A simple mean field equation for condensates in the BEC-BCS crossover regime

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by  Cheng Chin

A simple mean field equation for condensates in the BEC-BCS crossover regime
Cheng Chin
Institut fu¨r Experimentalphysik, Universit¨at Innsbruck, Technikerstr. 25, 6020 Innsbruck, Austria
(Dated: February 2, 2008)
Wepresentameanfieldapproachbasedonpairsoffermionicatomstodescribecondensatesinthe
5
BEC-BCS crossover regime. By introducing an effective potential, the mean field equation allows
0
us to calculate the chemical potential, the equation of states and the atomic correlation function.
0
The results agree surprisingly well with recent quantum Monte Carlo calculations. We show that
2
thesmoothcrossoverfromthebosonicmeanfieldrepulsionbetweenmoleculestotheFermipressure

n among atoms is associated with the evolution of theatomic correlation function.
a
J PACSnumbers: 03.75.Hh,05.30.Fk,34.50.-s,39.25.+k

4

] Recentstudies onultracoldFermigasesandmolecular different internal states can pair via s-wave interaction.
n
condensates[1]addressanintriguingtopic,the crossover For simplicity, we assume the interaction range is zero.
o
from a Bose-Einstein condensate of composite bosons to In the absence of many-body effects, the center-of-mass
c
- a fermionic Bardeen-Cooper-Schrieffer superfluid (BEC- motion of an atom pair Ψ0(R~) is decoupled from the in-
pr BCS crossover) [2]. By magnetically tuning the interac- ternal relative atomic motion ψ0(~r) = (4πr2)−1/2ψ0(r)
u tion strength near a Feshbach resonance [3], a molecular with r = |~r| the atomic separation. Given the atomic
s BECcanbe smoothlyconvertedintoadegenerateFermi scattering length a, ψ0(r) satisfies Schr¨odinger’s equa-
t. gasandviceversa. Experimental[4,5,6]andtheoretical tion,
a research [7, 8] into the quantum gases in the crossover
m
regime are highly active and may provide new insights
~2
d- into other strongly interacting Fermi systems. − ψ0′′(r)=−Ebψ0(r) (1)
m
n In contrast to weakly interacting atomic BECs, for
o whichasimplemeanfielddescriptionbasedontheGross- with the boundary condition ψ0(0) = −aψ0′(0). Here m
c Pitaevskiiequationhasbeenverysuccessful[9],theoreti- is the atomic mass, 2π~ is Planck’s constant, and E is
b
[ calmodelsonthecondensatesinthecrossoverregimeare the molecular binding energy.
2 generally very sophisticated and require expertise bor- For positive scattering lengths a > 0, the bound state
v rowed from condensed matter theory. The difficulty in is described by ψ0 =(2/a)1/2e−r/a with Eb =~2/(ma2).
9 providingasimplemodelforthe fermionicsystemcomes The size of the molecule is given by hri = a/2. For
8 from, firstly, the lack of a small expansion parameter. negative scattering lengths a < 0, the bound state does
4
The full range of atomic scattering length a should be not exist and the ground state energy is −E =0.
9 b
taken into account to describe the crossover. Secondly, Now considera condensate ofpairswith a density dis-
0
4 quantum many-body correlations are intrinsically more tributionn(R~)inaslow-varyingpotentialwellV(R~). We
0 complicatedfor fermionic systems thanfor bosonic ones. introduce the many-body wave function to include the
t/ TheBEC-BCScrossover,however,suggestsanalterna- condensate of the bosonic pairs Ψ(R~) = n(R~)1/2 as well
a tive approachto model the stronglyinteracting fermions as the internal atomic correlation ψ(r). The mean field
m
basedoncompositebosons. ThisispossiblesinceaFermi
equation for the composite bosons is then
- gasinthecrossoverregimeconstitutesthesamequantum
d
phase as of a condensate of interacting pairs. Recent ex-
n perimentsonthe wavefunctionprojection[5]andonthe (−~2∇2R − ~2∂r2 +V +Uˆ)Ψ(R~)ψ(r)=µ Ψ(R~)ψ(r),(2)
o 4m m m
c pairing gap [10] indeed indicate that near the Feshbach ψ(0)=−a∂ ψ(0). (3)
r
: resonance,a large fraction of fermionic atoms are paired
v
at low temperatures. From these observations, we pro- Here µ is the chemical potential, Uˆ = gˆ|Ψ(R~)|2 is the
i m
X pose a bosonic mean field equation, complementary to mean field interaction and gˆ is the interaction term.
r thefermion-basedBCSapproaches,todescribetheatom In conventional approaches, gˆ is given by the scatter-
a
pairs in the crossover regime. Our mean-field approach ing length of the bosons. For pairs of fermions, scatter-
is relativelysimple andwell-behavednearthe resonance. ing length a is determined by that of the constituent
m
We obtain analytic expressions for the chemical poten- atoms as a = 0.60a, resulting from an effective repul-
m
tial and the equation of states, which agree very well sive potential between molecules [14]. This dependence
withothercalculations. Inparticular,wefindthechemi- can be understood in a simple picture. Low-energy col-
calpotentialinthe unitarity limitis∼0.4357times that lision with a repulsive interactions acquires a scattering
in the BCS limit, in excellent agreement with the recent length which is proportional to the size of the scatterer.
quantumMonteCarlocalculationsof0.42∼0.44[12,13]. For pairs of atoms, we have a ∼hri=a/2.
m
We consider an ultracold gas of two-component From the above considerations, we hypothesize that
fermionic atoms. At low temperatures, only atoms in the interaction term gˆ is effectively proportional to the
2
interatomic separation r as since the pair is compressed. This result also provides
a simple picture to understand the augmentation of the
molecular binding energy reported in Ref. [10].
~2
gˆ=g(r)=c r, (4) We extend the mean field model to the crossover and
m
the BCS regime, where the atom pairs strongly overlap.
where c is a dimensionless constant. Althoughitbecomeslessclearifthemeanfieldapproach
To proceed with minimum algebra,we consider a uni- canfullycapturetheFermionicnatureofthegas,ouraim
form gas with a density |Ψ(R~)|2 =n. Eq. (2) becomes hereistodetermineaneffectivepotentialwhichcanbest
describe the system in the BEC-BCS crossoverregime.
In this regime, the four-body calculation of a =0.6a
~2 m
2 is no longer valid, and we determine the mean field in-
(− ∂ +gˆ)ψ(r)=µ ψ(r). (5)
m r m teraction Uˆ from the properties of the Fermi gas. First
of all, in the dilute gas limit, we still expect the inter-
Todeterminec,weconsidertheBEClimit(na3 ≪1),
m action to be proportional to the square of the bosonic
wherethe meanfieldtermcanbe treatedperturbatively. field,Uˆ =gˆ|Ψ(R)|2. Secondly,weexploittheasymptotic
That is, the expectation value of Uˆ based on the bare behaviorofthegasintheweakcouplinglimitna3 →0−,
molecular wave function ψ0(r) = (2/a)1/2e−r/a should where the system approaches an ideal degenerate Fermi
yield the molecular mean field shift 4π~2a n/2m,
m gas with the chemical potential
∞ 2π~2a n
∗ m
Z0 ψ0(r)gˆnψ0(r)dr = m . (6) lim µm =2EF = (6π2n)2/3~2, (10)
na3→0− m
Using a =0.60a, we find Eq. (6) can indeed hold for
m
arbitraryscatteringlengthsam basedonthelinearmean where EF = ~2kF2/2m is the Fermi energy and kF =
field potential in Eq. (4). We determine c = 4πam/a ≈ (6π2n)1/3 is the Fermi wave number.
7.5.
Based on Eq. (5), we find that the above density de-
From Eq.(3), (4), and (5), the exact solution of the
pendence µ ∝ n2/3 can be satisfied only when the in-
pair wave function is given by m
teraction term gˆ is again a linear function of r. Taking
the limit of a = 0− and assuming g(r) = c′(~2/m)r,
ψψ((0r)) == −NaA∂i(cψ1/(30n),1/3r−c−2/3µm/E0) ((87)) wEqe.(c8a)nassoµlvme t=heαcch′2e/m3Eic0a,lwphoetreent−iaαl f≈rom−2E.3q3.8(7)isatnhde
r first zero of the Ai(x) function. Equating µm to 2EF
yields c′ = 6π2α−3/2 ≈ 16.56. This value is about twice
whereN isthenormalizationconstant,Ai(x)isAiry’sAi
function, and E0 = ~2n2/3/m. Notice that the chemical as large as c.
potential µ in Eq. (7) is determined from Eq. (8). We first test the equation in the unitarity limit a =
m
In the weak interaction limit 0 < na3 ≪ 1, the wave ±∞. Fermigasesinthislimithavebeenextensivelystud-
m
function ψ(r) obtained from Eq.(7) is identical to the ied, for which a universal and fermionic behavior is ex-
unperturbed one ψ0(r) for r ≪ n−1/3. For r ≫ n−1/3, pected[8]. Duetothedivergenceofthescatteringlength,
ψ(r) is exponentially smaller than ψ0(r) and approaches weexpecttheonlyenergyscaleinthesystemistheFermi
s∼epra−r1a/t4ioexnpi(s−e32xrp3/e2c)t.edTshiinscesutphpereisnstieornacfotironlaregneeragtyominic- egn=ergcy′(E~2F/.mF)rro,mwtehedebtoeurnmdianreytchoendchiteimonic∂arlψp(o0t)e=nti0aalnads
creases when the pairs start overlapping. As a conse- µm = α′c′2/3E0, where −α′ ≈ −1.019 is the first zero
quence, the pair wave function ψ(r) is compressed to a of the Ai′(x) function. Given c′ = 6π2α−3/2, we get
smaller size than that of a bare molecule. Similar effect µm/2 = (α′/α)EF ≈ 0.4357EF. This result agrees ex-
is also discussed in Ref. [15] cellently with recent quantum Monte Carlo calculations
The distortion of the pair wave function can be char- which gives µm/2 = 0.44(1)EF[12] and 0.42(1)EF [13],
acterized by an effective shift in the binding energy E . and the measurements [4, 17], where the uncertainties
b
In the weak interaction limit, the shift can be defined as are larger. We, however, cannot exclude this agreement
is coincidental. Near the unitarity limit, we have
∞ ~2∂2
ψ∗(r)(− r)ψ(r)dr =−E (1+o(na3)). (9)
b
Z0 m µ α′ α−1/2 1
m
= − +O( ). (11)
The binding energy correction o(na3) is positive. This 2EF α α′kFa kF2a2
effect is absent in the calculations for point-like bosons
[16] since it originates comes from the internal degree Next, we investigate the BEC-BCS crossover regime.
of freedom. This increase in binding energy is expected Rewritting Eq. (7) and (8) using c′ =6π2α−3/2, we get
3

1
1.0 BEC limit
0
2EF -1 gnt 0.8
/ m -2 one
m xp BCS and
e unitarity limit
-3 0.6
-4
-2 -1 0 1 2 -2 -1 0 1 2
1/ k a 1/ k a
F F
FIG.1: Chemical potentialµm inthecrossover regime(solid FIG.2: Exponentγ fortheequationofstatesfrom themean
line). Forlarge 1/kFa,thechemical potentialµm approaches fieldcalculation(solidline),theBCScalculation(dashedline)
theenergyofthemolecularstate−Eb(dottedline). Thesolid [19]andthefittothequantumMonteCarlocalculation(open
dots and the open dots show the Monte Carlo calculations circles) [13, 20]. The unitarity and BCS limit γ = 2/3 and
from Ref. [12] and Ref. [13], respectively. theBEC limit γ =1 are shown in dotted lines.

1.5
k a=1/2
F
µ
ψ(r) = NAi(α−1/2kFr−α2EmF) (12) 1/2 )) 1.0
kFa = −Ai(−αµm/2EF). (13) p/4F kFa=2
The chemαi1c/a2l potentAiail′(µ−mαµcmal/c2uElaFte)d from Eq.(13) is (r) ( k ( 0.5 kFa=¥
shown in Fig. 1. We see that µm approaches 2EF in the Y
BCSlimitand−E intheBEClimit,asexpected. Inthe
b
crossover regime, the values agree well with the Monte
0.0
Carlo calculation from [13]. We can also evaluate the 0 1 2 3 4
equation of states µ +E ∝ nγ, where the exponent γ r (k-1)
m b F
playsacrucialroleinthecollectiveexcitationfrequencies
[9, 18]. From Eq.(12) and Eq.(13), we obtain FIG. 3: Pair wave functions in the crossover regime. Wave
functionsψ(r) at kFa=1/2, kFa=2and kFa=±∞,shown
in solid lines, are calculated based on Eq.(12). In the former
dln(µ +E )
γ = m b (14) two cases, the bare molecular wave functions ψ0(r) (dotted
dlnn lines) are shown for comparison.
= 2 1+ Eb −1 1+ 2α−3/2kFaEF2/µ2m .(15)
3(cid:18) µm(cid:19) (cid:18) kF2a2+2EF/µm (cid:19) hri ≈ 2/kF ≈ 0.5n−1/3 suggests the size of the pairs is
Theexponentγ (seeFig.2)showstheexpectedbehav- about half of the mean molecular spacing.
ior: γ =1 in the BEC limit and γ =2/3in the BCS and Thedistortionofthewavefunctionsleadstosignificant
unitarity limits. In the range of 1 < kFa < ∞, γ shows consequencesfor the quantum gas. Given the mean field
a dramaticvariation. Inthe following,we showthatthis energy as hUˆi∝nhri, the evolutionof the pair size from
dramatic variation is directly linked to the crossoverna- hri∝aintheBECregimetohri∝n−1/3intheunitarity
ture of the quantum gas and is a result of the distortion limit underlies the crossover nature of the interactions
of the pair wave function ψ(r). from the bosonic mean field repulsion hUˆi ∝ na to the
FromEq.(12),wecalculateψ(r)forkFa=1/2,2,and Fermi pressure hUˆi ∝ n2/3. This explains the variation
±∞, shown in Fig. 3. For kFa = 1/2, we see very small of the exponent γ in Fig. (2). From these observations,
deviation of ψ(r) from the bare molecular wave function we can qualitatively define the BCS regime to be kFa <
ψ0(r). For kFa = 2, ψ(r) is clearly different from ψ0(r) 0, crossover regime 1/2 < kFa < ∞, and BEC regime
with a higher probability amplitude for r < kF−1 and a 0 < kFa < 1/2. The use of c′ ≈ 16.56 in the mean field
−1
lower amplitude for r > k . This is the compression term is appropriate in the BCS and crossover regimes
F
effect we discussed. In the unitarity limit kFa = ±∞, and c≈7.5 in the BEC regime.
the atomic pairing is fermionic since bare molecules dis- The pair wavefunctions canbe directly probedexper-
sociate at this point. The mean atomic separation of imentally by radio-frequency (rf) excitations as demon-
4

1 number and the scattering phase shift of the outgoing
2
atoms, respectively.
Kpk /EF1
1/E)F0.1 0-2 -1 1/k0a 1 2 regTimo ec,awlceurlaeptelaFceraψn0c(kr-)Cboynψdo(rn)faancdtoarsssuinmeththeecartoosmsosvienr
K) ( F tshheowoutthgaotinthgecFhraannncekl-dCoonndotoninftaecrtaocrtsδdi=sp0la.yInaFreisgo.n4a,nwcee
F(f structure in the crossover regime. The location of the
peakFranck-CondonfactorK providesasensitivemea-
0.01 pk
sureoftheatomiccorrelationlength. IntheBECregime,
4
0 1 2 3 4 Kpk approaches 3Eb ≫ EF [21] and suggests that the
K (E ) atomic separation is small compared to the intermolec-
F
ular distance. In the crossover regime, K approaches
pk
FIG.4: Bound-freeFranck-CondonfactorsFf(K)ofthepairs a small fraction of EF. The persistence of the resonance
for (from bottom to top) kFa = 1/2, kFa = 1, kFa = 2, structure at unitarity and in the BCS regime indicates
kFa=±∞(unitaritylimit)andkFa=0− (BCSlimit,dotted the correlation of the atoms in momentum space. This
line). Thearrows mark the peak positions K . In theinset,
pk dependence is recently reported in [10, 11]. A quantita-
Kpkisplottedasafunctionof1/kFa(solidline)togetherwith tive comparison with the measurements, however, must
theK for baremolecules (dashed line).
pk
include the effects of the trapping potential and the fi-
nite temperature [23], which is outside the scope of this
paper.
stratedinRef. [10, 22]. In these experiments,rfphotons
excite the bound pairs into another spin state in which
In conclusion, we provide a simple mean field model
no bound state exists. The excited pairs then dissociate
todescribethe BEC-BCScrossover. Fromthis equation,
into free atoms. Theoretical calculation based on bare
manypropertiesofthestronglyinteractinggascanbean-
molecules show that the excitation rate constant, or the
alytically calculated with high accuracy. Our model can
bound-freeFranck-CondonfactorF (K),reflectsthepair
f also be easily generalized to include the external poten-
wave function in the momentum space [21],
tial and to study crossover effects in systems with lower
or higher dimension.
∞ 2
m WethankG.V.Shlyapnikovforstimulatingdiscussions
Ff(K) = π~2k (cid:12)(cid:12)Z0 sin(kr+δ)ψ0(r)dr(cid:12)(cid:12) , (16) and the members in R. Grimm’s Li group in Innsbruck
(cid:12) (cid:12) for their support. C.C. is a Lise-Meitner research fellow
whereK =~2k2/m,k,a(cid:12)ndδaretheenergy,rela(cid:12)tivewave of the Austrian Science Fund (FWF).
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