Dimensional crossover in a strongly interacting ultracold atomic Fermi gas

2 MB·

by  Umberto Toniolo

Dimensional crossover in a strongly interacting ultracold atomic Fermi gas
Umberto Toniolo,1 Brendan C. Mulkerin,1 Chris J. Vale,1 Xia-Ji Liu,1,2 and Hui Hu1
1Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne 3122, Australia
2Kavli Institute for Theoretical Physics, UC Santa Barbara, USA
(Dated: January 31, 2017)
We theoretically explore the crossover from three dimensions (3D) to two (2D) in a strongly in-
teracting atomic Fermi superfluid through confining the transverse spatial dimension. Using the
gaussian pair fluctuation theory, we determine the zero-temperature equation of state and Lan-
dau critical velocity as functions of the spatial extent of the transverse dimension and interaction
7
strength. In the presence of strong interactions, we map out a dimensional crossover diagram from
1
the location of maximum critical velocity, which exhibits distinct dependence on the transverse di-
0
mensionfrom2Dtoquasi-2D,andto3D.Wecalculatethedynamicstructurefactortocharacterize
2
thelow-energyexcitationsofthesystemandproposethattheintermediatequasi-2Dregimecanbe

n experimentally probed using Bragg spectroscopy.
a
J PACSnumbers: 03.75.Ss,03.70.+k,05.70.Fh,03.65.Yz

0
3
Recentbreakthroughsinunderstandingstronglyinter-

] actingultracoldatomicFermigasesatthecrossoverfrom 2D quasi-2D
s 11
a Bose-Einstein condensates (BEC) to Bardeen-Cooper-
g Schrieffer (BCS) superfluids [1–4] have attracted enor-
3D
- mous attention from diverse fields of physics [5–7]. Due x 00
t a
uan tsoionthaelituynparnedceidnetnetreadtoamccicurianctyerianctcioonntr[o8l,lin9g],tshiegndiifimcaennt- )3Dmvc 11 1111....2424 kkF2FD//kkFF33DD
q progress has been made to realize systems in the 2D /a −− 3DF 11
at. laimniutm[1b0e–r2o6f].inIttrtihguusinpgrolvowid-edsimaenneswiopnaarlapdhigemnotmoeenxap,loinre- (lz 22 k/kF00..88
−−
m 00..66
cludingtheabsenceofatruelong-rangeorderatnonzero MF 2D quasi-2D
- temperature [27, 28], the existence of quasi-condensates 33 GPF 00..4411 1100
d
−−
due to the Berezinskii-Kosterlitz-Thouless mechanism
n 11 1100
o [29–31], the disruptive role of pair fluctuations around η
c the mean-field (MF) [32–40], and the possible observa-
[
tion of exotic imbalanced superfluidity [41–43]. These
Figure 1: (color online). The dimensional crossover diagram,

1 unusual features lie at the heart of many technologically tunedbythedimensionparameterη(inlogarithmicscale)and
v interestingmaterialssuchashigh-temperaturesupercon- the value of the interaction strength, (lz/a3D)vcmax, at which
7 ductors[44],wherethedimensionalcrossoverfrom3Dto the Landau critical velocity peaks. The red solid line (with
47 2D is dictated by the ratio of the Cooper pair size to the cthireclGesP)FanadnbdluMeFdatshheeodrileins,ersehsopwec(tlizv/elay3.D)Tvcmhaexirpdreisdtiicntcetddbey-
thickness of the superconducting layer.
8 pendencesonηenablesustoidentifythe2Dand3Dregimes,
0 Despiterapidexperimentaladvances,thefundamental andthequasi-2Dregimeinbetween. Theinsetshowsthedi-
.
1 criteriaforreachingthestrict2DregimeattheBEC-BCS mensional crossover diagram in the non-interacting case, de-
0 terminedfromthefreeFermigasnumberequation(seetext).
crossoverarestillnotwellunderstood. Experimentally,a
7
2DFermigasisrealizedbyfreezingtheatomicmotionin
1
: the transverse direction using a single highly-oblate har-
v
monictrap[13,24]oratightone-dimensionalopticallat- from 3D to 2D is challenging due to the strong correla-
i
X tice[10,11,16,17,20]. Intheabsenceofinteractions,the tions [45]. To date, an interacting quasi-2D Fermi gas
r 2D condition is easy to clarify within the single-particle has only been studied in the highly imbalanced polaron
a
picture: the chemical potential µ and temperature k T limit [46] or by using mean-field approach that is known
B
of the system should be smaller than the characteristic to break down in the 2D limit [47, 48].
energy scale (cid:126)ω along the transverse direction, so that In this Letter, we determine the dimensional crossover
z
all atoms stay in the lowest transverse mode [13]. With diagram (see Fig. 1), by considering a uniform strongly
strong interactions, the situation is less clear. Indeed, a interacting quasi-2D Fermi gas with periodic boundary
recent measurement of time-of-flight expansion indicates condition (PBC) in the tightly confined transverse di-
thatitisdifficulttodisplaythestrict2Dkinematicswhen rection. This configuration is motivated by the recent
the interaction becomes stronger [22]. Theoretically, the successful production of a box trapping potential that
dimensionalcrossoverofastronglyinteractingFermigas leadstoauniformBoseorFermigasinbulk[49,50]. We
2
apply a gaussian pair fluctuation (GPF) theory to ob- needs to be regularized and related to a physical observ-
tain the zero-temperature equation of state (Fig. 2) [39] able of the system. We achieve this by relating the bare
and Landau critical velocity (Fig. 3) at the dimensional interaction strength g to the bound state energy B [57],
0
crossover. For a given dimensional parameter η k3Dl ,
≡ F z 1 1
where lz is the periodic length of the confining poten- = , (3)
tial in the transverse direction and kF3D ≡ (3π2n)1/3 is g k(cid:88),kz 2((cid:15)k+(cid:15)kz)+B0
the three-dimensional (3D) Fermi momentum of the gas
with density n, we determine the interaction strength at where (cid:15)k = (cid:126)2k2/(2M) and the sums on (k,kz) carry a
which there is a maximum of the Landau critical veloc- volumefactorthatgoesto(2π)2lz atthethermodynamic
ity[51],(lz/a3D)vmax,wherea3Disthe3Ds-wavescatter- limit. Inordertorecoverthe3Dlimit,werequirethetwo-
inglength. AFermc isuperfluidismostrobusttoexternal bodyT-matrixinthedimensionalcrossoverbeequivalent
excitationsatthismaximum,whichisfound,in3D,close to its 3D counterpart in the limit lz . This implies
→ ∞
tounitarity[52–54]. Weobtainaregimewherethemax- that the binding energy, B0, can be analytically related
imum of the critical velocity in the BEC-BCS crossover to the 3D scattering length a3D, according to [57, 58],
depends on the logarithm of η for η < 2, denoting a
(cid:126)2 elz/(2a3D)
2D regime (i.e., the long-dashed line in Fig. 1). Also, B =4 arcsinh2 . (4)
0 Ml2 2
(lz/a3D)vmax depends linearly on η for η >8, denoting a (cid:18) z(cid:19) (cid:20) (cid:21)
c
3Dregime. Theregionthatlinkstheseregimesisdefined
It is also possible to define a 2D binding energy, (cid:15)2D
asquasi-2Dandhaspropertiesdistincttothe2Dand3D (cid:126)2/(Ma2 ), find the equivalence between the scattBerin≡g
limits. 2D
T-matrix and the 2D T-matrix as l 0, and show an-
z
Theoreticalframework. —Westartbydefiningvarious alytically that B =(cid:15)2D in the 2D lim→it.
Fermi momenta. We consider a s-wave two-component 0 B
Wesolvethemany-bodyHamiltonianEq. (2)byusing
Fermi gas at zero temperature where the transverse di-
the zero-temperature GPF theory, which provides rea-
rection is confined with periodic length l , implying the
z sonable quantitative predictions for equation of state in
discretization of momentum in the z-direction, k =
z both 2D [38] and 3D [59, 60]. The theory takes into
2πn /l ,foranyintegern . TheFermimomentumk of
z z z F account strong pair fluctuations at the gaussian level
the dimensional crossover system can then be defined as
on top of mean-field solutions [55, 56] and hence we
themaximallyallowedmomentumintheaxialdirection:
separate the thermodynamic potential into two parts:
Ω=Ω +Ω . The mean-field part is [38],
1 nmax 2πn 2 MF GF
n= k2 z , (1)
2πlz nz=(cid:88)−nmax(cid:34) F −(cid:18) lz (cid:19) (cid:35) ΩMF = ∆g2 + (ξk,kz −Ek,kz), (5)
wheren isthelargestintegersmallerthank l /(2π). k(cid:88),kz
max F z
It is useful to first examine the dimensional crossover where ξk,kz =(cid:15)k+(cid:15)kz −µ, Ek,kz = ξk2,kz +∆2, and
diagram for an ideal Fermi gas, as shown in the inset the order parameter ∆ is determined self-consistently
(cid:113)
of Fig. 1. At large lz (or η), kF approaches the 3D using the mean-field gap equation, ∆ k,kz[((cid:15)k +(cid:15)kz +
Fermi momentum k3D, as anticipated. In the limit of B /2)−1 E−1 ] = 0, ensuring the gapless Goldstone
F 0 − k,kz (cid:80)
small lz, instead, kF coincides with a 2D Fermi momen- mode [52]. The pair fluctuation part is given by (Q
tum kF2D ≡ √2πn2D = 2η/(3π)kF3D, where the column (q,qz,ω)) [38], ≡
density n nl . An ideal 2D Fermi gas is thus real-
2D z
tizheedlowwheesnt ktrF≡an=svkerF2sDeomr(cid:112)oηde<is√3o6cπcu(cid:39)pie5d.7.,Tfohriswshimicphleon2lDy ΩGF = (cid:119)∞d2ωπ ln M11(MQ)CM(1Q1)(M−QC)(−MQ)212(Q) ,
condition is not applicable in the presence of strong in- q(cid:88),qz 0 (cid:20) 11 11 − (cid:21)
(6)
teractions, a situation that we shall consider below. A
with the matrix elements,
strongly interacting Fermi gas with contact interactions
between unlike fermions can be described by a single- 1 u2u2 v2v2
M (Q) = + + − + − ,
channel Hamiltonian density [32, 38, 55, 56], 11 g ω E E − ω+E +E
k(cid:88),kz(cid:18) − +− − + −(cid:19)
= ψ¯σ(r) 0ψσ(r) gψ¯↑(r)ψ¯↓(r)ψ↓(r)ψ↑(r), (2) M (Q) = u+u−v+v− + u+u−v+v− ,
H H − 12
−ω E E ω+E +E
σ(cid:88)=↑,↓ k(cid:88),kz(cid:18) − +− − + −(cid:19)
where ψσ(r) are the annihilation operators for each spin MC (Q) = 1 + u2+u2− . (7)
wstiatthe,aHto0mi=c m−a(cid:126)s2s∇M2/,(2µMi)s−thµe cishetmheicafrleepoHteanmtiialtlo,nainand 11 g k(cid:88),kz ω−E+−E−
g >0denotesthebareinteractionstrength. Thecontact Here, we use the notations E E , u2 =
± ≡ k±q/2,kz±qz/2 ±
potentialisaconvenientchoiceofinteraction,howeverit (1+ξ /E )/2 and v2 = 1 u2
k±q/2,kz±qz/2 k±q/2,kz±qz/2 ± − ±
3
1
1 (a) 3 η=4
η=3
εF2 D 0.8 vc/vF3D η=2
Bµ+/ε0F2000)...468 η ∆/01 −3−1l.z5/a03D1.5 3 3D3/vc/v,sFF00..46 {csη/e=vxF3pD1.
( 6 vc 0.2
4
0.2
3
2.5 1.33 0
0 2 1 4 3 2 1 0 1 2 3
− − − −
4 3 2 1 0 1 2 3 l /a
− − − −l /a z 3D
z 3D
0.6 2D4 32D0 Figure3: (coloronline). Landaucriticalvelocityvccompared
/εF 0).4 321 3D/εF0).7 468 vawaifltuuhnetschtaieotnslpaoerfgedelzηo/fam3sDoauyanbtdedccisffoaemrceprnoatsrsevdtahlwueeiBtshEotCfh-ηeB.eCTxSpheecrritomhsseeoonvrteeatrli,craeas-l
B02 B020.4 sults of the critical velocity, obtained by Weimer et al. [61]
+ + fora3DtrappedFermigaswith(cid:15)3D/(cid:126)ω 4.2,whichinour
µ 0.2 µ F z
( ( dimensional crossover units roughly corres(cid:63)ponds to a dimen-
(b) 0.1 (c) sion parameter η = kF3DlzHO = (cid:112)(cid:126)(cid:15)3FD/(Mωz) = 2.9. Here,
0 v3D =(cid:126)k3D/M is the 3D Fermi velocity.
2 1 0 1 2 2 1 0 1 2 F F
−ln(− B /(2ε )) − (−k3Da ) 1
0 F F 3D −
√ system is effectively in the 3D limit for the entire BEC-
Figure2: (coloronline). (a)Thedimensionlessshiftedchemi- BCS crossover. Thus, we see a distinct quasi-2D regime
calpotential,(µ+B /2)/ε ,asafunctionofl /a atvarious
0 F z 3D forthedimensionparameter2<η <8. Thisobservation
dimension parameters (η =1 6), from the quasi-2D to 2D
∼ is confirmed below by the calculation of Landau critical
regime. The inset shows the order parameter ∆/ε . (b) The
F
velocity.
chemical potential near the 2D limit, replotted as a function
of ln((cid:112)B /(2ε )). (c) The chemical potential near the 3D Landau critical velocity. — Within the GPF theory,
0 F
limit in units of ε3D is shown as a function of 1/(k3Da ). we can calculate the critical velocity of the superfluid
F F 3D
through both the BEC-BCS and dimensional crossover.
Once we know the dispersion of in-plane (q =0) collec-
z
[38, 55, 56]. The chemical potential is found by solving tive modes ω0(q = q), which corresponds to the poles
| |
the number equation, n=−∂Ω/∂µ. of[M11(Q)M11(−Q)−M212(Q)]−1, foragivensetofthe
Equation of state. — In Fig. 2(a), we report the di- parametersη andlz/a3D,wecomputethespeedofsound
of the superfluid, c = lim ω (q)/q, and the pair-
mensionless shifted chemical potential (µ + B /2)/ε , s q→0 0
0 F
whereεF =(cid:126)2kF2/(2M)istheFermienergy,attheBEC- breaking velocity vpb = [(cid:126)2( ∆2+µ2 −µ)/M]1/2 [52].
BCS crossover tuned by l /a and at the dimensional According to Landau’s criterion, the critical velocity in
z 3D (cid:112)
crossover tuned by η = 1 6. For all values of η, the the BEC-BCS crossover is then given by,
∼
dependence of the chemical potential on η remains sim-
ω (q)
0
ilar to the typical decreasing slope found in 3D [55, 56]. vc =min =min cs,vpb . (8)
q≥0 q { }
However, as η decreases the curves shift towards nega-
tive values of l /a . The inset plots the order param- InFig. 3,wepresentthespeedsofsoundc andcritical
z 3D s
eter, ∆/ε , and we see a similar behavior to the chem- velocities v for dimensions η = 1 4 as a function of
F c
∼
ical potential as we decrease η. As η approaches the theinteractionstrengthl /a . Thecriticalvelocityofa
z 3D
2D limit, we can compare the magnitude of the chemi- 3DFermisuperfluidattheBEC-BCScrossoverhasbeen
cal potentials with the 2D case through the interaction experimentallymeasuredinaharmonictrap[54,61]and
parameter ln( B /(2ε )), as shown in Fig. 2(b). We can be compared with our results using the transverse
0 F
plot a range of dimensions, η = 1 4, and the 2D re- harmonicoscillatorlength,lHO = (cid:126)/(mω ),asinputto
(cid:112) ∼ z z
sult(blackdashed),andseeacleartrendofthechemical determineη. InRef. [61],the3Dregimeisapproximately
potential approaching the 2D result for η (cid:46) 2. In Fig. reachedwith(cid:15)3D/((cid:126)ω ) 4.2that(cid:112)correspondstoη 2.9
F z (cid:39)
2(c), we compare the chemical potential to the 3D result and the data match qua(cid:63)litatively well with the predicted
(black short dashed), where we plot the chemical poten- Landau critical velocity (i.e., at η =3).
tial in units of the 3D Fermi energy ε3D as a function In 3D the BCS regime displays a large speed of sound
F
of 1/(k3Da ). We find excellent agreement in the BEC and a smaller pair-breaking velocity that limits the crit-
F 3D
limit for η (cid:38) 4 and by η > 8 the dimensional crossover ical velocity [52, 53]. On the BEC side, close to the 3D
4
12 ωth(3.2kF,0) 0.3 ωth(0.6kF,0) 3 ωth(0,2π/lz)
S(3.2k,0,ω)S(3.2k,0,ω)FF 48 ωth(3.2kF,4π/lz) (0.6k,0,ω)(0.6k,0,ω)FF0.15 S(0,2π/l,ω)S(0,2π/l,ω)zz 12 ωth(4π/lz,2π/lz)
0 SS 0
0
0.5 1
2
ω/(ωar) 11.5 2 4 2 0lz/a3−D2 −4 ω(/bω)r 4 6 4 2 0lz/a3D−2 −4 ω/(ωc)r 2 3 4 4 2 0lz/a3−D2 −4
Figure4: (coloronline). ThedensitydynamicstructurefactorS(Q ,ω)scaledbytheratioω /N,whereω istherecoilenergy
r r r
and N the particle number, in the quasi-2D regime for η =4 at various interaction strengths l /a , with the in-plane recoil
z 3D
momentum Q =(3.2k ,0) (a), Q =(0.6k ,0) (b) and transverse recoil momentum Q =(0,2π/l ) (c). The spectral width
r F r F r z
of the Bogoliubov-Anderson phonon peak is illustrated by the height of the delta function.
unitarity,thepair-breakingvelocitybecomesequaltothe we have set Q = (q ,q ), a combination of an in-plane
r r z
speed of sound, which is referred to as the most robust momentumq ,andatransverseexcitation,q =2πn /l
r z z z
configuration of the BEC-BCS crossover [52]. Beyond for fixed integer n . We note that the calculation of the
z
this point the speed of sound becomes the critical veloc- dynamic structure factor S(Q ,ω) within the GPF the-
r
ity,markingthesystemundergoingmacroscopicconden- oryisnotoriouslydifficult[66],soweinsteadusetheran-
sation. The 2D and quasi-2D critical velocities behave dom phase approximation within the mean-field frame-
similarly to the 3D case. However, the tuning point of work [67].
theBEC-BCScrossover(lz/a3D)vmax –atwhichthecrit- In Fig. 4, we plot the dynamic structure factor in
c
ical velocity peaks – shows a non-trivial dependence on the quasi-2D regime at η = 4, normalized by the num-
the dimensional parameter η. This enables us to char- ber of particles N and recoil energy ω = (cid:126)Q2/(2M)
r r
acterize the dimensional crossover diagram in the pres- for three different recoil momenta, (a) Q = (3.2k ,0),
r F
ence of strong interactions, as shown in Fig. 1. In the (b) Q = (0.6k ,0) and (c) Q = (0,2π/l ). One ob-
r F r z
region 0 η < 2, the 2D regime, we see the logarith- serves in Figs. 4(a)-(b) that the response is similar to
≤
mic dependence of the critical velocity maximum with the 3D case [67], showing the characteristic peaks in the
respecttoη,withthepeakofthecriticalvelocityin2Dat continuumspectrumforω >ω ,andthepresenceofthe
th
ln(kF2Da2D)(cid:39)1.08[51,62]. Moreover,alinearbehavioris phononmode. Wenotetheappearanceofasecondpeak,
observed in the nearly 3D regime with η >8 placing the markedbyω (q ,4π/l )(greendashed)inFig. 4(a),cor-
th r z
peak of the critical velocity in 3D at 1/(kF3Da3D)(cid:39)0.056 responding to the generation of a transverse excitation.
[51, 52]. In between (2 < η < 8,), the maximum of the The response at ω (q ,2π/l ) is absent, due to the need
th r z
critical velocity lies in the interval 1 < lz/a3D < 0.67 of the system to excite two modes along z with opposite
−
and (lz/a3D)vmax varies non-monotonically with η. We momenta,inordertoconservethetotalmomentum. The
c
identify this as the quasi-2D regime, consolidating the same structure, present in Fig. 4(b), is not resolved due
previous conclusion made from equation of state. to the energy required at this momentum.
Probing the quasi-2D regime. — A practical way to The dynamic response of the system, for a trans-
measure both the speed of sound, cs, and the order pa- verse recoil momentum Qr = (0,2π/lz), is shown in
rameter,∆,isviaBraggspectroscopy. Thespectroscopic Fig. 4(c), and has a specific structure due to the
response probes the dynamic structure factor [63–65], quasi-2D regime. Conservation of total momentum
which in the case of a Fermi superfluid exhibits a peak forces in-plane excitations to place the second contin-
correspondingtotheBogoliubov-Andersonphononmode uum peak at ω (4π/l ,2π/l ) and gives no response at
th z z
andacontinuumofparticle-holeexcitations[52]. Dueto ω (2π/l ,2π/l ),whichwouldbreakmomentumconser-
th z z
the presence of a pairing gap in the excitation spectrum, vation. We expect this to be a signature of the quasi-
an external excitation of momentum Qr is collective if it 2D regime, as in 3D the pairing gap between box modes,
does not break pairs when it excites states with energy 2π/l ,goestozeroandtheisolatedpeaksmergeinacon-
z
below the threshold, tinuousstructure,whilein2D,thepeakω (4π/l ,2π/l )
th z z
moves too far away from the main spectrum.
2∆ µ>0 and (cid:126)2Q2 8Mµ
ω (Q )= r ≤ , Conclusions. — In summary, we have examined the
th r 2 µ2 +∆2 otherwise
(cid:40) Qr role of dimension in a strongly interacting Fermi super-
(cid:113) (9) fluid by treating the transverse confinement with PBC.
where µ = µ (cid:126)2Q2/(2M), and for our dimensional We have mapped out a dimensional crossover diagram
Qr − r
crossoversystemwithfinitetransverseperiodiclengthl , from the zero-temperature equation of state and have
z
5
quantitatively determined the boundaries between 2D, [18] V. Makhalov, K. Martiyanov, and A. Turlapov, Phys.
quasi-2D,and3DfromthelocationofmaximumLandau Rev. Lett. 112, 045301 (2014).
criticalvelocity. Thissetsaframeworkforcharacterizing [19] W. Ong, C. Cheng, I. Arakelyan, and J. E. Thomas,
Phys. Rev. Lett. 114, 110403 (2015).
the BCS-BEC crossover in quasi-2D, where the different
[20] M.G.Ries,A.N.Wenz,G.Zu¨rn,L.Bayha,I.Boettcher,
regimes of the superfluid can be experimentally probed
D. Kedar, P. A. Murthy, M. Neidig, T. Lompe, and S.
using Bragg spectroscopy. Our results are directly appli-
Jochim, Phys. Rev. Lett. 114, 230401 (2015);
cable to an interacting dimensional crossover Fermi gas [21] P. A. Murthy, I. Boettcher, L. Bayha, M. Holzmann, D.
realizedbyimposingaboxtrappingpotentialinthetight Kedar,M.Neidig,M.G.Ries,A.N.Wenz,G.Zu¨rn,and
confinementdirection[50], andweexpectourfindingsto S. Jochim, Phys. Rev. Lett. 115, 010401 (2015).
be qualitatively similar under harmonic transverse con- [22] P. Dyke, K. Fenech, T. Peppler, M. G. Lingham, S.
Hoinka,W.Zhang,S.-G.Peng,B.Mulkerin,H.Hu,X.-J.
finement.
Liu,andC.J.Vale,Phys.Rev.A93,011603(R)(2016).
This research was supported under Australian Re-
[23] K. Martiyanov, T. Barmashova, V. Makhalov, and A.
search Council’s Discovery Projects funding scheme Turlapov, Phys. Rev. A 93, 063622 (2016).
(project numbers DP140100637 and DP140103231) and [24] K. Fenech, P. Dyke, T. Peppler, M. G. Lingham, S.
Future Fellowships funding scheme (project numbers Hoinka, H. Hu, and C. J. Vale, Phys. Rev. Lett. 116,
FT130100815 and FT140100003). XJL was supported 045302 (2016).
in part by the National Science Foundation under Grant [25] I.Boettcher,L.Bayha,D.Kedar,P.A.Murthy,M.Nei-
dig,M.G.Ries,A.N.Wenz,G.Zurn,S.Jochim,andT.
No. NSF PHY11-25915, during her visit to KITP. All
Enss, Phys. Rev. Lett. 116, 045303 (2016).
numerical calculations were performed using Swinburne
[26] C. Cheng, J. Kangara, I. Arakelyan, and J. E. Thomas,
new high-performance computing resources (Green II). Phys. Rev. A 94, 031606(R) (2016).
[27] N.D.MerminandH.Wagner,Phys.Rev.Lett.17,1133
(1966).
[28] P. C. Hohenberg, Phys. Rev. 158, 383 (1967).
[29] V. L. Berezinskii, Sov. Phys. JETP 34, 610 (1972).
[1] D. M. Eagles, Phys. Rev. 186, 456 (1969). [30] J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181
[2] A. J. Leggett, in Modern Trends in the Theory of Con- (1973).
densed Matter, edited by A. Pekalski and J. Przystawa [31] L. Salasnich, P. A. Marchetti, and F. Toigo, Phys. Rev.
(Springer Verlag, Berlin, 1980), p.14. A 88, 053612 (2013).
[3] P.Nozi`eresandS.Schmitt-Rink,J.LowTemp.Phys.59, [32] M. Randeria, J.-M. Duan, and L.-Y. Shieh, Phys. Rev.
195 (1985). Lett. 62, 981 (1989).
[4] C.A.R.SadeMelo,M.Randeria,andJ.R.Engelbrecht, [33] S. Schmitt-Rink, C. M. Varma, and A. E. Ruckenstein,
Phys. Rev. Lett. 71, 3202 (1993). Phys. Rev. Lett. 63, 445 (1989).
[5] The Many-Body Challenge Problem (MBX) formulated [34] J.R.EngelbrechtandM.Randeria,Phys.Rev.Lett.65,
byG.F.Bertschin1999;G.A.Baker,Jr.,Phys.Rev.C 1032 (1990).
60,054311(1999);Int.J.Mod.Phys.B15,1314(2001). [35] R.Watanabe,S.Tsuchiya,andY.Ohashi,Phys.Rev.A
[6] S.Giorgini,L.P.Pitaevskii,andS.Stringari,Rev.Mod. 88, 013637 (2013).
Phys. 80, 1215 (2008). [36] M. Bauer, M. M. Parish, and T. Enss, Phys. Rev. Lett.
[7] M.RanderiaandE.Taylor,Annu.Rev.Condens.Matter 112, 135302 (2014).
Phys. 5, 209 (2014). [37] F. Marsiglio, P. Pieri, A. Perali, F. Palestini, and G. C.
[8] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. Strinati, Phys. Rev. B 91, 054509 (2015).
80, 885 (2008). [38] L. He, H. Lu¨, G. Cao, H. Hu, and X.-J. Liu, Phys. Rev.
[9] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. A 92, 023620 (2015).
Mod. Phys. 82, 1225 (2010). [39] B. C. Mulkerin, K. Fenech, P. Dyke, C. J. Vale, X.-J.
[10] K. Martiyanov, V. Makhalov, and A. Turlapov, Phys. Liu, and H. Hu, Phys. Rev. A 92, 063636 (2015).
Rev. Lett. 105, 030404 (2010). [40] G. Bighin and L. Salasnich, Phys. Rev. B 93, 014519
[11] M. Feld, B. Fro¨hlich, E. Vogt, M. Koschorreck, and M. (2016).
Ko¨hl, Nature (London) 480, 75 (2011). [41] G. J. Conduit, P. H. Conlon, and B. D. Simons, Phys.
[12] B. Fro¨hlich, M. Feld, E. Vogt, M. Koschorreck, W. Zw- Rev. A 77, 053617 (2008).
erger,andM.Ko¨hl,Phys.Rev.Lett.106,105301(2011); [42] S. Yin, J.-P. Martikainen, and P. Torma, Phys. Rev. B
[13] P. Dyke, E. D. Kuhnle, S. Whitlock, H. Hu, M. Mark, 89, 014507 (2014).
S. Hoinka, M. Lingham, P. Hannaford, and C. J. Vale, [43] U. Toniolo, B. C. Mulkerin, X.-J. Liu, and H. Hu, Phys.
Phys. Rev. Lett. 106, 105304 (2011). Rev. A 95, 013603 (2017).
[14] A.A.Orel,P.Dyke,M.Delehaye,C.J.Vale,andH.Hu, [44] P.A.Lee,N.Nagaosa,andX.-G.Wen,Rev.Mod.Phys.
New J. Phys. 13, 113032 (2011). 78, 17 (2006).
[15] M.Koschorreck,D.Pertot,E.Vogt,B.Fro¨hlich,M.Feld, [45] J. Levinsen and M. M. Parish, Annual Review of Cold
and M. Ko¨hl, Nature (London) 485, 619 (2012). Atoms and Molecules 3, 1 (2015).
[16] A.T. Sommer, L.W. Cheuk, M.J.H. Ku, W.S. Bakr, [46] J.LevinsenandS.K.Baur,Phys.Rev.A86,041602(R)
and M.W. Zwierlein, Phys. Rev. Lett. 108, 045302 (2012).
(2012). [47] H. Hu, Phys. Rev. A 84, 053624 (2011).
[17] Y. Zhang, W. Ong, I. Arakelyan, and J. E. Thomas, [48] A.M.FischerandM.M.Parish,Phys.Rev.B90,214503
Phys. Rev. Lett. 108, 235302 (2012). (2014).
1
[49] A. L. Gaunt, T. F. Schmidutz, I. Gotlibovych, R. P. 025302 (2014).
Smith,andZ.Hadzibabic,Phys.Rev.Lett.110,200406 [58] D. S. Petrov, and G. V. Shlyapnikov, Phys. Rev. A 64,
(2013). 012706 (2001).
[50] B. Mukherjee, Z. Yan, P. B. Patel, Z. Hadzibabic, T. [59] H. Hu, P. D. Drummond, and X.-J. Liu, Nat. Phys. 3,
Yefsah,J.Struck,andM.W.Zwierlein,arXiv:1610.10100 469 (2007).
(2016). [60] H. Hu, X.-J. Liu, and P. D. Drummond, New J. Phys.
[51] See Supplemental Material at [URL will be inserted by 12, 063038 (2010).
publisher] for alternative characterization of the dimen- [61] W.Weimer,K.Morgener,V.P.Singh,J.Siegl,K.Hueck,
sionalcrossoverandtheasymptoticbehavioroftheLan- N. Luick, L. Mathey, and H. Moritz, Phys. Rev. Lett.
dau critical velocity peak in the 3D and 2D regimes. 114, 095301 (2015).
[52] R.Combescot,M.Y.Kagan,andS.Stringari,Phys.Rev. [62] H.Shi,S.Chiesa,andS.Zhang,Phys.Rev.A92,033603
A 74, 042717 (2006). (2015).
[53] R. Sensarma, M. Randeria, and T.-L. Ho, Phys. Rev. [63] A. Brunello, F. Dalfovo, L. Pitaevskii, S. Stringari, and
Lett. 96, 090403 (2006). F. Zambelli, Phys. Rev. A 64, 063614 (2001).
[54] D.E.Miller,J.K.Chin,C.A.Stan,Y.Liu,W.Setiawan, [64] G. Veeravalli, E. Kuhnle, P. Dyke, and C. J. Vale, Phys.
C.Sanner,andW.Ketterle,Phys.Rev.Lett.99,070402 Rev. Lett. 101, 250403 (2008).
(2007). [65] S.Hoinka,M.Lingham,K.Fenech,H.Hu,C.J.Vale,J.
[55] H.Hu,X.-J.Liu,andP.D.Drummond,Europhys.Lett. E. Drut, and S. Gandolfi, Phys. Rev. Lett. 110, 055305
74, 574 (2006). (2013).
[56] R.B.Diener,R.Sensarma,andM.Randeria,Phys.Rev. [66] L. He, Ann. Phys. (N.Y.) 373, 470 (2016).
A 77, 023626 (2008). [67] P.Zou,E.D.Kuhnle,C.J.Vale,andH.Hu,Phys.Rev.
[57] M. T. Yamashita, F. F. Bellotti, T. Frederico, D. V. Fe- A 82, 061605(R) (2010).
dorov, A. S. Jensen, and N. T. Zinner, J. Phys. B 48,
Supplemental Material for
“Dimensional crossover in a strongly interacting ultracold atomic Fermi gas”
I. ALTERNATIVE CHARACTERIZATIONS OF THE DIMENSIONAL CROSSOVER
The dimensional crossover is tuned by the quasi-2D BCS-BEC crossover parameter, l /a , computed at the
z 3D
position where the Landau critical velocity has a maximum. Here, we present alternative chacterizations by using
the ratio between the pairing order parameter ∆ and the chemical potential µ, or the ratio between the pairing order
parameter and the Fermi energy ε .
F
C
11 E
B
00
D S
3 C
a B
/ 11
z −−
l vmax,GPF
c
22 ∆/µ=∆03D/µ03D
−− ∆/ε =1
F
33 2D quasi-2D 3D
−−
11 1100
η
Figure S1: The evolution of the tuning parameter l /a , computed at different conditions, as a function of the dimensional
z 3D
parameter η (in logarithmic scale). l /a is computed (i) when the Landau critical velocity has a maximum (red line with
z 3D
circles), (ii) when the ratio between the order parameter ∆ and the chemical potential µ reproduces the 3D typical values
∆0 =0.46ε3D and µ0 =0.4ε3D [55] (purple dot-dashed line), and (iii) when ∆=ε (solid line at the top right).
3D F 3D F F
In Fig. S1 we plot the critical values of l /a , across the dimensional crossover, when, (i) the Landau critical
z 3D
velocity has a maximum (circles), (ii) the ratio ∆/µ is equal to the 3D case (dashed-dotted) and (iii) when the order
parameter, ∆, is equal to the Fermi energy, ε . We observe that as expected, the ratio ∆/µ approaches the 3D limit
F
2
11 00
vmax,MF
(a) BEC BEC vmcax,GPF
c
00 ) 11 fit
) )F −−
D ε
3 2
Da 11 /( 22 (b)
3kF −− BCS B0 −− BCS
1/( −−22 vcmvcmaxa,x,GMPFF n( √33
fit l −−
33 2D quasi-2D 3D 2D quasi-2D 3D
−−
11 1100 11 1100
η η
Figure S2: The dimensional crossover diagram tuned by the dimensional parameter, η, and (a) the typical 3D BCS-BEC
crossover parameter, 1/(k3Da ), and (b) the typical 2D BCS-BEC crossover parameter, ln((cid:112)B /(2ε )). The fitted lines
F 3D 0 F
(tiny dotted) s(cid:112)how (a) the expected constant behaviour [1/(kF3Da3D)]vcmax (cid:39) 0.056 at η → ∞, and (b) the expected constant
behaviour [ln( B0/(2εF))]vcmax (cid:39)−1.08 at η→0.
for η , while the condition ∆ = ε has meaning only in the far 2D limit. Indeed, the condition ∆ = ε can be
F F
→ ∞
reached in 3D only at very large values of the tuning parameter 1/(k3Da ), in the deep BEC regime. We remark
F 3D
that the choice of the Landau critical velocity, as the most useful condition to characterize the crossover, allows a
complete independent description from both the 3D and 2D regimes, since the interaction effect is fully taken into
account in vmax.
c
II. LANDAU CRITICAL VELOCITY IN THE 3D AND 2D LIMITS
A. The 3D limit
We observe that, for a 3D Fermi gas, the MF theory predicts the critical velocity to be slightly on the BEC side
at approximately 1/(kF3Da3D)vcmax (cid:39)0.07 [52]. Figure 1 is expected to predict this behaviour when we restore the 3D
limit case for η . We consider the most general choice to descibe the BCS-BEC tuning parameter in this limit,
→∞
∞
l
z = a ηn. (S1)
n
a
(cid:18) 3D(cid:19)vcmax n=(cid:88)−∞
The limit of the proper 3D tuning parameter, 1/(k3Da ), is then given by
F 3D
∞
l 1
lim z = a ηn−1. (S2)
n
η→∞(cid:18)a3D(cid:19)vcmax η n=(cid:88)−∞
Every coefficient a for n<1 is negligible when η is large enough, while each term a =0, for n>1, would lead to a
n n
(cid:54)
divergence in the definition of the 3D peak for the critical velocity and it is therefore discarded. We fit the far right
hand side of Fig. 1 via a linear function,
l
z
=a +a η, (S3)
0 1
(cid:18)a3D(cid:19)vcmax(cid:12)η≥8
(cid:12)
and we included the a term due to the proximity of dat(cid:12)a to the quasi-2D regime when η 8. This leads to
0
(cid:39)
1
=a 0.056. (S4)
k3Da 1 (cid:39)
(cid:18) F 3D(cid:19)vmax
c
The behaviour in the 3D regime is shown in Fig. S2(a) that provides the same results of Fig. 1 with a change of scale
in the vertical axis from l /a to 1/(k3Da ).
z 3D F 3D
3
B. The 2D limit
In the 2D limit, we denote that
l
z
= (η), (S5)
a F
(cid:18) 3D(cid:19)vcmax
where the approximate form of (η) is to be determined. We observe from Fig. 2(a)-(b) that the proper BCS-BEC
F
crossover tuning parameter becomes ln[ B /(2ε )], where k = k2D, for η π61/3, and lim B /(2ε ) =
0 F F F ≤ η→0 0 F
1/(k2Da ). We consider the relation between the 2D and the quasi-2D tuning parameters,
F 2D (cid:112) (cid:112)
B √3π 1 l
0 z
= arcsinh exp . (S6)
2(cid:15) η3/2 2 2a
(cid:114) F (cid:20) (cid:18) 3D(cid:19)(cid:21)
ItisreasonabletoassumethatatthepositionwheretheLandaucriticalvelocitytakesthemaximumvalue, wewould
have,
B
0
lim =A=0. (S7)
η→0(cid:114)2(cid:15)F (cid:54)
By using the above three equations, we find that,
Aη3/2 2A
(η)=2ln sinh 2ln +3lnη+ (η3). (S8)
η=0
F √3π ∼ √3π O
(cid:20) (cid:18) (cid:19)(cid:21) (cid:18) (cid:19)
Therefore, our numerical results in the 2D limit could be fitted with the function
(η)=W +Zlnη. (S9)
F
We obtain the values W = 3.02 and Z = 2.97 from the fitting. The latter value Z 3 confirms our theoretical
(cid:39)
anticipation of the 2D limit, within a relative error of a few percents. Using the former value of W, we can compute
the position of the Landau critical velocity peak in the 2D limit being,
B W √3π
0
limln =lnA= +ln 1.08, (S10)
η→0 (cid:32)(cid:114)2(cid:15)F(cid:33)vmax 2 2 (cid:39)−
c
as represented in the dimensional crossover diagram of Fig. S2(b) that provides the same results of Fig. 1 with a
change of scale in the vertical axis from l /a to ln[ B /(2ε )]. This extracted position of the peak of the Landau
z 3D 0 F
critical velocity in the 2D limit is consistent with the position of the peak of the contact, ln(k2Da ) 1, obtained
(cid:112) F 2D ∼
recently via auxiliary-field Monte Carlo simulations for a 2D interacting Fermi gas [62].