Vortex pairs in a spin-orbit coupled Bose-Einstein condensate

2.1 MB·

by  Masaya Kato

Vortex pairs in a spin-orbit coupled Bose-Einstein condensate
Masaya Kato,1 Xiao-Fei Zhang,2,3 and Hiroki Saito1
1Department of Engineering Science, University of Electro-Communications, Tokyo 182-8585, Japan
2Key Laboratory of Time and Frequency Primary Standards,
National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China
3University of Chinese Academy of Sciences, Beijing 100049, China
(Dated: January 18, 2017)
7
1 Static and dynamic properties of vortices in a two-component Bose-Einstein condensate with
0 Rashba spin-orbit coupling are investigated. The mass current around a vortex core in the plane-
2 wave phase is found to be deformed by the spin-orbit coupling, and this makes the dynamics of
the vortex pairs quite different from those in a scalar Bose-Einstein condensate. The velocity of
n
a vortex-antivortex pair is much smaller than that without spin-orbit coupling, and there exist
a
stationary states. Two vortices with the same circulation move away from each other or unite to
J
form a stationary state.
7
1

I. INTRODUCTION one without SOC. The dynamics of a vortex-vortex pair
]
s with the same circulation are also quite different from
a those without SOC; the vortices move away from each
g Topological excitations in superfluids originate from
other, or they approach each other and unite to form a
- the intertwining between internal and external degrees
t stationary state.
n of freedom in the order parameters. The simplest exam-
This paper is organized as follows. The problem is
a ple is a quantized vortex in a scalar superfluid, in which
u formulated in Sec. II. The static properties of a single
the complex order parameter with the U(1) manifold
q vortex are studied in Sec. III. The dynamics of a vortex-
winds around the vortex core, producing azimuthal su-
. antivortexpairandthoseofavortex-vortexpairwiththe
t perflow[1,2]. Fortheorderparameterswithspindegrees
a samecirculationareinvestigatedinSecs.IVAand IVB,
m of freedom, a rich variety of topological excitations are
respectively. Conclusions are presented in Sec. V.
possible; these include skyrmions [3, 4], monopoles [5],
-
d half-quantum vortices [6], and knots [7]. Because of the
n closerelationshipbetweenthe spinandmotionaldegrees
II. FORMULATION OF THE PROBLEM
o of freedom in the topologicalexcitations, we expect that
c
their static and dynamic properties are significantly al-
[ We consider a two-dimensional (2D) quasispin-1/2
tered if there exists coupling between them, that is, if
BEC in a uniform systemwith Rashba SOC.Within the
1 there exists spin-orbit coupling (SOC).
v framework of mean-field theory, the system can be de-
1 Recently, Bose-Einstein condensates (BECs) of ultra- scribed by the order parameter Ψ(r) = [ψ (r),ψ (r)]T,
1 2
9 cold atomic gases with SOC have been realized exper- where T denotes the transpose. The kinetic and SOC
5 imentally [8–12]; in these experiments, the atomic spin energies are given by
4 orquasispinwascoupledwiththeatomicmomentumus-
0 ing Raman laser beams. Numerous theoretical studies p2 ~k
. E [Ψ]= drΨ† 0p σ Ψ, (1)
1 have been performed to evaluate the static properties of 0 2m − m · ⊥
0 topologicalexcitations in spin-orbit(SO) coupledBECs, Z (cid:18) (cid:19)
7 e.g., vortex arrays [13], vortices in rotating systems [14– where m is the atomic mass, k is the strength of the
0
1 17], half-quantum vortices [18, 19], skyrmions [20–24], SOC, and σ = (σ ,σ ) are the 2 2 Pauli matrices.
⊥ x y
: ×
v topologicalspintextures[25–29],dipole-inducedtopolog- The s-wave contact interaction energy is written as
i icalstructures[30–32],andsolitonswithvortices[33,34].
X
However,therehavebeen onlya few studies ontheir dy- g 2
ar namics in SO-coupled BECs. The dynamics of a single Eint[Ψ]= dr 20 |ψj|4+g12|ψ1|2|ψ2|2, (2)
quantized vortex in a harmonic trap was considered in Z j=1
X
Refs. [35, 36].  
where g and g are the intra- and inter-component in-
In this paper, we investigate the dynamics of a quan- 0 12
teraction coefficients, respectively. The total energy is
tized vortex pair in a quasispin-1/2 BEC with Rashba
given by
SOC.Whenasinglyquantizedvortexiscreatedinauni-
formplane-wavestate,thephasedistributionaroundthe E[Ψ]=E [Ψ]+E [Ψ]. (3)
0 int
vortex core is significantly altered by the SOC; this in-
dicates that the mass current around the vortex is quite In this paper, we consider an infinite system in which
different from that without SOC and affects the dynam- the atomic density Ψ†Ψ far from vortices is a constant,
ics of a vortex pair. As a result, a vortex-antivortexpair n . In the following, we normalize the length, veloc-
0
will be stationary or will travel much more slowly than ity, time, and energy by the healing length ~/√mg0n0,
2
the sound velocity g n /m, the characteristic time
scale ~/(g n ), and the0ch0emical potential g n . The (a) density 1 (b) phase
dimensionl0ess0 couplepd Gross-Pitaevskii (GP) e0qu0ations, 5 5
i∂Ψ/∂t=δE[Ψ]/δΨ, have the form 0 0 0
-5 -5
∂ψ 1 0
i ∂t1 =−2∇2ψ1+iκ∂−ψ2+ |ψ1|2+γ|ψ2|2 ψ1, (4a) (c) spin 1 (cid:1)(cid:7)(cid:2)(d) dependence of
4
(cid:0) (cid:1)
(cid:1)(cid:6)
∂ψ 1 2
i 2 = ∇2ψ +iκ∂ ψ + γ ψ 2+ ψ 2 ψ , (4b) (cid:1)(cid:5)
2 + 1 1 2 2
∂t −2 | | | | 0 0
(cid:1)(cid:4)
(cid:0) (cid:1)
where ∂± = ∂/∂x±i∂/∂y, κ = ~k0/√mg0n0, and the -2 (cid:1)(cid:3)
ratiobetweentheinter-andintra-componentinteractions
is γ = g /g . The ground state is the plane-wave state -4 (cid:1)(cid:2)
12 0 -1 (cid:1)(cid:2) (cid:1)(cid:2)(cid:8)(cid:4) (cid:1)(cid:2)(cid:8)(cid:6) (cid:1)(cid:7)(cid:8)(cid:3) (cid:1)(cid:7)(cid:8)(cid:5)
forγ <1andthestripestateforγ >1[13],whichbreaks
the rotational symmetry of the system. In the following
discussion,we willfocus onthe miscible case, γ <1, and
FIG.1. (a)-(c)Stablestationarystateofasinglevortexwith
the ground state is given by the plane-wave state,
counterclockwise circulation for κ = 1 and γ = 0.8. Pan-
els (a) and (b) show the density and phase profiles of each
1 eiκx
Ψ(r)= , (5) component, where the unit of density is n0. In (b), δ is the
√2 eiκx distance between the phase defects in the two components.
(cid:18) (cid:19)
Panel(c)showsthespindistributionS(r)definedinEq.(7).
where the wavevector is chosento be in the x direction. The arrows indicate the transverse direction of the spin vec-
The velocity field is useful for understanding the dy- tor, and the background color indicates the value of Sz. The
namics of vortices. From the equation of continuity dashed square region in (a) is shown magnified in (c). (d) κ
∂ρ/∂t+∇ (ρv)=0withatomicdensityρ= ψ 2+ ψ 2, dependence of the vortex shift δ. The solid curve shows 1/κ
1 2
· | | | |
we obtain the velocity field as for comparison.
1
v (r)= Ψ†(r) Ψ(r) Ψ(r)T Ψ∗(r)
ξ 2iρ(r) ∇ξ − ∇ξ where Φ(r) = tan−1(y/x). After sufficiently long
κS (r(cid:2)), (ξ =x,y) (cid:3)(6) imaginary-time propagation, we obtain the stable sta-
ξ
−
tionary state, as shown in Fig. 1. Figures 1(a) and 1(b)
where
show the density and phase distributions of the station-
1 arystate. We notethat the phasedefect incomponent1
S (r)= Ψ(r)†σ Ψ(r) (ξ =x,y,z) (7)
ξ ρ(r) ξ (2) is shifted in the +y ( y) direction. We define the
−
distance between the phase defects as δ. The vortex
is the pseudospin density. The first term in Eq. (6) cor- core in each component is occupied by the other com-
responds to the canonical part related to the superfluid ponent. This structure can therefore be regarded as a
velocity, and the second term corresponds to the gauge pair of half-quantum vortices; nevertheless, we will refer
part induced by the SOC. The velocity field vanishes for to it as a “single vortex” in this paper. In the absence
thevortex-freegroundstateinEq.(5),sincethefirstand of SOC, such a pair of half-quantum vortices repel each
second terms in Eq. (6) cancel each other. other and cannot form a stationary state [37]. A similar
We numerically solve Eq. (4) by the pseudospectral structure is also found in a one-dimensional SOC sys-
method with the fourth-order Runge-Kutta scheme. In tem[36]. Figure1(c)showsthespindistribution,andwe
the imaginary-timepropagation,on the left-hand side of can see a spin vortex near the origin. The dependence
Eq. (4), i is replaced with 1. The numerical space is of the vortex shift δ on the SOC strength κ is shown in
−
taken to be 400 400, which is sufficiently large, and Fig. 1(d), which implies δ 1/κ.
× ≃
the effect of the periodic boundary condition can be ne- Figure 2 shows the velocity field v(r) of the single-
glected. vortexstate. Thevelocityfieldisgreatlydeformedbythe
SOC,comparedwiththerotationallysymmetricvelocity
field without SOC. We note that the deformation of the
III. SINGLE VORTEX velocity field extends over a wide range, and the upper
region (y > 10) exhibits a uniformly leftward velocity
Webeginwithasinglevortexstate,inwhicheachcom- field, while∼the lower region (y < 10) is rightward. In
ponent contains a singly quantized vortex. The initial these regions, v < 0.01, which∼is much smaller than
state of the imaginary-time propagationis that without S|O|C∼, v = 1/r. This effect of SOC is
| |
also seen in Fig. 1(b), where the phase in the upper and
1 ei[Φ(r)+κx] lowerregionsisalmost eiκx,i.e.,the 2π phaserotation
Ψ(r)= √2 ei[Φ(r)+κx] , (8) aroundthevortexcorei∝sstronglycompressedaroundthe
(cid:18) (cid:19)
3
Substitution of this wave function into Eq. (1) yields
1 1 ∂χ
E = dr (∇χ)2+ (∇φ)2+κ
0
2 8 ∂x
Z (cid:20)
∂χ ∂χ
κ +κ cosφ+κ sinφ , (11)
− ∂x ∂y
(cid:18) (cid:19) (cid:21)
where χ = (Φ +Φ )/2, φ = Φ Φ , and the constant
1 2 1 2
−
term is neglected. The first and second lines in Eq. (11)
correspondtothekineticandSOCenergies,respectively.
From the numerical results that the cores are shifted by
δ and that the 2π phase rotation around the vortex core
is compressed in the y direction, the phases in Eq. (10)
are assumed to be
y δ /2
Φ (r)=tan−1 λ − y , (12a)
1
x δ /2
(cid:18) − x (cid:19)
y+δ /2
Φ (r)=tan−1 λ y , (12b)
2
x+δ /2
(cid:18) x (cid:19)
whereλandδ arevariationalparameters. Wesubstitute
these phases into Eq. (11) and integrate with respect to
θ. Because of the complicated structure near the vortex
cores,weconsidertheregioninwhichr 1. Theenergy
FIG. 2. Velocity field v(r) of the single-vortex state for is ≫
γ = 0.8 and (a) κ = 0.5 and (b) κ = 1. The arrows indi-
cate the directions of the velocity, and the background color π
indicates the value of |v|. The regions in the dashed squares E0 = rdr(−2πκ2+ 2λr2
are magnified in the right-hand panels. The red arrows indi- Z
cate thedirection of thevortex. πκ2 1 2
+ δ2+λ2 δ +O(r−3) , (13)
2r2 " x (cid:18) y − κ(cid:19) # )
x-axis. The velocity field near the vortex core exhibits which is minimized by δx = 0 and δy = 1/κ. Thus, the
complicated structures containing multiple circulations, energyisloweredbythedisplacementofthevortexcores
as shown in the right-hand panels in Fig. 2. intheydirection,andthedisplacementδy isestimatedto
be 1/κ; this is in good agreementwith the numerical re-
Due to the symmetry of the GP equation in Eq. (4),
sultsshowninFig.1(d). TheenergyinEq.(13)decreases
the single-vortex state with clockwise circulation can be
as λ increases, and this accounts for the compressed 2π
obtained from that with counterclockwise circulation by
phase rotation. A better variational wave function will
the following transformation:
allow us to determine the value of λ. We note that the
term δ in Eq. (13) originates from the last term in
y
∝
ψ1(x,y) ψ2(x, y), ψ2(x,y) ψ1(x, y). (9) theintegrandofEq.(11),whichthusplaysanimportant
→ − → −
role in the vortex deformation due to the SOC.
Bythistransformation,thewindingnumberofthevortex
is inverted without changing the direction of the plane
IV. VORTEX PAIR
wave eiκx. Applying the transformation to the state
shown in Fig. 1, we find that the vortex core in com-
First,forclarity,wedefinethepositionsofthevortices
ponent 1 (2) shifts in the +y ( y) direction also for
− andthedistancesbetweenthem,asshowninFig.3. The
the clockwise vortex. The velocity field and the pseu-
position of the phase defect of the jth vortex in compo-
dospin density are transformed as v (x,y) v (x, y),
x x
→ − nent i is denoted by (x ,y ). As shown in Fig. 1(b),
v (x,y) v (x, y), S (x,y) S (x, y), and ij ij
y y x x
→ − − → − in each vortex, the cores in the two components are
S (x,y) S (x, y).
y y
→− − shifted by δ in the y direction, and then x = x and
1j 2j
For a better understanding of the numerical results,
y y = δ. The position of the single vortex is de-
1j 2j
we perform variational analysis. The variational wave −
fined by (x ,y ) = ((x + x )/2,(y + y )/2). For
j j 1j 2j 1j 2j
function is
a vortex pair, the index j is taken in such a way that
y > y . The distance between the vortices is defined
1 2
1 ei[Φ1(r)+κx] by (d ,d ) = (x x ,y y ) and d = (d2 +d2)1/2.
Ψ(r)= √2 ei[Φ2(r)+κx] . (10) The cxentyer of th1e−vor2tex1p−air2is defined byx(xc,yyc) =
(cid:18) (cid:19)
4
vortex 1
vortex pair
vortex shift
single vortex
vortex of
vortex 2
vortex of
plane wave
center of vortex pair
FIG.3. Schematicillustration ofthevortexpairforh1,−1i.
Theblackpointsarethevortexpositions(x1,y1)and(x2,y2),
whicharedefinedasthemidpointsbetweenthephasedefects
in the two components. The open circle is the center of the
vortex pair (xc,yc)=((x1+x2)/2,(y1+y2)/2).
FIG.4. Stablestationarystatesofvortex-antivortexpairsfor
κ=1andγ =0.8. Thewindingcombinationsare(a)h1,−1i
and (b) h−1,1i. Panels (a1) and (b1) show the density pro-
((x +x )/2,(y +y )/2). The winding numbers of the
1 2 1 2 files, and (a2) and (b2) show the spin distributions. The
first and second vortices are denoted by n ,n . In the
1 2 regions indicated by dashed squares in (a1) and (b1) corre-
h i
following subsections, we will consider the vortex pairs spondto(a2)and(b2),respectively. Thearrowsindicatethe
1, 1 and 1, 1 , which we call vortex-antivortex directions of the transverse spin vector, and the background
h± ∓ i h± ± i
pairs and vortex-vortexpairs, respectively. color indicates thevalue of Sz.
A. Vortex-antivortex pair (cid:1)(cid:2)(cid:3)(cid:6)(cid:9)(cid:1) (a) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:10) (b)
(cid:1)(cid:2)(cid:3)(cid:6)(cid:1)(cid:5) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:9)
gy (cid:1)(cid:2)(cid:3)(cid:6)(cid:1)(cid:8) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:8)
IntheabsenceofSOC,avortex-antivortexpairtravels er (cid:1)(cid:2)(cid:3)(cid:6)(cid:1)(cid:3) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:7)
n
ataconstantvelocityorisannihilated[38,39]. Avortex- E (cid:1)(cid:2)(cid:3)(cid:6)(cid:1)(cid:7) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:5)
(cid:1)(cid:2)(cid:3)(cid:6)(cid:1)(cid:1) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)
antivortexpairisstationaryonlyinatrappotential[40],
(cid:1)(cid:2)(cid:3)(cid:3)(cid:4)(cid:5) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:3)
and there is no stationary state in a uniform system. (cid:10)(cid:9)(cid:1) (cid:10)(cid:6) (cid:11)(cid:1) (cid:11)(cid:6) (cid:11)(cid:9)(cid:1) (cid:11)(cid:12)(cid:1) (cid:11)(cid:5) (cid:13)(cid:1) (cid:13)(cid:5) (cid:13)(cid:12)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:9) (c) barrier (cid:1)(cid:2)(cid:3)(cid:4)(cid:4)(cid:4)(cid:9) (d)
In the presence of the SOC, our numerical results (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:4) (cid:1)(cid:2)(cid:3)(cid:4)(cid:4)(cid:4)(cid:8)
show that stable stationary vortex-antivortex pairs can ergy(cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:8) (cid:1)(cid:2)(cid:3)(cid:4)(cid:4)(cid:4)(cid:7)
be formed with a proper choice of the distance between En(cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:7) (cid:1)(cid:2)(cid:3)(cid:4)(cid:4)(cid:4)(cid:6)
vortices d; an example is shown in Fig. 4. We prepare (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:5) local minumum (cid:1)(cid:2)(cid:3)(cid:4)(cid:4)(cid:4)(cid:4)
the initial state in Eq. (8) with (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:6) (cid:1)(cid:2)(cid:3)(cid:4)(cid:4)(cid:4)(cid:5)
(cid:1)(cid:10)(cid:2) (cid:1)(cid:8) (cid:11)(cid:2) (cid:11)(cid:8) (cid:11)(cid:10)(cid:2) (cid:1)(cid:10)(cid:2) (cid:1)(cid:6) (cid:11)(cid:2) (cid:11)(cid:6) (cid:11)(cid:10)(cid:2)
2
y y
Φ(r)= n tan−1 − j, (14)
j FIG.5. Totalenergyofavortex-antivortexpairasafunction
x x
j
Xj=1 − ofthedistancebetweenvorticesdy for(a)κ=0,(b)κ=0.5,
(c) κ = 1, and (d) κ = 1.5. In (b)-(d), local energy minima
where,forthisexample,n = 1,n = 1,x =x =0, appear; these correspond to the stationary states shown in
1 2 1 2
± ∓
and y = y = d /2 with d being the initial distance Fig. 4.
1 2 i i
−
between vortices. From this initial state, the imaginary-
time propagation is performed sufficiently. The sta-
tionary state is always reached if the initial distance is
d >20. ItcanbeseeninFig.4thatthedistancebetween with phases
i
vor∼ticesinthestationarystateisd 4.1for 1, 1 and
y
≃ h − i
d 6.5 for 1,1 .
y y δ/2 y δ/2 d
T≃o understha−nd thie stabilization mechanismof the sta- Φ1(r)=tan−1 − tan−1 − − y ,
x − x
tionaryvortex-antivortexpairs,wecalculatethetotalen- (cid:18) (cid:19) (cid:18) (cid:19)
(16a)
ergy using a model function given by
y+δ/2 y+δ/2 d
Φ (r)=tan−1 tan−1 − y ,
ρ (r)ei[Φ1(r)+κx] 2 x − x
Ψ(r)= 1 , (15) (cid:18) (cid:19) (cid:18) (cid:19)
ρ (r)ei[Φ2(r)+κx] (16b)
(cid:18)p 2 (cid:19)
p
5
(a)
center of vortex pair
(a) (b) (cid:7)(cid:2)(cid:3)(cid:2)(cid:5)
5 (cid:7)(cid:2)(cid:3)(cid:2)(cid:6)
(cid:7)(cid:2)
0
(cid:1)(cid:2)(cid:3)(cid:2)(cid:6)
-5 (cid:1)(cid:2)(cid:3)(cid:2)(cid:5)
(cid:1)(cid:2)(cid:3)(cid:2)(cid:4)
(cid:7)(cid:2) (cid:7)(cid:8) (cid:7)(cid:9)(cid:2) (cid:7)(cid:9)(cid:8) (cid:7)(cid:6)(cid:2)
(c) (d)
(b)
5
(cid:7)(cid:2)(cid:3)(cid:2)(cid:5)
(cid:7)(cid:2)(cid:3)(cid:2)(cid:6)
0
(cid:7)(cid:2)
-5
(cid:1)(cid:2)(cid:3)(cid:2)(cid:6)
(cid:1)(cid:2)(cid:3)(cid:2)(cid:5)
unstable
(cid:1)(cid:2)(cid:3)(cid:2)(cid:4)
(cid:7)(cid:2) (cid:7)(cid:8) (cid:7)(cid:9)(cid:2) (cid:7)(cid:9)(cid:8) (cid:7)(cid:6)(cid:2)
FIG.6. Trajectoriesofvortex-antivortexpairsforκ=1and
γ = 0.8. Red and blue circles indicate the positions of the
vortexcoresinψ1 andψ2,respectively. Thedirectionsofthe
circulationsofthevorticesareindicatedbyblackarrows. The FIG. 7. Velocity vx versus the distance dy of a vertically
initial distancebetween vortices isd=10. Black circles indi- aligned vortex-antivortex pair for (a) κ = 1 and (b) κ = 0.5
cate the center of the vortex pairs (xc,yc) at t=0, 400, and with γ = 0.8. The red and blue plots are for h1,−1i and
800, and green arrows indicate the direction of motion. See h−1,1i, respectively. The configurations of the vortex pairs
theSupplementalMaterial for movies of thedynamics [41]. areillustratedintheinsets. In(b),thereisanunstableregion
(see text).
and densities
follows. We first prepare the state in Eq. (8) with the
1
ρ (r)= (r)ν(x,y δ/2)ν(x,y δ/2 d ), phase in Eq.(14), andthen we allow the imaginary-time
1 y
2N − − − evolution for a short period (typically, t 80). From
≃
(17a) this state, the real-time evolution begins. Figures 6(a)
1 and 6(b) show the dynamics of the vertically aligned
ρ (r)= (r)ν(x,y+δ/2)ν(x,y+δ/2 d ),
2 2N − y vortex pair; the distance d 10 is larger than that
≃
(17b) of the stationary states shown in Fig. 4. The vortex-
antivortex pair moves in the x and +x directions at
−
where (r)isthenormalizationfactortoensureρ (r)+ constant velocity with a fixed distance between vortices.
1
ρ (r)=N1 and ν(x,y)=(x2+y2)/(x2+y2+w2). Thesedirectionsforthepropagationagreewiththosefor
2
We set δ = 1/κ, and from the numerical results, the a scalar BEC. However, the velocities v 0.006 in
x
≃ −
radius of the vortex w is estimated to be 2w 1/κ. Fig. 6(a) and v 0.011 in Fig. 6(b) are much slower
x
≃ ≃
Figure5showsthetotalenergyasafunctionofd ;thisis than v =1/d 0.1, which is that seen in a scalar BEC
y x
≃
obtained by substituting Eq.(15) into Eq. (3). It can be for the same d. Figures 6(c) and 6(d) show the cases of
seen in Fig. 5 that localenergy minima appear on either oblique and horizontal alignments. The propagation di-
sideoftheglobalminimumandformtheenergybarriers, rectionsofthesevortexpairsaredifferentfromthoseina
that stabilize the vortex-antivortex pair. We note that scalarBEC.Thiscanbeunderstoodbyinspectingtheve-
without SOC, there are no such barriers, as can be seen locityfieldshowninFig.2. Forexample,onthenegative
in Fig. 5(a) for κ = 0. We also note that the barriers x-axisintheleft-handpanelofFig.2(b),thevelocityfield
do not appear for uniform densities ρ = ρ = 1/2, and is towards the lower right, which indicates that a vortex
1 2
theinhomogeneousdensitiesofEq.(17)arenecessaryfor locatedontheleft-handsideofthecounterclockwisevor-
the barriersto form. Hence, we conclude that this is the tex will feel a mass current in this direction. Similarly,
combined effect of SOC and the nonlinear interaction. a vortex located on the right-hand side of the clockwise
We now turn our attention to the dynamics of the vortex will feel a mass current towards the lower right;
vortex-antivortex pair. Figure 6 shows the trajectories this results in the dynamics shown in Fig. 6(d).
of the vortex cores,where the initial state is preparedas Figure7showsthe velocityv ofthe verticallyaligned
x
6
center of vortex pair
(a) (b)
5
0
-5
(c) (d)
5
0
-5
FIG.9. Trajectories ofvortex-vortexpairsforκ=1andγ =
0.8. The initial vortex distance is (a) d = 8.0, (b) d = 11.6,
(c)d=18.0, and(d)d=12.9. Redandbluecirclesshowthe
FIG. 8. Time evolution of the (a) density and (b) phase of
the unstable vortex-antivortex pair for κ = 0.5 and γ = 0.8, positionsofthevortexcoresinψ1 andψ2,respectively. Black
arrows indicate the direction of circulation. Black circles are
where the vertical gauges indicate the distance dy between
the vortex cores. See the Supplemental Material for a movie the center of the vortex pairs (xc,yc) at t = 0, 400, and
800, andgreen arrowsshowthedirectionsofmotion. Seethe
of the dynamics[41].
SupplementalMaterial for a movie of thedynamics [41].
vortex pair (i.e., d =0) as a function of the vortex dis-
x
phase in Eq. (14) with d >10. There is no stable state
tance d , which is obtained by a method similar to that y
y
for d = 9.6 according to Fig. 7(b). As the vortex pair
used to obtain Fig. 6. Such vortex pairs always travel y
travels in the x direction, the distance d decreases,
in the x direction. The dy dependence of the velocity − y
± and eventually the pair settles into a stable state with
is quite different from that in a scalar BEC. For κ = 1
d 5.7; the excess energy is released from the vortex
(Fig. 7(a)), the velocity vx of the 1, 1 pair (red cir- y ≃
h − i pair as density and spin waves.
cles) changes from negative to positive as d increases,
y
and v = 0 at d 5, which corresponds to the sta-
x y
≃
tionary state seen in Fig. 4(a). The velocity v of the
x
1,1 pair (blue circles) also crosses the v =0 axis at B. Vortex-vortex pair
x
h− i
d 7, which corresponds to the stationary state seen
y
in F≃ig. 4(b). For 6 < dy < 8, the velocity changes from Ina scalarBEC,twoquantizedvorticeswiththe same
negative to positive∼and f∼rom positive to negative as dy circulationmove around eachother. In contrast,the dy-
increases. For a relatively large distance between vor- namics of vortex-vortexpairs with SOC are significantly
tices (d > 10), the propagation directions are the same different from those in a scalar BEC. Figure 9 shows the
as those∼of a scalar BEC, but the dy dependence of vx trajectories of vortices for the 1,1 pair, where the ini-
h i
is weak; this can be understood from the fact that the tial state is prepared by the same method as in Fig. 6.
velocity field is almost uniform far from the vortex core, When the initial positions are those shown in Fig. 9(a),
as shown in Fig. 2. The velocity vx is always smaller theymoveawayfromeachother. InthecaseofFig.9(b),
| |
than that in a scalar BEC for both 1, 1 and 1,1 the two vortices pass each other. The dynamics shown
h − i h− i
pairs. There is no stable vortex-antivortexpair for small inFigs.9(c)and9(d)aremoreinteresting. The twovor-
dy; the vortices are unstable against pair annihilation. tices approacheachother and unite to forma stationary
The 1,1 pair exhibits interesting dynamics when κ state,andtheexcessenergyisreleasedaswaves. There-
h− i
issmall. As showninFig.7(b), thereisnostable 1,1 sultantstationarystateisstableandremainsatrest,and
pair in the region 5.5 < d < 9.6. Figure 8 shohw−s thei the two vortices lie in a line perpendicular to the plane
y
dynamicsofthe 1,1 ∼pairw∼iththeinitialdistanced = wave. In all cases, the center of the pair initially moves
y
h− i
9.6,where the initial state is preparedby the imaginary- in the direction of +x. The transformation in Eq. (9)
time propagation for a short duration from the initial gives the dynamics of 1, 1 .
h− − i
7
plane-wavestate. We found that the static and dynamic
(a) (b)
propertiesofvorticesaresignificantlydifferentfromthose
of a scalar BEC.
For a single vortex state, we found that the vortex
cores in two components are shifted in the y directions
±
by 1/κ (Fig. 1). We also found that the phase distri-
≃
bution and velocity field around the vortex are greatly
deformed compared with those of a scalar BEC (Figs. 1
and 2), which affects the dynamics of the vortex pairs.
center of vortex pair
plane wave Thevortex-antivortexpairshavestablestationarystates
velocity of center
at rest (Fig. 4), and this is in marked contrast to the
relative velocity
vortex-antivortex pairs in a scalar BEC, which always
travel. The stationary states can be explained by varia-
tional analysis (Fig. 5). Other than when in a station-
FIG. 10. Schematic illustration of the dynamics of (a)
ary state, the vortex-antivortexpair travels ata velocity
vortex-antivortex pairs and (b) vortex-vortex pairs when the
much slower than that for a scalar BEC with the same
distance is d = 10. The red arrows indicate the direction
of the velocity of (xc,yc), when one vortex is located at the vortex distance. The dependence of the velocity and
origin. In (a), the relative position of the vortices remains moving direction on the vortex location is also quite dif-
nearly constant. In (b), the relative velocity (the velocity of ferentfromthat in the caseof a scalarBEC (Figs.6 and
the vortex in the moving frame in which the other vortex is 7). The vortex-vortex pair exhibits interesting dynam-
fixed tothe origin) is indicated by bluearrows. ics: the vorticespass andmove awayfromeachother,or
approacheachother and combine into a stationarystate
(Fig. 9).
Figures 10(a) and 10(b) summarize the directions of In experiments, the vortex states shown in this pa-
the vortexmotion when d 10 for the 1, 1 and 1,1 per may be produced by the phase imprinting tech-
≃ h − i h i
pairs,respectively. InFig.10(a),themotionofthecenter nique [42, 43] and the ensuing relaxation. The dynam-
of the vortex-antivortex pair (xc,yc) is indicated by the ics of vortices can be observed by the destructive imag-
red arrows, and the relative position of the two vortices ing [44] or the real-time imaging [45]. We hope that our
is nearly constant. In Fig. 10(b), the relative motion of numericalresults presented in this paper can provide in-
thevortex-vortexpairisindicatedbythebluearrow,and sight into a range of topics in the nonlinear dynamics of
the centerofthe pairalwaysshiftsinthedirectionofthe SO-coupled BECs.
plane wave.
ACKNOWLEDGMENTS
V. CONCLUSIONS
This work was supported by JSPS KAKENHI Grant
We have investigated the behaviors of quantized vor- NumbersJP16K05505,JP26400414,andJP25103007,by
tices in quasispin-1/2BECswith RashbaSO coupling in the NMFSEID under Grant No. 61127901, and by the
a uniform 2D system, where the atomic interactions sat- Youth Innovation Promotion Association of CAS under
isfy the miscible condition and the ground state is the Grant No. 2015334.
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