Pairing Symmetry in the Anisotropic Fermi Superfluid under p-wave Feshbach Resonance

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by  Chi-Ho Cheng

Pairing symmetry in the anisotropic Fermi superfluid under p-wave Feshbach
Resonance
Chi-Ho Cheng and Sung-Kit Yip
Institute of Physics, Academia Sinica, Taipei, Taiwan
(Dated: February 6, 2008)
6 Theanisotropic Fermisuperfluidofultra-coldFermiatomsunderthep-waveFeshbachresonance
0 isstudiedtheoretically. Thepairingsymmetryof theground stateisdeterminedbythestrength of
0 theatom-atommagneticdipoleinteraction. Itiskz forastrongdipoleinteraction;whileitbecomes
2 kz −iβky, up to a rotation about zˆ, for a weak one (Here β < 1 is a numerical coefficient). By
changingtheexternalmagneticfieldortheatomicgasdensity,aphasetransitionbetweenthesetwo
n
states can be driven. Wediscuss how thepairing symmetry of theground state can be determined
a
in the time-of-flight experiments.
J

0
PACSnumbers: 03.75.Ss,05.30.Fk,34.90.+q
2

]
r I. INTRODUCTION state. This has been confirmed in [24].
e
However, in the atomic gas system, as demonstrated
h
t The trapped Bose or Fermi atomic gases [1, 2] are and explained by Ticknor et al. [19] in the context of
o usually weakly interacting since the inter-particle dis- 40K, the magnetic dipole interaction, which breaks rota-
.
t tances betweenthe atomsaretypicallymuchlargerthan tionalsymmetry,maybeimportant. (AnAlkaliatomhas
a
m theirscatteringlengths. However,ithasbeenrecognized one additional electron beyond closed electronic shell(s),
that the inter-particle interaction can in fact be tuned henceitpossessesamagneticmomentmainlyduetothis
-
d via Feshbach resonances [3, 4]. The interaction changes extra electron, though the precise value of this moment
n from weakly to strongly attractive by varying the exter- depends on the hyperfine interaction and the external
o
nal magnetic field across the Feshbach resonance. For a magnetic field). They show that, due to this magnetic
c
[ two-componentFermigas,the groundstateevolvesfrom dipole interaction, for magnetic field along zˆ, the l = 1,
aBardeen-Cooper-Schrieffer(BCS)superfluidwithlong- m = 0 resonance occurs at a higher magnetic field than
1
ranged (compared with inter-particle distances) Cooper the m = 1 ones (see also [21]) For a given magnetic
v ±
1 pairing to a Bose-Einstein condensate (BEC) of tightly field, the induced effective interaction is anisotropic. In
6 bound pairs [5, 6, 7]. This cross-over has recently been our previous paper [28] (see also [29]), we discussed the-
4 a subject of intensive theoretical [8, 9, 10, 11, 12] and oreticallytheexpectedgroundstateorderparameterun-
1 experimental [13, 14, 15, 16, 17] investigations. derthiscircumstance. Sincetheinteractionofthem=0
0
Most previous investigations deal with s-wave Fesh- channel is more attractive than that of the m = 1,
6 ±
0 bach resonance. Therefore, both the Cooper pairs and in the case that the m = 0 and m = 1 resonances
±
/ theboundstatesbetweentwofermionshaves-wavesym- aresufficiently far apart,weshowedthat the pairing can
t
a metries. Recent experiments demonstrated p-waveFesh- only occur in the m = 0 channel, and the ground state
m bach resonances [18, 19, 20, 21]. It thus raises the pos- becomes Y10(kˆ) kz. The system is either the BCS
∝
- sibility of p-wave Fermi superfluid and BEC of p-wave statewithkz CooperpairingortheBECstateofbosonic
d
“molecules” [22, 23, 24] (see also [25]). “molecules” in the m=0 channel, depending on the de-
n
o To begin,we firstrecallthe well-knownsuperfluid3He tuning. Fermion pairing or bosons in m = ±1 channel
c [26]. 3He has (nuclear) spin 1/2, in general not spin- do not exist. The orbital symmetry of the pairing is the
: polarized and basically spatially isotropic, that is, the same as the “polar” phase in the 3He literature [26].
v
i interaction between two atoms is dependent of the ori- Forthecasethattworesonancesinm=0andm= 1
X ±
entation of their relative distance. Back in the 60’s, An- channels are closed to each other, we showed that the
r derson and Morel [27] investigated theoretically the su- groundstatesymmetryis k iβk uptorotationabout
a z y
−
perfluid state for this system by assuming that the pair- zˆ,wherezˆisalongthemagneticfielddirectionandβ <1.
ing exists only between the same (say ) species. They Thisstateisthusintermediatebetweentheaxial(β =1)
↑
showedthatthegroundstatecorrespondstoCooperpair- the the polar (β =0) phases.
ing in the l = 1, m = 1 channel, that is, the pair- This paperprovidessomedetails ofourearlierinvesti-
ing wavefunction has the symmetry of the spherical har- gation [28], as well as some additional information. It is
monic Y11(kˆ) (kˆx +ikˆy) (or its spatial rotation, such organized as follows. We first study the Cooper pairing
∝
askˆ ikˆ ). Thisorbitalsymmetryisrealizedinthe3He symmetry of a weak-coupling BCS model, where the p-
z y
−
A-phase and is known as the “axial” state [26]. One can wave pairing interaction is anisotropic. We show in that
thus suspect that for a spin-polarizedbut otherwise spa- case how the analogous results cited above arise. Then
tially isotropic system, the pairing is again in the Y (kˆ) we return to the slightly more involved case of Feshbach
11
2
resonance and determine the general phase diagram for l=1 partial waves,
this case. We first provide the results for the case where
thesystemcanbedescribedbyaneffectivesinglechannel ∆~k = ∆f(kˆ)
model,i.e.,purelyintermsofatomsinteractingwitheach m=1
otherviaatwo-bodyinstantaneousinteraction. Thenwe = ∆ fmY1m(kˆ) (7)
study the necessary modifications to these results when m= 1
X−
this approximation is relaxed. Finally a probe of the
with normalization f 2 =1. Substituting Eqs.(6)-
pairingsymmetrybasedonthe time-of-flightexperiment m| m|
(7) into Eq.(5), we have
is discussed. P
Y (kˆ )f(kˆ )
II. LESSONS FROM ANISOTROPIC BCS MODEL f(kˆ)=πX~k′ Xm Vm(k,k′)Y1m(kˆ)[(ǫk′ −µ1∗)m2+′∆2|f′(kˆ′)|2]21
(8)
In the spirit of the work of Anderson and Morel
[27], we first consider a weak-coupling BCS model of a Suppose further thatVm(k,k′)=Vm is independent ofk
Fermisystemwithonespinspeciesundertheinteraction and k′, Eq.(8) becomes
V . (WeshallhoweverstudythecasewherethisV is
a−ni~sko−t~kr′opic, see below). The Hamiltonian of our system f =πV Y1∗m(kˆ)f(kˆ) (9)
is H =Hf +HV, where m mX~k [(ǫk−µ)2+∆2|f(kˆ)|2]12
Hf = (ǫk−µ)a~†ka~k (1) We perform the integration in~k in the following way,
X~k
Λ
ρ dǫ dΩ (10)
0
→
HV =− V~k ~k′aq†~+~k′aq†~ ~k′aq~ ~kaq~+~k X~k Z−Λ Z
~kX,~k′,q~ − − − whereρ0 isthedensityofstatearoundtheFermisurface,
(2) Λ is the energy cutoff, and dΩ is solid angle. Then we
get totally 3 nonlinear equations,
Hereǫ =~2k2/2M,andµisthechemicalpotential,and
k
av~ekcitsorth~ke.anSntaihnidlaatridonBoCpSertahteoorrfyorparoFpeorsmesiotnhewiftohllowwaivneg- fm =2πρ0Vm fmlog(2∆Λ)− dΩY1∗m(kˆ)f(kˆ)log|f(kˆ)|
(cid:20) Z (cid:21)
trial ground state wavefunction (11)
for m= 1,0,1.
|Gi= (u~k+v~ka~†ka†~k)|0i (3) Wewo−uldliketofindthesolutiontoEq. (11)withthe
k −
Y
anisotropy characterized by the ratio V /V . Particular
1 0
with normalizationcondition u 2+ v 2 =1. Minimiz- solutions to Eq.(11) can be easily found at two limits.
| ~k| | ~k|
ing the ground state energy hG|H|Gi with respect to u~k When V1/V0 = 0, ∆~k ∝ Y10(kˆ) ∝ kz because no other
and v~k gives the excitation spectrum pairing can be possible. At the other limit, V1/V0 = 1,
the system is again isotropic the ground state pairing is
E~k = (ǫk−µ)2+|∆~k|2 (4) ∆by~kf∝’Ys1,1f(okˆr),Vor/Vits=rot0a,tifons=. I1n,fterm=sfof th=e l0an(guupagtoe
q m 1 0 0 1 −1
where the energy gap ∆~k satisfies asolguatuiogne itsrafns=for1m/√at2io,fn).=Ffor V=1/1V/02 s=uc1h,tahaptarticular
0 1 1
−
∆
∆ = V ~k′ (5) 1 1
~k ~k ~k′2E ∆ Y (kˆ)+ (Y (kˆ)+Y (kˆ))
X~k′ − ~k′ ~k ∝ √2 10 2 11 1,−1
k ik (12)
In case the interaction is p-wave, only the l = 1 compo- ∝ z − y
nent survive in V~k−~k′. We write NotethatitdoesnotcontradictwiththepairingY11(kˆ)∝
k +ik since these states are degenerate in an isotropic
x y
m=1
V~k ~k′ =2π Vm(k,k′)Y1m(kˆ)Y1∗m(kˆ′) (6) lpimariatmVe1ter=0 V<−1V=/VV0<. 1W, iwthe wgrilaldsuhaollwy btuenloiwngththaet
− 1 0
m= 1
X− ∆(kˆ) k iβk up to a z-axis rotation with β <1.
z y
∝ −
We shall study the situation where V = V V . We BeforesolvingthegapequationEq.(11)forthegeneral
1 1 0
− ≤
also decompose the energygapin sphericalharmonicsof intermediate values of the pairing 0 < V /V < 1, it is
1 0
3
helpful to first identify the critical V /V such that f
1∗ 0 1
andf starttodeviatefromzero(i.e.,whenthepairing 1.00
1
starts−to deviate from Y10). Let us take the gauge where 00..011
f0 is real. Linearizing Eq.(11) in f1 and f 1 around the 0.95 01..50
− 5.0
point (f ,f ,f )=(1,0,0), we have 10.0
0 1 1
− 0.90
1 f A A f
1 = 1 2 1 (13)
2πρ0V1∗ (cid:18)f−∗1 (cid:19) (cid:18)A2 A1 (cid:19)(cid:18)f−∗1 (cid:19) f00.85
whereA1 =log(2Λ/∆) dΩY11 2log Y10 1/2,A2 = 0.80
− | | | |−
1/2. Thus when we increase V , the first non-trivial so-
1
R
lution for f1 and f 1 is obtained when (f1,f∗1) gives 0.75
the largest eigenval−ue of the matrix. It can −be easily
seenthatthis correspondsto thesolutionf =f ,with 0.70
1 ∗1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
eigenvalue A +A . The critical value V thus s−atisfies V/V
1 2 1∗ 1 0
1 2Λ
= log( ) dΩY 2log Y (14)
11 10
2πρ V ∆ − | | | |
0 1∗ (cid:20) Z (cid:21) FIG. 1: f0 as a function of V1/V0 for ρ0V0’s given in the
On the other hand, at this point (from Eq. (11) with legend.
m=0),
H isforthemoleculesinthe“closed”channels,withb
1 2Λ b q~,m
= log( ) dΩY10 2log Y10 (15) the annihilation operator for a Boson with angular mo-
2πρ V ∆ − | | | |
0 0 (cid:20) Z (cid:21) mentum l = 1 and z-axis projection m with momentum
Eliminating log(2Λ/∆) in Eqs.(14)-(15) gives ~q,
1 ǫ
V1∗/V0 = 1+2πρ0V0 (16) Hb = 2q +δm−2µ bq†~,mbq~,m (18)
Xq~ (cid:16) (cid:17)
For V /V <V /V <1, f becomes finite. Provided
1∗ 0 1 0 1
no additional phase transiti±ons occur, we expect then Here δm’s are the (bare) detuning of the l = 1, m res-
the relation f1 = f∗1 remains valid. Furthermore, our onance. Hα represents the Feshbach coupling, which is
system is rotationa−lly invariant around zˆ. Under this a generalizationof the ones already commonly employed
rotation, f f eiφ but f f e iφ where φ is the [8, 30]fors-waveFeshbachresonancestothe caseofsev-
1 1 1 1 −
rotational an→gle. By suita−ble→cho−ice of φ we can thus eral l =1, m=0, 1 closed channels [23, 24]:
±
make both f and f = f real and positive. We shall
confine ourse1lves to−th1is ca1∗se without loss of generality. Hα = ϕ∗m(~k)bq†~,ma ~k+q~/2a~k+q~/2+h.c. (19)
−
Under this choicethe orderparameterhasthe formkz mX,q~,~k
−
iβk .
y
Hence, when V /V < V /V < 1, we search for the whereϕ (~k)= i√4πkY (kˆ)α . ThefactorY (kˆ)re-
1∗ 0 1 0 m −L3/2 1m m 1m
solution f > 0,f = f > 0 under the normaliza- flectsthesymmetryofthel,mboundstateandthelinear
0 1 1
tion condition f2 + 2f2 −= 1. The solution f (where factor in k arises from the small momentum approxima-
0 1 0
f = f = (1 f2)/2 ) for different V /V and ρ V tion for the coupling. α is the corresponding coupling
is1prese−n1ted in Fi−g.10. For given ρ V , the1 pa0iring is0k0 constant, it has dimensiomn [E][L]5/2. More precisely α
p 0 0 z m
(f =0)atsmallV /V until the latter exceedsthe criti- can be an analytic function of k. For low energy (den-
0 1 0
calV /V . BeyondthiscriticalV /V ,f decreasesgrad- sity) phenomena, we shall however only need its value
1∗ 0 1∗ 0 0
ually with increasing V /V . At V /V = 1, f = 1/√2 α α (k =0).
1 0 1 0 0 m ≡ m
for all ρ V ’s meaning that the pairing becomes k ik We shall apply mean-field theory to the Hamiltonian
0 0 z y
−
(corresponding to β =1). H. In this approximation, we discard all terms that in-
volve finite momentum Bosons b with ~q = 0, and re-
q~,m
6
gardb as c-numbers. It is then convenientto intro-
q~=0,m
III. P-WAVE FESHBACH PAIRING duce
i√4π
We return to the case of p-wave Feshbach resonance.
D = α b (20)
WebeginwiththetotalHamiltonianH =Hf+Hb+Hα m − L23 m 0,m
where again Hf is for the free fermions, ∆~k = DmkY1m(kˆ) (21)
m
Hf = (ǫk−µ)a~†ka~k . (17) X
Hence the mean-field Hamiltonian for the Fermions be-
X~k
4
comes 0.03
2 (a) m = 0 (b)
Hmf = (ǫk−µ)a~†ka~k+(∆~∗ka ~ka~k+h.c.) (22) m = 1
X~k − 1 0.025
Tmhaitsiocna,nwbhhaei−csh~koalgv~kieivde=sv−ia2t[h(ǫekf−amµi)l∆i2a~k+r B|∆og~ko|2li]u12bov trans(f2o3r)- -5-3- 1 / v (10a)m0-01 -1- c (a)m000.0.0125
Minimization with respect to the c-numbers b give
q~=0,m
-2
b0,m(δm−2µ)=− ϕ∗m(~k)ha ~ka~ki (24) 195 200 205 0.01195 200 205
X~k − B (G) B (G)
which can be written as,
(δm−2µ)Dm = L2π3 X~k (ǫ|αkm−|2µk)Y21∗+m(|∆kˆ)~k∆|2~k 12 (25) FR|v9aIe/nGf2.i[,.s1h−927]a:/tfo2triPhh−eloy1rtpe/esvoormffinnaatnehntsedtfia−efitletdc.mstvBo1fom∗=rt,h4vw0e−Kit1ehxaapnBtoed0∗mric>ms1ei=Bnn±t∗cta1−hl.e1.r|ef−s,u1ml/tfvsimo=’sf
TheR.H.S.ofthisequation(cid:2) isformallydiverg(cid:3)ent(This
divergenceisonlyformal: thesummayactuallyconverge
(p~,ω) is the Greens function for free Fermions at
ifthekdependencesofα (k)areincluded)Nevertheless, f
m G
wavevector p~ (p has dimension of [L] 1), frequency ω,
itcanbe”renormalized”byre-expressingthisequationin −
terms of parameters that enter the two-body scattering and (m)(E) is the Greens functions of free m-Boson at
Gb0
amplitude (c.f. [24]andalso the theoreticalreferencesin zero momentum ~q = 0 and energy E. Here we have al-
thes-wavecase[5,9,10,11,12].) Letusthenconsiderthe ready made use of a simplification due to the fact that
scattering of two particles in vacuum. We denote f~k(~k′) Π(mm′) vanishes for m 6= m′ (see Eq. (29)), hence there
asthescatteringamplitudefortwoincidentparticleswith are no cross-terms between different m’s in Eq. (28).
relative momentum ~k and out-going momentum ~k . We Substituting Eqs.(28)-(29) into Eq.(27), and using the
′
can define f (k) (do not confuse this f (k) with those definition (26), we get
m m
in Sec II) via M k2 2ǫ δ 1 p2
k m
= −
f~k(~k′)= fm(k)(4π)Y1m(kˆ′)Y1∗m(kˆ) (26) − 4π~2fm(k) |αm|2 − L3 Xp~ 2ǫk−2ǫp+i0+
Xm (30)
This equation is the generalization of the corresponding
Thesumoverp~intheR.H.S.formallydiverges. However,
one in scattering theory to the case of anisotropic inter-
we are interested only on its dependence on k at small
action (but still with rotational invariance around zˆ). It
k’s. We thus expand the R.H.S. in powers of k:
reduces to those in standard quantum mechanics text-
books[31]iffm(k)is independentofm. f~k(~k′)is related 2ǫk δm 1 p2
to the T-matrix by α−2 − L3 2ǫ 2ǫ +i0+
m k p
| | p~ −
ML3 X
f~k(~k′)=−4π~2T~k′,~k(E =2ǫk+i0+) (27) = −( αδm2 − M~2 L13 1)
m
The T-matrix can be evaluated in the standard way: | | Xp~
~2 M 1 1 M
T~k′,~k(E) = ϕm(~k′)Gb(0m)(E)ϕ∗m(~k) +(M αm 2 + ~2 L3 p2)k2−i4π~2k3 (31)
Xm (cid:2) | | Xp~
+ϕ (~k ) (m)(E)Π(mm)(E) (m)(E)ϕ (~k)+
m ′ Gb0 Gb0 ∗m ···
= ϕ∗m(~k′)ϕm(~k) (28(cid:3))theIngtehneerlaolwr-eesnuelrtgfyorlimthiet,lEowq-.(e3n0e)r-g(y31s)cashttoeurlidngrebdeutwceeetno
E δ Π(m)(E)
m − m− a pair of particles. Using the same notation adopted in
X
where Ref.[19], the scattering amplitude is parametrized as
Π(mm′)(E) = Xp~,ω ϕ∗m(p~)Gf(p~,ω)Gf(−p~,E−ω)ϕm′(p~) fm(k)= vm−1+ckm2k2 ik3 (32)
− −
= ϕ∗m(p~)ϕm′(p~) . (29) wherethedimensionsofvm andcmare[L]3and[L]−1,re-
E 2ǫ
p spectively. Themagneticfielddependentparametersv ,
p~ − m
X
5
c areinprincipleavailableexperimentally. Anexample where
m
is as shown in Fig.2. Our renormalization scheme is to
absorb the bare parameters δ and α into the physi-
m m
cal (renormalized) parameters v , c by identifying the
m m
equivalence between Eqs.(30)-(31) and Eq.(32). Hence
1 4π~2 δ M 1
m
= ( 1) (33)
− v M α 2 − ~2 L3
m m
| | p~
X
4π~2 ~2 M 1 1
c = ( + ) (34)
m − M M α 2 ~2 L3 p2
m
| | p~
X
In Eq. (33) and (34), if the p dependence of α ’s
m
had been included, the sums involving p~ should have
the extra factors |αm(p)|2/|αm(0)|2. When treating the h(~k) k2 k2(1+ µ |∆~k|2)
m”dainveyr-gbeondty” psurombslemar,e aasgawine shallαse(ep)b2e/loαw,(t0h)e2 oannldy ≡ (ǫk−µ)2+|∆~k|2 1/2 − ǫk ǫk − 2ǫ2k
α (p)2/α (0)2p2. We shp~a|llmappl|y E|qm. (33|) and (cid:2) (cid:3) (36)
p~| m | | m | P
(34) to eliminate these sums in favor of v and c . Af-
m m
P
terthisprocedure,lowenergyphenomena(involvingonly
small p’s) can be parameterized entirely by v , c and
m m
α (0). Tosimplifyourpresentation,weshallnotinclude
m
explicitly the factors α (p)2/α (0)2 in our calcula-
m m
| | | |
tions below.
Returning to Eq.(25), we rewrite its R.H.S as
2π |αm|2kY1∗m(kˆ)∆~k
L3 (ǫ µ)2+ ∆ 2 21
X~k k− | ~k|
= 2π|αm|2(cid:2) Dm′Y1∗m(kˆ)Y(cid:3)1m′(kˆ)k2
L3 (ǫ µ)2+ ∆ 2 12 The sum involving h(~k) in Eq. (35) is convergent. For
~kX,m′ k− | ~k|
the rest of the terms, their divergences are the same as
= 2π|Lα3m|2 D(cid:2)m′Y1∗m(kˆ)Y1m′(kˆ)(cid:3) in either Eqs. (33) and (34). These sums can thus be
~kX,m′ expressedintermsofthephysicalparametersvm andcm.
Using Eq.(35), the ”gap equation” Eq.(25) can therefore
k2 µ ∆ 2
h(~k)+ (1+ | ~k| ) (35) be rewritten as
× ǫ ǫ − 2ǫ2
(cid:26) k k k (cid:27)
M 1 2Mc µ 3M M2c 1
4πD0(−v0 + ~20 )− 10π(4π~40 + α0 2) D0(2|D1|2+3|D0|2+2|D−1|2)−2D0∗D1D−1
| |
(cid:2) 2π (cid:3)
= D Y (kˆ)Y (kˆ)h(~k) (37)
L3 m 1∗0 1m
~Xk,m
M 1 2Mc µ 3M M2c 1
D ( + 1 ) ( 1 + ) D (3D 2+2D 2+6D 2) D D2
4π 1 −v1 ~2 − 10π 4π~4 α1 2 1 | 1| | 0| | −1| − −∗1 0
| |
(cid:2) 2π (cid:3)
= D Y (kˆ)Y (kˆ)h(~k) (38)
L3 m 1∗1 1m
~Xk,m
and a corresponding equation with m = 1 m = 1. Eq.(37)-(38) are to be solved under the constraint from
↔ −
6
the number equation into account higher order terms in k2 in denominator of
f (k).
m
1
n= L3( ha~†ka~ki+2 |b0,m|2) (39) Moreover, as explained in Ref.[19], due to the dipole
X~k Xm interaction,B0∗ >B1∗ =B∗1, ascanbe seenagaininFig
2. Thusinthefieldrange−ofinterest, 1/v = 1/v >
1 1
The first term is from open-channel atoms whereas the − − −
1/v . We can say that, at a given field, the effective
0
second term is from the closed-channel molecules. From −
interaction between the Fermions is less attractive for
BCS theory,
relative angular momentum projections m = 1 than
±
m=0.
1 ǫ µ
k
ha~†ka~ki= 2 1− [(ǫk−µ)2−+|∆~k|2]21! meSnitnucme tphreoijnetcetrioanctsiomn=is le1ss,faotrtrsaucffiticvieenftolryalnarggueladriffmeor--
±
ence between 1/v and 1/v we expect (and verify
The first term in Eq. (39) is thus again formally diver- − 0 − ±1
gent: ha~†ka~ki∼ |∆4ǫ~k2k|2 atlargek. However,thissumitcan bwealvoew.)Wtheatthuthsefiprsatirbinegginisoeunrtiarnelaylyisnistbhye amss=um0inpgarthtiaatl
betreatedinanalogousmannerasEq. (35)byemploying only D is non-vanishing.
0
again Eq. (34). The result is
1 ǫ µ ∆ 2
n = (1 k− | ~k| )
2L3 X~k − [(ǫk−µ)2+|∆~k|2]21 − 2ǫ2k 1
1 (M2cm + 1 )D 2+ 1 |Dm|2 0.8 ~µ
−4π 4π~4 α 2 | m| 2π α 2
m m
Xm | | Xm | | 0.6
(40) ~
0.4 D0
Eqs.(37), (38), and (40) are our principal equations,
0.2
with parameters characterizing the Feshbach resonances
entirelyintermsofv ,c and1/α 2. Theseequations
m m m 0
| |
determinetheorderparametersD andchemicalpoten-
m
tial µ for given density n and “interaction parameters” -0.2
v , c and 1/α 2.
m m | m| -0.4
-400 0 400~ 800 1200
-1 / v
IV. BEC-BCS CROSSOVER AND QUANTUM 0
PHASE TRANSITIONS
GenerallywithFeshbachresonanceforthesub-channel FIG.3: Thedimensionlessparameters D˜0 andµ˜ asfunctions
m, 1/v is field dependent, vanishing at the resonant of −1/v˜0. c˜0 = c˜1 = −100 and D˜±1 = 0 in this case. The
field Bm. In contrast, c has a definite sign (c.f. Fig. dashed line represents −ǫb/2ǫF.
m∗ m
2). For the ease of discussions, we shall assume that
cm < 0 and field independent, 1/vm is an increasing For simplicity, we shall first drop the terms with ex-
−
tfuhnecctaiosne ooff4fi0eKld. (−T1h/isvmcor>res(p<o)n0dfsortoBth>e c(a<se)Bwm∗h,eraesδimn (p3li7c)i,t(13/8|)α,m(|420)faacrtoerssa.mIfetahsisaiscovarrlieds,ptohnedninogurmeoqduealtcioonns-
is an increasing function of field and αm weakly field sisting only of Fermions interacting among each other
dependent, c.f. Eqs.(33) and (34)). For B < Bm∗ , a with an interaction such that the corresponding two-
bound state appears. The energy of this bound state body scattering amplitude parameters are given by the
is given by −ǫb,m = −~2κm2/M with k = iκm being a same cm and vm. In analogy with the s-wave literature,
pole for fm(k), that is, −1/vm −cmκ2m −κ3m = 0. For we shall call this the single-channelapproximation(note
given m and small detuning below that resonance,κm is therearestillthree(m= 1,0,1)Feshbachresonances).
small and is given by κ2m = 1/[(−cm)(vm)]. Since 1/vm The effect from finite 1/|−αm|2 will be discussed at the
should be roughly linear in B near the resonance, ǫb,m end of this Section.
increases linearly with (B B) (in contrast to s-wave,
m∗ − With these simplifications, Eq. (37) and (40) become
whereitisquadratic;seealso[24])Theseresultsforsmall
detuning apply provided κ << ( c ), or equivalently
t1h/ivsmre≪gim(e−fcomr)3n.egWatieveshdaelmtluanliwnagys−s(ecvmoennfinwehoeunrsweelvwesrittoe 4Mπ(−v10+2M~2c0µ)+409πM2~4(−c0)|D0|2 = L2π3 ~Xk,m|Y10(kˆ)|2h(~k)
B B ). Forlargernegativedetuningsweneedtotake (41)
≪ m∗
7
and We have also performed calculations for other values
of c˜ . The size of the crossover region is roughly pro-
0
n = 1 (1 ǫk−µ |∆~k|2) portional to the value of c˜0 (which is in turn inversely
2L3 X~k − [(ǫk−µ)2+|∆~k|2]21 − 2ǫ2k p(nrootposrhtoiownna)l,ttohtehceodrerensspitoyn)d.iFnogrreexsaumltsplcea,nfobrec˜0ca=pt−u2r0ed0
+(4πM)22~4(−c0)|D0|2 (42) bwyel1l/b√y2reipnlaFciign.g3.the x-axis by −1/2v˜0 and dividing D˜0
The above behavior applies only to sufficiently large
roeussplyecstiimveillya.rEtoqst.(h3e7)s-awnadve(4c0a)scea.nBbyegsoaluvgeedisnivmaurlitaanncee-, s−mv˜a±−l11l,−D˜(−v˜w0−i1ll)b>ec0o.mWefihneintet.hSisimdiilffaerrteontcheeistrseuaffitmcieennttilny
we shall choose D0 to be real without loss of general- Section I±I1for the anisotropic weak-couplingBCS model,
ity. It is convenient to express the results in dimension-
less form. We thus define µ˜ µ/ǫ , D˜ D /~v , we first identify the critical v˜∗−11 such that D1 and/or
vc˜mF ≡≡~nk−F/1/M3c,ma,nadnkdF3v˜≡m6≡π2nnv.≡mThwehreFerseuǫltFms ≡a≡re~a2skmF2s/h2oMwFn, rDgeo−sou1dltfiasrpsbpterlsootwaxrimtwtaeotsidhoeanvllaiatttaelkeefarscot0mf=oz±rec4r±0o1K..FTnoehraisrsimitshpselaircteiistsfiyo,enidannttocheaes
in Fig. 3 (for the case c˜ =c˜ = 100,see below for the
0 1 − (see Fig 2). We linearize Eqs.(37)-(38) (and the latter
reason of this choice).
with m=1 m= 1) in D and D . We obtain
1 1
In the BCS regime, corresponding to “large” external ↔ − −
lamenfatdghnD˜aent0idc≃fisied0le.doBWf Ee≫qc.Ba(4n0∗1)itg.hnuTosrhe−isv˜teh0−qe1u→taetriom∞n,∝twhee|nDh0are|v2deuoµ˜cne≃stht1oe, − 4πMv1∗ (cid:18)DD−∗11 (cid:19)=(cid:18)AA12 AA21 (cid:19)(cid:18)DD−∗11 (cid:19) (44)
the correspondingoneinSec II withthe effectivepairing where
interaction V in the m = 0 channel there proportional
0 2π 2π
to (−1/v0 + 2Mc0µ/~2)−1 = (−1/v0 + c0kF2)−1. This A1 = L3 |Y11|2h(~k)− L3 D02|Y10|2|Y11|2g(~k)
combinationfollows from the factor (δ0 2µ) on the left X~k X~k
−
hand side of Eq. (25). M2c 3M
1(µ D2) (45)
s−tav˜I0n−nt1.t→hIne−EB∞qE.,C(µ41r≃e)gw−imeǫbec,,0a/nc2oi,rgarnenosdrpeoD˜an0ldliatnpegprmrtoosaecBhxpesl≪ictoitBain0∗co∆on~kr- A2 = −L2π32π~2D02|−Y101|02π|Y~121|20g(~k)− 430Mπ23~c14D02 (46)
or D0. Writing κ=(2M µ)1/2 and performing the sum X~k
opstvaaertrien,~kg,wtwheeisfifinwndidthtµhtaht=eκeǫqoub|a/et2y|isoan−sf1oc/lrav0itmh−eedct.0wκo2In=boEdκqy3..bCo(o4um2n)d-, g(~k) ≡ k24([(ǫk−µ)2+D102|Y10|2kz2]3/2 − ǫ13k) (47)
b0
| |
w(ǫekc+an|µe|)x−pa|n∆d~kt|2h/e2s(qǫkua+re|µ-r|o).otW[(eǫkth−enµ)ge+t|∆~k|2]−1/2 ≈ H0.ereWhe(~kca)nisvaesridfyefitnheadtiAn2E>q. 0(3.6)TbhuertewfoirtehtohnelylaDrg0e6=st
eigenvalue for the A-matrix, hence the smallestvalue for
M2 3 v , belongs to the class D = D . The corresponding
n= D 2 ( c ) κ (43) 1∗ 1 ∗1
(4π)2~4| 0| (cid:20) − 0 − 2 (cid:21) critical value is then given by −4−πMv1∗ =A1+A2. On the
other hand, from Eq.(37), we have
where κ was defined above. For small detuning κ
≪
( c ), i.e. 1/v˜ c˜ 3, a relation well-satisfied in the M 2π M2c 9M
(r−3a2nπg0e/3i)n1/F3(ig c˜300)a−n≪1d/2o|.t0h|ers below, we then obtain D˜0 ≃ − 4πv0 = L3 X~k |Y10|2h(~k)− 2π~20(µ− 20π~2D02)
− (48)
The “cross-over” behavior in Fig.3 is analogous to
the s-wave case, where the corresponding x-axis is x =
Combining these two equations and re-writing it in di-
1/(n1/3a ) where a is the s-wave scattering length.
− s mensionless form, we have
Note here D has the dimension of [E][L] 1 and behaves
0 −
rdaiffteesrernattlhyerfrtohmanthcoenst-iwnuaveeto∆inicnrethaseeB. RECigolrimouits:lyitspseaatuk-- −v˜1 + v˜1 = 3(65ππ2)23D˜02(−c˜0)
ing, we actually expect a (quantum) phase transition at 1∗ 0
the pointwhere µcrosseszero [7, 22]: the systemis gap- ∞ 1 x4(1 3y2)
+9π dx dy −
less for µ>0 but gapful if µ<0. However,we found no Z0 Z−1 [(x2−µ˜)2+ π3D˜02x2y2]1/2
evidentchangesofslope in µand∆ when this transition (49)
is crossed, and thus expect this phase transition is very
weak thermodynamically. Similar results were found in Inthe BCS limit, D˜ 0,the firstterm in the R.H.S.
0
→
Ref [24] for the axial (D =0, D =D =0 state). ofEq.(49)isnegligiblewhereasthesecond,whichweshall
1 0 1
6 −
8
call K , is finite. The dominant contribution to K oc- 34
1 1
curs near x √µ˜, where the integrand diverges if D˜ 0
were zero exa≃ctly. To evaluate K1 in the limit D˜0 00, 00..0024
we add and subtract the integral → 33 0.06
~v0 0.08
K2 ≡9πZ−ξξ00dξZ−11dy2[ξ2(1+−π33D˜y022y)2]12 (50) ~1 / v + 1 / 132 0.1
-
where ξ x2 µ˜, and ξ D˜ is an arbitrary cut-off. 31
0 0
≡ − ≫
The integrand for K K now has no divergence near
1 2
x2 µ˜ 0, and their−difference vanishes in the D˜ 0
0
lim−it du≈e to the y integral. K2 can be evaluated by→first 30 -600 -400 -200 - 1 0/ ~v 200 400 600
integrating with respect to ξ, we get 0
1 ξ0 (1 3y2)
9π dy dξ − FIG. 4: Contour plot of β as a function of −1/v˜1+1/v˜0 and
Z−1 1 Z−ξ0 2[ξ2+ π3D˜02y2]21 t−o1t/hv˜e0cforirtic˜c0al=vac˜l±u1e=−1−/v˜11∗00+. 1T/hv˜e0.line for β = 0 corresponds
= 18π dy (1 3y2)logy
− −
Z−1
= 12π (51)
34
Ingoingfromthefirsttothesecondline,wehaveusedthe 0
0.02
fact that 1 dy (1 3y2)=0,hence the terms involving 0.04
1 − 33 0.06
ξtsht0aeIannBnttdCh≃SeD˜Bl0R(i−m3Ed2Cirπto/plwi3me)o1iuo/ttb,3(tri−aneicc˜nta0hl)−le−inv˜1ξg1∗/0−2t≫1hwa+thD˜v˜eD˜00r−e01laiams→piµp˜t1.r2oiπTsach=hleaerr3seg7faeo.7rc.eaonnind- ~~1 / v + 1 / v1032 00..018
negative, we find that the second term in the RHS of -
Eq. (49) is negligible compared with the first (provided 31
κ ( c )). Using the value for D˜ , Eq. (49) yields
0 0
−rtieov˜g≪n1∗im−o1ef,−+−thv˜v˜0−e0−1n1uism→sehroi4wc8anπl/are5sstu=hltest3fh0oi.rc2k.−bv˜l1I∗an−c1kt+hlienv˜e0−in(1tD˜ear0sm=aefd0ui)naitcne- 30 -1200 -800 -400 - 1 0/ v~0 400 800 1200
Fig.4.
We now solve for the order parameters D when
m FIG. 5: Contour plot of β as a function of −1/v˜1+1/v˜0 and
−v˜1−1 +v˜0−1 is less than the critical value. As already −1/v˜0 for c˜0 =c˜±1 =−200.
mentioned, we have already made use of gauge invari-
ance to choose D to be real. Under this choice, the
0
solutions we found belong to the class D =D . Writ-
1 ∗1
ing D = D eiχ, ∆ then has the angula−r depen- For c˜0 = 200, the results are qualitatively the same
1 | 1| ~k providedwe−double the value of 1/v˜ . The correspond-
dence ∝ D0kˆz + i√2|D1|kˆ · aˆ ∝ (kˆz − iβkˆ · aˆ) where ing plot is shown in Fig.5. − 0
aˆ = (cosχ)yˆ+ (sinχ)xˆ is a unit vector perpendicular
In the intermediate splitting regime, we thus predict
to zˆand β = √2D /D . A particular solution is given
1 0
| | the state is k iβk on the BCS side, whereas it is
bythecasewhereD1andD 1arebothrealwhereaˆ=yˆ. z − y
Other solutions are simply r−elated to this one by a rota- kz on the BEC side. For large positive detuning, the
splitting should be less relevant and the pairing state
tion about zˆ. Without loss of generality, we shall there-
should resemble more that of the isotropic system. On
fore assume that D ’s are all real below.
m
the BEC side, the system should be closer to a Bose
The contour plot of β (by solving Eq. (37)-(40) with-
condensate of lowest energy molecules (kˆ ).
out the 1/α 2 terms) is shown in Fig.4. (see Fig. 2 of z
m
| |
[28] for a plot of D ) β monotonically increases down-
1
±
wards or towardsthe rightin this diagram. In the β =0
(D =0)phase,thestateisrotationallyinvariantabout
1
±
zˆ, whereas this symmetry is broken in the β = 0 phase.
6
Thereisa(quantum)phasetransitionbetweenthesetwo
phases when one crosses the critical line −v˜1∗−1+v˜0−1.
9
For the 40K case studied in Ref.[19], the Feshbach res- In the above treatment, we have assumed that the
onances are at B 198.8G and B 198.4G. There, terms with explicit 1/α 2 factors in Eqs.(37)-(40) can
0∗ ≈ 1∗ ≈ | m|
c and c are both only weakly field dependent and are be dropped. The validity of this assumption has to be
1 0
approximatelygivenby−0.02a−01(seeFig2). Ourchoice investigated by first-principle calculations of two-atom
of c˜ = 100 above corresponds to a density of roughly collisions. Following Ref [37], we define the quantity
1g0o−fr1o1am−0−03 =to61.7co×rr1e0s1p3ocnmd−s3t.oTrohuegrhalnyg0e.5oGf−if1/wv˜e0tfaokreµ˜thtoe M2 αm 2 1
η | | . (52)
samegasdensity. (Thisfieldrangeisproportionalton2/3 m ≡ ~4L3 p2
). Near the resonantfields, −v1−1+v0−1 ≈2.1×10−8a−03 Xp~
and is roughly field independent. Thus the density de- η quantifies the relative strength of coupling between
m
termines the values for both c˜0,1 and −v˜1−1+v˜0−1 while the m closed-channel and the continuum. With this no-
varying the magnetic field corresponds roughly to mov- tation, Eq. (34) becomes
ing alongahorizontalline onourphasediagramofFig.4
(with increasing field towards the right and the distance 4π~4
ofthe line fromthe horizontalaxisproportionalto n−1). cm =−M2 αm 2(1+ηm) . (53)
For the density cited above, −v˜1−1+v˜0−1 ≈ 2000, hence | |
we expect only the k phase to be observed. To observe The single channel approximation used above corre-
z
thephasetransition,weneeda40K gasofamuchhigher sponds to η . We now investigate the effect of
m
→ ∞
density, or another gas species with much smaller split- finite η on the many-body problem. Expressing the
m
ting between the m=0 and m= 1 resonances. Eqs.(37),(38),and(40)byη insteadof1/α 2,weget
m m
± | |
M 1 2Mc µ η 3M3c
D ( + 0 ) 0 0 D (2D 2+3D 2+2D 2) 2D D D
4π 0 −v0 ~2 − 1+η040π2~4 0 | 1| | 0| | −1| − 0∗ 1 −1
(cid:2) 2π (cid:3)
= D Y (kˆ)Y (kˆ)h(~k) (54)
L3 m 1∗0 1m
~Xk,m
M 1 2Mc µ η 3M3c
D ( + 1 ) 1 1 D (3D 2+2D 2+6D 2) D D2
4π 1 −v1 ~2 − 1+η140π2~4 1 | 1| | 0| | −1| − −∗1 0
(cid:2) 2π (cid:3)
= L3 DmY1∗1(kˆ)Y1m(kˆ)h(~k) (55)
~Xk,m
n = 1 (1 ǫk−µ |∆~k|2) The value of η0 mainly affects the results on the BEC
2L3 X~k − [(ǫk−µ)2+|∆~k|2]21 − 2ǫ2k siniddee.penIndenfatcto,f dηe,epasinextpheectBedCSsinlcimeiatllthateomressualrtes baare-
0
M2 2+η
m D 2c (56) sically in the open-channels. µ˜ is still basically linear
−(4π)2~4 m 1+ηm| m| m in 1/v˜0 with the same slope, and thus the width of
X
the ”crossover” region remains to be determined by c˜
0
We note here that the last term of Eq.(56) involves a only (but not η0). In the BEC limit, when η0 decreases
factor (2+η )/(1+η ). This term actually arises from from , the contribution from bound state molecules
m m ∞
two contributions. The first one, involving 2/(1+η ), becomes more important (see in particular Eq. (56)
m
is due to the closed-channel molecules (the last term in and the discussions below it). The chemical potential
Eq.(40)), and a second one, involving η /(1+η ) due µ˜ becomes closer to the two-body value for decreasing
m m
to the open-channel atoms. η0 as shown in Fig. 6(b). D˜0 has the limiting value
(32π/3)1/3( c˜ ) 1/2[(1 + η )/(2 + η )]1/2, and thus
0 − 0 0
≃ −
decreases with decreasing η .
0
Thephasetransitionbetweenthek stateandthek
z z
−
iβk state can be investigated as before. The value for
y
resTuoltsshfoowr Dt˜heaenffdecµ˜t oinf fithneitestηa0t,e wD˜e plo=t t0heingeFniegr.6a.l B−Cv˜1∗S−l1im+itv˜.0−I1nisthneoBtEaffCeclitmedit,biyf ηthe=vηalue ofηη,mw’esoinbttahine
0 1 0 1
± ± ≡
10
0.4 (a) η0 = 8 (b) ter αs in Eq.(19) satisfies ~4/(|αs|2M2) ≪ 1/n1/3. This
1.0 0 caseisoftenalsoreferredtothe”wideresonance”regime.
0.5
Though for both s and p-wave resonances, the single-
0.2
0.1 channel approximation would be valid for sufficiently
0.3
-0.1 strong coupling (large α) between the closed and open
channels,fors-wavetheconditionforasingle-channelap-
~D0 ~µ
proximationisalwayssatisfiedwhenthegasissufficiently
-0.2 dilute. In contrast, for p-wave the condition η 1 is
0.2 ≫
independent of the density of the gas. This difference
can be understood by noting that the dimension of α
s
-0.3 is [E][L]3/2, whereas that of α here is [E][L]5/2. The
m
0-.11000 -500 0 500 1000 -500 -400 -300 -200 -100 0 effective range rs for s-wave Feshbach resonance [32, 33]
- 1 / v~ - 1 / v~ isproportionalto~4/(α 2M2),andtheadditionalterm
0 0 s
| |
(1/p2) in Eq. (34) does not arise.
∝
Forthe s-wavecase,the terms ”single-channelapprox-
FIG. 6: Plots of (a) D˜0 and (b) µ˜ as functions of −1/v˜0 imPation” and ”wide-resonance” have been used inter-
at c˜0 = c˜±1 = −100, D˜±1 = 0 for different η0’s. η0 = ∞ changably. However,we should note that for our p-wave
corresponds that of Fig 3. The dashed line in (b) represents
case, the condition η 1 is not necessarily in conflict
to thetwo-body result. ≫
withthefactthatthethepeakinscatteringcross-section
discussed in Fig.1 of Ref.[19] is narrow. This p-wave
−v˜1∗−1+v˜0−1 →(30.2)η/(2+η). This critical value thus siscagtitveerninbgycross-section is proportional to |fm|2, which
decreases with decreasing η. Its behavior is as shown in
Fig 7. The η dependence of this transition line has also
k4
been investigated in Ref [29]. f 2 = (57)
| m| ( 1 + c k2)2+k6
vm | m|
For B > B , where 1/v < 0, the cross-section
m∗ m
thus has a peak at finite energy E = ~2k2/M where
o
k2 = 1/(c v ). The denominator can be written as
o | m m|
c2 (k2 k2)2+k6. ForB &B , f 2 asafunctionofk2
m − o m∗ | m|
then peaks very near k2 = k2 and has a width approx-
o
35 imately given by k3/c = 1/(c 5/2( v )3/2) k2,
o | m| | m| − m ≪ o
and therefore a very narrow (in energy) resonance. This
30
isentirelyaresultofthespecialformoff forfiniteangu-
η =
0 8 lar momentum. This narrownessof the peak is therefore
25 1.0
~v0 0.5 not directly connected with the assumption on the mag-
+ 1 / 20 00..21 itude of η made here.
~*1 / v1 15 neIcnessaadrdiliytioinn,cotnhfleicctonwdiitthiotnhefoarssηum≫pt1iohneorefais“nalasroronwo”t
-
10 resonance in the context discussed by Gurarie et al [29].
Theydefinedaparameterγ α2 M2n1/3,andcalledthe
5 ∝ m
case γ 1 to be a narrow resonance. Their condition
≪
is fulfilled always at sufficiently low densities. Thus the
0
-600 -400 -200 - 1 0/ ~v 200 400 600 resonance can be narrow in their language whereas the
0
single-channelapproximationcan still be valid if n1/3
~4/(α M)2 c . ≪
m m
FIG.7: Thetransitionline−v˜1∗−1+v˜0−1betweenthekz phase ≪| |
and the kz −iβky phase for η’s in the legend. c˜0 = c˜±1 =
−100.
V. EXPERIMENTAL PROBE OF PAIRING
Weconcludewithadiscussionontheconditionη 1, SYMMETRY
orequivalently~4/(α M)2 c ,forthevalidityo≫fthe
m m
≪| |
effectivesingle-channelapproximation. Itisworthnoting It was suggested by Altman et al. [38] that the mea-
thatthecorrespondingconditionforthes-waveFeshbach surement of the atom shot noise in the time-of-flight
resonance superfluidity is quite different. For this latter (TOF) absorptionimage canrevealthe properties of the
case, the single-channel approximation near resonanace many-body states, including the symmetry of the pair-
is valid if [32, 33, 34, 35, 36] the corresponding parame- ing state. Later the proposal was carriedout by Greiner