p-Wave superfluid and phase separation in atomic Bose-Fermi mixture

0.19 MB·English

by  Kazunori Suzuki

p-wave superfluid and phase separation in atomic Bose-Fermi mixture
Kazunori Suzuki1, Takahiko Miyakawa2, and Toru Suzuki1
1Department of Physics, Tokyo Metropolitan University,
1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan
2Department of Physics, Faculty of Science, Tokyo University of Science,
1-3 Kagurazaka, Shinjuku, Tokyo 162-8601, Japan
(Dated: February 3, 2008)
We consider a system of repulsively interacting Bose-Fermi mixtures of spin polarized uniform
8
atomic gases at zero temperature. We examine possible realization of p-wave superfluidity of
0
fermions due to an effective attractive interaction via density fluctuations of Bose-Einstein con-
0
densate within mean-field approximation. Wefind the ground state of thesystem by direct energy
2
comparison of p-wave superfluid and phase-separated states, and suggest an occurrence of the p-

n wave superfluid for a strong boson-fermion interaction regime. We study some signatures in the
a p-wavesuperfluidphase,suchasanisotropicenergygapandquasi-particleenergyintheaxialstate,
J
thathavenotbeenobservedinspinunpolarizedsuperfluidofatomicfermions. Wealsoshowthata

8 Cooper pair is a tightly bound state like a diatomic molecule in thestrong boson-fermion coupling
2 regime and suggest an observable indication of the p-wavesuperfluid in the real experiment.

] PACSnumbers: 03.75.Ss,03.75.Mn
n
o
c
- I. INTRODUCTION they are trapped by magneto-optical trap together with
r bosons. The fermion-fermion interaction has been neg-
p
ligible since the s-wave scattering amplitude between
u Magnetically tunable Feshbach resonances have
s fermions vanishes because of Pauli exclusion principle,
opened up a new field of research in the physics of ul-
. and the other higher partial waves do not contribute at
t tracold atomic gases that exhibits exciting phenomena.
a lowtemperatures. Inaboson-fermionmixedsystem,bo-
m In two-component fermi gases, for instance, the conden-
son density fluctuations may give rise to an attractive
sate of atompairs in the BCS-BEC crossoverregime has
- interaction between fermions as in the electron-phonon
d been intensively studied [1, 2, 3, 4, 5, 6]. In an ultra-
system, and may induce a superfluid transition [29, 30].
n cold atomic system, the p-wave Feshbach resonance was
o also found, which allows one to tune p-wave interactions This mechanism has been studied in the liquid 3He-4He
c mixtures [31]. If this mechanism is strong enough in the
between atoms in the spin-polarized fermi system [7]. It
[ ultracold atomic mixture, p-wave Cooper pairs will be
has now become possible to produce p-wave molecules
formed between spin-polarized fermions [32, 33, 34]. We
1 between 40K atoms using this technique [8].
v should note, however, that a realization of the super-
Atomic boson-fermion mixed systems, too, are ex-
3 fluid depends on a balance of different mechanisms as
9 pected to show many interesting phenomena. In this mentioned above. A sufficiently strong attractive boson-
1 mixed system, it has been reported the observation of fermion interaction, for instance, may cause a collapse
4 simultaneous quantum degeneracy of Bose-Einstein con-
of the system rather than the superfluid state. On the
. densate(BEC)andDegenerateFermigas[9,10,11],and
1 other hand, a strong boson-fermion repulsion may favor
0 the collapse of fermions in the attractively interacting a phase separationof bosonsand fermions insteadof the
8 mixture [12, 13]. On the theoretical side a number of superfluid transition.
0 studiesonthestaticanddynamicalpropertiesofthemix-
v: ture have been made [14, 15, 16, 17, 18]. We note here The purpose of the present paper is to investigate
i thatthestronglycoupledboson-fermionpairmaybehave a possibility of p-wave superfluid transition in the re-
X
as a heteronuclear molecule or resonance, and its role in pulsively interacting spin-polarized Bose-Fermi mixture
ar the mixture has been studied [19, 20, 21, 22, 23]. Re- at zero temperature. We consider superfluid transi-
cent experiments show an existence of the Feshbach res- tioninduced by a Bogoliubovphonon-mediatedfermion-
onances between bosons and fermions [24, 25], and quite fermion interaction, and compare energies of the p-wave
recently a formation of the boson-fermion heteronuclear superfluid state and the boson-fermion phase-separated
moleculeintheopticallatticehasbeenreported[26]. As state. Inviewofthe recentdevelopmentin the tuning of
in the case of fermionic system, the existence of the Fes- boson-fermion interaction via Feshbach resonances, we
hbachresonanceallows one to controlthe boson-fermion study the system in a broad range of the interaction
interaction [27, 28]. Tuning the interaction one may parameters. Earlier study in this direction shows that
induce a collapse of fermions or a phase separation of the p-wave superfluid transitionis hardto occur, i.e., its
bosons and fermions, depending on the sign of the inter- transition temperature is too low to be attainable, be-
action [14, 16, 17, 29]. cause the phonon-mediatedattractiveinteractionis very
In many experiments of atomic Bose-Fermi mixture, weak [32]. A stronger repulsive interaction would cause
all atomic fermions have the same spin components as an instability of the system towards phase separation,
2
and then the induced attractive interaction reduces. We experimentalsituation[24,25,27,28]. Hereafter,wetake
show, however, that for a very strong repulsive interac- a > 0, leading to repulsively interacting Bose-Fermi
BF
tion it is possible to realize the p-wave superfluid tran- mixtures.
sition. In the superfluid phase, anisotropic energy gap We considertwo types ofquantum phasesof fermions,
appears like that of p-wave superfluid of 3He and heavy- that is, the normal phase and the superfluid phase. On
fermionsystems. We alsoshowthatthe Cooperpairis a the onehand, a phase separationofbosons andfermions
tightlyboundstatelikeadiatomicmoleculeinthestrong occurs in the normal state of fermions, as discussed in
boson-fermion coupling regime [35, 36]. Ref. [29] for a large positive U . On the other hand,
BF
Main content of the present paper is as follows. Sec.II forastronglyrepulsiveboson-fermioncoupling,aneffec-
derivesgroundstateenergiesofthesysteminthe p-wave tive fermion-fermion interaction via density fluctuations
superfluid phase due to the phonon-mediated attractive of BEC becomes strong and is expected to lead to a su-
interaction and in the normal phase by the method of perfluid state of fermions. In the following we calculate
Ref.[29]. Sec.III presents phase diagram of the system ground state energies of these two types of phases, sepa-
bydirectenergycomparisonofthep-wavesuperfluidand rately [37].
phase-separated states. The momentum dependence of
the energy gap and other properties of the superfluid
state are shown. Finally, Sec. IV is a summary. A. Superfluid phase : p-wave pairing
Weassumethatallbosonsarecondensateandthefluc-
II. MODEL
tuationsaretreatedbyexcitationsofBogoliubovphonon
of energy: ~ωb = ǫb(ǫb +2U n ) where n is a con-
We consider a spin-polarized uniform mixture of k k k BB B B
atomic bosons of mass m and fermions of mass m at densate density. Tqhus the bosonic part of the Hamilto-
B F
zerotemperature. ThesystemisdescribedbytheHamil- nian, Eq.(3), is transformed to
tonian
Hˆ = µ n +U n2/2+ ~ωbβ†β ,
Hˆ =Hˆ +Hˆ +Hˆ , (1) B − B B BB B k k k
F B int k=6 0
X
where and the boson-fermion interaction Hamiltonian, Eq.(4),
is also transformed to
Hˆ = (ǫf µ )c†c , (2)
F k− F k k
k Hˆ U n c†c
X U int ≃ BF B k k
HˆB = (ǫbk−µB)b†kbk+ 2BB b†p+qb†k−qbpbk, (3) Xk
Xk kXpq +U √n ǫbk β c† c +β†c†c ,
Hˆint =UBF b†p+qc†k−qbpck. (4) BF BkX=6 0,ps~ωkb (cid:16) k k+p p k p k+p(cid:17)
kpq (5)
X
Here ck(†) and bk(†) are the annihilation (creation) opera- whereβ(†)isaBogoliubovphononannihilation(creation)
torsforthefermionicandbosonicatomsofmomentumk, k
operator. In these transformations, we have neglected
respectively. The corresponding momentum states have higher order terms of β(†). We regard the first term of
kineticenergiesofbosonsǫb =~2k2/2m andoffermions k
k B Eq.(5) as an energy shift of the fermionic chemical po-
bǫfkos=on~s2akr2e/2dmenFo.teTdhbeychµemaincadlµpot.eTnthiaelbsoosfofner-bmoisoonnsaanndd tential µrF ≡µF−UBFnB.
F B Byapplyingthe second-orderperturbationtheory,the
boson-fermion collisions are described by the interaction
phonon-mediatedinteractionbetweentwofermionsisde-
strengths U =4π~2a /m and U =2π~2a /m ,
BB BB B BF BF r rived as one-phonon exchange process,
respectively, where a and a are the corresponding
BB BF
sis-wthaeveresdcuatcteedrimngaslse.ngTthhesealnadstimcrfe=rmmioBnm-feFr/m(mionBs+-wmaFve) Hˆint =−21 UFBF(p′,q)c†p+qc†p′−qcp′cp,
scattering for spin-polarized fermions is absent because pp′q
X
of Pauli exclusion principle.
with
In order to give a realistic estimate of the phase di-
agram, we consider a system of 87Rb-40K mixture for
2ǫb
1th0e−2a5tokmgiacndbomsoFn=s a0n.6d49fe×rm10io−n2s5,kwgh,earnedmaBBB==19.84.1998a×0 UFBF(p′,q)=UB2FnB(~ωqb)2−(ǫfpq′−q−ǫfp′)2.
with Bohr radius a . As for the boson-fermion interac-
0
tion, we assume the scattering length to be tunable via We assume that Fermi velocity v = ~k /m is
F F F
Feshbach resonance. This can be realized in the current much smaller than Bogoliubov sound velocity s =
3
U n /m , i.e. v s, so the phonon-mediated and the requirement of the mean number of fermions
BB B B F
≪
effective interaction can be written as in Ref.[32],
p 1 ξ
k
U2 1 nF = 1 , (11)
UFBF(q)= UBBBF 1+(~q /2mBs)2. (6) 2Xk (cid:18) − Ek(cid:19)
| |
determinetheenergygap∆(k)andchemicalpotentialof
This interaction is equivalent to an effective interac-
fermionsµr. Concreteformsofthe gapwillbe discussed
tion between fermions induced by density fluctuations F
in the next section.
of backgroundof BEC when the retardationeffect is ne-
Once we have the optimized value of the energy gap
glected[29, 30]. The effective Hamiltonianof fermions is
and chemical potential, we obtain the ground state en-
then given by
ergy per volume in the superfluid phase as
Hˆeff = (ǫf µr)c†c
F k− F k k 1 ∆(k)2
Xk ESF = 2 ξk−Ek+ | 2E |
1 U (k k′)c† c† c c . Xk (cid:18) k (cid:19)
− 2 FBF − P/2+k P/2−k P/2−k′ P/2+k′ 1
PXkk′ (7) + µrFnF+ 2UBBn2B+UBFnBnF. (12)
The third and fourth terms in Eq. (12) are mean-field
Sinceweconsiderspin-polarizedfermions,theeffective
energies of the boson-boson and boson-fermion interac-
interaction in the channel with even angular momentum
tions.
l is absentdue toantisymmetrizationofthe orbitalwave
function in the relative coordinate. Furthermore for the
effectiveinteractiondescribedbyEq.(6)thecontribution
B. Normal phase : phase separation
totheinteractionforhigherlcanbenegligible[30]. Thus
weextractthedominantl=1componentofthephonon-
mediated interaction Nowweconsideranormalphaseofrepulsivelyinteract-
ingBose-Fermimixtures. InRef.[29],Viveritetal. stud-
UFpBF(k,k′)=3Uipnd(k,k′) kˆikˆi′ (8) iedaphasediagramoftheuniformmixturesatzerotem-
i=x,y,z peratureandpredictedthree types ofequilibrium states:
X
(A)asingleuniformmixedphase,(B)apurelyfermionic
with
phase coexisting with a mixed phase, and (C) a purely
Up (k,k′)= UB2F 2m2Bs2 fermionic phase coexisting with a purely bosonic one.
ind U ~2kk′ × By the method to find equilibrium states in Ref. [29],
BB
k2+k′2+(2m s/~)2 (k+k′)2+(2m s/~)2 weobtainthegroundstateenergyofthesystemofafinite
B B
4kk′ ln (k k′)2+(2m s/~)2 −1 , volumeV andtotalnumberofbosonsNBandoffermions
(cid:18) (cid:12) − B (cid:12) (cid:19) N . Suppose that the system is composed of two phases
(cid:12) (cid:12) F
where k = k and kˆi = (kˆ/(cid:12)(cid:12) k)i. In this approxim(cid:12)(cid:12)ation, ofvolumeV1 andV2 whereV =V1+V2. Thenumbersof
only p-wave| p|airing is possib|le| to realize. bosons and fermions are given by NBi and NFi for i-th
LetusconsideraCooperpairwithzerocenter-of-mass phase, respectively, yielding to NB = NB1 + NB2 and
momentum and introduce a p-wave pair energy gap as NF =NF1+NF2. ThusthetotaldensitiesofbosonsnB =
N /V and of fermions n = N /V are given by n =
B F F B
∆(k)= UFpBF(k,k′)hc−k′ck′i nB1v+nB2(1−v)andnF =nF1v+nF2(1−v),respectively,
k′ where v =V1/V. Total energy per volume V is given by
X
where denotes an expectation value in the ground E =E v+E (1 v), (13)
state. Ihnithe standardBCStheorythe effectiveHamilto- N 1 2 −
nian Eq.(7) can be diagonalized by Bogoliubov transfor- where the energy per volume V of i-th phase has
i
mation α =u c v c† ,
k k k− k −k 3 1
E = ǫ n + U n2 +U n n . (14)
1 ∆(k)2 i 5 Fi Fi 2 BB Bi BF Bi Fi
HˆFeff = Ekα†kαk+2 ξk−Ek+ | 2E | , (9)
k k (cid:18) k (cid:19) In Eq. (14), the first term is the kinetic energy of
X X
fermions where ǫ = ~2(6π2n )2/3/2m is Fermi en-
Fi Fi F
where the quasi-particle energy is E = ξ2 + ∆(k)2
k k | | ergy of i-th phase. The second and third terms cor-
with ξk = ǫfk − µrF, and uk and vkpare given by respond to the mean-field energies of the boson-boson
u = (1+ξ /E )/2 and v = (1 ξ /E )/2, re- and boson-fermion interactions. In i-th phase, the pres-
k k k k k k
−
spectively. The gap equation sure is P = ∂(E V )/∂V and the chemical potentials
i i i i
p p −
of bosons and fermions are given by µ = ∂E /∂n
∆(k′) Bi i Bi
∆(k)= Up (k,k′) , (10) and µFi = ǫFi = ∂Ei/∂nFi, respectively. The equilib-
k′ FBF 2Ek′ rium conditions of the pressure and chemical potentials
X
4
of bosons and fermions between two phases, P = P , (a)
1 2
µ = µ , and ǫ = ǫ , determine V , n , and n . 25
B1 B2 F1 F2 i Bi Fi
As a result, the three types of equilibrium states can be
20
realized. Pure boson
(C) and pure fermion phase
Figure 1(a) shows a phase diagram for the 87Rb-40K UBB 15
mnBix=ed1s0y1s4tcemm−3inanthdeUnBFB/n=B5v.1s5.7 UB1F0/−U51BB[Jmsp3a].ceTfhoer U / BF 10 (B)
×
lower and upper solid lines in Fig. 1(a) represent the
5 Single uniform phase
boundaries between the phases (A)-(B) and phases (B)- (A)
(C), respectively. The result shows that a phase sepa- 0
0.0001 0.001 0.01 0.1 1 10 100 1000
ration of bosons and fermions is preferred for a higher
n / n
fraction of fermions and for a stronger boson-fermion F B
interaction strength compared to the boson-boson one. (b)
The dashed line corresponds to the critical curve above
0.80
whichtheuniformmixtureisunstableagainstsmallden-
sity fluctuations. 0.75
Figure 1(b) shows total energy per volume of the 0.70
tgfuarnloucetnniedorngstyoafptUeerBEFvNo/lUcuaBmlBceufoloafrttenhdFefg=rroomnuBnEd=qss1t.0a(1t1e43)sccmaan−led3d.(b1Ty4h)ǫeFatsnoFa- Total energy 00..6650
withǫ =~2(6π2n )2/3/2m isplottedagainsttheinter-
F F F 0.55
actionratioU /U . Theenergyvalueincreasesasthe
BF BB
0.50
repulsiveboson-fermioninteractionbecomesstrongerun- 0 2 4 6 8 10
til UBF/UBB ≃6 above which fermions start to separate UBF / UBB
from the bosonic cloud. For U /U & 8 the ground
BF BB
state is the completely phase-separated phase (C), and
the energy becomes independent of the interaction ratio FIG.1: (a) Phase diagram of normal phase of the 87Rb-40K
UBF/UBB. mixtures. There exist three types of equilibrium states: (A)
asingleuniformmixedphase,(B)apurelyfermionicphaseco-
existingwithamixedphase,and(C)apurelyfermionicphase
III. RESULTS coexisting with a purely bosonic one. The lower and upper
solid lines in Fig. 1(a) represent the boundaries between the
phases(A)-(B)andphases(B)-(C),respectively. Thedashed
In this section, we examine phase transition between
line corresponds to a critical curve above which the uniform
normal state and superfluid state of fermions. To do
mixture is unstable against small density fluctuations. (b)
so, we assume two typical types of energy gaps, an ax-
Total energy per volume of the ground state EN scaled by
ial state and a polar state in the superfluid phase, and ǫFnF as a function of UBF/UBB for nF =nB =1014cm−3.
calculate ground state energies in the two phases and
directly compare them. We obtain phase diagram of
87Rb-40K mixtures showing that phase transition from Weassume,however,justtwotypicaltypesofpairenergy
phase-separatedstate in the normal phase to p-wave su- gaps, so-called an axial state [38]
perfluidstateoccursasrepulsiveboson-fermioncoupling
∆ax(k)=∆ax(k) (kˆ +ikˆ ), (15)
becomes stronger. The unique characters of the p-wave x y
pairing, such as momentum dependence of energy gap, and a polar state
momentum distribution of fermions, and quasi-particle
excitation spectrum, in a strong boson-fermion coupling ∆pl(k)=∆pl(k) kˆz, (16)
regime are also discussed.
and regard a lower energy state of them as the ground
state in superfluid phase. Note that there is no symmet-
rical solution due to the spin-polarization of fermions in
A. Transition from phase separation to p-wave
this system. This fact is in stark contrast to unpolar-
superfluidity
ized Fermi systems such as 3He and heavy-fermion sys-
tems [39]. Amplitudes of the axial and polar states have
In order to obtain ground state energy of superfluid
∆ax(k) = ∆ax(k) sinθ and ∆pl(k) = ∆pl(k) cosθ ,
fermions,weneedtocalculatethetotalenergydescribed | | | | | | | |
corresponding to anisotropic energy gaps in momentum
by Eq. (12). In general, the p-wave pair energy gap can
spacewith zero energygaponthe northandsouthpoles
be expanded by spherical harmonics
forthe axialstateandonthe equatorforthe polarstate,
respectively.
1
∆(k)= ∆m(k)Y (θ,φ). Figure2(a)showstotalenergiespervolumeoftheaxial
1,m
state(tetragon)andpolarstate(asterisk)inunitofǫ n
m=−1 F F
X
5
(a)
50
1.5
Superfluid phase
40
1.0
gy 0.5 UBB 30
al ener 0.0 U / BF 20 Panudre p buores ofenrmion phase
ot Polar (C)
T -0.5 Axial 10 (B)
Normal /
-1.0 Single uniform phase (A)
0
-1.5 0.01 0.1 1 10 100
0 10 20 30 40 50 n / n
F B
U / U
BF BB
(b) FIG.3: Phasediagramwithconsideringthesuperfluidtransi-
tionatnB =1014cm−3. Circlesdenotetheboundarybetween
2 thep-wavesuperfluidphasewithaxialstateandnormalphase
with phaseseparated state.
al 1
nti
e
cal pot 0 cthreaansetsh.aWt oefsteheethpaotlatrhseteanteerignytohfethsteroanxgiablsotsaotne-fiesrlmowioenr
mi Polar
he Axial coupling regime. It is caused by the angle dependence
C -1
of the energy gap. The axial state has anisotropic en-
ergy gap in momentum space with zero energy gap on
-2 the north and south poles. For the polar state, on the
0 10 20 30 40 50
other hand, the energy gap vanishes on the equator in
U / U
BF BB momentum space. So the former state contributes to
the decrease of the total energy more than the latter
case. Figure 3 shows phase diagramof the system in the
sFcIaGle.d2b:yǫ(Fan)FTaontdal(ben)ecrhgeympicearl vpooltuemnteiaolfµrFthsecaslyesdtebmyǫEFSaFs UBF/UBB-nF/nB plane for nB = 1014cm−3. The lower
a function of UBF/UBB at nF =nB =1014cm−3. Each mark and upper lines are boundaries of the phases (A)-(B)
ofatetragonandanasteriskdenotestheaxialstateandpolar and the phases (B)-(C) in normal phase as already ex-
one. plainedinFig.1(a). Thecirclecorrespondstothecritical
points below and above which the phase-separatedstate
andaxialstaterealizes,respectively. Thedependence on
as a function of UBF/UBB for nF = nB = 1014cm−3. It nF/nB of the phase boundary is not strong. Thus we
showsthat the totalenergiesper volume ofboth ofaxial conclude that p-wave superfluidity can be observed at
and polar states increase as the boson-fermion coupling sufficientlystrongboson-fermioninteractioncomparedto
becomes stronger until U /U 30. It is caused by boson-boson interaction. This is the main result in the
BF BB
≃
the mean-field energy of boson-fermion repulsive inter- present paper. Using typical experimental parameters,
action. At sufficiently strong boson-fermion interaction we estimate the s-wave scattering length between boson
strength,however,theeffectofthechemicalpotentialµr and fermion simply from the rate of coupling constant.
F
is significant and the total energy turns to decrease. We It is estimated as aBF/aBB 23 that is tunable by the
≃
plot the behavior of the chemical potential of the super- technique of Feshbach resonance [27, 28].
fluid states µr as a function of U /U in Figure 2(b).
F BF BB
In the weak boson-fermion coupling regime, the relation
µr ǫ holds as in the ordinary BCS theory. However B. p-wave pairing state in strong boson-fermion
inFt≃he Fstrong boson-fermion coupling regime, µr starts coupling regime
F
to decrease, eventually becoming negative with increas-
ing UBF/UBB. Negative values of the chemical potential We have just seen that p-wave superfluid state arises
indicate formation of bound pairs [35, 36]. atsufficiently strongboson-fermioninteractionstrength.
The dot-dashed line in Fig. 2(a) corresponds to total Properties of the strong boson-fermion coupling super-
energy per volume of the ground state in normal phase, fluid state such as energy gap, quasi-particle, and mo-
which is the same as the result in Fig. 1(b). The direct mentum distribution areexpected to differ fromthose of
comparisonbetween the ground states in the two phases the weak boson-fermion coupling one. In the following
shows that phase transition from the phase-separated we will show some aspects of the p-wave pairing in the
state in normal phase to the axial state in superfluid strong boson-fermion coupling regime.
phase occurs when the interaction ratio U /U in- Figure 4 shows the radial component of energy gap
BF BB
6
1.2
4
1.0
Energy gap 321 Population number 000...864
0.2
0.0
0 0.0 0.5 1.0 1.5 2.0
0 1 2 3 4
k / k
k / k F
F
FIG. 4: Momentum |k| dependence of the radial gap of the FIG.6: Populationnumbersoffermions,vk2 =(1−ξk/Ek)/2,
axial state ∆ax(k) scaled by ǫF at nF = nB = 1014cm−3. in momentum space. Solid, dashed, dashed-dotted, and dot-
Solid, dashed,dashed-dotted,anddotted lines denoteresults ted lines denote UBF/UBB =10, 20, 40 and 50, respectively.
for UBF/UBB =20, 30, 40 and 50, respectively.
4 molecule with the binding energy 2µr in the strong
− | F|
boson-fermion coupling regime.
3 Finally, the momentum distribution function of
Quasi-particle energy 21 furleeersrdesmu6cii.omensApssosfirUogtrnBaniFdfit/ic.ffUaeBnrTBtelnhyitnecaUmrneBdoaFms/eFUese,nBrttmBuhimevoastlcduacietusistpsrtaiiisbtciussothnitoouownrfnnfasetirnmtloaFioribnggees-
U /U is expected to be proportional to momentum
BF BB
0 distribution of l = 1 bound state of two fermions. The
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
behavior of the distribution at k =0 is explained as fol-
k / k
F lows. From Eq.(11) the population number of fermions
at k = 0 becomes v2 = (1 + µr/µr )/2. In the
k=0 F | F|
U /U = 10, 20, and 40 cases, the population num-
FIG.5: Quasi-particleenergyintheaxialstateEk scaledby beBrFof BfeBrmions at k = 0 becomes v2 = 1 because of
aǫFt nasFa=funnBct=ion10o1f4ckm/k−F3.wDitahshtheed,andgaushlaerd-cdoomttpeodn,eanntdθd=otπte/d2 µrF > 0 shown in Fig. 2(b). In the Uk=BF0/UBB = 50 case
withµr <0,however,thepopulationnumberoffermions
lines denote UBF/UBB = 20, 40, and 50. And a solid line F
denotes dispersion relation of ideal fermi gases. vanishes atk =0. This resultcanbe alsounderstoodby
noting the following fact. For µr > 0, as indicated by
F
Fig. 4, fermions of extremely small momentum are al-
of the axial state ∆ax(k) in unit of Fermi energy ǫ as most non-interacting and occupy as free particles. For
a function of k/kF where kF = (6π2nF)1/3. The enFergy µrF < 0, the population number distribution tends to
havethe formofmomentumdistributionofsinglebound
gappeaksataroundtheFermisurface,andaCooperpair
canbeformedeveninhighmomentumregionof k >k . pair wave function [36] and vanishes at k =0 because of
F
| | l = 1 bound state. The particle distribution is observ-
The gap also becomes larger, and the width around the
able in the real experiments, for example, by using the
peak gets broader with increasing U /U . The gap
BF BB
time-of-flightmethod. Thusweconsiderthisresultasan
vanishes at k =0, and it originates from the property of
indication for the observation of the p-wave superfluid.
the interaction with angular momentum l = 1 between
fermions.
Since ∆(k) corresponds to a pairing potential that af-
fects on the pairing state of (k, k), the gap crucially IV. SUMMARY
−
influences quasi-particle spectrum, see Figure 5. The
quasi-particleenergyscaledbyFermienergyǫ isplotted WeconsideredasystemofrepulsivelyinteractingBose-
F
as a function of k/k . The solid line denotes dispersion Fermimixturesofspinpolarizeduniformatomicgasesat
F
relationofafreefermion. Intheweakboson-fermioncou- zerotemperature. Weinvestigatedthepossibilityofreal-
pling regime, the quasi-particle energy differs from that izationofp-wavesuperfluidityoffermionsduetoeffective
of a free fermion only around the Fermi surface. In the attractive interaction via density fluctuations of BEC in
strong boson-fermioncoupling regime, the quasi-particle thecaseofstrongboson-fermioninteractionwithinmean-
spectrum changes dramatically. For µr < 0, the gap field approximation. By direct energy comparison be-
F
opens at zero momentum and corresponds to molecular tween p-wave superfluid and phase-separated states, we
binding energy [35, 36]. We can see that a Cooper pair found that p-wave superfluidity can be observed at suf-
is now a tightly bound state of l = 1 like a diatomic ficiently strong boson-fermion interaction compared to
7
boson-boson interaction. in the real experiment.
We also discussed unique features of the ground state
in strong coupling p-wave superfluidity. We calculated The present analysis was made by using the standard
the quasi-particle energy and found that a Cooper pair mean-field approach with BCS wave function [35]. The
is a tightly bound state like a diatomic molecule in the results at very strong coupling would be modified by
strong boson-fermion coupling regime. We also calcu- fermionic self-energy correction as well as by dynamical
lated the momentum distribution function of fermions. screening of the effective interaction [40]. The consider-
In the strong boson-fermion coupling regime, we showed ation of these effects is left for a future study. Neverthe-
the property of the single bound pair wave function and less, the mean-field approaches are expected to provide
suggested an observable indication of p-wave superfluid good ways of building up intuition.
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