Spinor dipolar Bose-Einstein condensates; Classical spin approach
Spinor dipolar Bose-Einstein condensates; Classical spin approach
M. Takahashi1, Sankalpa Ghosh1,2, T. Mizushima1, K. Machida1
Department of Physics, Okayama University, Okayama 700-8530, Japan and
Department of Physics, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi 110016, India
(Dated: October 26, 2018)
Magnetic dipole-dipole interaction dominated Bose-Einstein condensates are discussed under spin-
ful situations. We treat the spin degrees of freedom as a classical spin vector, approaching from
large spin limit to obtain an effective minimal Hamiltonian; a version extended from a non-linear
sigma model. By solving the Gross-Pitaevskii equation we find several novel spin textures where
the mass density and spin density are strongly coupled, depending upon trap geometries due to the
long-range and anisotropic natures of the dipole-dipole interaction.
PACS numbers: 03.75.Mn, 03.75.Hh, 67.57.Fg
Bose-Einstein condensates (BEC) with internal de-
grees of freedom, the so-called spinor BEC have attract
much attention experimentally and theoretically in re-
cent years [1]. Spinor BEC opens up a new paradigm
where the order parameter of condensates is described
by a multi-component vector [2, 3]. This can be possi-
ble by optically trapping cold atoms where all hyperfine
states are liberated, while magnetic trapping freezes its
freedom. So far 23Na (the hyperfine state F = 1), and
87Rb (F = 2) are extensively investigated.
Griesmaier et al. [4] have recently succeeded in achiev-
ing BEC of 52Cr atom gases whose magnetic moment per
atom is 3 µB (Bohr magneton). There has been already
emerging [5] several novel aspects associated with larger
magnetic moment in 52Cr atom even in this magnetic
trapping, where all spin moments are polarized along an
external magnetic field. Namely the magnetic dipole-
dipole (d-d) interaction, which is proportional to F 2 is
expected to play an important role in a larger spin atom.
It is natural to expect realization of BEC with still
larger spin atomic species under the spinful situations
by optical trapping or control the d-d interaction via the
Feshbach resonance relative to other interaction channels.
There has already been existing a large amount of theo-
retical studies for dipolar BEC [6]. Most of them treat
the polarized case where the dipolar moments are aligned
along an external field. The intrinsic anisotropic or ten-
sorial nature of the d-d interaction relative to the polar-
ization axis manifests itself in various properties. The
head-to-tail moment arrangement due to the d-d interac-
tion is susceptible to a shape instability by concentrating
atoms in the central region. We have seen already that
tensorial and long-ranged d-d interaction is responsible
for this kind of shape dependent phenomenon where the
mass density is constrained by the polarization axis.
In contrast the theoretical studies of the spinor dipolar
BEC are scarce, and just started with several impressive
works [7, 8, 9, 10]. They consider either the F = 1 spinor
BEC by taking into account the d-d interaction or F = 3
for 52Cr atom gases in a realistic situation. Here one must
handle a 7-component spinor with 5 different interaction
channels g0, g2, g4, g6, and gd. The parameter space to
hunt is large and difficult enough to find a stable con-
figuration. The situation becomes further hard towards
a larger F where the d-d interaction is more important
and eventually dominant one among various channels.
Here we investigate generic properties of the spinor
dipolar BEC under an optical trapping where the d-d
interaction dominates other interactions except for the
s-wave repulsive channel. A proposed model Hamilto-
nian is intended to capture essential properties of the
spinor BEC system. We note that this long-ranged and
anisotropic d-d interaction has fascinated researches for
a long time, for example, Luttinger and Tisza in their
seminal paper [11] theoretically discussed the stable spin
configurations of a spin model on a lattice where classi-
cal spins with a fixed magnitude are free to rotate on a
lattice. The present paper is designed to generalize this
lattice spin model to a dipolar BEC system. Here we are
interested in the interplay between the spin degrees of
freedom and the mass density through the d-d interac-
tion.
We approach this problem from atomic species with
large magnetic moment. This spinor dipole BEC with the
hyper fine state F (Fz = −F,−F+1, · · · , F ) is character-
ized by 2F+1 components Ψα(r). In general the number
of the interaction channels are F + 1. For example, the
F = 1 spinor BEC [2, 3] is characterized by the scattering
lengths a0 and a2, leading to the spin independent repul-
sive interaction g0 = 4πh̄
2(a0 + 2a0)/3m and the spin
dependent exchange interaction g2 = 4πh̄
2(a2 − a0)/3m.
Since a0 and a2 are comparable, g2 is actually small;
|g2|/g0 ∼ 1/10 for 23Na [12, 13] and ∼ 1/35 for 87Rb
[14, 15]. This tendency that, except for the dominant re-
pulsive part g0, other spin-dependent channels are nearly
cancelled is likely to be correct for other F ’s [16].
We can take a view in this paper that instead of work-
ing with ~Ψ(r) full quantum mechanical 2F + 1 compo-
nents (ΨF ,ΨF−1, · · · ,Ψ−F ) with the interaction param-
eters g0, g2, g4, ..., and g2F , the order parameter can be
simplified to ~Ψ(ri) = ψ(ri)~S(ri) where ~S(ri) is a clas-
sical vector with |~S(ri)|2 = 1. Namely we can treat it
http://arxiv.org/abs/0704.0122v2
as the classical spin vector whose magnitude |ψ(ri)|2 is
proportional to the local condensate density. In other
words, we focus on long-wavelength and low energy tex-
tured solutions of a dipolar system which will manifest
the interplay between the mass and spin density degrees
of freedom.
We start with the following minimal model Hamilto-
d3ri~Ψ
†(ri)H0(ri)~Ψ(ri)
d3rid
rjVdd(ri, rj)|ψ(ri)|2|ψ(rj)|2,(1)
H0 = −
∇2i + Vtrap(ri)− µ+
|~Ψ(ri)|2, (2)
Vdd(ri, rj) =
~Si · ~Sj − 3(~Si · ~eij)(~Sj · ~eij)
, (3)
where ~eij ≡ (ri − rj)/rij with rij = |ri − rj |. The uni-
axially symmetric trap potential is given by Vtrap(r) =
mω2{γ(x2 + y2) + z2} with γ being the anisotropy pa-
rameter. µ is the chemical potential. The repulsive
(g > 0) and the dipole-dipole (gd) interaction are intro-
duced. The classical spin vector ~Si ≡ ~S(ri) characterizes
the internal degrees of freedom of the system at the site
i and is denoted by spherical coordinates (ϕ(ri), θ(ri))
with |~Si|2 = 1. A dimensionless form of this Hamiltonian
may be written as
|∇ψ(ri)|2 + ni
∇θ(ri)
+ sin2 θ(ri)
∇ϕ(ri)
+ γ2(x2 + y2) + z2
− 2µni + gn2i
d3rid
~Si · ~Sj − 3(~Si · ~eij)(~Sj · ~eij)
ninj , (4)
with |ψ(ri)|2 = ni. We note that the spin gradient term
in the first line is a Non-linear sigma model[17]. Here it
is extended to include the dipole-dipole interaction be-
tween the different parts of the spin density. The energy
(length) is measured by the harmonic frequency ω (har-
monic length d ≡ 1/
mω) with h̄ = 1 The functional
derivatives with respect to ψ∗(ri), ϕ(ri) and θ(ri) lead
to the corresponding Gross-Pitaevskii equations.
In this paper under a fixed repulsive interaction
(g/ωd3 = 0.01) we vary the d-d interaction gd in a range
of 0 ≤ gd ≤ 0.4g, beyond which the system is unsta-
ble. We consider two types of the confinement: A pan-
cake (γ = 0.2) and a cigar (γ = 5.0) to see the shape
dependence of the d-d interaction, which is long-ranged
and anisotropic. The total particle number ∼ 104. The
three dimensional space is discretized into the lattice sites
∼ 2.5 × 104. Using the imaginary time (τ) evolution of
Gross-Pitaevskii equations e.g. ∂ψi/∂τ = −δH/δψ∗i , we
obtain stable configurations for spin and particle densi-
ties by starting with various initial patterns.
We start with the pancake shape (γ = 0.2). Figure 1
shows a stereographic image of the particle density and
spin distributions. We call it spin current texture, where
the spin direction circulates around the origin and is con-
fined into the x-y plane without the third component,
that is, a coplanar texture. It is seen that the particle
density distribution is strongly coupled to the spin one;
FIG. 1: Stereographic view of the spin current texture, dis-
playing simultaneously the number and spin densities. The
pancake (γ = 0.2) is distorted and at the center the number
density is depleted to give a doughnut like shape. gd = 0.2g.
All spins lie in the x-y plane, i.e. a coplanar spin structure,
circulating around the origin O. The length of the arrow is
proportional to its number density. Inset shows the schematic
spin configuration on z = 0 plane.
In the central region the particles are depleted over the
coherent length ξd of the d-d interaction. In the present
case ξd ∼ 2.0ξc (ξc is the ordinary coherent length of the
FIG. 2: The r-flare texture. Left (right) column shows the
cross-sectional density plots of the particle number (the corre-
sponding spin structure). The circular profile in the x-y plane
is spontaneously broken. gd = 0.2g, γ = 0.2.
repulsive interaction).
This spin current texture can be readily explained in
the following way: (1) Locally, along the stream line of
the spin current the head-to-tail configuration minimizes
the energy. (2) Globally, the spins at A and B which are
situated far apart about the origin O shown in inset of
Fig. 1 are orientated anti-parallel to minimize the d-d in-
teraction. (3) When the two antiparallel spins at A and
B come closer towards the origin O, the kinetic energy
due to the spin modulation increases. To avoid this en-
ergy loss, the particle number is depleted in the central
region at the cost of the harmonic potential energy.
For an alternative explanation of the spin current
texture we rewrite the d-d interaction as vdd(rij) ∝
Y2µ(cos θ)Σµ(ij) with Σµ(ij) being a rank 2
tensor consisting of the two spins at i and j sites, and
Y2µ(cos θ) a spherical harmonics [18]. θ is the polar angle
in spherical coordinates of the system. The spin current
texture shown in inset of Fig. 1 picks up the phase factor
e2iϕ when winding around the origin. This is coupled to
Y2±2(cos θ) ∝ sin2 θ, meaning that this orbital moment
dictates the number density depletion at the pancake cen-
ter. The spin-orbit coupling directly manifests itself here.
The total angular momentum consisting of the spin and
orbit ones is a conserved quantity of the present axis-
symmetric system, leading to the Einstein-de Haas effect
[7]. The spin current texture is stable for the wide range
of anisotropy γ: 0.01 ≤ γ ≤ 0.6, beyond which it becomes
unstable.
Figure 2 displays another stable configuration in a sim-
ilar situation. The left (right) column shows the den-
sity plots of the particle number (the corresponding spin
structure). The spins are almost parallel to the x-axis,
but at the outer region they bent away. We call it r-flare
texture, which is a non-coplanar spin arrangement. It
is clearly seen that the axis-symmetry in the x-y plane,
FIG. 3: Cross sections of the particle number in Fig. 2 along
the x and y-axis compared with Thomas-Fermi (TF) profile
for gd = 0. The profile is elongated (compressed) along the x
(y)-axis.
which was originally circular, is spontaneously broken so
that the circular shape is elongated along the x-axis and
compressed along the y-axis. Figure 3 displays the x
and y-axis cross-sections of the particle density, com-
pared with the Thomas-Fermi (TF) profile for gd = 0
with the same particle number. Because of the d-d in-
teraction which favors the head-to-tail arrangement, the
particle number is increased at the center. The bending
tendency at the circumference increases with increasing
gd. Beyond a certain critical value gd ∼= 0.27g for ∼ 104
particles, the r-flare texture becomes unstable, indicating
a quantum phase transition. Upon increasing the total
particle number the r-flare is replaced by the spin current
texture. We also note that the z-flare texture in which
the polarization points to the z-axis is equally stable as
we explain shortly.
Let us turn to the cigar shape case elongated along
the z-axis with the trap anisotropy γ = 5.0. The stable
configuration we obtain is shown in Fig. 4 where the spin
structure is basically a flare spin texture which is a non-
coplanar spin arrangement. Namely, the bending occurs
radially so that the spin texture is a three dimensional
object, but keeps axis-symmetry around the z-axis. The
particle density is modified from the TF profile for gd = 0,
elongated along the z direction and compressed to the z-
axis.
This can be understood by seeing Fig. 4 (b). The up-
spin density near the center exerts the d-d force so as to
align the outer spins parallel to the vector connecting the
center and its position, taking the head-to-tail configu-
ration. This results in a non-coplanar structure, but the
axis-symmetry about the z-axis is preserved. This spin
texture is stable for gd ≤ 0.3g and robust for different
aspect ratios: γ = 0.2 and 1.5. The bending angle of
FIG. 4: (a) The z-flare spin texture in the cigar trap along
the z-axis. The spins almost point to the z direction. In
the outer regions they bent. The bright region in background
corresponds to high number density. gd = 0.2g, γ = 5.0. (b)
Schematic figure to explain this spin configuration due to d-d
interaction.
FIG. 5: (a) The two-z-flare spin texture under the same
parameter set (gd = 0.2g, γ = 5.0.) as in Fig. 4 with different
initial spin configuration. The bright region in background
corresponds to high number density. (b) Schematic figure to
explain this spin structure. At the z = 0 plane two oppositely
aligned spins meet and the number density is depleted.
the flare spin texture increases and the elongation along
the z direction becomes larger as gd increases (= 0.1 and
0.2).
Finally we display an example to show how the model
Hamiltonian admits many subtle spin textures with com-
parable energies. Figure 5 (a) shows the two-z-flares op-
positely polarized stacked back to back. This configura-
tion is stabilized starting with a hedgehog spin config-
uration, or skyrmion at the center from which all the
spins point outward from the origin. In the end the
two-z-flares oppositely polarized become stable, but at
the central z = 0 plane the antiparallel spins meet as
seen from Fig. 5 (b). To avoid drastic changes of the
spin direction, or the spin kinetic energy loss, the parti-
cle density decreases there. As a result even though the
harmonic potential energy is minimal there, the two-z-
flare spin textures oppositely polarized are stacked back
to back, but two objects are almost split. This example
illustrates strong coupling between the particle number
and spin densities through the d-d interaction.
These spin textures can be observed directly via a novel
phase-sensitive in situ detection [1] or indirectly via con-
ventional absorption imaging for the number density. It
is interesting to examine the vortex properties under ro-
tation. For the spin current texture, the vortex entry
into a system should be easy because in the central region
the mass density is already depleted. We point out that
the collective modes might be also intriguing because the
mass density is tightly coupled with the spin degrees of
freedom. These problems belong to future work.
In summary, we have introduced a model Hamiltonian
to capture the essential nature of dipolar spinor BEC
where the spin magnitude is large enough, focusing on
long wavelength and low energy textured solutions. We
show several typical stable configurations by solving the
Gross-Pitaevshii equation where the spin and mass den-
sities are strongly coupled due to the dipole-dipole inter-
action. The shape of the harmonic potential trapping is
crucial to determine the spin texture. The model Hamil-
tonian is a minimal extension of the Non-linear sigma
model with the d-d interaction, and yet complicate and
versatile enough to explore further because it is expected
that there are many stable configurations with compara-
ble energies. Finally the model Hamiltonian is applica-
ble literally for electric dipolar systems without further
approximation. We expect that BEC of hetero-nuclear
molecules with permanent electric dipole moment might
be realized in near future [19] where the formation of such
textures may be possible.
We thank Tarun K. Ghosh and W. Pogosov for useful
discussions in the early stage of this research. This work
of S. G. was supported by a grant of the Japan Society
for the Promotion of Science.
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