Superradiance for Atoms Trapped along a Photonic Crystal Waveguide
A. Goban,
1,2
C.-L. Hung,
1,2
,
†
J. D. Hood,
1,2
S.-P. Yu,
1,2
J. A. Muniz,
1,2
O. Painter,
2,3
and H. J. Kimble
1,2
,*
1
Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, California 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
3
Thomas J. Watson, Sr., Laboratory of Applied Physics 128-95, California Institute of Technology, Pasadena, California 91125, USA
(Received 14 March 2015; published 5 August 2015)
We report observations of superradiance for atoms trapped in the near field of a photonic crystal
waveguide (PCW). By fabricating the PCW with a band edge near the
D
1
transition of atomic cesium,
strong interaction is achieved between trapped atoms and guided-mode photons. Following short-pulse
excitation, we record the decay of guided-mode emission and find a superradiant emission rate scaling as
̄
Γ
SR
∝
̄
N
Γ
1
D
for average atom number
0
.
19
≲
̄
N
≲
2
.
6
atoms, where
Γ
1
D
=
Γ
0
¼
1
.
0
0
.
1
is the peak single-
atom radiative decay rate into the PCW guided mode, and
Γ
0
is the radiative decay rate into all the other
channels. These advances provide new tools for investigations of photon-mediated atom-atom interactions
in the many-body regime.
DOI:
10.1103/PhysRevLett.115.063601
PACS numbers: 42.50.Ct, 37.10.Gh, 42.70.Qs
Interfacing light with atoms localized near nanophotonic
structures has attracted increasing attention in recent years.
Exemplary experimental platforms include nanofibers
[1
–
3]
, photonic crystal cavities
[4]
, and waveguides
[5,6]
.
Owing to their small optical loss and tight field confinement,
these nanoscale dielectric devices are capable of mediating
long-range atom-atom interactions using photons propagat-
ing in their guided modes. This new paradigm for strong
interaction of atoms and optical photons offers new tools
for scalable quantum networks
[7]
, quantum phases of light
and matter
[8,9]
, and quantum metrology
[10]
.
In particular, powerful capabilities for dispersion and
modal engineering in photonic crystal waveguides (PCWs)
provide opportunities beyond conventional settings in
atomic, molecular and optical physics within the new field
of
waveguide QED
[2,3,6,11
–
13]
. For example, the edge of a
photonic band gap aligned near an atomic transition strongly
enhances single-atom emission into the one-dimensional
(1D) PCW due to a
“
slow-light
”
effect
[14
–
16]
. Because
the electric field of a guided mode near the band edge
approaches a standing wave, optical excitations can be
induced in an array of trapped atoms with little propagation
phase error,resulting inphase-matchedsuperradiantemission
[17,18]
into both forward and backward waveguide modes of
the PCW. Superradiance has important applications for
realizing quantum memories
[19
–
23]
, single-photon sources
[24,25]
, laser coolingbywayofcooperative emission
[26,27]
,
and narrow linewidth lasers
[28]
. Related cooperative effects
are predicted in nanophotonic waveguides absent an external
cavity
[29]
, including atomic Bragg mirrors
[30]
and self-
organizing crystals of atoms and light
[31
–
33]
.
Complimentary to superradiant emission is the collective
Lamb shift induced by proximal atoms virtually exchang-
ing off-resonant photons
[34
–
37]
. With the atomic tran-
sition frequency placed in a photonic band gap of a PCW,
real photon emission is largely suppressed. Coherent atom-
atom interactions then emerge as a dominant effect for
QED with atoms in band-gap materials
[38
–
43]
. Both the
strength and length scale of the interaction can be
“
engineered
”
by suitable band shaping of the PCW, as
well as dynamically controlled by external lasers
[42,43]
.
Explorations of many-body physics with tunable and strong
long-range atom-atom interactions are enabled
[42,43]
.
In this Letter, we present an experiment that cools,
traps, and interfaces multiple atoms along a quasi-one-
dimensional PCW. Through precise band-edge alignment
and guided-mode (GM) design, we achieve strong radiative
coupling of one trapped atom and a guided mode of the
PCW, such that the inferred single-atom emission rate into
the guided mode is
Γ
1
D
=
Γ
0
¼
1
.
0
0
.
1
, where
Γ
1
D
is the
peak single-atom radiative decay rate into the guided mode,
and
Γ
0
is the radiative decay rate into all the other channels.
With multiple atoms, we observe superradiant emission in
both time and frequency domains. We infer superradiant
coupling rate
̄
Γ
SR
that scales with the mean atom number
̄
N
as
̄
Γ
SR
¼
η
̄
N
Γ
1
D
over the range
0
.
19
≲
̄
N
≲
2
.
6
atoms,
where
η
¼
0
.
34
0
.
06
.
We stress
“
waveguide
”
and not
“
cavity
”
QED because
the dominant effects in our experiment are a result of the
combination of atom-light localization within an area
A
w
comparable to the free-space atomic cross section
A
w
∼
λ
2
and an enhancement in the atom-field coupling due to band
structure, namely, a group index
n
g
≈
11
that dominates
over the enhancement
E
I
∼
4
from a weak external cavity
with
“
mirror
”
reflectivity
R
≈
0
.
48
. By contrast, in conven-
tional cavity QED, an atom interacts with a cavity mode
with area
A
c
≫
λ
2
,
n
g
≈
1
and an enhancement
E
I
∼
A
c
=
λ
2
>
10
5
(i.e., mirror reflectivity
R>
0
.
99999
)
[22,23]
.
Stated differently, in our PCW, we are able to engineer
the real and imaginary parts of the Green
’
s function for
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115,
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=
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=
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© 2015 American Physical Society
atom-light interactions (e.g., Fig. SM3 in Ref.
[44]
)in
ways that are not possible within the setting of conventional
cavity QED. For example, our alligator photonic crystal
waveguide (APCW) enables many-body physics that is both
nontrivial and inaccessible by using conventional optical
cavities
[42]
. Our observation of superradiance demon-
strates at least two important criteria toward the exploration
of such new physics, namely (i) the capability to trap
stably multiple atoms along an APCW and, therefore, to
build few-body quantum systems extending to
N
∼
20
atoms with improved trapping and (ii) the ability to achieve
relative band-edge alignment to better than
5
×
10
−
4
, which
enables
n
g
≫
1
with high-contrast Bloch modes.
Our experimental platform is based on trapped cesium
atoms near a 1D APCW. The APCW consists of 150
identical unit cells with lattice constant
a
¼
371
nm and is
terminated at either end by 30 tapered cells for mode
matching to parallel nanobeams without corrugation.
Design principles and device characterization can be found
in Refs.
[5,6,44]
. For the APCW used here, we align the
band edge of the TE-like fundamental guided mode near
the cesium
D
1
line at 894.6 nm, with a mode-matched
transverse electric (TE) input field
E
in
tuned around the
6
S
1
=
2
;F
¼
3
→
6
P
1
=
2
;F
0
¼
4
transition. Near the band
edge, the atom-photon coupling rate is significantly
enhanced by the group index
n
g
≃
11
and by reflections
from the tapering regions corresponding to an intensity
enhancement
E
I
∼
4
[44]
.
To trap atoms along the APCW, we create tight
optical potentials using the interference pattern of a side-
illumination (SI) beam and its reflection from the surface of
the APCW
[4]
. The polarization of the SI beam is aligned
parallel to the
x
axis to maximize the reflected field.
Figure
1(b)
shows the calculated near-field intensity dis-
tribution in the
y
-
z
plane
[50]
. With a red-detuned SI beam,
cold atoms can be localized to intensity maxima [e.g.,
positions
z
−
1
;z
1
;z
2
in Fig.
1(b)
]. However, because of the
exponential falloff of the GM intensity, only those atoms
sufficiently close to the APCW can interact strongly with
guided-mode photons of the input field
E
in
shown in
Fig.
1(c)
. The trap site with the strongest atom-photon
coupling is located at
ð
y
1
;z
1
Þ¼ð
0
;
220
Þ
nm, and
Δ
z
∼
120
nm is the distance from the plane of the upper surfaces
of the APCW. Other locations are calculated to have
coupling to the guided mode less than 1% of that for site
z
1
.
Along the
x
axis of the APCW, the dipole trap
U
ð
x;
0
;z
1
Þ
is insensitive to the dielectric corrugation within a unit cell
and is nearly uniform within
<
2%
around the central
region of the APCW. By contrast, atomic emission into the
TE-like mode is strongly modulated with
Γ
1
D
ð
x;
0
;z
1
Þ
≃
Γ
1
D
cos
2
ð
kx
Þ
due to the standing-wave-like coupling rate
near the band edge (
k
≈
π
=a
), as shown in the inset of
Fig.
1(a)
. Thus, even for atoms uniformly distributed along
the
x
axis, only those close to the center of a unit cell can
strongly couple to the guided mode. We choose a
50
μ
m
waist for the SI beam to provide weak confinement along
the
x
axis, with atoms localized near the central region
(
Δ
x
≃
10
μ
m) of the APCW for the estimated temper-
ature
∼
50
μ
K from a time-of-flight measurement in
free space.
Cold atoms from a magneto-optical trap (MOT) that
surrounds the APCW
[6]
are loaded into the dipole trap
during an optical molasses phase (
∼
5
ms) and then optically
pumped to
6
S
1
=
2
,
F
¼
3
(
∼
1
ms). Atoms are held in the
dipole trap for time
t
hold
relative to the end of the loading
sequence, and then free-space absorption imaging is initiated
over the interval (
t
hold
;t
hold
þ
Δ
t
m
)with
Δ
t
m
¼
0
.
2
ms. We
introduce the measured time
t
m
¼
t
hold
þ
Δ
t
m
=
2
centered
in the measurement window. As shown in Fig.
2(a)
,we
measure a trap lifetime
τ
fs
¼
54
5
ms and a peak
density
ρ
0
≈
2
×
10
11
cm
−
3
near the APCW.
To determine the lifetime for trapped atoms along the
APCW, we again hold atoms for
t
hold
and then launch
E
in
as
a resonant GM probe with
Δ
t
m
¼
5
ms. From the trans-
mitted signals, we compute
T=T
0
, where
T
0
is the trans-
mission without atoms. During the probe period, we also
apply free-space repump beams tuned to the
D
2
,
6
S
1
=
2
;F
¼
4
→
6
P
3
=
2
;F
0
¼
4
resonance to remove population in
6
S
1
=
2
,
F
¼
4
since the probe excites an open transition.
Figure
2(b)
shows
T=T
0
gradually recovering to
T=T
0
¼
1
as
t
m
increases, with a fit to the data giving a
1
=e
time of
y
(
μ
m
)
z
(
μ
m
)
−1
0
1
−1
0
1
10
-3
10
-2
10
-1
1
10
−1
01
−1
0
1
0.2
0.4
0.6
0.8
1
0
y
(
μ
m
)
z
(
μ
m
)
SI
E
in
rE
in
tE
in
APCW
(a)
(b)
(c)
z
1
z
2
z
-1
1D
x
500 nm
0
x
y
z
FIG. 1 (color online). Trapping and interfacing atoms with an
APCW. (a) A SI beam is reflected from an APCW (gray shaded
structure) to form a dipole trap to localize atoms. The red shaded
region represents trapped atoms along the APCW. An incident
field
E
in
excites the TE-like mode and trapped atoms couple to
this guided mode. The inset shows a SEM image of the APCW
and corresponding single-atom coupling rate
Γ
1
D
along the
x
axis
at the center of the gap (
y
¼
0
). (b) Normalized intensity cross
section of the total intensity
I
tot
resulting from the SI beam and its
reflection. Trap locations along the
z
axis at
y
¼
0
are marked by
z
i
. Masked gray areas represent the APCW. (c) The single-atom
coupling rate into the TE-like guided mode
Γ
1
D
ð
0
;y;z
Þ
normal-
ized to
Γ
0
where
Γ
0
is the Einstein-
A
coefficient for free space.
PRL
115,
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063601-2
τ
GM
¼
28
2
ms. Effects leading to
τ
GM
<
τ
fs
are dis-
cussed in Ref.
[44]
.
Our principal investigation of superradiance involves
observation of the transient decay of both forward and
backward emission from atoms trapped along the APCW.
We excite superradiance by employing weak and short
excitation pulses (FWHM 10 ns) with an average photon
number
N
p
≪
1
per pulse, ensuring a small degree of
excitation can be uniformly shared among all ground-state
atoms. For a collection of
N>
1
atoms, superradiance is
heralded by a total decay rate
Γ
tot
¼
Γ
SR
þ
Γ
ð
1
Þ
tot
that is
enhanced beyond the total decay rate for one atom
Γ
ð
1
Þ
tot
¼
Γ
1
D
þ
Γ
0
.
Γ
SR
is the
N
-dependent superradiant rate
operationally determined from
Γ
tot
and
Γ
ð
1
Þ
tot
. Here,
Γ
0
is the
radiative decay rate into all channels other than the TE-like
guided mode. We numerically evaluate
Γ
0
=
Γ
0
≈
1
.
1
for an
atom at the trap site
z
1
in Fig.
1(b)
along the APCW where
Γ
0
is the Einstein-
A
coefficient for free space
[15]
.
We record the temporal profiles of either forward or
backward atomic emission into the guided mode following
short-pulse (
∼
10
ns FWHM) resonant excitations via
E
in
.
After a time
t
hold
, the excitation cycle is repeated every
500 ns for
Δ
t
m
¼
6
ms, and detection events are accumu-
lated for the reflected intensity. We consider decay curves
of GM emission at
15
ns
<t
e
<
70
ns after the center of
the excitation pulse
[44]
. The total decay rate
̄
Γ
tot
is extracted
by exponential fits as shown in the inset of Fig.
3(a)
.
An enhanced total decay rate with an increasing atom
number is evidenced in Fig.
3(a)
, where the atom number is
adjusted by varying the trap hold time
t
hold
prior to the
measurement. At the shortest measurement time
t
m
¼
3
ms
with
t
hold
¼
0
ms, the measured total decay rate is largest at
̄
Γ
tot
=
Γ
0
≈
2
.
9
.At
t
m
¼
63
ms, much longer than the trap
lifetime
τ
GM
¼
28
2
ms, the total decay rate settles to
̄
Γ
tot
=
Γ
0
≈
2
.
0
. This asymptotic behavior suggests that
̄
Γ
tot
at
long hold time corresponds to the single-atom decay rate
̄
Γ
ð
1
Þ
tot
.
To determine quantitatively the superradiant and single-
atom emission rates, we present two different analyses that
yield consistent results. First is a simple and intuitive
analysis applied to Fig.
3(a)
in which we employ an
empirical exponential fit,
̄
Γ
tot
ð
t
m
Þ¼
̄
Γ
SR
e
−
t
m
=
τ
SR
þ
̄
Γ
ð
1
Þ
tot
,
with the maximum superradiant
̄
Γ
SR
, single-atom
̄
Γ
ð
1
Þ
tot
,
and
τ
SR
characterizing decay of superradiance due to the
atom loss. The fit yields
̄
Γ
SR
=
Γ
0
¼
1
.
1
0
.
1
with
τ
SR
¼
17
3
ms, as shown by the red curve in Fig.
3(a)
.
The asymptote
̄
Γ
ð
1
Þ
tot
=
Γ
0
¼
2
.
0
0
.
1
gives the total
single-atom decay rate. With
Γ
0
=
Γ
0
≈
1
.
1
, we deduce
̄
Γ
1
D
=
Γ
0
¼
0
.
9
0
.
1
.
To substantiate this empirical model, our second analysis
is a detailed numerical treatment based upon transfer matrix
calculations
[44]
. Decay curves are generated for a fixed
number of atoms
N
distributed randomly along the
x
axis
of the APCW with spatially varying coupling
Γ
1
D
ð
x
Þ
≃
Γ
1
D
cos
2
ð
kx
Þ
. These
N
-dependent spatially averaged decay
curves are further averaged over a Poisson distribution with
mean atom number
̄
N
. Fitting to this model, we extract
Γ
1
D
=
Γ
0
¼
1
.
1
0
.
1
for measurements at long hold time
t
m
¼
63
ms in Fig.
3(a)
. Since the GM emission is spatially
modulated by cos
4
ð
kx
Þ
, only an atom near the center of the
unit cell can strongly couple to the guided mode, resulting
in the small difference between averaged
̄
Γ
1
D
and peak
Γ
1
D
.
Also, the decay curve for GM emission at
t
m
¼
3
ms is
0.0
0.2
0.4
0.6
0.8
1.0
020406080100
0 20406080
0.7
0.8
0.9
1.0
(a)
(b)
FIG. 2. Lifetime of trapped atoms near the APCW. (a)
1
=e
lifetime of
τ
fs
¼
54
5
ms is determined using free-space
absorption imaging of the trapped atom cloud. (b)
1
=e
lifetime
of
τ
GM
¼
28
2
ms is observed from the normalized trans-
mission
T=T
0
of a resonant GM probe.
2.0
2.2
2.4
2.6
2.8
3.0
0 102030405060
0.0
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.2
2.4
2.6
2.8
3.0
0 204060
0.01
0.1
1
(a)
(b)
FIG. 3 (color online). Decay rate and atom number. (a) Fitted
total decay rate
̄
Γ
tot
normalized with
Γ
0
(circles) as a function of
measurement time
t
m
. The solid line is an exponential fit to
determine the superradiant decay rate
̄
Γ
SR
=
Γ
0
¼
1
.
1
0
.
1
and the single-atom decay rate
̄
Γ
ð
1
Þ
tot
=
Γ
0
¼
2
.
0
0
.
1
with
τ
SR
¼
17
3
ms. The inset shows the normalized temporal profiles of
backward emission
I
p
. Exponential fits (solid curves):
t
m
¼
3
ms
(red), 13 ms (green), and 63 ms (blue). The black dashed curve
shows exponential decay with
Γ
0
. (b) Fitted total decay rate
̄
Γ
tot
=
Γ
0
as a function of the mean number of trapped atoms
̄
N
.We
adjust
̄
N
by changing the trap hold time (red) or atom loading
time (blue). The black line is a linear fit to the combined data
sets giving
̄
Γ
SR
¼
η
̄
N
Γ
1
D
with
η
¼
0
.
34
0
.
06
.
PRL
115,
063601 (2015)
PHYSICAL REVIEW LETTERS
week ending
7 AUGUST 2015
063601-3
well fitted with
̄
N
¼
2
.
6
0
.
3
atoms
[44]
. The red points
in Fig.
3(b)
display the total decay rate
̄
Γ
tot
as a function of
̄
N
extracted from the model fits, which shows that super-
radiance emission rate is proportional to
̄
N
.
Γ
1
D
=
Γ
0
¼
1
.
0
0
.
1
from our measurements agrees rea-
sonably well with the theoretical value
Γ
1
D
=
Γ
0
≈
1
.
1
deter-
mined by finite-difference time-domain calculations
[44,51]
despite several uncertainties (e.g., locations of trap minima).
The agreement validates the precision of our fabricated
samples as well as the power of the theoretical tools
[15,42,43]
.
We confirm that the variation of
̄
Γ
tot
in Fig.
3(a)
is not due
to the heating of atomic motion during the trap hold time. To
see this, we adjust
̄
N
via different MOT loading times and
measure the decay rate at the shortest hold time (
t
m
¼
3
ms),
as shown by the blue points in Fig.
3(b)
. These observations
are consistent with those from varying the trap hold time [red
points in Fig.
3(b)
] and lead to an almost identical single-
atom decay rate
̄
Γ
ð
1
Þ
tot
=
Γ
0
¼
2
.
0
0
.
1
at the shortest loading
time corresponding to
ρ
=
ρ
0
¼
0
.
16
and
̄
N
≪
1
.
The data and our analysis related to Fig.
3
strongly
support the observation of superradiance for atoms trapped
along the APCW. Assuming
̄
Γ
tot
¼
̄
Γ
SR
þ
̄
Γ
ð
1
Þ
tot
and fitting
̄
Γ
tot
linearly with
̄
N
, as shown in Fig.
3(b)
, we find that the
superradiant rate is given by
̄
Γ
SR
¼
η
̄
N
Γ
1
D
with
η
¼
0
.
34
0
.
06
. For a motivation of the scaling of the
superradiant coupling rate and a physical interpretation of
the constant
η
, see Ref.
[44]
.
This observation of superradiance is complemented by
line broadening for steady-state transmission spectra
T
ð
Δ
Þ
measured at
t
m
¼
3
ms with
Δ
t
m
¼
5
ms in Fig.
4
. The
measured linewidths
̄
Γ
m
are significantly broader than the
free-space width
Γ
0
=
2
π
¼
4
.
56
MHz
[52]
. We also mea-
sure
T=T
0
≃
0
.
30
at line center for maximum density
ρ
0
,
due to strong atom-photon coupling.
No clear density-dependent shift is observed in Fig.
4(a)
,
in support of our neglect of cooperative energy shifts
j
H
dd
j
[44]
. The shift in line center for
T
ð
Δ
Þ
from
Δ
¼
0
in free
space to
Δ
¼
14
MHz for trapped atoms is induced by the
dipole trap. Furthermore, trapped atoms should suffer small
inhomogeneous broadening in the spectra, since the light
shift induced by the dipole trap is small (
<
1
MHz) for the
6
P
1
=
2
;F
¼
4
0
excited state, and atoms are well localized
around the trap center due to their low temperature
T
∼
50
μ
K corresponding to a small range of light shifts
≲
1
MHz for the ground state.
In Fig.
4(b)
, we plot the linewidths
̄
Γ
m
extracted from
T
ð
Δ
Þ
as a function of
ρ
=
ρ
0
.
̄
Γ
m
=
Γ
0
≈
3
.
4
is largest at
ρ
=
ρ
0
¼
1
and reduces to
̄
Γ
m
=
Γ
0
≈
2
.
1
at
ρ
=
ρ
0
¼
0
.
06
.
From linear extrapolation, the single-atom linewidth is
estimated to be
̄
Γ
ð
1
Þ
m
=
Γ
0
¼
2
.
1
0
.
1
. Absent inhomo-
geneous broadening, we expect
̄
Γ
ð
1
Þ
m
¼
̄
Γ
1
D
þ
Γ
0
. With
Γ
0
=
Γ
0
≈
1
.
1
, the single-atom coupling rate is deduced as
̄
Γ
1
D
=
Γ
0
≈
1
.
0
0
.
1
. A simple estimate of the maximum
mean number of atoms follows from
̄
N
m
¼½
̄
Γ
m
ð
ρ
0
Þ
−
Γ
0
=
̄
Γ
1
D
≃
2
.
4
0
.
4
atoms
[53]
.
In conclusion, we have demonstrated superradiance for
atoms trapped along the APCW and a peak single-atom
coupling rate
Γ
1
D
=
Γ
0
¼
1
inferred, where
Γ
0
=
Γ
0
≈
1
.
1
is the
radiative decayrate into all the other channels.Our weaktrap
along the APCW is a promising platform to study opto-
mechanical behavior induced by the interplay between
sizable single-atom reflectivity and large optical forces
and investigations of spin-motion coupling
[32,33]
.By
optimizing the power and detuning of an auxiliary GM
field, it should be possible to transport these trapped atoms
into trap sites centered within the vacuum gap
[15]
and
achieve stable trapping and ground-state cooling
[54,55]
.
We expect
Γ
1
D
to increase by more than fivefold
[15]
.
Opportunities for new physics in the APCW arise by
fabricating devices with the atomic resonance inside the
band gap to induce long-range atom-atom interactions
[41
–
43]
enabling investigations of novel quantum transport
and many-body phenomena.
We gratefully acknowledge the contributions of D. J.
Alton, D. E. Chang, K. S. Choi, J. D. Cohen, J. H. Lee, M.
Lu, M. J. Martin, A. C. McClung, S. M. Meenehan, R.
Norte, and L. Peng. Funding is provided by the IQIM, an
NSF Physics Frontiers Center with support of the
Moore Foundation and by the DOD NSSEFF program
(H. J. K.), the AFOSR QuMPASS MURI, NSF Grant
0 102030
0.4
0.6
0.8
1.0
0.00.20.40.60.81.0
2.0
2.5
3.0
3.5
(a)
(b)
FIG. 4 (color online). Steady-state transmission spectra
T
ð
Δ
Þ
and fitted atomic linewidth
̄
Γ
m
. (a)
T
ð
Δ
Þ
with
Δ
¼
0
corresponding to the free-space line center. The three sets of
points are measured at relative densities
ρ
=
ρ
0
¼
0
.
12
(black),
0.24 (blue), and 1 (red). Solid curves are Lorentzian fits to
determine the linewidth
̄
Γ
m
. (b) Fitted linewidths (circles)
normalized to
Γ
0
as a function of
ρ
=
ρ
0
. The solid line is a linear
fit with intercept of
̄
Γ
ð
1
Þ
m
=
Γ
0
¼
2
.
1
0
.
1
.
PRL
115,
063601 (2015)
PHYSICAL REVIEW LETTERS
week ending
7 AUGUST 2015
063601-4
No. PHY-1205729 (H. J. K.), and the DARPA ORCHID
program. A. G. is supported by the Nakajima Foundation.
S.-P. Y. and J. A. M. acknowledge support from the
International Fulbright Science and Technology Award.
A. G., C.-L. H., J. D. H., and S.-P. Y. contributed equally
to this research.
*
To whom all correspondence should be addressed.
hjkimble@caltech.edu
†
Present address: Purdue University, West Lafayette, IN
47907, USA.
[1] D. E. Chang, V. Vuleti
ć
, and M. D. Lukin,
Nat. Photonics
8
,
685 (2014)
.
[2] E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and
A. Rauschenbeutel,
Phys. Rev. Lett.
104
, 203603 (2010)
.
[3] A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M.
Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble,
Phys.
Rev. Lett.
109
, 033603 (2012)
.
[4] J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V.
Akimov, M. Gullans, A. S. Zibrov, V. Vuleti
ć
, and M. D.
Lukin,
Science
340
, 1202 (2013)
.
[5] S.-P. Yu, J. D. Hood, J. A. Muniz, M. J. Martin, R. Norte,
C.-L. Hung, S. M. Meenehan, J. D. Cohen, O. Painter, and
H. J. Kimble,
Appl. Phys. Lett.
104
, 111103 (2014)
.
[6] A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H.
Lee,M. J.Martin,A. C.McClung,K. S.Choi,D. E.Chang,O.
Painter, and H. J. Kimble,
Nat. Commun.
5
, 3808 (2014)
.
[7] H. J. Kimble,
Nature (London)
453
, 1023 (2008)
.
[8] M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio,
Nat.
Phys.
2
, 849 (2006)
.
[9] A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L.
Hollenberg,
Nat. Phys.
2
, 856 (2006)
.
[10] P. Kómár, E. M. Kessler, M. Bishof, L. Jiang, A. S.
Sørensen, J. Ye, and M. D. Lukin,
Nat. Phys.
10
, 582 (2014)
.
[11] D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D.
Lukin,
Nat. Phys.
3
, 807 (2007)
.
[12] A. L. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A.
Blais, and A. Wallraff,
Science
342
, 1494 (2013)
.
[13] J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A.
Wallraff,
Nat. Commun.
5
, 5186 (2014)
.
[14] T. Baba,
Nat. Photonics
2
, 465 (2008)
.
[15] C.-L. Hung, S. M. Meenehan, D. E. Chang, O. Painter, and
H. J. Kimble,
New J. Phys.
15
, 083026 (2013)
.
[16] P. Lodahl, S. Mahmoodian, and S. Stobbe,
Rev. Mod. Phys.
87
, 347 (2015)
.
[17] R. H. Dicke,
Phys. Rev.
93
, 99 (1954)
.
[18] M. Gross and S. Haroche,
Phys. Rep.
93
, 301 (1982)
.
[19] L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller,
Nature
(London)
414
, 413 (2001)
.
[20] C. H. van der Wal, M. D. Eisaman, A. André, R. L.
Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin,
Science
301
, 196 (2003)
.
[21] A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou,
L.-M. Duan, and H. J. Kimble,
Nature (London)
423
, 731 (2003)
.
[22] B. Casabone, K. Friebe, B. Brandstätter, K. Schüppert, R.
Blatt, and T. E. Northup,
Phys. Rev. Lett.
114
, 023602 (2015)
.
[23] R. Reimann, W. Alt, T. Kampschulte, T. Macha, L.
Ratschbacher, N. Thau, S. Yoon, and D. Meschede,
Phys.
Rev. Lett.
114
, 023601 (2015)
.
[24] C. W. Chou, S. V. Polyakov, A. Kuzmich, and H. J. Kimble,
Phys. Rev. Lett.
92
, 213601 (2004)
.
[25] A. T. Black, J. K. Thompson, and V. Vuletic,
Phys. Rev.
Lett.
95
, 133601 (2005)
.
[26] H. W. Chan, A. T. Black, and V. Vuletic,
Phys. Rev. Lett.
90
,
063003 (2003)
.
[27] M. Wolke, J. Klinner, H. Keßler, and A. Hemmerich,
Science
337
, 75 (2012)
.
[28] J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland,
and J. K. Thompson,
Nature (London)
484
, 78 (2012)
.
[29] F. Le Kien, S. D. Gupta, K. P. Nayak, and K. Hakuta,
Phys.
Rev. A
72
, 063815 (2005)
.
[30] D. E. Chang, L. Jiang, A. V. Gorshkov, and H. J. Kimble,
New J. Phys.
14
, 063003 (2012)
.
[31] I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D.
Phillips,
Phys. Rev. A
52
, 1394 (1995)
.
[32] D. E. Chang, J. I. Cirac, and H. J. Kimble,
Phys. Rev. Lett.
110
, 113606 (2013)
.
[33] T. Grießer and H. Ritsch,
Phys. Rev. Lett.
111
, 055702 (2013)
.
[34] A. Svidzinsky and J.-T. Chang,
Phys. Rev. A
77
, 043833 (2008)
.
[35] M. O. Scully,
Phys. Rev. Lett.
102
, 143601 (2009)
.
[36] R. Röhlsberger, K. Schlage, B. Sahoo, S. Couet, and R.
Rffer,
Science
328
, 1248 (2010)
.
[37] J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes, D.
Sarkisyan, and C. S. Adams,
Phys. Rev. Lett.
108
, 173601 (2012)
.
[38] S. John and J. Wang,
Phys. Rev. Lett.
64
, 2418 (1990)
.
[39] G. Kurizki,
Phys. Rev. A
42
, 2915 (1990)
.
[40] P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S.
Bay,
Rep. Prog. Phys.
63
, 455 (2000)
.
[41] E.Shahmoonand G.Kurizki,
Phys.Rev.A
87
, 033831(2013)
.
[42] J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov,
H. J. Kimble, and D. E. Chang,
Nat. Photonics
9
, 326
(2015)
.
[43] A. Gonzlez-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac,
and H. J. Kimble,
Nat. Photonics
9
, 320 (2015)
.
[44] See the Supplemental Material at
http://link.aps.org/
supplemental/10.1103/PhysRevLett.115.063601
for a
detailed discussion about ACPW, lifetime measure-
ments, and decay rate analyses, which includes
Refs. [5,11,15,30,31,45
–
49].
[45] J. D. Hood
et al.
(to be published).
[46] G. S. Agarwal,
Phys. Rev. A
12
, 1475 (1975)
.
[47] S. Y. Buhmann, L. Knöll, D.-G. Welsch, and H. T. Dung,
Phys. Rev. A
70
, 052117 (2004)
.
[48] S. Y. Buhmann and D.-G. Welsch,
Phys. Rev. A
77
, 012110
(2008)
.
[49] R. J. Thompson, Q. A. Turchette, O. Carnal, and H. J.
Kimble,
Phys. Rev. A
57
, 3084 (1998)
.
[50]
COMSOL
,
http://www.comsol.com
.
[51] A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D.
Joannopoulos, and S. G. Johnson,
Comput. Phys. Commun.
181
, 687 (2010)
.
[52] R. J. Rafac, C. E. Tanner, A. E. Livingston, and H. G. Berry,
Phys. Rev. A
60
, 3648 (1999)
.
[53]
̄
N
m
is defined from
̄
Γ
m
ð
ρ
Þ¼
̄
N
m
̄
Γ
1
D
þ
Γ
0
for an approxi-
mate estimate of the number of atoms. Empirically, we find
̄
N
m
∼
̄
N
.
[54] J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuleti
ć
, and
M. D. Lukin,
Phys. Rev. Lett.
110
, 133001 (2013)
.
[55] A. M. Kaufman, B. J. Lester, and C. A. Regal,
Phys. Rev. X
2
, 041014 (2012)
.
PRL
115,
063601 (2015)
PHYSICAL REVIEW LETTERS
week ending
7 AUGUST 2015
063601-5