Aspects of superfluid cold atomic gases in optical lattices

Journal of the Korean Physical Society, Vol. 63, No. 4, August 2013, pp. 839∼857 Review Articles

Aspects of Superfluid Cold Atomic Gases in Optical Lattices

Gentaro Watanabe∗

Asia Pacific Center for Theoretical Physics, Pohang 790-784, Korea,Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea, and

Nishina Center, RIKEN, Saitama 351-0198, Japan

Sukjin Yoon

Asia Pacific Center for Theoretical Physics, Pohang 790-784, Korea

(Received 21 January 2013, in final form 26 April 2013)

We review our studies on Bose and Fermi superfluids of cold atomic gases in optical lattices atzero temperature. Especially, we focus on superfluid Fermi gases along the crossover between theBardeen-Cooper-Schrieffer (BCS) and the Bose-Einstein condensate (BEC) states, which enableus to study the Bose and the Fermi superfluids in a unified point of view. We discuss basicstatic and long-wavelength properties (such as the equation of state, incompressibility, and effectivemass), energetic stability, and energy band structures of the superfluid Fermi gases in an opticallattice periodic along one spatial direction. The periodic potential causes pairs of atoms to bestrongly bound, and this can affect the static and long-wavelength properties and the stabilityof the superflow. Regarding the band structure, a peculiar loop structure called “swallowtail” canappear in superfluid Fermi gases and in the Bose case, but the mechanism of emergence in the Fermicase is very different from that in bosonic case. Other quantum phases that the cold atomic gasesin optical lattices can show are also briefly discussed based on their roles as quantum simulators ofHubbard models.

PACS numbers: 03.75.Ss, 03.75.Kk, 03.75.LmKeywords: Cold atomic gas, Optical lattice, Superfluid, Hubbard modelDOI: 10.3938/jkps.63.839

I. INTRODUCTION

In 1995, the atomic physics community witnessed thecoolest state of matter realized in the laboratory by thattime. With lasers and electromagnetic fields, atomicBose gases were trapped and cooled to within a micro-Kelvin to make them coalesce into a single quantumwave, known as a Bose-Einstein condensate (BEC) state[1]. With the development and upgrading of technologyand tools, finally about 10 years later, physicists suc-ceeded in cooling atomic Fermi gases to even lower tem-peratures and in providing conclusive evidence that theinteracting atomic Fermi gas could be controlled to be ina state from a Bardeen-Cooper-Schrieffer (BCS) super-fluid of loosely-bound pairs to a BEC of tightly-bounddimers [2]. This quantum wonderland of cold atomicgases has been proven to be rich in physics, and it hasalso shown us new realms of research [3–7]. Unprece-dented high-precision control over the cold atomic gaseshas made it possible to mimic various quantum systems.It has also paved a way to new physical parameter region

∗E-mail: gentaro [email protected]

that had not been attainable in other quantum systemsand, therefore, had not been well thought about.

In a cold Fermi gas with an equal mixture of twohyperfine states (for simplicity, called spin up/down),wide-range control of the short-range interatomic inter-action (characterized by an s-wave scattering length as)via Feshbach resonances [8] has enabled the mapping ofa new landscape of superfluidity, known as “BCS-BECcrossover,” smoothly connecting the BCS superfluid tothe BEC superfluid [9]. For a weakly-attractive inter-action (1/kFas � −1, with kF conventionally being theFermi momentum of a free Fermi gas of the same den-sity), the gas shows a superfluid behavior originatingfrom the many-body physics of Cooper pairs that areformed from weak pairings of atoms of opposite spin andmomentum and whose spatial extent is larger than theinterparticle distance. For a strongly-attractive interac-tion (1/kFas � 1), the gas shows a superfluid behaviorthat can be explained by the Bose-Einstein condensationof tightly-bound bosonic dimers made up of two fermionsof opposite spin. In the crossover region/unitary region(1/kF|as| � 1) bridging the two well-understood regionsabove, the gas is strongly interacting, which means thatthe interaction energy is comparable to the Fermi energy

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of a free Fermi gas of the same density and defies per-turbative many-body techniques. Especially, the physicsin the so-called “unitary limit” (|as| = ∞) is universalbecause only one remaining length scale, 1/kF, which isapproximately the interparticle distance, appears in theequation of state of the gas, while the interaction effectappears as a universal dimensionless parameter. Becausethe physics at unitarity is universal, i.e., independent ofits constituents and is non-perturbative due to the stronginteraction, the unitary Fermi gas has been the test-bedfor the various theoretical techniques developed so far(see, e.g., Sec. VB and V C in Ref. 6).

Cold atomic gases also provide insights into otherforms of matter. Inside neutron stars, there might bequark matter made of different ratios of quarks, andthis imbalance might result in new types of superfluid,such as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)phase, which has spatially non-uniform order parame-ter [10]. Even though the electric-charge neutrality ofatoms might change the picture, we might get some ideasfrom investigations on imbalanced atomic Fermi gases[11]. Another example is the quark-gluon plasma (QGP)created in the Relativistic Heavy Ion Collider (RHIC). AQGP with a temperature of about 1012 K was producedby smashing nuclei together. Measurements on its ex-pansion after its creation show that the QGP is a nearlyperfect fluid with very small shear viscosity. A cloud ofatomic Fermi gases at unitarity shows the same strangebehavior while the origins of the similarity between hotand cold matter are to be speculated [12].

Cold atoms in periodic potentials are more intrigu-ing because of their possible connection to the solidstate/condensed matter physics (see, e.g., Refs. 5, 13,and 14 for reviews). The artificial periodic potentialof light, so-called “optical lattice,” is configured withpairs of counter-propagating laser fields that are detunedfrom atomic transition frequencies enough that they actas practically free-of-defect, conservative potentials thatthe atoms experience via the optical dipole force. Thegeometry and the depth of the lattice potential can becontrolled by orienting the directions and by changingthe intensities of the laser beams, respectively. Moreover,the macroscopically-large lattice constant of the opticallattice compared with the lattice potential in solids isa great advantage for experimental observation and ma-nipulation; nowadays, in-situ imaging and addressing ofstrongly-correlated systems at the single-site level havebeen realized (e.g., Ref. 15). The high controllabilityof both the optical lattice properties via laser fields andthe interatomic interaction via the Feshbach resonanceallows cold atomic gases to serve as quantum simulatorsfor various models of condensed matter physics. Manyphenomena of solid state/condensed matter physics havealready been observed or realized in cold atomic gases onthe optical lattices, including a band structure, a Blochoscillation, Landau-Zener tunneling, and a superfluid-Mott insulator transition [16,17]. Precision control andmeasurement at the level of one lattice site enable the

cold atomic gases on optical lattices to be applied ininventing new manipulation techniques and novel de-vices, such as matter-wave interferometers, optical lat-tice clocks, and quantum registers [18,19].

In this review article, we mainly explore the super-fluid properties of cold atomic gases in an optical lat-tice at zero temperature. Superfluidity is the most well-known quantum phase of ultracold atomic gases andis prevalent in many other systems, such as supercon-ductors, superfluid helium, and superfluid neutrons in“pasta” phases (see, e.g., Ref. 20) of neutron star crusts.Knowing its properties is also important both for judg-ing whether the superfluid state is approached in experi-ments and applying its properties in controlling the sys-tem. Depending on the physical conditions, cold atomicgases in optical lattices show various interesting phasesother than the superfluid phase, for example, simulatingHubbard models and some other quantum phases willbe explained very briefly in Sec. VI. In our discussion,we mainly consider atomic Fermi gases partly becauseweakly-interacting atomic Bose gases can be consideredas an extreme limit in the BCS-BEC crossover of theatomic Fermi gases and partly because the physics ofthe cold Fermi gases is richer. Bose gases are discussedseparately in the cases where we cannot find the cor-responding phenomena in the Fermi gases. This doesnot mean that the cold Bose gases are less interestingand less important. The tools and the techniques de-veloped for creating the BEC were the building blocksfor those for creating superfluid Fermi gases, and super-fluid properties of the Bose gases might be more usefulin some applications due to their higher superfluid tran-sition temperature.

In Sec. II, our system and the theoretical frameworksare presented. There, we explain the mean-field theoryand the hydrodynamic theory, as well as the validity re-gion of each theory. Using cold atomic gases, variousparameter regions become accessible, and the mean-fieldtheory in the continuum model is a powerful tool that al-lows us to study superfluid states covering such a wide re-gion. Hydrodynamic theory, though its validity region islimited, can provide precise analytic predictions in somelimits, which are complementary to mean-field theory.Based on these methods, we study various aspects of thesuperfluid state in a lattice potential periodic along onespatial direction. This setup is the simplest, but containsessential effects of the lattice potential.

Section III discusses basic static and long-wavelengthproperties of the cold Fermi gases in an optical lattice.Specifically, the incompressibility and the effective massare obtained from the equation of state (EOS). Focusingon the unitary Fermi gas as a typical example, here weshow that these properties in the lattice can be stronglymodified from those in free space [21]. Formation ofbound pairs is assisted by the periodic potential, andthis results in a qualitative change of the EOS.

In Sec. IV, we examine the stability of the superfluidflow in the optical lattice. Mainly, the energetic instabil-

Aspects of Superfluid Cold Atomic Gases in Optical Lattices – Gentaro Watanabe and Sukjin Yoon -841-

ity and the corresponding critical velocity of the super-flow are discussed. The stability of the superflow is a fun-damental problem of the superfluidity, and cold atomicgases allow us to study this important problem for vari-ous parameter regions for both the Bose and the Fermisuperfluids. Unlike an obstacle in uniform superfluids,the periodic potential can modify the excitation spec-trum, and this manifests itself in the behavior of thecritical velocity [22,23].

In Sec. V, we investigate the energy band structuresof cold atomic Fermi gases in an optical lattice along theBCS-BEC crossover. By tuning the interatomic interac-tion strength, we can make the nonlinear effect of theinteraction energy to dominate the external periodic po-tential. In this parameter region, “swallowtail” energyband structures emerge, and the physical properties ofthe system are affected by their appearance [24]. This isone example of cold atomic gases showing physical phe-nomena inaccessible in other systems.

In Sec. VI, we discuss quantum phases of cold atomicgases in optical lattices and show how they play theirrole as “quantum simulators” of Hubbard models. Be-fore the advent of cold atomic gases in optical lattices,the Hubbard model was considered as an approximatetoy model of more complicated real systems and was oneof the central research topics in solid-state/condensed-matter physics. High-precision control of cold atomicgases in optical lattices now allows Hubbard models tobe realized in experiments, so open questions of quan-tum magnetism and high Tc superconductivity can beaddressed. Finally, Sec. VII contains a summary andpresents some perspectives on the cold atomic gases inoptical lattices.

II. THEORETICAL FRAMEWORKS

2.1. Setup of the System and the Periodic Po-tential

In the present article, we discuss superfluid flow madeof either fermionic or bosonic cold atomic gases subjectto an optical lattice. For concreteness, we consider one ofthe most typical cases: the system is three dimensional(3D), and the potential is periodic in one dimension withthe following form:

Vext(r) = sER sin2 qBz ≡ V0 sin2 qBz. (1)

Here, V0 ≡ sER is the lattice height, s is the latticeintensity in dimensionless units, ER = �

2q2B/2m is the

recoil energy, m is the mass of atoms, qB = π/d is theBragg wave vector, and d is the lattice constant. Forsimplicity, we also assume that the supercurrent is in thez direction; thus, the system is uniform in the transverse(i.e., x and y) directions. Throughout the present article,we set the temperature T = 0.

Before giving a detailed description of the theoreticalframework, it is useful to summarize the scales in thissystem. The periodic potential is characterized by twoenergy scales: one is the recoil energy ER, which is di-rectly related to the lattice constant d, and the other isthe lattice height V0 = sER. Regarding Bose-Einsteincondensates of bosonic atoms, a characteristic energyscale is the interaction energy gn, where g = 4π�

2as/mis the interaction strength and n is the density of atoms.On the other hand, in the case of superfluid Fermigases, the total energy is on the order of the Fermi en-ergy EF = �

2k2F/(2m), with the Fermi wave number

kF ≡ (3π2n)1/3 corresponding to that of a uniform non-interacting Fermi gas with the same density n. There-fore, the relative effect of the lattice strength is given bythe ratio ηheight = V0/gn for bosons and ηheight = V0/EF

for fermions. Likewise, the relative fineness of the lat-tice (compared to the healing length) is characterized bythe ratio ηfine = (gn/ER)−1 ∼ (ξ/d)2 for bosons andηfine = (EF/ER)−1 ∼ (kFd)−2 for fermions, which is∼ (ξ/d)2 near unitarity. Here, ξ is the healing lengthgiven as ξ = �/(2mgn)1/2 for Bose superfluids and asξ ∼ k−1

F for Fermi superfluids at unitarity, which is con-sistent with the BCS coherence length ξBCS = �vF/∆,where vF = �kF/m and ∆ is the pairing gap. We canalso say that the validity of the local density approxi-mation (LDA) is characterized by 1/ηfine � (d/ξ)2 � 1corresponding to a lattice with a low fineness ηfine � 1.In the present article, we shall consider a large parameterregion covering weak to strong lattices, ηheight = O(10−2)– O(10) (s ∼ 0.1 – 5), and low to high fine lattices,ηfine ∼ 0.1 – 10.

2.2. Mean-field Theory in the ContinuumModel

For the study the superfluidity in such various regionsin a unified manner, one of the most useful theoreticalframeworks is the mean-field theory in the continuummodel. Because our system is a superfluid in three di-mensions and the number of particles in each site is in-finite in our setup, the effects of quantum fluctuations,which are not captured by the mean-field theory, may besmall. We also note that widely-used tight-binding mod-els are invalid in the weak lattice region of ηheight � 1.

For dilute BECs at zero temperature, the system iswell described by the Gross-Pitaevskii (GP) equation [4,25,26]:

− �2

2m∂2

zΨ + Vext(z)Ψ + g|Ψ|2Ψ = µΨ, (2)

where Ψ(z) =√

n(z) exp[iφ(z)] is the condensate wavefunction, φ(z) is its phase, and µ is the chemical po-tential. The local superfluid velocity is given by v(z) =(�/m)∂zφ(z).


For superfluid Fermi gases, we consider a bal-anced system of attractively-interacting (pseudo)spin1/2 fermions, where the density of each spin componentis n/2. To describe the BCS-BEC crossover at zero tem-perature, we use the Bogoliubov-de Gennes (BdG) equa-tions [6,27](

H ′(r) ∆(r)∆∗(r) −H ′(r)

)(ui(r)vi(r)

)= εi

(ui(r)vi(r)

), (3)

with H ′(r) = −�2∇2/2m + Vext − µ, ui(r) and vi(r) are

quasiparticle wave functions, which obey the normaliza-tion condition

∫dr [u∗

i (r)uj(r) + v∗i (r)vj(r)] = δi,j , andεi are the corresponding quasiparticle energies. The pair-ing field ∆(r) and the chemical potential µ in Eq. (3) areself-consistently determined from the gap equation

∆(r) = −g∑

i

ui(r)v∗i (r) , (4)

together with the constraint on the average number den-sity

n =2V

∑i

∫|vi(r)|2 dr =

1V

∫n(r) dr, (5)

with n(r) ≡ 2∑

i |vi(r)|2. Here, g is the coupling con-stant for the contact interaction, and V is the volume ofthe system. The average energy density e can be calcu-lated as

e =1V

∫dr

[�

2

2m

(2∑

i

|∇vi|2)

+Vext n(r)+1g|∆(r)|2

].

(6)

For contact interactions, the right-hand side of Eq. (4)has an ultraviolet divergence, which has to be regularizedby replacing the bare coupling constant g with the two-body T -matrix related to the s-wave scattering length[28–31]. A standard scheme [28] is to introduce the cut-off energy EC ≡ �

2k2C/2m in the sum over the BdG

eigenstates and to replace g by the following relation:

1g

=m

4π�2as−∑

k<kC

1

2ε(0)k

, (7)

with ε(0)k ≡ �

2k2/2m.In the presence of a supercurrent with wavevector Q =

P/� (P is the quasi-momentum for atoms rather thanfor pairs; thus, it is defined in the range of |P | ≤ �qB/2)in the z direction, one can write the quasiparticle wave-functions in the Bloch form as ui(r) = ui(z)eiQzeik·r andvi(r) = vi(z)e−iQzeik·r, leading to the pairing field

∆(r) = ei2Qz∆(z). (8)

Here, ∆(z), ui(z), and vi(z) are complex functions withperiod d, and the wave vector kz (|kz| ≤ qB) lies in the

first Brillouin zone. This Bloch decomposition trans-forms Eq. (3) into the following BdG equations for ui(z)and vi(z) :

(HQ(z) ∆(z)∆∗(z) −H−Q(z)

)(ui(z)vi(z)

)= εi

(ui(z)vi(z)

), (9)

where

HQ(z) ≡ �2

2m

[k2⊥ + (−i∂z + Q + kz)

2]+Vext(z)−µ .

Here, k2⊥ ≡ k2

x + k2y, and the label i represents the wave

vector k, as well as the band index.

2.3. Hydrodynamic Theory

When the local density approximation (LDA) is valid,such that the typical length scale of the density varia-tion given by d is much larger than the healing length ξof the superfluid, hydrodynamic theory in the LDA canbe useful [22]. In hydrodynamic theory, we describe thesystem in terms of the density field n(z) and the (quasi-)momentum field P (z) [or the velocity field v(z)]. TheLDA assumes that, locally, the system behaves like a uni-form gas; thus, the energy density e(n, P ) can be writtenin the form

e(n, P ) = nP 2/2m + e(n, 0), (10)

and one can define the local chemical potential µ(n) =∂e(n, 0)/∂n. The density profile of the gas at rest inthe presence of the external potential can be obtainedfrom the Thomas-Fermi relation µ0 = µ[n(z)] + Vext(z).If the gas is flowing with a constant current densityj = n(z)v(z), the Bernoulli equation for the stationaryvelocity field v(z) is

µj =m

2

[j

n(z)

]2+ µ(n) + Vext(z), (11)

where µj is the z-independent value of the chemical po-tential.

In the following two typical cases, the uniform gas hasa polytropic equation of state,

µ(n) = αnγ : (12)

1) a dilute Bose gas with repulsive interaction, whereγ = 1 and α = g = 4π�

2as/m, and 2) a diluteFermi gas at unitarity, where µ(n) = (1 + β)EF =[(1 + β)(3π2)2/3

�2/(2m)]n2/3, i.e., γ = 2/3 and α =

(1 + β)(3π2)2/3�

2/(2m). Here, β is a universal parame-ter, which is negative, and its absolute value is of orderunity, accounting for the attractive interatomic interac-tions [6,32].


Fig. 1. (Color online) Coarse-grained density profiles ofa trapped unitary Fermi gas, n(r⊥ = 0, z) for s = 0 and 5in units of the central density n(0) = 0.0869q3

B calculated fors = 0 (this local density corresponds to EF/ER = 1.88). The

quantity R(0)z is the axial Thomas-Fermi radius for s = 0. The

inset shows the density dependence of the chemical potentialof unitary Fermi gases. Here, µ0 is the chemical potential inthe limit of n = 0. This figure is taken from Ref. 21

Using the equation of state, one can write

mc2s(z) = n

∂

∂nµ(n) = γµ(n) , (13)

where cs(z) is the local sound velocity, which dependson z through the density profile n(z). In a uniformgas of density n, the sound velocity is given by c

(0)s =

[γµ(n)/m]1/2.

III. EQUATION OF STATE,INCOMPRESSIBILITY, AND

EFFECTIVE MASS

In this section, focusing on superfluid Fermi gases atunitarity, we discuss the effects of the periodic poten-tial on the macroscopic and the static properties of thefluid, such as the equation of state, the incompressibil-ity, and the effective mass [21]. The important pointis that the periodic potential favors the formation ofbound molecules in a two-component Fermi gas evenat unitarity [33] (see also, e.g., Refs. 34 and 35). Theemergence of the lattice-induced bound states drasti-cally changes the above macroscopic and static proper-ties from those of uniform systems in the strong latticeregion of ηheight � 1 [21]. Such an effect is absent in idealFermi gases and BECs of repulsively-interacting bosonicatoms, which can be considered as two limits in the BCS-BEC crossover.

3.1. Basic Equations

At zero temperature, the chemical potential µ (theequation of state) and the current j of a superfluid Fermigas in a lattice are given by the derivatives of the aver-age energy density e = E/V with respect to the av-erage (coarse-grained) density n [36] and the averagequasi-momentum P of the bulk superflow, respectively(hereafter, for notational simplicity, we omit “ ¯ ” for thecoarse-grained quantities, which should not be confusedwith the local quantities):

µ =∂e(n, P )

∂n, j =

∂e(n, P )∂P

. (14)

The incompressibility (or inverse compressibility) κ−1

and the effective mass m∗ are given by the second deriva-tives of e with respect to n and P :

κ−1 =n∂2e(n, P )

∂n2= n

∂µ(n, P )∂n

, (15)

1m∗ =

1n

∂2e(n, P )∂P 2

=1n

∂j(n, P )∂P

. (16)

We calculate these quantities for P = 0, i.e., for a gas atrest, in the periodic potential.

In the absence of the lattice potential (s = 0), thethermodynamic properties of unitary Fermi gases showa universal behavior: the only relevant length scale is theinterparticle distance fixed by kF. Due to translationalinvariance, one can write e(n, P ) = e(n, 0) + nP 2/2mso that j = nP/m and m∗ = m. Furthermore, theenergy density at P = 0 can be written as e(n, 0) =(1 + β)e0(n, 0), where e0(n, 0) ≡ (3/5)nEF ∝ n5/3 is theenergy density of the ideal Fermi gas. Thus, we haveµ = (1 + β)EF + P 2/2m and κ−1 = (2/3)(1 + β)EF.

3.2. Equation of State and Density Profile

When the lattice height s is large, the periodic po-tential favors the formation of bound molecules. In thestrong lattice limit ηheight = sER/EF � 1, the systemtends to be a BEC of lattice-induced bosonic molecules.Therefore, in this region, the chemical potential showsa linear density dependence, µ ∝ n, as shown by thered solid line in the inset of Fig. 1 calculated for uni-tary Fermi gases in a lattice with s = 5. This is clearlydifferent from the density dependence of the chemical po-tential in the uniform system (s = 0), µ ∝ n2/3, as shownby the blue dashed line in the same inset. We also notethat, for s � 1, this linear density dependence persistseven at relatively large densities so that EF/sER ∼ 1(e.g., nq−3

B = 0.1 corresponds to EF/ER 2.1; thus,EF/sER 0.41 in the case of this figure) because thesystem effectively behaves like a 2D system in this den-sity region due to the bandgap in the longitudinal degreeof freedom.


Fig. 2. (Color online) Incompressibility κ−1 and effectivemass m∗ of unitary Fermi gases for s = 1 (red), 2.5 (blue),and 5 (green). Asymptotic expressions, Eqs. (18) and (19),obtained by using the hydrodynamic theory are shown by thedotted lines. Open circles in panel (b) show m∗ from Ref. 33which was obtained by solving the Schrodinger equation forthe two-body problem. The s = 1 results for m∗ are alsoshown in the inset on the linear scale. This figure is adaptedfrom Ref. 21.

This drastic change of EOS manifests itself as a changeof the coarse-grained density profile when a harmonicconfinement potential Vho(r) is added to the periodicpotential. The coarse-grained density profile, n(r), iscalculated using the LDA for µ:

µ0 = µ[n(r)] + Vho(r). (17)

Here, µ[n] is the local chemical potential as a function ofthe coarse-grained density n obtained by using the BdGcalculation for the lattice system and µ0 is the chemicalpotential of the system fixed by the normalization con-dition

∫drn(r) = N . Figure 1 clearly shows that, for

s = 5, the profile takes the form of an inverted parabola,reflecting the linear density dependence of the chemicalpotential (see inset). In this calculation, we assume anisotropic harmonic potential, Vho(r) = mω2r2/2, where

Fig. 3. (Color online) Sound velocity cs of unitary Fermi

gases in a lattice in units of the sound velocity c(0)s =

[(2/3)(1 + β)EF/m]1/2 for a uniform system. As in Fig. 2,red, blue and green lines correspond to s = 1, 2.5, and 5,respectively. This figure is taken from Ref. 21

ω is the trapping frequency, �ω/ER = 0.01, and the num-ber of particles N = 106; these parameters are close tothe experimental ones in Ref. 37.

3.3. Incompressibility and Effective Mass

The formation of molecules induced by the lattice alsohas important consequences for κ−1 and m∗. Due to thelinear density dependence of the chemical potential inthe strong lattice region (ηheight � 1 or EF/ER � s;or low density limit for a fixed value of s), κ−1 is alsoproportional to n and κ−1/κ−1(s = 0) ∝ n1/3 → 0 forEF/ER → 0 [see Fig. 2(a)]. This means that the gasbecomes highly compressible in the presence of a stronglattice. This is in strong contrast to the ideal Fermigas corresponding to the BCS limit, which gives nonzerovalues of κ−1/κ−1(s = 0) ∼ 1 even in the same limit(see Fig. 2 in Ref. 21). On the other hand, in the weak-lattice limit (or high-density limit for a fixed value ofs), the system reduces to a uniform gas. By using anhydrodynamic theory, which is valid when EF/ER �1, and expanding with respect to the small parametersER/EF, we obtain κ−1 of unitary Fermi gases in thisregion as [21]

κ−1 23(1 + β)EF

[1 +

132

(1 + β)−2

(sER

EF

)2]

+ O[(sER/EF)4

]. (18)

This is shown by dotted lines in Fig. 2(a). Note that,in this region, κ−1/κ−1(s = 0) > 1, and it decreases tounity with increasing EF/ER. Therefore, κ−1/κ−1(s =0) should take a maximum value larger than unity in theintermediate region of EF/ER ∼ 1, as can be seen in


Fig. 4. (Color online) Critical velocity vc for energeticinstability of superfluids in a 1D periodic potential. Panel(a) is for superfluids of dilute Bose gases, and panel (b) is forsuperfluids of dilute Fermi gases at unitarity. The critical ve-locity is given in units of the sound velocity of a uniform gas,

c(0)s , and is plotted as a function of the maximum of the ex-

ternal potential in units of the chemical potential µj=0 of thesuperfluid at rest. Thick solid lines: prediction of the hydro-dynamic theory within the LDA, as calculated from Eq. (33).Symbols: results obtained from the numerical solutions of theGP equation [panel (a)] and the BdG equations [panel (b)].The thinner black solid lines are the tight-binding prediction,Eq. (35). Dashed lines are guides for the eye. This figure isadapted from Ref. 22.

Fig. 2(a), which is mainly caused by the bandgap in thelongitudinal degree of freedom.

Because the tunneling rate between neighboring sites,which is related to the (inverse) effective mass 1/m∗, isexponentially suppressed with increasing mass, the for-mation of molecules induced by the lattice can yield adrastic enhancement of m∗ for s � 1 in the low-densitylimit [Fig. 2(b)]. This enhancement makes m∗ muchlarger than it is for ideal Fermi gases (see Fig. 2 inRef. [21]) and for BECs of repulsively-interacting Bosegases (see Fig. 4 in Ref. 38) with the same mass m. AsEF/ER increases, the effective mass exhibits a maximumat EF/ER ∼ 1 due to the bandgap in the longitudinal

degree of freedom; then, it decreases to the bare mass,m∗ = m. Hydrodynamic theory can explain the behaviorof m∗ of unitary Fermi gases for small sER/EF [21]:

m∗

m 1+

932

(1+β)−2

(sER

EF

)2

+O[(sER/EF)4

]. (19)

The numerical factor in the second term shows that theeffect of the lattice is stronger for m∗ than for κ−1. Itis worth comparing the results with the case of bosonicatoms, where m∗ decreases monotonically with increas-ing density because the interaction broadens the conden-sate wave function and favors the tunneling [38].

In Fig. 3, we show the sound velocity of the unitaryFermi gases in a lattice,

cs =

√κ−1

m∗ , (20)

calculated from the above results for κ−1 and m∗. Itshows a significant reduction compared to the uniformsystem, mainly due to the larger effective mass m∗/m >1 except for the low-density (more precisely, strong lat-tice) limit in which κ−1 and, thus, cs show abrupt reduc-tions. Using Eqs. (18) and (19), we obtain the expressionof the sound velocity in the weak lattice limit as

c2s c(0)

s

2

[1 − 1

4(1 + β)−2

(sER

EF

)2

+ O[(sER/EF)4

]],

(21)

where c(0)s ≡ [(2/3)(1+β)EF/m]1/2 is the sound velocity

for a uniform system.

IV. STABILITY

4.1. Landau Criterion

Stability of a superfluid flow is one of the most fun-damental issues of superfluidity and was pioneered byLandau [39]. He predicted a critical velocity vc of the su-perflow above which the kinetic energy of the superfluidwas large enough to be dissipated by creating excitations(see, e.g., Refs. 39–42). This instability is called the Lan-dau or energetic instability, and its critical velocity is theLandau critical velocity.

The celebrated Landau criterion for energetic instabil-ity of uniform superfluids is given by [39–42]

v > vc = min(

ε(p)p

), (22)

where v is the velocity of the superflow, ε(p) is the exci-tation spectrum in the static case (v = 0), and p is themagnitude of the momentum p of an excitation in the


comoving frame of the fluid. Here, vc is determined bythe condition for which there starts to exist a momen-tum p at which the excitation spectrum in the comovingframe of a perturber (it can be an obstacle moving inthe fluid or a vessel in which the fluid flows) is zero ornegative.

For superfluids of weakly interacting Bose gases, theexcitation spectrum is given by the Bogoliubov disper-sion relation (e.g., Refs. 40–43)

ε(p) =

√p2

2m

(p2

2m+ 2gn

)= csp

√1 +

(p

2mcs

)2

, (23)

with

cs =√

gn

m(24)

being the sound velocity defined by Eq. (20) [note thatµ = gn and, thus, κ−1 = gn, and m∗ = m for uni-form BECs]. Thus, from Eqs. (22) and (24), the Landaucritical velocity is easily seen to be given by the soundvelocity vc = cs for p = 0. This means that, in su-perfluids of dilute Bose gases, the energetic instability iscaused by excitations of long-wavelength phonons. Wealso note that BECs of non-interacting Bose gases can-not show superfluidity in a sense that vc = 0 and theycannot support a superflow.

In superfluid Fermi gases, another mechanism cancause the energetic instability: fermionic pair-breakingexcitations. In the mean-field BCS theory, the quasipar-ticle spectrum of uniform superfluid Fermi gases is givenby

ε(p) =

√(p2

2m− µ

)2

+ ∆2 . (25)

Thus, the Landau critical velocity due to the pair-breaking excitations is given by [44]

vc =

√1m

(√µ2 + ∆2 − µ

). (26)

In the deep BCS region, where µ EF � ∆, we obtainvc ∆/pF with pF ≡ �kF.

In the BCS-BEC crossover of superfluid Fermi gases,where the above two kinds of excitations exist, the Lan-dau critical velocity is determined by which of themgives smaller vc. In the weakly-interacting BCS re-gion (1/kFas � −1), the pairing gap is exponentiallysmall, ∆ ∼ EFe−π/2kF|as|, and vc is set by the pair-breaking excitations. On the other hand, in the BECregion (1/kFas � 1), where the system consists ofweakly-interacting bosonic molecules, creation of long-wavelength superfluid phonon excitations causes an en-ergetic instability. In the unitary region, both mecha-nisms are suppressed, and the critical velocity shows amaximum value [44–46].

4.2. Stability of Superflow in Lattice Systems

4.2.1. Energetic instability and dynamical instability

For the onset of an energetic instability, energy dis-sipation is necessary in general; i.e., in closed systems,v > vc is not a sufficient condition for a breakdown of thesuperflow, so the flow could still persist even at v > vc.The energetic instability corresponds to the situation inwhich the system is located at a saddle point of the en-ergy landscape (i.e., there is at least one direction inwhich the curvature of the energy landscape is negative).

In the presence of a periodic potential, another typeof instability, called dynamical (or modulational) insta-bility, can occur in addition to the energetic instabil-ity. The dynamical instability means that small pertur-bations on a stationary state grow exponentially in theprocess of (unitary) time evolution without dissipation.Similar to energetically-unstable states, dynamically-unstable states are also located at saddle points in theenergy landscape, but a difference from energetically-unstable states is that there are kinematically-allowedexcitation processes that satisfy the energy and (quasi-)momentum conservations. This means that an energeticinstability is a necessary condition for dynamical insta-bility (see, e.g., Refs. 47 and 48 for bosons and Ref. 49for fermions); therefore, the critical value of the (quasi-)momentum for the dynamical instability should alwaysbe larger than (or equal to) that for the energetic insta-bility. For such a reason, we shall focus on the energeticinstability hereafter [50].

4.2.2. Determination of the critical velocity

If the energetic instability is caused by long-wavelength superfluid phonon excitations, the critical ve-locity can be determined by using a hydrodynamic anal-ysis of the excitations [22,23,41,55–57] (this should notbe confused with the LDA hydrodynamics discussed inSec. 2.3). This analysis is valid provided that the wave-length of the excitations that trigger the instability ismuch larger than the typical length scale of the densityvariation, i.e., the lattice constant d.

We consider the continuity equation and the Eulerequation for the coarse-grained density n and the coarse-grained (quasi-)momentum P averaged over the lengthscale larger than the lattice constant:

∂n

∂t+ ∇ · j =

∂n

∂t+

∂

∂z

∂e

∂P= 0, (27)

∂P

∂t+

∂µ

∂z=

∂P

∂t+

∂

∂z

∂e

∂n= 0, (28)

where e = e(n, P ) is the energy density of the super-fluid in the periodic potential for the averaged densityn and the averaged (quasi-)momentum P . Linearizing


with respect to the perturbations of n(z, t) = n0+δn(z, t)and P (z, t) = P0 + δP (z, t) with δn(z, t) and δP (z, t) ∝eiqz−iωt, we obtain the dispersion relation of the long-wavelength phonon,

ω(q) =∂2e

∂n∂Pq +

√∂2e

∂n2

∂2e

∂P 2|q| . (29)

Here, �ω and q are the energy and the wavenumber ofthe excitation, respectively. In the first term (so-calledDoppler term), ∂n∂P e ≥ 0. Thus, the energetic insta-bility occurs when ω(q) for q = −|q| becomes zero ornegative:

∂2e

∂n∂P≥√

∂2e

∂n2

∂2e

∂P 2. (30)

Using this condition, we determine the critical quasi-momentum Pc at which the equality of Eq. (30) holds,and finally, we obtain the critical velocity from

vc =1n

(∂e

∂P

)Pc

. (31)

We note that, for calculating Pc and vc using Eqs. (30)and (31), what we only need is the energy density of thestationary states as a function of n and P . This can beobtained by solving, e.g., the GP or the BdG equationsfor the periodic potential.

If the energetic instability of Fermi superfluids iscaused by pair-breaking excitations, the critical veloc-ity can be determined by using the quasiparticle energyspectrum εi obtained from the BdG equations. The en-ergetic instability due to the pair-breaking excitationsoccurs when some quasiparticle energy εi becomes zeroor negative:

εi ≤ 0. (32)

We obtain a critical velocity for the pair-breaking ex-citations from Eq. (31) evaluated at the critical quasi-momentum determined by this condition.

4.2.3. Critical velocity of superfluid Bose gases andsuperfluid unitary Fermi gases in a lattice

First, we consider the situation where the LDA is valid;i.e., the lattice constant d is much larger than the heal-ing length ξ. This condition corresponds to gn/ER � 1for superfluid Bose gases and EF/ER � 1 for super-fluid Fermi gases at unitary (see discussion in Sec. 2.1).In the framework of the LDA hydrodynamics explainedin Sec. 2.3, the system is considered to become ener-getically unstable if there exists some point z at whichthe local superfluid velocity v(z) is equal to or largerthan the local sound velocity cs(z). If the external po-tential is assumed to have a maximum at z = z0 [i.e.,

V (z0) = V0 for our periodic potential], then at the samepoint, the density is minimum, cs(z) is minimum, andv(z) is maximum due to the current conservation; i.e.,j = n(z)v(z) being constant. This means that the super-fluid first becomes unstable at z = z0. Using Eq. (13), wecan write the condition for the occurrence of the insta-bility as m[jc/nc(z0)]2 = γµ[nc(z0)] = γαnγ

c (z0), wherenc(z) is the density profile calculated at the critical cur-rent [58]. By inserting this condition into Eq. (11), wecan obtain the following implicit relation for the criticalcurrent [22]:

j2c =

γ

mα2/γ

[2µjc

2 + γ

(1 − V (z0)

µjc

)] 2γ +1

. (33)

It is worth noticing that this equation contains only z-independent quantities. It is also independent of theshape of the external potential: the only relevant pa-rameter is its maximum value V (z0). Moreover, it canbe applied to both Bose gases and unitary Fermi gases(see also Refs. 59–61 for bosons).

In Fig. 4, we plot as thick solid red lines the criticalvelocity in a lattice obtained from the hydrodynamic ex-pression in Eq. (33) for BECs [panel (a)] and unitaryFermi gases [panel (b)]. In both cases, the critical veloc-ity vc = jc/n0 (n0 is the average density) is normalizedto the value of the sound velocity in the uniform gas,c(0)s , and is plotted as a function of V0/µj=0. The limit

V0/µj=0 → 0 corresponds to the usual Landau criterionfor a uniform superfluid flow in the presence of a smallexternal perturbation, i.e., a critical velocity equal tothe sound velocity of the gas. In this hydrodynamicscheme, as mentioned before, the critical velocity de-creases when V0 increases mainly because the densityhas a local depletion and the velocity has a correspond-ing local maximum, so that the Landau instability occursearlier. When V0 = µjc

, the density exactly vanishes atz = z0; hence, the critical velocity goes to zero.

Next, we discuss the critical velocity of the energeticinstability beyond the LDA [22]. Here, we use the energydensity e(n, P ) of superfluids in a periodic potential cal-culated by using the mean-field theory in the continuummodel, i.e., the GP equation for bosons and the BdGequations for fermions. Based on the energy density, wedetermine the critical quasi-momentum and the criticalvelocity from Eqs. (30) and (31).

For Bose superfluids, we plot vc for various values ofgn/ER in Fig. 4(a). We can clearly see that these re-sults approach the LDA prediction for gn/ER � 1, asexpected. We also note that vc exhibits a plateau forgn/ER � 1 (i.e., ξ � d) and small V0. This can beunderstood as follows: If the healing length ξ is largerthan the lattice spacing d and V0/gn is not too large,the energy associated with quantum pressure, which isproportional to 1/ξ2, acts against local deformations ofthe order parameter, and the latter remains almost unaf-fected by the modulation of the external potential. Thisis the region of the plateau in Fig. 4. In terms of Eq. (30),


this region occurs when the left-hand side is P/m andthe right-hand side is c

(0)s , so that the critical quasi-

momentum obeys the relation Pc/m = c(0)s , which is the

usual Landau criterion for a uniform superfluid in thepresence of small perturbers. With increasing V0, thisregion ends when µ ∼ ER.

If we further increase V0, the chemical potential µ be-comes larger than ER, the density is forced to oscillate,and vc/c

(0)s starts to decrease. The system eventually

reaches a region of weakly-coupled superfluids separatedby strong barriers, which is well described by the tight-binding approximation (also for gn/ER � 1, the systementers this region when V0 is sufficiently large). There,the energy density is given by a sinusoidal form withrespect to P as

e(n, P ) = e(n, 0) + δJ [1 − cos (πP/Pedge)] . (34)

Here, Pedge is the quasi-momentum at the edge of thefirst Brillouin zone; i.e., Pedge = �qB for superfluids ofbosonic atoms (Pedge = �qB/2 for those of fermionicatoms). The quantity δJ = nP 2

edge/π2m∗ correspondsto the half width of the lowest Bloch band. Becauseof the sinusoidal shape of the energy density and thelarge effective mass in the tight-binding limit, the crit-ical quasi-momentum is around Pedge/2. Thus, we seethat, from Eq. (31), the critical velocity is determinedby the effective mass as

vc 1π

Pedge

m∗ . (35)

The values of vc obtained from Eq. (35), with m∗ ex-tracted from the GP calculation of e(n, P ), are plottedby thinner black solid lines in Fig. 4(a) for gn/ER = 0.4and 1 in the region of V0/µj=0 � 2.

We also calculate the critical velocity by using anothermethod based on a complete linear stability analysis forthe GP energy functional as in Refs. 47,55, and 62 (seealso Refs. 41 and 48). We have checked that the re-sults agree with those obtained from Eq. (30) based onthe hydrodynamic analysis to within 1% over the wholerange of gn/ER and V0 considered in the present work.This confirms that the energetic instability in the pe-riodic potential is triggered by long-wavelength excita-tions, because the excitation energy of the sound modeis the smallest in this limit.

For superfluid unitary Fermi gases, we plot vc for var-ious values of EF/ER in Fig. 4(b). We observe qualita-tively similar results compared to those for bosons plot-ted in Fig. 4(a). For EF/ER � 1, the critical velocityvc shows a plateau at small V0, and it decreases from c

(0)s with increasing V0. For larger EF/ER > 1, vc

approaches the LDA result (thick solid red line). In theregion of large V0 such that V0/µ is sufficiently large, vc iswell described by the tight-binding expression in Eq. (35)plotted by thinner black solid lines in Fig. 4(b). Here,we use m∗ calculated from the BdG equations. We also

Fig. 5. (Color online) Critical velocity vc of the energeticinstability for EF/ER = 0.1 with s = 1 and 5 in the BCS-BECcrossover. Open circles and filled squares show the criticalvelocity due to long-wavelength phonons and fermionic pair-breaking excitations, respectively. The horizontal dotted line

represents the value of the sound velocity c(0)s of a uniform

system at unitarity, c(0)s /vF = (1 + β)1/2/

√3 � 0.443. The

red solid line is a guide for the eye.

note that the pair-breaking excitations are irrelevant tothe energetic instability at unitarity except for very lowdensities such that EF/ER � 1 and small, but nonzero,V0. These excitations are important on the BCS side ofthe BCS-BEC crossover, as discussed in Sec. 4.2.4.

Finally, we would like to point out that, in the LDAlimit, while the quantities discussed in Sec. III approachthe value in the uniform system, the critical velocity ofthe energetic instability is most strongly affected by thelattice. Figure 4 clearly shows that, as we increase thelattice height, the critical velocity decreases from that inthe uniform system most rapidly in the LDA limit. Themain reason for this opposite tendency is that the criticalvelocity is determined only around the potential maximawhile the quantities discussed in Sec. III are determinedby contributions from the whole region of the system.

4.2.4. Along the BCS-BEC crossover

Here, we extend our discussion on the Landau criticalvelocity to the BCS-BEC crossover region [23]. As inthe uniform systems, both long-wavelength phonon ex-citations and pair-breaking excitations can be relevantto the energetic instability, depending on the interactionparameter 1/kFas. However, there is an additional effect


due to the lattice: when the lattice height is much largerthan the Fermi energy, the periodic potential can causepairs of atoms to be strongly bound even in the BCSregion, so the pair-breaking excitations are suppressed[23].

In Fig. 5, we plot vc for superfluid Fermi gases inthe BCS-BEC crossover for different values of the latticeheight with s = 1 and 5. Here, we set EF/ER = 0.1 as anexample. The open circles show vc of the energetic insta-bility caused by long-wavelength superfluid phonon exci-tations, and the filled squares show that caused by pair-breaking excitations. For a moderate lattice strength ofs = 1, the result for vc is qualitatively the same as that ofthe uniform system. On the BCS side, the smallest vc isgiven by the pair-breaking excitations while, on the BECside, vc is set by the long-wavelength phonon excitations,and around unitarity vc shows a maximum. For a largervalue of s = 5, however, the result is very different fromthe uniform case. Due to the lattice-induced binding,pair breaking is suppressed so that it does not cause anenergetic instability even on the BCS side. (Note thatthere is no filled square for s = 5. This means thatwe do not have a negative quasiparticle energy for anyvalue of P in the whole Brillouin zone. This point willbe discussed later.) Throughout the calculated region of−1 ≤ 1/kFas ≤ 1, vc is determined only by the long-wavelength phonon excitations, and vc decreases mono-tonically with increasing 1/kFas.

The effect of the lattice-induced molecular formationcan be clearly seen in the quasiparticle energy spectrumεi and in the enhancement of the order parameter ∆(z).In Fig. 6, we show εi and |∆(z)| in the case of s = 5,EF/ER = 0.1, and 1/kFas = −1. The spectrum for P =0 shows a quadratic dependence on kz with a positivecurvature around kz = 0, and there are no minima atkz �= 0. Even though the figure represents a case inthe BCS region, the structure of εi is consistent withthe formation of bound pairs. We can also see that thespectrum never becomes negative for any values of P inthe first Brillouin zone. In the inset of the same figure,we show the amplitude |∆(z)| of the order parameter atP = 0. Here, we see a large enhancement of |∆(z)| nearz = 0 compared to the uniform system, which showsthe formation of bosonic bound molecules. We also notethat the minimum value of |∆(z)| at z/d = ±1 is smallerthan, but still comparable to, the value of |∆| in theuniform case, suggesting that the system is, indeed, inthe superfluid phase.

4.2.5. Experiments

In closing this section, we briefly summarize experi-mental studies on the stability of a superfluid. Usingcold atomic gases, stability of the superfluid and its crit-ical velocity was first experimentally studied in Ref. 63and further examined in Ref. 64. These experiments used

Fig. 6. (Color online) Lowest band of the quasiparticleenergy spectrum εi for large lattice height with s = 5 andEF/ER = 0.1 (i.e., EF/V0 = 0.02) in the BCS region at1/kFas = −1. Here, we show the first radial branch withk2⊥ ≡ k2

x + k2y = 0, which always gives the smallest values of

εi in this case. The inset shows the amplitude |∆(z)| of theorder parameter at P = 0. The horizontal dotted line showsthe amplitude of the order parameter for a uniform system atthe same value of 1/kFas = −1. This figure is adapted fromRef. 23.

a large (diameter � ξ) and strong (height � µ) vibrat-ing circular potential in a BEC. However, it has beenconcluded that what was observed in these works wasnot likely to be the energetic instability: dynamical nu-cleation of vortices by vibrating potential [64]. Recently,Ramanathan et al. performed a new experiment on thestability of a superflow with a different setup [65]. Theyused a BEC flowing in a toroidal trap with a tunableweak link (width of the barrier along the flow directionmuch larger than ξ). In that experiment, they obtaineda critical velocity consistent with the energetic instabil-ity due to vortex excitations [66]. For 2D Bose gases,the stability of the superfluidity and the critical velocitywere also studied recently [67].

Regarding the superflow in optical lattices, its stabil-ity was first experimentally studied in Ref. 68. In thatexperiment, they used a cigar-shaped BEC that under-went a center-of-mass oscillation in a harmonic trap inthe presence of a weak 1D optical lattice, and they mea-sured the critical velocity. Although their original con-clusion was that they had measured the Landau criticalvelocity for the energetic instability, further careful ex-


Fig. 7. (Color online) (a) Energy E per particle as afunction of the quasi-momentum P for various values of1/kFas, and (b) half width of the swallowtails along the BCS-BEC crossover. These results are obtained for s = 0.1 andEF/ER = 2.5. The quasi-momentum Pedge = �qB/2 fixes theedge of the first Brillouin zone. The dotted line in (b) is thehalf width in a BEC obtained by solving the GP equation; itvanishes at 1/kFas � 10.6. This figure is taken from Ref. 24.

perimental [69] and theoretical [47,62] follow-up studiesclarified that the instability observed in Ref. 68 was adynamical instability. In this follow-up experiment byDe Sarlo et al. [69], they employed an improved setup,i.e., a 1D optical lattice moving at constant and tunablevelocities instead of an oscillating BEC in a static lattice.With this new setup, they succeeded in observing bothenergetic [69] and dynamical [69, 70] instabilities, andobtained a very good agreement with theoretical pre-dictions [62,69,70]. It is worthwhile to stress that thisunderstanding was finally obtained through a continuouseffort over a few years by the same group.

Experimental study on the stability of superfluidFermi gases in optical lattices was carried out by Milleret al. [37]. Similar to the experiment of Ref. 69, theyalso used a 1D lattice moving at constant and tunablevelocities. A different point is that, instead of imposinga periodic potential on the whole cloud, they produced alattice potential only in the central region of the cloud.They measured the critical velocity of the energetic in-stability in the BCS-BEC crossover and found that itwas largest around unitarity. They also did a systemat-ical measurement of the critical velocity at unitarity forvarious lattice heights. However, there is a significantdiscrepancy from a theoretical prediction [22], so furtherstudies are needed.

V. ENERGY BAND STRUCTURE

In this section, we discuss how the superfluidity can af-fect the energy band structures of ultracold atomic gasesin periodic optical potentials. Starting with the nonin-teracting particles in a periodic potential, we obtain the

well-known sinusoidal energy band structure. When theinterparticle interaction is turned on in such a way thatthe superfluidity appears, we see a drastic change in theenergy band structure. For BECs, it has been pointedout that the interaction can change the Bloch band struc-ture, causing the appearance of a loop structure called“swallowtail” in the energy dispersion [71–74]. This isdue to the competition between the external periodicpotential and the nonlinear mean-field interaction: theformer favors a sinusoidal band structure while the lattertends to make the density smoother and the energy dis-persion quadratic. When the nonlinearity wins, the effectof the external potential is screened, and a swallowtailenergy loop appears [75]. This nonlinear effect requiresthe existence of an order parameter; consequently, theemergence of swallowtails can be viewed as a peculiarmanifestation of superfluidity in periodic potentials.

Qualitatively, one can argue in a similar way on theexistence of the swallowtail energy band structure inFermi superfluids in optical lattices; the competition be-tween the external periodic potential energy and the non-linear interaction energy (g|∑i ui(r)v∗i (r)|2) determinesthe energy band structure. However, the interaction en-ergy in the Fermi gas is more involved, and the unifiedview along the crossover from the BCS to the BEC statesis a nontrivial and interesting problems in itself. To an-swer the questions of 1) whether or not swallowtails existin Fermi superfluids and 2) whether unique features thatare different from those in bosons exist, we solve the BdGequations [see Eq. (9)] of a two-component unpolarizeddilute Fermi gas subject to a one-dimensional (1D) opti-cal lattice [24].

5.1. Swallowtail Energy Spectrum

The energy per particle in the lowest Bloch band as afunction of the quasi-momentum P for various values of1/kFas is computed [76]. The results in Fig. 7(a) showthat the swallowtails appear above a critical value of1/kFas where the interaction energy is strong enough todominate the lattice potential. In Fig. 7(b), the half-width of the swallowtails from the BCS to the BECside is shown. It reaches a maximum near unitarity(1/kFas = 0). In the far BCS and BEC limits, the widthvanishes because the system is very weakly interactingand the band structure tends to be sinusoidal. Whenapproaching unitarity from either side, the interactionenergy increases and can dominate over the periodic po-tential, which means that the system behaves more like atranslationally-invariant superfluid and the band struc-ture follows a quadratic dispersion terminating at a max-imum P larger than Pedge.

The emergence of swallowtails on the BCS side forEF/ER � 1 is associated with peculiar structures of thequasiparticle energy spectrum around the chemical po-tential. In the presence of a superflow moving in the z


Fig. 8. (Color online) (a) Lowest three Bloch bands of the quasiparticle energy spectrum at k⊥ = 0 for P = Pedge and1/kFas = −0.62. Thin black dashed lines labeled by l’s show the approximate energy bands obtained from Eq. (36) by usingµ � 2.66ER and |∆| � |∆(0)| � 0.54ER. (b) Bloch band of the quasiparticle energy spectrum around the chemical potentialfor P/Pedge = 0 (black dotted), 0.5 (green dashed), and 1 (red solid) at k⊥ = 0 and 1/kFas = −0.62 in the case of s = 0.1 andEF/ER = 2.5. Each horizontal line denotes the value of the chemical potential for the corresponding value of P . This figure isadapted from Ref. 24.

direction with wavevector Q (≡ P/�), the quasiparticleenergies are given by the eigenvalues in Eq. (9). Becausethe potential is shallow (s � 1), some qualitative resultscan be obtained even when ignoring Vext(z) except forits periodicity. With this assumption, we obtain

εk≈ (kz+2qBl)Qm

+

√[k2⊥+(kz+2qBl)2+Q2

2m−µ

]2+|∆|2 ,

(36)

with l being integers for the band index. If Q =0, the l = 0 band has the energy spectrum√

[(k2⊥ + k2

z)/2m − µ]2 + |∆|2, which has a local maxi-mum at kz = k⊥ = 0. When Q = Pedge/�, the spectrumis tilted, and the local maximum moves to kz qB/2provided |∆| � EF (and EF/ER � 1). In the absence ofthe swallowtail, the full BdG calculation, indeed, givesa local maximum at kz = qB/2, and the quasiparticlespectrum is symmetric about this point, which reflectsthat the current is zero. As EF/ER increases, the bandbecomes flatter as a function of kz and narrower in en-ergy.

In Fig. 8(a), we show the quasiparticle energy spec-trum at k⊥ = 0 for P = Pedge. When the swallowtailis on the edge of appearing, the top of the narrow bandjust touches the chemical potential µ [see the dotted el-lipse in Fig. 8(a)]. Suppose 1/kFas is slightly larger thanthe critical value so that the top of the band is slightlyabove µ. In this situation, a small change in the quasi-momentum P causes a change of µ. In fact, when Pis increased from P = Pedge to larger values, the bandis tilted, and the top of the band moves upwards; thechemical potential µ should also increase to compensatefor the loss of states available, as shown in Fig. 8(b).This implies ∂µ/∂P > 0. On the other hand, because

Fig. 9. (Color online) Incompressibility κ−1 at P = Pedge

around the critical value of 1/kFas ≈ −0.62 where the swal-lowtail starts to appear. The quantity κ−1

0 is the incompress-ibility of the homogeneous free Fermi gas of the same averagedensity. In both panels, we have used the values s = 0.1 andEF/ER = 2.5. This figure is adapted from Ref. 24.

the system is periodic, the existence of a branch of sta-tionary states with ∂µ/∂P > 0 at P = Pedge implies theexistence of another symmetric branch with ∂µ/∂P < 0at the same point, thus suggesting the occurrence of aswallowtail structure.

5.2. Incompressibility

A direct consequence of the existence of a narrowband in the quasiparticle spectrum near the chemicalpotential is a strong reduction of the incompressibility


Fig. 10. (Color online) Profiles of the pairing field |∆(z)| and the density n(z) at 1/kFas = −0.8 (blue dotted), −0.6 (greendashed), and −0.4 (red solid) for P = Pedge in the case of s = 0.1 and EF/ER = 2.5. The swallowtail starts to appear at acritical value of 1/kFas ≈ −0.62. This figure is taken from Ref. 24.

κ−1 = n∂µ(n)/∂n close to the critical value of 1/kFas

in the region where swallowtails start to appear in theBCS side (see Fig. 9). A dip in κ−1 occurs in the sit-uation where the top of the narrow band is just aboveµ for P = Pedge (1/kFas is slightly above the criticalvalue). An increase in the density n has little effect on µin this case because the density of states is large in thisrange of energy and the new particles can easily adjustthemselves near the top of the band by a small increaseof µ. This implies that ∂µ(n)/∂n is small and that theincompressibility has a pronounced dip [77]. It is worthnoting that on the BEC side, the appearance of the swal-lowtail is not associated with any significant change inthe incompressibility. In fact, for a Bose gas with an av-erage density nb0 of bosons, the exact solution of the GPequation gives κ−1 = nb0g near the critical conditionsfor the occurrence of swallowtails, being a smooth andmonotonic function of the interaction strength.

5.3. Profiles of Density and Pairing Fields

Both the pairing field and the density exhibit inter-esting features in the range of parameters where theswallowtails appear. This is particularly evident at theBrillouin zone boundary, P = Pedge. The full profilesof |∆(z)| and n(z) along the lattice vector (z direction)are shown in Fig. 10. In general, n(z) and |∆(z)| takemaximum (minimum) values where the external poten-tial takes its minimum (maximum) values. By increasingthe interaction parameter 1/kFas, we find that the orderparameter |∆| at the maximum (z = ±d/2) of the lat-tice potential exhibits a transition from zero to nonzerovalues at the critical value of 1/kFas at which the swal-lowtail appears. Note that here we plot the absolutevalue of ∆; the order parameter ∆ behaves smoothlyand changes sign.

In Fig. 10, we show the magnitude of the pairing field|∆(z)| and the density n(z) calculated at the minimum(z = 0) and at the maximum (z = ±d/2) of the lat-tice potential. The figure shows that |∆(d/2)| remainszero in the BCS region until the swallowtail appears at1/kFas ≈ −0.62. Then, it increases abruptly to valuescomparable to |∆(0)|, which means that the pairing fieldbecomes almost uniform at P = Pedge in the presenceof swallowtails. As regards the density, we find that theamplitude of the density variation, n(0) − n(d/2), ex-hibits a pronounced maximum near the critical value of1/kFas.

In contrast, on the BEC side, the order parameterand the density are smooth monotonic functions of theinteraction strength even in the region where the swal-lowtail appears. At P = Pedge, the solution of the GPequation for bosonic dimers gives the densities nb(0) =nb0(1 + V0/2nb0gb) and nb(d/2) = nb0(1 − V0/2nb0gb),with V0/2nb0gb = (3π/4)(sER/EF)(1/kFas), where gb ≡4π�

2ab/mb, and ab and mb are the scattering length andthe mass of bosonic dimers, respectively [71,78,79]. Nearthe critical value of 1/kFas, unlike the BCS side, thenonuniformity just decreases all the way even after theswallowtail appears. The local density at z = d/2 is zerountil the swallowtail appears on the BEC side while itis nonzero on the BCS side irrespective of the existenceof the swallowtail. The qualitative behavior of |∆(z)|around the critical point of 1/kFas is similar to that ofnb(z) because nb(r) = (m2as/8π)|∆(r)|2 [80] in the BEClimit.

VI. QUANTUM PHASES OF COLD ATOMICGASES IN OPTICAL LATTICES

So far, we have discussed some superfluid features ofcold atomic gases in an optical lattice by mainly focusingon our works. In this section, we discuss some important


Fig. 11. (Color online) Profiles of (a) the pairing field |∆(z)| and (b) the density n(z) for changing 1/kFas for P = Pedge inthe case of s = 0.1 and EF/ER = 2.5. The values of |∆(z)| and n(z) at the minimum (z = 0, blue �) and at the maximum(z = ±d/2, red ×) of the lattice potential are shown. The vertical dotted lines show the critical value of 1/kFas above whichthe swallowtail exists. The dotted curve in (a) shows |∆| in a uniform system. This figure is taken from Ref. 24.

topics regarding various quantum phases that the coldatomic gases on optical lattices can show/simulate. Inthis section, strong enough periodic potentials to ignorethe next-nearest hopping and small average numbers ofparticles per lattice site will be assumed so that the sys-tem can be described by the Hubbard lattice model.

6.1. Quantum Phase Transition from a Super-fluid to a Mott Insulator

At a temperature of absolute zero, cold atomic gases inoptical lattices can undergo a quantum phase transitionfrom superfluid to Mott insulator phases as the interac-tion strength between atoms is tuned from the weakly- tothe strongly-interacting region [81]. The quantum phasetransition of an interacting boson gas in a periodic latticepotential can be captured by the following Bose-HubbardHamiltonian:

H = −J∑〈i,j〉

(b†i bj + h.c.) +U

2

∑i

ni(ni − 1) ,

where bi and b†i are annihilation and creation operatorsof a bosonic atom on the i-th lattice site and ni = b†i bi

is the atomic number operator on the i-th site. Thefirst term is the kinetic energy term whose strength ischaracterized by the hopping matrix element J betweenadjacent sites 〈i, j〉, and U(> 0) in the second term is thestrength of a short-range repulsive interactions betweenbosonic atoms.

In the limit where the kinetic energy dominates, themany-body ground state of N atoms and M lattice sitesis a superposition of delocalized Bloch states with lowest

energy (quasi-momentum k = 0):

|ΨSF〉U/J=0 ∝ (b†k=0)N |0〉 ∝

(M∑i=1

b†i

)N

|0〉 , (37)

where |0〉 is the empty lattice. This state has perfectphase correlation between atoms on different sites whilethe number of atoms on each site is not fixed. Thissuperfluid phase has gapless phonon excitations.

In the limit where the interactions dominate (so-called,“atomic limit”), the fluctuations in the local atom num-ber become energetically unfavorable, and the groundstate is made up of localized atomic wavefunctions witha fixed number of atoms per lattice site. The many-bodyground state with a commensurate filling of n atoms perlattice site is given by

|ΨMI〉J/U=0 ∝M∏i=1

(b†i )n|0〉 . (38)

There is no phase correlation between different sites be-cause the energy is independent of the phases of thewavefunctions on each site. This Mott insulator state,unlike the superfluid state, cannot be described by us-ing a macroscopic wavefunction. The lowest excitedstate can be obtained by moving one atom from onesite to another, which gives an energy gap of ∆ =(U/2)[(n + 1)2 + (n − 1)2 − 2n2] = U .

As the ratio of the interaction term to the tunnel-ing term increases (U/J can be controlled by chang-ing the depth V0 of the optical lattice even without us-ing the Feshbach resonances), the system will undergoa quantum phase transition from a superfluid state toa Mott insulator state accompanying the opening ofthe energy gap in the excitation spectrum. Greineret al. realized experimentally a quantum phase tran-sition from a Bose-Einstein condensate of 87Rb atoms


with weak repulsive interactions to a Mott insulator ina three-dimensional optical lattice potential [17]. No-tably, they could induce reversible changes between thesetwo ground states by changing the strength V0 of theoptical lattice. The superfluid-to-Mott-insulator tran-sitions were also achieved in one- and two-dimensionalcold atomic Bose gases [82]. The Mott insulator phasesof atomic Fermi gases with repulsive interactions on athree-dimensional optical lattice have been realized, andthe entrance into the Mott insulating states was observedby verifying vanishing compressibility and by measuringthe suppression of doubly-occupied sites [83,84].

6.2. Quantum Phase Transition from a Para-magnet to an Antiferromagnet

Sachdev et al. showed that the one-dimensional Mottinsulator of spinless bosons in a tilted optical lattice canbe mapped onto a quantum Ising chain [85]. The Bose-Hubbard Hamiltonian for a tilted optical lattice takesthe form

H = −J∑

i

(bib†i+1+b†i bi+1)+

U

2

∑i

ni(ni−1)−E∑

i

ini ,

where E is the lattice potential gradient per lattice spac-ing. For a tilt near E = U , the energy cost of movingan atom to its neighbor (from site i to site i + 1) is zero.If we start with a Mott insulator with a single atom persite, an atom can resonantly tunnel into the neighbor-ing site to produce a dipole state (at the link) consist-ing of a quasihole-quasiparticle pair on nearest neigh-bor sites. Only one dipole can be created per link, andneighboring links cannot support dipoles together. Thisnearest-neighbor constraint is the source of the effectivedipole-dipole interaction that results in a density waveordering. If a dipole creation operator d†i = bib

†i+1/

√2 is

defined, the Bose-Hubbard Hamiltonian is mapped ontothe dipole Hamiltonian:

H = −√

2J∑

i

(d†i + di) + (U − E)∑

i

d†i di,

with the constraint d†i di ≤ 1 and d†

i did†i+1di+1 = 0. If

the dipole present/absent link is identified with a pseu-dospin up/down, Sz

i = d†i di − 1/2, the pseudospin-1/2

Hamiltonian takes the form of a quantum Ising chain:

H = JS

∑i

Szi Sz

i+1 − 2√

2J∑

i

Sxi + (JS − D)

∑i

Szi

= JS

∑i

(Szi Sz

i+1 − hxSxi + hzS

zi ) ,

where D = E − U and∑

i JS(Szi + 1/2)(Sz

i+1 + 1/2),with JS → ∞ being added to the Hamiltonian for im-plementing the constraint d†i did

†i+1di+1 = 0. The dimen-

sionless transverse and longitudinal fields are defined ashx = 23/2J/JS and hz = 1 − D/JS , respectively.

Thus, a quantum phase transition from a paramag-netic phase (D < 0) to an antiferromagnetic phase(D > 0) can be studied by changing the lattice potentialgradient E between adjacent sites. With 87Rb atoms,a quantum simulation of antiferromagnetic spin chainsin an optical lattice was done, and a phase transitionto the antiferromagnetic phase from the paramagneticphase was observed [86].

VII. SUMMARY AND OUTLOOK

In this article, we have focused on some superfluidproperties of cold atomic gases in an optical lattice peri-odic along one spatial direction: basic macroscopic andstatic properties, the stability of the superflow, and pe-culiar energy band structures. As a complement, somephases other than the superfluid phase have been dis-cussed in the last section. Considering the rapid growthand interdisciplinary nature of the research on coldatomic gases in optical lattices, it is practically impossi-ble to cover all perspectives. We conclude by mentioninga few of them.

The high controllability of the lattice potential opensmany possibilities: An optical disordered lattice can beconstructed to study the problem of the Anderson lo-calization of matter waves and the resulting phases [87].The time modulation of the optical lattice is shown totune the magnitude and the sign of the tunnel couplingin the Hubbard model, which allows us to study variousphases [88]. Cold atomic gases in optical lattices mayuncover many exotic phases that are still under debateor even lack solid state analogs.

Due to their long characteristic time scales and largecharacteristic length scales, cold atomic gases are goodplaygrounds for the experimental observation and con-trol of their dynamics. Particularly, a sudden quenchcan be realized experimentally. The nonequilibrium dy-namics after sudden quenches can be studied with highprecision in experiments to discriminate between can-didate theories. Long-standing problems such as ther-malization, its connection with non-integrability and/orquantum chaos are actively sought-after topics in thisdirection [89].

The charge neutrality of atoms seems to limit the useof this system as a quantum simulator. However, theinternal states of an atom, together with atom-laser in-teractions, can be exploited for the atom to gain geomet-ric Berry’s phases, which amounts to generating artificialgauge fields interacting with these charge-neutral parti-cles [90]. In this way, interactions with electromagneticfields, spin-orbit coupling, and even non-Abelian gaugefields can be emulated to open the door to the studyof the physics of the quantum Hall effects, topologicalsuperconductors/insulators, and high energy physics byusing cold atoms in the future.

Cold atomic gases provide a promising platform for


controlling dissipation and for engineering the Hamilto-nian, e.g., by controlling the coupling with a subsystemacting as a reservoir, by using external fields that inducelosses of trapped atoms, etc. This possibility will allowus to use the cold atomic gases as quantum simulatorsof open systems. In addition, the controlled dissipationwill offer an opportunity to study the quantum dynamicsdriven by dissipation and its steady states, and to studythe non-equilibrium phase transitions among the steadystates determined by the competition between the co-herent and the dissipative dynamics. Furthermore, con-trolling the dissipation will pave the way to the design ofLiovillian and to the dissipation-driven state-preparation(e.g., Ref. 91).

Last, but not least, cold atomic gases in optical latticesare also useful for the precision measurements; the “op-tical lattice clock” consists of millions of atomic clockstrapped in an optical lattice and working in parallel.The large number of simultaneously-interrogated atomsgreatly improves the stability of the clock, and state-of-the-art optical lattice clocks outperform the primaryfrequency standard of Cs clocks (Ref. 18 and referencestherein). Bloch oscillations of cold atomic gases in opti-cal lattices offer a promising way of measuring forces at aspatial resolution of few micrometers (e.g., Ref. 92). Pre-cision measurement devices/techniques will enable high-precision tests of time and space variations of the fun-damental constants, the weak equivalence principle, andNewton’s gravitational law at short distances.

ACKNOWLEDGMENTS

We acknowledge Mauro Antezza, Franco Dalfovo, Elis-abetta Furlan, Giuliano Orso, Francesco Piazza, LevP. Pitaevskii, and Sandro Stringari for collaborations.This work was supported in part by the Max PlanckSociety, by the Korean Ministry of Education, Scienceand Technology, by Gyeongsangbuk-Do, by Pohang City[support of the Junior Research Group (JRG) at theAsia Pacific Center for Thoretical Physics (APCTP)],and by Basic Science Research Program through the Na-tional Research Foundation of Korea (NRF) funded bythe Ministry of Education, Science and Technology (No.2012R1A1A2008028). Calculations were performed bythe RIKEN Integrated Cluster of Clusters (RICC) sys-tem, by WIGLAF at the University of Trento, and byBEN at ECT*.

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Aspects of superfluid cold atomic gases in optical lattices