VOLUME
75,
NUMBER
9
PH
YS
ICAL REVIEW LETTERS
28 AUGUST 1995
Evidence of
Bose-Einstein
Condensation
in
an Atomic Gas
with
Attractive Interactions
C.C.
Bradley,
C.A.
Sackett,
J.
J.
Tollett,
and R.
G.
Hulet
Physics
Department
and Rice
Quantum
Institute, Rice
University, Houston,
Texas
77251-1892
(Received
25
July
1995)
Evidence for
Bose-Einstein
condensation of
a
gas
of
spin-polarized
Li atoms is
reported.
Atoms
confined
to a permanent-magnet
trap
are laser cooled to
200
p,
K and are then
evaporatively
cooled
to
lower
temperatures.
Phase-space densities consistent with
quantum
degeneracy
are measured for
temperatures
in the
range
of
100
to
400 nK. At these
high
phase-space
densities,
diffraction
of a
probe
laser beam is observed.
Modeling
shows that this diffraction is a
sensitive
indicator
of
the
presence
of
a
spatially
localized condensate.
Although
measurements of
the number of condensate
atoms have
not been
performed,
the
measured
phase-space
densities
are
consistent with
a
majority
of
the atoms
being
in
the condensate,
for
total
trap
numbers
as
high
as
2
&
10 atoms. For
Li,
the
spin-triplet
s-wave
scattering
length
is known to be
negative,
corresponding
to an attractive interatomic interaction.
Previously,
Bose-Einstein
condensation was
predicted
not to
occur in such
a
system.
PACS numbers:
03.75.
Fi,
05.30.
Jp,
32.80.
pj
Bose-Einstein condensation
(BEC)
of
an
ideal
gas,
pre-
dicted
more than
70
years
ago
[1],
is
the
paradigm
of quan-
tum statistical
phase
transitions. This
phase
transition is
manifested
by
an
abrupt growth
in
the
population
of the
ground
state of the
potential confining
the
gas.
BEC
oc-
curs
when the mean
particle separation
is
comparable
to
the thermal
de
Broglie
wavelength A.
Although
BEC-like
phase
transitions have
been
observed in
liquid
He
and in
excitonic
systems
[2],
BEC of
a
weakly
interacting
gas
has
been
a
long
sought,
but elusive
goal
[3].
Recently
this
goal
was
realized in
a
rarified
gas
of
spin-polarized,
ultra-
cold
s
Rb
atoms
[4].
Because
of
the
extraordinarily
low
temperatures
that can
be achieved in
atomic
gases,
BEC
can be studied in the
low
density
regime
where interatomic
distances are much
greater
than the distance scale of
atom-
atom interactions. In this
regime,
two-body
interactions
take
a
simple
form,
and three-body interactions can be
ne-
glected.
It is
hoped
that
investigations
of
these
relatively
simple
quantum
degenerate
systems
will enable
a
better
understanding
of more
complex
Bose
systems,
such
as
su-
perconductors,
where
particle
interactions are
significant.
It is
possible
that even the weak interactions between
atoms
of an ultracold
gas
can
result in
behavior in
the
quantum
degenerate
regime
that
departs
significantly
from that of an ideal
gas.
At low
temperatures
and
densities,
only
two-body
s-wave
scattering
is
important.
If the
temperature
is
so
low that
the
magnitude
of
the
s-wave
scattering
length
a
is
much
less than
A,
then
the exact
shape
of the
atom-atom
interaction
potential
is
unimportant
[5].
For
a
purely
repulsive
potential,
a
is
positive,
while for
a
purely
attractive
potential
that
supports
no bound
states, a
is
negative.
The
spin-triplet
interaction
potential
between
alkali-metal
atoms
supports
bound states. In such
a
case,
a
can
be
either
positive
or
negative
depending
on the
proximity
of the least bound
state to the dissociation limit
[6].
From recent
analysis
of
the
photoassociative
spectrum
of ultracold
Rb,
it is
believed that a is
positive
for
Rb
[7].
The
scattering
length
is
accurately
known for the
triplet
potential
of
Li;
direct two-photon
photoassociation
spectroscopy
of the
least bound
triplet ground
state of
Li2
combined with
analysis
of
previous
Li2
spectra
[8]
give
a
=
(
—
27.
3
~
0.
8)an,
where
ao
is the Bohr radius
[9].
The condensate wave
function
can be described
by
a
nonlinear
Schrodinger
equation,
in
which
the
interatomic
interactions are
represented
by
a
mean-held
pseudopoten-
tial
[5,
10].
For a
spatially
homogeneous
gas
with
a
)
0,
there are stable
solutions to this
equation, and,
for low
den-
sities,
the
properties
of
the
gas
are
given
by
an
expansion
in
the small
parameter
(na3)'I,
where n is the
density
[11].
The
situation
is
drastically
different for
a
(
0,
because
the
gas
is
not
mechanically
stable
against
collapse
[10].
It
was
predicted
that for an atom with
negative
a
BEC would
be
preempted
by
a
first-order
gas-liquid
or
gas-solid
phase
transition
[12].
However,
we
and
others
suggested
that the
situation
may
be
quite
different in the
spatially
inhomoge-
neous environment of an atom
trap.
In a
potential,
it is
possible
for the zero-point
energy
to exceed the
attractive
interaction
for
a
limited number
of condensate
atoms,
sta-
bilizing
the
gas
against collapse.
This
suggestion
has been
developed formally
by
solving
the nonlinear
Schrodinger
equation
with
a
Hamiltonian that includes
the
confining
potential
[13].
However,
the
solution
is
valid
only
for
temperature
T
=
0,
and
the
theory
does not
address
the
dynamical
issue of whether
a condensate can
form for
a
(
0.
Consequently,
the
question
of whether a
weakly
interacting
Bose
gas
with attractive interactions
could
un-
dergo
BEC has remained
open.
In this
Letter,
we
report
evidence
for
BEC of a
gas
of
spin-polarized
Li
~
The
apparatus
was described in detail in
a
previous
pub-
lication
[14].
The atoms are confined in a
magnetic
trap
that is
produced
by
six
permanent
magnet
cylinders.
The
magnets
are
arranged
to
produce
a
minimum
at
the
trap
center near
which
the field
strength
varies
quadratically.
The minimum is offset
by
a
uniform bias field of
823 G
along
the
z
axis.
This field distribution was used
because
0031-9007/95/75(9)/1687(4)$06.
00 1995
The American
Physical Society
1687
VOLUME
75,
NUMBER
9
PHYS ICAL REVIEW
LETTERS
28 AuGUsT
1995
the bias field
prevents
the loss of atoms
due to
nonadiabatic
spin-Hip
transitions that can occur
at
the center
of mag-
netic
traps
that do
not
have
a bias
field
[15,
16].
The
trap
oscillation
frequencies
of
v,
=
117
~
6
Hz and
v
~
=
163
~
15 Hz were determined
by
a
combination
of spec-
troscopic
observations, direct
magnetic
field
measurement,
and
the observed
spatial
asymmetry
of
the
trapped
atom
cloud
[17].
These values determine the
critical number
for
the BEC
phase
transition
to be
N,
=
3.
5
X
10
(T/p,
K)
[18].
The
trap
is loaded from
a
laser-slowed
atomic
beam
that
passes through
the center of the
trap.
Three
pairs
of
counterpropagating
laser
beams,
tuned to
near the
25]/2,
F
=
2,
mp
=
2
2P3/~,
F
=
3,
mp
=
3
transi-
tion
frequency
at the bias
magnetic
field
(A
=
671
nm
wavelength),
are directed
along
three axes
passing
through
the
trap
center.
Through
Doppler
cooling,
these beams
provide
the
dissipation necessary
for
trap
loading.
Each
of the six
trap
loading
beams is
linearly polarized
and
has
an
intensity
of
17 mW/cm .
Loading
typically
results in
—
2
X
10
trapped
atoms.
Upon
completion
of
loading,
the
intensity
of the
trap
loading
beams is reduced
by
a
fac-
tor of 50 to
cool
the atoms to
—
200
p,
K,
which is near the
Doppler
cooling
limit of
140
p,
K. The
peak
density
after
Doppler
cooling
is
—
7
&&
10'
cm .
The
lifetime for
the atoms
to
remain
trapped,
limited
presumably
by
col-
lisions
with
background
gas
atoms,
exceeds 10 min.
By
extrapolation,
the
background
gas
pressure
is estimated
to
be near
10
'
Torr,
and is achieved
using
both an ion and
a
titanium sublimation
pump
on
a
chamber with
only
a
nar-
row,
conduction-limiting tube
connecting
it to the
slow-
atom source
[14].
The
atoms are cooled further
by
evaporation,
a
process
whereby
the hottest atoms are
selectively
removed
and the
remainder
are
allowed
to rethermalize via elastic two-body
collisions
[19].
This
technique
was demonstrated
sev-
eral
years
ago
using
magnetically
trapped
spin-polarized
H
[20],
and
recently
using
laser cooled
alkali-metal
atoms
[16,
21].
Given
the initial
temperature,
density,
and elastic
scattering
cross
section
of
5.0
X
10
'3
cm
[9],
the
num-
ber of
rethermalizing
collisions
per
trap
lifetime is
—
1000,
which
is
sufficient for efficient
evaporative
cooling.
A
mi-
crowave field is tuned to
selectively
drive
the
electron
spin-
flip
transition
for
atoms at
a
particular
value
of
magnetic
field,
and
therefore
at
a
unique
trap
radius
[16,
22].
The
mi-
crowave field
is
produced
by
a
frequency
synthesizer
and
coupled
to
a
loop
antenna inside
the vacuum
chamber.
The
frequency
vs
time
trajectory
that
optimizes phase-space
density
was
determined
by
modeling
the time-dependent
energy
distribution
[23],
including
losses due to
back-
ground
gas
collisions
and two-body
dipolar
relaxation
col-
lisions
[24].
The calculated
trajectory
for
our initial
con-
ditions crosses the
BEC
phase
transition
point
at
T
=
300
nK
and
N
=
1
X
10,
after
—
5
min of
evaporation.
An
absorption
probe
is used to
measure the
number
and
temperature
of the
trapped
atoms. This
probe
laser
beam is directed
through
the
atom
cloud,
and its
near-
field
absorption
field
is
imaged
by
a
12
cm focal
length
1688
compound
lens
positioned
outside
the chamber
at
a
distance of 17
cm
from
the atoms. The
image
is
recorded
by
a
slow-scan
charge-coupled-device
(CCD)
camera. An
aperture
stop
of 3 cm diameter
is located
at
the
lens to
reduce
spherical
aberration.
The
probe
is
a
collimated
Gaussian
beam
with a waist
(e
radius)
of
1.
5
mm
and
a
power
of
1 mW. The
probe
frequency
is
typically
detuned
by
12
—
18
MHz
(2
—
3
natural
linewidths)
below
the atomic resonance
frequency
at the bottom
of the
trap,
so that the
probe
absorption
is
relatively
weak
and also
so that differences in
Zeeman shifts
among
the
atoms
are
negligible.
The
probe
beam
propagates
along
the
x
=
y
=
z
body-diagonal axis of the
trap
and is
linearly
polarized
perpendicular
to the
trap
bias field.
Because
the
probe
polarization
does not
decompose
into
pure
o
+
polarization
in the
trap
coordinates,
not all
of the
light
can be
absorbed
by
the
atoms. Given that
the
probe
propagates
at an
angle
of 54.
7
with
respect
to the
bias
field, a
maximum
of 75% of the
probe
beam
intensity
is
absorbable.
In the
following discussion,
probe
absorption
is
expressed
relative to this maximum value.
An
image
is recorded
by
pulsing
on the
probe
for
30
p,
s.
Probing
heats the
sample,
so the
trap
must be
reloaded and the
evaporative
cooling
cycle
repeated
for
each measurement.
By
comparing
the
image
to one taken
with
no atoms in the
trap,
the
spatial absorption
profile
is
obtained. The total
number of
missing
photons,
that
is,
the total
absorption, provides
a
measure of N. For low
N,
the
uncertainty
is
2
X
10"
due
to
the
intensity
noise
of the
probe.
For
larger
numbers, the
uncertainty
is
20%
due to the 1.
5
MHz
uncertainty in the
probe
detuning.
The
temperature
of the
atoms
is
determined
by
the
spatial
size
of the
absorption
image.
Because the
trap
spring
constants
are
asymmetric,
the
image
has an
elliptical
shape,
so a
profile
of
absorption
vs radius is obtained
by
scaling
the
axes
of the
ellipse
and
performing
an
angular
average
about the center of the
cloud.
Assuming
a Boltzmann
energy
distribution, the radius at which
the
optical density
of the
cloud has fallen to e
'
of
its
peak
value determines
the
temperature,
with
an
accuracy
of 25%. In
addition,
the
peak
optical density
of
the cloud
(accurate
to
5%)
combined
with the e
'
radius
provides
a second value
for
N;
these values
typically
agree
to within their uncertainties.
The
peak
optical
density,
the
e
'
radius,
and the
trap
oscillation
frequencies
can also be used to calculate the
critical
parameter
N/N,
with
an
accuracy
of 50%.
Figure
1(a)
shows
an
image
in
which the atom cloud
has
an
e radius of 36
p,
m
in the
trap
x-y direction,
and
a
peak
absorption
of 22%. These values
correspond
to 1.2
X
10 atoms with
a
temperature
of
590
nK,
giv-
ing
N/N,
=
0.
2.
Figure
1(b)
shows the result
of a
sep-
arate load-evaporate
cycle,
giving
N/N,
)
1,
in
which
a
distinct
ringlike
halo
surrounding
the
absorption
peak
is observed.
Figure
2(a)
shows
a
cross-sectional
profile,
and
Fig.
2(b)
shows the
angular
average
of
these
data.
The halo
is evident as a
region
of
negative absorption
surrounding the
peak.
All of the 16
images
of clouds with
VOLUME
75,
NUMBER
9 PH YS
ICAL
REVIEW
LETTERS
28
AvovsT
1995
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00-
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0.20
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Distance
(p,
rn)
200
FIG.
1
(color).
False-color
images
of
trapped
atom clouds
resulting
from
different
load-evaporate
cycles.
Images
are
obtained
by
passing
a
probe
laser
beam
through
the
atom cloud
and
imaging
its
near-field
absorption
profile
on
a CCD camera.
These
images
are then
subtracted from
a
background
image
taken without
atoms
present.
The
spatial
scale
corresponds to
250
p,
m
along
each axis
at the location of
the atoms. In
these
images,
the
diagonal
extending
from
the lower
right
corner
to
the
upper
left corner is
aligned
with
the
trap
x-y
direction.
The
color scale
above the
images
maps
the
colors
to
probe
absorption.
The
scale varies
linearly from
—
15%
(violet)
to
20%
(red)
absorption, where
negative
values
indicate
areas
that
are
brighter
in the
presence
of atoms.
(a)
corresponds to
1.
2
X
105
atoms
at
590
nK,
and
(b)
to the nominal
values of
2
X
10 atoms
at 100 nK. The
halo
surrounding
the
absorption
peak
visible in
(b)
is attributed
to
diffraction from the
spatially
localized Bose-Einstein
condensate.
N/N,
)
1.5 exhibit a
halo,
while
none
of
the
images
with
N/N,
(
0.
7 do.
The
remaining 5
images
are consistent
with either
N/N,
(
1
or
N/N,
)
1,
given
the
measure-
ment
uncertainty.
Quantum
degenerate
conditions
were
attained
and halos
observed for
temperatures as
high
as
400
nK,
and total numbers
as
great
as 2
X
10
atoms.
We
interpret
these halos
as
being
due to diffraction
of
the
probe
beam from
the small Bose-Einstein
condensate.
The
probe
beam diffracts
from
atom clouds
at
any
temperature,
but,
as
long
as the entire
diffraction
pattern
is
captured
by
the
lens,
the
near-field
image
recorded
by
the
camera will
be
an
accurate
depiction
of the
absorption
profile
at the
atoms and will
appear
smooth and
nearly
Gaussian. The
observation
of
a
diffraction
pattern
in an
image
indicates that
not all of
the
probe
light
is
captured
by
the
lens. For
our
imaging system,
this
can
only
occur
if
the scatterer is
smaller than
or
comparable
to the
3
p,
m
ground
state of the
trap.
By
Babinet's
principle,
the
shadow
produced
by
an
optically
thin
Gaussian cloud will
itself
propagate
as
a Gaussian beam.
If the
e
'
radius of
the cloud
is
ro
and
the
aperture
has diameter
D located
a
distance
L from
the
atoms,
the fraction
of the shadow
blocked
by
the
aperture
is
—
(~Dr, /wl.
)'
Even
for
ro
=
10
p,
m,
corresponding
to
a
nondegenerate
distribution
at 50
nK,
this
fraction is
negligible.
The
asymmetry
evident in
the halo
of
Fig.
1(b)
is due
to
the
asymmetry
of the
trap
harmonic
coefficients:
The
diffraction
angle
is
larger
in the
direction
of
tighter
confinement.
0.15
0.
10
o
oo
0.
00
-0.
05-
I ~ ~
~
30
60
90
Radius
(tltn)
120
FIG.
2. Cross
section
(a)
and
angular
average
(b)
of the
absorption
image
shown in
Fig.
1(b).
The
cross section
in
(a)
is
along
the
trap
x-y
axis,
and each solid
circle
corresponds
to
a
pixel
on
the CCD camera. The
profile
in
(b)
is obtained
by
scaling
the
image
data
according
to the
asymmetry
of the
trap,
and
averaging
around the center
of the
trap
for each
radius. The location
of each
solid circle
corresponds to the
mean
radius of the annulus
being
averaged.
The
halo evident
in
Fig.
1(b)
manifests itself as areas
of
negative
absorption
beside the
peak.
The
probe detuning
is 18
~
1.
5
MHz below
the atomic
resonance
frequency.
In
order to test this
interpretation,
the
experiment was
modeled
numerically. A
Gaussian
wave
front
was
atten-
uated
according
to the
Bose-Einstein
density
distribution
for
a harmonic
oscillator, and then
propagated
through
an
appropriate
aperture
and lens
to
the
image
plane.
Because
of the numerical
intensity
of
the
calculation, the
beam
profile
was
approximated
as one dimensional
rather
than
cylindrical.
The results
of the
calculation are
consistent
with
observation, in that
diffraction
features are not
seen
in the
image
plane
when
N/N,
(
1,
but
appear
abruptly
for
larger
numbers.
This
point
was verified
for tempera-
tures
as
low
as
50
nK.
The
shape
and size
of the diffraction
features
given
by
the model in the
presence
of a condensate
did not
agree
quantitatively
with
the halos
observed experimen-
tally.
This
disagreement
may
be attributed
to
unusual
light
scattering
properties
of the
dense
condensate, or to the
one-
dimensional
nature of
the model.
However,
the
light
scat-
tering
properties
of the
noncondensed
part
of the
gas
are
more
certain,
because
multiple
scattering
and
cooperative
scattering processes
cannot
occur.
Multiple
photon
scat-
tering
is ruled out
by
the low
optical absorption.
The
peak
absorption
is
as
low
as 20Vo for
some of
the data with
low values
of T and
high
values of
N/N,
.
Cooperative
scattering,
such
as
superfIuorescence, is
important
when
the
density
is such
that
n
~
(27r/A)
=
8
X
10'
cm
At 400
nK,
the critical
density
of the noncondensed
gas
is
only
2
X
10'
cm,
and
at
lower
temperatures
it
de-
creases as T
1689
VOLUME
75,
NUMBER
9
PH YS ICAL REVIEW
LETTERS
28 AUGUsT
1995
While our estimates
of the
temperature
and number are
relatively
accurate when
N/N,
(
I,
the
diffraction
by
the condensate
changes
both the size and
shape
of the
absorption
signal.
The narrowest
peaks,
such as the one
shown in
Fig.
1(b),
are not
Gaussian, as
can
be
seen
in
the
profiles
of the
absorption
signal
shown in
Fig.
2. The
e
'
radius of the
peak
implies
a
temperature
of
—
100
nK,
but,
given
the uncertainties in
measuring
small radii and
the
presence
of diffraction
effects,
the
temperature
may
be
as
high
as 150
nK or
as low as
zero. This
temperature
range
is
comparable
to the
trap
zero-point
energy
of
10
nK.
Similarly,
the
number
of atoms
is
difficult to determine
because not a11 of the diffracted
light
is
imaged
and
the
scattering
properties
of a Bose condensate are
uncertain
[25].
However, the
total
absorption
indicates N
=
2
X
10,
while the
peak
absorption
indicates N
=
4
X
10 .
Five
cycles
resulted in
images
similar to
Fig.
1(b),
with
nominal
temperatures
of
—
100
nK and numbers from
2
X
104
to
2
X
105.
At 100
nK,
N,
is
only
3500
atoms,
suggesting
that a
majority
of the atoms reside in the
condensate.
The numbers of condensate atoms
implied
by
these measurements is
significantly greater
than the
maximum
of
1500 atoms
predicted
for a
stable condensate
by
mean-field
theories
[13],
though
a
recent
study
of
three-
body
interactions
suggests
that condensates
with more
atoms
may
be
possible
[26].
However,
we caution that
due to the
uncertainties
described
above,
more
quantitative
measurements are needed.
We
have
presented
evidence
that a
gas
with
negative
scattering length
can
undergo
BEC. Our observations
im-
ply
that the
gas
does
not form a
liquid
or solid in tens of
seconds
that
elapse
between
entering
the
quantum
degen-
erate
regime during
the final
stages
of
evaporative
cooling,
and the
subsequent
absorption
image
exposure.
Otherwise,
an
abrupt
loss
of
atoms
would be observed as
the latent
heat from
the
ordinary
phase
transition was
released. In
the
future,
we
plan
to use
a lower f-number
optical
sys-
tem and
microwave
spectroscopy
to
more
directly
detect
the condensate,
and to
quantitatively
measure its occupa-
tion
number,
spatial size,
and
lifetime.
Many
intriguing
possibilities
await further
exploration,
including gaseous
superfluidity,
anomalous
light
scattering
[25],
and
experi-
ments
with
the
fermionic
atom Li.
The authors are
grateful
to
M.
Holland for a
helpful
discussion. We
thank
Hewlett-Packard for
loaning
us
the
microwave
synthesizer
used
in this
experiment.
This
work
is
supported
by
the
National Science
Foundation and
the Welch
Foundation.
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1690
