσt(t0) i−n wbBeChiSd
heotreidsrme
roinpnea(cid:28)drr.ammeAedtsenrau,mwreeesruai
lntatliol
yfi.ptahTteehIe(cid:29)Jrmaeatfxodr∝eis,t∆rtihFbeFuLtsOipo(anxt0ioa)f,l
H
L
the average refers to the unperturbed systems and inhomogeneityofFFLOphases
anbepre
iselydete
ted
H
R
. Two
ontributions
an be easily identi(cid:28)ed: the byJosephsone(cid:27)e
tatvaryingpositionsoftheweaklink.
normal single-parti
le
urrent and the Josephson
ur- Wearenowin positionto dis
ussthe experimentalre-
rent of Cooper pairs. The expli
it expression of the lat- alization of the atomi
Josephson e(cid:27)e
t. The tunneling
4
a g = 2~2ω a /(1−Aa /a )
3d 1d ⊥ 3d 3d ⊥
jun
tion of two 1D Fermi gases sket
hed in Fig. 1a is iang=lengt~h/mω, A≃1.0326 , where
⊥ ⊥
most easily set up on an atom
hip using a
ombination and . The denominator indi-
of stati
and radio-frequen
y magneti
(cid:28)elds that forms
atesap
on(cid:28)nement-indu
edFeshba
hresonan
ethato
-
a ∼a
3d ⊥
anadiabati
doublewellpotential. Thiste
hniquehasal-
urs6when [18℄,aaso≃bs1e1rv4edexperimaen∼tal1l3y0[19℄.
3d ⊥
ready been applied su

essfully to
reate a matter-wave For Li,theba
kgωrou∼nd2π×105 nm,and nm
⊥
interferometry of 1D Bose gases [14℄. A narrow weak
orrespondingto Hz. Therefore,by (cid:28)nely
a
⊥
linkbetweenwellsmightbebuiltupbythe additionofa tuning the shape of double well potenital and hen
e ,
γ
tight dipole dimple mi
rotrap [15℄ at a spe
i(cid:28)
position. the
oupling
onstant may be varied at will.
Bytiltingthebias(cid:28)eldtodispla
ethe
enteroftheaxial
In summary, we have proposed a s
heme to realize
traps, the position of the link
an be e(cid:27)e
tively moved.
∼ 100 µ the atomi
Josephson jun
tion to dete
t the existen
e
Further, the inter-well distan
e m is su(cid:30)
ient
of FFLO phases. Our proposal opens the possibility for
largeto allowanindependent adjustmentofthespin po-

reating ultra
old Fermi gasesfor pra
ti
al appli
ations,
larization in ea
h well with a radio-frequen
y sweep [2℄.
su
h as pre
ision measurement and interferometry.
ToobservetheJosephsonos
illations,itisne
essaryto
ful(cid:28)lltwo
onditions: (i)thenumberoffermionsinvolved We a
knowledge fruitful dis
ussions with Professor P.
in the os
illations, to be measured by phase-
ontrast D. Drummond. This work is supported by the Aus-
imaging [2℄, should be larger than the (cid:28)nite opti
al res- tralianResear
hCoun
ilCenterofEx
ellen
eandbythe
olution of the imaging system, but small enough to en- National S
ien
e Foundation of China under Grant No.
sure the validity of the linear response theory; and (ii) NSFC-10574080andtheNationalFundamentalResear
h
∆x
the width of the weak link should be m2uπ
h~/sqmaller Program under Grant No. 2006CB921404.
than the period of FFLO order parameter FFLO.
Typi
ally, thNe t∼ota1l03number of atoms in one well would
be around , and the frequen
y of axial trap
ω ∼ 2π ×1
ω⊥H∼z, 2wπhi×
h1i0s5mu
h smaller than the radial [1℄ For reviews, see R. Casalbuoni and G. Nardulli, Rev.
fNreωqu≤enω
y 6Hz so that the 1D
ondition Mod. Phys.76, 263 (2004).
⊥
hold2sπ~[1/4q℄. For∼L2iπa~tovm/s,∆wit∼h t3h0esµe param- [2℄ M. W. Zwierlein et al., S
ien
e 311, 492 (2006); Nature
F 0
eters we (cid:28)nd FFLO m at an (Lodon) 442, 54 (2006); Y. Shinet al.,Phys. Rev. Lett.
γ = 1.6 v ∆
intera
tion
oupling , where F and 0 are re- 97, 030401 (2006); G. B. Partridge et al., S
ien
e 311,
spe
tively the Fermi velo
ity and BCS gap at the trap 503 (2006).
[3℄ H.HuandX.-J.Liu,Phys.Rev.A73,051603(R)(2006).

enter. This period is mu
h larger than the width of
∆x [4℄ P.FuldeandR.A.Ferrell,Phys.Rev.135,A550(1964);
dimple potential that is ∆abxou≪t se2vπe~r/aql mi
rometer, A. I. Larkin and Y. N. Ov
hinnikov. Zh. Eksp. Teor.
and therefore the
ondition FFLO is well Fiz. 47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1965)℄;
Jsat∼is(cid:28)1e0d−.3~Oωn the other hand, with a tunneling energy T. Mizushima, K. Ma
hida, and M. I
hioka, Phys. Rev.
⊥
,Ithe m∼axσimum∆ J∼ose1p0h3so−n1
urrent is es- Lett. 94, 060404 (2005).
timated to be J,max NN 0 s . Assuming a [5℄ H.MütherandA.Sedrakian,Phys.Rev.Lett.88,252503
∼0.1
times
aleofos
illation s,thenumberoftransferred (2002).
[6℄ W. V. Liu and F. Wil
zek, Phys. Rev. Lett. 90, 047002
fermionsisthenaroundhundreds,anorderofmagnitude
(2003);G.Sarma,J.Phys.Chem.Solids24,1029(1963).
smaller than the total number of atoms. This number is
[7℄ P. F. Bedaque, H. Caldas, and G. Rupak, Phys. Rev.
withinthepresentexperimentaldete
tionlimits,i.e., the
Lett. 91, 247002 (2003).
Josephsonos
illationin aBose-Einstein
ondensatewith [8℄ H. A. Radovan et al., Nature (London) 425, 51 (2003);
similarnumberofatomsandtimes
alehasalreadybeen A.Bian
hi et al.,Phys.Rev.Lett.91,187004 (2003); C.
observed [16℄. Martin et al.,Phys. Rev. B 71, 020503(R) (2005).
Before
on
luding two remarks are in order
on
ern- [9℄ B.D.Josephson, Phys.Lett.1,251(1962); K.Yangand
D. F. Agterberg, Phys. Rev. Lett. 84, 4970 (2000).
ing the temperature and intera
tions. (i) In
ontrast to
[10℄ H. Hu, X.-J. Liu, and P. D. Drummond,
the 3D
ase, the 1D FFLO state is notably stable in

ond-mat/0610448.
response to a nonzero temperature. In magnitude the
[11℄ K. Yang, Phys.Rev. B 63, 140511(R) (2001).

riti
al temperature of the FFLO state is at the same [12℄ P. de Gennes, Super
ondu
tivity of Metals and Alloys
order (i.eT., a h∼alf4o.5r4oen−eπ2t/h2iγrTd) of its unTpolarized
oun- (Addison-Wesley,NewYork, 1966).
terpart, BCS F, where F is the Fermi [13℄ For more details, see G. D. Mahan, Many-Parti
le
γ =1.6
Physi
s (Kluwer A
ademi
,NewYork, 2000), 3nd ed.
temperature. Given an intermediate intera
tion ,
TFFLO ∼ 0.10TF [14℄ T. S
hummet al., Nature Physi
s 1, 57 (2005).
we estimate a
riti
al temperature at
[15℄ D. M. Stamper-Kurn et al., Phys. Rev. Lett. 81, 2194
the trap
enter, whi
h is well above the lowest tempera-
(1998).
turereportedsofar[17℄. Weanti
ipatealowertransition [16℄ M. Albiezet al.,Phys.Rev. Lett. 95, 010402 (2005).
temperatureinsidetheweaklinkbe
auseofredu
edden- [17℄ G.B.Partridgeetal.,Phys.Rev.Lett.97,190407(2006).
γ
sity. However, it
an be mu
h enhan
ed by in
reasing . [18℄ M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).
g
(ii) In pra
tise, 1d is parameterized by a 3D s
atter- [19℄ H. Moritz et al.,Phys. Rev. Lett. 94, 210401 (2005).