Spin-orbit Coupled Bose-Einstein Condensates in Spin-dependent Optical Lattices

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by  Wei Han

Spin-orbit Coupled Bose-Einstein Condensates in Spin-dependent Optical Lattices
Wei Han,1,2 Suying Zhang,2 and Wu-Ming Liu1
1Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
2Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
Weinvestigatetheground-statepropertiesofspin-orbitcoupledBose-Einsteincondensatesinspin-
dependentopticallattices. Thecompetitionbetweenthespin-orbitcouplingstrengthandthedepth
3
oftheopticallatticeleadstoarichphasediagram. Withoutspin-orbitcoupling,thespin-dependent
1
opticallatticesseparatethecondensatesintoalternatingspindomainswithoppositemagnetization
0
directions. With relatively weak spin-orbit coupling, the spin domain wall is dramatically changed
2
fromN´eelwalltoBlochwall. Forsufficientlystrongspin-orbitcoupling,vortexchainsandantivortex

n chainsareexcitedinthespin-upandspin-downdomainsrespectively,correspondingtotheformation
a ofalatticecomposedofmeron-pairsandantimeron-pairsinthepseudospinrepresentation. Wealso
J discuss how to observethese phenomena in real experiments.

0
1 PACSnumbers: 03.75.Lm,03.75.Mn,05.30.Jp,67.85.Fg

]
s
Introduction.—In recent years, the experimental con- entation of the spins in the domain walls, causing the
a
g trolonultracoldatomicgaseshasreachedtrulyunprece- transformationfromN´eelwalltoBlochwall. Sufficiently
- dented levels. By employing two lasers with different strong SO coupling excites meron-pairs and antimeron-
t
n frequenciesandpolarizationsplusanon-uniformvertical pairs in the spin-up and spin-down domains respectively
a magneticfield,experimentalistshaveproducedspin-orbit and generates a meron-pair lattice. This is essentially
u
(SO) coupling, which couples the internal states and the differentfromthe mechanismofgeneratinga meron-pair
q
. orbit motion of the atoms [1–4]. The SO coupled ultra- lattice by bulk rotation [33]. Our findings provide a new
t
a cold atomic gases have attracted great interests of re- way to create and manipulate topological excitations in
m searchers[5–15]. Ithas beenindicatedthatthe interplay SO coupled systems.
- among SO coupling, interatomic interaction and exter- Energy band structure.—We consider SO coupled
d
nal potential leads to rich ground-state phases, such as BECs confined in the combined potential of a quasi-2D
n
plane wave, density stripe, fractional vortex and various harmonic trap and 1D spin-dependent optical lattices.
o
c vortexlattices[16–26]. TheSOcoupledultracoldatomic The Hamiltonian of this system is given by
[ gases open a new window for quantum simulation, and
v3 pinroavihdiegholpypcoorntturnoiltliaebsletoimstpuudryityS-Ofreceoeunpvliinrognpmheennto.mena H = Z drhΨ†(−¯h22m∇2 +Vso)Ψ+V1(r)n↑+V2(r)n↓
97 All the existing studies on SO coupled Bose-Einstein +g11n2↑+g22n2↓+g12n↑n↓i, (1)
condensates (BECs) only refer to the case that different
0
2 internal states of the atoms are trapped in an identical where Ψ = [Ψ↑(r),Ψ↓(r)]T denotes the two-component
. externalpotential. However,byusing twocounterpropa- wave functions and is normalized as drΨ†Ψ=N with
1
gatinglaserswiththesamefrequencybutdifferentpolar- R
1
2 izations, the experimentalists have been able to produce
(a) (b)
1 spin-dependent opticallattices that allowdifferent inter- 900 100
: nal states of the atoms experience drastically different
v 750
80
i external potentials [27–31]. The spin-dependent optical ]⊥ ]⊥ density stripe
X ω 600 ω
ar la[a2nt7dt],ichceaosvobelripninoggteamnntdoiartelhaceporpmmlipocmlaicteaitotrneyds[ig2ne8qo],muaaenntdtruyqmoufacconomtnupdmuentsasiatmitouens- hunitsof¯ 450 I IIA IIB hunitsof¯ 4600 triangular lattice
[ 300 [
lation [29–32]. It is natural to ask what new structures V0 V0
20
150
can be formed due to the competition between the SO rectangular lattice
coupling and the spin-dependent optical lattices. 0 0
0 5 10 15 20 25 30 0 2 4 6 8 10
In this Letter, we investigate the ground-state phase κ[unitsofahω⊥] κ[unitsofahω⊥]
diagram of SO coupled BECs in spin-dependent optical
FIG. 1: (Color online) (a) Single-particle phase diagram
lattices. In the absence ofSO coupling,the groundstate spanned by the SO coupling strength κ and the depth V0 of
of the system is characterized by the formation of al- theopticallattice. (b)Many-bodyphasediagramspannedby
ternating spin domains. However, such a structure can κ and V0 with the effective interaction parameter g˜ = 6000.
be dramatically changed due to the SO coupling effects. Thedashedlinein(b)indicatesthephaseboundarybetween
phase I and phase II in (a) for comparison.
Relatively weak SO coupling basically changes the ori-
2
FIG. 2: (Color online) Band structures induced by the competition between the SO coupling strength κ and the depth V0 of
the optical lattice corresponding to phase I (a), phase IIA (b), and phase IIB (c) of the single-particle phase diagram in Fig.
1(a). The superposition of themomenta at theminima of thebands producesa density stripe in (a), a lattice in (b) and (c).
N thetotalparticlenumber. n =Ψ∗Ψ andn =Ψ∗Ψ IIB) [See Figs. 2(b) and 2(c)].
↑ ↑ ↑ ↓ ↓ ↓
describe the particle number density ofeach component. The single-particle ground state in phase I is nonde-
gij = 4π¯h2aij/m (i,j = 1,2) represent the interatomic generate and can be expressed as a linear superposition
interaction strengths characterized by the s-wave scat- of the plane waves with wave vectors (k K ,k = 0).
x 1 y
tering lengths aij and the atomic mass m. This yields alternating spin domains with∈opposite mag-
WeconsideraRashbaSOcoupling so = i¯hκ(σx∂x+ netization directions (density stripe). In both phase IIA
V −
σy∂y), where σx,y are the Pauli matrices and κ de- andphaseIIB,thesingle-particlegroundstateisdouble-
notes the SO coupling strength. The combined exter- degenerate. Eachdegeneratestate can be expressedas a
nal potential Vi(r) = VH(r) + VOLi(x), where VH = linearsuperpositionoftheplanewaveswithwavevectors
1mω2[(x2 + y2) + λ2z2] is the harmonic trapping po- (k K,k = δ) or (k K,k = δ), where K = K
2 ⊥ x y x y 1
tential with λ = ωz/ω⊥ 1, and VOL1 = V0sin2(νx) for p∈hase IIA and K = ∈K2 for pha−se IIB. For an arbi-
≫
and VOL2 = V0cos2(νx) describe the 1D spin-dependent trarynonzerosuperpositionofthetwodegeneratestates,
optical lattice potentials, which are experienced by the lattice will be formedas the single-particlegroundstate.
two components, respectively. Approximating the z de- Phase diagram.—By using the imaginary time evolu-
pendence of the wave functions by the single-particle tion method, we can solve Eq. (1) to obtain the many-
ground state in a harmonic potential, one can obtain body ground state. Considering that the spin-exchange
the 2D dimensionless effective interaction parameters integrationsareveryweakintypicalexperiments,wejust
g˜ij =2√2πλNaij/ah, with ah = ¯h/(mω⊥) [34]. discuss the case that g˜ij = g˜. The many-body phase di-
The single-particle energy bandps are critically impor- agram spanned by κ and V with g˜ = 6000 is presented
0
tant to understand the ground-state properties of the in Fig. 1(b). We find that the competition between the
condensates. Without considering the harmonic trap, SOcouplingstrengthandthedepthoftheopticallattice
the 2D single-particle wave functions in the k-space leads to three distinct phases—density stripe, triangular
obey the secular equations 2h¯m2[(kx+2sν)2 + ky2]Us − vortex lattice and rectangular vortex lattice.
V40 (Us+1−2Us+Us−1)+h¯κ(kx+2sν−iky)Ws = εUs Inthedensitystripephase,thespin-upandspin-down
and h¯2[(k +2sν)2+k2]W +V0 (W +2W +W )+ components are arranged alternately and form alternat-
2m x y s 4 s+1 s s−1
¯hκ(k +2sν+ik )U = εW , where U = ing spin domains [See Figs. 3(a1-a6)]. Comparing the
x y s s s
U(k +2sν,k ) and W = W(k +2sν,k ) with phase diagrams in Figs. 1(a) and 1(b), we find that the
x y s x y
s = 0, 1, 2,..., represent the wave functions at interaction has no significant influence on the phase re-
the poin±t (k± +2s±ν,∞k ). Typically, we choose ν = a−1 gion of the density stripe.
x y h
for our present discussion. By numerical exact diagonal- In the vortex lattice phases, both the triangular
ization, we can solve the secular equations and obtain and rectangular lattices are composed of alternately ar-
the energy band structure. ranged vortex and antivortex chains, which are excited
We findthatthereexistthreedifferentkindsofenergy in the spin-up and spin-down domains respectively [See
band structures depending on the competition between Figs. 3(b1-b6) and 3(c1-c6)]. The only difference is that
the SO coupling strength κ and the depth V of the op- the vortices of the neighboring chains are staggered for
0
tical lattices. Fig. 1(a) presents the single-particle phase the triangular lattice, but are parallel for the rectan-
diagramspannedby κ andV . Inphase I,the minima of gular lattice [See Figs. 3(b3) and 3(c3)]. These two
0
the energy bands locate in a set of k points with k = 0 different arrangements of vortices correspond to odd-
y
and k K = ν, 3ν, 5ν,... [See Fig. 2(a)]. In parity and even-parity distributions of the particles in
x 1
∈ {± ± ± }
phase II, the minima of the energy bands locate in a set k direction of the k-space. [See Figs. 3(b6) and 3(c6)].
x
ofk points with k = δ 0<δ mκ ,andk K for These correspond to the single particle band structures
y ± ≤ h¯ x ∈ 1
phase IIA (k K = (cid:0)0, 2ν, 4ν,(cid:1) 6ν,... for phase described in Figs. 2(c) and (b), where the minima of
x 2
∈ { ± ± ± }
3
a1 a2 a3 a4 a5 a6
10 10 10 2π 2π 6
k
a]h y[u
nitsof 0 0 0 0 0 0 nitsof
u a
[ h−
y ]1
−10 −10 −10 −2π −2π −6
−10 0 0 −10 0 0 −10 0 0 −2π 0 2π −2π 0 2π −6 0 6
x[unitsofah] x[unitsofah] x[unitsofah] x[unitsofah] x[unitsofah] kx[unitsofa−h1]
b1 b2 b3 b4 b5 b6
10 10 10 2π 2π 6
k
a]h y[u
y[unitsof−100 −100 −100 −2π0 −2π0 −06 hnitsof]a1−
−10 0 0 −10 0 0 −10 0 0 −2π 0 2π −2π 0 2π −6 0 6
x[unitsofah] x[unitsofah] x[unitsofah] x[unitsofah] x[unitsofah] kx[unitsofa−h1]
c1 c2 c3 c4 c5 c6
10 10 10 2π 2π 6
k
a]h y[u
nitsof 0 0 0 0 0 0 nitsof
[u ha−
y−10 −10 −10 −2π −2π −6 ]1
−10 0 0 −10 0 0 −10 0 0 −2π 0 2π −2π 0 2π −6 0 6
x[unitsofah] x[unitsofah] x[unitsofah] x[unitsofah] x[unitsofah] kx[unitsofa−h1]
FIG. 3: (Color online) Ground state as a function of the SO coupling strength κ and the depth V0 of the optical lattice with
κ=4ahω⊥, V0 =40h¯ω⊥ (a1-a6), κ=4ahω⊥, V0 =20h¯ω⊥ (b1-b6), κ=4ahω⊥, V0 =15h¯ω⊥ (c1-c6). The effective interaction
parameter is fixed at g˜=6000. The spin-up, spin-down, and total density profiles are shown in (a1, b1, c1), (a2, b2, c2), and
(a3, b3, c3), respectively. The phases of thespin-up and spin-down wave functions, with values ranging from −2π to 2π (blue
to red), are shown in (a4, b4, c4) and (a5, b5, c5). Themomentum distributions are depict in (a6, b6, c6).
thebandsalsoshowodd-parityandeven-paritydistribu- wall. In the N´eel wall spin flip occurs in a plane, while
tions respectively, although their phase regions are not in the Bloch wall the spin flip occurs by tracing a he-
consistent due to the influence of the interatomic inter- lix [See Figs. 4(b1) and 4(b2)]. An intriguing finding
actions [See Figs. 1(a) and 1(b)]. As discussed above, of the present work is that the SO coupling dramati-
the single-particle ground state in phase II is double- cally changed the domain wall from N´eel wall to Bloch
degenerate. From Figs. 3(b6) and 3(c6), we can see that wall. Figs. 4(a1) and 4(a2) show the spin density vector
the interaction removes the degeneracy and chooses an S = Ψ†σΨ without and with SO coupling. We can see
|Ψ|2
equal weighted linear superposition of the two degener-
that in the absence of SO coupling, the spin density vec-
ate states as the many-body ground state.
tor acrossthe opposite domains forms a N´eel wall, while
The alternating arrangement of the vortex and an- in the presence of SO coupling it forms a Bloch wall.
tivortexchainsleadstoalternating-directionplanewaves, This phenomenon can be understood as follows. The
which propagating on two sides of each chain [See directionofthespinflipinthedomainwallonlydepends
Figs. 3(b4,b5) and 3(c4,c5)]. The vortex line density on the relative phase, and can be represented by an az-
nv and the wave number of the plane waves ky satisfy imuthalangleα=arctan(Sy/Sx)=θ↓ θ↑,whereθ↑and
nv = kπy. Numericalsimulationsindicatethatforagiven θ↓ are the phases of the wave function−s. When the SO
SO coupling strengthκ, asthe lattice depth V0 increases coupling is absent, there is a constant phase difference 0
from 0, ky gradually decreases from mh¯κ and eventually or π [See the solid line of Fig. 4(c)], so the spin in the
becomes 0 on the boundary of the vortex lattice phase. wallsjustflipsalongthex-directionandformsN´eelwalls.
This implies that by adjusting the depth of the optical When the SO coupling is present, the phase difference is
lattice, one cancontinuously controlthe vortexline den- changed into π [See the dashed line of Fig. 4(c)], so
sity from 0 to mκ. the spin in the±w2alls just flips along the y-direction and
πh¯
Spin domain wall.—The separation between the spin- forms Bloch walls.
upandspin-downdomainsis notsharp,butrequiresthe Meron-pair lattice.—The regular triangular or rectan-
spin density vector varying gradually across the oppo- gularvortexlatticeobtainedinFig.3canbeequivalently
site domains andforming aspin domainwall[35]. There described by the spin density vector S in the pseudospin
aretwobasictypes ofdomainwalls,N´eelwallandBloch representation. Fig.5(a)presentsthevectorialplotsofS
4
(a) (b)

π π
π/2 π/2
nits of a]h 0 c2 nits of a]h 0
u u
y [−π/2 c1 y [−π/2
−π −π

−π −π/2 0 π/2 π −π −π/2 0 π/2 π
x [units of a] x [units of a]
h h
(c1) meron pair (c2) antimeron pair

(b1)
Néel wall

circular meron hyperbolic meron circular antimeron hyperbolic antimeron
(b2)
Bloch wall
FIG. 5: (Color online) (a) The vectorial plots of the pseu-
(c) dospin S projected onto the x-y plane under a pseudo-spin
units of rad] π/π02 rmdoeetnansttiistoyinnqσ((rxa))→.. (−c)σzThanedamσzp→lificσaxt.io(nb)ofTthheetotwpoolokgiincdasl cohfaerlgee-
[↓−π/2
θ
−
↑
θ −3π −2π −π 0 π 2π 3π
x [units of ah] trappingfrequenciesω⊥ 2π 40Hzandωz 2π 200
≈ × ≈ ×
Hz. It is convenient to produce spin-dependent optical
FIG. 4: (Color online) (a) The vectorial plots of the pseu-
lattices with a large lattice spacing by using a CO laser
dospin S projected onto the x-y plane with g˜ = 6000, V0 = 2
operated at a wavelength of 10.6 µm. Under typical
40h¯ω⊥, and κ=0 (a1), κ=4ahω⊥ (a2). The colors ranging
experimental conditions, the s-wave scattering lengths
from bluetoreddescribethevaluesoftheaxialspinSz from
−1 to 1. (b) 3D renderings of N´eel wall and Bloch wall. (c) aij 100aB (aB is the Bohr radius). Based on these
≈
Section views of the relative phase θ↑ −θ↓ along the x axis experimentalparameters,wecancalculatethattheeffec-
with κ=0 (solid line) and κ=4ahω⊥ (dashed line). tive interactionparameter g˜ 6000and the wavevector
ν = a−1, which are consisten≈t with our present calcula-
h
tion.
underapseudo-spinrotation,whichcorrespondingtothe For a given SO coupling strength κ = 4 ¯hω /m,
⊥
state represented in Figs. 3(c1-c6), and the correspond- adjusting the lattice depth from V = k p80 nK to
0 B
ing topological charge density q(r)= 81πǫijS·∂iS×∂jS V0 = kB × 40 nK, then to V0 = kB × 30×nK (kB is
is plotted in Fig. 5(b). One can see that the spin tex- the Boltzmann’s constant), one can directly observe the
ture in Fig. 5(a) represents a lattice composed of meron phasetransitionsfromdensitystripetotriangularvortex
pairs and antimeron pairs [36, 37]. Either a meron pair lattice, then to rectangle vortex lattice by monitoring in
or an antimeron pair has a “circular-hyperbolic” struc- situ the density profile. And for a given lattice depth
ture [See Figs. 5(c1) and 5(c2)], and the only difference V = k 80 nK, adjusting the SO coupling strength
0 B
×
is thattheyhaveexactlyoppositespinorientations. The from κ = 0 to κ = 4 ¯hω /m, one may indirectly ob-
⊥
spatial integral of q(r) indicates that a meron pair just serve the transition opf the spin domain wall from N´eel
carriestopologicalcharge1,whileanantimeronpaircar- walltoBlochwallbydualstateimagingtechnique,which
ries topological charge 1. Previous studies indicated can spatially resolve the relative phase [38].
−
thatstablemeron-pairlatticecanbeobtainedinarotat- Conclusion.—In summary, we have investigated the
ingsystem[33]. Ourresultsshowthatmeron-pairlattice ground-state phase diagram of spin-orbit coupled BECs
can also be stabilized in alternating spin domains by SO in spin-dependent optical lattices. We observe novel
coupling without rotation. spin-orbit coupling effects on alternating spin domains
Experimental proposal.—In real experiments, we produced by the spin-dependent optical lattices. Actu-
can choose a two-level 87Rb BEC system with ally,alternatingspindomainsexistinmanyphysicalsys-
F =1,m =1 and F =1,m = 1 . About N = tems, such as magnetic materials and condensed matter
f f
| i | − i
1.7 105 atomsareconfinedinaharmonictrapwiththe systems [39, 40], thus similar spin-orbit coupling effects
×
5
would be discoveredin those systems. We hope that our (2011).
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We are grateful to Shih-Chuan Gou for helpful dis-
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cussions. This work was supported by the NKBRSFC Rev. Lett.108, 010402 (2012).
under Grants No. 2011CB921502, No. 2012CB821305, [22] B.Ramachandhran,B.Opanchuk,X.J.Liu,H.Pu,P.D.
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