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arXiv:1110.0558v2 [cond-mat.quant-gas] 7 Oct 2011

Spin-orbit-coupled dipolar Bose-Einstein condensates

Y. Deng1,2, J. Cheng3, H. Jing2, C.-P. Sun1, and S. Yi1

1Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China

2Department of Physics, Henan Normal University, Xinxiang 453007, China and

3Department of Physics, South China University of Technology, Guangzhou 510640, China

(Dated: October 10, 2011)

We propose an experimental scheme to create spin-orbit coupling in spin-3 Cr atoms using Raman

processes. Employing linear Zeeman effect and optical Stark shift, two spin states within the ground

electronic manifold are selected, which results in a pseudo-spin-1/2 model. We further study the

ground state structures of a spin-orbit-coupled Cr condensate. We show that, in addition to the

stripe structures induced by the spin-orbit coupling, the magnetic dipole-dipole interaction gives

rise to the vortex phase, in which spontaneous spin vortex is formed.

PACS numbers: 37.10.Vz, 03.75.Mn

Over the past few years, there have been rapidly grow-

ing interests in engineering Abelian and non-Abelian arti-

ficial gauge fields in ultracold atomic gases [1–6]. Particu-

larly, the non-Abelian gauge field, or more specifically the

spin-orbit (SO) coupling, is of fundamental importance in

many branches of physics. Fascinating examples include

the quantum spin-Hall effect and the topological insu-

lators in condensed matter physics [7]. With enormous

tunability of interaction and geometry, ultracold atomic

gases may offer a tremendous opportunity for studying

exotic quantum phenomena in many-body systems with

SO coupling [8–16].

In their pioneer experiments, the NIST group have re-

alized the light-induced vector potentials [17], the syn-

thetic magnetic fields [18], and the electric forces [19] in

ultracold Rb gases through Raman processes [4], which

differs from most dark-state based theoretical propos-

als [20] on that the linear Zeeman shift is compensated

by the two-photon detuning. More remarkably, they also

created a two-component SO-coupled condensate of Rb

atoms and observed the phase transition from spatially

mixed to separated states [21]. An important ingredient

in this experiment is that the quadratic Zeeman shift is

employed to separate two desired spin states from the re-

maining one. Hence, this scheme is inapplicable to atoms

without nuclear spin, such as certain isotopes of Cr and

Dy, in which the quadratic Zeeman effect is absent.

In this Letter, we propose an experimental scheme to

create SO coupling in spin-352Cr atoms by selecting two

internal states from the J = 3 ground electronic mani-

fold. Similar to NIST group’s scheme, ours also relies on

Raman processes. However, we utilize optical Stark shift

to compensate the linear Zeeman shift so that the lowest

two levels are near degenerate and well separated from

other levels, which leads to a pseudo spin-1/2 model. The

proposed scheme has the advantages that only a moder-

ate magnetic field strength is required and it also applies

to atoms without nuclear spin.

An interesting feature of Cr atom is that it possesses

a large magnetic dipole moment, which makes scalar Cr

condensate an important platform for demonstrating the

dipolar effects [22]. Moreover, when atom’s spin degree of

(a)

(b)

FIG. 1: (color online). (a) Scheme for creating SO coupling

in Cr atom. Two Raman beams, propagating along ˆ x+ ˆ y and

−ˆ x + ˆ y with frequency difference ∆ωL, are linearly polarized

along ˆ z and ˆ x + ˆ y, respectively. A bias field B0 is applied

along negative z axis, which generates a Zeeman shift ωZ in

the ground state manifold. (b) Level diagram for the Raman

coupling within the |J = 3? ground state manifold by utilizing

the |J′= 2? excited state.

freedom becomes available, magnetic dipole-dipole inter-

action (MDDI) also couples the spin and orbital angular

momenta, which is responsible for the Einstein-de Haas

effects [23, 24] and the spontaneous spin vortices [25, 26]

in spinor condensates. Unfortunately, in spin-3 Cr con-

densates, contact interaction also contains spin-exchange

terms which is much larger than the strength of the

MDDI [27]. Therefore, the spin related dipolar effects

are yet to be observed. In the pseudo spin-1/2 Cr con-

densate, we show that only the MDDI contains spin-

exchange terms and spontaneous spin vortex is readily

observable.

We consider a condensate of52Cr atoms subjected to

a bias magnetic field B0along negative z-axis. The Zee-

man shift within the ground state manifold is ?ωZ =

Page 2

2

gsµB|B0| with gs = 2 being the electron spin g-factor

and µBthe Bohr magneton. Here, the quadratic Zeeman

shift is zero because of the absence of the nuclear spin.

As shown in Fig. 1, atoms are illuminated by a pair of

linearly polarized Raman beams which propagate along

ˆ x+ ˆ y and −ˆ x+ ˆ y with frequencies ωL+∆ωLand ωL, re-

spectively. The ground- (7S3) to excited-state (7P2) tran-

sitions are coupled by the Rabi frequencies Ω1eik1·rand

Ω2eik2·r, where k1= kL(ˆ x+ ˆ y) and k2= kL(−ˆ x+ ˆ y) are

the wave vectors of the Raman beams with kL=√2π/λ

and λ being the wave length of the lasers. For simplic-

ity, Ω1,2are assumed to be real. If the frequency of the

lasers are far detuned from the ground- to excited-state

transition, i.e. |Ω1,2/∆| ≪ 1 with ∆ being the detuning,

the excited states can be adiabatically eliminated to yield

the atom-light interaction Hamiltonian

?

U2

ΩRXU2T

ΩRX∗

U2T∗

∆c+ U1+ U2

ΩR(X∗+ XT∗) 2∆c+ U1+ 2U2

U2T∗

ΩR(X + X∗T)U2T

ΩR(X + X∗T)U2T

ΩR(X∗+ XT∗) 3∆c+ U1+ 2U2

U2T∗

ΩR(X + X∗T)U2T

ΩR(X∗+ XT∗) 4∆c+ U1+ 2U2 ΩR(X + X∗T)

U2T∗

ΩR(X∗+ XT∗) 5∆c+ U1+ U2

U2T∗

U2T

ΩRX∗T

6∆c+ U2

ΩRXT∗

, (1)

where ∆c = ωZ + ∆ωL is the two-photon detuning,

ΩR = −Ω1Ω2/∆ is the Rabi frequency for the Raman

coupling, U1,2 = −Ω2

induced by the laser fields Ω1and Ω2, respectively, and

T(t) ≡ e2i∆ωLtand X(x) ≡ e2ikLxare introduced for

short-hand notation. The physical significance of Eq. (1)

can be readily understood [28] using the level diagram,

Fig. 1(b).

From Hamiltonian (1), it is apparent that, under the

conditions ∆c+ U1 ≈ 0 and |U2|,|ΩR| ≪ |∆c|, the en-

ergy levels mJ = 3 and 2 can be separated from other

levels due to the large Zeeman shift. These conditions

can be satisfied by choosing ωZ = Ω2

that |∆ωL/ωZ| ≪ 1 and |Ω2/Ω1| ≪ 1, which eventually

leads to an effective two-level Hamiltonian:

p2

2M

ΩRe−2ikLx

1,2/∆ are the optical Stark shifts

1/∆ and assuming

ˆh =

ˆI + ?

?

−∆ωL/2ΩRe2ikLx

∆ωL/2

?

, (2)

for pseudo spin-up | ↑? = |mJ = 3? and -down | ↓? =

|2?, whereˆI is the identity matrix and a constant term,

−(U2+∆ωL/2)ˆI, has been added to obtain Eq. (2). We

note that the atom-light interaction term inˆh can be

intuitively treated as an effective magnetic field,

Beff= ?(gsµB)−1(2ΩRcos2kLx,−2ΩRsin2kLx,−∆ωL).

Unlike the NIST group’s scheme [21], here, an optical

Stark shift −Ω2

man shift, so that only the levels mJ = 3 and 2 are

Raman coupled near resonance (∆ωL≈ 0).

To proceed further, let us focus on the motion of an

atom along x axis by freezing its y and z degrees of free-

dom. Applying a simple gauge transform [12], the single-

particle Hamiltonian can be recast into

??2q2

1/∆ is used to compensate the linear Zee-

ˆh′

x=

2M

+ EL

?

ˆI + 2κqˆ σz+ ?ΩRˆ σx−?∆ωL

2

ˆ σz, (3)

where q

?2k2

L/(2M) is the single-photon recoil energy, ˆ σx,y,zare

the Pauli matrices, and κ = EL/kL is the SO coupling

strength. Even though κ is independent of Raman cou-

pling strength, SO coupling strength is still tunable by

varying the relative angle of the Raman beams [21]. It

can be readily shown that, after dropping the constant

ELterm, the eigenenergies of Eq. (3) are

?

=px/? is the quasi-momentum, EL

=

E±(q) =?2q2

2M

±

?2Ω2

R+

?

2κq −?∆ωL

2

?2

, (4)

in analogy to those in the spin-1 Rb condensate. In par-

ticular, on the lower branch E−(q), there exist two local

minima at q± ≃ ±kL

2EL and ?∆ωL ? EL. The corresponding energies are

E−(q±) ≃ −EL− ?2Ω2

with quasi-momenta ?q− and ?q+ (labeled as | ↑′? and

| ↓′?, respectively) represent the dressed spin states in

which atoms condense in the absence of the interactions.

Here, we would like to discuss the experimental feasi-

bility of our scheme. Firstly, the transition wavelength

from the ground- to excited-state is 429.1nm, which cor-

responds to a recoil energy EL/? ≃ (2π)10kHz. Other

laser parameters can be set up as follows. Since the lin-

ear Zeeman shift ωZin our proposal plays the role of the

quadratic Zeeman shift in the NIST experiment [21], we

may set ?ωZ= 3.8EL, which implies the laser intensity,

|Ω1|2= 3.8EL|∆|/?, is about the same order of magni-

tude as that used in the experiment. To allow the Raman

coupling ΩRto vary from 0 to EL, which covers the the

most interesting parameter region in the experiment, the

maximum value of |Ω2| can be chosen as 0.26|Ω1|. Con-

sequently, the maximum value of U2is less than 0.26EL,

which justifies the neglecting of U2in Eq. (1). Finally, we

point out that the SO coupling strength κ in our scheme

?1 − ?2Ω2

R/(4EL) ± ?∆ωL/2. The states

R/(4E2

L) when ?ΩR ?

Page 3

3

is 3.14 times larger than that in the Rb experiment due

to the smaller mass and the shorter transition wavelength

of Cr atom.

Now we turn to study the many-body effect in a SO-

coupled Cr condensate. To this end, we first write down

the single-particle Hamiltonian, which, in the second

quantized, takes the form

?

where V (r) = Mω2

symmetric harmonic trap with ω⊥being the radial trap

frequency and γ the trap aspect ratio, µ is the chem-

ical potential, andˆΨ(r) = [ˆψ↑(r),ˆψ↓(r)]Tis the field

operator for the bare spin states.

can also be expressed in terms of dressed spin states

by using the transformˆψ↑(r) ≃ˆψ↑′(r) − εe2ikLxˆψ↓′(r)

and ˆψ↓(r) ≃ −ˆψ↓′(r) + εe−2ikLxˆψ↑′(r), where ε ≃

?ΩR/(4EL+ ?∆ωL) ≪ 1 in the weak Raman coupling

limit ?ΩR/EL≪ 1.

In terms of the bare spin states, the collisional inter-

action takes the form

?

+2g6ˆψ†

ˆH0=drˆΨ†(r)

?ˆh + V (r) − µ

⊥(x2+ y2+ γ2z2)/2 is an axially

?

ˆΨ(r),(5)

We note that ˆH0

ˆHc =

1

2

dr

?

g6ˆψ†

↑ˆψ†

↑ˆψ↑ˆψ↑+5g4+ 6g6

11

ˆψ†

↓ˆψ†

↓ˆψ↓ˆψ↓

↑ˆψ†

↓ˆψ↓ˆψ↑

?

,(6)

where g4,6 = 4π?2a4,6/M with a4 = 58aB and a6 =

112aBbeing the s-wave scattering lengths for the colli-

sional channel with total spin angular momentum j = 4

and 6, respectively [29]. This result can be understood

as follows. The collision between two mJ = 3 atoms

can only happen in the total spin j = 6 channel; con-

sequently, it has a scattering length a6. For collisions

between mJ= 3 and 2 atoms, the projection of the total

spin along z-axis is mj = 5, which is conserved during

collision. Therefore, this collision also happens in the

j = 6 channel. While, when two mJ = 2 atoms collide

with each other, both j = 6 and 4 channels will con-

tribute. Apparently, the spin-up and -down states are

immiscible.

The MDDI for the pseudo spin-1/2 system can be de-

composed intoˆHd=ˆH(1)

d

+ˆH(2)

?4π

5

−4ˆψ†

?

5

+2ˆψ†

+

d

with

ˆH(1)

d

= gd

?

drdr′

|r − r′|3Y2,0(ˆ e)

↓ˆψ↓− 12ˆψ†

9π

drdr′

|r − r′|3

↓ˆψ′

?

−9ˆψ†

↑ˆψ′†

↑ˆψ′

↑ˆψ↑

?

↑ˆψ′†

↓ˆψ′†

↓ˆψ′

↑ˆψ′†

↓ˆψ′

↓ˆψ↑+ 3ˆψ†

?

↑ˆψ′†

↑ˆψ′†

↓ˆψ′

?

↓ˆψ↓+ h.c.

↑ˆψ↓

,(7)

ˆH(2)

d

= −gd

?

2Y2,−1(ˆ e)3ˆψ†

↑ˆψ′

↑ˆψ↓

?

↑ˆψ′†

↓ˆψ↓

?

√6Y2,−2(ˆ e)ˆψ†

↑ˆψ′

,(8)

here, gd= µ0g2

meability and µBthe Bohr magneton, ˆ e = (r−r′)/|r−r′|

is an unit vector, and we have adopted the notations

sµ2

B/(4π) with µ0being the vacuum per-

x(µm)

y(µm)

−24024

−24

0

24

FIG. 2: Integrated densities (columns 1 and 3 for spin-up

and -down, respectively) and phases of the condensate wave

functions on the z = 0 plane (columns 2 and 4 for spin-up

and -down, respectively). From the first to the fourth rows,

the frequency differences are ?∆ωL/EL = 0.01, 0.04, 0.0875,

and 0.1, respectively.

ˆψα ≡ˆψα(r) andˆψ′

first three terms ofˆH(1)

species dipolar interactions in a mixture of mJ= 3 and 2

atoms, and the last term is the exchange dipolar interac-

tion.ˆH(2)

d

is of particular interest. It represents the SO

coupling containing in the MDDI and does not conserve

the atom number in individual spin state. However, the

total angular momentum is conserved byˆH(2)

In this spin-1/2 model, interactions have a much sim-

pler form compared to those in the spin-3 system. In

particular, here, only the MDDI contains spin-exchange

terms. As will be shown, even gdis much smaller than

g4,6, the spin associated dipolar effect can be readily de-

tected in pseudo spin-1/2 Cr condensates.

We now investigate the ground state structures of the

SO-coupled dipolar condensate using the mean-field the-

ory. To this end, the field operatorsˆψαare replaced by

the condensate wave function ψα= ?ˆψα?, which can be

obtained by numerically minimizing the free energy func-

tional F[ψ↑,ψ↓] = ?ˆH0+ˆHs+ˆHd?. Specifically, we con-

sider a Cr condensate with N = 106atoms. The parame-

ters for trapping potential is chosen as ω⊥= (2π)100Hz

and γ = 6, representing a three-dimensional pancake-

shaped trap. Furthermore, the Rabi frequency for Ra-

man coupling is fixed at ?ΩR= −0.01EL. Since ε ≪ 1 is

satisfied, we shall only discuss the ground state in terms

of the bare spin states.

In Fig. 2, we plot the the integrated density, ¯ nα(x,y) =

α≡ˆψα(r′) with α =↑ and ↓. The

d

represent the intra- and inter-

d.

Page 4

4

x(µm)

y(µm)

−8−4048

−8

−4

0

4

8

FIG. 3: (color online). Vector plot of the transverse compo-

nents of s(r) on z = 0 plane for ?∆ωL = 0.0875EL. The

grayscale indicates the integrated density ¯ n↑(x,y).

00.030.060.090.12

0

0.2

0.4

0.6

0.8

1

¯ h∆ωL/EL

? Nα

(a)

00.03 0.06 0.090.12

−1

−0.8

−0.6

−0.4

−0.2

0

¯ h∆ωL/EL

¯Lz/¯ h

(b)

PPVPPP

↑

↓

FIG. 4: (color online). Reduced atom number? Nα (a) and av-

erage orbital angular momentum¯Lz (b) as functions of ∆ωL.

N−1?dz|ψα(r)|2, and the phases of the condensate wave

first row in Fig. 2), the single-particle energies of the two

pseudo spin states are nearly degenerate such that the

ground state structure is mainly determined by the inter-

actions. Apparently, bothˆHcandˆHd(in pancake-shaped

trap) favor the spin-down state. Therefore, | ↓ ? becomes

dominantly populated, which we refer to as the polar-

ized phase (PP). As shown in the fourth row of Fig. 2,

PP also occurs when the frequency difference ∆ωL (or,

equivalently, the z component of the effective magnetic

field Beff) is sufficient large, under which | ↑? becomes

dominantly occupied.

The common feature of the PPs is that the wave func-

tion of the highly populated state is structureless, as any

structure developed in the high density spin state would

cost too much kinetic energy. On the other hand, striped

structure forms in both the density and phase of the

less populated spin state. The phase stripe can be in-

tuitively understood as follows. To lower the energy, the

pseudo spin density, s(r) =

anti-parallel to the local effectively magnetic field, which

requires the relative phase of the condensate wave func-

tion to take the form arg(ψ↑)−arg(ψ↓) ∼ π−2ikLx. Since

the phase of the highly populated state is a constant, the

phase of the other spin state is then periodically mod-

functions for various ∆ωL’s. When ∆ωL is small (the

?

αβψ∗

αˆ σαβψβ, has to be

ulated along x direction. The density stipe in the less

populated spin state is caused by the immiscible nature

of the two-component condensate.

More remarkably, we observe a vortex phase (VP) for

intermediate ∆ωL values. As shown in the second and

third rows of Fig. 2, a singly-quantized vortex appears in

spin-up state due to the SO coupling in theˆH(2)

of the MDDI. In the VP, the atom numbers in spin-up

and -down states become comparable [Fig. 4(a)], which

allows the spin s(r) of the atom to form significant trans-

verse components. As shown in Fig. 3, since the MDDI

is minimized with a head-to-tail spin configuration, the

transverse component of s(r) are forced to form a spin

vortex. Consequently, the wave function ψ↑develops a 2π

phase winding, representing a vortex state. The reason

that the vortex state only appears on the spin-up com-

ponent is due to the immiscibility of our two-component

system, which results a density depletion at the center

of ψ↑. Therefore, forming a vortex in spin-up state costs

less kinetic energy. Moreover, in the VP, the phase stripes

also appear in the low density regions of both spin states,

which is the manifestation of the SO coupling induced by

the light fields.

d

term

To determine the phase boundaries, we plot the

∆ωL dependences of the reduced atom number,? Nα =

α

−i?(x∂

lar momentum. As can be seen, the phase boundaries are

marked by two critical ∆ωL values, ∆ω∗

and ∆ω∗∗

L

= 0.088EL/?.For ∆ω∗

the condensate lies in the VP; otherwise, it is in the PP.

Within the VP, atom numbers and orbital angular mo-

mentum change dramatically as one varies ∆ωL.

N−1?dr|ψα|2, and the average orbital angular momen-

∂y− y∂

tum,¯Lz= N−1?

?drψ∗

αˆLzψα, in Fig. 4, whereˆLz=

∂x) is the z-component of the orbital angu-

L= 0.031EL/?

L< ∆ωL < ∆ω∗∗

L,

In conclusion, we have proposed an experimental

scheme to generate SO coupling in spin-3 Cr condensates

via Raman processes. Optical Stark shift is employed to

selecting two spin states from atom’s ground electronic

manifold. The proposed scheme should be readily re-

alizable experimentally. Subsequently, the ground state

structures of a SO-coupled Cr condensate have been in-

vestigated. We show that the interplay between the light

fields and the MDDI gives rise to the polarized and vortex

phases. In particular, spontaneous spin vortex is formed

in VP. Compared to its spin-3 counterpart, the pseudo

spin-1/2 Cr condensate has the advantage of a much sim-

pler form of contact interaction, such that the spin vor-

tex is readily observable. Finally, we point out that our

scheme should also apply to the Dy atom [30], which has

an even larger dipole moment.

This work was supported by the NSFC (Grant Nos.

11025421, 11174084, and 10935010) and National 973

program. We thank the helpful discussions with Ruquan

Wang.

Page 5

5

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