Three-dimensional matter-wave vortices in optical lattices

0.31 MB·English

by  Tristram J. Alexander

Three-dimensional matter-wave vortices in optical lattices
Tristram J. Alexander, Elena A. Ostrovskaya, Andrey A. Sukhorukov, and Yuri S. Kivshar
Nonlinear Physics Centre and Australian Centre of Excellence for Quantum-Atom Optics,
Research School of Physical Sciences and Engineering,
Australian National University, Canberra, ACT 0200, Australia
5 We predict the existence of spatially localized nontrivial vortex states of a Bose-Einstein con-
0 densate with repulsive atomic interaction confined by a three-dimensional optical lattice. Such
0 vortex-likestructures include planar vortices, their co- and counter-rotating bound states, and dis-
2 tinctlythree-dimensionalnon-planarvortexstates. Wedemonstratenumericallythatmanyofthese
vortexstructuresareremarkablyrobust,andthereforecanbegeneratedandobservedinexperiment.
n
a
PACSnumbers: 03.75.Lm
J

5
2 The importanceofthree-dimensional(3D) vortexcon- tions [5], there is promise for the future experimental

figurations in physics was recognized long ago by Sir study of these highly nontrivial vortex phenomena.
]
r W. Thompson when he suggested that robust vortex We model the macroscopic dynamics of the Bose-
e
rings can be employed to understand the structure of Einsteincondensateina3Dopticallatticebytherescaled
h
t the atom[1]. Since then topologicallystable vortexlines mean-field Gross-Pitaevskii(GP) equation:
o
havebeeninvestigatedinmanyphysicalcontexts,includ-
.
t ing cosmology, hydrodynamics, optics, and condensed ∂Ψ 2
a i +∆Ψ−V(r)Ψ+γ|Ψ| Ψ=0, (1)
m matter physics. New experimental opportunities for the ∂t
studyofvorticeshaveappearedintherapidlydeveloping
- where Ψ is the complex matter-wave wavefunction, r ≡
d field of Bose-Einsteincondensates (BECs) [2] as conden- (x,y,z),∆≡∂2/∂x2+∂2/∂y2+∂2/∂z2,andγ =−1for
n sates provide an excellent test system for the study of
o repulsive atomic interactions. The characteristic spatial
nucleation,dynamics,andtheinteractionofvortexlines.
c scale is aL = d/π, where d = λ/2 is the lattice period,
[ Oneoftheoutstandingphysicalproblemsisthebehav- andλisthewavelengthofthelightproducingthelattice.

1 ior of vortices in systems with reduced symmetry, such We consider a cubic lattice potential:
as periodic structures. BECs loaded into periodic po-
v
0 tentials of optical lattices provide an excellent example V(r)=V0(cid:0)sin2x+sin2y+sin2z(cid:1), (2)
9 ofsucha system, being relativelyeasyto manipulate ex-
5 perimentally. Inatwo-dimensional(2D)geometry,stable with the depth V0 scaled by the lattice recoil energy
01 vortices localized in an optical lattice were predicted to EL = ~2/2ma2L. The characteristic time scale is given
5 exist in the form of matter-wave gap vortices [3]. These by ωL = ~/EL. We have neglected any additional trap-
0 vortices are spatially localized states of the condensate ping potential as we are interestedin spatiallocalization
/ with a phase circulation that exist due to the interplay due to the optical lattice and nonlinearity.
t
a of the repulsive nonlinearity and the anomalous diffrac- WelookforthestationarysolutionsofEq.(1)usingthe
m
tionofthe matter wavesdue to their Braggscatteringin ansatz Ψ(r,t)=ψ(r)exp(−iµt), where µ is the chemical
- thelattice. Although2D(planar)vorticeshavealsobeen potential. At low atomic densities, the condensate dy-
d
investigated in 3D discrete lattice models [4], nothing is namicsinthelatticeisdescribedbythelinearizedEq.(1)
n
o known about nonlinear dynamics of the vortex states in wherethelasttermisneglected. Duetotheperiodicityof
c a fully 3D continuous lattice geometry. the potential,the correspondingstationarystatescanbe
:
v InthisLetterwenotonlyshowthatplanar2Dmatter- expressedasasuperpositionofthe Floquet-Blochmodes
i wave gap vortices persist in the presence of the third satisfying the condition, ψk(r) = Bk(r)exp(ikr), where
X
lattice dimension, but also predict the existence of com- Bk(r)=Bk(r+d),andkisthequasi-momentumwithin
r
a pletely new vortex states localized inside the lattice. In the first Brillouin zone of the lattice (see Fig. 1). Mat-
particular we show that the circulation of particles can ter waves can propagate through the lattice if k is real-
break the traditional 2D vortex symmetry and form valued, i.e. when the chemical potential of the Floquet-
states which circulate simultaneously in multiple sym- Bloch linear states belongs to one of the bands (shaded
metry planes. Furthermore we show that the 3D lattice in Fig. 1). In the 3D potential (2), there always exists
provides a unique opportunity to study the coherent in- a semi-infinite spectral gap extending to µ → −∞, and
teraction of multiple vortices, and demonstrate that an additionalgapsappearonlywhenthe potentialdepthV0
interacting stack of vortices containing an extended vor- exceeds the threshold value Vth ≈ 2.24. In what follows
tex line, which is known to be unstable in homogeneous we choose the value V0 =6.
systems, is stabilized by the lattice. As 3D optical lat- Athigheratomicdensities,theeffectofrepulsiveatom-
tices provide improved control over condensate excita- atom interactions described by the nonlinear term in
2
15
µ
10
5 Γ X M R Γ
FIG. 1: Matter-wave spectrum µ(k) in a three-dimensional
optical lattice (2) at V0 = 6. Shaded and open areas show
bands and gaps, respectively. Inset shows the first Brillouin
zone and the high-symmetrypoints in thereciprocal lattice.
FIG.2: (coloronline)Gapsolitonina3Dopticallatticewith
Eq.(1) becomes important. In free space,repulsive non-
V0=6. (a)NormofthewavefunctionN vs. µinsidethegap;
linearitytends toacceleratethe spreadingofthe conden-
point C (at µ = 7.9) corresponds to the state shown in (b)
sate. However, in a lattice the effective mass of the con- and (c). Line at left band edge calculated in free-space GP
densate becomes negative near the lower gap edges [8], limit. Dotted line an interpolation. (b) An isosurface of the
allowing for the formation of localized states inside the solitondensity,|ψ|2,atthelevel|ψ|2 =0.2,withphaseinthe
spectral gaps [6]. These nonlinear states - gap solitons z=0 plane (semi-transparent) shown below. (c) The cutsof
ψ (dashed) and |ψ|2 (solid) along the line y = z = 0. Thin
- are localized in all three dimensions. Figure 2 shows
dashedlinecorrespondsto|ψ|2 =0.2. Shadedareasshowthe
an example of a gap soliton, and the dependence of its
norm, N = |Ψ|2dr, on the chemical potential within lattice minima (partsof thepotential less than V0/2).
R
thefirstgapcalculatedusinganumericalfunctionalmin-
imizationtechnique[7]. Despiteanontrivialphaseofthe found as Ψ ≃ PMm=1ψs(r − rm)exp(iφm − iµt). For-
wavefunction[seeFig.2(c)],characteristicoflinearBloch mation of a stationary vortex is possible when all the
wave tails, there is no particle flow in the soliton. incoming and outgoing flows are balanced and hence [9]:
Nearthe lowergapedgethesolitonspreadsovermany
M
lattice sites and may be described by the free-space GP
equation for an effective envelope of the correspond- X cnmsin(φm−φn)=0. (3)
m=1
ing Bloch state [8]. In this limit, the 3D gap soliton
is expected to be unstable according to the Vakhitov- Eachterm in this sum defines the flow between gap soli-
Kolokolov criterion since ∂N/∂µ < 0 [see Fig. 2(a)]. tons numbered n and m, which depends on the rela-
Such aninstability may resultin the transformationof a tive phases and the nonlinear coupling constants cnm ≡
weaklylocalizedatomiccloudintoahighlyconfinedstate cmn =γR ψs3(r−rn)ψs(r−rm)dr/R ψs2(r)dr.
deeperinsidethegap[suchasshowninFig.2(b,c)],where The sign of the coupling depends on the sign of the
∂N/∂µ>0. We have investigated the dynamical stabil- overlapbetweenthewavefunctionsofthegapsolitonsand
ity of the strongly localized gap solitons numerically by the sign of the nonlinearity. The former may be positive
introducing 5% random phase and amplitude perturba- ornegativedue to the oscillatingtailsofthe gapsolitons
tions. No evidence of instability can be seen over the [Fig. 2(c)]. As the latter is negative (γ = −1), positive
evolution time t=103, which for 87Rb in an optical lat- coupling occurs when the overlapping parts of the wave-
tice created with a 0.25µm period is equivalent to 17ms. functions are π out-of-phase. Solitons in the cluster can
Spatially localized stationary states of the repulsive be positioned in the “nearest neighbor” configuration,
condensate with a phase circulationcan also exist in the whereby they are separated by a single lattice period in
gaps of the linear spectrum. These are localized vortices thex,y orz direction,orina“nextneighbor”configura-
with stationary density distribution, but nonvanishing tion,wherebythesolitonsareplacedin-plane,diagonally
particle flow around a closed path, which can be charac- across a lattice cell (e.g. in the direction Γ → M in Fig.
terized by the local current density, j = −2Im(Ψ∇rΨ∗). 1). The “nearest neighbor” coupling is positive, whereas
The gap vortex states in a lattice can be thought of as the “next neighbor” coupling is negative. Importantly,
clusters of gap solitons ψs having different positions on forapositivecoupling,theflowisdirectedfromasoliton
the lattice rm and phases φm. The vortex flow between with phase φm = 0 to one with π/2 to π etc., while for
such solitons occurs due to a nontrivial phase structure negative coupling the flow is reversed. Below we discuss
of the full matter-wave function which is approximately the physical origins of this reversal.
3
FIG.3: (coloronline)Examplesofplanar(a)on-siteand(b)
off-site vortices (V0 = 6, µ = 7.9). The orientation in (a) is FIG. 4: (color online) Examples of interacting vortex soli-
the(x−y)plane,while in (b)it isthe(y−z)plane. Arrows
tons at V0 = 6, µ=7.9.(a) In-phase co-rotating vortex pair.
show directions of particle flow.
(b) Counter-rotating vortex pair. Lines correspond to pri-
mary vortex lines, with different shadings showing opposite
topological charges.
A 3D lattice can support vortex states of different ge-
ometries. First and the most obvious type are the gap
vortices localized in a plane which can be considered as of([0,1],[1,3],[1,0],[3,1]), while the hybrid vortexpair
2 2 2 2
a generalization of 2D gap vortices [3]. Such 2D states has a phase structure ([0,0],[1,3],[1,1],[3,1]). In an
2 2 2 2
were shown to have two basic symmetries relative to the equivalent hybrid state the second vortex is shifted in
lattice sites: on-site and off-site vortices [3]. The off-site phasebyπ. Theparticleflowsofthein-phaseandout-of-
vortexhasalatticemaximumatitscenterand,therefore, phase vortices are co-rotating,and hence the pair shares
it may consist of four lobes (atomic density maxima) in a singular vortex line at the center [Fig. 4(a)]. In con-
anearestneighborconfiguration[Fig.3(a)]. Asthenear- trast,the hybridvortexstatesareequivalentto interact-
est neighbor coupling is positive, the flow circulation is ingcounter-rotatingvortices. Their vortexlineshaveop-
defined in the direction 0 → π/2 → π → 3π/2 → 0 of posite charge and collide, producing two additional vor-
the phases of the composite solitons. The on-site vor- tex lines splintering away from the interaction area [see
tices are centered on a lattice site and consist of four Fig. 4(b)], but conserving the total vortex charge. The
lobesinthenextneighborconfiguration[Fig.3(b)]. With algebra of these vortex lines and the exciting possibility
the nearest neighbor coupling normalized to +1, the on- of controlling their direction will be discussed elsewhere.
site vortex has an inter-lobe next neighbor coupling of Stability of the planar vortex pairs depends critically
−0.11, and the particle flow is opposite to that expected onthecouplingbetweenvortices. Wefindnoexamplesof
fromthe compositesolitonphases. At the physicallevel, stable out-of-phase co-rotating and the counter-rotating
as particle flow is always directed along the phase gradi- pairs, whereas the in-phase co-rotating vortices can be
entthisseeminglyanomalousflowactuallyarisesthrough very robust. The in-phase configuration,due to the pos-
theinterference of multiplephase singularitiesproducing itive coupling between nearest-neighbor lobes, is equiva-
a counter-flowoppositeto thatofthe centralsingularity. lenttoaπ-phasejumpbetweenthewavefunctions,dueto
We have examined the dynamical stability of the pla- the nature of the linear Bloch waves. Such phase “twist-
narvorticesshowninFigs.3(a,b)numericallyandfound ing” has a stabilizing effect as previously discovered in
3
that they are stable, at least up to t = 10 . An im- the frameworkofdiscrete nonlinear systems withattrac-
portant consequence of the 3D geometry is that we may tive nonlinearities [10]. The actual region of stability for
orient these planar vortices in any of the three symme- the co-rotating vortex states depends on the strength of
try planes of the lattice. Additional planes exist in the the coupling (which depends on the values of µ and V0),
three-dimensional lattice, however they are asymmetric in agreementwith the results for 2Dvorticeswith an at-
with respect to the inter-site coupling, and we expect tractive nonlinearity [11]. The state shown in Fig. 4(a)
such planes to host asymmetric planar vortices [9]. is dynamically stable to t=103.
A new feature of 3D optical lattices is the possibil- Most remarkably,we find that vortex “stacks”of mul-
ity of studying the coherent interaction of multiple pla- tiple co-rotating vortices (somewhat analogous to the
nar vortices. From the analysis of the discrete equations stacks of “pancakes” [12]), forming an extended discrete
(3), we have identified three basic types of coupled pairs vortexline,canalsobehighlyrobust,inboththe off-site
of planar vortices: in-phase, out-of-phase, and hybrid- [Fig. 5(a)] and on-site configurations [Fig. 5(b)]. These
phase. These may be described by the phases of the resultsdemonstratethata3Dopticallatticecanprovide
vortexlobes. The in-phasepairhas aphasestructure,in transverse stabilization of vortex lines. This effect can
1 1 3 3
multiplesofπ,correspondingto([0,0],[ , ],[1,1],[ , ]) be exploredexperimentally, since without the transverse
2 2 2 2
where we have used the notationof [vortex1,vortex2] for periodicity,vortexlinesexhibitstrong‘snake-type’insta-
eachofthe fourlobes. Theout-of-phasepairhasaphase bilities [12]. The possibility of long wavelength modula-
4
FIG. 5: (color online) Example of a stable off-site vortex
stack at t=0 (a) and at t=103 (b) (µ=7.9). (c) Unstable
on-site vortex stack at t=0 and t=150 (µ=7.5)(d).
FIG. 6: (color online) Novel 3D vortices at V0 =6, µ=7.9.
tionalinstabilitycannotberuledouthowever,especially
(a) Folded vortex shown in (b) relative to the positions of
at long time scales [13]. Our numerical simulations per- the lattice minima. (c) Crossed-dipole vortex and (d) vortex
formedwith stacksup to 16lattice sites long havefound diamond. Thicklinescorrespondtotheprimaryvortexlines.
no trace of such an instability at t = 103 [Fig. 5(a)].
When the instability is present, e.g. due to stronger
inter-lobe coupling, it develops rapidly, leading to the In conclusion, we have predicted novel classes of spa-
spectacular “explosion”of the vortexline [see Fig. 5(b)]. tiallylocalizedvortexstructuresinBECsloadedinto3D
Apart from the generalizations of planar 2D gap vor- optical lattices. Such structures include planar vortices,
tices, 3D lattices can support vortices which have no vortex stacks, folded vortices and vortex diamonds. We
2D analogs. The “truly” 3D vortex with the strongest have demonstrated numerically that many of these vor-
coupling between lobes is the four-site state with simple tex structures are remarkablyrobust, and therefore they
phases (0,1,1,3) (in multiples of π) shown in Fig. 6(b). may be generated and observed in experiment. Our re-
2 2
We term this configuration a “folded vortex”, which sults open up the possibility of studying more complex
can be viewed as a combination of two dipoles (0,1) condensate structures such as localized vortex rings and
and (1,3) that are not equivalent in inter-site coupling knots in periodic potentials.
2 2
strengths. Similar coupling structure also occurs for a
planar rhomboid vortex [9]. The simplest fully symmet-
ric 3D vortex is that of the out-of-plane coupled dipole
state of Fig. 6(c). As can be seen by the particle flow
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