Landau's criterion for an anisotropic Bose-Einstein condensate

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by  Zeng-Qiang Yu

Landau’s criterion for an anisotropic Bose-Einstein condensate
Zeng-Qiang Yu∗
Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
In this work, we discuss the Landau’s criterion for anisotropic superfluidity. To this end, we
considerapoint-likeimpuritymovinginauniformBose-Einsteincondensatewitheitherinterparticle
dipole-dipole interaction or Raman induced spin-orbit coupling. In both cases, we find that the
Landau critical velocity v is generally smaller than the sound velocity in the moving direction.
c
Beyond v , the energy dissipation rate is explicitly calculated via a perturbation approach. In the
c
7 plane-wave phase of a spin-orbit coupled Bose gas, the dissipationless motion is suppressed by the
1 Raman coupling even in the direction orthogonal to the recoil momentum. Our predictions can be
0 tested in the experiments with ultracold atoms.
2

r
a One of the manifestations of superfluidity is the abil- Critical velocity (2) can be derived in the standard
M ity to support dissipationless motion. An impurity im- way similar to the isotropic case [2]. Consider a heavy
mersed in a superfluid can move without friction when point-likeimpuritymovinginauniformsuperfluidatzero

7 its velocity is below a critical value. This phenomenon temperature. Theimpuritycanloseitskineticenergyby
1 is explained by the famous Landau’s criterion [1, 2], ac- creating one or more excitations in the superfluid, and
cording to which, the critical velocity is given by the onset of dissipation only involves the processes of
]
s creating a single excitation [18]. Such processes conserve
(cid:18) (cid:19)
ga vc =min Eqq , (1) energy and momentum, thus, satisfy the relation
-
t q·v=Eq, (3)
n
with E being the elementary excitation spectrum of
a q
the superfluid. The Landau critical velocity had been where v is the velocity of the impurity, q is the momen-
u
q observed in liquid He-II and Bose-Einstein condensate tum of the excitation, and we have assumed that the
. (BEC) of a dilute gas, where the onset of dissipation is massofimpurityisverylargecomparedtothatofatoms
t
a due to the creation of roton [3] and phonon [4], respec- in the superfluid. If condition (3) cannot be satisfied by
m tively. any q, no excitation will be created. As a result, when
- Recently, two new types of BEC were realized in the v <vc,themotion(inthevˆ direction)isdissipationless.
d ultracold atomic gases: One was achieved in magnetic Thedifferencebetween(1)and(2)isnotinsignificant.
n
o atoms of 52Cr [5], 164Dy [6] and 168Er [7] with inter- In an isotropic system, the minimum value of Eq/(q·vˆ)
particle dipole-dipole interaction (DDI); the other was always appears at a momentum in the vˆ direction, con-
c
[ achieved in alkali atoms of 87Rb [8] and 23Na [9] with sequently, vc reduces to its well-known form (1). In an
synthetic spin-orbit (SO) coupling. Remarkably, the ex- anisotropic system, however, the minimization in (2) is
2
v citation spectra of these superfluids are anisotropic and complicated, and in general, vc is not determined by the
6 can be tuned by an external magnetic field or laser field. dispersionrelationinthemovingdirection(seeexamples
8 While anisotropic dynamics and collective excitations below). Physically, (1) and (2) correspond to distinct
6 have been experimentally observed in these systems [10– scenarios of creating excitations at the threshold veloc-
8 17], the critical velocity has not been measured so far. ity. In an isotropic system, the excitation created at vc
0 The application of the Landau’s criterion to an propagates in the same direction as vˆ. In contrast, in
1. anisotropic superfluid is not a trivial problem. In this an anisotropic system, the excitation could propagate in
0 work,weshowthattheanisotropiccriticalvelocity,along a different direction other than vˆ. Geometrically, for a
7 a certain direction vˆ, is given by given vˆ, one can determine vc as follows: Plot the set
1 of curves of E versus q·vˆ along different directions of
q
v: v =min(cid:18) Eq (cid:19), (2) momentum; then, from the origin, draw tangent lines to
i c q·vˆ each of the curves and find the line with the smallest
X
positive slope; this slope is just equal to v .
c
ar where the minimum is calculated over all the values of ForthesystemwithGalileaninvariance, thecondition
momentum q under the constraint q·vˆ >0. The condi- v < vc also ensures the stability of a supercurrent state,
tion v <v , with v given by (2), can be recognized as a as first proposed by Landau [1, 2]. Suppose a fluid flows
c c
generalizedLandau’scriterionforanisotropicsuperfluid- at a constant velocity in the −vˆ direction. The appear-
ity,anditsconsequenceisdiscussedlaterinbothdipolar ance of an excitation at momentum q costs an energy
and SO coupled Bose gases. Eq +q·vflow. Therefore, the flow suffers an energetic
instability once v exceeds v . It should be noted that
flow c
if the Galilean invariance is broken, the maximum veloc-
ity of the stable superflow would be different from the
∗Electronicaddress: [email protected] critical velocity (2).
2
√
Beyondv ,themotionoftheimpuritybecomesdissipa- Thesoundvelocityreachesitsminimumc =c 1−(cid:15)
c ⊥ √0 dd
tive. Tocalculatetheenergydissipationrate, weassume at θ =π/2 and reaches its maximum c =c 1+2(cid:15)
q (cid:107) 0 dd
theinteractionbetweentheimpurityandthecondensate at θ =0 or π, where c =(cid:112)gn/m is the sound velocity
q 0
is weak and treat it as a perturbation (set (cid:126)=1), with a pure s-wave interaction. Below, we restrict our
discussion in the regime 0 < (cid:15) < 1 and g > 0 to make
(cid:88)N (cid:90) d3q dd
H = U(r −vt)= ρ†U e−iq·vt (4) sure the excitation spectrum is stable.
pert j (2π)3 q q Since the system possesses a rotation symmetry about
j=1
the dipole axis, the anisotropic critical velocity only de-
whereU(r)istheinteractionpotentialbetweentheimpu-
pends on the polar angle of the moving direction θ .
rity and the atoms of the superfluid, ρq = (cid:80)Nj=1e−iq·rj From the dispersion relation (7), it is readily shown thavt
is the density fluctuation operator, N is the atoms num- theonsetofdissipationisduetothecreationofaphonon,
ber,andUq =(cid:82) d3rU(r)e−iq·r. Forconvenience,wetake and the critical velocity (2) can be written as
U(r) as a contact potential and set U = U . The dissi-
q 0
pationrateoftheimpurityP equalstotheenergytrans- (cid:20) c(θ ) (cid:21)
v (θ )=min q , (9)
ferredtothecondensateintheperunittimedESF/dt. In c v cos(θq−θv)
the linear response regime, we have [1]
U2n (cid:90) (cid:90) wheretheminimumiscalculatedoverallthevaluesofθq
P = 0 d3q dωωS(q,ω)δ(ω−q·v) (5) under the constraint θ −π/2<θ <θ +π/2. After a
v q v
4π2N
straightforwardalgebra,wefindthatthephononcreated
where n is the average density of atoms, and S(q,ω) = at v carries a momentum with θ satisfying
c q
(cid:12)(cid:12)(cid:104)Φq|ρ†q|Φ0(cid:105)(cid:12)(cid:12)2δ(ω−Eq)isthedynamicstructurefactorof c2 tanθ =c2tanθ . (10)
thesuperfluid,with|Φ0(cid:105)and|Φq(cid:105)beingthegroundstate ⊥ q (cid:107) v
and the excited state respectively. Since S(q,ω) is non-
Clearly,exceptforthespecialcasesofθ =0,π/2andπ,
zeroonlyattheresonantfrequenciesω =E , theenergy v
q
the propagating direction of the phonon is different from
dissipation vanishes for v < v , which is consistent with
c
the moving direction of the impurity, in accordance with
the Landau’s criterion. In experiments, the dissipation
the scenario we discussed previously. Substituting (10)
can be observed through the heating of the superfluid.
into (9), we obtain the critical velocity
The dissipation rate (5) can be also written as P =
−F·v, where F = U2n/(4π2N)(cid:82) d3qS(q,q·v)q is the
0 (cid:16) (cid:17)−1/2
drag force acting on the impurity [19]. The drag force v (θ )=c c sin2θ c2+cos2θ c2 . (11)
c v (cid:107) ⊥ v (cid:107) v ⊥
arises for v >v . In an isotropic superfluid, from a sym-
c
metry argument, one can show that F is always in the
Unlikethecondensatewithapures-waveinteraction,the
direction of −vˆ. This conclusion no longer holds in an
critical velocity of the dipolar BEC is generally smaller
anisotropicsystem[21]. Nevertheless,onlytheprojection
than the sound velocity in the moving direction [24]
F·vˆ contributes to the energy dissipation.
Now we apply the above general results to a dipolar v (θ )(cid:54)c(θ ), (12)
c v v
BEC [22]. The interparticle interactions in a dipolar gas
consistoftwoparts: theshortrangevanderWaalsinter- where the equality holds only for the motion in the di-
action and the long-range DDI. Assuming all the dipoles rectioneitherparallelorperpendiculartothedipoleaxis.
aligned along the ˆz axis, the effective interaction poten- Thedifferencebetweenthevaluesofv andcisillustrated
c
tial reads [23] in Fig. 1. For a given (cid:15) , the ratio v /c takes the mini-
dd c
V (r)=g(cid:2)δ(r)+3(cid:15) (1−3cos2θ)/(4πr3)(cid:3), (6) mum value at θv =π/4.
int dd
Beyond v , the dissipation rate (5) can be written as
c
wheremistheatomicmass,g isthecouplingconstantof
thes-wavecontactinteraction,θistheanglebetweenthe U2n(cid:90) q2
P = 0 d3q δ(E −q·v), (13)
dipoleaxisandtheinteratomicdistancer,and(cid:15)ddisadi- 8π2 m q
mensionless parameter quantifying the relative strength
(cid:82)
of the DDI. In experiments, (cid:15) can be tuned via the where we have used the f-sum rule dωωS(q,ω) =
dd
technique of Feshbach resonance. Nq2/(2m) to simplify the expression. For (cid:15) = 0,
dd
The Bogoliubov excitation spectrum of a polarized Eq. (13) recovers the previous result with a pure s-wave
dipolar BEC is given by [23] interaction,P =U2nm3(v2−c2)2Θ(v−c )/(πv)[19,20].
0 0 0
(cid:113) For(cid:15)dd >0,thedissipationrateisnumericallycalculated
Eq =q q2/(4m2)+c2(θq), (7) as a function of velocity v, and the results for θv = π/4
aredisplayedinFig.2. Obviously,theenergydissipation
where c(θ ) is the anisotropic sound velocity depending
q arises even in the subsonic regime (v < c), which is in
on the polar angle of the momentum q,
contrast to isotropic weakly interacting bosons. In the
(cid:113) limit(cid:15) →1,v approachestozeroduetothevanishing
c(θ )= c2cos2θ +c2 sin2θ . (8) dd c
q (cid:107) q ⊥ q c . However, in this case, the dissipation rate increases
⊥
3
FIG. 2: (a) Dissipation rate P of a moving impurity in a
polarized dipolar BEC as a function of v and (cid:15) . The dash-
dd
dotted line shows the critical velocity; and the dashed line
showsthesoundvelocityinthemovingdirection. (b)Velocity
dependenceofP for(cid:15) =0.8. Forbothplots,θ =π/4,and
dd v
FIG. 1: (a) Excitation spectrum E of a dipolar BEC as a P is in units of U2nm3c3.
q 0 0
function of q·vˆ, where vˆ is in the direction with θ = π/4
v
(see the inset). The solid line is for the momentum q in the
direction with θ determined by Eq. (10); the dashed-dotted
q in which the spin-dependent interaction is very weak.
line is for θ =θ ; the dotted line is tangent to the solid line
q v
intheq→0limit,anditsslopeisequaltov . (b)Anisotropic ThegroundstatephasediagramofaSOcoupledBose
c
critical velocity as a function of θ (solid line). The dashed gas has been studied previously [8, 27, 28]. When the
v
line shows the sound velocity in the moving direction. For interparticle interaction is spin-independent, the ground
both plots, (cid:15) =0.8. state is the plane-wave (PW) phase for Ω < Ω and the
dd c
zero-momentum (ZM) phase for Ω > Ω , where Ω =
c c
4E , and E = k2/(2m) is the recoil momentum. In the
r r r
veryslowlyatsmallv,becausethephasespacecontribut- PW phase, the ground state spontaneously breaks the
ing to the integral in (13) is quite limited. Z symmetry, and the condensate occupies a non-zero
2
Next, we turn to another anisotropic superfluid – a momentum with a finite spin magnetization. Without
SO coupled BEC. The synthetic SO coupling had been loss of generality, we assume the magnetization to be
successfully realized in ultracold atoms using the tech- positive. IntheZMphase,bosonscondenseatp=0with
niqueofthetwo-photonRamantransition[25]. Here,we abalancedspinpopulation. Thetransitionbetweenthese
considerthespin-halfbosonssubjectedtotheequalcon- two phases is of second-order nature with a divergent
tributions of Rashba and Dresselhaus SO coupling. The magnetic susceptibility at the critical point Ω [29].
c
single-particle Hamiltonian is given by [8]
TheexcitationspectrumofaSOcoupledBECconsists
of two branches [30, 31], and the critical velocity is de-
(p−k σ )2 Ω δ
H = r z + σ + σ , (14) termined by the lower branch. In the long wave-length
SO 2m 2 x 2 z limit,thelowerbranchexcitationisaphononmodewith
anisotropic sound velocity. Remarkably, the orientation
where k is the recoil momentum of the Raman lasers, Ω
r
dependenceofthesoundvelocityissimilartothedipolar
is the Raman coupling strength, δ is the detuning of the
BEC, and it can be written in the form of (8) [32]. In
Ramantransition, andσ andσ arethePaulimatrices.
x z (cid:112)
Below, we assume kr to be oriented along theˆz axis and the PW phase, c(cid:107) = c0 1−Ω2/Ω2c; in the ZM phase,
(cid:112)
set the detuning δ equal to zero. The term ∼ p·k σ c =c 1−Ω /Ω; and in both phases, c =c is inde-
r z (cid:107) 0 c ⊥ 0
represents the SO coupling, which breaks the symme- pendent of Ω. Following the procedure as shown before,
try of Galilean invariance [26]. The interactions between we find the critical velocity obtained in (11) is also valid
atoms can be written as H = g/2(cid:82) d3rn2(r), where for the SO coupled BEC as long as the onset of dissipa-
int
wehaveassumedtheequalcouplingconstantsinthedif- tion is due to the emission of a phonon. This is indeed
ferentspin-channelswithg >0. Suchachoiceisrelevant the case of the ZM phase, and it is also true in the PW
to the current experiments with 87Rb atoms [8, 15–17], phase when Ω is close to Ω . At the critical point Ω , c
c c (cid:107)
4
FIG. 4: (a) Dissipation rate P of a moving impurity in a
SO coupled BEC as a function of v and Ω. The dash-dotted
lineshowsthecriticalvelocity;andthedashedlineshowsthe
sound velocity in the moving direction. (b) Velocity depen-
dence of P for Ω=2.5E . For both plots, vˆ is perpendicular
r
FIG.3: (a)LowerbranchexcitationspectrumE inthePW tok (i.e. θ =π/2),gn=1E ,andP isinunitsofU2nm3c3.
q r v r 0 0
phase of a SO coupled Bose gas as a function of q·vˆ, where
vˆ is in the direction with θ =π/2 (see the inset). The solid
v
line is for the momentum q in the direction that E /(q·vˆ)
q excitationspectruminthisdirectionremainsunchanged.
reaches the global minimum with θ (cid:39) 0.91π; the dashed-
q
AsshowninFig.4,fortheintermediatevaluesofΩ,visi-
dotted line is for θ = θ ; the dotted line is tangent to the
q v
solid line at the roton minimum, and its slope is equal to bledissipationarisesatvelocitiesbetweenvc andc0. For
vc. (b) Anisotropic critical velocity as a function of θv (solid Ω (cid:46) 0.5Er, the dissipation in the subsonic regime be-
line). Thedashed-lineshowsthesoundvelocityinthemoving comes negligibly small. This is because, in this case, the
direction. For both plots, Ω=2.5Er, and gn=1Er. roton exhibits a dominant spin-flip nature, and it pro-
duces a very weak response to the density fluctuation
[see Eq. (5)]. In the limit Ω → 0, while v vanishes due
c
vanishes, and v equals zero in all the directions except totheclosingoftherotongap,dissipationonlyarisesfor
c
θv = π/2 [33], which is similar to the situation of the v >c0 [37].
dipolar BEC when (cid:15) →1 [34]. Finally, we make some remarks on the normal fluid
dd
For the smaller values of Ω, roton-like excitations density, which concerns another aspect of superfluidity.
emerge [30, 31, 35] and give rise to a dramatic influ- For He-II and ordinary Bose gases, the dissipationless
ence on the anisotropic superfluidity. As an example, motion below vc is associated with the fact that nor-
in Fig. 3(a), we show how the critical velocity in the di- mal fluid density ρn vanishes at zero temperature. More
rectionperpendiculartokr isaffectedbytherotonstruc- precisely, a finite value of vc rules out the existence of
ture. Thenon-monotonicbehaviorofE onlyappearsfor low-energyexcitationsoftransversenature[1]. Thiscon-
q
θ closetoπ. Forthemotioninthedirectionwithasmall clusionalsoholdsfortheanisotropicsystemsinvestigated
q
θ , v is not affected by the roton structure because the here. Ontheotherhand,foraSOcoupledBEC,thehigh-
v c
minimization in (2) is under the constraint q·vˆ >0. In energy branch of the excitations has a nontrivial contri-
Fig. 3(b), v is plotted as a function of θ for Ω=2.5E , butiontothetransversecurrentresponsefunction,which
c v r
and there is a kink at θ(cid:63) (cid:39)0.47π. For θv <θ(cid:63), vc is de- givesrisetoafiniteρn evenatzerotemperature[38,39].
termined by the sound velocity of the phonon [Eq. (11)]; In summary, in this work, we discussed the Landau’s
for θ > θ , v is determined by the dispersion near the criterionforanisotropicsuperfluidityandinvestigatedits
v (cid:63) c
roton. consequence in both dipolar and SO coupled Bose gases.
Beyond v , the dissipation rate (5) is numerically cal- Our predictions can be tested in the experiments with
c
culated within the Bogoliubov approximation [36]. Re- ultracold atoms by moving a microscopic object through
markably, when the roton-like excitation emerges, the a condensate [4]. The general result for critical velocity
dissipationless motion is suppressed by the Raman cou- can be also applied to fermionic superfluids.
plingeveninthedirectionorthogonaltok ,althoughthe The author thanks Hui Zhai, Wei Yi, Bo Liu and Lan
r
5
Yin for helpful discussions. This work is supported by NSFC under Grant No. 11674202.
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