Swallowtail Band Structure of the Superfluid Fermi Gas in an Optical Lattice

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by  Gentaro Watanabe

Swallowtail Band Structure of the Superfluid Fermi Gas in an Optical Lattice
Gentaro Watanabe,1,2,3 Sukjin Yoon,1 and Franco Dalfovo4
1Asia Pacific Center for Theoretical Physics (APCTP), Pohang, Gyeongbuk 790-784, Korea
2Department of Physics, POSTECH, Pohang, Gyeongbuk 790-784, Korea
3Nishina Center, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
4INO-CNR BEC Center and Department of Physics, University of Trento, 38123 Povo, Italy
(Dated: January 6, 2012)
2 Weinvestigatetheenergy bandstructureofthesuperfluidflowof ultracold diluteFermigases in
1 aone-dimensionalopticallatticealongtheBCStoBECcrossoverwithinamean-fieldapproach. In
0 each side of the crossover region, a loop structure (swallowtail) appears in the Bloch energy band
2 of the superfluid above a critical value of the interaction strength. The width of the swallowtail is
largest near unitarity. Across the critical value of the interaction strength, the profiles of density
n
and pairing field change more drastically in the BCS side than in the BEC side. It is found that
a
alongwiththeappearanceoftheswallowtail, thereexistsanarrowbandinthequasiparticleenergy
J
spectrum close to the chemical potential and the incompressibility of the Fermi gas consequently
5
experiences a profound dip in theBCS side, unlikein theBEC side.

]
s PACSnumbers: 03.75.Ss,67.85.De,67.85.Hj,03.75.Lm
a
g
- Ultracold atoms in optical lattices attract much inter- existence and the conditions for emergence of swallow-
t
n est because the controllabilityof both the lattice geome- tails in Fermi superfluids and presenting the unique fea-
a
tryandtheinteratomicinteractionissuchthattheyserve tures which make them different from those in bosons.
u
as testing beds for various models [1]. For Bose-Einstein Weconsideratwo-componentunpolarizeddiluteFermi
q
. condensates (BECs), it has been pointed out that the gas made of atoms of mass m interacting with s-wave
t
a interaction can change the Bloch band structure drasti- scattering length a and subject to a one-dimensional
s
m cally, causing the appearance of a loop structure called (1D) optical lattice of the form V (r)=sE sin2q z ≡
ext R B
- “swallowtail”in the energy dispersion [2, 3]. This is due V0sin2qBz. Here, V0 ≡sER is the lattice height, s is the
d to the competition between the external periodic poten- lattice intensity in dimensionless units, E = ~2q2/2m
n R B
tial and the nonlinear mean-field interaction: the former is the recoil energy, q = π/d is the Bragg wave vector
o B
c favorsa sinusoidalband structure,while the latter tends and d is the lattice constant. We compute the energy
[ to make the density smoother and the energy dispersion band structure of the system by solving the Bogoliubov-

quadratic. When the nonlinearity wins, the effect of the de Gennes (BdG) equations [10]:
2
v external potential is screened and a swallowtail energy
H(r) ∆(r) u (r) u (r)
42 ltoeonpceaopfpaenarosrd[4e]r.pTahraismneotnerlinaneadr,ceoffnecsetqrueeqnutilrye,sththeeemexeisr-- (cid:18)∆∗(r) −H(r)(cid:19)(cid:18) vii(r) (cid:19)=ǫi(cid:18) vii(r) (cid:19) , (1)
0
0 gence of swallowtails can be viewed as a peculiar mani- where H(r) = −~2∇2/2m + Vext(r) − µ, ui(r) and
8. festation of superfluidity in periodic potentials. vi(r) are quasiparticle wavefunctions, and ǫi is the cor-
responding quasiparticle energy. The chemical poten-
0
The problem of swallowtails can be even more impor-
1 tial µ is determined from the constraint on the parti-
1 tant in Fermi superfluids due to the possible wide impli- cle number N = 2 |v (r)|2dr and the pairing field
: cations for various systems in condensed matter physics ∆(r) should satisfy aiself-cionsistency condition ∆(r) =
v
Xi aanctdivnituiecsleaarrepdheyvsoitcesd. tIontdheeedsi,meuxltaetniosinveofrseocleidntstraetseesarucsh- −g iui(r)vi∗(r),wPherRegisthecouplingconstantforthe
s-wave contact interaction which needs to be renormal-
r ingcoldFermigasesandthe behaviorofcoldfermionsin izedP. In the presence of a supercurrent with wavevec-
a
optical lattices can lead to interesting analogies with su- tor Q = P/~ (|P| ≤ P ≡ ~q /2) moving in the
edge B
perconductors and superconductor superlattices. In ad-
z-direction [11], one can write the quasiparticle wave-
dition,ourworkmayhaveimplicationsalsoforsuperfluid functions in the Bloch form as u (r) = u˜ (z)eiQzeik·r
i i
neutrons in neutron stars, especially those in “pasta” and v (r) = v˜(z)e−iQzeik·r leading to the pairing field
i i
phases (see, e.g., Ref. [5] and references therein) in neu- as ∆(r) = ei2Qz∆˜(z). Here ∆˜(z), u˜ (z), and v˜(z) are
i i
tronstarcrusts[6],wherenucleiformacrystallinelattice
complex functions with period d and the wave vector k
z
in which superfluid neutrons can flow. However, unlike
(|k |≤q )liesinthefirstBrillouinzone. ThisBlochde-
z B
the Bose case [2–4, 7–9], little has been studied in this
composition transforms Eq. (1) into the following BdG
problem so far and a fundamental question of whether
equations for u˜ (z) and v˜(z) :
i i
or not swallowtails exist along the crossover from the
Bardeen-Cooper-Schrieffer (BCS) to BEC states is still H˜ (z) ∆˜(z) u˜ (z) u˜ (z)
Q i =ǫ i , (2)
open. In this context, our work is aimed at showing the (cid:18) ∆˜∗(z) −H˜−Q(z)(cid:19)(cid:18) v˜i(z) (cid:19) i(cid:18) v˜i(z) (cid:19)
2
0.5 0.5
(a) (b)
R
=0)] / N E 00..34 1 / 0k F as h (P )edge 00..34
−P) E(P 0.2 ---001..565 Half widt 0.2
E( 0.1 0.1
[
0 0
0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5
P / P 1 / k a
edge F s FIG. 2: (Color online) The parameter region (EF/ER,
1/kFas) where the swallowtails appear for s = 0.1. Sym-
FIG.1: (Coloronline)(a)EnergyEperparticleasafunction bols on the vertical line at EF/ER = 2.5 correspond to the
of the quasimomentum P for various values of 1/kFas; (b) cases shown in Fig. 1(a).
half-width of the swallowtails along the BCS-BEC crossover.
These results are obtained for s = 0.1 and EF/ER = 2.5.
ThequasimomentumPedge =~qB/2fixestheedgeofthefirst
shouldbe,thedifferencebetweenthetwocurvesbecomes
Brillouin zone. The dotted line in (b) is the half-width in
a BEC obtained by solving the GP equation; it vanishes at vanishingly small in the deep BEC regime. With our set
1/kFas≃10.6. of parameters (s = 0.1 and EF/ER = 2.5), the width of
theswallowtailpredictedbytheGPequationvanishesat
1/k a ≃10.6.
F s
where
Whether the swallowtail appears or not depends on
~2
H˜Q(z)≡ k⊥2 +(−i∂z+Q+kz)2 +Vext(z)−µ. three parameters: the interactionparameter 1/kFas, the
2m lattice intensity s, and the ratio between the Fermi
h i
Here, k⊥2 ≡ kx2 +ky2 and the label i represents the wave energy and the recoil energy EF/ER. In Fig. 2, we
vector k as well as the band index. fix s = 0.1 and show the parameter region (EF/ER,
In the following, we mainly present the result for s = 1/kFas) where we find swallowtails in the BCS side of
V0/ER = 0.1 and EF/ER = 2.5 as an example, where the crossover (1/kFas < 0) [14]. We see that for weaker
EF = ~2kF2/(2m) and kF = (3π2n0)1/3 are the Fermi interaction (i.e., larger values of 1/kF|as|) higher den-
energy and momentum of a uniform free Fermi gas of sities (larger values of EF) are required to create swal-
density n . These values fall in the range of parameters lowtails, as expected. The s-dependence of critical val-
0
of feasible experiments [12]. ues of 1/kFas in the BCS side is much weaker than the
We first compute the energy per particle in the low- 1/s scaling behavior in the BEC side. In the BCS side,
est Bloch band as a function of the quasimomentum P as far as we have checked in 0.1 ≤ s ≤ 0.5, the criti-
for various values of 1/kFas. The results in Fig. 1(a) cal value of 1/kFas changes within 30% at EF/ER = 5
show that the swallowtails appear above a critical value and the change gets smaller with increasing EF/ER.
of 1/k a where the interaction energy is strong enough This weak dependence of the swallowtail region on s in
F s
to dominate the lattice potential. In Fig. 1(b), the half- the BCS side is due to the Fermi statistics: Provided
widthoftheswallowtailsfromtheBCStotheBECsideis EF/V0 =(EF/ER)/sis sufficientlylargerthanunity, the
shown. It reaches a maximum near unitarity (1/k a = flow of the BCS condensate formed from fermions near
F s
0). In the far BCS and BEC limits, the width vanishes the Fermi surface is not very sensitive to the presence of
because the system is very weakly interacting and the the lattice potential, while the flow of the BEC formed
band structure tends to be sinusoidal. When approach- frombosonicdimersallatthebottomoftheenergylevels
ing unitarity from either side, the interaction energy in- is more sensitive to it.
creases and can dominate over the periodic potential, Boththe pairingfield andthe density exhibit interest-
which means that the system behaves more like a trans- ing features in the range of parameters where the swal-
lationallyinvariantsuperfluidandthebandstructurefol- lowtails appear. This is particularly evident at the Bril-
lows a quadratic dispersion terminating at a maximum louin zone boundary, P = P . In Fig. 3, we show the
edge
P larger than P . In the BEC side, we compare the magnitudeofthepairingfield|∆(z)|andthedensityn(z)
edge
results of our BdG calculations with those of the Gross- calculatedatthe minimum (z =0)andat the maximum
Pitaveskii (GP) equation for bosons of mass m = 2m (z =±d/2) of the lattice potential. In general, n(z) and
b
interactingwithscatteringlengtha =2a [13]in anop- |∆(z)|takemaximum(minimum)valueswheretheexter-
b s
ticallattice2V (z): −(~2/4m)∇2Φ(r)+2V (z)Φ(r)+ nal potential takes its minimum (maximum) values (for
ext ext
(8π~2a /2m)|Φ(r)|2Φ(r) = µ Φ(r), where Φ(r) is a sin- thefullprofiles,seeSupplementalMaterial[15]). Thefig-
s B
gle macroscopic wavefunction describing the BEC. As it ure shows that |∆(d/2)| remains zero in the BCS regime
3
1.4 (a) |∆(d/2)| 1.3 (b) n(d/2) (a) 3 µ (b) 0.8
∆||(z) / EF 0001 ....14682 |∆(0)| n(z) / n0 011 ...1912 n(0) ε / Ek ; k =0R⊥z 012 l = 0 l = 1l = -1 -1-1κκ / 0 000 ...0246
0.2 0.8 -1
-0.2
0-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.7-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 -1 -0.5 0 0.5 1 -0.7 -0.6 -0.5 -0.4
1 / kF as 1 / kF as k z / qB 1 / kFas
FIG. 3: (Color online) Profiles of (a) the pairing field |∆(z)| FIG. 4: (Color online) (a) Lowest three Bloch bands of the
and (b) the density n(z) along the change of 1/kFas for P = quasiparticle energy spectrum at k⊥ = 0 for P = Pedge and
Pedge in the case of s = 0.1 and EF/ER = 2.5. The values 1/kFas = −0.62. Thin black dashed lines labeled by l’s
of |∆(z)| and n(z) at the minimum (z = 0, blue (cid:3)) and at show the approximate energy bands obtained from Eq. (3)
the maximum (z = ±d/2, red ×) of the lattice potential are using µ ≃ 2.66ER and |∆| ≃ |∆(0)| ≃ 0.54ER. (b) Incom-
shown. The vertical dotted lines show the critical value of pressibility κ−1 at P = P around the critical value of
edge
1/kFas above which theswallowtail exists. The dotted curve 1/kFas ≈−0.62 where the swallowtail starts to appear. The
in (a) shows |∆| in theuniform system. quantityκ−1 istheincompressibilityofthehomogeneousfree
0
Fermi gas of the same average density. In both panels, we
haveused the valuess=0.1 and EF/ER =2.5.
untiltheswallowtailappearsat1/k a ≈−0.62. Thenit
F s
increasesabruptly tovalues comparableto |∆(0)|,which
means that the pairing field becomes almost uniform at with l being integers for the band index. If Q =
P = Pedge in the presence of swallowtails. As regards 0, the l = 0 band has the energy spectrum
the density, we find that the amplitude of the density [(k2 +k2)/2m−µ]2+|∆|2 which has a local maxi-
⊥ z
variation, n(0) − n(d/2), exhibits a pronounced maxi- mpum at kz =k⊥ =0. When Q=Pedge/~, the spectrum
mum near the critical value of 1/kFas. In contrast, in istiltedandthelocalmaximummovestokz ≃qB/2pro-
the BEC side, the order parameter and the density are vided|∆|≪E (andE /E &1). Intheabsenceofthe
F F R
smooth monotonic functions of the interaction strength swallowtail,the full BdG calculationindeed gives a local
even in the region where the swallowtail appears. At maximum at k = q /2 [see Fig. 4(a), where the green
z B
P = Pedge, the solution of the GP equation for bosonic dashedlineintheregion−1<kz/qB <−0.5andthered
dimers gives the densities nb(0) = nb0(1 + V0/2nb0U0) solid line in the region −0.5 < kz/qB < 1 correspond to
and nb(d/2) = nb0(1 − V0/2nb0U0) with V0/2nb0U0 = the l = 0 band] and the quasiparticle spectrum is sym-
(3π/4)(sER/EF)(1/kFas), where nb0 is the average den- metric about this point, which reflects that the current
sity of bosons and U0 = 4π~2ab/mb [2, 16, 17]. Near is zero. As EF/ER increases, the band becomes flatter
the critical value of 1/kFas, unlike the BCS side, the as a function of kz and narrower in energy.
nonuniformity just decreases all the way even after the
In Fig. 4(a), we show the quasiparticle energy spec-
swallowtailappears. The localdensity atz =d/2 is zero
trum at k⊥ = 0 for P = Pedge. When the swallowtail is
until the swallowtail appears in the BEC side while it
ontheedgeofappearing,thetopofthenarrowbandjust
is nonzero in the BCS side irrespective of the existence
touches the chemical potential µ [see the dotted ellipse
of the swallowtail. The qualitative behavior of |∆(z)|
in Fig. 4(a)]. Suppose 1/k a is slightly larger than the
F s
around the critical point of 1/k a is similar to that of
F s criticalvalue,sothatthetopofthebandisslightlyabove
n (z)becausen (r)=(m2a /8π)|∆(r)|2 [18]intheBEC
b b s µ. In this situation a small change of the quasimomen-
limit.
tumP causesachangeofµ. Infact,whenP isincreased
Thequasiparticleenergyspectrumplaysanimportant
from P = P to larger values, the band is tilted and
edge
roleindeterminingthe propertiesoftheFermigas. Here
the top of the band moves upwards; the chemical po-
we show that the emergence of swallowtails in the BCS
tential µ should also increase to compensate for the loss
sideandforE /E &1isassociatedwithpeculiarstruc-
F R of states available. This implies ∂µ/∂P > 0. (See also
tures of the quasiparticle energy spectrum around the
Supplemental Material [15].) On the other hand, since
chemicalpotential. Inthepresenceofasuperflowmoving
the system is periodic, the existence of a branch of sta-
in the z direction with wavevector Q, the quasiparticle
tionary states with ∂µ/∂P >0 at P =P implies the
edge
energies are given by the eigenvalues in Eq. (2). Since
existence of another symmetric branch with ∂µ/∂P < 0
the potential is shallow(s≪1), some qualitative results
at the same point, thus suggesting the occurrence of a
can be obtained even ignoring V (z) except for its pe-
ext swallowtail structure.
riodicity. With this assumption we obtain
A direct consequence of the existence of a narrow
band in the quasiparticle spectrum near the chemical
ǫk≈(kz+m2qBl)Q+s(cid:20)k⊥2+(kz+22mqBl)2+Q2−µ(cid:21)2+|∆|2 ,(3) pκo−t1en=tina∂lµis(na)/s∂trnonclgosreedtouctthioencroitfictahlevainlucoemofp1re/sksFibaislitiny
4
the region where swallowtails exist in the BCS side [see
Fig. 4(b)]. The dip of κ−1 occurs in the situation where
the top ofthe narrowband is just aboveµ for P =P
edge [1] O. Morsch and M. Oberthaler, Rev.Mod. Phys.78, 179
(1/k a is slightly above the critical value). An increase
F s (2006); M.Lewenstein et al.,Adv.Phys.56,243(2007);
ofthedensitynhaslittleeffectonµinthiscase,because I. Bloch et al.,Rev.Mod. Phys.80, 885 (2008).
the density of states is large in this range of energy and [2] B. Wu, R. B. Diener, and Q. Niu, Phys. Rev. A 65,
the new particles can easily adjust themselves near the 025601 (2002).
top of the band by a small increase of µ. This implies [3] D. Diakonov et al., Phys.Rev.A 66, 013604 (2002).
that ∂µ(n)/∂n is small and the incompressibility has a [4] E. J. Mueller, Phys. Rev.A 66, 063603 (2002).
[5] G.Watanabeetal.,Phys.Rev.Lett.103,121101(2009).
pronounced dip [19]. It is worth noting that in the BEC
[6] G. Watanabeet al.,Phys.Rev. A 83, 033621 (2011).
side the appearance of the swallowtail is not associated
[7] M. Machholm, C. J. Pethick, and H. Smith, Phys. Rev.
with any significant change of incompressibility. In fact, A 67, 053613 (2003).
the exactsolutionofthe GPequationgivesκ−1 =nb0U0 [8] B.T.Seaman,L.D.Carr,andM.J.Holland,Phys.Rev.
near the critical conditions for the occurrence of swal- A 71, 033622 (2005); ibid. 72, 033602 (2005).
lowtails, being a smooth and monotonic function of the [9] I. Danshita and S. Tsuchiya, Phys. Rev. A 75, 033612
interaction strength. (2007).
[10] Neither the hydrodynamic theory nor the tight-binding
Swallowtailsmayproduce observableeffects in the be-
modelisappropriatesincetheycannotdescribetheswal-
havior of Bloch oscillations [3, 20]. Since Bloch oscilla-
lowtailscorrectlyalongtheBCS-BECcrossover: thefor-
tions have various important applications, such as preci- mer always yields the dispersion of the quadratic form
sion measurements of forces [21] and controlling the mo- without termination and the latter gives the sinusoidal
tion of a wave packet [22, 23], a better understanding of band.
swallowtails in bosonic and fermionic gases is certainly [11] Nonzero values of P can be obtained, for instance, by
orienting the periodic potential in the direction of the
useful in these contexts. One may also exploit the tun-
gravity field or by imposing a suitable time-dependent
abilityoftheinteractionandthepeculiardynamicsofthe
phase to theoptical lattice.
superfluidinthelatticefordifferentapplications. Forin-
[12] D. E. Miller et al.,Phys.Rev. Lett.99, 070402 (2007).
stance, by periodically sweeping a magnetic field across [13] Here we use ab = 2as in the GP equation for consis-
thecriticalregionfortheappearanceofswallowtails,one tency with the mean-field BdG theory. The exact value
canproduceatimemodulationoftheshapeofthelowest in the BEC limit would be ab = 0.6as [D. S. Petrov et
energy band between a sinusoidal form and a quadratic- al., Phys. Rev. Lett. 93, 090404 (2004)], and the cor-
responding width of the swallowtail obtained in experi-
like form. Since the absolute value of the group velocity
ments would besmaller in this limit.
∂ [E(P)/N]ofthelatterisalwayslargerthanthatofthe
P [14] The energetic stability of swallowtails can be addressed
former[seeFig.1(a)],onecouldrealizeadirectedmotion
by considering the fermionic pair-breaking excitations
ofthegasbysynchronizingtheperiodofthismodulation and long-wavelength phonon excitations [see, e.g., G.
withtheperiodoftheBlochoscillations(seealsoSupple- Watanabeet al.,Phys.Rev.A80,053602 (2009)]. From
mentalMaterial[15]). This may be experimentally more theexcitationspectrumonecanextractthecriticalquasi-
accessible in the BCS side than in the BEC side because momentum Pc below which the system is stable. If Pc
exceeds P , there exists an energetically stable re-
thecriticalvalueof1/k |a |isoforder1forawiderange edge
F s
gion for the swallowtail. In the case of s = 0.1 and
of s and E /E . This new method would complement
F R EF/ER = 2.5, we find stable swallowtails in the range
other proposals for realizing directed motion of atomic
−0.2 . 1/kFas < 0.92. However, even outside of this
wave packets in 1D optical lattices [23, 24]. region, the energetic instability is not a serious obstacle
In summary, we have predicted the existence of swal- for the experimental observation since the breakdown of
lowtails in the energy band of superfluid fermions in a superfluidityduetotheenergeticinstabilityrequiresdis-
lattice and have pointed out some key features which sipation, which is inefficient at low temperatures. As far
asthedynamicalinstabilityisconcerned,weexpectthat
make these swallowtails different from those in a BEC.
The results are obtained within a range of parameters the range of 1/kFas, where the system is dynamically
stable, is larger than that for theenergetic instability as
compatible with current experiments [12]. We hope our
known tohappen in thecase of BEC.
predictionsstimulateexperimentsaimedtoobserveswal- [15] See Supplemental Material for the profiles of |∆(z)| and
lowtails with Fermi gases. n(z), the dependence of the quasiparticle energy spec-
We acknowledge C. J. Pethick, Y. Shin, and T. Taki- trum on P, and more discussion about the directed mo-
tion.
moto for helpful discussions. This work was supported
[16] J. C.Bronskiet al.,Phys.Rev.Lett.86,1402 (2001); J.
by the Max Planck Society, the Korea Ministry of Edu-
C. Bronski et al., Phys.Rev.E 63, 036612 (2001).
cation,Science andTechnology,Gyeongsangbuk-Do,Po-
[17] Thisresultisexactonlywhentheswallowtailsexist,oth-
hang City, for the support of the JRG at APCTP, and erwise it is approximate, but qualitatively correct.
ERC through the QGBE grant. Calculations were per- [18] P. Pieri and G. C. Strinati, Phys. Rev. Lett. 91, 030401
formedonRICCinRIKENandWiglafattheUniversity (2003).
of Trento. [19] With the parameters used in Fig. 4(b), the incompress-
5
ibility takes negative values in a small region around
1/kFas = −0.55, which means that the system might
bedynamicallyunstableagainstlong-wavelengthpertur-
bations.Bychoosingappropriateparameters,thisregion
disappears.
[20] B. Wu and Q. Niu,Phys. Rev.A 61, 023402 (2000).
[21] G. Roati et al., Phys. Rev. Lett. 92, 230402 (2004); G.
Ferrari et al., Phys. Rev. Lett. 97, 060402 (2006); M.
Fattori et al.,Phys. Rev.Lett. 100, 080405 (2008).
[22] A.Alberti et al.,Nature Phys.5, 547 (2009).
[23] E. Haller et al.,Phys.Rev.Lett. 104, 200403 (2010).
[24] C. Mennerat-Robilliard et al., Phys. Rev. Lett. 82, 851
(1999); Q. Thommen, J. C. Garreau, and V. Zehnl´e,
Phys. Rev. A 65, 053406 (2002); M. Schiavoni et al.,
Phys.Rev.Lett.90,094101(2003);L.Sanchez-Palencia,
Phys. Rev. E 70, 011102 (2004); C. E. Creffield, Phys.
Rev. Lett. 99, 110501 (2007); R. Gommers et al., Phys.
Rev.Lett.100, 040603 (2008); K.Kudoand T.S.Mon-
teiro, Phys. Rev. A 83, 053627 (2011); C. E. Creffield
andF.Sols, Phys.Rev.A84,023630 (2011); F.Zhanet
al.,Phys.Rev. A 84, 043617 (2011).
6
Supplemental Material for Swallowtail Band Structure of the Superfluid Fermi Gas in an Optical Lattice
Gentaro Watanabe, Sukjin Yoon, and Franco Dalfovo
Full profiles of the pairing field |∆(z)| and the interaction strengths are short, just enough, and suffi-
density n(z) cientlylargefordevelopingtheswallowtails,respectively.
InFig.3 ofthe paper,the values of|∆(z)| andn(z)at
theminimumandatthemaximumofthelatticepotential
are shown. The full profiles of |∆(z)| and n(z) along Behavior of the Bloch Band near the value of the
the lattice vector (z-direction) are given in the following chemical potential with the change of the
quasimomentum
Fig. 5. By increasing the interaction parameter 1/k a ,
F s
we find that the order parameter |∆| at the maximum
(z = ±d/2) of the lattice potential exhibits a transition InFig. 4(a)ofthepaper,thelowestthreeBlochbands
fromzerotononzerovaluesatthecriticalvalueof1/kFas of the quasiparticle energy spectrum for P = Pedge at
atwhichtheswallowtailappears[seealsoFig.3(a)ofthe k⊥ = 0 and 1/kFas = −0.62 are given. To visualize the
paper]. Note that here we plot the absolute value of ∆; behavior of the second band (the red solid line in that
the order parameter ∆ behaves smoothly and changes figure) near the value of the chemical potential with the
sign across zero. change of the quasimomentum P, we show the cases of
P/P = 0 (black dotted), 0.5 (green dashed), and 1
edge
0.7 (red solid) in the following Fig. 6. Notice that the sharp
1/k a
0.6 -0F.4s minima in the curves for P/Pedge = 0 and −0.5 are due
-0.6 to avoided crossings with other bands.
F 0.5 -0.8
E
/ 0.4 3
|
)
(z 0.3
∆ 2.5
| R
0.2 E
/
0 .01 -0.4 -0.2 0 0.2 0.4 ε k ; k =0⊥z 1 .25 P /0 Pedge
z / d 0.5
1
1.2 1
-1 -0.5 0 0.5 1
1
k / q
z B
n0 0.8
/
(z) 0.6 1/k F as FarIoGu.nd6:thBelcohcehmbicaanldpootfentthiaelqfouraPsi/pParticle=e0n(ebrglayckspdeoctttreudm),
n -0.4 edge
0.4 0.5 (green dashed), and 1 (red solid), respectively, at k⊥ =0
-0.6
-0.8 and1/kFas=−0.62inthecaseofs=0.1andEF/ER =2.5.
0.2 The each horizontal line denotes the value of the chemical
potential for thecorresponding value of P.
0
-0.4 -0.2 0 0.2 0.4
z / d
Application of the swallowtail band structure to a
FIG. 5: Profiles of the pairing field |∆(z)| and the density
Directed motion
n(z) at 1/kFas = −0.8 (blue dotted), −0.6 (green dashed),
and −0.4 (red solid) for P =P in thecase of s=0.1 and
edge
EF/ER = 2.5. The swallowtail starts to appear at a critical InFig.7,weshowaschematicdiagramforourproposal
valueof 1/kFas≈−0.62. ofthedirectedmotionofanatomicwavepacket. Theba-
sicideaismakinguseofthedifferenceinthegroupveloc-
The blue dotted, green dashed, and red solid ityv betweenthequadratic-likebandwithaswallowtail
G
lines correspond to the cases where the interparticle and the sinusoidal-like band. By modulating 1/k a in
F s
7
such a way that the dispersion is quadratic when v is
G
positive and sinusoidal when v is negative, the net dis-
G
placement over one period of modulation of 1/k a is
F s
G positive and a directed motion can be produced.
0
v
0 1
t / T
mod
FIG. 7: Schematic diagram of the group velocity vG during
one period (T ) of modulation of the magnetic field (ac-
mod
cordingly, 1/kFas). The green (blue) dashed line shows the
groupvelocityalongthetimewhen1/kFas issettothevalue
foraquadratic(sinusoidal)bandstructureasdepictedinFig.
1(a) of thepaper. When 1/kFas is switched betweenthetwo
values in a proper way, the group velocity of a wave packet
will follow the black solid line. In real situation, the black
linearoundt=T /2isasmoothcurveratherthanasharp
mod
drop due to a continuous variation of 1/kFas connecting the
two values.