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 . 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 , the syn-
thetic magnetic fields , and the electric forces  in
ultracold Rb gases through Raman processes , which
differs from most dark-state based theoretical propos-
als  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 . 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 . Moreover, when atom’s spin degree of
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 . 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
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 =
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
∆c+ U1+ U2
ΩR(X∗+ XT∗) 2∆c+ U1+ 2U2
ΩR(X + X∗T)U2T
ΩR(X + X∗T)U2T
ΩR(X∗+ XT∗) 3∆c+ U1+ 2U2
ΩR(X + X∗T)U2T
ΩR(X∗+ XT∗) 4∆c+ U1+ 2U2 ΩR(X + X∗T)
ΩR(X∗+ XT∗) 5∆c+ U1+ U2
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  using the level diagram,
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:
1,2/∆ are the optical Stark shifts
1/∆ and assuming
ˆI + ?
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,
Unlike the NIST group’s scheme , 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 , the single-
particle Hamiltonian can be recast into
1/∆ is used to compensate the linear Zee-
ˆI + 2κqˆ σz+ ?ΩRˆ σx−?∆ωL
ˆ σz, (3)
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 . It
can be readily shown that, after dropping the constant
ELterm, the eigenenergies of Eq. (3) are
=px/? is the quasi-momentum, EL
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 , 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
L) when ?ΩR ?
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
?ˆh + V (r) − µ
⊥(x2+ y2+ γ2z2)/2 is an axially
We note that ˆH0
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 . 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
The MDDI for the pseudo spin-1/2 system can be de-
|r − r′|3Y2,0(ˆ e)
|r − r′|3
here, gd= µ0g2
meability and µBthe Bohr magneton, ˆ e = (r−r′)/|r−r′|
is an unit vector, and we have adopted the notations
B/(4π) with µ0being the vacuum per-
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-
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
represent the intra- and inter-
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).
00.03 0.06 0.090.12
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
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
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.
To determine the phase boundaries, we plot the
∆ωL dependences of the reduced atom number,? Nα =
lar momentum. As can be seen, the phase boundaries are
marked by two critical ∆ωL values, ∆ω∗
= 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-
αˆLzψα, in Fig. 4, whereˆLz=
∂x) is the z-component of the orbital angu-
L< ∆ωL < ∆ω∗∗
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 , which has
an even larger dipole moment.
This work was supported by the NSFC (Grant Nos.
11025421, 11174084, and 10935010) and National 973
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