BCS-BEC crossover in a quasi-two-dimensional Fermi gas

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by  Andrea M. Fischer

BCS-BEC crossover in a quasi-two-dimensional Fermi gas
Andrea M. Fischer1,2 and Meera M. Parish1,2
1Cavendish Laboratory, JJ Thomson Avenue, Cambridge, CB3 0HE, United Kingdom
2London Centre for Nanotechnology, Gordon Street, London, WC1H 0AH, United Kingdom
(Dated: January 23, 2013)
Weconsideratwo-componentgasoffermionicatomsconfinedtoaquasi-two-dimensional(quasi-
2D) geometry by a harmonic trapping potential in the transverse direction. We construct a mean
fieldtheoryoftheBCS-BECcrossoveratzerotemperaturethatallowsustoextrapolatetoaninfinite
number of transverse harmonic oscillator levels. In the extreme BEC limit, where the confinement
lengthexceedsthedimersize,werecover3Ddimersconfinedto2Dwithweakrepulsiveinteractions.
However, even when the interactions are weak and the Fermi energy is less than the confinement
frequency, we find that the higher transverse levels can substantially modify fermion pairing. We
arguethatrecentexperimentsonpairinginquasi-2DFermigases[Y.Zhangetal.,Phys.Rev.Lett.
108, 235302 (2012)] have already observed the effects of higher transverse levels.
3
1
0 Recent experimental advances have made it possible theory that generalizes the 2D results of Randeria et
2 to study quasi-two-dimensional (quasi-2D) atomic Fermi al. [14] to quasi-2D. We expect the mean-field approxi-

n gases in a very controlled manner [1–8]. Such simple mationtobereasonablesinceitappearstobeconsistent
a quasi-2D systems may provide useful insights into struc- with recent experiments in the 2D limit (ε (cid:28)(cid:126)ω ) [7].
F z
J
turally complicated unconventional superconductors like The BCS regime of the quasi-2D Fermi gas was previ-
2
the cuprates, where superconductivity originates in the ously studied in Ref. [16] using a mean-field Bogoliubov-
2
copper oxide planes [9]. Quasi-2D Fermi gases are also deGennescalculationthatincludedthelowestthreehar-

] of fundamental interest, since they are the marginal case monic oscillator levels. Our approach, however, allows
s
of the Mermin-Wagner theorem and thus have modified us to extrapolate to an infinite number of levels and
a
g superfluid properties [10, 11]. In addition, the quasi-2D thus explore the entire BCS-BEC crossover. In the limit
- geometry can strongly affect fermion pairing within the ε (cid:29) (cid:126)ω , we find that our calculation recovers weakly
t B z
n superfluid, as we investigate in this paper. repulsive3Dbosonsconfinedtoquasi-2D.However,even
ua Incold-gasexperiments,atomsmaybeconfinedtoone forweakinteractions,εB (cid:28)(cid:126)ωz,wefindthathigherhar-
ormorequasi-2D“pancake”structuresusinga1Doptical monic levels can substantially modify fermion pairing as
q
. lattice. Thesystemcanthenbetunedfrom3Dto2Dby we perturb away from pure 2D and εF approaches (cid:126)ωz.
t
a increasingtheconfininglatticepotential. Forsufficiently We determine the radio frequency (RF) spectra for the
m strong lattices, the confining potential for a single quasi- quasi-2D Fermi gas and show that recent measurements
- 2Dlayercanbemodelledasaharmonicoscillatorpoten- of pairing in the quasi-2D regime εF ∼ (cid:126)ωz [8] are con-
d
tial,V(z)= 1mω2z2,inthetransversezdirection,where sistent with effects due to higher transverse levels.
n 2 z
m is the atom mass. When the temperature k T (cid:28)(cid:126)ω In the following, we consider a two-component (↑, ↓)
o B z
c and the Fermi energy εF (cid:28) (cid:126)ωz, the atoms will reside Fermi gas interacting close to a broad s-wave Feshbach
[ in the lowest harmonic oscillator level in the absence of resonance in 3D. Under harmonic confinment in the z-

1 interactions, and the gas is considered to be kinemat- direction, themotionofeachatomcanbeparameterized
v ically 2D. The short-range interatomic interactions are byits2Dmomentumkinthex-yplaneandtheharmonic
6 alsorenormalizedtoyieldaneffective2Ds-wavescatter- oscillatorquantumnumbern inthetransversedirection.
3 ingamplitudeandassociatedtwo-bodyboundstatewith The many-body Hamiltonian is thus (setting the system
2
abindingenergyε thatdependsonboththe3Dscatter- volume to 1):
5 B
inglengthandtheconfinement[12,13]. Thus,byvarying
01. theratioεB/εF,onecanexplorethecrossoverin2Dfrom Hˆ = (cid:88)((cid:15)kn−µ)c†knσcknσ (1)
weakBCSpairing(ε /ε (cid:28)1)totheBoseEinsteincon- k,n,σ
3 B F
:1 deveenrs,atthioenin(tBeEraCc)tioonfsdaimlseorsm(ixεBin/εhFig(cid:29)her1)ha[1rm4,o1n5ic].leHvoewls-: + (cid:88) (cid:104)n1n2|gˆ|n3n4(cid:105)c†kn1↑c†q−kn2↓cq−k(cid:48)n3↓ck(cid:48)n4↑,
v k,n1,n2
i forinstance,intheBEClimit,dimerswillbesmallerthan k(cid:48),n3,n4
X the confinement length l =(cid:112)(cid:126)/mω once ε >(cid:126)ω , so q
z z B z
r thattheyessentiallybecome3Dbosonsconfinedtoquasi- where(cid:15) =k2/2m+nω arethesingleparticleenergies
a kn z
2D. Here, we are interested in how the confinement can relative to the zero-point energy of the n = 0 state (we
impactpairingandleadtoadeparturefrom2Dbehavior now set (cid:126) = 1), and µ is the chemical potential. Note
throughout the BCS-BEC crossover. that we assume the mass and chemical potential (and
We focus on zero temperature, where there is a well- thus the particle density) are the same for each spin σ.
defined condensate in 2D, and we construct a mean-field Since the short-range interactions only depend on the
2
relative motion, we obtain the interaction matrix ele- spectively given by
ments (cid:104)n n |gˆ|n n (cid:105) by switching to relative and center
1 2 3 4 (cid:88)
of mass harmonic oscillator quantum numbers, ν and N γk†n↑ = (ukn(cid:48)nc†kn(cid:48)↑+vkn(cid:48)nc−kn(cid:48)↓) (7)
respectively. This yields n(cid:48)
(cid:88)
(cid:104)n n |gˆ|n n (cid:105)=g(cid:88)f (cid:104)n n |Nν(cid:105)f(cid:48)(cid:104)Nν(cid:48)|n n (cid:105) γ−kn↓ = (ukn(cid:48)nc−kn(cid:48)↓−vkn(cid:48)nc†kn(cid:48)↑), (8)
1 2 3 4 ν 1 2 ν 3 4 n(cid:48)
N
(cid:88) where the real amplitudes u, v only depend on the mag-
≡g VNn1n2VNn3n4, (2) nitude k ≡ |k| and satisfy (cid:80) (|u |2+|v |2) = 1.
N n(cid:48) kn(cid:48)n kn(cid:48)n
Note that while they have a well defined spin and mo-
where f = (cid:80) φ˜ (k ), with φ˜ the Fourier transform mentum, they involve a superposition of different har-
of the νν-th harkmzonνic zoscillator eνigenfunction. It is eas- monic oscillator levels. The corresponding BCS wave
ily seen that f2ν+1 =0 and f2ν = (−ν1!)ν(m2ωπz)1/4(cid:113)(222νν)!. fuuunmctsiotante|ΨfoMrFt(cid:105)he∝ba(cid:81)rekonpσeγrkantoσr|0s(cid:105)c, wh.eWree|t0h(cid:105)enismthineimvaizce-
knσ
TSihnecec(cid:104)hnannge|Nofν(cid:105)b∼asiδs coefficientasndarνe,νg(cid:48)ivaerneeinvenR,enf. +[17n]. (cid:104)HˆMF(cid:105)=(cid:80)k,n((cid:15)kn−µ−Ekn)−∆g20 withrespectto∆0 at
1 2 N+ν,n1+n2 1 2 fixedµtoobtainthegroundstate. Thevalueofµischo-
mustequal,modulo2,n3+n4 toobtainanon-zerointer- sentokeepthedensityofparticlesρ=2(cid:80) |v |2,
actionmatrixelement. The3Dcontactinteractiong can k,n(cid:48),n kn(cid:48)n
and thus the Fermi energy ε , constant throughout the
be written in terms of the binding energy ε of the two- F
B crossover [23].
body bound state which always exists in the quasi-2D
For the case where there are only two levels n = 0,1,
geometry:
the problem can be solved analytically. Here, Eq. (6) is
−1 = (cid:88) fn21+n2|(cid:104)n1n2|0 n1+n2(cid:105)|2 . (3) ginretahtelynsi=mp0liafineddnsin=ce1thleevreelsis. nTohpeaqiruiansgipbaerttwiceleendaistpomers-
g (cid:15) +(cid:15) +ε
k,n1,n2 kn1 kn2 B sionsarethenEkn =(cid:112)((cid:15)kn−µ)2+(V0nn∆0)2. Onecan
now minimize (cid:104)Hˆ (cid:105) by simply using ∂(cid:104)Hˆ (cid:105)/∂∆ =0.
Here, we simply take N = 0 since ε is independent of MF MF 0
B
Combining this with Eq. (3) yields the implicit equation
the center of mass motion. One can also determine ε
B
as a function of the 3D scattering length a [12, 13] from
(cid:16) (cid:17) s (cid:32)(cid:112)∆2+µ2−µ(cid:33)4 ε +2ω
Eq. (3) using 1 = m 1 − 2Λ , where Λ is a UV cutoff = B z ,
for the 3D momg ent4uπm tahsat cπan be sent to infinity at the εB ω −µ+(cid:113)∆2 +(ω −µ)2
z 4 z
end of the calculation. (9)
Now if we define the superfluid order parameter where we have defined ∆ = ∆ V00, the pairing gap in
0 0
the lowest level. Also, the density ρ = −∂(cid:104)Hˆ (cid:105)/∂µ,
(cid:88) MF
∆qN =g VNn1n2(cid:104)cq−kn2↓ckn1↑(cid:105), (4) and in the regime εF ≤ωz where εF =πρ/m, this gives
k,n1,n2
(cid:112) (cid:112)
2ε = ∆2+µ2+ ∆2/4+(ω −µ)2+2µ−ω . (10)
andassumefluctuationsaroundthisaresmall,weobtain F z z
the mean-field Hamiltonian,
WeseethatEqs.(9)and(10)reducetothe2Dmean-field
Hˆ = (cid:88)((cid:15) −µ)c† c (5) equations [14] in the limit ωz → ∞, as expected. They
MF kn knσ knσ also yield the lowest order correction to the 2D result
k,n,σ due to confinement (ε /ω (cid:54)= 0): in the BCS regime
(cid:18) F z
+(cid:88) ∆ (cid:88) Vn1n2c† c† εB/εF (cid:28)1, we have:
qN N kn1↑ q−kn2↓
q,N k,n1,n2 ∆ (cid:114)2ε (cid:18) ε (cid:19) µ ε (cid:18) ε (cid:19)
+∆∗qN (cid:88) VNn3n4cq−k(cid:48)n3↓ck(cid:48)n4↑− |∆qgN|2(cid:19). εF (cid:39) εFB 1+ 8ωFz , εF (cid:39)1− 2εBF 1+ 4ωFz .
k(cid:48),n3,n4 Thus, ∆ is enhanced by the confinement in this regime
We further assume that the ground state has a uniform while µ is suppressed. This trend is also observed in the
orderparameterwithoutnodessothat∆ =δ δ ∆ . full calculation involving many levels (see Figs. 1 and 2).
qN q0 N0 0
In this case, Eq. (5) only contains a single unknown pa- Perturbing away from the 2D limit, we must now in-
rameter ∆0, so it can be diagonalized to yield cludemultiplelevelsoftheconfinement. Ingeneral,HˆMF
must be diagonalized numerically to obtain E and the
kn
Hˆ =(cid:88)((cid:15) −µ−E )−∆20+(cid:88) E γ† γ , (6) quasiparticle amplitudes for a given µ and ∆. Equiva-
MF kn kn g kn knσ knσ lently,onecansolvetheBogoliubov-deGennesequations
k,n k,n,σ
self-consistently,butitisconsiderablyfastertominimize
where E are the quasiparticle excitation energies. The the energy (cid:104)Hˆ (cid:105) directly and it also allows us to take
kn MF
quasiparticle creation and annihilation operators are re- intoaccountupto100levels. Indeed,wefindthathigher
3
2 orderparameter∆throughouttheBCS-BECcrossoverin
2D 1
3D BEC a quasi-2D Fermi gas, as depicted in Figs. 1 and 2. The
(cid:161) /(cid:116) = 0.1 lowest density we consider (ε /ω = 0.1) corresponds
F z F z
1.6 (cid:161) /(cid:116) = 0.5 approximatelytotheexperimentsofRef.[3–6]. Notethat
F z
(cid:161) /(cid:116) = 1.0 in the extreme BCS and BEC limits, we must have µ→
F F z
(cid:161) ε and µ → −ε /2, respectively, a feature which holds
/2]/B1.2 0.1 1 10 aFcrossalldimensBions. IntheBCSregime,thebehavioris
+ (cid:161) in qualitative agreement with the two-level calculation:
µ 0.8 µissuppressedand∆isenhancedwithrespecttothe2D
[
result, with the deviation from 2D being increased with
increasing ε /ω . However, multiple levels are required
F z
0.4
tocorrectlycapturethedependenceonε /ε asε shifts
B F F
away from zero. We also see in Fig. 1 that the relative
chemicalpotentialµ+ε /2exhibitsaverysteepgradient
0 B
0 1 2 3 4 as ε → 0 for ε /ω (cid:38) 0.5. This illustrates how higher
(cid:161) /(cid:161) B F z
B F levels can lead to a strong deviation from 2D even when
ε <ω and ε /ε (cid:28)1.
FIG. 1: (Color online) Chemical potential µ measured with F z B F
For larger ε /ε , ∆ eventually becomes suppressed
respect to half the binding energy ε for several values of B F
B
compared to the 2D result and appears to approach the
the Fermi energy ε /ω . The dashed line is the 2D mean-
F z
fieldresult[14],whilethedottedcurveisthe3DBECresult, 3Dmean-fieldcurveintheBEClimit(Fig.2). Thechem-
µ + εB (cid:39) −2√2εF(cid:113)εF. Inset: Asymptotic behaviour in icalpotential,however,alwaysremainslowerthanthe2D
2 3π εB result, and has a behavior in the BEC regime that is in-
the BEC regime ε /ε > 1 plotted on a logarithmic scale.
F B
termediate between 2D and 3D mean field. In the limit
The solid straight lines are straight line fits to the data with
gradient −1/3. Error bars for the numerical data are within εB/εF (cid:29)1, the pairing gap is no longer given by ∆ and
symbol size. thepropertiesoftheBosesuperfluidareinsteadencoded
in µ. In particular, the relative quantity µ+ε /2 yields
B
the mean-field energy for the repulsion between dimers.
3
ReferringtoFig.1(inset),weseethatittendstozeroas
2.5 a power law with increasing ε /ε , similarly to 3D and
B F
incontrasttothe2Dmean-fieldresult. Thisisconsistent
2 withadimensionalcrossoverto3Ddimersonceε (cid:38)ω .
B z
F However, we obtain a power of −1/3 rather than −1/2
(cid:161)1.5
(cid:54)/ as expected from 3D mean field theory. Indeed, a power
1 of −1/2 is also expected for weakly interacting bosons
confined to quasi-2D since their mean-field energy scales
0.5 as a /l ∼ (cid:112)ω /ε [18]. The discrepancy is likely due
s z z B
to the fact that our mean-field approximation does not
0
0 1 2 3 4 allowforthescatteringofdimersinthetransversedirec-
(cid:161) /(cid:161)
B F tionsincetheyareconstrainedtobeintheN =0center
of mass mode. The repulsion between dimers has also
FIG. 2: (Color online) Behavior of the order parameter been discussed in the context of a two-channel model for
∆ throughout the BCS-BEC crossover for different values
the quasi-2D system [19].
of ε /ω . The dashed curve is the 2D mean-field result
F√z Deviations from 2D behavior will also be apparent in
∆ = 2ε ε [14], while the dotted curve is the 3D BEC
B F
(cid:113) (cid:16) (cid:17)1 experimental probes of the quasi-2D superfluid. Typi-
result, ∆(cid:39)εF 31π6 2εεBF 4. The key for the numerical data cally, investigations of pairing have exploited RF spec-
is the same as in Fig. 1. troscopy [20], where atoms in one hyperfine spin state
(e.g. ↓) are transferred via an RF pulse to another hy-
perfinestatethatisinitiallyunoccupied. Intheidealsce-
harmoniclevelsareimportantevenforweakinteractions nario where the final state is non-interacting, the mean-
once ε /ω shifts away from zero. For the values of ε field transition rate or RF current is given by
F z B
and ε considered in this paper, we can in fact extrapo-
F (cid:88)
I (ω)∝ |v |2δ((cid:15) −µ+E −ω) (11)
latetheresults forµ and∆toan infinite numberofhar- RF kn(cid:48)n kn(cid:48) kn
monic levels since we find that they both scale linearly k,n(cid:48),n
with the inverse of the number of levels in this limit. whereω isthefrequencyshiftrelativetothebaretransi-
By incorporating an infinite number of levels, we can tion frequency between hyperfine states. Here, the onset
determine the evolution of the chemical potential µ and frequencyoftheRFspectrumcorrespondstoE −
k=0,n=0
4
µandisassociatedwiththepairinggapofthesuperfluid. 1.6
2D
In the 2D case, (11) reduces to IRF(ω) ∝ ∆ω22Θ(ω−εB) 1.4 (cid:161)(cid:161)F//(cid:116)(cid:116)z == 00..15
and thus the RF pairing gap is simply εB, as noted by (cid:161)F/(cid:116)z = 1.0
Sommer et al. [7]. Perturbing away from 2D, we find 1.2 F z
(cid:161) /(cid:116) = 1.5
that the RF spectrum can be substantially modifed by F z
B
the higher confinement levels. In Fig. 3, we see that the )/(cid:161) 1
µ
strongest effects are in the BCS regime, where the pair- -
=00.8 2
ieinnngghaggnaacppemciasenninteixtiisasltnlyoetveensnuhrawpnhrciesenidntgchoegmrivepeiansrnethdoatttwo,oiεn-Bb3o.DdyS,uabcophuanairdn- (Ek=0 n0.6 E/(cid:116)knz1.51 n = 2
state. However, once εF (cid:38) ωz, the pairing gap drops 0.4 0.5 n = 0
below ε and even becomes negative for small enough n = 1
B 0
ε /ε . This is because the coupling between n=0 and 0.2 0 0.4 0.8 1.2 1.6
B F k/k
n = 2, and the associated level repulsion (see inset of F
0
Fig. 3), reduces the energy E . In this case, the 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4
k,n=0
(cid:161) /(cid:161)
lowest energy quasiparticle contains a smaller fraction of B F
the n = 0 harmonic level and the RF peak is instead
FIG. 3: (Color online) The pairing gap measured using RF
dominated by the n = 2 quasiparticle. Thus, the onset
spectroscopy as a function of ε /ε for a range of differ-
frequencyisnolongeranaccuratemeasureofpairing, as B F
ent ε /ω . In the BCS regime, it deviates substantially
F z
we can see in Fig. 4. Note that the deep lattices used in
from the 2D result with increasing ε /ω . In the BEC
F z
Ref.[7]correspondtoεF/ωz ≈0.03andthusthepairing limit, Ek=0,n=0−µ must always approach εB, regardless of
gap will lie very close to the 2D result, as was observed. ε /ω . Inset: Lowestquasiparticledispersionsforε /ω =1,
F z F z
ε /ε =0.1. Notethatthereislevelrepulsionbetweenn=0
B F
On the other hand, the experiments in Ref. [8] cor- and n = 2, but no avoided crossing between the n = 0 and
n=1 dispersions.
respond to ε /ω (cid:39) 1.5 and thus the RF spectrum in
F z
the BCS regime will be strongly affected by the confine-
ment. In particular, we see in Fig. 4 that the RF peak 1
is shifted to higher frequencies compared to the 2D case quasi-2D (cid:161)F/(cid:116)z = 1.5
and develops more structure at lower frequencies. Fur- 0.8 2D
thermore, the pairing gap in the BEC regime appears to
)(cid:116) 0.6
be less sensitive to confinement and closer to the 2D re- ( (cid:161) /(cid:116) = 0.5
F F z
sult (Fig. 3) since it is dominated by two-body physics. IR0.4
These features are all consistent with the experimental
observations [8]. We note that finite temperature may 0.2
also play a role in these experiments — see Ref. [21] for
0
an alternative explanation based on fermionic polarons. 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5
(cid:116) (cid:116)
However,theshiftduetoconfinementappearstobesub-
stantialatT =0,withadirectionthatisconsistentwith
FIG. 4: Radio frequency spectra of the quasi-2D Fermi gas
experiment, and therefore it cannot be disregarded. In- in the BCS regime with ε /ε = 0.2 for ε /ω = 0.5,1.5.
B F F z
deed, it has also been shown that effects due to confine- TheRFcurrentI isscaledsothatthepeakvalueisalways
RF
ment can be significant in spin-polarized quasi-2D Fermi 1. The peak is strongly shifted towards higher frequencies
gases [22]. compared to the 2D result when εF/ωz (cid:39)1.5. Note that the
spectra are calculated assuming that the final state is non-
interacting.
To conclude, we have constructed a mean-field theory
for the quasi-2D Fermi gas that is able to capture the
deviationsfrom2Dbehaviorresultingfromconfinement.
We expect it to provide a benchmark for further inves-
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