Page 1

arXiv:0809.3532v3 [cond-mat.supr-con] 7 Jun 2010

Spontaneous generation of a half-quantum vortex in spin-orbit coupled Bose-Einstein

condensates

Congjun Wu1and Ian Mondragon-Shem1,2

1Department of Physics, University of California, San Diego, CA 92093

2Instituto de F´ ısica, Universidad de Antioquia, AA 1226, Medell´ ın, Colombia

According to the “no-node” theorem, many-body ground state wavefunctions of conventional

Bose-Einstein condensation (BEC) are positive-definite, thus time-reversal symmetry cannot be

spontaneously broken. We find that multi-component bosons with spin-orbit coupling provide an

unconventional type of BEC beyond this paradigm. In a harmonic confining trap, the condensate

spontaneously generates a half-quantum vortex combined with the skyrmion type of spin texture.

This phenomenon can be realized for both the cold atom systems with artificial spin-orbit coupling

generated from atom-laser interaction, and for exciton condensates in semi-conductor systems.

PACS numbers: 71.35.-y, 73.50.-h, 03.75.Mn, 03.75.Nt

The conventional many-body ground state wavefunc-

tions of bosons satisfy the celebrated “no-node” theorem

in the absence of rotation, as pointed out by Feynman

[1], which means that they are positive-definite in the

coordinate representation.

time-reversal (TR) symmetry cannot be spontaneously

broken.It applies to various quantum ground states

of bosons including Bose-Einstein condensates (BEC),

Mott-insulating states, density-wave states, and super-

solid states, thus making this a very general statement.

It would be interesting to search for novel types of

quantum states of bosons beyond this paradigm, such

as unconventional BECs with complex-valued wavefunc-

tions. The “no-node” theorem is only a ground state

property, hence it does not apply to excited states. Re-

cently, a meta-stable state of bosons has been realized,

using cold atoms in optical lattices, by pumping atoms

into high orbital bands (e.g. p-bands) with a long life

time [2, 3]. Interactions among orbital bosons are char-

acterized by an orbital Hund’s rule [4, 5], which gives

rise to a class of complex superfluid states by develop-

ing onsite orbital angular momentum, thus breaking TR

symmetry spontaneously [3–7].

The “no-node” theorem does not apply to spinful

bosons with spin-orbit (SO) coupling either. The linear

dependence on momentum introduced by the SO cou-

pling invalidates Feynman’s proof. Although alkali atoms

are too heavy to exhibit relativistic SO coupling, artificial

SO coupling can be induced by laser-atom interactions

[8–13]. The BECs in such systems have been studied,

and have been shown to exhibit frustrations of superflu-

idity [14]. On the other hand, excitons in semiconductors

[15–18] with small effective mass can exhibit SO coupling

in the center-of-mass motion [19–21]. In particular, ex-

citing progress has been made in indirect exciton systems

in coupled quantum wells where electrons and holes are

spatially separate [22–25]. The extraordinarily long life

time of indirect excitons provides a wonderful opportu-

nity to investigate the exotic state of matter of exciton

condensation [16].

This theorem implies that

In this article, we show that bosons with Rashba SO

coupling provide a new opportunity to investigate un-

conventional BEC beyond the “no-node” theorem. We

find that Rashba SO coupling leads to fragmented con-

densation in homogeneous systems. Furthermore, with

an external harmonic potential trap, TR symmetry is

spontaneously broken. In this case, the ground state

condensate develops orbital angular momentum, which

can be viewed as a half-quantum vortex. Moreover, the

spin density distribution exhibits a cylindrically symmet-

ric spiral pattern as skyrmions. The experimental real-

ization, in both cold atom and semiconductor exciton

systems, is discussed.

We begin with a 3D two-component boson system with

Rashba SO coupling in the xy-plane and with contact

interaction, described by

H =

?

d3? r ψ†

α[−?2∇2

2ψ†

2M

− µ]ψα+ ?λRψ†

α[−i∇yσx

+ i∇xσy]ψβ+g

αψ†

βψβψα, (1)

where ψα is the boson operator; the pseudospin indices

α =↑,↓ refer to two different internal components of

the bosons; λRis the SO coupling strength; g describes

the s-wave scattering interaction. Although Eq. (1) is

for bosons, it satisfies a suitably defined TR symmetry

T = iσ2C satisfying T2= −1, where C is the complex

conjugate operation and σ2operates on the boson pseu-

dospin degree of freedom.

In the homogeneous system, the single particle states

are the helicity eigenstates of ? σ · (?k × ˆ z) which have a

dispersion relation given by ǫ±(?k) =

where k0 =

?

. The energy minima are located on

the lower branch along a ring with radius k0 in the

plane of kz = 0.The corresponding two-component

wavefunction ψ+(?k) with |k| = k0 can be solved as

ψT

angle of the projection of?k in the xy-plane. The in-

teraction part in Eq. (1) in the helicity basis can be

?2

2M((k −k0)2+k2

z),

MλR

+(?k) =

1

√2(e−iφk/2,ieiφk/2), where φk is the azimuth

Page 2

2

−k

k

0

0

B

A

q

−q

q

−q

I

I

II

II

−k

k

Λ

FIG. 1: The degeneracy of the single particle ground states

along a circle with the radius k0 in momentum space. We

assume that the fragmented condensation occurs at points A

(k0ˆ ex,0,0) and B (−k0ˆ ex,0,0) with spin polarizations along

±ˆ ey, respectively. The low energy excitations within |k−k0| <

Λ, |kz| < Λ, and Λ/k0 ≪ 1 are classified into two regimes I

(inside two cylinders centering around A and B with radius

of Λ) and II (outside).

represented as

Hint =

g

2

?

λµνρ

?

p1p2q

?? p1+ ? q;λ|? p1;ρ??? p2− ? q;µ|? p2;ν?

× ψ†

λ(? p1+ ? q)ψ†

µ(? p2− ? q)ψν(? p2)ψρ(? p1), (2)

where the Greek indices λ,ν,µ,ρ are the helicity indices

±; the matrix elements denote the inner product of spin

wavefunctions of two helicity eigenstates at different mo-

menta, e.g., ?? p1+ ? q;λ|? p1;ρ? =1

At the Hartree-Fock level, the kinetic energy can be

minimized through coherent or fragmented condensation

in the linear superpositions of the single particle ground

states.For the interaction energy, the exchange part

among bosons is positive within the s-wave scattering ap-

proximation where g is independent of momentum, thus

the ground state favors the zero exchange energy. The

spin polarizations at?k and −?k in the plane of kz= 0 are

orthogonal to each other. Thus, without loss of gener-

ality, the fragmented condensate defined below satisfies

this requirement

2[1 + λρei(φp1−φp1+q)].

Φ0=

1

√NA!NB![ψ†

+(?kA)]NA[ψ†

+(?kB)]NB|0?, (3)

where?kA= (−k0,0,0) and?kB= (k0,0,0); and (NA,NB)

is the particle number partition satisfying NA+NB= N0,

with N0the total particle number in the condensate. In

homogeneous systems, fragmented condensates with dif-

ferent partitions do not mix because of momentum con-

servation.

An “order from disorder” calculation on the low energy

excitation spectra is needed to determine the optimal

particle number partition. The momentum cutoff ?Λ is

defined as ?2Λ2/(2M) ≈ gn, where gn is the interaction

energy scale and n is the particle density. We only con-

sider the limit of strong SO coupling, k0≫ Λ, and leave

the general case for later study. The low energy Bogoli-

ubov excitations that significantly mix particle and hole

operators mainly lie in the regime |

and |kz| < Λ, which can be further divided into two parts

I and II as depicted in Fig.1. Part I is inside two cylin-

ders with the radius of Λ centering around A and B, and

part II is outside these two cylinders.

For the low energy states in part I, we define

the bosonic operators in the lower branch as a? q =

ψ+(−k0ˆ ex+? q) and b? q= ψ+(k0ˆ ex+? q), and ?aq=0? =√Na

and ?bq=0? =√Nb respectively. The relative phase be-

tween two condensates can be set to zero due to a suit-

able translation operation. The low energy excitations in

this region have been calculated in Ref. [14]. By defin-

ing γ†

1(? q) =

√Nab†

the quadratic level of q, can be represented as

?

k2

x+ k2

y− k0| < Λ

1

N0(√Naa†

q+√Nbb†

q), γ†

2(? q) =

1

N0(√Nba†

q−

q), the mean-field Hamiltonian in region I, up to

HMF,1 =

?

q

E(? q)?γ†

?γ†

x+ q2

1(? q)γ(? q) + γ†

2(? q)γ2(? q)?

+ gN0

1(? q)γ†

1(−? q) + h.c.?,

z)/(2M).

(4)

where E(±? q) = ?(q2

ter the Bogoliubov diagonalization yields the spectra

?E? q(E? q+ 2gN0) ≈

the phonon mode describing the overall density fluctua-

tions, which exhibits linear dispersion relation for ? q in the

xz-plane and becomes soft for ? q along ˆ ey. The γ2mode

represents the relative density fluctuations between two

condensates which can also be considered as spin density

fluctuations. This mode remains a free particle spectrum

E(? q). Both of the γ1,2modes, at q ≪ k0, only depend on

the total condensation density NA+ NB = N0. Hence,

the contribution from part I does not lift the degeneracy

between different partitions of (NA,NB) to the quadratic

order of q.

Now, we turn to the low energy excitations in part

II where the dispersion of ψ?k,+is degenerate with the

state of ψ−?k,+but not with ψ2?kA,B−?k,+. The mean field

Hamiltonian reads

?ǫ(k) +g

+ g?ψ†

where ∆N = NA− NB. The Bogoliubov spectra can be

solved as HMF,2=?

?

g∆N

2

The γ1 mode af-

ω(? q) =

?

?gN0

M

?q2

x+ q2

z. This is

HMF,2 =

?

k

ψ†

?k,+ψ?k,+

2(N0− ∆N cosφk)?

NaNbcosφ?keiφ?k+ h.c.?, (5)

?k,+ψ†

−?k,+

?

?k

?ω(?k)(γ†

3(?k)γ3(?k) +1

2)?with

ω(k) =

ǫ?k(ǫ?k+ gN0) +g2

4[N2

0sin2φk+ (∆N)2cos2φk]

+

cosφk. (6)

After summing over?k around the ring, the second term

in Eq. 6 disappears. Thus the ground state zero point

Page 3

3

012345

0.02

0.04

0.06

0.08

r ? ?

Density

Spin down component

Spin up component

Total

A?

: Spin density vector

B?

?1

?0.50 0.51

x ? ?

FIG. 2:

down components, and the total density distribution in the

unit of N0 at α = 4 and β = 40 in a harmonic trap. B) The

spin density distribution along the x-axis which spirals in the

z-x plane at an approximate wavevector of 2k0, whose value

at the origin is normalized to 1. The spin density in the whole

plane exhibits the skyrmion configuration.

A) The radial density distribution of spin up and

motion from Eq. 6 selects the optimal partition of ∆N =

0, i.e., an equal partition between two condensates.

Fragmentization of BECs is usually unstable against

perturbations [26, 27].It can be removed by non-

conservation of momentum induced by spatial inhomo-

geneity. Now we consider a realistic system with confin-

ing trap. Our main interest is the interplay between the

confining potential and the Rashba SO coupling which

only lies in the xy-plane. Thus we neglect the effect of

the trapping potential along the z-axis, and assume that

the condensate wavefunction is uniform along this axis.

We consider Vex(r) =

characteristic SO energy scale for the harmonic trap as

Eso= ?λR/l where l =??/(MωT) is the length scale of

the trap; correspondingly, we also define the dimension-

less parameter α = Eso/(?ωT) = lk0. Due to the 2D ro-

tational symmetry, the single particle wavefunctions can

be denoted by the total angular momentum jz= m+1

The ground state single particle wavefunctions form a

Kramer doublet as represented in polar coordinates as

1

2Mω2

T(x2+ y2), and define the

2.

ψ 1

2(r,φ,z) =

?

f(r)

g(r)eiφ

?

, ψ−1

2(r,φ,z) = Tψ 1

2. (7)

where both f(r) and g(r) are real functions.

Let us gain some intuition about f(r) and g(r) by con-

sidering the strong SO limit α ≫ 1. The Fourier com-

ponents of ψ±1

around the low energy ring. They can be approximately

2in momentum space mainly distribute

represented as

|ψ±1

2? =

?

dφk

?

dk w(k) e±i1

2φk|ψ+(?k)?, (8)

where the weight w(k) mainly distributes along the low

energy ring with a broadening at the order of 1/l. In

other words, the harmonic potential, which reads Vex=

1

2Mω2

?A(?k) = i?ψ+(?k)|∂?k|ψ+(?k)?. For the Rashba coupling, the

field strength of?A(?k) corresponds to a π-flux located at

?k = (0,0). Vexbreaks the degeneracy of the plane wave

states with |k| = k0 by quantizing the orbital motion

along the ring as

T(i∂?k−?A(?k))2in momentum representation, where

∆Em+1

2=1

2Mω2

T(m +1

2

k0

)2=

1

2α2?ωT(m +1

2)2. (9)

The ground state forms the Kramer doublets correspond-

ing to m +1

2, and the low energy azimuthal ex-

citations are with m +1

in the radial direction of |k| change the node structures

of f(r) and g(r), which cost the energy at the order of

?ωT and thus are much higher energy states than the

azimuthal excitations at α ≫ 1. For the strong SO cou-

pling α ≫ 1, the single particle ground doublet states

have nearly equal weight in the spin up and down com-

ponents, i.e.

?drdφ r|f(r)|2≈?drdφ r|g(r)|2, thus the

spin moment averages to zero. When the bosons con-

dense into one of the TR doublets, the average orbital an-

gular momentum per particle is ?/2, i.e., one spin compo-

nent stays in the s-state and the other one in the p-state.

This is a half-quantum vortex configuration [28–30], thus

spontaneously breaking TR symmetry.

Next we turn on interactions and define the character-

istic interaction energy scale Eint = gN0/(πl2) and the

dimensionless parameter β = Eint/(?ωT). We numeri-

cally solve the Gross-Pitaevskii equation

2= ±1

2= ±3

2,±5

2.... The excitations

?

−?2∇2

2M

+ ?(−i∇yσx,αβ+ i∇xσy,αβ) + g(ψ∗

1

2Mω2

γψγ)

+

Tr2?

ψβ(r,φ) = Eψα(r,φ)(10)

in the harmonic trap with parameters α = 4 and β = 40.

The interaction effectively weakens the harmonic poten-

tial and does not change the orbital partial wave struc-

ture of the wavefunctions. We show the radial density

profiles of both spin components |f(r)|2and |g(r)|2in

Fig. 2 A. Each of them exhibits oscillations at a pitch

value of approximately 2k0, which originate from the low

energy ring structure and, thus, are analogous to the

Friedel oscillations in fermion systems.

The spindensity,defined

ψ∗

exhibits

logical spin texture configuration. Let us first look at its

distribution along the x-axis where the supercurrent is

as

interesting

?S(r,φ)=

α(r,φ)? σαβψβ(r,φ), an topo-

Page 4

4

along the y-direction and the spin lies in the xz-plane.

Explicitly, we express Sz(r,φ) =1

Sx(r,φ) = f(r)g(r). The radial oscillations of |f(r)|2

and |g(r)|2have an approximate π phase shift, which

arises from the different angular symmetries.

result,?S spirals along the x-axis as plotted in Fig. 2 B

at the pitch value of the density oscillations. The spin

density distribution in the whole space can be obtained

through a rotation around the z-axis, which exhibits the

skyrmion configuration.

The recent research focus of the “synthetic gauge

fields” in cold atom systems provides a promising method

to observe the above proposed exotic spontaneous half-

quantum vortex [10–12, 14].

have been synthesized by light-atom interaction to gen-

erate vortices experimentally [12]. Similar methods to

generate SO coupling have been proposed [10, 13, 31],

and are currently under experimental implementation. A

convenient method is the “tripod scheme”: three atomic

degenerate hyperfine ground states are coupled to an ex-

cited state by laser fields to create a pair of degener-

ate dark states denoted with pseudospin indices | ↑? and

| ↓?. The spatially dependent laser fields couple the pseu-

dospin and the momentum of the atoms. A typical set of

parameter values are provided in Ref. [10] for a trap fre-

quency ωT= 2π×10 Hz, a characteristic Rabi frequency

Ω = 2π×107Hz, and a detuning ∆ = 2π×1011Hz, which

satisfy ωT ≪ Ω ≪ ∆ and ωT ≪ Ω2/∆. The value of α

can vary from 0 up to the order of several tens. With the

typical particle number N0= 106, the interaction energy

scale Eintis around 100nK, or 2kHZ [32], and thus β is

of the order of several tens. Hence, the parameters used

in the Fig. 2 A and B are realistic in real experiments.

The experimental detection would be straightforward as

performed in previous time-of-flight imaging in vortex

experiments [33, 34]. By separately imaging the density

profiles of the pseudospin up and down components, the

radial oscillation of the Sz component can be directly

seen.

Another class of boson systems exhibiting SO coupling

is the indirect excitons in 2D coupled double quantum

well systems. We only need to consider the heavy hole

(hh) band with the effective mass m∗

which is separated from the light hole band with a gap of

the order of 10meV. The Hamiltonians of the electron-

hole system reads He= −?2

Hhh= −

m∗

the Rashba SO coupling strength of the conduction elec-

tron; ε is the dielectric constant; and d is the distance

between the double wells.

of excitons in the BEC limit can be separated from the

electron-hole relative motion. Similarly to Ref. [35], the

effective Hamiltonian of the 4-component hh excitons de-

noted as (se,jhh) = (±1

2(|f(r)|2− |g(r)|2) and

As a

Artificial magnetic fields

hhand jz = ±3

2?,

2m∗

e∇2

e+i?λR(∂e,xσy−∂e,yσx),

e2

ε√

?2

2m∗

hh∇2

h, and He−ph= −

|? re−? rhh|2+d2, where

eis the effective mass of the conduction electrons; λRis

The center-of-mass motion

2,3

2),(±1

2,−3

2) can be represented

by the matrix form as

Hex=

Eex(?k)

H∗

0

0

Hso(?k)00

0

so(?k) Eex(?k) + ∆(?k)

W∗(?k)

W(?k)

Eex(?k) + ∆(?k) Hso(?k)

H∗

0

so(?k)Eex(?k)

,

where?k is the center-of-mass momentum; M = m∗

is the total mass of the exciton; Eex(?k) = ?2k2/(2M);

Hso(?k) = −m∗

and W(?k) has the d-wave structure as (kx+ iky)2, both

of which are exponentially suppressed by the tunneling

barrier between electron and hole layers, and can be ne-

glected. Consequently, Hexbecomes block-diagonalized.

We consider using circularly polarized light to pump the

exciton of (−1

with heavy hole spin jz=3

the 2D version of the Hamiltonian given by Eq. 1.

In summary, we find that bosons with SO coupling ex-

hibit a novel type of BEC beyond Feynman’s argument.

In the limit of large SO coupling, the degeneracy of the

single particle ground states in momentum space leads to

two fragmented condensations with equal particle num-

ber partition in homogeneous space. In the presence of a

harmonic trap, the condensate spontaneously breaks TR

symmetry and develops the half-quantum vortex config-

uration with the skyrmion type topological spin density

configurations.

C. W. is supported by NSF under No. DMR- 0804775,

and ARO-W911NF0810291.

e+m∗

hh

e

MλR(ky+ikx); ∆(?k) is the exchange integral

2,3

2), and then focus on the left-up block

2. We will then end up with

[1] R. P. Feynman, Statistical Mechanics, A Set of Lec-

tures (Addison-Wesley Publishing Company, ADDRESS,

1972).

[2] T. Mueller, S. Foelling, A. Widera, and I. Bloch, Phys.

Rev. Lett. 99, 200405 (2007).

[3] A. Isacsson and S. M. Girvin, Phys. Rev. A 72, 053604

(2005).

[4] W. V. Liu and C. Wu, Phys. Rev. A 74, 13607 (2006).

[5] C. Wu, W. V. Liu, J. E. Moore, and S. Das Sarma, Phys.

Rev. Lett. 97, 190406 (2006).

[6] A. B. Kuklov, Phys. Rev. Lett. 97, 110405 (2006).

[7] V. M. Stojanovic, C. Wu, W. V. Liu, and S. D. Sarma,

Phys. Rev. Lett. 101, 125301 (2008).

[8] G. Juzeliunas et al., Phys. Rev. Lett. 100, 200405 (2008).

[9] J. Y. Vaishnav and C. W. Clark, Phys. Rev. Lett. 100,

153002 (2008).

[10] T. D. Stanescu, C. Zhang, and V. Galitski, Phys. Rev.

Lett. 99, 110403 (2007).

[11] Y.-J. Lin et al., Phys. Rev. Lett. 102, 130401 (2009).

[12] Y.-J. Lin et al., Nature 462, 628 (2009).

[13] I. B. Spielman, Phys. Rev. A 79, 063613 (2009).

[14] T. Stanescu, B. Anderson, and V. Galitski, Phys. Rev. A

78, 023616 (2008).

[15] D. W. Snoke, J. P. Wolfe, and A. Mysyrowicz, Phys. Rev.

B 41, 11171 (1990).

Page 5

5

[16] L. V. Butov, J. Phys.: Cond. Matt. 16, R1577 (2004).

[17] L. V. Butov, J. Phys.: Cond. Matt. 19, 295202 (2007).

[18] L. A. V. Timofeev V. B., Gorbunov A. V., J. Phys.:

Cond. Matt. 19, 295209 (2007).

[19] T. Hakioglu and M. Sahin, Phys. Rev. Lett. 98, 166405

(2007).

[20] M. A. Can and T. Hakioglu, arXiv:0808.2900, 2008.

[21] W. Yao and Q. Niu, Phys. Rev. Lett. 101, 106401 (2008)

[22] L. V. Butov et al., Phys. Rev. Lett. 73, 304 (1994).

[23] L. V. Butov and A. I. Filin, Phys. Rev. B 58, 1980 (1998).

[24] L. V. Butov et al., Phys. Rev. Lett. 86, 5608 (2001).

[25] L. V. Butov, A. C. Gossard, and D. S. Chemla, Nature

418, 751 (2002).

[26] E. J. Mueller, T. L. Ho, M. Ueda, and G. Baym, Phys.

Rev. A 74, 33612 (2006).

[27] T. L. Ho and S. K. Yip, Phys. Rev. Lett. 84, 4031 (2000).

[28] M. M. Salomaa and G. E. Volovik, Phys. Rev. Lett. 55,

1184 (1985).

[29] C. Wu, J. P. Hu, and S. C. Zhang, Int. J. Mod. Phys. B

V24, 311 (2010).

[30] F. Zhou, Int. Jour. Mod. Phys.B, 17 17, 2643 (2003).

[31] J. Larson and E. Sj¨ oqvist, Phys. Rev. A 79, 043627

(2009).

[32] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001).

[33] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dal-

ibard, Phys. Rev. Lett. 84, 806 (2000).

[34] B. P. Anderson, P. C. Haljan, C. E. Wieman, and E. A.

Cornell, Phys. Rev. Lett. 85, 2857 (2000).

[35] M. Z. Maialle, E. A. de Andrada e Silva, and L. J. Sham,

Phys. Rev. B 47, 15776 (1993).