Spinor Dynamics in an Antiferromagnetic Spin-1 Condensate
A. T. Black, E. Gomez, L. D. Turner, S. Jung, and P. D. Lett
Joint Quantum Institute, University of Maryland and
National Institute of Standards and Technology, Gaithersburg, Maryland 20899
(Dated: October 30, 2018)
We observe coherent spin oscillations in an antiferromagnetic spin-1 Bose-Einstein condensate
of sodium. The variation of the spin oscillations with magnetic field shows a clear signature of
nonlinearity, in agreement with theory, which also predicts anharmonic oscillations near a critical
magnetic field. Measurements of the magnetic phase diagram agree with predictions made in the
approximation of a single spatial mode. The oscillation period yields the best measurement to date
of the sodium spin-dependent interaction coefficient, determining that the difference between the
sodium spin-dependent s-wave scattering lengths af=2−af=0 is 2.47 ± 0.27 Bohr radii.
PACS numbers: 03.75.Mn, 32.80.Cy, 32.80.Pj
Atomic collisions are essential to the formation of Bose-
Einstein condensates (BEC), redistributing energy dur-
ing evaporative cooling. Collisions can be coherent and
reversible, leading to diverse phenomena such as super-
fluidity [1] and reversible formation of molecules [2] in
BECs with a single internal state. When internal de-
grees of freedom are included (as in spinor condensates),
coherent collisions lead to rich dynamics [3, 4] in which
the population oscillates between different Zeeman sub-
levels. We present the first observation of coherent spin
oscillations in a spin-1 condensate with antiferromagnetic
interactions (in which the interaction energy of colliding
spin-aligned atoms is higher than that of spin-antialigned
atoms.)
Spinor condensates have been a fertile area for the-
oretical studies of dynamics [5, 6, 7, 8], ground state
structures [9, 10], and domain formation [11]. Extensive
experiments on the ferromagnetic F=1 hyperfine ground
state of 87Rb have demonstrated spin oscillations and co-
herent control of spinor dynamics [3, 12]. Observation of
domain formation in 23Na demonstrated the antiferro-
magnetic nature of the F=1 ground state [13] and de-
tected tunneling across spin domains [14]; no spin oscilla-
tions have been reported in sodium BEC until now. The
F=2 state of 87Rb is thought to be antiferromagnetic, but
a cyclic phase is possible [15, 16]. Experiments on this
state have demonstrated that the amplitude and period
of spin oscillations can be controlled magnetically [4].
At low magnetic fields, spin interactions dominate the
dynamics. The different sign of the spin dependent in-
teraction causes the antiferromagnetic F=1 case to differ
from the ferromagnetic one both in the structure of the
ground-state magnetic phase diagram and in the spinor
dynamics. Both cases can exhibit a regime of slow, an-
harmonic spin oscillations; however, this behavior is pre-
dicted over a wide range of initial conditions only in the
antiferromagnetic case [8]. The spin interaction energies
in sodium are more than an order of magnitude larger
than in 87Rb F =1 for a given condensate density [3],
facilitating studies of spinor dynamics.
The dynamics of the spin-1 system are much simpler
than the spin-2 case [4, 15, 16], having a well-developed
analytic solution [8]. This solution predicts a divergence
in the oscillation period (not to be confused with the
amplitude peak observed in 87Rb F=2 [4] oscillations).
This Letter reports the first measurement of the
ground state magnetic phase diagram of a spinor con-
densate, and the first experimental study of coherent
spinor dynamics in an antiferromagnetic spin-1 conden-
sate. Both show good agreement with the single-spatial-
mode theory [10]. To study the dynamics, we displace
the spinor from its ground state, observing the resulting
oscillations of the Zeeman populations as a function of
applied magnetic field B. At low field the oscillation pe-
riod is constant, at high field it decreases rapidly, and at a
critical field it displays a resonance-like feature, all as pre-
dicted by theory [8]. These measurements have allowed
us to improve by a factor of three the determination of
the sodium F = 1 spin-dependent interaction strength,
which is proportional to the difference af=2 − af=0 in
the spin-dependent scattering lengths.
The state of the condensate in the single-mode ap-
proximation (SMA) is written as the product φ(r)ζ of a
spin-independent spatial wavefunction φ(r) and a spinor
ζ = (
eiθ− ,
iθ0 ,
iθ+). We use ρ
, ρ0, and
ρ+ (θ−, θ0, and θ+) to denote fractional populations
(phases) of the Zeeman sublevels mF = −1, 0, and 1,
so that
ρi=1. The spinor’s ground state and its non-
linear dynamics may be derived from the spin-dependent
part of the Hamiltonian in the single-mode and mean-
field approximations, subject to the constraints that to-
tal atom number N and magnetization m≡ ρ+−ρ− are
conserved [8]. The “classical” spinor Hamiltonian E is a
function of only two canonical variables: the fractional
population ρ0 and the relative phase θ ≡ θ+ + θ− − 2θ0.
It is given by
E = δ(1−ρ0) + cρ0
(1−ρ0) +
(1−ρ0)2−m2 cos θ
where δ = h × (2.77× 1010Hz/T2)B2 is the quadratic
http://arxiv.org/abs/0704.0925v2
Zeeman shift [8] with h the Planck constant. (The linear
Zeeman shift has no effect on the dynamics.) The spin-
dependent interaction energy is c= c2 〈n〉, where 〈n〉 is
the mean particle density of the condensate and
(af=2 − af=0) (2)
is the spin-dependent interaction coefficient [8, 17]. Here
M is the atomic mass. af=2 and af=0 are the s-wave
scattering lengths for a colliding pair of atoms of total
spin f = 2 and f = 0, respectively; Bose symmetry en-
sures there are no s-wave collisions with total spin of
1. If c2 is positive (negative), the system is antiferro-
magnetic (ferromagnetic). The spinor ground state and
spinor dynamics are determined by Eq. (1).
The apparatus is similar to that described previ-
ously [18]. We produce a BEC of 105 23Na atoms in
the F=1 state, with an unobservably small thermal frac-
tion, in a crossed-beam 1070nm optical dipole trap. The
trap beams lie in the horizontal xy plane, so that the trap
curvature is nearly twice as large along the vertical z axis
as in the xy plane. By applying a small magnetic field
gradient with the MOT coils (less than 10mT/m) during
the 9 s of forced evaporation, we fully polarize the BEC:
all atoms are in mF =+1. Conservation of spin angular
momentum ensures that the magnetization remains con-
stant once evaporation has ceased; a state with ρ+ = 1
persists for the lifetime of the condensate, about 14 s.
We then turn off the gradient field and adiabatically
apply a bias field B of 4 to 51µT along x̂, leaving the
BEC in the ρ+ = 1 state. To prepare an initial state,
we apply an rf field resonant with the linear Zeeman
splitting; typically the frequency is tens to hundreds of
kilohertz. Rabi flopping in the three-level system is ob-
served [19], and controlling the amplitude and duration
of the pulse can produce any desired magnetization m,
which also determines the population ρ0. The flopping
time is less than 50µs, much shorter than the character-
istic times for spin evolution governed by Eq. (1). Using
this Zeeman transition avoids populating the F=2 state,
thus avoiding inelastic losses, which are much greater for
23Na than for 87Rb.
We measure the populations ρi of atoms in the three
Zeeman sublevels by Stern-Gerlach separation and ab-
sorption imaging [20]. The Stern-Gerlach gradient is par-
allel to the bias field ~B, while the imaging beam propa-
gates in the ẑ direction. The phase θ is not measured.
To measure the ground state population distribution
as a function of magnetization and magnetic field, we
first set the magnetization using the rf pulse. We then
ramp the field to a desired final value over 1 s, wait 3 s
for equilibration, and measure the populations as above.
Figure 1(b) displays the measured ground-state mag-
netic phase diagram. The theoretical prediction in
Fig. 1(a) is the population ρ0 that minimizes the energy,
B (µT)
B (µT)
FIG. 1: a) Theoretical prediction of the ground-state frac-
tional population ρ0 as a function of magnetization m and
applied magnetic field B, assuming a spin-dependent interac-
tion energy c=h×20.5 Hz. The thick line lying in the ρ0 = 0
plane indicates the boundary between the ρ0 = 0 and the
ρ0 > 0 regions. b) Experimental measurement. The surface
plot is produced by interpolation of data points.
Eq.(1). Such minima always occur at θ = π for antifer-
romagnetic interactions. The measurements agree well
with the prediction, which is made for spin interaction
energy c= h×20.5Hz (determined by spin dynamics as
described below).
The first term of Eq. (1) depends on the external mag-
netic field and tends to maximize the equilibrium ρ0
population. The second, spin dependent, term has the
same sign as c2 and in the antiferromagnetic case tends
to minimize the equilibrium ρ0 population. The phase
transition indicated by the thick line in Fig. 1a arises
at the point where these opposing tendencies cancel for
ρ0 = 0. Along the transition contour, ρ0 rapidly falls
to zero. By contrast, the ferromagnetic phase diagram
has ρ0 = 0 only at m = 1. In the region B < 15µT
and m > 0.6, there should be virtually no population in
mF = 0 for antiferromagnetic interactions, and popula-
tions up to ρ0 = 0.34 for ferromagnetic interactions (as-
suming the same magnitude of c). For our equilibrium
data, the reduced χ2 with respect to the antiferromag-
netic (ferromagnetic) prediction in this region is 2 (20).
This demonstrates that sodium F =1 spin interactions
are antiferromagnetic, as previously shown by the misci-
bility of spin domains formed in a quasi-one-dimensional
trap [13].
Across most of the phase diagram, the scatter in the
population is consistent with measured shot-to-shot vari-
ation in atom number. This variation is 20%, implying
an 8% variation in the mean condensate density accord-
ing to Thomas-Fermi theory. The variance of results is
not due to the magnetic field (calibrated to a precision of
0.2µT), nor to residual field variations across the BEC
(less than 250pT). Uncertainties in setting the magne-
tization are obviated, as the magnetization is measured
for each point as the difference in fractional populations
m= ρ+−ρ−. Discrepancies between theory and exper-
iment at low magnetic fields may be attributed to the
field dependence of the equilibration time. We observe
equilibration times (see below) ranging from 200ms at
high fields to several seconds at low fields, by which time
atom loss is substantial.
If the spinor is driven away from equilibrium, the full
coherent dynamics of the spinor system Eq.(1) are re-
vealed. We initiate the spinor dynamics with the rf tran-
sition described above, but now look at the evolution over
millisecond timescales.
The spinor dynamics are described by the Hamilton
equations for Eq. (1) [8]:
ρ̇0 = −
and θ̇ =
The system is closely related to the double-well “bosonic
Josephson junction” (BJJ) [21, 22] and exhibits a regime
of small, harmonic oscillations and, near a critical field
Bc, is predicted to display large, anharmonic oscilla-
tions. At Bc the period diverges (where δ(Bc) = c[(1 −
ρ0) +
(1 − ρ0)2 −m2 cos θ], with ρ0 and θ taken at
t = 0) [8]. The critical value corresponds to a transi-
tion from periodic-phase solutions of Eq. (3) to running-
phase solutions. At the critical value it is predicted that
the population is trapped in a spin state with ρ0=0. This
phenomenon is related to the macroscopic quantum self-
trapping that has been observed in the BJJ [22]. How-
ever, very small fluctuations in field or density will drive
ρ0 away from 0. Observing a ten-fold increase in the pe-
riod above its zero-field value would require a technically
challenging magnetic field stability of better than 100 fT.
Figure 2 plots the period and amplitude of oscillation
as a function of magnetic field. An example of the os-
cillating populations is shown in the inset. The spinor
condensate is prepared with initial ρ0=0.50± 0.01 1 and
m = 0.00 ± 0.02, and a plot of ρ0 versus time is taken
at each field value. Qualitatively, the period is nearly
independent of magnetic field at low fields, with a small
peak at a critical value Bc = 28µT, followed by a steep
decline in period. The amplitude likewise shows a max-
imum at Bc. Oscillations are visible over durations of
40ms to 300ms. Beyond these times, the amplitude of
the shot-to-shot fluctuations in ρ0 is roughly equal to
the harmonic amplitude. This indicates dephasing due
to shot-to-shot variation in oscillation frequency, proba-
bly associated with the variations in magnetic field and
condensate density, rather than any fundamental damp-
ing process. At even longer times, we observe damping
and equilibration to a new constant ρ0; the damping time
varies with magnetic field from 200ms to 5 s.
For the theoretical prediction in Fig. 2, the initial value
of ρ0 and m are obtained experimentally. We treat only
c and θ(t = 0) as free parameters; c is also predicted
by prior determinations of c2 and our knowledge of the
condensate density. The initial relative phase is not the
equilibrium value θ=π, due to our rf preparation. For a
three-level system driven in resonance with both transi-
tions, the relative phase is θ=0 at all times during the rf
transition, as we derive from Ref. [19]. Small deviations
from initial θ=0 could be caused by an unequal splitting
between the levels, from e.g., the quadratic Zeeman shift.
The best fit to the data in Fig. 2a and b is obtained
by using c=h×(21± 2)Hz and θ(t=0)=0.5± 0.3 (with
no other free parameters). Away from the critical field
Bc, agreement with theory is good. The fitted value of
c implies that Bc is 27µT, in reasonable agreement with
the apparent peak observed at 28µT. Our ability to ob-
serve strong variations in period near Bc is limited by
density fluctuations (8%) and magnetic field fluctuations
(0.2µT). Near Bc, typically only one cycle is visible be-
fore dephasing is complete. Such rapid dephasing can,
itself, be taken as evidence of a strongly B-dependent
period, as expected near the critical field.
To include the known fluctuations in density and mag-
netic field in our model, we perform a Monte Carlo sim-
ulation of the expected signal, based on measured, nor-
mally distributed shot-to-shot variations in values of c,
δ, m and ρ0(t = 0). At each value of B in Fig. 2, we
generate 80 simulated time traces, with each point in the
time trace determined from Eq. 3. We fit the simulated
traces using sine waves and record the mean and stan-
dard deviation of the amplitude and period of the fits.
The results (shaded regions in Fig. 2) show a less sharp
peak in the period. The smoothing of the peak at Bc is
consistent with our data.
1 All uncertainties in this paper are one standard deviation com-
bined statistical and systematic uncertainties
Magnetic Field (µT)
0 10 20 30 40 50
Time (ms)
0 40 80
FIG. 2: Period (a) and amplitude (b) of spin oscillations as a
function of applied magnetic field, following a sudden change
in spin state. The solid lines are theoretical predictions from
solving Eq. (3). The theoretical prediction of the period goes
to infinity at about 27µT. The shaded regions are ±1 stan-
dard deviation about the mean values predicted by the Monte
Carlo simulation. Inset: Fractional Zeeman population (solid
dots) and magnetization (open circles) as a function of time
after the spinor condensate is driven to ρ0 = 0.5, m = 0.
B=6.1µT. The solid line is a sinusoidal fit.
It is clear in Fig. 2 that the oscillation period is insensi-
tive to the magnetic field at low values of the field. In this
regime, the period is sensitive only to the spin interaction
c2 and the density of the condensate 〈n〉. Measuring this
period allows us to determine the difference in scattering
lengths af=2 − af=0. The trace inset in Fig. 2 was taken
in this regime, at a magnetic field of B = 6.1µT, and
shows harmonic oscillations with period 24.6 ± 0.3ms.
Here the predicted period dependence on magnetic field,
14µs/µT, is indeed weak and the oscillations dephase
only slightly over the duration shown. Using this mea-
surement of the period (in which much more data was
taken than for each point making up Fig. 2 (a) and (b)),
and including uncertainties in initial θ, ρ0, and m, we
obtain the spin interaction energy c=h× (20.5± 1.3)Hz.
Finding af=2 − af=0 requires a careful measurement
of the condensate density. We take absorption images
with various expansion times to find the mean field en-
ergy. The images yield the column density in the xy
plane, and the distribution in the z direction can be
inferred from our trap beam geometry. We find that
the mean density of the condensate under the conditions
of the inset to Fig. 2 is 〈n〉 = 8.6 ± 0.9× 1013 cm−3.
From this we calculate af=2 − af=0 = (2.47 ± 0.27)a0,
where a0 = 52.9pm is the Bohr radius. This is consis-
tent with a previous measurement, from spin domain
structure, of af=2 − af=0 = (3.5 ± 1.5)a0 [13] and is
smaller than the difference between scattering lengths
determined from molecular levels, af=2 = (55.1 ± 1.6)a0
and af=0=(50.0± 1.6)a0 [23]. A multichannel quantum
defect theory calculation gives af=2 − af=0 = 5.7a0 [24].
Finally, we consider the validity of the spatial single-
mode approximation. The SMA was clearly violated in
previous work on 23Na [13] and 87Rb [3] F =1 spinor
condensates where spatial domains formed. Spatial de-
grees of freedom decouple from spinor dynamics when the
spin healing length ξs=2π~/
2m|c2|n is larger than the
condensate. From our density measurements we find typ-
ical Thomas-Fermi radii of (9.4, 6.7, 5.7)µm. The spin
healing length, based on our measurements of c, is typi-
cally ξs =17µm. We therefore operate within the range
of validity of the SMA. Furthermore, Stern-Gerlach ab-
sorption images show three components with identical
spatial distributions after ballistic expansion, indicating
that domain formation does not occur.
In conclusion, we have studied both the ground state
and the spinor dynamics of a sodium F=1 spinor conden-
sate. Both agree well with theoretical predictions in the
SMA. By measuring the spin oscillation frequency at low
magnetic field, we have determined the difference in spin-
dependent scattering lengths. The observed peak in oscil-
lation period as a function of magnetic field demonstrates
that the spinor dynamics are fundamentally nonlinear.
It also suggests the existence of the predicted regime of
highly anharmonic spin oscillations at the center of this
peak, which should be experimentally accessible with suf-
ficient control of condensate density and magnetic field.
Observation of anharmonic oscillations, as well as popu-
lation trapping and spin squeezing effects, could be aided
by a minimally destructive measurement of Zeeman pop-
ulations [25] to reduce the effects of magnetic field drifts
and shot-to-shot density variations.
We thank W. Phillips for helpful discussions, and ONR
and NASA for support. ATB acknowledges an NRC
Fellowship. LDT acknowledges an Australian-American
Fulbright Fellowship.
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  We observe coherent spin oscillations in an antiferromagnetic spin-1
Bose-Einstein condensate of sodium. The variation of the spin oscillations with
magnetic field shows a clear signature of nonlinearity, in agreement with
theory, which also predicts anharmonic oscillations near a critical magnetic
field. Measurements of the magnetic phase diagram agree with predictions made
in the approximation of a single spatial mode. The oscillation period yields
the best measurement to date of the sodium spin-dependent interaction
coefficient, determining that the difference between the sodium spin-dependent
s-wave scattering lengths $a_{f=2}-a_{f=0}$ is $2.47\pm0.27$ Bohr radii.

Spinor Dynamics in an Antiferromagnetic Spin-1 Condensate
A. T. Black, E. Gomez, L. D. Turner, S. Jung, and P. D. Lett
Joint Quantum Institute, University of Maryland and
National Institute of Standards and Technology, Gaithersburg, Maryland 20899
(Dated: October 30, 2018)
We observe coherent spin oscillations in an antiferromagnetic spin-1 Bose-Einstein condensate
of sodium. The variation of the spin oscillations with magnetic field shows a clear signature of
nonlinearity, in agreement with theory, which also predicts anharmonic oscillations near a critical
magnetic field. Measurements of the magnetic phase diagram agree with predictions made in the
approximation of a single spatial mode. The oscillation period yields the best measurement to date
of the sodium spin-dependent interaction coefficient, determining that the difference between the
sodium spin-dependent s-wave scattering lengths af=2−af=0 is 2.47 ± 0.27 Bohr radii.
PACS numbers: 03.75.Mn, 32.80.Cy, 32.80.Pj
Atomic collisions are essential to the formation of Bose-
Einstein condensates (BEC), redistributing energy dur-
ing evaporative cooling. Collisions can be coherent and
reversible, leading to diverse phenomena such as super-
fluidity [1] and reversible formation of molecules [2] in
BECs with a single internal state. When internal de-
grees of freedom are included (as in spinor condensates),
coherent collisions lead to rich dynamics [3, 4] in which
the population oscillates between different Zeeman sub-
levels. We present the first observation of coherent spin
oscillations in a spin-1 condensate with antiferromagnetic
interactions (in which the interaction energy of colliding
spin-aligned atoms is higher than that of spin-antialigned
atoms.)
Spinor condensates have been a fertile area for the-
oretical studies of dynamics [5, 6, 7, 8], ground state
structures [9, 10], and domain formation [11]. Extensive
experiments on the ferromagnetic F=1 hyperfine ground
state of 87Rb have demonstrated spin oscillations and co-
herent control of spinor dynamics [3, 12]. Observation of
domain formation in 23Na demonstrated the antiferro-
magnetic nature of the F=1 ground state [13] and de-
tected tunneling across spin domains [14]; no spin oscilla-
tions have been reported in sodium BEC until now. The
F=2 state of 87Rb is thought to be antiferromagnetic, but
a cyclic phase is possible [15, 16]. Experiments on this
state have demonstrated that the amplitude and period
of spin oscillations can be controlled magnetically [4].
At low magnetic fields, spin interactions dominate the
dynamics. The different sign of the spin dependent in-
teraction causes the antiferromagnetic F=1 case to differ
from the ferromagnetic one both in the structure of the
ground-state magnetic phase diagram and in the spinor
dynamics. Both cases can exhibit a regime of slow, an-
harmonic spin oscillations; however, this behavior is pre-
dicted over a wide range of initial conditions only in the
antiferromagnetic case [8]. The spin interaction energies
in sodium are more than an order of magnitude larger
than in 87Rb F =1 for a given condensate density [3],
facilitating studies of spinor dynamics.
The dynamics of the spin-1 system are much simpler
than the spin-2 case [4, 15, 16], having a well-developed
analytic solution [8]. This solution predicts a divergence
in the oscillation period (not to be confused with the
amplitude peak observed in 87Rb F=2 [4] oscillations).
This Letter reports the first measurement of the
ground state magnetic phase diagram of a spinor con-
densate, and the first experimental study of coherent
spinor dynamics in an antiferromagnetic spin-1 conden-
sate. Both show good agreement with the single-spatial-
mode theory [10]. To study the dynamics, we displace
the spinor from its ground state, observing the resulting
oscillations of the Zeeman populations as a function of
applied magnetic field B. At low field the oscillation pe-
riod is constant, at high field it decreases rapidly, and at a
critical field it displays a resonance-like feature, all as pre-
dicted by theory [8]. These measurements have allowed
us to improve by a factor of three the determination of
the sodium F = 1 spin-dependent interaction strength,
which is proportional to the difference af=2 − af=0 in
the spin-dependent scattering lengths.
The state of the condensate in the single-mode ap-
proximation (SMA) is written as the product φ(r)ζ of a
spin-independent spatial wavefunction φ(r) and a spinor
ζ = (
eiθ− ,
iθ0 ,
iθ+). We use ρ
, ρ0, and
ρ+ (θ−, θ0, and θ+) to denote fractional populations
(phases) of the Zeeman sublevels mF = −1, 0, and 1,
so that
ρi=1. The spinor’s ground state and its non-
linear dynamics may be derived from the spin-dependent
part of the Hamiltonian in the single-mode and mean-
field approximations, subject to the constraints that to-
tal atom number N and magnetization m≡ ρ+−ρ− are
conserved [8]. The “classical” spinor Hamiltonian E is a
function of only two canonical variables: the fractional
population ρ0 and the relative phase θ ≡ θ+ + θ− − 2θ0.
It is given by
E = δ(1−ρ0) + cρ0
(1−ρ0) +
(1−ρ0)2−m2 cos θ
where δ = h × (2.77× 1010Hz/T2)B2 is the quadratic
http://arxiv.org/abs/0704.0925v2
Zeeman shift [8] with h the Planck constant. (The linear
Zeeman shift has no effect on the dynamics.) The spin-
dependent interaction energy is c= c2 〈n〉, where 〈n〉 is
the mean particle density of the condensate and
(af=2 − af=0) (2)
is the spin-dependent interaction coefficient [8, 17]. Here
M is the atomic mass. af=2 and af=0 are the s-wave
scattering lengths for a colliding pair of atoms of total
spin f = 2 and f = 0, respectively; Bose symmetry en-
sures there are no s-wave collisions with total spin of
1. If c2 is positive (negative), the system is antiferro-
magnetic (ferromagnetic). The spinor ground state and
spinor dynamics are determined by Eq. (1).
The apparatus is similar to that described previ-
ously [18]. We produce a BEC of 105 23Na atoms in
the F=1 state, with an unobservably small thermal frac-
tion, in a crossed-beam 1070nm optical dipole trap. The
trap beams lie in the horizontal xy plane, so that the trap
curvature is nearly twice as large along the vertical z axis
as in the xy plane. By applying a small magnetic field
gradient with the MOT coils (less than 10mT/m) during
the 9 s of forced evaporation, we fully polarize the BEC:
all atoms are in mF =+1. Conservation of spin angular
momentum ensures that the magnetization remains con-
stant once evaporation has ceased; a state with ρ+ = 1
persists for the lifetime of the condensate, about 14 s.
We then turn off the gradient field and adiabatically
apply a bias field B of 4 to 51µT along x̂, leaving the
BEC in the ρ+ = 1 state. To prepare an initial state,
we apply an rf field resonant with the linear Zeeman
splitting; typically the frequency is tens to hundreds of
kilohertz. Rabi flopping in the three-level system is ob-
served [19], and controlling the amplitude and duration
of the pulse can produce any desired magnetization m,
which also determines the population ρ0. The flopping
time is less than 50µs, much shorter than the character-
istic times for spin evolution governed by Eq. (1). Using
this Zeeman transition avoids populating the F=2 state,
thus avoiding inelastic losses, which are much greater for
23Na than for 87Rb.
We measure the populations ρi of atoms in the three
Zeeman sublevels by Stern-Gerlach separation and ab-
sorption imaging [20]. The Stern-Gerlach gradient is par-
allel to the bias field ~B, while the imaging beam propa-
gates in the ẑ direction. The phase θ is not measured.
To measure the ground state population distribution
as a function of magnetization and magnetic field, we
first set the magnetization using the rf pulse. We then
ramp the field to a desired final value over 1 s, wait 3 s
for equilibration, and measure the populations as above.
Figure 1(b) displays the measured ground-state mag-
netic phase diagram. The theoretical prediction in
Fig. 1(a) is the population ρ0 that minimizes the energy,
B (µT)
B (µT)
FIG. 1: a) Theoretical prediction of the ground-state frac-
tional population ρ0 as a function of magnetization m and
applied magnetic field B, assuming a spin-dependent interac-
tion energy c=h×20.5 Hz. The thick line lying in the ρ0 = 0
plane indicates the boundary between the ρ0 = 0 and the
ρ0 > 0 regions. b) Experimental measurement. The surface
plot is produced by interpolation of data points.
Eq.(1). Such minima always occur at θ = π for antifer-
romagnetic interactions. The measurements agree well
with the prediction, which is made for spin interaction
energy c= h×20.5Hz (determined by spin dynamics as
described below).
The first term of Eq. (1) depends on the external mag-
netic field and tends to maximize the equilibrium ρ0
population. The second, spin dependent, term has the
same sign as c2 and in the antiferromagnetic case tends
to minimize the equilibrium ρ0 population. The phase
transition indicated by the thick line in Fig. 1a arises
at the point where these opposing tendencies cancel for
ρ0 = 0. Along the transition contour, ρ0 rapidly falls
to zero. By contrast, the ferromagnetic phase diagram
has ρ0 = 0 only at m = 1. In the region B < 15µT
and m > 0.6, there should be virtually no population in
mF = 0 for antiferromagnetic interactions, and popula-
tions up to ρ0 = 0.34 for ferromagnetic interactions (as-
suming the same magnitude of c). For our equilibrium
data, the reduced χ2 with respect to the antiferromag-
netic (ferromagnetic) prediction in this region is 2 (20).
This demonstrates that sodium F =1 spin interactions
are antiferromagnetic, as previously shown by the misci-
bility of spin domains formed in a quasi-one-dimensional
trap [13].
Across most of the phase diagram, the scatter in the
population is consistent with measured shot-to-shot vari-
ation in atom number. This variation is 20%, implying
an 8% variation in the mean condensate density accord-
ing to Thomas-Fermi theory. The variance of results is
not due to the magnetic field (calibrated to a precision of
0.2µT), nor to residual field variations across the BEC
(less than 250pT). Uncertainties in setting the magne-
tization are obviated, as the magnetization is measured
for each point as the difference in fractional populations
m= ρ+−ρ−. Discrepancies between theory and exper-
iment at low magnetic fields may be attributed to the
field dependence of the equilibration time. We observe
equilibration times (see below) ranging from 200ms at
high fields to several seconds at low fields, by which time
atom loss is substantial.
If the spinor is driven away from equilibrium, the full
coherent dynamics of the spinor system Eq.(1) are re-
vealed. We initiate the spinor dynamics with the rf tran-
sition described above, but now look at the evolution over
millisecond timescales.
The spinor dynamics are described by the Hamilton
equations for Eq. (1) [8]:
ρ̇0 = −
and θ̇ =
The system is closely related to the double-well “bosonic
Josephson junction” (BJJ) [21, 22] and exhibits a regime
of small, harmonic oscillations and, near a critical field
Bc, is predicted to display large, anharmonic oscilla-
tions. At Bc the period diverges (where δ(Bc) = c[(1 −
ρ0) +
(1 − ρ0)2 −m2 cos θ], with ρ0 and θ taken at
t = 0) [8]. The critical value corresponds to a transi-
tion from periodic-phase solutions of Eq. (3) to running-
phase solutions. At the critical value it is predicted that
the population is trapped in a spin state with ρ0=0. This
phenomenon is related to the macroscopic quantum self-
trapping that has been observed in the BJJ [22]. How-
ever, very small fluctuations in field or density will drive
ρ0 away from 0. Observing a ten-fold increase in the pe-
riod above its zero-field value would require a technically
challenging magnetic field stability of better than 100 fT.
Figure 2 plots the period and amplitude of oscillation
as a function of magnetic field. An example of the os-
cillating populations is shown in the inset. The spinor
condensate is prepared with initial ρ0=0.50± 0.01 1 and
m = 0.00 ± 0.02, and a plot of ρ0 versus time is taken
at each field value. Qualitatively, the period is nearly
independent of magnetic field at low fields, with a small
peak at a critical value Bc = 28µT, followed by a steep
decline in period. The amplitude likewise shows a max-
imum at Bc. Oscillations are visible over durations of
40ms to 300ms. Beyond these times, the amplitude of
the shot-to-shot fluctuations in ρ0 is roughly equal to
the harmonic amplitude. This indicates dephasing due
to shot-to-shot variation in oscillation frequency, proba-
bly associated with the variations in magnetic field and
condensate density, rather than any fundamental damp-
ing process. At even longer times, we observe damping
and equilibration to a new constant ρ0; the damping time
varies with magnetic field from 200ms to 5 s.
For the theoretical prediction in Fig. 2, the initial value
of ρ0 and m are obtained experimentally. We treat only
c and θ(t = 0) as free parameters; c is also predicted
by prior determinations of c2 and our knowledge of the
condensate density. The initial relative phase is not the
equilibrium value θ=π, due to our rf preparation. For a
three-level system driven in resonance with both transi-
tions, the relative phase is θ=0 at all times during the rf
transition, as we derive from Ref. [19]. Small deviations
from initial θ=0 could be caused by an unequal splitting
between the levels, from e.g., the quadratic Zeeman shift.
The best fit to the data in Fig. 2a and b is obtained
by using c=h×(21± 2)Hz and θ(t=0)=0.5± 0.3 (with
no other free parameters). Away from the critical field
Bc, agreement with theory is good. The fitted value of
c implies that Bc is 27µT, in reasonable agreement with
the apparent peak observed at 28µT. Our ability to ob-
serve strong variations in period near Bc is limited by
density fluctuations (8%) and magnetic field fluctuations
(0.2µT). Near Bc, typically only one cycle is visible be-
fore dephasing is complete. Such rapid dephasing can,
itself, be taken as evidence of a strongly B-dependent
period, as expected near the critical field.
To include the known fluctuations in density and mag-
netic field in our model, we perform a Monte Carlo sim-
ulation of the expected signal, based on measured, nor-
mally distributed shot-to-shot variations in values of c,
δ, m and ρ0(t = 0). At each value of B in Fig. 2, we
generate 80 simulated time traces, with each point in the
time trace determined from Eq. 3. We fit the simulated
traces using sine waves and record the mean and stan-
dard deviation of the amplitude and period of the fits.
The results (shaded regions in Fig. 2) show a less sharp
peak in the period. The smoothing of the peak at Bc is
consistent with our data.
1 All uncertainties in this paper are one standard deviation com-
bined statistical and systematic uncertainties
Magnetic Field (µT)
0 10 20 30 40 50
Time (ms)
0 40 80
FIG. 2: Period (a) and amplitude (b) of spin oscillations as a
function of applied magnetic field, following a sudden change
in spin state. The solid lines are theoretical predictions from
solving Eq. (3). The theoretical prediction of the period goes
to infinity at about 27µT. The shaded regions are ±1 stan-
dard deviation about the mean values predicted by the Monte
Carlo simulation. Inset: Fractional Zeeman population (solid
dots) and magnetization (open circles) as a function of time
after the spinor condensate is driven to ρ0 = 0.5, m = 0.
B=6.1µT. The solid line is a sinusoidal fit.
It is clear in Fig. 2 that the oscillation period is insensi-
tive to the magnetic field at low values of the field. In this
regime, the period is sensitive only to the spin interaction
c2 and the density of the condensate 〈n〉. Measuring this
period allows us to determine the difference in scattering
lengths af=2 − af=0. The trace inset in Fig. 2 was taken
in this regime, at a magnetic field of B = 6.1µT, and
shows harmonic oscillations with period 24.6 ± 0.3ms.
Here the predicted period dependence on magnetic field,
14µs/µT, is indeed weak and the oscillations dephase
only slightly over the duration shown. Using this mea-
surement of the period (in which much more data was
taken than for each point making up Fig. 2 (a) and (b)),
and including uncertainties in initial θ, ρ0, and m, we
obtain the spin interaction energy c=h× (20.5± 1.3)Hz.
Finding af=2 − af=0 requires a careful measurement
of the condensate density. We take absorption images
with various expansion times to find the mean field en-
ergy. The images yield the column density in the xy
plane, and the distribution in the z direction can be
inferred from our trap beam geometry. We find that
the mean density of the condensate under the conditions
of the inset to Fig. 2 is 〈n〉 = 8.6 ± 0.9× 1013 cm−3.
From this we calculate af=2 − af=0 = (2.47 ± 0.27)a0,
where a0 = 52.9pm is the Bohr radius. This is consis-
tent with a previous measurement, from spin domain
structure, of af=2 − af=0 = (3.5 ± 1.5)a0 [13] and is
smaller than the difference between scattering lengths
determined from molecular levels, af=2 = (55.1 ± 1.6)a0
and af=0=(50.0± 1.6)a0 [23]. A multichannel quantum
defect theory calculation gives af=2 − af=0 = 5.7a0 [24].
Finally, we consider the validity of the spatial single-
mode approximation. The SMA was clearly violated in
previous work on 23Na [13] and 87Rb [3] F =1 spinor
condensates where spatial domains formed. Spatial de-
grees of freedom decouple from spinor dynamics when the
spin healing length ξs=2π~/
2m|c2|n is larger than the
condensate. From our density measurements we find typ-
ical Thomas-Fermi radii of (9.4, 6.7, 5.7)µm. The spin
healing length, based on our measurements of c, is typi-
cally ξs =17µm. We therefore operate within the range
of validity of the SMA. Furthermore, Stern-Gerlach ab-
sorption images show three components with identical
spatial distributions after ballistic expansion, indicating
that domain formation does not occur.
In conclusion, we have studied both the ground state
and the spinor dynamics of a sodium F=1 spinor conden-
sate. Both agree well with theoretical predictions in the
SMA. By measuring the spin oscillation frequency at low
magnetic field, we have determined the difference in spin-
dependent scattering lengths. The observed peak in oscil-
lation period as a function of magnetic field demonstrates
that the spinor dynamics are fundamentally nonlinear.
It also suggests the existence of the predicted regime of
highly anharmonic spin oscillations at the center of this
peak, which should be experimentally accessible with suf-
ficient control of condensate density and magnetic field.
Observation of anharmonic oscillations, as well as popu-
lation trapping and spin squeezing effects, could be aided
by a minimally destructive measurement of Zeeman pop-
ulations [25] to reduce the effects of magnetic field drifts
and shot-to-shot density variations.
We thank W. Phillips for helpful discussions, and ONR
and NASA for support. ATB acknowledges an NRC
Fellowship. LDT acknowledges an Australian-American
Fulbright Fellowship.
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