HAL Id: hal-00949944
https://hal.archives-ouvertes.fr/hal-00949944
Submitted on 12 Jan 2015
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Observation of Suppression of Light Scattering Induced
by Dipole-Dipole Interactions in a Cold-Atom Ensemble
J Pellegrino, R Bourgain, Stephan Jennewein, Yvan R. P. Sortais, Antoine
Browaeys, S. D. Jenkins, J Ruostekoski
To cite this version:
J Pellegrino, R Bourgain, Stephan Jennewein, Yvan R. P. Sortais, Antoine Browaeys, et al.. Ob-
servation of Suppression of Light Scattering Induced by Dipole-Dipole Interactions in a Cold-Atom
Ensemble. Physical Review Letters, American Physical Society, 2014, 113, pp.133602 �10.1103/Phys-
RevLett.113.133602�. �hal-00949944�
Observation of Suppression of Light Scattering Induced by Dipole-Dipole Interactions
in a Cold-Atom Ensemble
J. Pellegrino, R. Bourgain, S. Jennewein, Y. R. P. Sortais, and A. Browaeys
Laboratoire Charles Fabry, Institut d’Optique, CNRS, Université Paris Sud, 2 Avenue Augustin Fresnel,
91127 Palaiseau cedex, France
S. D. Jenkins and J. Ruostekoski
Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom
(Received 18 February 2014; published 26 September 2014)
We study the emergence of collective scattering in the presence of dipole-dipole interactions when we
illuminate a cold cloud of rubidium atoms with a near-resonant and weak intensity laser. The size of the
atomic sample is comparable to the wavelength of light. When we gradually increase the number of atoms
from 1 to ∼450, we observe a broadening of the line, a small redshift and, consistently with these, a strong
suppression of the scattered light with respect to the noninteracting atom case. We compare our data to
numerical simulations of the optical response, which include the internal level structure of the atoms.
DOI: 10.1103/PhysRevLett.113.133602 PACS numbers: 42.50.Ct, 03.65.Nk, 32.80.Qk, 42.50.Nn
When resonant emitters, such as atoms, molecules,
quantum dots, or metamaterial circuits, with a transition
at a wavelength λ, are confined inside a volume smaller
than λ
3
, they are coupled via strong dipole-dipole inter-
actions. In this situation, the response of the ensemble to
near-resonant light is collective and originates from the
excitation of collective eigenstates of the system, such
as super- and subradiant modes [1–3]. Dipole-dipole inter-
actions affect the response of the system and the collective
scattering of near-resonant light differs from the case of
an assembly of noninteracting emitters [4]. It has even
been predicted to be suppressed for a dense gas of cold two-
level atoms [5].
Following the recent measurement of the collective
Lamb shift [6] in a Fe layer [7], in a hot thermal vapor
[8], and in arrays of trapped ions [9], it was pointed out [10]
that the collective response of interacting emitters is
different between ensembles exhibiting inhomogeneous
broadening, such as solid state systems or thermal vapors,
and those free of it, such as cold-atom clouds. In particular,
inhomogeneous broadening suppresses the correlations
induced by the interactions between dipoles, leading to
the textbook theory of the optical response of continuous
media [10,11]. In the absence of broadening, however, this
theory fails and should be revisited to include the light-
induced correlations [12–19]. Several recent experiments
aiming at studying collective scattering with identical
emitters used large and optically thick ensembles of cold
atoms [20–23]. However, the case of a cold-atom ensemble
with a size comparable to the optical wavelength has not
been studied experimentally, nor has the transition between
the well-understood case of scattering by an individual
atom [24] to collective scattering. In particular, the sup-
pression of light scattering when the number of atoms
increases in a regime of collective scattering has never been
directly observed.
Here, we study—both experimentally and theoretically—
the emergence of collective effects in the optical response
of a cold-atom sample due to dipole-dipole interactions, as
we gradually increase the number of atoms. To do so, we
send low-intensity near-resonant laser light onto a cloud
containing from 1 to ∼450 cold
87
Rb atoms, with a size
comparable to the wavelength of the optical transition at
λ ¼ 780 nm. Starting from one atom, we observe a broad-
ening of the line as the number of atoms increases, as well
as a small redshift and a strong suppression of the amount
of scattered light with respect to the case of noninteracting
atoms. We show that this suppression is consistent with the
measured broadening and shift. We finally compare our
measurements to a numerical simulation of the response of
the system in the low excitation limit, including the internal
level structure of the atoms.
The suppression of light scattering by resonant
dipole-dipole interactions can be understood qualitatively
as follows. Consider a laser radiation with frequency ω
impinging on an ensemble of classical radiating dipoles
with resonance frequency ω
0
¼ 2πc=λ (see Fig. 1). When
the dipoles interact through the dipole-dipole potential
V
lβ
jα
¼ −V
dd
½p
αβ
ðikr − 1Þþq
αβ
ðkrÞ
2
e
ikr
; ð1Þ
the system features collective modes with various eigen-
frequencies and decay rates. Here, j and l denote two
dipoles separated by a distance r, V
dd
¼ 3Γ=4ðkrÞ
3
,
k ¼ 2π=λ, Γ is the radiative decay rate in the absence of
interactions, and the angular functions p
αβ
and q
αβ
depend
on the polarizations α and β and the relative orientations of
the dipoles j and l [25,26]. In our experiment, the geometry
PRL 113, 133602 (2014)
PHYSICAL REVIEW LETTERS
week ending
26 SEPTEMBER 2014
0031-9007=14=113(13)=133602(5) 133602-1 © 2014 American Physical Society
of the atomic system and its orientation with respect to the
excitation are such that the incident laser best couples to
only a few modes with decay rates Γ
c
larger than Γ
(superradiant modes), leading to a broader excitation
spectrum. The excitation rate, and therefore the amount
of scattered light, should thus be reduced by a factor
ðΓ=Γ
c
Þ
2
. The effect is stronger when the average distance
between dipoles hri is smaller than λ=2π.
To study the collective scattering by an ensemble of
atoms coupled via resonant dipole-dipole interactions we
use the setup depicted in Fig. 1(a). We prepare small clouds
containing up to 450 atoms at a temperature ∼100 μK,
confined in a microscopic dipole trap [27], and illuminate
them with laser light nearly resonant with the atomic
transition at λ ¼ 780 nm. The Doppler width of the sample
(150 kHz) is much smaller than the atomic linewidth
Γ=2π ¼ 6 MHz, making inhomogeneous broadening neg-
ligible. The anisotropy of the trap results in an elongated
cloud with calculated root-mean-square thermal sizes
σ
ρ
¼ 0.3λ and σ
z
¼ 2.4λ. The maximal density is ρ ¼
2.5 × 10
14
at.=cm
3
and the minimal average interatomic
distance hri¼ρ
−
1
3
¼ 0.2λ [Fig. 1(b)]. In this regime,
khri ∼ 1, leading to V
dd
∼ Γ, and the resonant dipole-dipole
interaction will therefore have an effect on the scattering.
Experimentally, we prepare the trapped atoms in the
F ¼ 2 hyperfine manifold with an efficiency better than
95%. We then release them in free space by switching off the
trapping light while exciting them with σ
þ
polarized light at a
frequency ω ¼ ω
0
þ Δ tuned near the (5S
1=2
, F ¼ 2)to
(5P
3=2
, F
0
¼ 3) transition (see Fig. 1). In this way we avoid
extra light shifts induced by the trapping beam that would
obscure the measurement of small collective shifts and
broadening. Also, we choose the intensity saturation I=I
sat
¼
0.1 to be in the low excitation limit (I
sat
¼ 1.6 mW=cm
2
).
We interleave excitation pulses with duration 125 ns and
recapture periods in the dipole trap with duration 1 μs. This
sequence is repeated 200 times using the same cloud of
atoms, in order to improve the duty cycle of the experiment.
Finally, we prepare a new atomic sample and repeat the set of
excitation pulses a few hundred times. The scattered light that
we collect in the z direction is therefore the result of an
average over many spatial configurations of the atoms. The
choice of the number of pulses (200) is a trade-off between
getting a good signal-to-noise ratio and avoiding light-
assisted losses [28] or heating of the cloud, both of which
would lower the density. We checked that both effects do not
exceed 5% over the entire set of pulses and that less than 5%
of the atoms are depumped in the (5S
1=2
, F ¼ 1)hyperfine
level during the excitation.
Figure 2 shows the number of photons n
z
ðN;ΔÞ detected
by the I-CCD as a function of the detuning Δ of the
excitation laser, for various numbers of atoms N.A
Lorentzian fit agrees well with the data for the range of
N explored here. As expected from the qualitative argument
described above, we observe that the full width at half
maximum (FWHM) increases with the number of atoms
[see Fig. 3(a)], since the interatomic distance then
decreases, leading to stronger dipole-dipole interactions.
We also measure a small redshift δω of the center frequency
[Fig. 3(b)]. For N ¼ 1, the FWHM is 1.35 0.15Γ,in
agreement with the short duration of the excitation pulses
(125 ns), which broadens slightly the resonance. Figure 2
laser
(c)
λ
= 780nm
(a)
x
y
z
P
I-CCD
L
B
laser
2.4λ
0.3λ
λ/2π
(b)
FIG. 1 (color online). (a) Experimental setup. The atoms
are initially confined in a microscopic single-beam dipole trap
(not shown) (wavelength 957 nm, depth 1 mK, and a waist
1.6 μm, oscillation frequencies ω
x
¼ ω
y
¼ 2π × 62 kHz and
ω
z
¼ 2π × 8 kHz). The excitation laser propagates along the
quantization axis x, set by a B ∼ 1 G magnetic field. We collect
the scattered light along z, after a polarizer P oriented at an angle
of 55° with respect to x, using a lens L with a large numerical
aperture (NA ¼ 0.5) and an image intensifier followed by a CCD
camera (I-CCD). (b) Simulation of the distribution of nearest
neighbors fo r a single stochastic realization of a cloud of
N ¼ 450 atoms. (c) Structure of
87
Rb atoms relevant to this
work. The excitation light at frequency ω is near resonant with the
transition at λ ¼ 2πc=ω
0
¼ 780 nm.
Δ
Δ=0)
Δ / Γ
FIG. 2 (color online). Amount of scattered light detected
n
z
ðN; ΔÞ, versus the detuning Δ of the excitation light for
numbers of atoms N ¼ 1, 5, 20, 50, 200, 325, 450 (from bottom
to top). The amplitudes of the curves are normalized to the
amount of light detected at resonance for a single atom,
n
z
ðN ¼ 1; Δ ¼ 0Þ. Solid lines: Lorentzian fits to the data. Typical
uncertainties: 10% (vertically) and 20% (horizontally).
PRL 113, 133602 (2014)
PHYSICAL REVIEW LETTERS
week ending
26 SEPTEMBER 2014
133602-2
also shows that the amount of light scattered in the z
direction at resonance does not increase linearly with the
number of atoms as one would expect for noninteracting
atoms, but actually increases more slowly. Figure 4(a)
indicates that this is also the case off resonance, where we
plot n
z
ðN; ΔÞ=n
z
ðN ¼ 1; ΔÞ for different atom numbers
and detunings. For noninteracting atoms this ratio is equal
to the number of atoms N (and is thus independent of the
detuning Δ), as we verified by collecting the scattered light
after letting the atomic cloud expand in free space for a
sufficiently long time [29]. By contrast, here we observe
that the amount of scattered light is strongly suppressed on
resonance as the number of atoms increases, and that we
gradually recover the behavior of noninteracting atoms as
we detune the laser away from resonance.
All the observations reported above can be reproduced
by a single functional form:
n
z
ðN; ΔÞ¼C
N
Γ
c
ðNÞ
2
þ 4½Δ − δω
c
ðNÞ
2
; ð2Þ
where C includes the detection efficiency of the imaging
system. This is illustrated in Fig. 4(b): we find that the
quantity RðN; ΔÞ=RðN ¼ 1; ΔÞ, where RðN; ΔÞ¼
n
z
ðN; ΔÞ½Γ
2
c
þ 4ðΔ − δω
c
Þ
2
and Γ
c
and δω
c
are, respec-
tively, the phenomenological fits of FWHM and the shift
(see Fig. 3), collapses on a single curve whatever the
detuning. For N ≲ 300, this curve is linear with N with a
slope of 1, in agreement with Eq. (2). It emphasizes that in
this regime, the scattered intensity is suppressed by a factor
ðΓ=Γ
c
Þ
2
at resonance, as expected from the qualitative
discussion earlier. We note that this scaling cannot be
explained by a model where the suppression would come
from an incoherent superposition of the intensities scattered
by each atom with resonant frequencies inhomogeneously
distributed over a distribution of width FWHM: that would
lead to a suppression that would scale as Γ=Γ
c
near
resonance, instead of the ðΓ=Γ
c
Þ
2
scaling observed here.
For N>300, the departure from the linear law indicates
that C depends on the number of atoms in this regime, and
that the simple Lorentzian form (2) becomes inaccurate, as
also found in the simulation (see below).
We have performed numerical simulations of the col-
lective dynamic response of the atomic sample to near-
resonant pulsed light in the low excitation limit. In this
model, each atom, located at position r
j
(j ¼ 1; …;N) and
with dipole d
j
, is driven by the incident laser field and by
the fields scattered by all the N − 1 other atoms, i.e., each
dipole is coupled to the N − 1 other dipoles via the resonant
interaction of Eq. (1). This classical electrodynamics
simulation incorporates all the interactions between an
ensemble of nonsaturated discrete dipoles. This approach
has been used to study dielectric media comprising two-
level or spatially averaged isotropic electric dipoles
[10,14,16,18,30,31] as well as magnetodielectric circuit
resonator systems [32]. Here, we also incorporate the
Zeeman level structure of the atoms [13] and the shifts
associated to the presence of the magnetic field. To
calculate the dipoles d
j
in our experimental configuration,
Γ
(a)
δω / Γ
(b)
FIG. 3 (color online). (a) FWHM and (b) line shift δω with
respect to the atomic frequency for N ¼ 1 atom, in units of Γ.
Filled symbols: data extracted from the Lorentzian fits shown in
Fig. 2, versus the number of atoms. Dashed lines: phenomeno-
logical fits of the FWHM and shift by, respectively,
Γ
c
=Γ ¼ 1.49ð6Þ × N
0.08ð1Þ
, and δω
c
=Γ ¼ 47ð9Þ × 10
−5
N. The
error bars are from the fits of Fig. 2. Green solid line: results
of the simulation (see text).
Δ
Δ)
(b)
Δ
Δ)
(a)
FIG. 4 (color online). (a) Scattered light detected in the z
direction, versus the number of atoms, for different detunings of
the laser: Δ ¼ 0 (red circles), Γ (up or down open triangles),
and Δ ¼2.5Γ (up or down filled triangles). The intensity for
each atom number is normalized to the single atom case at the
same detuning. Red line: result of the simulation (see text)
with widths of the cloud σ
ρ
and σ
z
. Black diamond: model with
widths 2σ
ρ
and 2σ
z
. (b) Ratio RðN; ΔÞ=RðN ¼ 1; ΔÞ versus the
number of atoms (see text). Dashed line in (a) and (b): case of
noninteracting atoms.
PRL 113, 133602 (2014)
PHYSICAL REVIEW LETTERS
week ending
26 SEPTEMBER 2014
133602-3
we stochastically sample the positions of the atoms
according to a 3-dimensional Gaussian density distribution
with root-mean-square sizes given by the thermal sizes of
the cloud along and perpendicular to the trap propagation
axis; each atomic position is treated as an independent and
identically distributed random variable. At each realization
the N atoms are fixed at positions r
j
(j ¼ 1; …;N) and we
stochastically sample the magnetic quantum number of the
Zeeman states m
j
of each atom j. The probability of atom j
being in state jg; mi (m ¼2, 1, 0) is the initial
population of that Zeeman state p
m
(0 <p
m
< 1;
P
m
p
m
¼ 1). The optical pumping used in the preparation
step before the excitation sequence, skews the initial
populations; here we use the values p
0
¼ p
1
¼ p
2
¼
1=3 and p
−1
¼ p
−2
¼ 0. We write the positive frequency
component of the dipole produced by each atom j that
oscillates at the laser frequency as d
j
¼ D
P
σ
ˆ
e
σ
C
ðσÞ
m
j
P
jσ
,
where the sum runs over the unit spherical polarization
vectors σ ¼1, 0. The amplitude of the atomic dipole j
associated to the optical transition jg; m
j
i → je; m
j
þ σi is
proportional to the reduced dipole matrix element D, the
atomic coherence P
jσ
, and the corresponding Clebsch-
Gordan coefficient C
ðσÞ
m
j
. The temporal evolution of the
coherences is given by the set of coupled equations
_
P
jα
− iðΔ
jα
þ iΓ=2ÞP
jα
¼ −iΩ
jα
ðtÞ − i
X
l≠j
X
β
C
ðβÞ
m
l
C
ðαÞ
m
j
V
lβ
jα
ðrÞP
lβ
; ð3Þ
where Ω
jα
ðtÞ and Δ
jα
¼ ω − ω
jα
are, respectively, the
time-dependent Rabi frequency and the detuning of the
driving laser with respect to the Zeeman shifted transi-
tion of the α-polarized atom j with frequency ω
jα
, and
β ¼1, 0. Here, we deduce Ω
jα
ðtÞ from the experimen-
tally measured temporal profile of the excitation pulse.
The last term in Eq. (3) couples the α-polarized dipole j to
the β-polarized dipole l separated by r ¼ r
j
− r
l
according
to Eq. (1). We have solved Eqs. (3) numerically in the
presence of a 1 G magnetic field to calculate the light field
amplitude that is scattered into the solid angle encompassed
by the aspherical lens in the far field. Finally, accounting
for the polarization-sensitive detection scheme, we calcu-
lated the measured light intensity.
The simulation predicts that the spectra n
z
ðN;ΔÞ should
present an increasing broadening and asymmetry (Fig. 5), a
negligible shift [Fig. 3(b)], as well as a suppression of the
scattered light [Fig. 4(a)] when the number of atoms
increases. These features are in good agreement with our
data for N ≲ 50. In this range, the simulated spectra are well
fitted by a Lorentzian for N ≲ 50, thus justifying our fitting of
the data by Eq. (2) and the collapse of the data shown in
Fig. 4(b).ForN ≳ 50, the agreement is only qualitative, as
the effects are found to be less pronounced experimentally.
We attribute these discrepancies to two possible reasons.
First, forces induced by the dipole-dipole interactions may
expel atom pairs with shortest interatomic distances, thus
breaking down the assumption that atoms have frozen
positions during the sequence of pulses excitations. This
is all the more likely as the number of atoms is large, and
could explain the evolution of the FWHM in Fig. 3(a).This
effect is hard to check experimentally since the sample is
smaller than the diffraction limit of our imaging system. We
found numerically, however, that an increase by a factor 2 in
the widths σ
ρ
and σ
z
already restores a nearly Lorentzian
profile close to the measured spectra (see Fig. 5), and yields
the observed suppression of light scattering [see Fig. 4(a)].
Second, the simulation predicts that the number of detected
photons increases by a factor of 2 when the initial distribution
of Zeeman state populations varies from p
0
¼ p
1
¼ p
2
¼
1=3 to p
2
¼ 1. For large numbers of atoms, optical pumping
during the set of excitation pulses may change the distribu-
tion of populations, an effect not accounted for in our model.
In conclusion, we have directly measured the suppression
of light scattering induced by dipole-dipole interactions in
an ensemble of cold atoms driven by a near-resonant weak
laser field and compared it with a time-dependent model
of coupled dipoles. The model reproduces the observed
trends. In the future, we plan to investigate to what extent
the observed collective scattering involves beyond-mean-
field scattering processes, i.e., is cooperative in nature.
Experimental investigations of the temporal response of the
system, and comparisons to the case of a single atom [24],
should also provide insight into the interplay between
dipole-dipole interactions and collective scattering.
We acknowledge support from the E.U. through the ERC
Starting Grant ARENA, from the Triangle de la Physique,
EPSRC, and Leverhulme Trust. We thank P. Pillet, J.-J.
Greffet, and J. Javanainen for fruitful discussions.
[1] R. H. Dicke, Phys. Rev. 93, 99 (1954).
[2] Y. Li, J. Evers, W. Feng, and Shi-Yao Zhu, Phys. Rev. A 87,
053837 (2013).
Δ)
/n
Z
(N=1,
Δ=0)
Δ / Γ
FIG. 5 (color online). Comparison between experiment and
theory for the number of detected photons n
z
ðN;ΔÞ (normalized
to the single atom case at resonance) for N ¼ 450. The red, blue,
and green lines correspond to samples with widths σ
ρ
and σ
z
,
multiplied by a factor 1, 1.44, and 2, respectively.
PRL 113, 133602 (2014)
PHYSICAL REVIEW LETTERS
week ending
26 SEPTEMBER 2014
133602-4
