Coherent dynamics of domain formation in the Bose Ferromagnet
Coherent dynamics of domain formation in the Bose Ferromagnet
Qiang Gu1,2, Haibo Qiu1,3
Department of Physics, University of Science and Technology Beijing, Beijing 100083, China
Institut für Laser-Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China
(Dated: November 20, 2018)
We present a theory to describe domain formation observed very recently in a quenched 87Rb
gas, a typical ferromagnetic spinor Bose system. An overlap factor is introduced to characterize
the symmetry breaking of MF = ±1 components for the F = 1 ferromagnetic condensate. We
demonstrate that the domain formation is a co-effect of the quantum coherence and the thermal
relaxation. A thermally enhanced quantum-oscillation is observed during the dynamical process
of the domain formation. And the spatial separation of domains leads to significant decay of the
MF = 0 component fraction in an initial MF = 0 condensate.
PACS numbers: 05.30.Jp, 03.75.Kk, 03.75.Mn
Very recently, the Berkeley group observed sponta-
neous symmetry breaking in 87Rb spinor condensates [1].
Ferromagnetic domains and domain walls were clearly
shown using an in-situ phase-contrast imaging. This ap-
pears the first image of the domain structure in a Bose
ferromagnet. Although ferromagnetism has been inten-
sively studied in the context of condensed matter physics
and is regarded as one of the best understood phenom-
ena in nature [2], the description of ferromagnetism is
not yet complete. The conventional ferromagnets being
considered are usually comprised of either classical parti-
cles (insulating ferromagnets) or fermions (itinerant fer-
romagnets) while Bose systems are seldom touched [3].
The realization of cold spinor 87Rb gases [4], a typical
ferromagnetic Bose system, has provided an opportunity
to study Bose ferromagnets and thus opens up a way to
a comprehensive understanding of ferromagnetism in all
kinds of condensed matters.
The ferromagnetic spinor Bose gas has attracted nu-
merous theoretical interests [5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15]. On one side, researchers expect that it will show
some general properties as conventional ferromagnets do.
Ho [5], Ohmi and Machida [6] pointed out that this sys-
tem has a spontaneous symmetry-broken ground state
and a normal spin-wave excitations spectrum at small
wave vector k, ωs = csk
2. On the other side, researchers
aim at exploring distinct features of the system. Studies
on thermodynamics and phase transitions have revealed
that the ferromagnetic spinor Bose gas displays a quite
surprising phase diagram. Its Curie point can be larger
by magnitudes than the energy scale of the ferromag-
netic interaction between bosons, and never below the
Bose-Einstein condensation point [7, 8]. It means that
once the Bose gas condenses, it is already spontaneously
magnetized.
A conventional ferromagnet usually has some domain
structure below the Curie point, as illustrated in Fig. 1a.
But whether it is true for a Bose ferromagnet is still some-
what controversial. One even questions whether there
exists a Curie point in ferromagntic Bose gases [9, 10],
as the cold atomic gas under experimental conditions is
usually not in the thermodynamic limit while the phase
diagram mentioned above is derived from the thermal-
equilibrium grand canonical ensemble [7, 8]. Never-
theless, a number of theoretical works have discussed
the possibility of the domain formation [3, 13, 14, 15].
Within the mean-field theory, Zhang et al. found out
that the ferromagnetic condensate has a dynamical in-
stability leading to spontaneous domain formation in an
initially magnetized Bose gas [13]. Moreover, Mur-Petit
et al. showed that a multi-spin-domain structure mani-
fests in 87Rb condensates at finite temperatures[14]. The
Berkeley experiment confirms that the Bose ferromagnet
can indeed form domain structures at least under certain
conditions [1]. Then new questions come, is the process
of the domain formation similar to that inside a conven-
tional ferromagnet, and how does the domain formation
affect spin dynamics? We attempt to answer these ques-
tions in the present letter.
We start with the Hamiltonian for an F = 1 system in
the following form [5]
∇ψ†a · ∇ψa − (µ− U)ψ†aψa
ψa′ψa +
Fab · Fa′b′ψb′ψb
where ψa(r) is the field annihilation operator for an atom
in state MF = a at point r, µ is the chemical potential
and U is the trapping potential. g0 and g2 are the spin-
independent and spin-dependent mean-field interaction,
respectively. For macroscopically occupied Bose systems
it is common to replace the field annihilation operator
for the ath spin component by its expectation value, i.e.
ϕa(r, t) ≡ 〈ψa(r, t)〉, which for spinor condensates is con-
http://arxiv.org/abs/0704.1995v2
(d)(c)
(b)(a)
FIG. 1: Schematic domain structure inside a Ferromagnet.
(a) The bulk material shows no magnetism because domains
are randomly oriented. (b) The multi-domain configuration is
often simplified to a two-domain structure with opposite sign
of magnetization for theoretical convenience. The gray region
denotes the domain wall. (c) Two-domain structure for a fer-
romagnetic spinor Bose-Einstein condensate. Dotted, dashed
and solid lines represent the normalized particle distributions
of MF = 1, MF = −1 and MF = 0 bosons respectively. (d) In
a homogeneous system, the normalized particle distribution
is taken as a constant in each domain; for MF = 0 bosons it
remains a constant through the whole system.
veniently expressed as
ϕa(r, t) =
Na(t)ηa(r, t)e
iφa(r,t). (2)
Here Na(t) is the number of condensed particles,
ηa(r, t) denotes the normalized particle distribution with
d3rηa = 1 and φa(r, t) the phase. The multiplier
of Na and ηa refers to the condensed particle density
of the a-component, and the total particle density is
ρ(r, t) = Na(t)ηa(r, t). We suppose that the particle
distribution can be different for distinct components in
our approach. Neglecting excitations, the Hamiltonian is
simplified to H = H0 +Hu +Hs, with
Naηa(∇φa)2
(U − µ)Naηa +
NaηaNa′ηa′
− − 2N+N−η+η−
+2N0η0(N+η+ +N−η−)
+ 4N0
N+N−η0
η+η−cosθ
, (3)
where θ = φ+ + φ− − 2φ0 is the relative phase.
In case that the ground state of the condensate is
symmetry-broken, certain magnetic domain structure is
formed spontaneously. Within each domain the atomic
magnetic moments are aligned in a preferential direc-
tion. A direct consequence of domain formation is that
the MF = 1 and MF = −1 components are spa-
tially separated, as portrayed in Fig. 1c. The integral
d3rη+η− measures the extent of overlap between the
two, which is called the overlap factor hereinafter. In
general, several “overlap factors” should be introduced
to Eqs.(3), e.g., α0± = V
d3rη2±, α1 = V
d3rη+η−,
α2± = V
d3rη0η± and α3 = V
d3rη0
η+η−, where V
is the volume of the system. Treating the relative phase
θ as a spatially independent constant as previous theory
did [11, 12, 13, 14, 15], the term of Hs in Eqs. (3) is
rewritten as
+ + α0−N
− − 2α1N+N−
+2N0(α2+N+ + α2−N−)
+ 4α3N0
N+N−cosθ
. (4)
According to their definition, these overlap factors are
not totally independent from each other and the number
of independent ones can be further reduced. For simplic-
ity, we consider a homogeneous spinor Bose gas with a
two-domain structure, as shown in Fig. 1d, and neglect
the domain wall. In this case, we derive, after some in-
tegration and algebraic manipulation, that there is only
one independent overlap factor and the above equation
is reduced to
(2− α)(N2+ +N2−)− 2αN+N−
+ 2N0(N+ +N−) + 4
N+N−cosθ
, (5)
where the reduced overlap factor is just given by α =
d3rη+η−. Our model allows the α-factor to vary from
one to zero, corresponding to the case that the two com-
ponents are from thoroughly mixed to completely sep-
arated. For a homogeneous system, the gradient term
H0 can be doped. The spin-irrelevant term Hu remains
a constant since the total density distribution is hardly
responsive to the evolution of domain structure [1, 15],
ρ(r, t) ≈ ρ(r, 0). Thus the dynamics of domain formation
is only determined by the Hamiltonian expressed in Eq.
The Berkeley experiment considered a pure spinor con-
densate initially prepared in the unmagnetized state. It
is important to emphasize that the total spin is con-
served in an atomic quantum gas under experimental
conditions[16, 17, 18]. Therefore the particle numbers of
MF = 1 and −1 component are always equal, N+ = N−.
Such a system is described by the Hamiltonian
Hs = −(1− α)(1 − n0)2 − 2n0(1− n0)
αcosθ
Here Hs = Hs/(N | g22 |) with total particle number N =
N+ + N− + N0; na = Na/N is the fraction of MF = a
component. n0 and θ form a pair of conjugate variables
0.0 0.2 0.4 0.6 0.8 1.0
-2 -1 0 1 2
-2 -1 0 1 2
FIG. 2: Contour plot of the energy surface in the θ-n0 plane
with the overlap factor α = 0.5 (a) and α = 0.1 (b). (c)
shows the fraction of MF = 0 atoms at the minima, n̄0, as a
function of α.
and their equations of motion are given by
n0 = −2
αn0(1− n0)sinθ, (7a)
θ = 2(1− α)(1 − n0)
−2(1− 2n0)
αcosθ
. (7b)
The population dynamics depicts the process of the do-
main formation; and the magnetization of magnetic do-
mains is defined as m = N+η+ −N−η−.
A number of theoretical works have investigated the
spin dynamics of the 87Rb spinor condensate [11, 12, 13].
We notice that previous theories usually treated the three
components being mixed as they share the same spatial
wave function, which is known as the single-mode approx-
imation. Therefore the domain structure is smeared out
and thus the ferromagnetic feature was not sufficiently
dealt with. Those theories correspond the special case of
α = 1 in this letter.
The dynamical behaviors of Eqs. (7) can be visual-
ized by the phase-space portrait with constant energy
lines. Figure 2a and 2b plot the contour lines of en-
ergy with the overlap factor α = 0.5 and 0.1, respec-
tively. In case of α = 1, there are energy minima along
n0 = 1/2 at θ = 2nπ [11, 12, 13] and all the contour
lines are closed loops around those minima. As domains
0 30 60 90 120
0 30 60 90 120
0 30 60 90 120
0 30 60 90 1200.0
0 30 60 90 1200.0
0 30 60 90 120
1.0  
(f)(e)
(d)(c)
(b)(a)
FIG. 3: Dynamics of domain formation for Bose systems with
the overlap factor α = 0.1 and TR = 20, 10, and 2 from top
to bottom. The left column shows the particle fractions of
MF = 0 (n0, solid lines) and MF = ±1 (n±, dotted lines)
components. The right column shows the magnetization of
magnetic domains, m. The initial state configuration is (n+ =
0.005, n0 = 0.99, n− = 0.005).
build up, the value of α drops down and two apparent
changes take place, seen in Fig. 2a and 2b. (i) There
exist two distinct regimes in the phase space diagram.
The newly appeared regime lies in the upper region of
the figures, which consists of a set of open curves. Each
line corresponds to a rotation type of solution, in which
the relative phase θ is ”running” with the time. The
closed orbits, lying in the lower region, represent libra-
tion type of solutions, in which θ oscillates around the
minimum. (ii) The positions of those minima move to-
wards to smaller values of n0. The minima points are
connected to the ground state of the system. Figure
2c shows the MF = 0 particle fraction at the minima,
n̄0. The less α is, the smaller n̄0 is. It means that
the domain formation tends to reduce the number of
MF = 0 atoms. This point has also been affirmed by
Mur-Petit et al. who obtained a state with equipartition
in populations, (n+ ≈ 1/3, n0 ≈ 1/3, n− ≈ 1/3), from
a starting state (0.005,0.99,0.005) [14]. We derive that
n0 = n+ = n− = 1/3 when α drops to 0.25. In the limit
case of α = 0, n0 = 0 and n+ = n− = 0.5.
The state of the quenched MF = 0 condensate is
viewed as a point lying in the upper region of the phase
space diagram (Fig. 2a or 2c), when Eqs. (7) yield a self-
trapping solution[19]. This motion reflects the quantum
mechanical nature of the Bose-Einstein condensate. The
oscillating amplitude of n0 is very small and the resulting
magnetization m is so low in magnitude that it is hard to
be probed experimentally during this stage[1]. The self-
trapping effect prevents the growth of magnetic domains.
This case is similar to the classical Larmor precession of
a spin around magnetic field: it is rotating all the time,
but the spin direction can be never parallel to the field
without energy dissipation.
Then one has to take into consideration the effect of
thermal agitation, which can change the energy of the
system and drive the system into thermal equilibrium.
Therefore, if n0 departs from its thermal equilibrium
value n̄0, it will relaxes to n̄0 exponentially with a charac-
teristic time scale TR, called the relaxation time. Assum-
ing that the relaxation velocity is proportional to n̄0−n0,
∂n̄0/∂t ∝ n̄0−n0, Eq. (7a) can be replaced with the fol-
lowing one [20],
n0 = −2
αn0(1− n0)sinθ +
n̄0 − n0
. (8)
TR scales qualitatively the thermal dissipation rate.
Longer TR denotes weaker thermal agitation.
Figure 3 displays the population of MF = 0 and ±1
components, as well as the magnetization m. The evo-
lution of n0 and m reflects the dynamical process of the
domain formation. As shown in Fig. 3a and 3b, n0 de-
creases oscillatorily with time driven by the the thermal
agitation, and meantime m arises. A very interesting re-
sult is that the oscillation amplitude increases as the sys-
tem relaxes. Generally, the thermal agitation suppresses
the macroscopic quantum coherence, and thus tends to
kill oscillations of the population, while here we see that
the oscillation is enhanced. Furthermore, the magnetiza-
tion persists in oscillating for a very long period of time
after the amplitude reaches its maximum. This result is
qualitatively consistent with the Berkeley group’s obser-
vation of the unstable magnetization mode[1].
If the thermal agitation gets stronger, the oscillation
will be enhanced first, and then suppressed, as shown in
Fig. 3c and 3d. Correspondingly, we divide the whole
process into two periods with respect to the domain for-
mation, the growing period and the stabilizing period. In
the latter period, n0 andm oscillate around their thermal
equilibrium values and the amplitudes decrease gradu-
ally, then we have a stable domain structure eventually.
If sketching the solution in the phase space diagram, one
can find that the growing period is represented by the
trajectory in the libration regime, and solutions for the
stabilizing period lie in the rotation regime. Based on
this understanding, Fig. 3a and 3b show only the grow-
ing period. Given the thermal agitation strong enough,
the quantum mechanical feature will be smeared out in
both periods, as Fig. 3e and 3f indicate.
According to the above discussions, the present model
can qualitatively describe the dynamic process of the do-
main formation in a quenched MF = 0 condensate. Sig-
nificantly, we show that the spatial separation of mag-
netic domains brings about much nontrivial effects on the
spin dynamics of the ferromagnetic condensate. To get a
quantitative description, more local details of the particle
distribution and the relative phase should be considered.
In conclusion, we have investigated the dynamics of do-
main formation in a ferromagnetic spinor Bose-Einstein
condensate, taking into account of the symmetry-
breaking of the MF = 1 and −1 components. Magnetic
domains develop with the separation ofMF = ±1 compo-
nents. Our results suggest that the MF = 0 component
in the condensate can significantly decay to a very small
value, far less than 1/2 as previous theories predicted.
The domain structure is formed and stabilized with the
help of the thermal dissipation. A thermally enhanced
quantum-oscillation is observed during the process.
This work is supported by the National Natural Sci-
ence Foundation of China (Grant No. 10504002), the
Fok Yin-Tung Education Foundation, China (Grant
No. 101008), and the Ministry of Education of China
(NCET-05-0098). Q.G. acknowledges helpful discussions
with K. Sengstock, K. Bongs and L. You and support
from the Deutsche Forschungsgemeinschaft through the
Graduiertenkolleg No. 463.
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