Superfluidity of Bose-Einstein condensates in ultracold atomic gases

0.96 MB·English

by  Qizhong Zhu

Superfluidity of Bose-Einstein condensates in ultracold atomic gases
Qizhong Zhu(朱起忠)1 and Biao Wu(吴飙)1,2
1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
2Collaborative Innovation Center of Quantum Matter, Peking University, Beijing 100871, China
(Dated: January 20, 2015)
Liquid helium 4 had been the only bosonic superfluid available in experiments for a long time.
This situation was changed in 1995, when a new superfluid was born with the realization of the
5
1 Bose-Einstein condensation in ultracold atomic gases. The liquid helium 4 is strongly interacting
0 and has no spin; there is almost no way to change its parameters, such as interaction strength and
2 density. The new superfluid,Bose-Einstein condensate (BEC), offers various aspects of advantages
overliquidhelium. Ontheonehand,BECisweaklyinteractingandhasspindegreesoffreedom. On
n
theotherhand,itisconvenienttotunealmostalltheparametersofaBEC,forexample,thekinetic
a
energybyspin-orbitcoupling,thedensitybytheexternalpotential,andtheinteractionbyFeshbach
J
resonance. Great efforts have been devoted to studying these new aspects of superfluidity,and the
7
resultshavegreatlyenrichedourunderstandingofsuperfluidity. Herewereviewthesedevelopments
1
byfocusingonthestabilityandcriticalvelocityofvarioussuperfluids. TheBECsystemsconsidered

include a uniform superfluid in free space, a superfluid with its density periodically modulated, a
]
s superfluidwithartificiallyengineeredspin-orbitcoupling,andasuperfluidofpurespincurrent. Due
a totheweakinteraction,theseBECsystemscanbewelldescribedbythemeanfieldGross-Pitaevskii
g theoryandtheirsuperfluidity,inparticularcriticalvelocities,canbeexaminedwithLandau’stheory
-
of superfluid. Experimental proposals to observe thesenew aspects of superfluidity are discussed.
t
n
a PACSnumbers: 05.30.Jp,03.75.Mn,03.75.Kk,71.70.Ej
u
q
t. I. INTRODUCTION dipolar superfluid [8]. All these are impossible with liq-
a
uid helium. Moreover,as most ultracoldgases are dilute
m
and weakly-interacting,controllable theoretical methods
Superfluidity, as a remarkable macroscopic quantum
- are available to study these superfluids in detail.
d phenomenon, was first discovered in the study of liq-
n uid helium 4 in the 1938 [1, 2]. Although it is found Inthisreviewwemainlydiscussthreetypesofbosonic
o
theoretically that superfluidity is a general phenomenon superfluids: superfluid with periodic density, superfluid
c
forinteractingbosonsystems [3], liquidheliumhadbeen withspin-orbitcoupling,andsuperfluidofpurespincur-
[
theonlybosonicsuperfluidavailableinexperimentsuntil rent. The focus is on the stability and critical velocities

1 1995. Inthisyear,thanksto the advanceoflasercooling ofvarioussuperfluids. Forauniformsuperfluid,whenits
v
of atoms, the Bose-Einstein condensation of dilute al- speedexceeds a criticalvalue, the systemsuffers Landau
3
kali atomic gases was realized experimentally [4]; a new instability andsuperfluidity is lost. When the superfluid
5
superfluid, Bose-Einstein condensate (BEC), was born. moves in a periodic potential, with large enough quasi-
1
4 This addition to the family of superfluids is highly non- momentum,newmechanismofinstability,i.e.,dynamical
0 trivial as BECs offer various aspects of advantages over instability comes in. This instability usually dominates
. liquid helium that can greatly enrich our understanding the Landauinstability asitoccursonamuchfastertime
1
0 of superfluidity. scale. Theperiodicdensityalsobringsanothertwist: the
5 Greatdeal ofworkhas been done to explorethe prop- superfluidity can be tested in two different ways, which
1 erties of liquid helium as superfluid [5]. However, these yield two different critical velocities. For a superfluid
v: effortshavebeenhinderedbylimitationsofliquidhelium. with spin-orbit coupling, a dramatic change is brought
i As a liquid, the superfluid helium is strongly-interacting in, namely the breakdown of Galilean invariance. As a
X
system,whichmakesthetheoreticaldescriptiondifficult. result, its critical velocity will depend on the reference
r Atthesametime,nosystemparameters,suchasdensity frame. The stability of a pure spin current is also quite
a
and interaction strength, can be tuned experimentally. striking. We find that the pure spin currentin generalis
And helium 4 has no spin degrees of freedom. notasuper-flow. However,itcanbestabilizedtobecome
a super-flow with quadratic Zeeman effect or spin-orbit
BECs are strikingly different. Almost all the param-
coupling. Related experimental proposals are discussed.
eters of a BEC can be controlled easily in experiments:
its kinetic energy, density, and the interaction between Thepaperisorganizedasfollows. InSec. II,wediscuss
atoms can all be tuned easily by engineering the atom- brieflythebasicconceptsrelatedtotheunderstandingof
laser interaction, magnetic or optical traps, and the Fes- superfluidity, including Landau’s theory of superfluidity,
hbachresonance [6]. Inaddition,bychoosingtheatomic meanfieldGross-Pitaevskiiequation,andBogoliubovex-
species andusing opticaltrapto release the spin degrees citations. These concepts areillustratedwith the special
of freedom, one can also realize various types of new case of a uniform superfluid. We then apply these gen-
superfluids, including the spinor superfluid [7] and the eralmethods tostudy indetailthe superfluidinperiodic
2
potentials in Sec. III, the superfluid with artificially en- its speed was smaller than the sound speed. This is in
gineered spin-orbit coupling in Sec. IV, and finally the fact nothing but the Cerenkov radiation [10, 11], where
superfluid of pure spin current in Sec. V. In these three a charged particle radiates only when its speed exceeds
superfluids, special attention is paid to their excitations, the speed of light in the medium.
stabilities, and critical velocities. We finally summarize
in Sec. VI.
II. BASIC CONCEPTS OF SUPERFLUIDITY
(a) (b)
A. Landau’s theory of superfluidity
v
Superfluid is a special kind of fluid which does not v
Superfluid Superfluid
suffer dissipation when flowing through a tube. It loses
its superfluidity only when its speed exceeds a certain
critical value. The superfluidity of liquid helium 4 was
first explained by L. D. Landau [9]. He considered a
superfluid moving inside a stationary tube with veloc-
Superfluid
ity v. Since the system is invariant under the Galilean v
transformation,thisscenarioisequivalenttoastationary
fluid inside a moving tube. If the elementary excitation
in a stationary superfluid with momentum q has energy (c)
ǫ (q),thentheenergyofthesameexcitationintheback-
0
groundofamovingfluidwithvisǫ (q)=ǫ (q)+v q. A
v 0
·
fluid experiences friction only through emitting elemen-
taryexcitations,anditisasuperfluidiftheseelementary
excitationsareenergeticallyunfavorable. Inotherwords,
FIG. 1: (a) A superfluid moves inside a stationary tube. (b)
a superfluid satisfies the constraint ǫ (q)>0. It readily
v Thesuperfluidisdraggedbyatubemovingatthespeedofv.
leadstothewell-knownLandau’scriterionforsuperfluid,
(c) An impurity moves at v in the superfluid. The two-way
arrow indicates theequivalence between different scenarios.
ǫ (q)
0
v <v = . (1)
c q
(cid:18) | | (cid:19)min These two different ways of derivation are equivalent
when the system has the Galilean invariance. By trans-
Here v is the critical velocity of the superfluid, which is
c
forming to another reference frame illustrated in Fig.
determined by the smallest slope of the excitation spec-
1(b), the superfluid can be viewed as being dragged by
trum of a stationary superfluid.
a moving tube. We then replace the moving tube with
Anotherwayofderivingtheformulaofcriticalvelocity
a macroscopic impurity moving inside the superfluid as
is fromthe pointview ofCerenkovradiation. Considera
shown in Fig. 1(c). Consequently, for systems with
macroscopicimpurity movinginthesuperfluidgenerates
Galilean invariance, the critical velocity of a superfluid
an excitation. According to the conservations of both
movinginsideatubewithoutexperiencingfrictionisjust
momentum and energy, we should have
thecriticalvelocityofanimpuritymovinginasuperfluid
m v = m v +q, (2) without generating excitations. For systems without the
0 i 0 f
Galileaninvariance,suchanequivalenceislostaswewill
m0vi2 = m0vf2 +ǫ (q), (3) see later with the spin-orbit coupled BEC.
0
2 2 For ultracold bosonic gases, the low energy excitation
is phonon, linear with respect to momentum q, thus the
where m is the mass of the impurity, v and v are the
0 i f
critical speed of this superfluid is just the sound speed.
initial and final velocities of the impurity, respectively.
Taking into account the non-uniformity of the trapped
The above two conservations (2) and (3) can not be sat-
gas, the critical velocity measured in experiments agrees
isfied simultaneously when
well with the value predicted by Landau’s theory [12].
ǫ (q) For liquid helium 4, as there is another kind of elemen-
v v v <v = 0 . (4)
≈| i|≈| f| c q taryexcitationscalledrotons,thecriticalspeedofsuper-
(cid:18) | | (cid:19)min fluid helium 4 is largely determined by the roton excita-
The critical velocity v here has the same expression as tion, much smaller than its sound speed. Nevertheless,
c
that obtainedfromLandau’s criterion. If the excitations the critical velocity measured in experiments is still one
are phonons, i.e., ǫ (q) = cq, then v < c. This means order of magnitude smaller than the value predicted by
0 c
| |
that the impurity could not generate phonons in the su- the theory [13]. So it is remarkable that Landau’s pre-
perfluid and would not experience any viscosity when diction of critical velocity was experimentally confirmed
3
with BEC almost six decades after its invention. into account the interaction between the condensed par-
One remark is warranted on Landau’s theory of su- ticles. The first order approximation is to take into ac-
perfluidity. Landau’s criterion (1) of critical speed does count the interaction between the condensed particles
notapplyformanysuperfluids. However,Landau’sener- and un-condensed particles. This is settled by the Bo-
getic argument for superfluidity is very general and can goliubovtheoryofelementaryexcitations[17]. Thestan-
be applied to all the cases considered in this review. We dard Bogoliubov theory is to diagonalize the mean field
shall use this argument to determine the critical speeds Hamiltonianbythe Bogoliubovtransformation. Herewe
of various superfluids. adoptanequivalentyetmoreconvenientwaytodealwith
thisproblemininhomogeneoussystems. Forastationary
state ψ(r,t) that satisfies the GP equation, we consider
B. Gross-Pitaevskii equation and Bogoliubov some small time dependent perturbations δψ(r,t) added
excitation tothisstationarystate. TheperturbedstateΨ=ψ+δψ
also satisfies the GP equation,
As liquid helium 4 is a strongly interacting system,
the calculation of its excitation spectrum theoretically i~∂Ψ(r,t) = ~2∇2Ψ+V(r)Ψ+cΨ2Ψ. (7)
is challenging. However, for a dilute ultracold bosonic ∂t − 2M | |
gases,convenientyetpreciseapproximationscanbemade
Expanding this equation to first order in the perturba-
to determine its excitation in theory. Due to the dilute
tion δψ, one arrives at the equation of motion of the
nature and short range interaction, the interaction be-
perturbation δψ(r,t),
tween atoms can be approximated by a contact interac-
tion. Furthermore, for low energy scattering of bosonic ∂δψ ~2
atoms,thes-wavescatteringchanneldominates. So,asa i~ = 2δψ+V(r)δψ
∂t −2M∇
good approximation, the actual complex interaction be-
+2cψ 2δψ+cψ2δψ∗, (8)
tween atoms is replaced by an effective s-wave contact
| |
interaction,i.e., U(r ,r )=4π~2aδ(r r )/M,wherea
1 2 1− 2 and its complex conjugate partner. Assume the state
is the s-wavescatteringlength. At zerotemperature,as-
before perturbation is a stationary state with the wave
sume allthe particlescondense into the same orbitψ(r), functionψ(r,t)= n(r)e−iµt/~,andwritetheperturba-
then its evolution is governed by the mean field Gross-
tions as δψ(r,t) = n(r)e−iµt/~[u(r)e−iωt v∗(r)eiωt],
Pitaevskii (GP) equation [14], p −
then we arrive at a pair of equations that u(r) and v(r)
p
∂ψ(r,t) ~2 2 satisfy,
i~ = ∇ ψ+V(r)ψ+cψ 2ψ, (5)
∂t − 2M | |
u(r) u(r)
=~ω , (9)
where V(r) is the trapping potential or other external M v(r) v(r)
potential,andc=4π~2a/M istheinteractionparameter (cid:18) (cid:19) (cid:18) (cid:19)
or coupling constant. The GP energy functional reads where the matrix is given by
M
E[ψ,ψ∗]= dr −2~M2 |∇ψ|2+V(r)|ψ|2+ 2c|ψ|4 . M= cnH(0r) −cnH(0r) , (10)
Z (cid:20) (cid:21) (cid:18) − (cid:19)
(6)
Note that here the wave function ψ is normalized to the withH0 =−2~M2 ∇2+V(r)−µ+2cn(r). Bydiagonalizing
total particle number N. The approximation we make , one obtains two sets of solutions, but only the solu-
here is the mean field approximation, and the interac- Mtion satisfying the constraint dr u(r)2 v(r)2 > 0
| | −| |
tionbetweenatomsisreplacedbyaneffectivemeanfield represents physical excitations. Note that since the
R (cid:0) (cid:1)
potential. The validity of the approximation usually de- characteristic matrix is not hermitian, its eigenval-
pends on the condensed fraction. For weakly interact- ues are not necessarilMy real. If its eigenvalue ǫ = ~ω
ing dilute bosonic gases near zero temperature, the con- has nonzero imaginary part, the perturbation δψ(r,t) =
densed fraction can be more than 90 percent. In these n(r)e−iµt/~[u(r)e−iωt v∗(r)eiωt] will grow exponen-
−
situations, the GP equation works well. It has been tiallywithtime,signalingthatthestatebeforeperturba-
p
used to calculate the collective excitations of trapped tion suffers dynamical instability. If its eigenvalue is real
BEC, as well as vortex dynamics. Pretty good agree- but negative, elementary excitations associated with the
ment is achieved between theory and experiment. With perturbations will be energetically favorable and super-
a stochastic term describing the effect of thermal atoms, fluidityislost,whichiscalledLandau instability. Forthe
the modified stochastic GP equation can also simulate excitations of repulsive Bose gases without external po-
BEC systems at finite temperature [15]. In addition, for tentials,thelowenergyexcitationsmustbenon-negative
atoms with spin, one can also derive a multicomponent and gapless from the Hugenholtz-Pines theorem [18].
GP equation to describe a spinor superfluid [16]. As a specific example, we apply the above formalism
The stationary state the GP equation describes is a to the uniform Bose gas, namely, putting V(r)=0. The
zeroth order approximation in some sense. It only takes wavefunctionbeforeperturbationisjustψ =√ne−iµt/~,
4
and the perturbation has this form u(r) = u eiq·r/√V, whereallthevariablesarescaledtobedimensionlesswith
q
v(r) = v eiq·r/√V. Plugging these wave functions into respect to a set of characteristic parameters of the sys-
q
the above equations, one immediately obtains the exci- tem, the atomic mass M, the wave number kL of the
tation spectrum for a uniform Bose gas, laser light generating the optical lattice, and the aver-
agedensityn oftheBEC.Thechemicalpotentialµand
0
the strength of the periodic potential v are in units of
ǫ = ǫ0 ǫ0+2cn , (11)
q q q 4~2k2/M, the wavefunction ψ is in units of√n , x is in
L 0
lwohnegrweaǫv0qe=len~g2tqh2/o2rMsmiasqlltmheo(cid:0)msinengtleumpaq(cid:1)r,titchleeesxpceicttartuiomn.hAats uacntiitosnocfoknLs/ta2n,tanisdgtiviesninbyuncit=s πofn0Mas//4k~L22k,L2w.heTrheeaisn>ter0-
is the s-wave scattering length.
the asymptotic form ǫ q ~2nc/M. This is nothing
q ∼ For non-interacting case (c = 0), diagonalizing the
but phonon excitation, andp nc/M is just the speed Hamiltonian will give the standard Bloch waves and en-
of sound. From Landau’s theory of critical velocity, we ergy bands. When the mean field interaction is turned
p
concludethatthecriticalvelocityofauniformsuperfluid on (c = 0), in principle the Hamiltonian allows for
Bose gas is just the sound speed. other ty6 pes of solutions which have no counterpart in
The method of mean field approximation and Bogoli- the non-interacting case [24, 25]. Here we focus on the
ubovtransformationisverygeneral,andappliesinother solutions which still have the form of Bloch waves, i.e.,
morecomplicatedsituations. Inthefollowingdiscussion, ψ (x)=eikxφ (x),whereφ (x)hasthesameperiodwith
k k k
we use this method to study three types of superfluids, theopticallattice. φ (x)canbefoundbyextremizingthe
k
superfluid in a periodic potential, superfluid with spin- Hamiltonian above [23]. The solution found in this way
orbit coupling and superfluid of pure spin current. should satisfy the stationary GP equation with periodic
potential,
III. PERIODIC SUPERFLUID 1 ∂2
ψ+vcosxψ+cψ 2ψ =µψ. (13)
− 2∂x2 | |
It is hard to change the density of helium 4 as it is
To determine the superfluidity of these Bloch states,
a liquid. In contrast, we can easily modulate the den-
we must consider elementary excitations around these
sity of a BEC which is a gas. When we put a BEC in
Bloch states, and check whether the excitation energy is
an optical lattice, we obtain a superfluid whose density
alwayspositive. Positiveexcitationenergyindicatesthat
is periodically modulated. One can even further period-
theBlochstateisalocalenergyminimum,anditisstable
ically modulate the interatomic interaction of the BEC
againstsmallperturbations. Duetotheperiodicityofthe
with optical Feshbach resonance [19]. Supersolid helium
Bloch wave, the perturbations can be decomposed into
4 may be also regardedas a periodic superfluid as it can
different decoupled modes labeled by q,
be viewed as some superfluid defects (most likely vacan-
cies)flowinginaheliumsolidlattice[20]. Inthissection, δφ (x,q)=u (x,q)eiqx+v∗(x,q)e−iqx, (14)
k k k
we use a BEC in an optical lattice as an example to ex-
amine the properties of a periodic superfluid. Compared whereq rangesbetween 1/2and1/2andthe perturba-
to the uniform superfluid in free space,a new type ofin- tion functions u and v −are of periodicity of 2π.
k k
stability,i.e.,thedynamicalinstabilityisfoundwhenthe Followingthe similar method inSec. IIB, we linearize
quasi-momentumk ofthesuperfluidislargerthanacrit- theGPequationabovetoobtaintheBogoliubovequation
ical value. Usually the dynamical instability dominates that u and v satisfy
k k
the accompanying Landau instability as it happens on a
muchfaster time scale [21]. The presence of the periodic u u
(q) k =ǫ (q) k , (15)
potential also brings along another critical velocity. Mk vk k vk
(cid:18) (cid:19) (cid:18) (cid:19)
where
A. Stability phase diagram
(k+q) cφ2
Mk(q)= Lcφ∗2 (− kk+q) , (16)
Nowwestudy the superfluidity ofa BECina periodic (cid:18) k −L − (cid:19)
potential[22,23],whichisprovidedbytheopticallattice
with
in cold atom experiments. For simplicity, we consider a
quasi-one-dimensional BEC, confined in a cigar-shaped 1 ∂ 2
(k)= +ik +vcosx µ+2cφ 2. (17)
trap. We treat the system with the mean field theory k
L −2 ∂x − | |
and obtain the grand-canonicalGP Hamiltonian (cid:18) (cid:19)
This eigenvalue equation has two sets of solutions, one
= ∞ dx ψ∗ 1 ∂2 +vcosx ψ+ c ψ 4 µψ 2 , corresponds to physical excitations, which is mostly
H −2∂x2 2| | − | | phonon excitation, and the other can be called anti-
Z−∞ (cid:20) (cid:18) (cid:19) (cid:21)
(12) phonon that is not physical. If the physical excitation
5
ǫ (q) is positive, the Bloch wave ψ is a local minimum, v=0.01 v=0.05 v=0.10
k k
2
and the system will have superfluidity. Otherwise, the /1 a.1 b.1 c.1
system suffers Landau instability or dynamical instabil-
ity, depending on whether ǫk(q) is real negative or com- 01
plex. q 4/1 =0.
In the case of free space v = 0, the Bloch state ψ c
k
becomes a plane wave eikx. Then the operator Mk(q) 0 ro
becomes 2 a.2 b.2 c.2
/1
q2/2+kq+c c 5
Mk(q)=(cid:18) c −q2/2−+kq−c (cid:19), (18) q 4/1 =0.0
c
and we recover the excitations in the uniform case 0
ro
2 a.3 b.3 c.3
ǫ±(q)=kq cq2+q4/4. (19) /1
±
k 0
p d 1
Orenael,iwmhmicehdimateealynssetehsatthathtethBeEexCcifltaotwiosninenferregeysipsaaclewaayres q 4/1 c=0.
always dynamically stable. 0
When there is periodic potential, the situation is dra- ro
2 a.4 b.4 c.4
maticallydifferent, wherethe excitationenergycanhave /1
imaginarypart,signalingthedynamicalinstabilityofthe 5
1
system. By numericallysolving the Bogoliubovequation q 4/1 =0.
above, we show the stability phase diagrams for BEC c
Bloch waves in the panels of Fig. 2, where different val-
uesofv andcareconsidered. Theresults havereflection 0
0 1/4 1/2 0 1/4 1/2 0 1/4 1/2
symmetry in k and q, so we only show the parameter
k k k
region, 0 k 1/2 and 0 q 1/2. In the shaded
≤ ≤ ≤ ≤
area (light or dark) of each panel of Fig. 2, the exci-
FIG. 2: Stability phase diagrams of BEC Bloch states in op-
tation energy is negative, and the corresponding Bloch tical lattices. k is the wave number of BEC Bloch waves;
states ψk are saddle points. For those values of k out- q denotes the wave number of perturbation modes. In the
side the shaded area, the Bloch states are local energy shaded (light or dark) area, the perturbation mode has neg-
minima and representsuperfluids. The superfluid region ative excitation energy; in the dark shaded area, the mode
expands with increasing atomic interaction c, and occu- grows or decays exponentially in time. The triangles in (a.1-
pies the entire Brillouin zone for sufficiently large c. On a.4) represent the boundary, q2/4+c = k2, of saddle point
regions at v =0. The solid dots in the first column are from
the other hand, the lattice potential strength v does not
theanalyticalresultsofEq.(20). Thecirclesin(b.1)and(c.1)
affect the superfluid region very much as we see in each
arebasedontheanalyticalexpression(21). Thedashedlines
row. Thephaseboundariesforv 1arewellreproduced
≪ indicate themost unstable modes for each Bloch wave ψk.
from the analytical expression k = q2/4+c for v =0,
which is plotted as triangles in the first column.
p
If ǫ (q) is complex, the system suffers dynamical in-
k
stability, which is shown by the dark-shaded areas in spectrum and the anti-phonon spectrum with the ones
Fig. 2. The dynamical instability is the result of the given in Eq. (19). By equating them, ǫ (q 1)=ǫ (q),
+ −
resonancecouplingbetweena phononmode andananti- forthedegeneracy,wefindthatthedynamic−alinstability
phononmodebyfirst-orderBraggscattering. Thematrix should occur on the following curves
(q)isrealinthe momentumrepresentation,meaning
k
M
thatitscomplexeigenvaluecanappearonlyinconjugate k = q2c+q4/4+ (q 1)2c+(q 1)4/4. (20)
− −
pairsand they must come froma pair ofrealeigenvalues
p p
that are degenerate prior to the coupling. Degeneracies These curves areplotted as solid dots in Fig.2, and they
or resonances within the phonon spectrum or within the fallrightinthemiddle ofthethindark-shadedareas. To
anti-phonon spectrum do not give rise to dynamical in- someextent,onecanregardthesethindark-shadedareas
stability; they only generate gaps in the spectra. Based asbroadeningofthe curves(20). Itis notedinRef. [26]
onthis generalconclusion,we considertwospecialcases, that the relation (20) is also the result of ǫ (q 1)+
+
−
which allow for simple explanations of the onset of dy- ǫ ( q) = 0, which involves only the physical phonons.
+
−
namical instability. Therefore, the physical meaning of Eq. (20) is that one
One case is the weak periodic potential limit v 1, can excite a pair of phonons with total energy zero and
≪
where we can approximate the boundary with the free withtotalmomentumequaltoareciprocalwavenumber
space case. This case corresponds to the first column of of the lattice.
Fig. 2. In this limit, we can approximate the phonon The other case, c v, is shown in the first row of
≪
6
Fig. 2. The open circles along the left edges of these
dark-shaded areas are given by (a)
E (k+q) E (k)=E (k) E (k q), (21)
1 1 1 1
− − −
where E (k) is the lowest Bloch band of
1
1 ∂2
H0 =−2∂x2 +vcos(x). (22) (b)
In this linear periodic system, the excitation spectrum
(phonon or anti-phonon) just corresponds to transitions
from the Bloch states of energy E (k) to other Bloch FIG. 3: (a) A macroscopic obstacle moves with a velocity of
1
v inside a superfluid residing in a periodic potential. The
states of energy E (k + q), or vise versa. The above
n superfluid and periodic potential are “locked” together and
equation is just the resonance condition between such
there is no relative motion between them. (b) The lattice
excitations in the lowestband(n=1). Alternatively, we
where a superfluid resides is slowly accelerated to a velocity
can write the resonance condition as
of v.
E (k)+E (k)=E (k+q)+E (k q). (23)
1 1 1 1
−
So, this condition may be viewed as the energy and mo-
in Fig. 3(b), where the lattice is slowly accelerated to a
mentum conservation for two particles interacting and
givenvelocity andthere is a relativemotion betweenthe
decaying into two different Bloch states E (k +q) and
1 superfluid and the lattice. For these two different situa-
E (k q). This is the same physical picture behind Eq.
1 tions, naturally arise two critical velocities.
−
(20).
In the first scenario, we consider a moving impurity
One common feature of all the diagrams in Fig.2 is that generates an excitation with momentum q and en-
that there are two critical Bloch wave numbers, kt and ergyǫ (q)intheBEC.Accordingtotheconservationsof
0
k . Beyond k the Bloch waves ψ suffer the Landau in-
d t k both momentum and energy, we should have
stability; beyond k the Blochwavesψ are dynamically
d k
unstable. The onset of instability at k always corre- m v = m v +q+n~G, (24)
d 0 i 0 f
sstpaotnedψsktofrqom=1k/=2.0Intootkhe=r w1/o2rdtsh,eiffiwrsetdurnivsetatbhleeBmloodche m0vi2 = m0vf2 +ǫ0(q), (25)
2 2
appearing is always q = 1/2, which represents period
±
doubling. Onlyfork >kdcanlongerwavelengthinstabil- where m0 is the mass of the impurity, vi and vf are the
ities occur. The growth of these unstable modes drives initial and final velocities of the impurity, respectively,
the system far away from the Bloch state and sponta- G is the reciprocal vector, and ǫ (q) is the excitation of
0
neouslybreaksthetranslationalsymmetryofthesystem. the BEC at the lowest Bloch state k = 0. Note that in
contrastto the conservation of momentum in free space,
here the total momentum of the impurity and the ex-
B. Two critical velocities citation is not exactly conserved due to the presence of
an optical lattice, and the momenta differing by integer
Besides inducing the dynamical instability, the pres- multiples of reciprocal lattice vector are equivalent. For
ence of the optical lattice has also non-trivial conse- phonon excitations, i.e., ǫ (q) = cq, these two conser-
0
| |
quencesonthe conceptofcriticalvelocity. Incontrastto vations canalwaysbe satisfiedsimultaneouslyno matter
the homogeneous superfluid which has only one critical howsmallthevelocityoftheimpurityis. Inotherwords,
velocity,therearetwodistinctcriticalvelocitiesforape- the critical velocity for this scenario is exactly zero.
riodic superfluid [27]. The first one, which we call inside In the other scenario,there is relativemotion between
critical velocity, is for an impurity to move frictionlessly the superfluid and the optical lattice, and the superfluid
inthe periodicsuperfluidsystem(Fig. 3(a)); thesecond, no longer resides in the k = 0 point of the Brillouin
which is called trawler critical velocity, is the largest ve- zone. We should examine the stability of Bloch waves
locityofthelatticeforthesuperfluiditytomaintain(Fig. with nonzero k, which is discussed in detail in the previ-
3(b)). We illustrate these two critical velocities with a ous subsection. The critical wave number k mentioned
t
BEC in a one-dimensional optical lattice. above corresponds precisely to the trawler critical veloc-
The presenceoftheopticallattice playsadecisiverole ity v here. As k is not zero, v is not zero.
t t t
in the appearance of the two critical velocities: two very Both critical velocities can be measured with BECs
different situations can arise. The first situation is de- in optical lattices. The inside critical velocity v can
i
scribed in Fig. 3(a), where one macroscopic impurity be measured with the same experimental setting as in
movesinside the superfluid. The keyfeature inthis situ- Ref. [28], where the superfluidity of a BEC was stud-
ation is that there is no relative motion between the su- ied by moving a blue-detuned laser inside the BEC. For
perfluidandthelattice. Theothersituationisillustrated the trawlercriticalvelocityv , one canrepeatthe exper-
t
7
iment in Ref. [21], where a BEC is loaded in a moving
opticallattice. Oneonlyneedstoshifthisattentionfrom (a) (b)
dynamical instability to superfluidity. The potential dif-
ficulty lies in that the Landau instability occurs over a
much larger time scale, which may be beyond the life
time of a BEC [21].
IV. SUPERFLUIDITY WITH SPIN ORBIT
COUPLING
The intrinsic spin-orbit coupling (SOC) of electrons
plays a crucial role in many exotic materials, such as
topological insulators [29]. In spintronics [30], its pres- (c)
ence enables us to manipulate the spin of electrons by
means of exerting electric field instead of magnetic field,
whichismucheasiertoimplementforindustrialapplica-
tions. However,as a relativistic effect, the intrinsic SOC FIG.4: (a)AsuperfluidwithSOCmoveswhilethetubeisat
rest. (b) The superfluid is dragged by a tube moving at the
doesnotexistorisveryweakforbosonsinnature. With
speedofv. (c)AnimpuritymovesatvintheSOCsuperfluid.
the method of engineering atom-laser interaction, an ar-
The reference frame is the lab. The two-way arrow indicates
tificialSOC has beenrealizedfor ultracoldbosonic gases
theequivalencebetweendifferentscenariosandthearrowwith
in [31–34]. A great deal of effort has been devoted to
a barindicates thenon-equivalence.
study many interesting properties of spin-orbit coupled
BECs [35–47].
AdramaticchangethattheSOCbringstotheconcept
A system is invariant under the Galilean transformation
ofsuperfluidityisthebreakdownofLandau’scriterionof if the following equation is satisfied (Ref. [48]),
critical velocity (1) and the appearance of two different
critical velocities. Laudau’s criterion of critical velocity ∂ ∂
(1) is based on the Galilean invariance. It is apparent G†(v,t) i~ H G(v,t)= i~ H . (30)
∂t − ∂t −
to many that the scenario where a superfluid flows in- (cid:20) (cid:21) (cid:20) (cid:21)
side a motionlesstube is equivalentto the otherscenario TheaboveconditionisclearlysatisfiedbyaHamiltonian
where a superfluid at rest is dragged by a moving tube. without SOC, e.g., H = p2/2m. However, for a Hamil-
If the flowing superfluid loses its superfluidity when its tonianwith SOC,e.g., H =p2/2m+γ(σ p σ p ),
soc x y y x
speed exceeds a critical speed vc, then the superfluid in it is easy to check that −
the other scenariowill be draggedinto motion by a tube
movingfasterthanvc. However,thisequivalenceisbased G†(v,t) i~∂ H G(v,t)= i~ ∂ H′ , (31)
on that the superfluid is invariant under the Galilean ∂t − soc ∂t − soc
transformation. As SOC breaks the Galilean invariance (cid:20) (cid:21) (cid:20) (cid:21)
of the system [48], we find that the two scenarios men- where H′ = H + mγ(σ v σ v ). Clearly there
tionedabovearenolongerequivalentasshowninFig. 4: is an addsoitcional tseorcm dependxenyt−onythxe velocity of the
thecriticalspeedforscenario(a)isdifferentfromtheone reference frame. This new term can be regarded as an
forscenario(b). Foreasyreference,the criticalspeedfor effective Zeeman effect and can not be gauged away;the
(a) is hereafter called the critical flowing speed and the Galilean invariance of the system is thus lost.
one for (b) the critical dragging speed.
Intheexperimentsofultra-coldatomicgases,theSOC
For ultra-cold atomic gases, the breakdown of the iscreatedbytwoRamanbeamsthatcoupletwohyperfine
Galilean invariance at the presence of SOC can be un-
states of the atom. Since the Galilean transformation
derstood both theoretically and experimentally. only boosts the BEC, not including the laser setup as a
From the theoretical point of view, we show in de-
whole, the moving BEC will experience a different laser
tail how a system with SOC changes under the Galilean field due to the Doppler effect, resulting a loss of the
transformation. We adopt the formalism in Ref. [48].
Galilean invariance.
The operator for the Galilean transformation is
G(v,t)=exp[iv (mr pt)/~], (26)
· − A. Bogoliubov excitations and definition of critical
which satisfies the definition velocities
G†(v,t)rG(v,t)= r vt, (27)
− We use the method introduced in Sec. IIB to study
G†(v,t)pG(v,t)= p mv, (28) thesuperfluidityofaBECwithSOCbycomputingitsel-
−
G†(v,t)σG(v,t)= σ. (29) ementaryexcitations[49]. Inexperiments,onlytheequal
8
combinationof Rashba and Dresselhaus coupling is real- stateφ withtheBogoliubovequationfordifferentvalues
k
ized. Here we use the Rashba coupling as an example as of k.
the main conclusion does not rely on the details of the Without loss ofgenerality,we choose k=kyˆwith k >
SOC type. 0. Following the standard procedure of linearizing the
We calculate how the elementary excitations change GP equation [22, 23], we have the following Bogoliubov
with the flow speed and manage to derive from these equation
excitations the critical speeds for the two different sce-
u u
narios shown in Fig. 4(a,b). We find that there are two 1 1
u u
branchesofelementaryexcitationsfora BECwith SOC: 2 =ǫ 2 , (34)
the lower branch is phonon-like at long wavelengths and M v1   v1 
v v
theupperbranchisgenerallygapped. Carefulanalysisof  2   2 
   
theseexcitationsindicatesthatthecriticalflowingveloc-
where
ityforaBECwithSOC(Fig. 4(a))isnon-zerowhilethe
critical dragging speed is zero (Fig. 4(b)). This shows H+ b 1C 1C
that the critical velocity depends on the reference frame bk H1+2 −12C1 −21C2
for a BEC with SOC and, probably, for any superfluid M= 12C211 21Ck2 −H2k−2 −b2341 , (35)
that has no Galilean invariance.  1C 1C b H− 
 2 2 2 1 43 k 
Specifically, we consider a BEC with pseudo-spin 1/2  
and the Rashba SOC. The system can be described by with H± = qx2+(qy±k)2 A, A = C1 k2 +γk, b =
the Hamiltonian [36, 39, 50, 51] γ(iq k+q +±k)+2C2, b±=γ(iq 2q − k2)+ C2, b12 =
− x y 2 21 x− y− 2 34
γ(iq q +k) C2, and b = γ(iq +q k) C2.
~2 2 x− y − 2 43 − x y − − 2
= dr ψ∗ ∇ +V(r) ψ As usual, there are two groups of eigenvalues and only
H Z (σX=1,2 σ(cid:18)− 2M (cid:19) σ tvhe2o=ne1sw(ih=ose1,c2o)rraerseppohnydsinicgale.igenvectorssatisfy|ui|2−
+γ[ψ1∗(ipˆx+pˆy)ψ2+ψ2∗(−ipˆx+pˆy)ψ1] | iI|n general there are no simple analytical results. We
+C1 ψ1 4+ ψ2 4 +C2 ψ1 2 ψ2 2 , (32) have numerically diagonalized M to obtain the elemen-
2 | | | | | | | | tary excitations. We find that part of the excitations
(cid:27)
(cid:0) (cid:1) are imaginary for BEC flows with k < γ. This means
where γ is the SOC constant, C1 and C2 are interaction that all the flows with k < γ are d|y|namically unstable
strengths between the same and different pseudo-spin and therefore do not ha|ve| superfluidity. For other flows
states, respectively. For simplicity and easy comparison with k γ, the excitations are always real and they are
with previous theory, we focus on the homogeneous case plotted≥in Fig. 5. One immediately notices that the ex-
V(r) = 0 despite that the BEC usually resides in a har- citations havetwobranches,whichcontacteachotherat
monic trap in experiments. Besides, we limit ourselves a single point. Closer examination shows that the upper
tothecaseC1 >C2,namely,intheplanewavephase. In branch is gapped in most of the cases while the lower
thefollowingdiscussion,forsimplicity,weset~=M =1 branch has phonon-like spectrum at large wavelength.
andignorethenon-essentialz direction,treatingthesys- These features are more apparent in Fig. 6, where only
temastwo-dimensional. WealsoassumetheBECmoves the excitations along the x axis and y axis are plotted.
inthe y direction,andthe criticalvelocityis foundto be In Fig. 5(c) and Fig. 6(c2), we notice that part of the
not influenced by the excitation in the z direction. excitations in the upper branch are negative, indicating
The GP equation obtained from the Hamiltonian (32) that the underlying BEC flow is thermodynamically un-
has plane wave solutions stable and has no superfluidity. In fact, our numerical
computation shows that there exists a critical value k :
c
φ = ψ1 = 1 eiθk eik·r−iµ(k)t, (33) when k > kc either part of the upper branch of excita-
k ψ2 √2 1 tions or part of the lower branch or both become nega-
(cid:18) (cid:19) (cid:18) − (cid:19)
tive. This means that the flows described by the plane
wheretanθk =kx/ky,µ(k)=|k|2/2−γ|k|+(C1+C2)/2. wave solution φk,− with |k| >kc suffer Landau instabil-
The solution φk is the ground state of the system when ity and have no superfluidity. Combined with the fact
k = γ. There are another set of plane wave solutions, that the flows with k < γ are dynamically unstable,
| | | |
which have higher energies and are not relevant to our we can conclude that only the flows with γ k k
c
≤ | | ≤
discussion. have superfluidity. The corresponding critical speed is
WestudyfirstthescenariodepictedinFig. 4(a),where v = k γ. We have plotted how the critical flowing
c c
−
the BEC flows with a given velocity. Since the system velocity v varies with the SOC parameter γ in Fig. 7.
c
is not invariant under the Galilean transformation, we We turn to another reference frame illustrated in Fig.
cannotuseLaudau’sargumenttofindthe excitationsfor 4(b), where the BEC can be viewed as being draggedby
theflowingBECfromtheexcitationofastationaryBEC. a moving tube. To simplify the discussion, we replace
We have to compute the excitations directly. This can themovingtubewithamacroscopicimpuritymovingin-
be done by computing the elementary excitations of the side the BEC as shown in Fig. 4(c). Correspondingly,
9
15
10 15
10
10
--21 5 --21 5 --21 5
qx01 0 qx01 0 qx01 0
2-3 -2 -q1y 0 1 2-3 -2 -q1y 0 1 2-4 -2qy 0
HaL HbL HcL
FIG. 5: (color online)Elementary excitations of a BEC flow with the SOC. (a) k =1; (b) k =3; (c) k =4. C1 =10, C2 =4,
γ =1.
Ha1L Hb1L Hc1L momentum and energy, we should have
ΕHLqx 2468 ΕHLqx 102468 ΕHLqx 11022468 mm00vvi2i == mm00vvff2++qǫ,(q), ((3376))
0 0 0 2 2 0
-2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2
as those for a BEC without SOC, where ǫ (q) is the ex-
qx qx qx 0
Ha2L Hb2L Hc2L citation of the BEC at k = γ. The critical dragging
ΕHLqy 110246802 ΕHLqy 110505 ΕHLqy 11205005 velocity derived fromvcE=qs.ǫ0(q(3q6),37) is.given by (38)
-4-3-2-10 1 2 -4-3-2-10 1 2 -4-3-2-10 1 2 (cid:12) | | (cid:12)min
(cid:12) (cid:12)
qy qy qy If the excitations were pu(cid:12)(cid:12)rely p(cid:12)(cid:12)honons, i.e., ǫ0(q)=c|q|,
these two conservations would not be satisfied simulta-
neously when v v v < c. This means that the
i f
≈ | | ≈ | |
impurity could not generate phonons in the superfluid
FIG. 6: Excitations along the x axis (the first row) and y
and would not experience any viscosity when its speed
axis (the second row) at different values of k. (a1,a2) k =1;
was smaller than the sound speed. Unfortunately, for
(b1,b2) k=3; (c1,c2) k=4. C1 =10, C2 =4, γ =1.
our BEC system, the elementary excitations ǫ (q) are
0
not purely phonons, as will be shown below.
When γ = 0, the excitations ǫ (q) also share two
(a) (b) 0
6
branches. Along the x axis, these two branches are
2.8 3
2.6 ǫ±(q )= s +s q2+ qx4 t +t q2+t q4+γ2q6,
0 x 1 2 x 4 ± 1 2 x 3 x x
vc2.4 C1>3C2 vc2.5 C1<3C2 r p (39)
where s =2γ4+γ2(C C ), s =2γ2+ 1C , t =s2,
2.2 1 1− 2 2 2 1 1 1
t = 2s s , and t = 2s + γ2+C /2 2. Along the y
2 1 2 3 1 2
2 2
0 2 4 6 0 2 4 6 axis, the excitations of the ground state are
γ γ
(cid:0) (cid:1)
C +C q4
ǫ−(q ) = 1 2q2+ y , (40)
0 y 2 y 4
r
FIG.7: Critical flowingvelocity vc ofaBECasafunctionof
the SOC parameter γ. (a) C1 = 11, C2 = 3; (b) C1 = 14, ǫ+(q ) = 2γq + 2s + s C2 q2+ qy4 .(41)
C2 =6. 0 y y s 1 2− 2 y 4
(cid:18) (cid:19)
Whenγ >0,theupperbranchǫ+(q )isalwaysparabolic
0 x
at small q with a gap √2s . When expanded to the
x 1
the question“whetherthe BECwillbe draggedalongby second order of q , the lower branch has the following
x
the moving tube?” is replaced by an equivalent question form
“whether the impurity will experience any viscosity?”.
C +C
Suppose that the moving impurity generates an excita- ǫ−(q ) q2 1 2 . (42)
tion in the BEC. According to the conservations of both 0 x ≈ xs 8γ2
10
This shows that ǫ−(q ) is parabolic at long wavelengths inferred. For the typical atomic density of 1014 1015
instead of linear a0s usxually expected for a boson system. cm−3 achievablein currentexperiments [53],and∼the ex-
This agreeswith the results in Ref. [41]. This parabolic perimentalsetupinRef.[31],the criticalflowingvelocity
excitation has a far-reaching consequence: according to is 0.2 0.6 mm/s, while the critical dragging velocity
∼
Eq. (38), the critical dragging velocity v is zero, very is still very small, about 10−3 10−2 mm/s. To further
c
∼
differentfromthecriticalflowingvelocityforaBECmov- magnifythedifferencebetweenthetwocriticalvelocities,
ing in a tube. This shows that the critical velocity for a one may use the Feshbach resonance to tune the s-wave
BECwithSOCisnotindependentofthereferenceframe, scattering length.
instarkcontrastwith ahomogeneoussuperfluidwithout
SOC. This result of course has the root in the fact that
the BEC described by the SOC Hamiltonian (32) is not
invariant under the Galilean transformation [48]. V. SUPERFLUIDITY OF SPIN CURRENT
We have also investigated the superfluidity with the
general form of SOC, which is a mixture of Rashba and
A neutralbosoncancarryboth massand spin; it thus
Dresselhaus coupling. Mathematically, this SOC term
can carry both mass current and spin current. However,
has the form ασ p βσ p . The essential physics is
x y − y x when a boson system is said to be a superfluid, it tra-
the same: the critical flowing speed is different from the
ditionally refers only to its mass current. The historical
criticaldraggingspeed,andthereforethecriticalvelocity
reason is that the first superfluid discovered in experi-
depends on the choice of the reference frame. However,
ment is the spinless helium 4 which carries only mass
the details do differ when α = β. The critical dragging
6 current. For a boson with spin, say, a spin-1 boson, we
speed is no longer zero. Without loss of generality, we
can in fact have a pure spin current, a spin current with
let α > β. The slope of the excitation spectrum for the
no mass current. This pure spin current can be gener-
ground state along the y axis is
ated by putting an unpolarized spin-1 boson system in
a magnetic field with a small gradient. Can such a pure
C C
v = 2α2(C C +2α2)+2α2+ 1− 2 2α, spin current be a super-flow? In this section we try to
y 1 2
− 2 −
r address this issue for both planar and circular pure spin
p (43)
currents by focusing on an unpolarized spin-1 Bose gas.
and the slope along the x axis,
The stability of a pure spin current in an unpolarized
spin-1 Bose gas was studied in [57]. It was found that
β2 C +C
v = 1 1 2 . (44) such a current is generally unstable and is not a super-
x s(cid:18) − α2(cid:19) 2 flow. We have recently found that the pure spin current
can be stabilized to become a super-flow [58]: (i) for a
The critical dragging velocity is the smaller one of the
planar flow,it can be stabilized by quadratic Zeemanef-
above two slopes, both of which are nonzero.
fect;(ii)foracircularflow,itcanbestabilizedwithSOC.
We shall discuss these results and related experimental
schemes in detail in the next subsections.
B. Experimental observation
There has been lots of study on the counterflow in a
two-species BEC [57, 59–69], which appears very similar
Spin-orbit coupled BECs have been realized by many
to the pure spin current discussed here. It was found
different groups [31–34] through coupling ultracold
that there is a critical relative speed beyond which the
87Rbatomswithlaserfields. ThestrengthoftheSOCin
counterflow state loses its superfluidity and becomes un-
the experimentscanbe tunedby changingthe directions
stable [59–67]. We emphasize that the counterflow in a
of the lasers [31–33] or through the fast modulation of
two-species BEC is not a pure spin current for two rea-
the laser intensities [52]. The interaction between atoms
sons. Firstly, although the two species may be regarded
canbeadjustedbyvaryingtheconfinementpotential,the
astwocomponentsofapsudo-spin1/2,theydonothave
atomnumberorthroughtheFeshbachresonance[6]. For
SU(2) symmetry. Secondly, it is hard to prepare experi-
thescenarioinFig. 4(b),onecanuseablue-detunedlaser
mentallyaBECwithexactlyequalnumbersofbosonsin
tomimictheimpurityforthemeasurementofthecritical
thetwospeciestocreateacounterflowwithnomasscur-
draggingspeedsimilartotheexperimentinRef.[53]. For
rent. There is also interesting work addressing the issue
the scenario in Fig. 4(a), there are two possible exper-
of spin superfluidity in other situations [70–72].
imental setups for measuring the critical flowing speed.
Inthe firstone,one generatesa dipole oscillationsimilar
to the experiment in Ref. [33] but with a blue detuned
laser inserted in the middle of the trap. The second one
A. Planar flow
ismorecomplicated: Atfirst,onecangenerateamoving
BEC with a gravitomagnetic trap [54]. One then uses
Bragg spectroscopy [55, 56] to measure the excitations We consider a spin-1 BEC in free space. The mean-
of the moving BEC, from which the superfluidity can be field wave function of such a spin-1 BEC satisfies the