Feshbach-Resonant Interactions in 40K and 6Li Degenerate Fermi Gases

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by  Matt Mackie

Feshbach-Resonant Interactions in 40K and 6Li Degenerate Fermi Gases
Matt Mackie1 and Jyrki Piilo2
1QUANTOP–Danish National Research Foundation Center for Quantum Optics,
Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark
2School of Pure and Applied Physics, University of KwaZulu-Natal, Durban 4041, South Africa
(Dated: February 2, 2008)
5
0 Wetheoretically examinea system of Fermidegenerate atoms coupled to bosonic molecules bya
0 Feshbach resonance, focusing on the superfluid transition to a molecular Bose-Einstein condensate
2 dressed by Cooper pairs of atoms. This problem raises an interest because it is unclear at present
whether bimodal density distributions observed recently in 40K and 6Li are dueto a condensate of
n bosonicmoleculesorfermionicatompairs. Asopposedto40K,wefindthatanymeasurablefraction
a of above-threshold bosonic molecules are necessarily absent for the 6Li system in question, which
J
strongly implicates Cooper pairs as the culprit behind its bimodal distributions.
0
2
PACSnumbers: 03.75.Ss

]
r
e Introduction.–Magnetoassociation creates a molecule leavesthe systemproneto pairing. Thequestioniswhat
h
from a pair of colliding atoms when one of the atoms role molecules play in this process.
t
o spinflipsinthepresenceofamagneticfieldtunedneara Our answer is outlined as follows. After introducing
t. Feshbach resonance [1]. Recently, ultracold [2] and con- themodel[11],weshowthatweak(strong)couplinggives
a densate [3] molecules have been created via magnetoas- alarge(negligible)fractionofabovethresholdmolecules,
m
sociationofa Fermigas ofatoms,inthe courseofefforts andthat,althoughcontrarytotwo-bodyphysics,thisre-
d- to create superfluid Cooper-paired atoms [4, 5] (see also sultmakesperfect sense interms of ourprevious[12, 13]
n Refs. [6]). Inparticular,foramagneticfieldtunedabove boson results. In particular, for atom-molecule cou-
o the two-body threshold for molecule formation, the ob- plingsmuchlargerthantheFermienergy,dissociationto
c servation of a bimodal density distribution for a system fermionic pairs should dominate the creation of bosonic
[
of 40K atoms was attributed to the existence of a Bose- molecules. We then find that bosonic molecules can be

1 EinsteincondensateoffermionicCooperpairs[4]. Never- ruledoutfortheobservationsinRef.[5],butnotforthose
v theless, a lone theoretical analysis suggests that the 40K inRef.[4]. Beforeconcludingwecontrastourresultswith
5
data[4]canbe understoodasaBose-Einsteincondensate the recent related theories [7, 14–17].
7
of molecules, since the presence of the Fermi sea shifts
4
1 the thresholdformolecularformationto thepointwhere
0 the molecular binding energy is equalto twice the Fermi
t/05 etbhnuoetruioggnyhsn[i7no]t,6Lcaoinnafitinromtmeersdp[,r5eb].tyaTtoihboenseprtuvhraapttioosnhesaosoffbtbheiiemsnLodbetaotllesdrteiisrstertdio-, εGap ()F 1 µ
a d
m demonstratethatsignificantfractionsofabove-threshold ui
molecular condensate are absent only when the atom- erfl0.75
- up
d molecule coupling is much larger than the Fermi energy. S
d
con Ionf aasmuorlperciuselarrevaenrdsafle,romuironreicsuBltosspe-oEinintsttoeiinntceornpdreetnastaitoenss, ential an 0.5
v: rceosnpdeecntsivaetley., instead of fermionic [4] and molecular [5] al Pot0.25 ∆
Xi mic
The mean-field theory of magnetoassociation of a he
r C
a Fermi gas of atoms leads to two types of instabilities
0
againstmoleculeformation. Firstisadynamicalinstabil- 0.1 0.2 0.3 0.4 0.5
Temperature (T )
ity,wherebythelargerstatespaceofthemolecules,owing F
somewhat to Pauli blocking, leaves the atoms prone to
FIG. 1: Example of the onset of the superfluid transition to
spontaneousassociationintomolecules[8]. Herewefocus
aBose-Einstein condensateofmoleculesdressedbyfermionic
on the thermodynamic instability of a Fermi sea against
atom pairs. The appearance of a non-zero superfluid gap
the formation of Cooper pairs [9], a trait of supercon- (∆) lowers the chemical potential (µ) compared to the nor-
ductors whose analog is passed on to Feshbach-resonant malstate(dashedcurve),signalingtheonsetofthesuperfluid
superfluids [10]. A thermodynamical instability occurs regime. As it happens, the coupling constant and the detun-
because pairing lowers the energy, similar to Fig. 1, and ing in this example are chosen to be exceptionally large (see
later text),similar to recent experiments [5].
socouplingtoareservoirwithalowenoughtemperature
2
2 1
Ideal Gas Model.–We model an ideal two-component ρ = 2Φ2+
gas of fermionic atoms coupled by a Feshbach resonance | | V Xk exp β 21ǫk+δ−2µ −1
to bosonic molecules. An ideal gas is chosen for simplic- (cid:2) (cid:0) (cid:1)(cid:3)
1 ω +µ ǫ +(ω µ+ǫ )exp( βω )
k k k k k
ity, and is justified by a collisional interactions strength + − − − .
V ω [1+exp( βω )]
that is negligible compared to the atom-molecule cou- Xk k − k
pling (see next-to-last section). In the language of (4b)
second-quantization, an atom of mass m and momen-
tum ~k is described by the annihilation operator ak,1(2), Renormalizationisviatheresonantself-energyΣ(0)[13],
and a molecule of mass 2m and similar momentum is meaning the physical detuning δ replaces the bare δ0.
described by the annihilation operator bk. All operators Weak vs. Strong Coupling.–Solving the algebraic sys-
obeytheir(anti)commutationrelations. Themicroscopic tem (4) self-consistently determines the chemical poten-
Hamiltonian for such a freely-ideal system is written: tial as a function of temperature. Intuitively the ap-
pearance of a non-zero gap lowers the chemical poten-
H
~ = Xk h(ǫk−µ)a†k,σak,σ+(21ǫk+δ0−µmol)b†kbki ttioalthoef tnhoermsuaplesrtflautied, BwEhiCch-paailrlodwrsesusesdtostactoenficrommptahraetd,
for a given temperature, the system is in the superfluid
−√KV kX,k′(cid:16)b†k+k′ak,1ak′,2+H.c.(cid:17), (1) rceleginmuem,bc.efr.,,FFiigg.s.12. A(al,obn)ginwdiitchattehseacognrsoeurnvdatsiotanteoftphaarttiis-
a molecular condensate dressed by dissociated fermionic
where repeated greek indices imply a summation (σ =
1,2). The free-particle energy is ~ǫ = ~2k2/2m, the pairs, an admixture adjustable according to the detun-
k
atom (molecule) chemical potential is ~µ , and the ing from threshold. In particular, for an atom-molecule
σ(mol)
coupling that is weak compared to the Fermi energy,
bare detuning δ is a measure of the binding energy of
the molecule (δ0 > 0 is taken as above threshold), the Ω = √ρ . ǫF [where ǫF = ~(3π2ρ)2/3/2m], we find
0 K
a large fractionof above-thresholdmolecular condensate
mode-independent atom-molecule coupling is , and V
K [Fig.2(a)],whereasΩ ǫ leadstoanabove-resonance
isthequantizationvolume. Wehavealreadyimposedthe F
≫
systemthatispredominantlyfermionicpairs[Fig.2(b)].
ideal conditions for superfluidity with µ = µ = µ, and
1 2
now we impose chemical equilibrium between the atoms We have confirmed that for strong coupling the frac-
and molecules with µ = µ +µ = 2µ. Diagonaliza- tionofmolecularcondensateremainsnegligible( 10−3)
mol 1 2 ∼
tion of the Hamiltonian (1) is achieved by the standard forδ ǫF,i.e.,theabsenceofalargenear-thresholdfrac-
∼
transformation to a dressed basis [11]: tionisnottheresultofchoosingΩasthefrequencyscale;
similarly, having properly renormalized the detuning, it
(cid:18)αα†−kk,1,2 (cid:19)=(cid:18)e−ciϕossθinkθk −ecioϕssθinkθk (cid:19)(cid:18)aa†−kk,1,2 (cid:19), iAsdndoittiodnuaelltyo, tahneytrsappu,riaolubseisthoifmtiottferde,scoannanaccetutahlrleysfhaovlodr.
(2a) the occurrence of superfluid pairing [25].
Below threshold (δ < 0), Fourier analysis delivers
βk =bk−√VΦδk,0, (2b) the binding energy, ~ωB < 0, of the Bose-condensed
molecules[8,13]: ω δ Σ′(ω )+iη=0,whereΣ′(ω )
B B B
where δk,0 is the Kronecker delta-function; hence, isthefiniteself-energ−yof−theBosemoleculesandη =0+.
Abovethetwo-bodythreshold(δ >0)givesanimaginary
H
~ = (δ0−2µ)V|Φ|2+ 12ǫk+δ0−2µ βk†βk ωB, and the bound state ceases to exist; nevertheless,
Xk (cid:0) (cid:1) Fig.2(a)showsalargefractionofmolecularcondensate,
+Xk h(ǫk−µ)+ωk(cid:16)α†k,1αk,1+α†k,2αk,2−1(cid:17)i. wFihgi.ch2 (dbr)o.pTshoeffsefoarppinacrreenatslyingcocnotruapdliinctgorsytrreensgutlhtsaasrepeinr
(3) fact consistent with the dynamical studies of rapid adi-
abatic passage in bosons [12, 13]: just above threshold,
Thecondensatemean-fieldis b /√V =eiϕ Φ,themix- the coupledatom-molecule systemcanhave a significant
0
h i | |
ing angle is tan2θ = ∆/(ǫ µ), the eigenfrequencies fraction of molecular condensate [12], but “rogue”disso-
k k
| | −
are ω2 =(ǫ µ)2+ ∆2, and the gap is ∆ = Φ . ciationtoatompairswithequalandoppositemomentum
k k− | | | | | |K
To determine the thermodynamic ground state, we is expected to dominate the formation of molecules for
first calculate the pressure from the partition function Ω ~ρ2/3/m ǫ [12, 13]. Physically,the rate for con-
F
≫ ≈
Ξ = Trexp( βH), which is then extremized with re- verting atoms into molecules is Ω, whereas the rate
− ∼
specttothemolecularamplitudeand,inturn,thechem- for rogue magnetodissociation is Γ Ω2 [18]; hence,
0
∝
ical potential, yielding [11]: for a strongenough coupling,rogue magnetodissociation
to fermionic pairs of atoms with equal-and-opposite mo-
2 1
(δ 2µ)=Σ(0)+ K tanh1βω . (4a) mentum will dominate the formation of molecular con-
− 2V Xk ωk 2 k densate.
3
x 10-4 On the other hand, for a 6Li gas of typical density [5],
0.3 8
the atom-molecule coupling is [19, 21] Ω = 87ǫ . Ap-
n n F
o o
acti0.2 (a) Weak acti6 (b) Strong proximating ∆µ = 2µ0, any molecular BEC can already
Fr Fr4 beruledoutofexperiments[5]basedonFig.2(b). Even
EC 0.1 EC without explicitly accounting for the somewhat larger
MB MB2 6Li coupling constant, Fig. 2 (b) would predict a near-

0 0 unitfractionoffermionicpairs. Since the atom-molecule
1 2 3 0 10 20 30
δ[Ω] δ[Ω] coupling in Fig. 1 is Ω 150ǫF and the detuning is
10-1 δ Ω,weestimatethecri∼ticaltemperatureT 0.2T ,
n0.05 n ≫ C ∼ F
o o which is close to the measured [5] large-detuning criti-
acti (c) 40K acti (d) 40K caltemperature(allthingsconsidered);hence,weexpect
C Fr0.03 C Fr10-3 T = 0.05TF to be far enough below the critical temper-
E E
B B ature so that what is not molecular condensate is most
M0.01 M
0 10-5 likely superfluid fermionic pairs, with only a small ther-
0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 mal fraction. Indeed, whereas initial experiments mea-
∆B[G] ∆B[G]
sure 80% condensate fractions [5], recent measurements
are in excess of 90% [22], consistent with expectation.
FIG. 2: Molecular condensate fraction as a function of the
We pause to briefly justify the ideal gas model. The
above-threshold detuning. For T = 0.05TF < TC, where
TC is the critical temperature for the superfluid transition, collisional interaction strength is Λ = 2π~ρa/m, where
weak coupling (Ω = ǫF) finds a large fraction of above- a is the off-resonant atomic s-wave scattering length.
thresholdmolecularcondensate(a),whereasastrongcoupling The 40K and 6Li scattering length are roughly an or-
(Ω=35ǫF) findsa largely absent MBEC (b),independentof der of magnitude apart: a = 176a [23] and a =
K 0 Li
the above-threshold detuning. In panel (c), the solid line is | |
2110a [21, 24], with a the Bohr radius. For a typical
tfohreTwe=ak0r.e0s8uTlFt fr<omTCpaannedl(Ωa)=sca4lǫeFd,baynd1/t4h2e[2d0a]s.hTedhelionpeeins density0 ρ ∼ 1013cm−3,0it turns out that magnitude of
the collisional coupling strengths, in units of the atom-
circlesareusedtoemphasizehowonlythefirstpointistaken
fromtheregionwheretheidealgasapproximationmaybreak molecule coupling, are roughly equal: Λ/Ω 10−3.
| | ≈
down. The actual results are markedly better than the scal- Collisions should therefore be broadly negligible.
ing,andinagreementwithobservation[4]exceptforashiftof Before closing, it is important to draw contrast with
about 0.2G.Inpanel(d)thesuperfluid40Kphaseboundary
thelatestworkofothers. First,itappearsthatthestabil-
is evident in a departurefrom exponential behavior.
ityoftheabove-thresholdmoleculesisamatterofcompe-
titionbetweenformationofmolecularcondensateandthe
subsequentmagnetodissociationtofermionicpairs,asop-
Recent Experiments and Theory.–The significance of posedto many-bodyeffects [7](see alsoRef.[14]). Next,
theaboveresultsisseenbycomparisonwithexperiments although the chemical potential for the near-resonant
in 40K [4] and 6Li [5] systems. Detunings are converted 40K system is undoubtedly in the so-called universal
into magnetic fields according to ∆B = ~δ/∆µ, where regime [15], and molecules are not expected to play role
the difference in magnetic moments between the atom insuchacase[15],wefoundhereinthatmoleculescannot
pair and a molecule is ∆µ, and where ∆B = B B0 be ruled out. Also, whereas a prominent single-channel
−
(with B0 the magnetic-field position of resonance). theory [26]–i.e., only atoms and their pairs, no explicit
Fora40Kgasofdensityρ=2 1013cm−3,thecoupling molecules–has shown good agreement with the 40K ex-
×
strength is [8, 19] Ω 4ǫ ; the difference in magnetic periments[16],thistheorywouldpresumablydeliversim-
F
≈
moments is ∆ 0.19µ [8] (with µ the Bohr magne- ilar answers for both experiments [4, 5], unlike our the-
µ 0 0
≈
ton). Off hand, we are tempted to scale [20] the results ory. Finally,itis entirelypossiblethatthe thresholdofa
ofFig.2(a),buttheresultisnotencouraging[Fig.2(c), strongly-coupledsystem is systematically shifted to neg-
dashedline]; nevertheless,a full recalculation[Fig.2 (c), ativedetunings,assuggestedrecently[17]–albeitwithout
solid line and open circles] of the solution to Eqs.(4) for recourse to the atom-molecule coupling strength, a mat-
Ω = 4ǫ yields results that, except for a roughly 0.2 G ter defered to elsewhen.
F
shift, agree embarrassingly well with the measured [4] Summary and Conclusions.–We have found that, in
bimodal distributions (not shown). The 0.2 G disagree- a Feshbach-resonant gas of Fermi atoms, significant
ment is understood more clearly given Fig. 2 (d), where fractions of molecular condensate are absent for atom-
thesuperfluidphaseboundaryindetuning,givenroughly moleculecouplingsthatarestrongcomparedtotheFermi
by the dashed line, is marked by a clear departure from energy. While itis perhapsa stretchcallthe 40K [4]sys-
exponential behavior. Overall, it seems that we agree tem weak, it is clear that bosonic molecules can not be
with Ref. [7] that bosonic molecules can not be ruled ruled out of its bimodal distributions. However, bosonic
out, by two-body physics or otherwise, as the culprit re- molecules can be ruled out for the 6Li [5] system, which
sponsible for the condensate footprints in 40K [4]. stronglysuggeststhatCooper pairing/fermionicconden-
4
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