Vortex Lattice Locking in Rotating Two-Component Bose-Einstein Condensates

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by  Ryan Barnett

Vortex Lattice Locking in Rotating Two-Component Bose-Einstein Condensates
Ryan Barnett1, Gil Refael1, Mason A. Porter2, and Hans Peter Bu¨chler3
1Department of Physics, California Institute of Technology, MC 114-36, Pasadena, California 91125
2Oxford Centre for Industrial and Applied Mathematics,
Mathematical Institute, University of Oxford, OX1 3LB, UK and
3 Institut fu¨r Theoretische Physik III, Universit¨at Stuttgart, 70550
(Dated: February 1, 2008)
Thevortexdensityof arotatingsuperfluid,dividedbyitsparticle mass, dictatesthesuperfluid’s
angular velocity through the Feynman relation. To find how the Feynman relation applies to su-
8 perfluidmixtures,we investigatea rotating two-component Bose-Einstein condensate, composed of
0 bosons with different masses. We findthat in the case of sufficiently strong interspecies attraction,
0 thevortexlatticesofthetwocondensateslockandrotateatthedrivefrequency,whilethesuperflu-
2 ids themselves rotate at two different velocities, whose ratio is the ratio between the particle mass
of the two species. In this paper, we characterize the vortex-locked state, establish its regime of
n
stability,andfindthatit suriveswithin adisksmaller than acritical radius,beyondwhich vortices
a
becomeunbound,andthetwoBose-gasringsrotatetogetheratthefrequencyoftheexternaldrive.
J

4 PACSnumbers:
2

] After the first experimental realization of Bose- Ω . Ω
el Einsteincondensates(BECs)ofalkaliatoms,theirstudy 1 2
-
r hasexperiencedenormousadvancements[1]. Among the : species 1 vortex
t
s major threads of investigation in BECs has been the : species 2 vortex
t. study of vortices both experimentally [2, 3, 4] and theo-
a
m retically[5]. ItisknownfromtheclassicworksofOnsager Fm1ag Frs1tr Frs2tr Fm2ag
and Feynman[6, 7] that superfluids rotate by nucleating (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
-
vortices. When there are several vortices present, they
d
n form a triangular Abrikosov vortex lattice [8], with den-
o sity given by
c
[
mΩ FIG.1: Schematicdiagramofboundvortexpairs(forspecies
ρ = , (1)
3 v π¯h withmassesm1 andm2 <m1)intherestframeofthevortex
v lattice in the limit of large interspecies interaction in which
6 where m is the mass of a constituent boson and Ω is the the two vortex lattices are locked. The Magnus force Fmag
42 rateatwhichthesuperfluid–whichrotateswiththevor- opposessuchlockingandisbalancedbyarestoringforceFrstr
dueto theinterspecies vortex-vortexinteraction.
2 tex lattice – is being rotated (see, for instance, Ref. [9]).
. The so-called“Feynman relation” (1) states that, on av-
5
0 erage, a uniform superfluid rotates like a rigid body. It proportional to their masses, m , such that:
1,2
7 has been shown that corrections to Eq. (1) due to the
0 typical experimental situation of nonuniform superfluid m1Ω1 =m2Ω2 (2)
:
v density resulting from a harmonic trap are small [10] while the vortex lattices of the two components lock at
i but experimentally observable [11]. Vortex physics be-
X the drive angular velocity Ωv, lying between Ω1 and Ω2
comes much more intriguing in multi-component BEC’s.
(see Fig. 1). Qualitatively, the attractive interspecies in-
r
a So far, the investigation of vortex lattices in multi- teraction leads to an attraction between vortices of the
component BECs utilized different hyperfine levels of
two flavors. If it is sufficiently strong, vortices pair, and
the constituent atoms to obtain multi-component con-
the lowest-energyvortexconfigurationthen occurs when
densates (e.g., [12, 13]). Thus the mass of all conden-
the vortex lattices of the two flavors are “locked” to-
sate components was identical, and the generalizationof
gether, rotate at the same rate, and have essentially the
Eq. (1) straightforward.
same density. Eq. (1) then reads:
In this paper we ask what are the consequences of the
m Ω m Ω
Feynman condition, Eq. (1), in a system of interact- ρ1 = 1 1 2 2 =ρ2 (3)
v π¯h ≈ π¯h v
ing two-component rotating condensates with different
masses. We find that for sufficiently large attractive in- where ρ1,2 are the vortex densities of the two flavors.
v
teractions,the Feynmanconditionleadsto anovelstate. This state is strongly related to experiments in Ref. 14,
The two components, rather than rotate together at the where a vortex lattice was locked to an optical lattice
drivefrequency,rotateatangularvelocitiesΩ inversely with a similar periodicity.
1,2
2
As we show below, this state survives within a finite of the positions of the vortices as
disk about the center of the rotating condensate. The
relative motion between the vortices and the condensate E = ¯h2πnα log ξα +h¯Ω nαπ (rα)2,
givesrisetoaMagnusforcethatopposestheinterspecies α m 0 rα rα v 0 i
α i=j | i − j|! i
vortexattraction. BeyondacriticaldistancetheMagnus X6 X
(6)
force (Fig. 1) becomes larger then the maximal pairing
force, and the vortices become unbound. In this region, wherewehavedroppedtermsthatdo notdependonthe
the twocondensatesandtheirvorticesrotatetogetherat positions of the vortices and have also neglected effects
the drive frequency Ωv; the vortex densities in the two due to the nonuniform superfluid density [10]. The first
flavors are no longer equal, but reflect themselves the term in Eq. (6) is the usual logarithmic interaction be-
mass ratio: ρ1v/m1 = ρ2v/m2. Below we derive the char- tween vortices, and the second is the centripetal energy,
acteristics and conditions for the vortex locking state. reflecting the fact that vortices toward the edge of the
Energetics. The energy of weakly interacting Bose- cloudcarrylessangularmomentumrelativetothecenter
Einstein condensates is well-described by the Gross- of the cloud. In a single-component rotating BEC, the
Pitaevskii functional for the condensate wave function balance of the two terms gives the Feynman condition
ψ (α=1,2)forthe twoatomicspecies [1]. We consider (1). Theequationsdescribechargedparticlesinteracting
α
the situation in which the two condensatesare stirred at in two dimensions with a uniform background charge of
the same rate Ω . Transforming to the rotating frame, opposite sign.
v
our problem becomes time independent. The energy of The energy E12 arising from the interspecies interac-
the two-componentsysteminthe rotatingframeis given tionenergyislessstraightforwardtoevaluate. Unlikethe
by E =E +E +E , where intraspecieslogarithmicinteraction,thisnonuniversalin-
1 2 12
teraction depends on the details of the short-distance
density variationsaroundthe vortexcores. Forinstance,
¯h2
E = d2r ψ 2+V (r)n in [15] to study the interaction of a vortex with an opti-
α α trap α
2m |∇ |
Z (cid:18) α cal lattice, a step function having the width of the BEC
+ gαn2 ¯hΩ ψ i ∂ ψ (4) coherencelengthwastaken. InthisworkwetakeaGaus-
2 α− v α∗ − ∂ϕ α sian depletion around the vortex core:
(cid:18) (cid:19) (cid:19)
describestheenergyforeachatomicspeciesandgα isthe nα(r)=nα0(1−e−|r−r0|2/ξα2) (7)
intraspecies coupling for bosons of flavor α. The inter-
so that the system will be amenable to analytic treat-
species interaction is given by
ment. Fora singlevortexthis depletiongivesthe correct
behavior at short distances, but not the long distance
behavior, in which the density due to a single isolated
E =g d2r[n (r)n (r)], (5)
12 12 1 2 vortex heals as ξ2/r2. As we show in the Appendix, the
Z
combined density variations due to the vortex lattice on
where n = ψ 2 is the density of flavor α, V is the scales larger than the inter-vortex separation combines
α α trap
| | only to change the chemical potential (which is propor-
external trapping potential, and the z-component of the
angular momentum operator is Lz = −i∂∂ϕ (where ϕ is jtuiosntarlefltoecttshethdeenksiniteyticcoernreecrtgiyona)s,sobcyiat2h¯em2dπw2ρit2vhr2u,nwifhoircmh
the azimuthal coordinate).
rotationof the condensate. This can be shownto have a
The energy E of one BEC component is minimized
α negligible effect on the vortex pairs, which are the focus
via the nucleation of a vortex lattice rotating with the
of this work. Neglecting this piece of the density fluctu-
external drive Ω . In the following, we assume that the
v ation is also consistent with neglecting the intraspecies
coherence length ξα = 2mαh¯g2αnα0 is much shorter than core-coreinteractionswhichisastandardapproximation
thecharacteristicdistanqcebetweenvortices. Eachvortex [5]. Indeed, the short distance region is the relevant one
isthenwell-describedbyasmallcoreregionofsizeξ ,at for vortex locking; two-species vortex pairs become un-
α
which the superfluid density drops to zero and its phase bound once their separation is comparable to the coher-
fieldaccountsfortheflowaroundthevortex. Thevortex encelengths,wheretheapproximateformwetakeforthe
core region gives rise to a small constant energy, and we density profiles is still valid. Evaluating the interaction
canaccountforthephasefieldbywritingψα =√nαeiθα, integral in Eq. (5), and keeping only the contributions
where θ determines a lattice ofvorticeswith unitwind- dueto theinteractionsbetweenpairsofvorticesbetween
α
ing number at the positions rα . We assume that we different species, gives
{ i}
canwriteθ asasumofthedifferentvortexcontributions
θ =θα+αθα+.... ξ2ξ2 |r1i−r2j|2
α 1 2 E12 =g12n10n20πξ21+2ξ2 e− ξ12+ξ22 . (8)
With these assumptions, Eα can be written in terms 1 2 ij
X
3
We note that to prevent phase separation, the criterion for radii r > r the unlocked phase applies, and after a
c
g g > g 2 must be satisfied. Equations (6) and (8) short healing region, the two condensates rotate at the
1 2 12
| | | |
now give the energetics of the system purely in terms of samefrequency. Anexpressionforr canbe obtainedby
c
the positions of the vortices. equating the Magnus force with the maximum possible
Forces. An understanding of the locked phases can be value for the restoring force:
obtainedbyconsideringtheforcesactingonthevortices.
Equation(6)leadstothewell-knownMagnusforceacting r = 1 |g12|m1n20+m2n10 ξ12ξ22 . (12)
on a vortex of species α [9]: c 2e¯hΩ m m (ξ2+ξ2)3/2
r v 1− 2 1 2
Fα =2π¯hnα(vα v ) ˆκ, (9) Note that (12) diverges when the masses are equal. In
mag 0 SF − v ×
addition, the bound pairs of vortices are pulled further
where ˆκ is a unit vector pointing out of the plane, nα is apart at increasing distances from the center of the con-
0
the equilibrium superfluid density for species α (evalu- densate. The interspecies vortex separation x satisfies
v
ated away from the vortex core), vα = h¯ θ is the
superfluidvelocityforspeciesα(withStFhevomrtαe∇xoαnwhich x2v r
the force operates excluded from θα), and vv is the ve- xve−ξ12+ξ22 = rc√2e ξ12+ξ22, (13)
locity ofthe vortex. Onthe otherhand, the forcearising q
from the energy in Eq. (8) provides an attractive force which is valid for r < rc. This introduces a small cor-
betweentwo vorticesof differentspecies. Ithas the form rectiontothe vortexdensityandcreatesasmallshearin
the motion of the two condensates.
ξ2ξ2 d2 For the vortex-lockedstate to be stable up to the crit-
Fαrstr =−2π|g12|n10n20(ξ12+1 ξ222)2e−ξ12+ξ22 d, (10) ical radius rc, the superfluid velocities in the rotating
frame of the vortex lattice, vα v , must not exceed
where d is the displacement vector between vortices. thecriticalvelocityofthesup|eSrflFu−id. vO|therwise,itwould
Let us first briefly consider the unlocked case where bepossibletocreateelementaryexcitationsfromtheflow
the vortex interspecies interaction force is small. Since of the superfluid around the vortices. For the two cou-
the force counteractingthe Magnus force is too small, to pled superfluids, the Bogoliubov elementary excitations
bind vortex pairs, we must have Fmag = 0 for any iso- are given by
lated vortex, which implies that the superfluid velocity
mustbethesameasthevortexvelocity. Thatis,thevor- (Ω )2 = 1(E1+E2) 1 (E1 E2)2+16g2 n1n2ε1ε2,
texlatticerotateswiththesuperfluid. Thus,becausethe k 2 k k ±2 k − k 12 0 0 k k
q (14)
two vortexlattices rotate atthe same frequency,the two
superfluids rotate together at that frequency. Accord- where Ekα = (εαk)2+2gαnα0εαk and εαk = h¯2m2kα2. It is
ingly, for this case, the vortex densities are not equal: thenstraightforwardtocomputethecriticalvelocityvc =
p
ρ1v = mm12ρ2v. mink(Ωh¯kk); one obtains
We next consider the other extreme, in which the two
g nα ¯h
vortex lattices are locked. Our approach is to consider v =min α 0 =min . (15)
c α α
the forces acting on a bound pair of vortices at dis- (cid:26)r mα (cid:27) (cid:26)√2mαξα(cid:27)
tance r from the center of the trap (see Fig. 1). As
The superfluid velocity of species 1 or 2 in the vortex
stated before, the Magnus force for the locked state is
lattice frame evaluated at r (where it is maximal) is
nonzero because the superfluids are rotating at differ- c
given by
ent rates. Balancing the forces gives Fα = Fα . Be-
mag rstr
cause the restoring force acting on either species has the 1 g ¯h ξ2
vsaαm=eΩmαargnanitdudvev,=weΩvorbt(awine aFrme1aagss=umFimn2gagt.haNtortiinsgmtuhcaht |vS{1F,2}−vv|= √2eg|{21,21|}2m{2,1}(ξ12+{1ξ,222})3/2 . (16)
larger than the distance between the two vortices) gives
The condition vα v < v must be checked so that
n10(Ωv −Ω1) = n20(Ω2 −Ωv). This, along with the con- the vortex-lock|edSFsta−te vis| stabcle against the creation of
dition m Ω =m Ω (from ρ1 = ρ2) gives the following
1 1 2 2 v v elementary excitations. For instance, it can be shown
relation between the angular velocities:
thatthesystemisstableagainstcreatingsuchelementary
(n1+n2)m (n1+n2)m excitations if the conditions 1 n01 10 and 1
Ω1 = m 0n2+0m n21Ωv <Ωv <Ω2 = m 0n2+0m n11Ωv. m1 10 are satisfied. 10 ≤ n02 ≤ 10 ≤
1 0 2 0 1 0 2 0 m2 ≤
(11) Numerical Simulations. Now that the expected types
Note that unlike the restoring force, the Magnus force ofphases have been discussed,we minimize the totalen-
grows linearly with distance from the center of the con- ergy E = E + E + E as a function of the vortex
1 2 12
densate. Thus, at some critical distance r , the pairs positions given by Eqs. (6) and (8). The ability to com-
c
of vortices invariably become unbound from each other. puteanalyticalexpressionsforthegradientsoftheenergy
4
γ=28 γ=17 γ=5
FIG.2: Vortexlatticefortwo-componentcondensate(with43vorticesofeachspecies)fordifferentvaluesofγ =|gab|n0/(h¯Ωv).
Circles are shown for thetheoretical prediction for thecritical radius Eq. (19) at which the vortices become unbound.
as a function of the vortex positions r E allow us to condensatemixture[16],whichhasamassratioofabout
∇{ i}
apply the steepest descent method for the minimization. 1.5. One has exquisite control over the self-scattering
Specifically, starting with an initial configurationfor the length of Cesium [17], and a Cs-Rb mixture is also ex-
vortex positions r(0) , we perform the one-dimensional pected to exhibit interspecies Feshbachresonances. This
{ i }
minimization of allows one to control the interaction g12 over a wide
range; such interspecies resonances have recently been
E {r(in)}−λ∇{ri}E (17) identified for Li-Na [18] and Rb-K [19] mixtures.
(cid:16) (cid:17) Conclusions. In this paper, we described a novelstate
over λ, where ∇{ri}E is evaluated at {rni}. The new of rotating interacting condensates with unequal masses
vortex positions are given by in the Thomas-Fermi regime. The possibility of lock-
ingthe twoindividualvortexlatticesyieldsaremarkable
{r(in+1)}={r(in)}−λmin∇{ri}E, (18) demonstration of the nonintuitive behavior of superflu-
ids: thetwogasses,ratherthanequilibratingtothesame
and the above procedure is repeated until it converges. speed, prefer to move at different angular velocities that
To simplify the analysis, we restrict our attention to are inversely proportional to their masses. The vortex-
the case in which the equilibrium densities and heal- locking of the different-mass condensates is also an ex-
ing lengths of the two condensate components are equal: ample of synchronization: a phenomenon that is ubiqui-
n0 ≡ n10 = n20, ξ ≡ ξ1 = ξ2. Motivated by the example tous in physics, biology, and other fields [20]. Already
of a 133Cs-87Rb condensate [16], we fix the mass ratio in single-mass mixtures, a rich variety of vortex dynam-
to be m1/m2 = 1.5. The vortex interaction strength is ics arises from the extra degrees-of-freedom, resulting in
parametrized by γ = |g12|n0, which we vary while keep- such effects as the formation of square vortex lattice, as
h¯Ωv
ingthe quantities h¯π 1 forα=1,2fixed. (The total wellastopologicallynontrivialdefects suchasskyrmions
energy has been sΩcavmleαdπbξy2 h¯Ω πn ξ2.) We set the ra- or hedgehogs [13, 21]. To investigate these effects in the
v 0
different-massmixtures,aswellastobetterestablishthe
tio of the “average”vortex lattice constant to the coher-
lockedstate we proposedin this manuscript,this system
ence length a /ξ=10 (which is consistent with typical
lat
experiments). We define a by 2 = m˜Ωv, where mustbenumericallyinvestigatedbysolvingtheappropri-
lat √3a2lat πh¯ ate dynamical Gross-Pitaevskii equations. Such numeri-
m˜ = 2m1m2, and consider a system with 43 vortices of
m1+m2 cal investigation will also allow one to find the preferred
eachspecies. Theresultsforsuchacalculationareshown
lattice geometry of the locked vortex-lattice. Other di-
in Fig. 2. We also plot our prediction for the critical
rections for future study involve dynamical aspects such
radius at which the vortex pairs become unbound [see
as Tkachenko modes [22]) of the locked state, as well as
Eq. (12)], which for equal densities is
whether a similar state survives in the Landauregime of
large vortex density.
γ m +m
1 2
r = ξ. (19)
c Acknowledgements. We thank the hospitality of the
4√em m
1 2
− KavliInstitute forTheoreticalPhysicswherepartofthis
This prediction agrees quite well with our numerical re- work was completed. We acknowledge support from the
sults. Sherman Fairchild Foundation (RB), the Gordon and
Experimental realization. A very promising candidate Betty Moore Foundation through Caltech’s Center for
for the realization of these locked states is a 133Cs-87Rb thePhysicsofInformation(MAP),andtheNationalSci-
5
ence Foundation under Grant No. PHY05-51164 (RB, of the 1/r2 corrections. Inserting this into the equation
GR, HPB). We also acknowledge useful discussions with for the density profile one finds:
Simon Cornish, Michael Cross, Peter Engels, and Erich
Mueller. ¯h2 ¯h2
2f + π2ρ2r2f +V f +gf3 =µf. (24)
− 2m∇ 2m v trap
APPENDIX Thus the combined vortex effect re-normalizes the trap-
ping potential and does not need to be explicitly taken
into account. To get an estimate for the magnitude of
Inthisappendixwediscussthechangeinthesuperfluid
such a renormalization, one can compare this term with
densityasaresultofavortexlattice,anditseffectonthe
the chemical potential
validityofapproximation(7)andtheresultingexpression
for the interspecies vortex attraction, Eq. (8) and (5).
First consider a single species which has the energy 2h¯m2π2ρ2vr2 ξ r 2 (25)
functional µ ∼ a a
(cid:18) lat lat(cid:19)
E = d2r ¯h2 ψ 2+V (r)ψ 2+ 1g ψ 4 . (20) where alat is the vortex lattice constant which is small
trap
2m|∇ | | | 2 | | for typical experiments.
Z (cid:18) (cid:19)
For a two-componentBEC, the situation is similar for
We write ψ = feiθ where f is real and θ contains infor-
vortices which are many coherence lengths away from
mation about the positions of the vortices as θ = θ .
i i each other. On the other hand, when the cores of the
By varying f we obtain the equation determining the
different types of overlap, their interaction needs to be
P
superfluid density which minimizes E
explicitly calculated, and the continuum approximation
cannotbeused. Thisisthecaseforpaired-vortexconfig-
¯h2 ¯h2
2f + θ 2f +V f +gf3 =µf (21) urations. Since the combined effect of far-away vortices
trap
− 2m∇ 2m|∇ |
on a locked pair is small, the locking depends only on
for the particular vortex configuration. First let us con- the short distance density profile taken. Had we used
siderasinglevortextakentobeattheorigin θ =zˆ rˆ. the step-function potential interaction between two vor-
∇ ×r
The long-distance behavior r ξ is obtained from tices of Ref. 15 the results would only differ from the
≫
the Thomas-Fermi approximation (neglecting the 2f GaussiandepletionEq.7bysmallquantitativeamounts.
∇
in Eq. 21) and we obtain
µ ¯h2 ξ2
f 1 =n 1 (22)
≈ g − 2mµr2 0 − r2
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