Supplemental material: 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
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA and
3
Thomas J. Watson, Sr., Laboratory of Applied Physics 128-95
(Dated: July 11, 2015)
We provide supporting information for our manuscript [1], including device characterization, finite different
time domain (FDTD) calculations for collective coupling rates, lifetime of trapped atoms along the APCW, and
model for superradiance of trapped atoms.
I. DEVICE CHARACTERIZATION
A schematic of the alligator photonic crystal waveguide (APCW) is illustrated in Fig. SM1(a). The waveguide is made from
200-nm thick stoichiometric SiN with refractive index
n
= 2
.
0
[2]. The APCW is formed by two parallel SiN nanobeams
separated by 238 nm with periodic corrugations at the outer edges (Fig. SM1). The dimensions of the nominal photonic crystals
are the following: lattice constant
a
= 371
nm, gap
g
= 238
nm, width
w
= 157
nm, and tooth amplitude
A
= 131
nm, as
shown in Fig. SM1(b). The nominal photonic crystal section consists of
N
cell
= 150
unit cells (length
L
'
55
.
7
μ
m), terminated
by 30 tapered cells on each side to provide ‘mode-matching’ to and from double nanobeams sections. Photons can be coupled
into and out of the APCW from conventional cleaved-fibers at either end of the structure.
The APCW is characterized by measuring the transmission spectrum
T
0
(
ν
)
without atoms. The resonant structure around
frequencies
ν
i
displayed in Fig. SM2 arises from reflections in the tapered sections at the two ends of the APCW. The free
spectral range
∆
ν
i
=
ν
i
+1
−
ν
i
between resonances decreases as the band edge frequency
ν
BE
is approached, which is a
signature of an increasing group
n
g
index near
ν
BE
, with
n
g
∝
1
/
∆
ν
i
for an ideal structure. The distance from the band edge
frequency
ν
BE
to the first resonance
ν
1
is about 6 part in
10
4
, resulting in a group index
n
g
∼
11
; see discussion below and Fig.
SM2.
Our experiment in Ref. [1] is operated around the frequency
ν
A
of the D
1
: 6
S
1
/
2
, F
= 3
→
6
P
1
/
2
, F
′
= 4
transition in
atomic Cs, with
ν
1
aligned near
ν
A
by controlling fabrication accuracy at a level of
10
−
3
. Fine tuning for
ν
1
=
ν
A
to 3ppm
accuracy is achieved by way of a guided-mode (GM) heating beam with a wavelength of 850 nm and optimum power, typically
P
≥
100
μ
W. In addition, we turn on a strong GM heating beam for 100 ms at the end of each experimental cycle in order to
keep the device clean by desorbing Cs from the APCW.
In order to estimate the group index
n
g
, single-taper reflectivity
R
t
, and intensity loss
e
−
2
ζ
, we use a model based on the
transfer matrix formalism for a periodic system to fit the transmission spectrum [3], which we now briefly describe. The
E
in
rE
in
tE
in
APCW
(a)
taper
taper
x
y
z
500 nm
(b)
FIG. SM1: (a) Schematic of the APCW. An incident field
E
in
excites the TE-like fundamental mode, and the intensities for the transmitted
tE
in
and reflected field
rE
in
are recorded for device characterization. (b) SEM image of APCW with lattice constant
a
, gap
g
, width
w
, and
tooth amplitude
A
.
†
These authors contributed equally to this research.
‡
Present address: Purdue University, West Lafayette, IN 47907, USA
∗
Correspondence and requests for materials should be addressed to HJK (hjkimble@caltech.edu.)
2
(a)
(b
)
332
333
334
335
336
0
0.12
0.10
0.08
0.06
0.04
0.02
0.14
334
334.5
335
335.5
0
0.2
0.4
0.6
0.8
1
0
2
4
6
8
10
12
14
FIG. SM2: (a) Measured transmission spectrum
T
0
(
ν
)
for the APCW (black) around the edge of the dielectric band and the model fit (red).
The dashed lines mark the resonant frequencies
ν
i
from reflections in the taper sections and the solid line marks the band edge frequency
ν
BE
.
(b) Estimated group index
n
g
(green) and taper reflection
R
t
(blue) from the fitted model. For the reference, the transmission spectrum
T
0
(
ν
)
is overlaid. At the first resonance
ν
1
marked by the dashed line, the group index is
n
g
≈
11
, and the taper reflection is
R
t
≈
0
.
48
.
dispersion relation for the wavevector
k
(
ν
)
near the band edge is approximated by the fitting function [3],
k
(
ν
) =
k
0
1
−
√
(
ν
0
−
ν
(1 +
iκ
))
2
−
∆
2
g
ν
2
F
−
∆
2
g
,
(1)
where the wavevector at the band edge is
k
0
=
π/a
. Here, fitting parameters are the frequency at the center of the band gap
ν
0
,
the size of the band gap
2∆
g
, the asymptotic group velocity far from the band edge
2
πν
F
/k
0
, and the loss parameter
κ
. The loss
parameter
κ
comes from using perturbation theory to add a small imaginary component to the dielectric constant of the material,
resulting in an imaginary propagation constant that is approximately given by
Im
[
k
(
ν
)]
≈
2
πν
v
g
κ.
(2)
It provides a convenient way to model losses that scale with inverse group velocity.
Next we consider the weak cavity formed by the taper reflections
R
t
. The single-pass phase accumulation
φ
and single-pass
power transmission
e
−
2
ζ
through the cavity are written by
φ
=
N
cell
a
Re
[
k
]
and
ζ
=
N
cell
a
Im
[
k
]
,
(3)
where
N
cell
is the number of unit cells of the APCW and
a
is the lattice constant. Then, the transmission through a symmetric
cavity with mirrors
R
t
is given by
T
cavity
=
1
1 +
L
+
F
sin
2
[
φ
]
,
(4)
where the coefficient
F
and loss coefficient
L
are given by
L
=
(
1
−
R
t
e
−
2
ζ
)
2
e
−
2
ζ
(1
−
R
t
)
2
−
1
and
F
=
4
R
t
(1
−
R
t
)
2
.
(5)
In order to fit this model to the measured transmission spectrum, first we use the dispersion model (Eq. (1) with
κ
= 0
) to fit
the positions of the cavity resonances to. Second, we fit Eq. (4) with no loss (
L
= 0
) to the transmitted spectrum by using the
fitted dispersion model to find
φ
and by using a fitting function for
F
[3], namely
(
F
)
−
1
/
2
=
A
1
(d
ν/
∆
g
) +
A
2
(d
ν/
∆
g
)
2
+
A
3
(d
ν/
∆
g
)
3
,
(6)
where
dν
is the distance in frequency from the band edge. Finally, we find the loss parameter
ζ
that makes the on-resonant peak
heights of the model best match our measurement.
3
3
3
2
3
3
4
3
3
6
3
3
8
-
0
.
5
0
.
0
0
.
5
1
.
0
1
.
5
2
.
0
2
1
(
1
)
t
o
t
/
0
,
J
(
1
)
/
0
a
(
T
H
z
)
(
a
)
0
1
0
2
0
3
0
4
0
5
0
0
.
0
0
.
5
1
.
0
1
.
5
2
.
0
(
b
)
d
d
/
0
|
x
|
/
a
FIG. SM3: (a) Single-atom decay rate
Γ
(1)
tot
(black circles) and excited state level shift
J
(1)
(red circles) at
r
a
= (0
,
0
,z
1
)
nm. Vertical dashed
lines mark the frequencies of the first two guided mode resonances,
ν
1
and
ν
2
, near the band gap (frequency range
ν
a
&
ν
BE
= 335
.
1
THz
where
Γ
tot
appears constant) that are supported by the finite length of the APCW. Horizontal dashed line indicates
Γ
′
/
Γ
0
= 1
.
1
, estimated
from the constant
Γ
(1)
tot
in the band gap region. (b) Dissipative coupling rate
Γ
dd
(
x
)
≡|
Γ(
r
a
,
r
a
+
x
ˆ
x
)
|
between two trapped atoms separated
by
x
, with their resonant frequencies at either
ν
a
=
ν
1
(solid circles) or
ν
a
= 336
THz
> ν
BE
inside the band gap (open circles), respectively.
Solid line is an analytical calculation considering actual finite size of the APCW (Fig. SM1).
Figure SM2 (a) shows the measured transmission spectrum (black curve), overlaid with the model fit (red curve). The fitted
parameters for the dispersion model are
2∆
g
= 14.44 THz,
ν
F
/ν
0
= 0.60, and
ν
0
= 342.8 THz. The fitted parameters for
F
are
A
1
= 9,
A
2
=-48, and
A
3
= 128. The fitted loss parameter is
κ
= 1
.
5
×
10
−
5
. At the first resonance
ν
1
, the model linewidth
is 55 GHz, in reasonable agreement with the measured linewidth of 66 GHz. The fitted dispersion relation is used to estimate
the group index, and the fitted cavity model is used to estimate
R
t
and the single pass transmission
e
−
2
ζ
, as shown in Fig. SM2
(b). At the first resonance, the group index is
n
g
≈
11
, the taper reflection is
R
t
≈
0
.
48
and the single-pass transmission is
e
−
2
ζ
≈
0
.
89
[3], resulting in a peak intensity enhancement
E
I
≈
4
.
1
. Since the propagation loss in the APCW is reasonably
small, we ignore the loss in our analysis in the following sections.
We note that the principal contribution to the enhanced decay rate for an atom coupled to the APCW is the enhancement
n
g
≈
11
from a reduced group velocity for operation near the band edge. The “parasitic cavity” enhancement is estimated to
be a factor of
E
I
≈
4
.
1
< n
g
. Moreover, our ability to excite Dicke superradiance benefits directly from the standing wave-like
electric field of the guide mode near the band edge
ν
BE
. Unlike conventional high-finess cavity, the parasitic cavity plays a minor
role in the observed superradiance and does not provide a standing-wave like coupling rate
Γ
1D
cos
2
(
kx
)
as in our experiment
near the bandedge of the APCW; without operating near the band edge, superradiant effect becomes less significant due to
uncontrolled optical phases seen by atoms randomly distributed along the APCW.
For the APCW reported in [1], cesium hyperfine splitting 9.2GHz is small compared to the linewidth (FWHM 66GHz) of
the first resonance as well as its detuning from the band edge
ν
BE
−
ν
1
∼
200
GHz. When aligning
ν
1
to
6
P
1
/
2
,F
= 4
→
6
S
1
/
2
,F
= 3
transition frequency. Both decay rates into
F
= 3
and
F
= 4
hyperfine ground states should be similarly enhanced
by factors that differ no more than
10%
.
II. FINITE DIFFERENT TIME DOMAIN CALCULATIONS FOR COLLECTIVE COUPLING RATES
Due to strong coupling to the TE-like GM in the APCW, trapped atoms experience both enhanced atomic decay rates as well
as collective Lamb shifts. To estimate the size of these effects, we perform FDTD calculations and Fourier analysis as described
in Ref. [4] to obtain the two-point Green’s tensor
G
(
r
1
,
r
2
,ω
)
for the APCW shown in Fig. SM1. We then evaluate dissipative
and coherent coupling rates, respectively, as [5–7]
Γ(
r
1
,
r
2
) =
2
μ
0
ω
2
a
~
d
·
Im[
G
(
r
1
,
r
2
,ω
a
)]
·
d
(7)
J
(
r
1
,
r
2
) =
−
μ
0
ω
2
a
~
d
·
Re[
G
sc
(
r
1
,
r
2
,ω
a
)]
·
d
,
(8)
where
d
is the transition dipole moment,
ω
a
the transition frequency,
μ
0
the vacuum permeability, and
~
Planck’s constant
divided by
2
π
. Here,
G
sc
=
G
−
G
0
is the scattering Green’s tensor, in which the vacuum contribution
G
0
is subtracted from
4
-1 0
0
10
20
30
0.7
0.8
0.9
1.0
1.1
0
20
40
60
80
0.7
0.8
0.9
1.0
(b)
(a)
FIG. SM4: (a) Normlaized transmission spectrum. Each point in the spectrum is an average over 10 experiment repetitions. Black curve shows
the fit to Eq. (9) at
t
m
= 2
.
5
ms with
θ
=
−
0
.
6
±
0
.
1
,
C
0
= 0
.
24
±
0
.
1
,
Γ = 8
.
2
±
0
.
6
MHz and
∆
0
= 9
.
3
±
0
.
3
MHz. (b) Normalized
transmission
T/T
0
as a function of the holding time, measured by the on resonant guided-mode probe with
∆ = 10
.
5
MHz. By fitting the
measured data to Eq. (9) with fitted parameters extracted in Fig. SM4 (a), we obtain the lifetime of
τ
GM
= 28
±
2
ms (black curve).
the total Green’s tensor
G
;
Im[
.
]
and
Re[
.
]
represent imaginary and real parts, respectively. The coupling rate
̃
Γ = Γ
/
2 +
iJ
controls collective excitation dynamics of trapped atoms along the APCW.
We obtain single-atom rates by setting
r
1
=
r
2
=
r
a
at the location of a trapped atom, and evaluate the single-atom total
decay rate
Γ
(1)
tot
(
ν
a
) = Γ(
r
a
,
r
a
,ν
a
)
and excited state level shift
J
(1)
(
ν
a
) =
J
(
r
a
,
r
a
,ν
a
)
. Figure SM3(a) shows the calculation
for
r
a
= (0
,
0
,z
1
)
nm at the center of the trap shown in Fig. 1(b) of [1]. Here the total decay rate
Γ
(1)
tot
= Γ
1D
+ Γ
′
(black curve)
includes the contribution from the GM of interest (
Γ
1D
), which strongly depends on the atomic resonant frequency
ν
a
=
ω
a
/
2
π
and position
r
a
, as well as the coupling rate to all other modes (
Γ
′
).
Γ
′
can be estimated from
Γ
(1)
tot
(
ν
a
)
inside the band gap (
ν
a
&
ν
BE
);
Γ
′
/
Γ
0
≈
1
.
1
remains constant over a broad frequency range. Coupling rate to the TE-like GM,
Γ
1D
= Γ
(1)
tot
−
Γ
′
,
can be obtained from this analysis with
Γ
1D
/
Γ
0
= 1
.
2
.
In Fig. SM3 (a), we calculate a small excited state level shift
|
J
(1)
|
/
Γ
0
<
0
.
4
over a frequency range around
ν
1
= 335
THz.
For our experimental configuration, with
ν
a
≈
ν
1
, we find
|
J
(1)
(
ν
a
)
|
/
Γ
0
∼
0
. This also suggests that the collective level shift
for two trapped atoms,
|
H
dd
(
x
)
| ≡ |
J
(
r
a
,
r
a
+
x
ˆ
x
)
|
(Γ
0
,
Γ
1D
)
, is negligible, where
x
is the atomic separation. Indeed, we
do not see clear evidence of
N
-dependent level shifts in the steady-state transmission spectra shown in Fig. 4(a) of [1].
Figure SM3 (b) shows
Γ
dd
(
x
)
≡ |
Γ(
r
a
,
r
a
+
x
ˆ
x
)
|
for two trapped atoms located at the center of unit cells (
x/a
∈
Z
) and
with resonant frequencies at
ν
a
=
ν
1
or
ν
a
> ν
BE
inside the band gap, where
Γ
(1)
tot
−
Γ
′
∼
0
. For
|
x
|
> a
,
Γ
dd
(
x
)
can be
used to estimate the dissipative coupling rate between two atoms. When
ν
a
=
ν
1
and
|
x
|
/a >
2
,
Γ
dd
(
x
)
slowly drops from
Γ
dd
(0)
−
Γ
′
= 1
.
2 Γ
0
to smaller values as
|
x
|
becomes comparable to the size of the APCW (black circles). This is caused by
interference with reflections from the tapering regions surrounding the APCW. Solid line in Fig. SM3(b) shows an analytical
calculation
Γ
dd
(
x
) = (Γ
dd
(0)
−
Γ
′
) cos(
π
|
x
|
/N
eff
a
)
that compares to the numerical result, where the fitted effective number
of cells
N
eff
= 162
±
9
is larger than
N
cell
due to the leakage of the fields into the taper regions. Small variations between
the analytical and numerical calculations are due to residual coupling via other channels. On the other hand, when
ν
a
> ν
BE
inside the band gap,
Γ
dd
(
x
)
quickly drops below
0
.
1Γ
0
at
|
x
|
/a >
2
. This is expected because, inside the band gap, atoms can
only cooperatively decay via photonic channels that contribute to
Γ
′
, which are either weakly-coupled or are lost quickly into
freepace within distances
|
x
|
<
2
a
.
III. LIFETIME OF TRAPPED ATOMS ALONG THE APCW
Here, we describe the lifetime measurements of trapped atoms by using the SI dipole trapping beams. The SI beam is
220 GHz red-detuned with respect to the D
2
line and has a total power of 50 mW for all measurements reported. To characterize
the lifetime of trapped atoms near the APCW, we measure the normalized transmission
T/T
0
as a function of the measurement
time
t
m
, as shown in Fig. 2 [1] and replotted in Fig. SM4. To ensure small population in the excited state, we choose a probe
intensity well below the saturation intensity (
I/I
sat
<
0
.
1
) for all of our measurements, including the decay rate measurements
in Section IV B. During the lifetime measurement, the frequency
ν
a
of the D
1
transition for the probe field
E
in
is located between
the first and second taper resonances, which leads to the dispersive spectrum shown in Fig SM4 (a). In order to estimate the
5
lifetime of the trap with off resonant cavity, we employ the steady state equation [8],
T/T
0
= (1 +
θ
2
)
/
[
(
1 +
2
C
(
t
m
)
1 +
δ
2
m
)
2
+
(
θ
−
2
C
(
t
m
)
δ
m
1 +
δ
2
m
)
2
]
,
(9)
where the normalized detuning from the light shifted resonance
∆
0
is
δ
m
=
∆
−
∆
0
Γ
, the cooperativity parameter is
C
(
t
m
) =
C
0
exp(
−
t
m
/τ
GM
)
with peak cooperatively
C
0
, lifetime
τ
GM
, and normalized detuning from taper resonance
θ
. First, we fit the
measured spectrum at
t
m
= 2
.
5
ms to Eq. (9) and obtain the fitted parameters,
θ
=
−
0
.
6
±
0
.
1
,
C
0
= 0
.
24
±
0
.
1
,
Γ = 8
.
2
±
0
.
6
MHz and
∆
0
= 9
.
3
±
0
.
3
MHz as shown in black curve in Fig. SM4 (a). Then, to estimate the lifetime
τ
GM
, the measured
data for
∆ = 10
.
5
MHz in Fig. SM4 (b) are fitted to Eq. (9) with fitted parameters from Fig. SM4 (a). We obtain the lifetime
of
τ
GM
= 28
±
2
ms shown in black curve in Fig. SM4 (b).
τ
GM
is consistently shorter than
τ
fs
from free-space imaging,
which might be attributed to increased heating from the stronger light intensity near the APCW, the effect of surface potentials,
or outgassing from the silicon chip and structures that support the APCW. These contributions are being investigated in more
detail.
IV. MODEL FOR SUPERRADIANCE OF TRAPPED ATOMS
A. A Simple Model
Here we provide a simple model that explains the the scaling of the superradiant emission observed in our experiment, and
in particular the principal features of Fig. 3 in our manuscript [1]. First of all we address the role of atom number fluctuations.
Consider the “textbook” case of a fixed number of atoms
N
= 1
,
2
,
3
,
···
,
all with identical coupling. In this case, the total
emission rate under weak excitation conditions as in our experiment would be
Γ
tot
(
N
) = (
N
−
1)Γ
1D
+ Γ
(1)
tot
, where
Γ
(1)
tot
=
Γ
1D
+ Γ
′
is the single-atom decay rate. Clearly, superradiance only occurs for atom number greater than unity, as reflected by
the superradiant rate
Γ
SR
(
N
) = (
N
−
1)Γ
1D
, applicable only for
N
greater than one (i.e., superradiance requires at least two
atoms).
In our experiment, the atom number
N
is random from trial to trial with
N
= 0
,
1
,
2
,
3
,
···
. We take the
unconditional
distribution of atom number to be Poisson with mean
̄
N
. Note that because the background level in our experiment is sufficiently
small as shown in Fig. SM5, any count registered heralds the presence of one or more atoms to a good approximation. Hence, in
the presence of atom number fluctuations, we require the
conditional
distribution
P
1
(
N,
̄
N
)
that gives the probability of
N
atoms
with
N
>
1
. For a Poisson distribution of mean
̄
N
, the probability distribution
P
1
(
N,
̄
N
)
for
N
>
1
is straightforward to derive.
Averaging
Γ
SR
(
N
)
with respect to
P
1
(
N,
̄
N
)
, we find that
Γ
SR
(
N
)
→
̄
Γ
SR
=
(
̄
N
1
−
e
−
̄
N
−
1
)
Γ
1D
. As expected, for
̄
N
1
,
Γ
SR
(
N
)
→
̄
Γ
SR
=
(
̄
N
−
1
)
Γ
1D
≈
̄
N
Γ
1D
. However, for
̄
N
→
0
, we have that
̄
Γ
SR
→
̄
N
2
Γ
1D
, which expresses a small
superradiant enhancement in the total decay rate for the rare trials when a second atom is present, namely
̄
Γ
tot
=
̄
N
2
Γ
1D
+ Γ
(1)
tot
.
The transition from
̄
Γ
SR
≈
̄
N
Γ
1D
for
̄
N
1
to
̄
Γ
SR
≈
̄
N
2
Γ
1D
for
̄
N
→
0
in the intermediate regime for
̄
N
is discussed in the
quantitative analysis in the Section IV B.
In addition to taking into account the random atom number, we must also consider the sharply peaked excitation (
∝
cos
2
(
kx
)
)
and emission (
∝
cos
2
(
kx
)
) couplings along the APCW, as illustrated by the inset for
Γ
1
D
(
x
)
in Fig. 1(a) of our manuscript
[1]. These two effects of excitation and emission result in an overall coupling that goes as
ζ
(
x
) = cos
4
(
kx
)
along the APCW.
Averaging over a unit cell along
x
gives
〈
ζ
(
x
)
〉
= 3
/
8
, since atoms are randomly distributed in the unit cell with uniform
probability density due to our uniform trap configuration along
x
of the APCW. Because the coupling is sharply peaked, we
could consider a simple model that replaces the exact form with coupling of strength unity over
3
/
8
of the unit cell and zero
coupling elsewhere. Roughly speaking then, randomly placed atoms over the entire unit cell would have an effective density
reduced by a factor of
3
/
8
relative to the case of constant coupling for all
x
, which is implicit in the above treatment of atom
number fluctuations. We could then consider as a rough estimate that the mean number
̄
N
should be rescaled to incorporate the
reduced interval of coupling over the unit cell, namely
̄
N
→
(3
/
8)
×
̄
N
. That is, the value
η
= 1
in our simple model should be
rescaled to become
η
simple
= 3
/
8
.
Overall, a simple model that incorporates both the random atom number and spatial variation of the coupling strength leads
to the prediction that
̄
Γ
SR
'
(
̄
Nη
simple
1
−
e
−
̄
Nη
simple
−
1
)
Γ
1D
and
̄
Γ
SR
∼
η
simple
̄
N
Γ
1D
for
̄
N
1
, which is in reasonable accord with
the results shown in Fig. (3b) of Ref. [1]. In the intermediate regime with
̄
N
∼
1
, the behavior is more complex, as discussed in
the following section.