© 1999 Macmillan Magazines Ltd
crater in 10
6
yr, but the clear lesson from the MOC data is that the
blankets are not uniformly distributed airfall but represent a net
sum of airfall, erosion, saltation cleaning and deposition. Other
small craters have ®ll with periodic bedforms, and erosional strip-
ping of layers and jointed sediments. The ubiquity of martian
aeolian material is emphasized by the ®lling of craters and structural
features high on Arsia Mons, more than 20 km above the reference
6.1-mbar level in the atmosphere; these deposits indicate transport
of considerable amounts of material at pressures under 1 mbar (ref.
29; unless there has been a geologically recent period of higher
atmospheric pressure on Mars, which is generally considered very
unlikely
21
).
The indications of distinctive materials, possibly sulphates,
moved in saltation on Mars expands the known complexity of the
martian sedimentary system, and implies that remnants of ancient
sedimentary and weathering processes have not been completely
homogenized and lost from remote or in situ studies. This variety
may assist some future sampling efforts, but the presence of wide-
spread mantling reduces the areas within which bedrock or sorted
remnants are available for sampling.
M
Received 16 September; accepted 21 December 1998.
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87±88 (1997).
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Geophys. Res. 84, 8167±8182 (1979).
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results. J. Geophys. Res. 82, 4625±4634 (1977).
13. Rieder, R. et al. The chemical composition of Martian soil and rocks returned by the mobile alpha
proton X-ray spectrometer: Preliminary results from the X-ray mode. Science 278, 1771±1774 (1997).
14. White, B. R. Soil transport by winds on Mars. J. Geophys. Res. 84, 4643±4651 (1979).
15. Warren, P. H. Petrologic evidence for low-temperature, possibly ¯ood evaporitic origin of carbonates
in the ALH84001 meteorite. J. Geophys. Res. 103, 16759±16773 (1998).
16. McKee, E. D. Structures of dune at White Sands National Monument, New Mexico. Sedimentology 7,
3±69 (1966).
17. Jones, D. J. Gypsum±oolite dunes, Great Salt Lake desert, Utah. Bull. Am. Assoc. Petrol. Geol. 37, 2530±
2538 (1938).
18. Thomas, P. C., Veverka, J., Gineris, D. & Wong, L. `Dust' streaks on Mars. Icarus 49, 398±415 1984:
19. Malin, M. C. et al. Early views of the Martian surface from the Mars Orbiter Camera of Mars Global
Surveyor. Science 279, 1681±1685 (1998).
20. Arvidson, R. E., Guinness, E. A. & Lee, S. Differential aeolian redistribution rates on Mars. Nature 278,
533±535 (1979).
21. Kieffer, H. H. & Zent, A. P. in Mars (eds Kieffer, H., Jakosky, B., Snyder, C. & Matthews, M.) 1180±
1220 (Univ. Arizona Press, Tucson, 1992).
22. Thomas, P. C. Present wind activity on Mars: Relation to large latitudinally zoned sediment deposits. J.
Geophys. Res. 87, 9999±10008 (1982).
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general circulation model results. J. Geophys. Res. 98, 3183±3196 (1993).
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fallout. Icarus 58, 331±338 (1984).
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Res. 100, 5381±5395 (1995).
26. Edgett, K. S. & Blumberg, D. G. Star and linear dunes on Mars. Icarus 112, 448±464 (1994).
27. Zimbelman, J. R. Spatial resolution and the interpretation of Martian morphology: Implications for
subsurface volatiles. Icarus 71, 257±267 (1987).
28. Breed, C. S., McCauley, J. F. & Davis, P. A. Ripple blankets: geomorphic evidence for regional sand
sheet deposits on Mars. Proc. Lunar Planet. Sci. Conf. 18, 127 (1987).
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589 (1999).
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Acknowledgements. We acknowledge the technical help of M. Caplinger, J. Warren, M. Ravine, M. Ryan,
M. Ockert-Bell, I. Dauber, R. Sullivan, K. Edgett, B. Carcich, A. Fox, A. Lowenkron & M. Roth.
Correspondence and requests for materials should be addressed to P.C.T. (e-mail: thomas@cuspif.tn.
cornell.edu).
letters to nature
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Figure 2 MOC image showing blanketing material ®lling several craters that are
200±400 m in diameter. Image scale, 4.1 by 7.3 m per pixel; 3.68 S, 340.58 W. The
crater rims are moderately well preserved and visible, indicating that aeolian
material has not simply been draped over the topography but has preferentially
®lled the lowest areas. The average depth of fresh 200-m craters is probably
,20 m. Rim heights above surroundings are typically much less than crater
depths; thus the ®ll inside these craters is probably deeper than in much of the
intervening plains.
Light speed reduction
to 17 metres per second
in an ultracold atomic gas
Lene Vestergaard Hau*
²
, S. E. Harris
³
, Zachary Dutton*
²
& Cyrus H. Behroozi*§
* Rowland Institute for Science, 100 Edwin H. Land Boulevard, Cambridge,
Massachusetts 02142, USA
²
Department of Physics, § Division of Engineering and Applied Sciences,
Harvard University, Cambridge, Massachusetts 02138, USA
³
Edward L. Ginzton Laboratory, Stanford University, Stanford, California 94305,
USA
.........................................................................................................................
Techniques that use quantum interference effects are being
actively investigated to manipulate the optical properties of
quantum systems
1
. One such example is electromagnetically
induced transparency, a quantum effect that permits the propaga-
tion of light pulses through an otherwise opaque medium
2±5
. Here
we report an experimental demonstration of electromagnetically
induced transparency in an ultracold gas of sodium atoms, in
which the optical pulses propagate at twenty million times slower
than the speed of light in a vacuum. The gas is cooled to
nanokelvin temperatures by laser and evaporative cooling
6±10
.
The quantum interference controlling the optical properties of
the medium is set up by a `coupling' laser beam propagating at a
right angle to the pulsed `probe' beam. At nanokelvin tempera-
tures, the variation of refractive index with probe frequency can
be made very steep. In conjunction with the high atomic density,
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this results in the exceptionally low light speeds observed. By
cooling the cloud below the transition temperature for Bose±
Einstein condensation
11±13
(causing a macroscopic population of
alkali atoms in the quantum ground state of the con®ning
potential), we observe even lower pulse propagation velocities
(17 m s
-1
) owing to the increased atom density. We report an
inferred nonlinear refractive index of 0.18 cm
2
W
-1
and ®nd that
the system shows exceptionally large optical nonlinearities, which
are of potential fundamental and technological interest for quantum
optics.
The experiment is performed with a gas of sodium atoms cooled
to nanokelvin temperatures. Our atom cooling set-up is described
in some detail in ref. 14. Atoms emitted from a `candlestick' atomic
beam source
15
are decelerated in a Zeeman slower and loaded into a
magneto-optical trap. In a few seconds we collect a cloud of 10
10
atoms at a temperature of 1 mK and a density of 6 3 10
11
cm
2 3
. The
atoms are then polarization gradient cooled for a few milliseconds
to 50 mK and optically pumped into the F 1 ground state with an
equal population of the three magnetic sublevels. We then turn all
laser beams off and con®ne the atoms magnetically in the `4 Dee'
trap
14
. Only atoms in the M
F
2 1 state, with magnetic dipole
moments directed opposite to the magnetic ®eld direction (picked
as the quantization axis), are trapped in the asymmetric harmonic
trapping potential. This magnetic ®ltering results in a sample of
atoms that are all in a single atomic state (state | 1i in Fig. 1b) which
allows adiabatic optical preparation of the atoms, as described
below, and minimal heating of the cloud.
Next we evaporatively cool the atoms for 38 s to the transition
temperature for Bose±Einstein condensation, T
c
. The magnetic
®elds are then adjusted to adiabatically soften the trap. The resulting
trapping potential has a frequency of f
z
21 Hz along the sym-
metry (z) axis of the 4 Dee trap, and transverse frequencies
f
x
f
y
69 Hz. The bias ®eld, parallel to the z axis, is 11 G.
When we cool well below T
c
, we are left with 1±2 million atoms
in the condensate. For these parameters the transition occurs at a
temperature of T
c
435 nK and a peak density in the cloud of
5 3 10
12
cm
2 3
.
We now apply a linearly polarized laser beam, the coupling beam,
tuned to the transition between the unpopulated hyper®ne states | 2i
and | 3i (Fig. 1b). This beam couples states | 2i and | 3i and creates a
quantum interference for a weaker probe laser beam (left circularly
polarized) which is tuned to the j1i!j3i transition. A stable
eigenstate (the `dark state') of the atom in the presence of coupling
and probe lasers is a coherent superposition of the two hyper®ne
ground states | 1i and | 2i. The ratio of the probability amplitudes is
such that the contributions to the atomic dipole moment induced
by the two lasers exactly cancel. The quantum interference occurs in
a narrow interval of probe frequencies, with a width determined by
the coupling laser power.
Figure 2a shows the calculated transmission of the probe beam as
a function of its detuning from resonance for parameters which are
typical of this work. In the absence of dephasing of the j1i!j2i
transition, the quantum interference would be perfect, and at line
centre, the transmission would be unity. Figure 2b shows the
refractive index for the probe beam as a function of detuning.
Due to the very small Doppler broadening of the j1i!j2i transition
in our nanokelvin samples, application of very low coupling
intensity leads to a transparency peak with a width much smaller
than the natural line width of the j1i!j3i transition. Correspond-
ingly, the dispersion curve is much steeper than can be obtained by
any other technique, and this results in the unprecedented low
group velocities reported here. The group velocity v
g
for a propagat-
ing electromagnetic pulse is
16±19
:
v
g
c
nq
p
q
p
dn
dq
p
<
~ce
0
2q
p
j
c
j
2
jm
13
j
2
N
1
-50 -25 0 25 50
-50
-25
0
25
50
-200 -100 0 100 200
-150
-100
-50
0
50
100
150
200
x
(µm)
y
(µm)
(µm)
z
(µm)
x
1.8 GHz
60 MHz
D
line
λ = 589 nm
ωω
c
p
|2 = |
F
=2,
M
= –2
〉
〉
F
|1 = |
F
=1,
M
= –1
〉
〉
F
|4 = |
F
=3,
M
= –2
〉
〉
F
|3 = |
F
=2,
M
= –2
〉
〉
F
CCD 1
CCD 2
PMT
Pinhole
Coupling beam
|2 ➞|3 ,
F
=2➞3, linear
Imaging beam
Probe pulse
B
x
^
z
^
y
^
Atom cloud
circular
linear
〉〉
|1 ➞|3 ,
〉〉
2
ii
i
a
b
Figure 1 Experimental set-up. A `coupling' laser beam propagates along the x
axis with its linear polarization along the 11-G bias ®eld in the z direction. The
`probe' laser pulse propagates along the z axis and is left-circularly polarized.
With a ¯ipper mirror in front of the camera CCD 1, we direct this probe beam either
to the camera or to the photomultiplier (PMT). For pulse delay measurements, we
place a pinhole in an external image plane of the imaging optics and select a small
area, 15 mm in diameter, of the probe beam centred on the atom clouds (as
indicated by the dashed circle in inset (i)). The pulse delays are measured with the
PMT. The imaging beam propagating along the y axis is used to image atom
clouds onto camera CCD 2 to ®nd the length of the clouds along the pulse
propagation direction (z axis) for determination of light speeds. Inset (ii) shows
atoms cooled to 450 nK which is 15 nK above T
c
. (Note that this imaging beam is
never applied at the same time as the probe pulse and coupling laser). The
position of a cloud and its diameter in the two transverse directions, x and y,are
found with CCD 1. Inset (i) shows an image of a condensate.
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|
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Here n(q
p
) is the refractive index at probe frequency q
p
(rad s
-1
),
j
c
j
2
is the square of the Rabi frequency for the coupling laser and
varies linearly with intensity, m
13
is the electric dipole matrix
element between states | 1i and | 3i, N is the atomic density, and
e
0
is the permittivity of free space. At line centre, the refractive index
is unity, and the second term in the denominator of equation (1)
dominates the ®rst. An important characteristic of the refractive
index pro®le is that on resonance the dispersion of the group
velocity is zero (see ref. 16), that is, d
2
n=dq
2
p
0, and to lowest
order, the pulse maintains its shape as it propagates. The established
quantum interference allows pulse transmission through our atom
clouds which would otherwise have transmission coef®cients of e
-110
(below T
c
), and creates a steep dispersive pro®le and very low group
velocity for light pulses propagating through the clouds.
We note that the centres of the curves in Fig. 2 are shifted by
0.6 MHz from probe resonance. This is due to a coupling of state | 2i
to state | 4i through the coupling laser ®eld, which results in an a.c.
Stark shift of level | 2i and a corresponding line shift of the 2 ! 3
transition. As the transparency peak and unity refractive index are
obtained at two-photon resonance, this leads to a refractive index at
the 1 ! 3 resonance frequency which is different from unity. The
difference is proportional to the a.c. Stark shift and hence to the
coupling laser intensity, which is important for predicting the
nonlinear refractive index as described below.
A diagram of the experiment is shown in Fig. 1a. The 2.5-mm-
diameter coupling beam propagates along the x axis with its linear
polarization parallel to the B ®eld. The 0.5-mm-diameter, j
-
polarized probe beam propagates along the z axis. The size and
position of the atom cloud in the transverse directions, x and y,are
obtained by imaging the transmission pro®le of the probe beam
after the cloud onto a charge-coupled-device (CCD) camera. An
image of a condensate is shown as inset (i). A 55 mW cm
-2
coupling
laser beam was present during the 10-ms exposure of the atoms to a
5mWcm
-2
probe beam tuned close to resonance. The f/7 imaging
optics are diffraction-limited to a resolution of 7 mm.
During the pulse delay experiments, a pinhole (placed in an
external image plane of the lens system) is used to select only the
part of the probe light that has passed through the central 15 mmof
the atom cloud where the column density is the greatest. The outline
of the pinhole is indicated with the dashed circle in inset (i).
Both coupling and probe beams are derived from the same dye
laser. The frequency of the coupling beam is set by an acousto-optic
modulator (AOM) to the j2i!j3i resonance. Here we take into
account both Zeeman shifts and the a.c. Stark shift described above.
The corresponding probe resonance is found by measuring the
transmission of the probe beam as a function of its frequency. We
apply a fast frequency sweep, across 32 MHz in 50 ms, and determine
resonance from the transmission peak. The sweep is controlled by a
separate AOM. The frequency is then ®xed at resonance, and the
temporal shape of the probe pulse is generated by controlling the r.f.
drive power to the AOM. The resulting pulse is approximately
gaussian with a full-width at half-maximum of 2.5 ms. The peak
power is 1 mW cm
-2
corresponding to a Rabi frequency of
p
0:20 A, where the Einstein A coef®cient is 6:3 3 10
7
rad s
2 1
.
To avoid distortion of the pulse, it is made of suf®cient duration that
its Fourier components are contained within the transparency peak.
Probe pulses are launched along the z axis 4 ms after the coupling
beam is turned on (the coupling ®eld is left on for 100 ms). Due to
the magnetic ®ltering discussed above, all atoms are initially in state
|1i which is a dark state in the presence of the coupling laser only.
When the pulse arrives, the atoms adiabatically evolve so that
the probability amplitude of state | 2i is equal to the ratio
p
=
2
p
2
c
1=2
, where
p
is the probe Rabi frequency. To establish
the coherent superposition state, energy is transferred from the
–30 –20 –10 0 10 20 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Transmission
a
–30 –20 –10 0 10 20 30
Probe detuning (MHz)
0.994
0.996
0.998
1.000
1.002
1.004
1.006
Refractive index
b
Figure 2 Effect of probe detuning. a, Transmission pro®le. Calculated probe
transmission as a function of detuning from the j1i!j3i resonance for an atom
cloud cooled to 450 nK, with a peak density of 3:3 3 10
12
cm
2 3
and a length of
229 mm (corresponding to the cloud in inset (ii) of Fig. 1a). The coupling laser is
resonant with the j2i!j3i transition and has a power density of 52 mW cm
-2
. b,
Refractive index pro®le. The calculated refractive index is shown as a function of
probe detuning for the same parameters as in a. The steepness of the slope at
resonance is inversely proportional to the group velocity of transmitted light
pulses and is controlled by the coupling laser intensity. Note that as a result of the
a.c. Stark shift of the j2i!j3i transition, caused by a coupling of states | 2i and | 4i
through the coupling laser ®eld, the centre of the transmission and refractive
index pro®les is shifted by 0.6 MHz. The shift of the refractive index pro®le results
in the nonlinear refractive index described in the text.
–2 0 2 4 6 8 10 12
Time (µs)
0
5
10
15
20
25
30
PMT signal (mV)
T
= 450 nK
τ
Delay
= 7.05 ± 0.05 µs
L
= 229 ± 3 µm
v
g
= 32.5 ± 0.5 m s
–1
Figure 3 Pulse delay measurement. The front pulse (open circles) is a reference
pulse with no atoms in the system. The other pulse (®lled circles) is delayed by
7. 0 5 ms in a 229-mm-long atom cloud (see inset (ii) in Fig.1a). The corresponding
light speed is 32.5 m s
-1
. The curves represent gaussian ®ts to the measured
pulses.
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front of the probe pulse to the atoms and the coupling laser ®eld. At
the end of the pulse, the atoms adiabatically return to the original
state | 1i and the energy returns to the back of the probe pulse with
no net energy and momentum transfer to the atomic cloud. Because
the refractive index is unity, the electric ®eld is unchanged as the
probe pulse enters the medium. As the group velocity is decreased,
the total energy density must increase so as to keep constant the
power per area. This increase is represented by the energy stored in
the atoms and the coupling laser ®eld during pulse propagation
through the cloud.
The pulses are recorded with a photomultiplier (3-ns response
time) after they penetrate the atom clouds. The output from the
photomultiplier is ampli®ed by a 150-MHz-bandwidth ampli®er
and the waveforms are recorded on a digital scope. With a `¯ipper'
mirror in front of the camera we control whether the probe beam is
directed to the camera or to the photomultiplier.
The result of a pulse delay measurement is shown in Fig. 3. The
front pulse is a reference pulse obtained with no atoms present. The
pulse delayed by 7.05 ms was slowed down in an atom cloud with a
length of 229 mm (see Fig. 1a, inset (ii)). The resulting light speed is
32.5 m s
-1
. We used a coupling laser intensity of 12 mW cm
-2
corresponding to a Rabi frequency of
c
0:56 A. The cloud was
cooled to 450 nK (which is 15 nK above T
c
), the peak density was
3:3 3 10
12
cm
2 3
, and the total number of atoms was 3:8 3 10
6
.
From these numbers we calculate that the pulse transmission
coef®cient would be e
-63
in the absence of the coupling laser. The
probe pulse was indeed observed to be totally absorbed by the atoms
when the coupling beam was left off. Inhomogeneous broadening
due to spatially varying Zeeman shifts is negligible (,20 kHz) for
the low temperatures and correspondingly small cloud sizes used
here.
The size of the atom cloud in the z direction is obtained with
another CCD camera. For this purpose, we use a separate 1 mW cm
-2
laser beam propagating along the vertical y axis and tuned 20 MHz
below the F 2 ! 3 transition. The atoms are pumped to the
F 2 ground state for 10 ms before the imaging which is performed
with an exposure time of 10 ms. We image the transmission pro®le
of the laser beam after the atom cloud with diffraction-limited f/5
optics. An example is shown in Fig. 1a, inset (ii), where the
asymmetry of the trap is clear from the cloud's elliptical pro®le.
We note that the imaging laser is never applied at the same time as
the coupling laser and probe pulse, and for each recorded pulse or
CCD picture a new cloud is loaded.
We measured a series of pulse delays and corresponding cloud
sizes for atoms cooled to temperatures between 2.5 mK and 50 nK.
From these pairs of numbers we obtain the corresponding propaga-
tion velocities (Fig. 4). The open circles are for a coupling power of
52 mW cm
-2
(
c
1:2 A). The light speed is inversely proportional
to the atom density (equation (1)) which increases with lower
temperatures, with an additional density increase when a conden-
sate is formed. The ®lled circles are for a coupling power of
12 mW cm
-2
. The lower coupling power is seen to cause a decrease
of group velocities in agreement with equation (1). We obtain a light
speed of 17 m s
-1
for pulse propagation in an atom cloud initially
prepared as an almost pure Bose±Einstein condensate (condensate
fraction is >90%). Whether the cloud remains a condensate during
and after pulse propagation is an issue that is beyond the scope of
this Letter.
Transitions from state | 2i to state | 4i, induced by the coupling
laser (detuned by 60 MHz from this transition), result in a ®nite
decay rate of the established coherence between states | 1i and | 2i
and limit pulse transmission. The dephasing rate is proportional to
the power density of the coupling laser and we expect, and ®nd, that
probe pulses have a peak transmission that is independent of
coupling intensity and a velocity which reduces linearly with this
intensity. The dephasing time is determined from the slope of a
semi-log plot of transmission versus pulse delay
19
. At a coupling
power of 12 mW cm
-2
, we measured a dephasing time of 9 ms for
atom clouds just above T
c
.
Giant Kerr nonlinearities are of interest for areas of quantum
optics such as optical squeezing, quantum nondemolition, and
studies of nonlocality. It was recently proposed that they may be
obtained using electromagnetically induced transparency
20
.Herewe
report the ®rst (to our knowledge) measurement of such a non-
linearity. The refractive index for zero probe detuning is given by
n 1 n
2
I
c
where I
c
is the coupling laser intensity, and n
2
the
cross phase nonlinear refractive index. As seen from Fig. 2b, the
nonlinear term (n
2
I
c
) equals the product of the slope of the
refractive index at probe resonance and the a.c. Stark shift of the
j2i!j3i transition caused by the coupling laser. We can then
express n
2
by the formula (see equation (1));
n
2
Dq
S
I
c
dn
dq
p
<
1
2p
Dq
S
I
c
l
v
g
2
where Dq
S
is the a.c. Stark shift, proportional to I
c
, and l the
wavelength of the probe transition. We measured an a.c. Stark shift
of 1: 3 3 10
6
rad s
2 1
for a coupling laser intensity of 40 mW cm
-2
.
For a measured group velocity of 17 m s
-1
(Fig. 4), we obtain a
nonlinear refractive index of 0.18 cm
2
W
-1
. This nonlinear index is
,10
6
times greater than that measured in cold Cs atoms
21
.
With a system that avoids the | 1i-| 2i dephasing rate described
above (which can be obtained by tuning to the D
l
line in sodium),
the method used here could be developed to yield the collision-
induced dephasing rate of the double condensate which is generated
in the process of establishing electromagnetically induced transpar-
ency (see also refs 22, 23). In that case, the square of the probability
amplitude for state | 3i could be kept below 10
-5
during pulse
propagation, with no heating of the condensate as a result. With
improved frequency stability of our set-up and lower coupling
intensities, even lower light speeds would be possible, perhaps of
the order of centimetres per second, comparable to the speed of
0
T
c
= 435 1000 1500 2000 2500
Temperature (nK)
10
20
50
100
200
500
v
g
(m s
–1
)
10
20
50
100
200
500
52 mW cm
–2
12 mW cm
–2
Figure 4 Light speed versus atom cloud temperature. The speed decreases with
temperature due to the atom density increase. The open circles are for a coupling
power of 52 mW cm
-2
and the ®lled circles are for a coupling power of 12 mW cm
-2
.
The temperature T
c
marks the transition temperature for Bose±Einstein
condensation. The decrease in group velocity below T
c
is due to a density
increase of the atom cloud when the condensate is formed. From imaging
measurements we obtain a maximum atom density of 8 3 10
13
cm
2 3
at a tem-
perature of 200 nK. Here, the dense condensate component constitutes 60% of all
atoms, and the total atom density is 16 times larger than the density of a non-
condensed cloud at T
c
. The light speed measurement at 50 nK is for a cloud with a
condensate fraction >90%. The ®nite dephasing rate due to state | 4i does not
allow pulse penetration of the most dense clouds. This problem could be over-
come by tuning the laser to the D
l
line as described in the text.
© 1999 Macmillan Magazines Ltd
letters to nature
598 NATURE
|
VOL 397
|
18 FEBRUARY 1999
|
www.nature.com
sound in a Bose±Einstein condensate. Under these conditions we
expect phonon excitation during light pulse propagation through
the condensate. By deliberately tuning another laser beam to the
j2i!j4i transition, it should be possible to demonstrate optical
switching at the single photon level
24
. Finally, we note that during
propagation of the atom clouds, light pulses are compressed in the z
direction by a ratio of c/v
g
. For our experimental parameters, that
results in pulses with a spatial extent of only 43 mm.
M
Received 3 November; accepted 21 December 1998.
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Acknowledgements. We thank J. A. Golovchenko for discussions and C. Liu for experimental assistance.
L.V.H. acknowledges support from the Rowland Institute for Science. S.E.H. is supported by the US Air
Force Of®ce of Scienti®c Research, the US Army Research Of®ce, and the US Of®ce of Naval Research.
C.H.B. is supported by an NSF fellowship.
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Luttinger-liquid behaviour
in carbon nanotubes
Marc Bockrath*, David H. Cobden*, Jia Lu*,
Andrew G. Rinzler
²
, Richard E. Smalley
²
, Leon Balents
³
& Paul L. McEuen*
* Department of Physics, University of California and Materials Sciences Division,
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
²
Center for Nanoscale Science and Technology, Rice Quantum Institute and
Department of Chemistry and Physics, MS-100, Rice University, PO Box 1892,
Houston, Texas 77251, USA
³
Institute for Theoretical Physics, University of California, Santa Barbara,
California 93106-4030, USA
.........................................................................................................................
Electron transport in conductors is usually well described by
Fermi-liquid theory, which assumes that the energy states of the
electrons near the Fermi level E
F
are not qualitatively altered by
Coulomb interactions. In one-dimensional systems, however,
even weak Coulomb interactions cause strong perturbations.
The resulting system, known as a Luttinger liquid, is predicted
to be distinctly different from its two- and three-dimensional
counterparts
1
. For example, tunnelling into a Luttinger liquid at
energies near the Fermi level is predicted to be strongly sup-
pressed, unlike in two- and three-dimensional metals. Experi-
ments on one-dimensional semiconductor wires
2,3
have been
interpreted by using Luttinger-liquid theory, but an unequivocal
veri®cation of the theoretical predictions has not yet been
obtained. Similarly, the edge excitations seen in fractional quan-
tum Hall conductors are consistent with Luttinger-liquid
behaviour
4,5
, but recent experiments failed to con®rm the pre-
dicted relationship between the electrical properties of the bulk
state and those of the edge states
6
. Electrically conducting single-
walled carbon nanotubes (SWNTs) represent quantum wires
7±10
that may exhibit Luttinger-liquid behaviour
11,12
. Here we present
measurements of the conductance of bundles (`ropes') of SWNTs
as a function of temperature and voltage that agree with predic-
tions for tunnelling into a Luttinger liquid. In particular, we ®nd
that the conductance and differential conductance scale as power
laws with respect to temperature and bias voltage, respectively,
and that the functional forms and the exponents are in good
agreement with theoretical predictions.
SWNTs are suf®ciently robust and long to allow electrical con-
nections to lithographically de®ned metallic electrodes, thereby
making it possible to probe the intriguing electrical properties of
Figure 1 The two-terminal linear-response conductance G versus gate voltage
V
g
for a bulk-contacted metallic nanotube rope at a variety of temperatures. The
data show signi®cant temperature dependence for energy scales above the
charging energy that cannot be explained by the Coulomb blockade model.
Inset: average conductance as a function of temperature T. The samples used in
these experiments are made in one of two ways. In both methods, SWNTs are
deposited from a suspension in dichloroethane onto a 1-mm-thick layer of SiO
2
that has been thermally grown on a degenerately doped Si wafer, used as a gate
electrode. Atomic force microscopy imaging reveals that the diameters of the
ropes vary between 1 and 10 nm. In the ®rst method
9
, chromium±gold contacts
are applied over the top of the nanotube rope using electron beam lithography
and lift-off. From measurements of these devices in the Coulomb blockade
regime, we conclude that the electrons are con®ned to the length of rope
between the leads. This implies that the leads cut the nanotubes into segments,
and transport involves tunnelling into the ends of the nanotubes (`end-con-
tacted'). In the second method
10
, electron-beam lithography is ®rst used to
de®ne leads, and ropes are deposited on top of the leads. Samples were selected
that showed Coulomb blockade behaviour at low temperatures with a single well-
de®ned period, indicating the presence of a single quantum dot. The charging
energy of these samples indicates a quantum dot with a size substantially larger
than the spacing between the leads, as found by Tans et al.
10
. Transport thus
occurs by electrons tunnelling into the middle, or bulk, of the nanotubes (`bulk-
contacted').
