K. Machida et al- Mermin–Ho Vortex in Spinor Bose–Einstein Condensates under Rotation

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1083

Laser Physics, Vol. 13, No. 8, 2003, pp. 1083–1090.Original Text Copyright © 2003 by Astro, Ltd.Copyright © 2003 by MAIK “Nauka /Interperiodica” (Russia).

1. INTRODUCTIONSince optical trapping has succeeded in producing

Bose–Einstein condensates (BEC) with internaldegrees of freedom [1, 2], much studies on the so-called spinor BEC have been done both experimentallyand theoretically. The spinor BEC is distinguished fromthe scalar BEC trapped magnetically [3–5] where theinternal degrees of freedom due to the atomic hypernestates are frozen. Considering its spin freedom, it isexpected that a variety of topological defects may exist;in particular, vortices stabilized under rotation mayhave a quite different structure from that in the scalarBEC [6–13].

So far two groups produce the spinor BEC: 23

Nawith the hyperne states F

= 1 and F

= 2 and 87

Rb with

F

= 1 and F

= 2 in an optical trap [1, 2]. We conne ourfollowing discussion to F

= 1. In 23

Na, the spin-depen-dent interaction g

s

> 0 is known to be antiferromagneticwhile in 87

Rb it is estimated to be g

s

< 0 [14]. Thus thesetwo systems provide us both types of the interactionparameter.

In the scalar BEC where the order parameter isdescribed by one component, exhaustive studies onvortices are performed [15–21]: (1) conrmation of quantization of circulation, (2) dynamical process of single vortex nucleation, (3) observation of a hexagonal

vortex lattice consisting of many single quantized vor-tices, (4) nding of curved vortices, etc.Here, we present our theoretical efforts to investi-

gate possible vortex structures in order to further accel-erate theoretical and experimental studies on spinorBEC.

The paper is arranged as follows. First, we give aformulation of the problem within Bogoliubov theoryextended to the spinor BEC (

F

= 1) described by three-component order parameters. We examine several com-

peting factors which govern the stability of possiblevortex structures, including axisymmetric and nonaxi-symmetric vortices. Mermin–Ho (MH) and Anderson–Toulouse (AT) vortices [22, 23], which are predicted tobe realized in superuid 3

He, whose order parameter isalso multicomponent and has not been identied yet[24], are found to be stabilized in a rather large area of the phase diagram only in the ferromagnetic case underexternal rotation. We illustrate an example of the vortexarray under further high external rotation where fourMH vortices are stabilized. The phase diagrams bothfor antiferromagnetic and ferromagnetic interactionsare determined in the plane of the total magnetization of the system and the external rotation frequency.

2. EXTENDED BOGOLIUBOV THEORY

2.1. Formulation

We focus on the spinor BEC with internal degrees of freedom F

= 1 for both ferromagnetic and antiferro-magnetic cases. We start with the standard Hamiltonianby Ohmi and Machida [6] and Ho [7]:

(1)

Here,

(2)

ˆ = r Ψi†

h r( ) μi–{ }Ψ jδij

ij

∑ gn

2----- Ψi

†Ψ j†Ψ jΨi

ij

∑+d ∫

+g s

2---- Ψi

†Ψ j†

F ˆ α( )ik F ˆ α( ) jl Ψk Ψl

ijkl∑

α∑ .

h r( )2∇

2

2m------------– V r( ) W–+ r p×( )⋅=

BOSE–EINSTEIN CONDENSATIONOF TRAPPED ATOMS

Mermin–Ho Vortex in Spinor Bose–Einstein Condensatesunder Rotation

K. Machida

1

, T. Mizushima

1

, T. Kita

2

, and T. Isoshima

3

1

Department of Physics, Okayama University, Okayama 700-8530, Japan

e-mail: [email protected]

2

Division of Physics, Hokkaido University, Sapporo 060-0810, Japan

3

Materials Physics Laboratory, Helsinki University of Technology,P.O. Box 2200 (Technical Physics), FIN-02015 HUT, Finland

Received October 4, 2002

Abstract

—It is shown theoretically that the Mermin–Ho vortex is stable in a spinor Bose–Einstein condensatewithin extended Bogoliubov theory. The phase diagrams for ferromagnetic and antiferromagnetic spinor BECin a plane of the magnetization of the system and external rotation frequency are calculated. There are severaltypes of vortices with axisymmetry and nonaxisymmetry. Multiple Mermin–Ho vortex conguration is shownto be stable under higher rotation. We also discuss how to create and how to detect the Mermin–Ho vortex.



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LASER PHYSICS

Vol. 13

No. 8

2003

MACHIDA et al

.

is a one-body Hamiltonian. The quantity V

(

r

) =

(2

πν

r

)

2

(

x

2

+ y

2

) is the external connement poten-

tial, such as an optical potential. The scattering lengths

a

0

and a

2

characterize collisions between atoms throughthe total spin 0 and 2 channels, respectively, g

n

=

is the interaction strength through the

“density” channel, and g

s

= is that through

the “spin” channel. The subscripts α

= (

x

, y

, z

) and i

, j

,

k

, l

= (0, ±

1) correspond to the above three species. Thechemical potentials for the three components μ

i

(

i

= 0,

±

1) satisfy μ

1

– μ

0

= μ

0

– μ

–1

. We introduce μ

= μ

0

and

μ

' = μ

1 – μ0. (α = x, y, z) is the spin matrices and canbe expressed as

(3)

Following the standard procedure, the extendedGross–Pitaevskii (GP) equation in rotation frame isobtained as

(4)

Here, we take the external rotation as W = andassume uniformity along the z direction.

We can also dene the “local” stability, whichmeans the linear stability for a small perturbation. Thisis done by solving the extended Bogoliubov equationsto the three components under the axisymmetric situa-

tion [25, 26]:

(5)

(6)

where

12---m

4π 2

m------------

a 0 2a 2+

3-------------------

4π 2

m------------

a 0 a 0–3

----------------

F ˆ α

F ˆ

x

1

2-------

0 1 0

1 0 10 1 0⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞

, F ˆ

y

i

2-------

0 1– 0

1 0 1–0 1 0⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞

,= =

F ˆ z

1 0 00 0 00 0 1–⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞

.=

h r( ) μi– gn φl 2

l∑+⎩ ⎭⎨ ⎬⎧ ⎫δij

+ g s F ˆ α( )ij φk * F ˆ α( )kl φl

kl∑

α∑ φ j 0.=

Ωz

A ij uq r j,( ) B ij v q r j,( )–{ } j∑ εqu q r i,( ),=

B ij*uq r j,( ) A ij* v q r j,( )–{ } j∑ εq v q r i,( ),=

(7)

uq(r , i ) and v q(r , i ) are the q th eigenfunctions with thespin i, and εq corresponds to the q th eigenvalue.

2.2. Calculated SystemThe actual calculations are carried out by assuming

uniform density along the z axis, namely, in the cylin-drical symmetric system. Then, the spinor order param-eter and quasiparticle wave functions can be written bythis symmetry reason as

(8)

We have performed an extensive search to nd sta-ble vortices, starting with various vortex congura-tions, covering a wide range of the ferromagnetic andthe antiferromagnetic interaction strength, gs / gn =−0.2–0.2, and examining various axisymmetric andnonaxisymmetric vortices (see [25, 26] for the classi-cation of possible vortices in the axisymmetric case).

We use the following parameters: the mass of a87

Rbatom m = 1.44 × 10–25 kg, the trapping frequency νr =200 Hz, and the particle number per unit length alongthe z axis n z = 2.0 ×103 / μm. The results displayed hereare for gs / gn = –0.02 (ferromagnetic case) and gs / gn =0.02 (antiferromagnetic case). The external rotationfrequency Ω is normalized by the harmonic trap fre-quency.

3. COMPETING EFFECTSIn order to nd the most stable vortex conguration

under the given total magnetization M / N ( N is the totalnumber) and external rotation Ω, we must take intoaccount three mutually competing effects:

(1) The spin-dependent interaction energy is written

as – gs |2φ1φ–1 – |2. For the antiferromagneticcase gs > 0, the so-called polar state ( φ1, φ0, φ–1) = (0, 1, 0)is stable in a uniform case. For the ferromagnetic casegs < 0, the full polarized state (1, 0, 0) is stabilized [6, 7].

(2) The harmonic conning potential which givesthe maximum density at the potential minimum isimportant when placing the vortex center; if the vortex

A ij h r( )δij μiδij– g n φk 2δij φiφ j*+

k ∑⎩ ⎭⎨ ⎬⎧ ⎫

+=

+ g s F ˆ α( )ij F ˆ α( )kl φk *φl F ˆ α( )il F ˆ α( )kj φk *φl+[ ],kl∑

α∑

B ij gnφiφ j g s F ˆ α( )ik φk F ˆ α( ) jl φl[ ],kl∑

α∑+=

φ j r( ) φ j r ( )e iβ jθ,=

u q r j,( ) u q r ( )e i qθ β j+( )θ,=

v q r j,( ) v q r ( )e i q θ β j–( )θ.=

φ02



LASER PHYSICS Vol. 13 No. 8 2003

MERMIN–HO VORTEX 1085

core where the condensates are empty is placed at thepotential minimum, the condensation energy is lostmaximally. Thus the vortex core seeks a lower densityregion, leading to the vortex spiraling out effect and

ultimately giving rise to the intrinsic vortex instabilityin the scalar BEC [13, 26–28].

(3) The energy term – W · L associated with theexternal rotation is also decisive, in particular, in ahigher rotation region. The larger angular momentumof the system, the more this energy is gained. It is notedthat the total angular momentum is maximal when thevortex core is placed at the potential minimum position,and thus the factor (2) and (3) are mutually competing.

4. POSSIBLE VORTEX TYPES

We can classify the possible vortex types into twocategories: axisymmetric and nonaxisymmetric vorti-ces.

4.1. Axisymmetric Vortices

For axisymmetric vortices, the combination of thewinding numbers β1, β0, β–1 of three components ( φ1,φ0, φ–1) is restricted to 2 β0 = β1 + β–1. Since each wind-ing number should be small because a vortex with alarger winding number decays spontaneously into mul-tiple vortices with a smaller winding number, the pos-sible axisymmetric vortices are enumerated as 1, 1, 1 ,

1

0

–4 –20 2 4 x , μm

y, μm

–4 –2

02

4

Density, 10 20 m3

1

0

–4 –20 2 4 x , μm

y, μm

–4 –2

02

4

Density, 10 20 m3

1

0

–4 –20

2 4 x , μm

y, μm

–4 –2

02

4

Density, 10 20 m3

1.6

–30

1.4

1.2

1.0

0.8

0.6

0.4

0.2

–4 –2 –1 0 1 2 3 4 x , μm

Density, 10 20 m3

+1 –1

(a)(b)

(c)

(d)

Fig. 1. Density proles of the axisymmetric 1, x, 0 vortex in the antiferromagnetic situation ( gs = 0.02 gn): at Ω= 0.35 and M / N =

0.65. 3D plots of (a) the total density , (b) |φ1|2, and (c) |φ–1|2. The cross section is shown in (d). The bold line denotes the

total density, and the thin lines show the densities of the three components.

φ j2

j∑



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MACHIDA et al .

1, 0, –1 , 1, x, 0 , and 2, 1, 0 , where x means that thiscomponent is absent.

The 1, x, 0 Alice vortex as a typical example of theaxisymmetric vortex is shown in Fig. 1, where the den-sity proles for two components |φ1(r )|2 and |φ–1(r )|2 aredisplayed in Figs. 1b and 1c. It is seen from these thatthe φ–1 component with no winding is tted in the vor-tex core region of the φ1 component with the unit circu-lation at the center (see Fig. 1d for the cross-sectionalstructure). The resulting total density prole in Fig. 1ashows an almost bell-shape structure, which is advanta-geous in gaining the condensation energy.

4.2. Nonaxisymmetric Vortices

There are several vortex types which break axisym-metry. We try to nd some of them by starting with anaxisymmetric vortex as an initial conguration of

numerical computation. In Fig. 2, we display an exam-ple where the vortex cores of the three componentsalign linearly along a line passing through the potentialminimum point. Each component has unit circulation.Since the cores of the φ1 and φ–1 components avoid thepotential minimum point, the condensation energy lossdue to placing the core at this point is saved while thetotal angular momentum is decreased. The cross-sec-

tional prole in Fig. 2e is depicted along the line wherethe vortex cores are aligned.

5. MERMIN–HO VORTEX

Among the axisymmetric vortices enumeratedbefore, the 2, 1, 0 or 0, 1, 2 vortex is nothing but theso-called Mermin–Ho vortex or the Anderson–Tou-louse vortex, which are predicted in connection withsuperuid 3He [24]. The density proles are shown in

1.6

–30

1.4

1.2

1.0

0.8

0.6

0.4

0.2

–4 –2 –1 0 1 2 3 4 x , μm

Density, 10 20 m3

+1 –1

(e)

1

0

–4 –20 2 4 x , μm

y, μm

–4 –2

02

4

Density, 10 20 m3 (a)

1

0

–4 –20 2 4 x , μm

y, μm

–4 –2

02

4

Density, 10 20 m3

1

0

–4 –20

2 4 x , μm

y, μm

–4 –2

02

4

Density, 10 20 m3

1

0

–4 –20

2 4 x , μm

y, μm

–4 –2

02

4

Density, 10 20 m3

0

(b)

(c) (d)

Fig. 2. Density proles of the axisymmetric 1, 1, 1 vortex in the ferromagnetic situation ( gs = –0.02 gn): at Ω= 0.35 and M / N = 0.

3D plots of (a) the total density, (b) |φ1|2, (c) |φ0 |2, and (d) |φ–1|2. The cross section is shown in (e).





Fig. 3. It is seen from these that the φ1 component withzero winding occupies the central region, the φ0 withunit winding is in the intermediate region, and the φ–1with winding 2 is in the outer region. In this axisym-metric structure, each component is situated concentri-cally. This structure is advantageous because (1) thepotential minimum region is lled in the condensates,(2) the total angular momentum can be large by havingthe higher winding number 2, and (3) the ferromagnetic

spin interaction term favors phase separation, yieldinga concentric layered structure. This MH vortex onlyappears in the ferromagnetic case. Note that in the anti-ferromagnetic case the doubly quantized vortex of theφ–1 component tends to become two singly quantizedvortices since the antiferromagnetic spin interactionfavors the mixture of the components exemplied bythe polar state in the uniform case.

The spin texture in the MH vortex is illustrated inFig. 4. As seen from Figs. 4a and 4b, where the spatial

dependences of the l vector, which is dened as lα ∝

(α = x, y, z), are displayed in three-dimensional manner (Fig. 4a) and two-dimensionalmanner (Fig. 4b), the l vector is frared out radically.The magnitude of the l z component decreases out-wardly to zero (a negative value) for the MH vortex (ATvortex) as shown in Fig. 4c.

We can control two vortices by merely changing thetotal magnetization (in superuid 3He, the boundarycondition for a bucket wall controls it). The dotted lineindicates the analytic form l z = cos β(r ) with β(r ) = πr / R( R = 2.85 μm) for comparison.

6. PHASE DIAGRAM

We construct the phase diagrams both for ferromag-netic (Fig. 5) and antiferromagnetic (Fig. 6) cases in theplane: Ωvs. M / N . There appear both axisymmetric and

φi* F ˆ α( )ij φ jij∑

1.6

–30

1.4

1.2

1.0

0.8

0.6

0.4

0.2

–4 –2 –1 0 1 2 3 4 x , μm

Density, 10 20 m3

+1

–1

(e)

1

0

–4 –20 2 4 x , μm

y, μm

–4 –2

02

4

Density, 10 20 m3 (a)

1

0

–4 –20 2 4 x , μm

y, μm

–4 –2

02

4

Density, 10 20 m3

1

0

–4 –20

2 4 x , μm

y, μm

–4 –2

02

4

Density, 10 20 m3

1

0

–4 –20

2 4 x , μm

y, μm

–4 –2

02

4

Density, 10 20 m3

0

(b)

(c) (d)

Fig. 3. Density proles of the axisymmetric 0, 1, 2 vortex in the ferromagnetic situation ( gs = –0.02 gn): at Ω= 0.35, M / N = 0, 3D

plots of (a) the total density, (b) |φ1|2, (c) |φ0 |2, and (d) |φ–1|2. The cross section is shown in (e).



1088


MACHIDA et al .

nonaxisymmetric vortices. We focus on the low-fre-quency region Ω< 0.36 where the single vortex is sta-ble in the scalar BEC. By comparing the energies forseveral vortex congurations, we select the lowestenergy vortex at a given Ω and M / N . In the ferromag-netic case (Fig. 5), the 0, 1, 2 vortex, namely, the MHvortex, is stable for a rather large area. The upperboundary denoted by the dotted line signies the

boundary above which the collective modes becomethe negative eigenvalue. Below this line, the MH vortexis stable globally and locally; that is, MH has the lowestenergy among the various vortex congurations and allthe collective modes have positive eigenvalues. Thelarge empty region in Fig. 5 means that there is no sta-ble conguration, indicating the tendency toward phaseseparation in the ferromagnetic case.

As for the antiferromagnetic case shown in Fig. 6,the entire region is covered by individual states, exclud-ing the empty region. Except for the narrow regionalong M / N ~ 0 where the nonaxisymmetric 1, 1, 1 vor-tex becomes stable, only the axisymmetric vorticesappear. The detailed local stability analyses are donefor these vortices [25, 26]. Basically, these are shown tobe locally unstable.

7. HIGHER ROTATIONIt is rather difcult to systematically investigate

the higher rotation region because the variety of thepossible multiple vortex congurations to be exam-ined increases further. Here, we pick up an examplewhich becomes stable under a set of particularparameter values.

Figure 7 shows the spin texture of the four MH vor-tices, which are arranged regularly to form a square lat-tice. The four MH vortices are seen to have different lvector structures. This spin texture with the squarearrayed MH vortices is similar to that proposed by

1

–2

0

–4

–4 –2

–10

12

3 4

–3

43210 –1 –2 –3 –4

x , μm y, μm

(a)

(b)

–2 0 2 4

–2

–4

2

4

x , μm

y ,

μ m

(c)

–3 –4 –1 1 2 3 4 –1

0

0

l z

M / N = 0.46

0

r , μm

Fig. 4. The spin textures of the 0, 1, 2 vortex are displayed.The l vector in the ( x, y) plane is shown in (a). The arrowscorrespond to the l-vectors, and the small circles are perpen-dicular to each l vector. (b) ( l x , l y) and (c) spatial depen-dence of the l z component along the radial direction.

0.22

0.20 0.4 0.6 0.8 1.0

0.240.26

0.28

0.30

0.32

0.34

0.36

0.20

0, 1, 2

1, 1, 1

1, 0, –1

M / N

Ω, trap unit

Fig. 5. Phase diagram for the ferromagnetic state ( gs = –0.02 gn).The dashed line denotes the boundary where the lowest eigen-value of the 0, 1, 2 vortex becomes negative.

0.22

0.20 0.4 0.6 0.8 1.0

0.240.26

0.28

0.30

0.32

0.34

0.36

0.20

1, 1, 1 spilit-(I)

M / N

Ω, trap unit

0, 0, 0

0, x , 1

1, 0, –1

1, 1, 1 triangle

1, x , 0

Fig. 6. Phase diagram for the antiferromagnetic state ( gs =0.02 gn). Nonaxisymmetric vortices are stable in the shadedarea near M / N ~ 0.





–4 0 4 –4

0 y , μ

m

x , μm

–6

6

0

6

0 –6

y ,

μ m

x , μm

–6

6

0

6

0 –6

y ,

μ m

x , μm

(a)

(b) (c)

–6

6

0

6

0 –6

y ,

μ m

x , μm

–6

6

0

6

0 –6

y ,

μ m

x , μm

(d) (e)

Fig. 7. Properties of the four Mermin–Ho vortices at n z = 1.0 ×104 / μm and Ω= 0.4. (a) The direction of the l vector in the ( x, y)

plane and contour plots of (b) the total density, (c) |φ1|2, (d) |φ0 |2, and (e) |φ–1|2 are displayed.

Fujita et al. [29] in connection with the superuid 3He–Aphase under rotation. The density topomaps are shownin Figs. 7b–7e. The density prole has a quite smoothbell shape in spite of the intricate density proles foreach component.

8. CONCLUSIONS AND SUMMARY

We have shown that the Mermin–Ho vortex andAnderson–Toulouse vortex can be realized in a spinorBEC of atomic clouds, such as 87Rb of hyperne state

4



1090


MACHIDA et al .

F = 1, which is considered to be the ferromagnetic case.These vortices with a nonsingular core are character-ized by calculating the spin textures. The stability of these vortices are examined by comparing other possi-ble vortex congurations (global stability) and by eval-uating collective modes whose eigenvalues are positivein their stable region of the phase diagram (linear sta-bility or local stability). The total angular momentum

per particle is given by L z / N = 1 – M / N for MH and ATvortices. The above calculations are performed withinthe framework of Bogoliubov theory extended to BECwith spin degrees of freedom, namely, the three-com-ponent BEC.

In response to our earlier proposal [30, 31] that avortex with the winding number 2 can be created byadiabatically reversing a magnetic eld utilizing theBerry phase change, Leanhardt et al. [20] have suc-ceeded in producing a vortex by this topological vortexformation. During the eld reversing process at exactlythe half-way point of this process, the MH vortex isformed (see Fig. 2b in [31]). In this sense, the MH vor-tex has already been created. In fact, this has been con-rmed [32].

In order to conrm the MH vortex, we point out the“Stern–Gerlach” experiment, which can probe the den-sity proles for each spin component separately. Thismethod is ideally suited for this case because, as is seenfrom Fig. 3, each component is arranged concentricallyaround the harmonic potential minimum.

We have shown that MH vortices become stable inhigher rotation. This study leads ultimately to ourinvestigation of many vortex congurations under veryhigh rotation [33].

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K. Machida et al- Mermin–Ho Vortex in Spinor Bose–Einstein Condensates under Rotation

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