of 9
PHYSICS
Two-dimensional photonic crystals for engineering
atom–light interactions
Su-Peng Yu
a,1
, Juan A. Muniz
a,1
, Chen-Lung Hung
b,c
, and H. J. Kimble
a,2
a
Norman Bridge Laboratory of Physics, California Institute of Technology, Pasadena, CA 91125;
b
Department of Physics and Astronomy, Purdue University,
West Lafayette, IN 47907; and
c
Purdue Quantum Center, Purdue University, West Lafayette, IN 47907
Contributed by H. J. Kimble, May 9, 2019 (sent for review December 31, 2018; reviewed by Antonio Badolato and Hannes Pichler)
We present a 2D photonic crystal system for interacting with
cold cesium (Cs) atoms. The band structures of the 2D pho-
tonic crystals are predicted to produce unconventional atom–light
interaction behaviors, including anisotropic emission, suppressed
spontaneous decay, and photon-mediated atom–atom interac-
tions controlled by the position of the atomic array relative
to the photonic crystal. An optical conveyor technique is pre-
sented for continuously loading atoms into the desired trapping
positions with optimal coupling to the photonic crystal. The
device configuration also enables application of optical twee-
zers for controlled placement of atoms. Devices can be fabri-
cated reliably from a 200-nm silicon nitride device layer using a
lithography-based process, producing predicted optical properties
in transmission and reflection measurements. These 2D pho-
tonic crystal devices can be readily deployed to experiments for
many-body physics with neutral atoms and engineering of exotic
quantum matter.
nanophotonics
|
quantum optics
|
quantum many-body
T
he introduction of nanophotonics to the field of quantum
optics and atomic physics greatly broadens the capabilities
of atom–photon systems (1). Interaction with photonic struc-
tures such as tapered optical fibers (2–5), microcavities (6–8),
and photonic crystal (PhC) waveguides (9–13) enables engineer-
ing of atom optical properties by modifying the local density of
state (LDOS) of the electromagnetic field that interacts with
the atoms. Such engineering capabilities have been actively
explored in various solid-state systems such as quantum dots
(14–16), color centers (17, 18), and embedded rare-earth ions
(19–21). Among the variety of quantum emitters available today,
the identical-particle nature of neutral atoms provides particu-
lar advantages in forming quantum many-body systems. When
excited near a photonic structure, an array of neutral atoms
can all radiatively couple to the shared photonic modes, exhibit-
ing collective superradiant (10, 22) and subradiant (22) decay
phenomena. A coherent collective Lamb shift has also been
observed in a PhC with trapped atoms off-resonantly coupled
to the band edge of a photonic band gap (11). Such collective
coupling to common photonic modes effectively introduces inter-
actions between neutral atoms, making it possible to engineer
long-range spin Hamiltonians (using trapped atoms as spins)
with controllable range and strength of interactions (23, 24), with
pairwise interaction tunability (25), and even with atomic spin
and motion coupling for creating novel crystalline phases (26).
Furthermore, quantum emitters coupled to 1D waveguides have
been proposed as a way to implement a basic universal quantum
computer (27, 28).
Parallel to existing experimental studies of atom–photon cou-
pling in quasi-linear waveguides and cavities, there has been
substantial interest in engineering atom–light interactions in 2D
photonic structures. One expects new phenomena to arise from
the rich band structures and photonic transport properties in
2D geometries, such as anisotropic or directional atomic emis-
sion in a 2D PhC (29, 30), frustrated magnetism (25, 31), and
topological physics (32, 33). Importantly, 1D and 2D atomic
emitter arrays have been proposed to be topologically protected
systems, where the photon propagation is robust against large
imperfections while losses associated with free space emission
are strongly suppressed (34–36). The array of capabilities of
atom–2D photonics systems provides building blocks for realiz-
ing exotic quantum many-body systems complementary to, for
example, ultracold atoms in optical lattices that rely solely on
atomic contact interactions and tunneling (37).
Here, we discuss design, fabrication, and characterization
of versatile 2D photonic structures tailored for atom–photon
coupling. The particular platform of interest here is the 2D
photonic crystal slab, where a periodic spatial modulation in
dielectric defects distribution is arranged over a dielectric slab.
The properties of a photonic crystal can be exploited to pro-
vide subwavelength-scale optical trapping and dispersion engi-
neering (24, 38, 39). In this article, we present two distinct
types of 2D photonic crystal slabs demonstrating photonic
properties, such as directional spontaneous emission, sponta-
neous emission suppression, engineering of coherent atom–
atom interactions, and topological properties in linear dielectric
systems. Optical trapping schemes for trapping neutral ultra-
cold Cs atoms in the vicinity of the photonic crystal slabs are
also devised.
The 2D photonic crystal slabs are fabricated using electron
beam lithography and standard etching processes, as reported
in ref. 40, with a single suspended device layer of silicon nitride
(SiN). They provide sufficient optical access to enable applica-
tion of laser cooling and trapping techniques, such as a magneto-
optical trap (MOT) and an optical dipole trap in close vicinity
to the photonic crystal structures. An overview of our photonic
Significance
Specialized 2D photonic crystals have been developed to
interact with ultracold atoms, which are identical particles
demonstrating quantum behavior both in their interaction
with photons and in their motional degrees of freedom. In
the system presented here, the quantum nature of atoms
is complemented with capabilities of 2D photonic crystals
to engineer optical dispersion, light emission patterns, and
photon-mediated coherent interactions. The combined system
enables atom–atom interactions mediated by photons in the
guided modes of the photonic crystals to provide additional
tools to engineer quantum many-body systems and create
exotic quantum matter.
Author contributions: S.-P.Y., J.A.M., C.-L.H., and H.J.K. designed research; S.-P.Y.,
J.A.M., C.-L.H., and H.J.K. performed research; S.-P.Y. and J.A.M. contributed new
reagents/analytic tools; S.-P.Y. and J.A.M. analyzed data; and S.-P.Y., J.A.M., C.-L.H., and
H.J.K. wrote the paper.
y
Reviewers: A.B., University of Ottawa; and H.P., Harvard University.
y
The authors declare no conflict of interest.
y
This open access article is distributed under
Creative Commons Attribution-NonCommercial-
NoDerivatives License 4.0 (CC BY-NC-ND)
.
y
1
S.-P.Y. and J.A.M. contributed equally to this work.
y
2
To whom correspondence may be addressed. Email: hjkimble@caltech.edu.
y
Published online June 12, 2019.
www.pnas.org/cgi/doi/10.1073/pnas.1822110116
PNAS
|
June 25, 2019
|
vol. 116
|
no. 26
|
12743–12751
crystal devices is shown in Fig. 1
A
, where on-chip waveguides
are connected to conventional optical fibers for efficient address-
ing to the guided modes of the 2D PhC slab by means of a
self-collimation scheme (39, 41, 42). This enables direct opti-
cal characterization of their properties. The two types of lattice
structure presented here are a square lattice of circular holes,
shown in Fig. 1
B
, where we explore the anisotropic emission of
excited-state atoms into the guided photonic modes, and a tri-
angular array of hexagonal holes, depicted in Fig. 1
C
, where we
focus on configurations with atomic resonances in the photonic
band gap.
Engineering Atom–Photon Interactions
The two photonic crystal slab structures display different regimes
for light–matter interactions. For both designed structures, we
use numerical tools to investigate how light–matter interactions
are affected by the presence of the patterned dielectric.
We design the photonic crystals by specifying the unit cell
geometry and then computing the band structures using a finite-
element method (FEM). The parameter space defining the
geometries is explored for useful optical properties such as a
flat landscape of group velocities and opening of photonic band
gaps, while satisfying practical requirements such as minimum
feature sizes and mechanical robustness. The classical electro-
magnetic Green’s tensor is then calculated using finite-difference
time-domain (FDTD) methods on a simulated finite-size pho-
tonic crystal slab. Finally, further properties such as the emission
characteristics of a dipole near the crystal can be obtained from
the Green’s tensor (43–45).
25
μ
m
100 nm
100 nm
A
BC
Fig. 1.
A suspended 2D PhC slab for atom–light interactions. (
A
) SEM image
of a photonic crystal slab structure, suspended by two single-beam SiN
waveguides on left and right edges and released from the substrate for
trapping and coupling with cold atoms directly transported from free space
(Fig. 5 and Fig. 5 legend). The slab contains multiple sections with smoothly
varying crystal parameters for guiding purposes (Fig. 6 and Fig. 6 legend).
The dielectric tabs spaced along the top/bottom of the slabs delineate the
boundaries between such sections. Irregular pattern in the background is an
aluminum stage for the SEM, seen through a through-window of the chip.
(
B
and
C
) Zoom-in SEM image of the square lattice of circular holes (
B
) and
triangular lattice of hexagonal holes photonic crystals (
C
).
The LDOS can be written in terms of the imaginary part of
the electromagnetic Green’s tensor evaluated at the location of
the dipole source itself,
Im(
G
(
~
r
,
~
r
,
ν
))
. The atomic decay rate
from an optically excited state
|
j
can be expressed in terms of
the imaginary Green’s tensor as (43, 45–47)
Γ
Total
=
8
π
2
μ
0
~
·
i
ν
2
ij
Tr [
D
ij
·
Im(
G
(
~
r
,
~
r
,
ν
ij
))],
[1]
where
ν
ij
stands for the transition frequency between the excited
state
|
j
and the ground state
|
g
i
,
D
ij
=
g
i
|
ˆ
~
d
|
j
〉〈
j
|
ˆ
~
d
|
g
i
stands
for the transition dipole matrix between the states in consid-
eration, and the summation is over all ground states
|
g
i
. To
mediate atom–atom interactions using guided mode light, atoms
in the vicinity of the photonic structure need to preferentially
emit photons into the guided modes of the photonic crystal,
instead of emitting into free space or other loss channels. Such
performance can be characterized by the ratio
Γ
2D
/
Γ
, where
Γ
2D
, Γ
are the atom decay rates into the guided modes of inter-
est and into any other channels, respectively, such that
Γ
Total
=
Γ
2D
+ Γ
. It is then useful to design structures that maximize the
ratio
Γ
2D
/
Γ
.
The richness of a 2D structure manifests when studying the
spatial profile of the emitted electric field by an atomic dipole
near the PhC slab. For example, in the presence of a band gap,
the field produced by the dipole excitation is highly localized
but yet inherits the symmetry of the dielectric pattern. If the
dipole frequency lies outside a band gap and thus radiates into
the propagating guided modes, highly anisotropic emission can
be observed over select frequency ranges (29, 30, 48). Further-
more, the vector character of the guided-mode electric field and
the tensor components of the atomic electric dipole operator can
further affect the spatial emission pattern.
Our structures were designed to engineer interactions between
Cs atoms and the transverse-electric (TE)-like guided modes,
where the electric field is polarized predominately along the
plane of the slabs. Due to the spin–orbit coupling in the first
excited state, Cs has two families of optical transitions, marked
by
D
1
and
D
2
lines, respectively, that can be used for opti-
cal trapping and studying light–matter interactions. The crystal
dimensions are chosen such that the frequencies at high-
symmetry points are aligned to the Cs
D
1
and/or
D
2
transi-
tions at 335 THz (894 nm) and 351 THz (852 nm), respec-
tively. We have thus far constrained our design to be based
on a 200-nm-thick silicon nitride slab, given its low optical
loss at near-infrared range and suitability for lithography and
mechanical stability.
Anisotropic Spontaneous Emission in a Square Lattice Photonic Crys-
tal Slab.
We first consider a PhC slab consisting of a square
lattice of circular holes, as shown in Fig. 1
B
. The geometry has
lattice constant
a
= 290
nm, hole radius
R
= 103
nm, thickness
t
= 200
nm, and refractive index
n
= 2
. This set of parameters
was chosen to allow both Cs
D
1
and
D
2
lines to couple to a
TE-like guided band. Specifically, Cs
D
2
resonance crosses a
region of flat dispersion near the
X
point, as shown in Fig. 2
A
.
In
~
k
space, the dispersion relation
ν
(
~
k
)
shows the effects of the
dielectric patterning on the PhC as seen in Fig. 2
B
, which mani-
fest as equi-frequency curves (EFCs) of constant guided-mode
frequencies. The group velocity
~
v
g
= 2
π
~
k
ν
(
~
k
)
is perpendic-
ular to the EFC that passes through a given
~
k
. As marked
in Fig. 2
B
, there is a region in
~
k
space where the EFCs are
approximately linear, around
ν
= 390
THz, indicating that the
group velocity points in the same direction (i.e., along
~
k
Γ
M
as in
Fig. 2
B
). Therefore, all excitations with those wave vectors prop-
agate approximately in the same direction. This gives rise to a
12744
|
www.pnas.org/cgi/doi/10.1073/pnas.1822110116
Yu et al.
PHYSICS
250
350
450
150
50
M
X
Y
a
R
x
y
X
M
Y
Γ
X
Γ
M
x (
μ
m)
y (
μ
m)
-2 -1
0
1
2
2
1
0
-1
-2
0
x (
μ
m)
y (
μ
m)
-2 -1
0
1
2
2
1
0
-1
-2
x (
μ
m)
y (
μ
m)
-2 -1
0
1
2
2
1
0
-1
-2
E
n
0
1
E
n
0
1
E
n
0
1
D
1
D
2
Γ
A
CDE
B
Fig. 2.
Anisotropic spontaneous atomic emission in a square lattice of holes in a dielectric slab. (
A
) The reduced band structures for a unit cell with lattice
constant
a
=
290 nm, hole radius
R
=
103 nm, thickness
t
=
200 nm, and refractive index
n
=
2. Dashed lines mark Cs D
1
and D
2
transition frequencies.
A
,
Inset
shows the real space (black arrows) and momentum space (red arrows) basis vectors. (
B
) EFCs in momentum space for the lowest band shown in
A
.
The dashed black rectangle shows the region with parallel
~
v
g
, around
ν
=
390
±
20 THz. The
~
k
Γ
M
direction is indicated. (
C–E
) Electric-field modulus
|
~
E
n
|
,
from a dipole at the center of the unit cell, normalized after removing the field in the immediate vicinity of the dipole, for three different situations.
The dipole position and polarization are indicated by the blue arrows. In
C
, the dipole frequency is
ν
=
320 THz, and the pattern has weak directional
features. When the emission frequency is at
ν
=
390 THz, the pattern is clearly directional, showing propagation along all
~
k
Γ
M
directions. In
D
, the dipole is
polarized along the
y
direction and emits along both diagonal directions. However, if the dipole is polarized along the diagonal direction, as in
E
, only one
branch remains.
self-collimation effect (38, 49) that leads to directional emission
(29, 30, 48, 50, 51).
In Fig. 2
C
E
we study the effect of engineered band structure
on the dipole emission pattern, using the EFC as a guide. We
consider a dipole placed at the center of a hole, polarized parallel
to the slab. For a dipole radiating at
ν
= 320
THz and polar-
ized along the
y
direction, as in Fig. 2
C
, the radiation pattern
is roughly isotropic and no preferred direction is found. How-
ever, if the radiation frequency is at
ν
= 390
THz, as in Fig. 2
D
, a
clear directional component along
~
k
Γ
M
is present. Both branches
are present due to the dipole polarization and folding symmetry
of the square lattice. Finally, we can select a single branch by
polarizing the dipole along the diagonal direction, as shown in
Fig. 2
E
. We note that by changing the lattice constant to 330 nm,
the center frequency of the flat dispersion region in the lattice
momentum space will correspond to the Cs
D
2
line transition
frequency at 351 THz.
Engineering Atom–Light Interaction in a Triangular Lattice Photonic
Crystal Slab.
A photonic crystal with a triangular lattice of hexag-
onal holes was designed to create a band gap for the TE-like
modes. As indicated in Fig. 3
A
, we parameterize the photonic
crystal unit cell by its lattice constant
a
and the width
t
of the
dielectric tether separating adjacent holes. The unit cell geom-
etry and the band structure for
a
= 405 nm and
t
= 180 nm
are plotted in Fig. 3
A
. The TE band gap spans a frequency
range from the
K
point at the lower band to the
M
point
at the higher band and covers the range of Cs
D
1
and
D
2
frequencies. We now study the emission properties of dipole
excitation in the vicinity of the photonic crystal slab, by calcu-
lating the Green’s tensor in the hole center of a unit cell of
a photonic crystal slab (Fig. 3
B
). The TE band gap efficiently
suppresses emission of a dipole emitter in its frequency range.
Numerical simulations show that for the two in-plane polariza-
tions, a suppression of up to 8 dB on the spontaneous emission
rate can be achieved. This behavior does not manifest in struc-
tures such as the square lattice of holes discussed before, where
there is not a complete TE-like band gap. Here we note that
G
xx
=
G
yy
is due to threefold rotation symmetry of the lattice.
The minor differences between the
x
and
y
traces are likely
numerical artifacts associated with the simulation volume.
G
zz
is not affected by the TE band gap, demonstrating instead a
smooth profile within the simulated bandwidth that varies pri-
marily with the slab thickness. This is because the
z
-polarization
direction corresponds principally to a TM guided mode. We also
note that multiple Fabry–Perot resonances are manifest in
G
xx
Yu et al.
PNAS
|
June 25, 2019
|
vol. 116
|
no. 26
|
12745
10
10
10
-1
1
0
300
340
380
K
M
200
400
a
t/2
x
y
M
K
x
y
z
==m
a
=(m+1)
a
y
>
~
=+
-2
-1
0
1
2
2
1
0
-1
-2
-2.5
0
2.5
x (
m)
y (
m)
x
,
y
z
a
D
1
D
2
a = 405nm
A
B
CD
F
E
G
Fig. 3.
Engineering atom–light interaction in a band gap of a triangular lattice of hexagonal holes in a dielectric slab. (
A
) The reduced band structure of
the hexagonal lattice photonic crystal. A 2D TE-like band gap (blue shaded area) manifests between the top of the lower band (red) and the bottom of
the higher bands (green, yellow). The dashed red circle indicates a crossing at the
K
point between higher bands (Fig. 4
A
). (
B
) The imaginary component
of the Green’s tensor, Im(
G
ii
(
~
r
,
~
r
,
ν
))
/
Im
(
G
0
ii
(
~
r
,
~
r
,
ν
) , normalized by the free-space tensor components,
i
= x (blue), y (red), z (green).
B
,
Inset
shows the
excitation dipole position and polarization. (
C
) The real part of the Green’s tensor component Re(
T
+
1
(
~
r
,
ν
0
)) in unit of the decay rate
Γ
Total
. The excitation
frequency,
ν
0
, is placed in the middle of the band gap. The green dot indicates the position of the emitting dipole, while the green arrows indicate its
polarization. (
D
) A superlattice of atoms associated with lattice vectors
~
A
1
,
~
A
2
is formed by placing atoms in selected sites in the photonic crystal. (
E–G
)
The following interaction parameters
J
i
can be engineered: (
E
)
J
1
=
J
2
=
J
3
, with
J
i
=
0
.
40
Γ
Total
<
0 for
m
11
=
m
22
=
m
=
3, and
J
i
=
0
.
11
Γ
Total
>
0 for
m
= 4; (
F
) lengthening the
~
A
2
vector by one unit lattice vector from the
m
= 4 case results in
J
1
=
0
.
02
Γ
Total
>
0,
J
2
=
0
.
10
Γ
Total
>
0,
J
3
=
0
.
10
Γ
Total
,
which has a weak coupling to the
J
1
site, forming an effective square-lattice–like interaction; and (
G
) stretching the superlattice perpendicularly to
~
A
1
from
the
m
= 2 case creates an anisotropic interaction of
J
2
=
1
.
31
Γ
Total
while
J
1
=
J
3
=
0
.
27
Γ
Total
.
and
G
yy
near the edges of the band gap in Fig. 3
B
. This is
due to finite reflection at the boundaries of the simulated finite-
size PhC, which is terminated abruptly into a uniform dielectric
background.
The excitation field pattern in the TE-like band gap is plotted
in Fig. 3
C
. The modes created by the dipole emitter are non-
propagating modes that are confined in the vicinity of the dipole
in an evanescent manner. It was proposed that evanescent modes
surrounding an atom can be exploited to create a coherent inter-
action between adjacent atoms with the interaction terms (11, 23,
24, 52, 53)
J
pq
=
4
π
2
μ
0
ν
2
ij
~
·
Tr [
D
ij
·
Re(
G
(
~
r
p
,
~
r
q
,
ν
ij
))],
[2]
where
D
ij
is the dipole matrix as defined previously, and now the
Green’s tensor evaluated at the
p
th atom position generated by
the dipole of the
q
th atoms
G
(
~
r
p
,
~
r
q
)
is considered. The evanes-
cent guided modes of the photonic crystal mediate atom–atom
interactions with the Hamiltonian,
ˆ
H
I
=
pq
J
pq
σ
p
ij
σ
q
ji
,
[3]
where
σ
p
ij
=
|
g
i
p
j
|
p
.
This interaction Hamiltonian forms a coupled system for
atoms
p
and
q
with transition between the states
|
g
i
and
|
j
.
For simplicity, the following discussion focuses on a two-state
atom system. As an example, such a two-state system can be real-
ized in our photonic crystal platform with Cs atoms constrained
in the
|
F
= 4,
m
f
= 4
to
F
= 5,
m
f
= 5
transition. Writing the
Green’s tensor in its spherical components in Eq.
2
, only the
term
T
+1
(
~
r
,
ν
) = (ˆ
e
+
·
G
(
~
r
,
~
r
0
,
ν
)
·
ˆ
e
)
is nonzero through
the trace operation, where
ˆ
e
±
=
ˆ
x
±
i
ˆ
y
2
and
~
r
0
is the position of
the emitting dipole. The term is calculated numerically from the
counterclockwise circular component of the electric field
ˆ
e
+
·
~
E
generated by a clockwise circular dipole source
p
ˆ
e
placed at
the origin,
~
r
0
. The profile of
Re(
T
+1
(
~
r
,
ν
0
))
, with
ν
0
within the
bandwidth of the 2D TE band gap, is plotted in Fig. 3
C
, in units
of the decay rate
Γ
Total
.
The geometric pattern of the Green’s tensor
Re(
G
)
can be
exploited to engineer the form of the interaction Hamiltonian.
The relative position of atoms on the photonic crystal deter-
mines the sign and strength of the atom–atom interaction as
facilitated by the evanescent field pattern. Assuming a site-wise
controlled placement of atoms can be realized with a technique
like the optical tweezers (54), atoms can be assembled into a
superlattice on the photonic crystal to form a quantum many-
body system with the desired atom–atom interaction. We define
such a superlattice by lattice vectors spanned by integer multiple
combinations of the base lattice vectors,
~
A
i
=
m
ij
·
~
a
j
. In Fig. 3
D
G
we investigate this capability by identifying configurations
that would create nearest-neighbor coupling of controlled sign
and engineered anisotropy.
It is worth noting that the band structure of the triangular
PhC enables reliable creation of symmetry-protected, purely cir-
cularly polarized mode pockets in a hexagonal lattice structure.
The mode crossing in the two upper bands at the
K
point,
encircled in red in Fig. 3
A
, is a degeneracy that is protected
by the sixfold rotation symmetry of the lattice (55) around the
z
axis perpendicular to the device plane. This degeneracy can
be lifted by perturbing the unit cell dielectric distribution, while
preserving a threefold rotation symmetry, as shown in Fig. 4
A
and
B
. At high-symmetry points, such as the center of the hole,
the guided-mode field pattern is necessarily purely circularly
polarized. Any defined in-plane linear direction is incompatible
with the threefold rotation symmetry. The circular polariza-
tion persists around the
K
point with gradually reducing purity
in a bandwidth comparable to the splitting of the degener-
acy. In Fig. 4
C
F
we plot the field profiles and their circular
12746
|
www.pnas.org/cgi/doi/10.1073/pnas.1822110116
Yu et al.